ProceedingIICMA2013 PDF

Published by Indonesian Mathematical Society
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The Organizing Committee is not responsible for any errors in the papers as these are
individual author responsibility.
April 2014
FOREWORDS
i
President of the Indonesian Mathematical Society
(IndoMS)
FOREWORDS
Assalamu’alaikum Warahmatullahi Wabarakatuh
Good morning and best wishes for all of us
It is my pleasure to say that the proceedings of the Second IndoMS

International Conference on Mathematics and Its Applications (IICMA)
2013 from November 6 to November 7 at Yogyakarta-Indonesia finally
published. The IICMA 2013 is the second IICMA, after IICMA 2009,
which is organized by t h e Indonesian Mathematical Society (IndoMS) in
collaboration with Department of Mathematics, Faculty of Mathematics and
Natural Sciences, Gadjah Mada University and funded by Directorate of
Research and Community Services, the Directorate General of Higher
Education, Ministry of Education and CultureRepublic of Indonesia.
IICMA 2013 is one of the activities of IndoMS period 2012-2014.

Organizing an IICMA 2013 is not only a continuing academic activity for
IndoMS, but it is also a good opportunity for discussion, dissemination of
the research result on mathematics including: Analysis, Applied
Mathematics, Algebra, Theoretical Computer Science, Mathematics
Education, Mathematics of Finance, Statistics and Probability, Graph and
Combinatorics, also to promote IndoMS as a non-profit organization which
has a member more than 1,400 people from around Indonesia area.
We would like to express our sincere gratitude to all of the Invited Speakers
from the Netherlands, Georgia, India, Germany, Singapore and also Indonesia
from Universities (ITB, UPI, University of Jember) and LAPAN Bandung,
all of the speakers, members and staffs of the organizing committee of IICMA
2013. Special thanks to the Secretary of International Mathematics Union
(IMU), the Directorate General of Higher Education, the Dean of Faculty of
Mathematics and Natural Sciences-Gadjah Mada University, the Head of
Department of Mathematics together with all staffs and students, also for
supporting of lecturers and staffs as an organizing committee from Indonesian
University, Padjadjaran University, University North of Sumatera, Sriwijaya
University and Bina Nusantara University. Finally, we also would like to to
give a big thanks for all reviewers who help us to review all papers which
are submitted after IICMA.
ii
With warmest regards,
Budi Nurani Ruchjana

President IndoMS 2012-2014
iii
Chair of the Committee IICMA 2013
On behalf of the Organizing Committee of IndoMS International

Conference on Mathematics and its Applications (IICMA) 2013, I would
like to thanks all participants of the conference. This conference was
organized by Indonesia Mathematical Society (IndoMS) and hosted by
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Universitas Gadjah M a d a , Yogyakarta, Indonesia, during 6-7 November
2013.
In IICMA 2013, there will be 122 talks which consists of 10 invited and 112
contributed talks coming from diverse aspects of mathematics ranging from
Analysis, Applied Mathematics, Algebra, Theoretical Computer Science,
Mathematics Education, Mathematics of Finance, Statistics and Probability,
Graph and Combinatorics. However, the number of paper which were sent
and accepted in this proceedings is 33 papers. We would also like to give
our gratitude to all
Prof. Dr. S. invited
Arumugamspeakers:
(Combinatorics, Kalasalingam University,
India)
 Prof. Dr. Bas Edixhoven (Algebra, Universiteit Leiden-the
Netherlands)
 Prof. Dr. Dr. h.c. mult. Martin Grotschel (Applied Math, Technische
Universitat Berlin, Germany and International Mathematics Union)
 Prof. Dr. Kartlos Joseph Kachiashvili (Statistics, Tbilisi State
University-Georgia)
 Prof. Dr. Berinderjeet Kaur (Mathematics Education, National
Institute of Education, Singapore)
 Prof. Hendra Gunawan, Ph.D (Analysis, ITB-Bandung, Indonesia)
 Prof. Dr. Edy Hermawan (Atmospheric Modeling, LAPAN
Bandung)
 Prof. H. Yaya S. Kusumah, M.Sc., Ph.D (Mathematics Education,
UPI-Bandung, Indonesia)
 Prof. Dr. Slamin (Combinatorics, Universitas Jember, Indonesia)
 Dr. Aleams Barra (Algebra, ITB-Bandung, Indonesia)
We thank all who sent the papers or proceedings of IICMA 2013. We also
would like to give our gratitude for all reviewers who worked hard for
making this proceedings done.
IndoMS conveys high appreciation for the Directorate General of Higher

Education (DGHE) for the most valuable support in organizing the
iv
conference. We also would like to give our gratitude to Universitas Gadjah
Mada, especially to Department of Mathematics, Faculty of Mathematics and
Natural Sciences for providing the places and staffs for this conference.
It remains to thank all members of Organizing Committee spread across 3

cities, Depok, Bandung and Yogyakarta who have worked very hard to make
this conference happens.
Yogyakarta, January 5th, 2014

On behalf of the Committee
Dr. Kiki Ariyanti Sugeng - Chair.
v
ACKNOWLEDGEMENT
The organizing Committee of the IICMA 2013 and Indonesian Mathematical
Society (IndoMS) wish to express their gratitude and appreciation to all
Sponsors and Donors for their help and support for the Program, either in
form of financial support, facilities, or in other form. The Committee
addresses great thank especially to:
a. Directorate General of Higher Education (DGHE)

b. The Rector of the Gadjah Mada University
c. The Dean of Faculty ofMathematics and Natural Sciences, Gadjah
Mada University.
d. The Head of the Department of Mathematics, Gadjah Mada
University.
e. All sponsors of the conference
The Committee extends its gratitude to all invited speakers, parallel session
speakers, and guests for having kindly and cordially accepted the invitation
and to all participants for their enthusiastic response.
Finally the Committee also would like to acknowledge and appreciate for the
support and help of all IndoMS members in the preparation for and the
running of the program
vi
CONTENTS
FOREWORDS ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ i
President of the Indonesian Mathematical Society (IndoMS) ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ii
Chair of the Committee IICMA 2013‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ iv
ACKNOWLEDGEMENT ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ v
INVITED SPEAKER
PROTECTION OF A GRAPH
S. Arumugam …………………………………………………………………………………………………………..1
COUNTING QUICKLY THE INTEGER VECTORS WITH A GIVEN LENGTH
Bas Edixhoven ………………………………………………………………………………………………………….3
INVESTIGATION OF CONSTRAINED BAYESIAN METHODS OF HYPOTHESES
TESTING WITH RESPECT TO CLASSICAL METODS
Kartlos Joseph Kachiashvili ……………………………………………………………………………………….4
MATHEMATICS EDUCATION IN SINGAPORE – AN INSIDER’S PERSPECTIVE
BERINDERJEET KAUR ………………………………………………………………………………………………..8
MODELING, SIMULATION ANDOPTIMIZATION: EMPLOYING MATHEMATICSIN
PRACTICE
Martin Groetschel ………………………………………………………………………………………………….10
STRONG AND WEAK TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL
OPERATORS ON GENERALIZED MORREY SPACES
Hendra Gunawan ……………………………………………………………………………………………………11
THE ENDLESS LONG‐TERM PROGRAMS OF TEACHER PROFESSIONAL
DEVELOPMENT FOR ENHANCING STUDENT’S ACHIEVEMENT IN
MATHEMATICS
Yaya S Kusumah ………………………………………………………………………….………………………….12
DIGRAPH CONSTRUCTION TECHNIQUES AND THEIR CLASSIFICATIONS
Slamin …………………………………………………………………………………………………………………….13
AN APPLICATION OF ARIMA TECHNIQUE IN DETERMINING THE RAINFALL
PREDICTION MODELS OVER SEVERAL REGIONS IN INDONESIA
EDDY HERMAWAN1 AND RENDRA EDWUARD2 …………………………………….…………………15
MAC WILLIAMS THEOREM FOR POSET WEIGHTS
Aleams Barra1, Heide Gluesing‐Luerssen2 ……………………………………………………………….17
PARALEL SESSIONS
ON FINITE MONOTHETIC DISCRETE TOPOLOGICAL GROUPS OF PONTRYAGIN
DUALITY
L.F.D. Bali1, Tulus2, Mardiningsih3 …………………………………………………………………………..18
LINEAR INDEPENDENCE OVER THE SYMMETRIZED MAX PLUS ALGEBRA
Gregoria Ariyanti1, Ari Suparwanto2, and Budi Surodjo 2 …………………………………………25
ON GRADED N‐ PRIME SUBMODULES
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Sutopo1, Indah Emilia Wijayanti2, sri wahyuni3 ……………………………………………………….33
REGRESSION MODEL FOR SURFACE ENERGY MINIMIZATION BASED ON
CHARACTERIZATION OF FRACTIONAL DERIVATIVE ORDER
Endang Rusyaman1, ema carnia2, Kankan Parmikanti3, ………………………………………… 37
THE HENSTOCK‐STIELTJES INTEGRA IN
Luh Putu Ida Harini1 and Ch. Rini Indrati2 ………………………………………………………………43
ON UNIFORM CONVERGENCE OF SINE INTEGRAL WITH CLASS p‐SUPREMUM
BOUNDED VARIATION FUNCTIONS
MOCH. ARUMAN IMRON1, Ch. RINI INDRATI2AND WIDODO3 ………………………………..59
APPLICATION OF OPTIMAL CONTROL OF THE CO2 CYCLED MODEL IN THE
ATMOSPHERE BASED ON THE PRESERVATION OF FOREST AREA
AGUS INDRA JAYA1, RINA RATIANINGSIH2, INDRAWATI3 …………………………………………72
COMPARISON OF SENSITIVITY ANALYSIS ON LINEAR OPTIMIZATION USING
OPTIMAL PARTITION AND OPTIMAL BASIS (IN THE SIMPLEX METHOD) AT
SOME CASES
1
Bib Paruhum Silalahi, 2 Mirna Sari Dewi ………………………………………………………….……82
APPLICATION OF OPTIMAL CONTROL FOR A BILINEAR STOCHASTIC MODEL IN
CELL CYCLE CANCER CHEMOTHERAPY
D. Handayani1, R. Saragih 2, J. Naiborhu 3, N. Nuraini 4…………………............................91
OUTPUT TRACKING OF SOME CLASS NON‐MINIMUM PHASE NONLINEAR
SYSTEMS
Firman1, Janson Naiborhu2, Roberd Saragih 3……………………………………………………..…103
AN ANALYSIS OF A DUAL RECIROCITY BOUNDARY ELEMENT METHOD
Imam Solekhudin1, Keng‐Cheng Ang2 ……………………………………………………………........111
AN INTEGRATED INVENTORY MODEL WITH IMPERFECT‐QUALITY ITEMS IN THE
PRESENCE OF A SERVICE LEVEL CONSTRAINT
nughthoh Arfawi Kurdhi1 And Siti Aminah2 ……………….………..………………………………..121
HYBRID MODEL OF IRRIGATION CANAL AND ITS CONTROLLER USING MODEL
PREDICTIVE CONTROL
Sutrisno ………………………………………………………………………………………………………………..138
FUZZY EOQ MODEL WITHTRAPEZOIDAL AND TRIANGULAR
FUNCTIONS USING PARTIAL BACKORDER
ELIS RATNA WULAN1, VENESA ANDYAN2 ………………………………………………………………146
A GOAL PROGRAMMING APPROACH TO SOLVE VEHICLE ROUTING
PROBLEM USING LINGO
ATMINI DHORURI1, EMINUGROHO RATNA SARI2, AND DWI LESTARI3 …………………..155
CLUSTERING SPATIAL DATA USING AGRID+
Arief fatchul huda1, adib pratama2 ……………………………………………………………………….162
CHAOS‐BASED ENCRYPTION ALGORITHM FOR DIGITAL IMAGE
eva nurpeti1, suryadi mt2 ………………………………………………………………………………………169
APPLICATION OFIF‐THEN MULTI SOFT SET
viii
rb. fajriya hakim ……………………………………………………………………………………………………178
ITERATIVE UPWIND FINITE DIFFERENCE METHOD WITH COMPLETED
RICHARDSON EXTRAPOLATION FOR STATE‐CONSTRAINED OPTIMAL CONTROL
PROBLEM
Hartono1, L.S. Jennings2, S. Wang3…………………………………………………………………………191
STABILITY ANALYZE OF EQUILIBRIUM POINTS OF DELAYED SEIR MODEL WITH
VITAL DYNAMICS
Rubono Setiawan …………………………………………………………………………………………………207
THE TOTAL VERTEX IRREGULARITY STRENGTH OF A CANONICAL
DECOMPOSABLE GRAPH, G = S(A,B)◦ tK1
D. Fitriani1, A.N.M. Salman2 ……………………………….…………………………………………………216
THE ODD HARMONIOUS LABELING OF kCn‐SNAKE GRAPHS FOR SPESIFIC
VALUES OF n, THAT IS, FOR n 4 AND n 8
Fitri Alyani, Fery Firmansah, Wed Giyarti,Kiki A.Sugeng ………………………………………..225
CONSTRUCTION OF , ‐VERTEX‐ANTIMAGIC TOTAL LABELINGS OF UNION
OF TADPOLE GRAPHS
PUSPITA TYAS AGNESTI, DENNY RIAMA SILABAN, KIKI ARIYANTI SUGENG ……………231
SUPER ANTIMAGICNESS OF TRIANGULAR BOOK AND DIAMON LADDER
GRAPHS
Dafik1, Slamin2, Fitriana Eka R3, Laelatus Sya’diyah4 ……………………………..………………237
STUDENT ENGAGEMENT MODEL OF MATHEMATICS DEPARTMENT’S
STUDENTS OF UNIVERSITY OF INDONESIA
1strianti setiadi1 ………………………………………………………………………………………………….…245
DESIGNING ADDITION OPERATION LEARNING IN THE MATHEMATICS OF
GASING FOR RURAL AREA STUDENT IN INDONESIA
RULLY CHARITAS INDRA PRAHMANA1 AND SAMSUL ARIFIN2…………………………………253
MEASURING AND OPTIMIZING MARKET RISK USING VINE COPULA
SIMULATION
Komang Dharmawan 1 ………………………………………………………………………………………….265
MONTE CARLO AND MOMENT ESTIMATION FOR PARAMETERS OF A BLACK
SCHOLES MODEL FROM AN INFORMATION‐BASED PERSPECTIVE (THE BS‐BHM
MODEL):A COMPARISON WITH THE BS‐BHM UPDATED MODEL
MUTIJAH1, SURYO GURITNO2, GUNARDI3 ……………………………………………………………..277
EARLY DRUGS DETECTION TENDENCY FACTOR’S MODEL OF FRESH STUDENTS
IN MATHEMATICS DEPARTMENT UI
DIAN NURLITA1, RIANTI SETIADI2 ………………………………………………………………………….287
COMPARISON OF LOGIT MODEL AND PROBIT MODEL ON MULTIVARIATE
BINARY RESPONSE
JAKA NUGRAHA ……………………………………………………………………………………………………294
MULTISTATE HIDDEN MARKOV MODEL FOR HEALTH INSURANCE PREMIUM
CALCULATION
Rianti Siswi Utami1 and Adhitya Ronnie Effendie2 …………………………………………………303
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THEORETICAL METODOLOGY STUDY BETWEEN MSPC VARIABLE REDUCTION
AND AXIOMATIC DESIGN
Sri Enny Triwidiastuti ……………………………………………………………………………………………314
AN APPLICATION OF ARIMA TECHNIQUE IN DETERMINING THE RAINFALL
PREDICTION MODELS OVER SEVERAL REGIONS IN INDONESIA
EDDY HERMAWAN1 AND RENDRA EDWUARD2 …………………………………………………….327
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INVITED SPEAKERS
Proceeding of IICMA 2013
Invited Speakers
PROTECTION OF A GRAPH
S. ARUMUGAM1,2,3
1Department National Centre for Advance Research in Discrete Mathematics (n-
CARDMATH)), Kasalingam University, Anand Nagar, Krishnankoil-626 126, India

2Department of Computer Science, Ball State University, USA
3Department of Computer Sceinec, Liverpool Hope University, Liverpool, UK
E-mail: s.arumugam.klu@gmail.com
Abstract. Let , be a graph and let 0 be an integer. A function

∶ → 0, 1, 2, . . . , is called a safe function if every vertex with 0
isadjacent to at least one vertex with 0. A vertex with 0
issaid to be defended by a vertex u with 0, if ∈ and if the
function ∶ → 0, 1, . . . , defined by 1, 1 and
for all ∈ , , is again a safe function.In this talk we
present several graph theoretic parameters based on safe functions.
Key words and Phrases: adjacent, defended, safe function.
References
[1] J. Alber, H. Fernau, and R. Niedermeier, Parametrized complexity:

Exponentialspeed-up for planar graph problems, J. Algorithms, 52 (2004), 26–56.
[2] S. Arumugam, Karam Ebadi and Mart´ın Manrique, Co-secure domination ingraphs,
Util. Math., (accepted).
[3] K. S. Booth, Dominating sets in chordal graphs, Research Report CS-80-34,Univ. of
Waterloo, 1980.
[4] K. S. Booth and J. H. Johnson, Dominating sets in chordal graphs, SIAM J.Comput.,
11 (1982), 191–199.
[5] A. P. Burger, E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H. VanVuuren,
and W. Winterbach, Finite order domination in graphs, J. Combin.Math. Combin.
Comput., 49 (2004), 159–175.
[6] A. P. Burger, E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H. VanVuuren,
and W. Winterbach, Infinite order domination in graphs, J. Combin.Math. Combin.
Comput., 50 (2004), 179–194.
[7] A.P. Burger, Michael A. Henning, and Jan H. van Vuuren, Vertex covers andsecure
domination in graphs, Quaest. Math., 31 (2008), 163–171.
[8] E. J. Cockayne, P. A. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi,Roman
domination in graphs, Discrete Math., 278 (2004), 11–22.
1
2
[9] E. J. Cockayne, Irredundance, secure domination and maximum degree intrees,

Discrete Math., 307 (2007), 12–17.
[10] E. J. Cockayne, O. Favaron, and C. M. Mynhardt, Secure domination, weakRoman
domination and forbidden subgraphs, Bull. Inst. Combin. Appl., 39(2003), 87–100.
[11] E. J. Cockayne, P. J. P. Grobler, W. R. Grundlingh, J. Munganga, and J.H. van
Vuuren, Protection of a graph, Util. Math., 67 (2005), 19–32.
[12] G. Gunther, B. L. Hartnell, L. Markus, and D. F. Rall, Graphs with uniqueminimum
dominating sets, Congr. Numer. ,101 (1994), 55–63.
[13] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Fundamentals ofDomination
in Graphs, Marcel Dekker, Inc. New York, 1998.
[14] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Domination inGraphs:
Advanced Topics, Marcel Dekker, Inc. New York, 1998.
[15] M. A. Henning and S. M. Hedetniemi, Defending the Roman Empire-A
newstrategy, Discrete Math., 266 (2003), 239–251.
[16] C. M. Mynhardt, H. C. Swart, and E. Ungerer, Excellent trees and securedomination,
Util. Math., 67 (2005), 255–267.
Invited Speakers
COUNTING QUICKLY THE INTEGER VECTORS WITH

A GIVEN LENGTH
BAS EDIXHOVEN
Mathematisch Instituut, Universiteit Leiden

P.O. Box 9512, 2300 RA Leiden
The Netherlands E-mail: edix@math.leidenuniv.nl
Abstract. The question is how one can compute the number of ways in which an
integer m can be written as a sum of n squares of integers, fast. There are
explicit formulas for these numbers for n even up to 10, due to Fermat,
Legendre, Gauss, Jacobi, Eisenstein, Smith and Liouville. I will explain that for
even n greater than 10 there are no such formulas anymore, but that one can still
compute these numbers, for n even and m given with its factorisation in prime
numbers, in a running time at most a power of n.log(m) (assuming the Riemann
hypothesis for number fields). This is an application of a generalisation by Peter
Bruin of joint work of the speaker with Jean-Marc Couveignes, Robin de Jong
and Franz Merkl.
Key words and Phrases: F modular forms, Theta Function, Identities, algorithms,
Galois symmetry.
References
[1] Computational aspects of modular forms and Galois representations. Edited with J-
M. Couveignes, and with contributions by Johan Bosman, Jean-Marc Couveignes,
Bas Edixhoven, Robin de Jong, and Franz Merkl. Volume 176 of “Annals of
Mathematics Studies”, Princeton University Press, 2011. Also free on-
line: http://www.math.u-bordeaux1.fr/~jcouveig/book.htm
[2] Edixhoven, B and Couveignes, J.M, 2012, Approximate computations with modular
curves. arXiv:1205.5896v1 [math.NT]
[3] P. Bruin. 2010, Modular curves, Arakelov theory, algorithmic applications. PhD
thesis,Leiden, http://www.math.leidenuniv.nl/nl/theses/220/.
3
Invited Speakers
INVESTIGATION OF CONSTRAINED BAYESIAN

METHODS OF HYPOTHESES TESTING WITH
RESPECT TO CLASSICAL METODS
KARTLOS JOSEPH KACHIASHVILI
Full professor of the Georgian Technical University, 77, st. Kostava, Tbilisi, 0175,
Georgia, kartlos55@yahoo.com, k.kachiashvili@gtu.edu.ge
Main scientific worker of I. Vekua Institute of Applied Mathematics of the Tbilisi State
University, 2, st. University, Tbilisi, 0179, Georgia
Abstract. The article focuses on the discussion of basic approaches to

hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a
new one proposed by the author of this paper and called the constrained
Bayesian method (CBM). Wald and Berger sequential tests and the test based on
CBM are presented also. The positive and negative aspects of these approaches
are considered on the basis of computed examples. Namely, it is shown that
CBM has all positive characteristics of the above-listed methods. It is a data-
dependent measure like Fisher’s test for making a decision, uses a posteriori
probabilities like the Jeffreys test and computes error probabilities Type I and
Type II like the Neyman-Pearson’s approach does. Combination of these
properties assigns new properties to the decision regions of the offered method.
In CBM the observation space contains regions for making the decision and
regions for no-making the decision. The regions for no-making the decision are
separated into the regions of impossibility of making a decision and the regions
of impossibilityof making a unique decision. These properties bring the
statistical hypotheses testing rule in CBM much closer to the everyday decision-
making rule when, at shortage of necessary information, the acceptance of one
of made suppositions is not compulsory. Computed practical examples clearly
demonstrate high quality and reliability of CBM. In critical situations, when
other tests give opposite decisions, it gives the most logical decision. Moreover,
for any information on the basis of which the decision is made, the set of error
probabilities is defined for which the decision with given reliability is possible.
Key words and Phrases: hypotheses testing, likelihood ratio, frequentist

approaches, Bayesian approach, constrained Bayesian method.
References
[1] Bauer P. at all., 1988, Multiple Hypothesenprüfung. (Multiple Hypotheses Testing.)

Berlin: Springer-Verlag (In German and English).
[2] Berger, J.O., 2003, Could Fisher, Jeffreys and Neyman have Agreed on Testing?
Statistical Science, 18, 1–32.
4
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[3] Berger J.0., Boukai B. and Wang Y., 1997, Unified Frequentist and Bayesian
Testing of a Precise Hypothesis, Statistical Science, 12, 3, 133-160.
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Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing.
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[7] Berger, J.O. and Sellke, T., 1987, Testing a point null hypothesis. The
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Statistician, May 2005, Vol. 59, No. 2, 121-126.
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Testing of Composite Hypotheses. Scandinavian Journal of Statistics, Vol. 30, No.
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multinomial distribution, with application to chi-squared tests of fit. Ann. Statist.,
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Invited Speakers
MATHEMATICS EDUCATION IN SINGAPORE – AN

INSIDER’S PERSPECTIVE
BERINDERJEET KAUR1
1National Institute of Education, Nanyang Technological University, Singapore

berinderjeet.kaur@nie.edu.sg
Abstract. Singapore’s Education System has evolved over time and so has
Mathematics Education in Singapore. The present day School Mathematics
Curricula can best be described as one that caters for the needs of every child in
school. It is based on a framework that has mathematical problem solving as its
primary focus. The developments from 1946 to 2012 that have shaped the
present School Mathematics Curricula in Singapore are direct consequences of
developments in the Education System of Singapore during the same period. The
curriculum, teachers, leaners and the learning environment may be said to
contribute towards Singapore’s performance in international benchmark studies
such as TIMSS and PISA.
Key words and Phrases: mathematics education, Singapore, differentiated

curriculum, TIMSS, PISA.
References
[1] Yip Yip, S.K.J., Eng, S.P. & Yap, Y.C.J. 1990. 25 Years of educational reform. In
J.S.K. Yip & W.K. Sim (Eds.), Evolution of educational excellence – 25 Years of
education in the Republic of Singapore (pp. 1-30). Singapore: Longman Singapore
Publishers (Pte) Ltd.
[2] Ministry of Education. (1979). Report on the Ministry of Education by Dr Goh and
his team. Singapore: Ministry of Education.
[3] Lee, K.Y. (1979). Letter in response to the report on the Ministry of Education by
Dr Goh and his team. In Ministry of Education, Report on the Ministry of Education
by Dr Goh and his team. Singapore: Ministry of Education.
[4] Economic Committee (1986). Report of the Economic Committee The Singapore
Economy: New Directions. (Chaired by BG Lee Hsien Loong). Singapore: Ministry
of Trade and Industry.
[5] Tan, T.K.Y. (1986). Speech delivered at the Nanyang Technological Institute, 22
July 1986.
[6] Ministry of Education. (1987). Towards Excellence in Schools. Singapore: Ministry
of Education.
8
9
[7] Tan, T.K.Y. (1987). Speech delivered at the First Schools Council Meeting.
Reported in Straits Times, 14 Jan 1987.
[8] Kaur, B. (2002). Singapore’s school mathematics curriculum for the 21st century. In
J. Abramsky, Reasoning, explanation and proof in school mathematics and their
place in the intended curriculum-Proceedings of the QCA International Seminar, 4-
6 October 2001, pp 166-177. London: Qualifications and Curriculum Authority, UK.
[9] Goh, C.T. (1997). Shaping our future: “Thinking Schools” and a “Learning Nation”.
Speeches, 21(3): 12-20. Singapore: Ministry of Information and the Arts.
[10] Ministry of Education (MOE). (1998). Mathematics Newsletter, 1(17). Singapore:
Curriculum Planning and Development Division, Ministry of Education.
[11] Ministry of Education (MOE). 1997. Towards Thinking Schools. Singapore:
Ministry of Education.
[12] Ministry of Education. (2000). Proceedings of MOE work plan seminar: Ability-
Driven Education – Making it Happen. Singapore: Ministry of Education.
[13] Wong, K.Y. (1991). Curriculum development in Singapore. In C. Marsh, & P.
Morris (Eds.), Curriculum development in East Asia (pp. 129-160). London: Falmer
Press.
[14] Lee, P.Y. & Fan, L. (2002). The development of Singapore mathematics curriculum
– understanding the changes in syllabus, textbooks and approaches. A talk given at
the Chongqing conference 17 – 20 August 2002. Unpublished.
[15] Ministry of Education. (2006). A guide to teaching and learning of O level
mathematics 2007. Singapore: Curriculum Planning & Development Division,
[16] Ministry of Education. (2006). A guide to teaching and learning of N level
(Academic) mathematics 2007. Singapore: Curriculum Planning & Development
Division, Ministry of Education.
[17] Ministry of Education. (2006). A guide to teaching and learning of N level
(Technical) mathematics 2007. Singapore: Curriculum Planning & Development
Division, Ministry of Education.
[18] Ministry of Education. (2006). A guide to teaching and learning of primary
mathematics 2007. Singapore: Curriculum Planning & Development Division,
[19] Ministry of Education. (1992). Mathematics syllabus-Secondary 1 and 2 Normal
(Technical) Course. Singapore: Curriculum Planning Division, Ministry of
Education.
[20] Ministry of Education (2012). O-level mathematics teaching and learning syllabus.
Singapore: Curriculum Planning & Development Division, Ministry of Education.
[21] Ministry of Education (2012). N(A)-level mathematics teaching and learning
syllabus. Singapore: Curriculum Planning & Development Division, Ministry of
Education.
[22] Ministry of Education (2012). N(T)-level mathematics teaching and learning
syllabus. Singapore: Curriculum Planning & Development Division, Ministry of
Education.
[23] Goh, C.T. (2001). Shaping lives, moulding nation. Speech at the Teachers’ Day
Rally, Friday 31st August 2001. Singapore: Ministry of Education.
[24] The National Institute of Education. (2002). Moulding lives, shaping tomorrow –
The NIE story. Singapore: The National Institute of Education.
[25] Ministry of Education. (undated). Enhanced Performance Management System.
Singapore: Ministry of Education.
[26] Mckinsey & Co. (2007). How the world’s best-performing school systems come out
on top. Mckinsey & Co Report.
Invited Speakers
MODELING, SIMULATION ANDOPTIMIZATION:

EMPLOYING MATHEMATICSIN PRACTICE
MARTIN GROETSCHEL
ZuseInstitute, Matheon, and Technische Universitaet Berlin,

Germany
E-mail: groetschel@zib.de
Abstract. Unlike engineering, physics, and chemistry, mathematics isalmost

invisible in daily life; even many mathematiciansare unaware that significant
applications of mathematicsare ubiquitous in our environment and activities.
In my lecture I will outline where and how mathematics isemployed in
practice today. A first step is the mathematicalmodeling of real-world
phenomena, engineering tasks,business processes, and the like. Using simulation
toolsthe "correctness" of the models designed is checked,and afterwards
optimization methods are employed to solvethe problems considered to
optimality or to "practicalsatisfaction".I call this way of addressing real-world
problems the"application driven approach" and view this as a major contribution
of mathematics to the industrial creation of value. Mathematics has, in the last
years, become a keyin the handling of complex systems and is playing an
importantrole as a production factor and amplifier for innovations.
In my lecture I will substantiate my claims by providingnumerous examples of
projects (of my own research group andmy colleagues in Berlin) with industry
and partners from othersciences to demonstrate successes (and some failures) of
thismethodology. A wide range of real applications will be coveredincluding
problems in transport, traffic and logistics, energy,telecommunication,
manufacturing, medicine and the life sciences.Mathematics alone does not
suffice, though. Close and faithfulcooperation with engineers, management and
computer scientists,as well as researchers and practitioners in various otherfields
are fundamental for significantsuccess in practice.
Key words and Phrases: Applied mathematics, mathematics in industry and

other sciences, modelling, simulation, optimization
10
Invited Speakers
STRONG AND WEAK TYPE INEQUALITIES FOR

FRACTIONAL INTEGRAL OPERATORS ON
GENERALIZED MORREY SPACES
HENDRA GUNAWAN
Department of Mathematics, Institut Teknologi Bandung,

Bandung 40132, Indonesia
E-mail: hgunawan@math.itb.ac.id
Abstract. Fractional integral operators were first studied by G.H. Hardy &
J.E.Littlewood in the 1920's and by S.L. Sobolev in the 1930's. They
proved the famous inequality for the operators on Lebesgue spaces,
known as the Hardy-Littlewood-Sobolev inequality. Much development has
been made since then, especially in the last two decades. In this
talk, some recent results on the strong and weak type inequalities for
the operators on generalized Morrey spaces (of non-homogeneous type)
will be presented. Some related results and basic techniques to obtain
the inequalities will be exposed.
Key words and Phrases: Fractional integral operators, strong and weak type
inequalities, generalized Morrey spaces.
References
[1] Eridani; Gunawan, H.; Nakai, E. and Sawano, Y., 2013, Characterizations for the
generalized fractional integral operators on Morrey spaces, Mathematics Inequalities
and Applications, to appear.
[2] Gunawan, H.; Hakim, D.I.; Sawano, Y. and Sihwaningrum, I., 2013, Weak type
inequalities for some singular integral operators on generalized non-homogeneous
Morrey spaces, submitted.
11
Invited Speakers
THE ENDLESS LONG-TERM PROGRAMS OF

TEACHER PROFESSIONAL DEVELOPMENT FOR
ENHANCING STUDENT’S ACHIEVEMENT IN
MATHEMATICS
YAYA S KUSUMAH
Department of Mathematics Education, Uniersitas Pendidikan Indonesia (Indonesia

University of Education) Jl. Dr. Setiabudi 229, Bandung 40154
E-mail: yskusumah@upi.edu, yayaskusumah@yahoo.com
Abstract. To be proficient in teaching mathematics, mathematics teachers need

to master mathematics, school mathematics, and mathematical pedagogy. They
have to comprehend the knowledge of student understanding and thinking, since
all these aspects will definitely guide teachers in constructing important
mathematical tasks. To develop their professional skills, teachers should be
given opportunities to attend some educational meetings where special and
structured programs are designed; to be involved in collegial interactions, where
they can develop and apply various methods and approaches; and to conduct
reflection sessions based on their knowledge, learning experience, and best
teaching practices. Basically, learning mathematics is an integrated process, and
so is learning about mathematics teaching and long-term program for
professional development. In this seminar, some activities related to
mathematics teachers professional development and some findings based on
observations during professional development programs on mathematics will be
analyzed and discussed.
Key words and Phrases: Mathematics teacher professional development,

teaching and learning process, collegial interactions, teaching proficiency.
12
Invited Speakers
DIGRAPH CONSTRUCTION TECHNIQUES AND THEIR

CLASSIFICATIONS
SLAMIN
Study Program of Information System

University of Jember, Indonesia.
slamin@unej.ac.id
Abstract. A communication network can be modelled as a graphor a directed

graph (digraph), where each processing element is represented by a vertex and
the connection between two processing elements is represented by an edge (or,
in case of directed connections, by an arc). The problem of constructing large
communication network, subject to the constraints that the number of
connections which can be attached to a processing element is limited and that a
short communication route between any two processing elements is required,
corresponds to the well-known fundamental problem called the degree/ diameter
problem: construct graphs with the largest possible number of vertices (order)
for given degree and diameter. The directed version of the problems differs only
in that `degree' is replaced by òut-degree' in the statement of the problems.
There are two mainstreams of research activities related to the {\it
degree/diameter problem}, namely, (a) proving the non-existence of graphs or
digraphs of order `close' to the Moore bound and so lowering the upper bound
on the order; and (b) constructing large graphs or digraphs and so incidentally
obtaining better lower bounds on the order.
There are many ways to construct a graph or a digraph, for example, drawing by
hand, using computer search, using an algebraic specification and making use of
existing graphs or digraphs to obtain new graphs using some particular
construction technique. In this presentation, we classify the construction
techniques of large directed graphs. We first presentthe construction techniques
that have been established, namely, the generalised de Bruijn
digraphs,generalised Kautz digraphs, line digraphs, digon reduction, generalised
digraphs on alphabets, partialline digraphs, digraphs constructed by the use of
voltage assignments, and vertex deletion scheme as well as a new construction
technique using adjacency matrix. Then we classify them according to the
general method of generating newdigraphs; the diregularity of generated
digraphs; and the range of orders of the generated digraphs.
Key words and Phrases: construction technique, directed graph.
13
14
References
[1] Baskoro, E.T., Brankovic, L., Miller, M., Plesnık, J., Ryan, J. and Siran, J., 1997,
Large digraphs with small diameter: A voltage assignment approach, JCMCC 24
(1997) 161–176.
[2] Baskoro E.T. & Miller, M.. 1996, A procedure for constructing a minimum diameter
digraph, Bulletin of ICA 17 (1996) 8–14.
[3] Bridges, W.G. and Toueg, S., 1980. On the impossibility of directed Moore graphs,
Journal ofCombinatorial Theory Series B 29 (1980) 339-341.
[4] Comellas, F., and Fiol, M.A., 1990, Using simulated annealing to design
interconnection networks, Technical report DMAT-05-0290, Universitat Politecnica
de Catalunya. Presented at the Fifth SIAM Conference on Discrete Mathematics,
Atlanta (1990).
[5] Fiol, M.A., Yebra, J.L.A. and Alegre, I.,1984, Line digraph iterations and the (d,k)
problem for directed Graphs, IEEE Transactions on Computers C-33 (1984), 400–
403.
[6] Fiol, M.A., Lladó, A.S. and Villar, J.L., 1988, Digraphs on alphabets and the (d,N)
digraph problem, Ars Combinatoria 25C (1988) 105–122.
[7] Fiol, M.A. and Llado, A.S., 1992, The partial line digraph technique in the design of
large interconnection networks, IEEE Transactions on Computers 41 (1992.) 848–
857.
[8] Hoffman, A.J. and Singleton, R., 1960, On Moore graphs with diameter 2 and 3,
IBM J. Res. Develop. 4, (1960) 497 – 504.
[9] Imase, M. and Itoh, M., 1981, Design to minimize diameter on building-block
network, IEEE Trans. on Computers C-30 (1981) 439–442.
[10] Imase, M. and Itoh, M., 1983, A design for directed graphs with minimum diameter,
IEEE Trans. on Computers C-32 (1983) 782–784.
[11] Miller, M. and Fris, I., 1988, Minimum diameter of diregular digraphs of degree
2,Computer Journal 31 (1988) 71–75.
[12] Miller, M., and Fris, I., 1992, Maximum order digraphs for diameter 2 or degree 2,
Pullman volume of Graphs dan Matrices, Lecture Notes in Pure dan Applied
Mathematics 139 (1992) 269-278.
[13] Miller, M. and Slamin, 2000, On the monotonicity of minimum diameter with
respect to order and maximum out-degree, Proceedings of COCOON 2000, Lecture
Notes in Computer Science 1858 (D.-Z. Du, P. Eades, V. Estivill-Castro, X. Lin
(eds.)) (2000) 193-201.
[14] Plesnik, J. and Znam, S. 1974. Strongly geodetic directed graphs, Acta F. R. N.
Univ. Comen. - Mathematica XXIX, (1974) 29-34.
Invited Speakers
AN APPLICATION OF ARIMA TECHNIQUE IN

DETERMINING THE RAINFALL PREDICTION
MODELS OVER SEVERAL REGIONS IN INDONESIA
EDDY HERMAWAN1 AND RENDRA EDWUARD2
1The Atmospheric Modeling Division of Atmospheric Science and Technology

Center of National Institute of Aeronautics and Space (LAPAN),
Jln. Dr. Djundjunan No. 133, Bandung 40173, Indonesia
E-mail: eddy_lapan@yahoo.com
2Geophysics and Meteorology Department of Bogor Agriculture University (IPB),
Jln. Raya Darmaga Kampus IPB Darmaga Bogor, Bogor 16680

E-mail: rendra_edwuard@rocketmail.com
Abstract. In this present study, we mainly concerned an application of ARIMA

(Auto-Regressive Integrated Moving Average) technique in determining the
rainfall prediction models over several regions in Indonesia. They are Lampung
(South Sumatera), Pandeglang (West Java), Indramayu (West Java), Banjarbaru
(South Kalimantan), and Sumbawa Besar (Nusa Tenggara Barat). We used the
monthly of surface rainfall data for period of January 1970 to December 2000.
This study is motivated by the importance of understanding the mechanism of
air and sea interaction between Monsoon and El-Niño event when they come
simultaneously as already recommended by the IPCC (Intergovernmental Panel
on Climate Change (IPCC) AR (Assessment Report) 4 and GEOSS (Global
Earth Observation System to System). By applying the Power Spectral Density
(PSD) and Wavelet analysis on those rainfall anomalies data, and also the
Monsoon global index data, represented by the AUSMI (Australian Monsoon
Index) and WNPMI (Western North Pacific Monsoon Index), we found a pre-
dominant peak oscillation of them is about 12 month period that we call as the
Annual Oscillation (AO). While, for the El-Niño event, represented by SST (Sea
Surface Temperature) Niño 3.4, we found 60 month period. Furthermore
analysis, we found more significant relationship between rainfall anomalies and
AUSMI, WNPMI, and SST Niño 3.4. Please note here, this study was undertaken
with assumption that Monsoon and El-Niño is interacted each other, and the
model is developed by using multivariate regression method that formulated by
Rainfall = a + b [AUSMI] + c [WNPMI] + d [SST Niño3.4]. By applying the
ARIMA technique, we found the rainfall prediction models. They ARIMA
(1,0,1)12, with model equation Zt = 0.9989Zt-12 - 0.9338at-12 + at (for AUSMI),
ARIMA (1,1,1)12, with model equation Zt = -0.0674Zt-12 + Zt-12 - Zt-24 -
0.9347a t-12 + a t, (for WNPMI), and ARIMA (2,0,2), with model equation Zt =
3.594Zt-1 - 0.8362Zt-2 – 1.634a t-1 - 0.1053a t-2 + a t (for SST Niño 3.4). By
applying these techniques, we can predict the rainfall behavior over those area
until the end of 2013.
Keywords: ARIMA Technique and Rainfall Prediction Models
15
16
References
[1] Harijono, S.W.B. 2008. Analisis Dinamika Atmosfer di Bagian Utara Ekuator
Sumatera Pada Saat Peristiwa El-Niño dan Dipole Mode Positif Terjadi Bersamaan,
Jurnal Sains Dirgantara (JSD), 5(2), 130 – 148.
[2] [2] IPCC. 2001 .Climate Change 2001 : Impact, Adaptation and Vulnerabi1ity.
Cambridge: Cambridge University Press.
[3] [3] Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis,
Forecasting and Control, 3rd ed. Prentice Hall, Englewood Clifs, NJ.
[4] [4] Chatfield, C. (1996). The Analysis of Time Series, 5th ed., Chapman & Hall,
New York, NY.
[5] [5] Brockwell, Peter J. and Davis, Richard A. (1987). Time Series: Theory and
Methods, Springer-Verlang..
[6] [6] Brockwell, Peter J. and Davis, Richard A. (2002). Introduction to Time Series
and Forecasting, 2nd. ed., Springer-Verlang.
[7] [7] Ljung, G. and Box, G. (1978). "On a Measure of Lack of Fit in Time Series
Models", Biometrika, 65, 297-303.
[8] [8] Hasan, M. Iqbal. 2003. Pokok-Pokok Materi Statistik 2 (Statistik Iterensif) Edisi
Kedua. Jakarta: Bumi Aksara.
Invited Speakers
MAC WILLIAMS THEOREM FOR POSET WEIGHTS
ALEAMS BARRA1, HEIDE GLUESING-LUERSSEN2
1Department of Mathematics, Institut Teknologi Bandung,

Bandung 40132, Indonesia, E-mail: barra@math.itb.ac.id
2University of Kentucky, USA, E-mail: heide.gl@uky.edu
Abstract. In this paper we introduce the notion of local-global property. We

show that a certain matrix group has a local global property and this leads for
the extension theorem of the poset weights. We prove that the poset wieght over
Frobenius ring satisfies the MacWilliams extension theorem if and only if the
poset is a hierarchical poset.
Key words and Phrases: MacWilliams extension theorem, partitions, poset

structures.
References
[1] Brualdi, R. A., Graves, J. S.\& Lawrence, K. M. (1995). Codes with a poset metric.
Discrete Mathematics, 147(1), 57-72.
[2] Claasen, H. L., & Goldbach, R. W. (1992). A field-like property of finite rings.
Indagationes Mathematicae, 3(1), 11-26.
[3] Goldberg, D. Y. (1980). A generalized weight for linear codes and a Witt-
MacWilliams theorem. Journal of Combinatorial Theory, Series A, 29(3), 363-367.
[4] Hirano, Y. (1997). On admissible rings. Indagationes Mathematicae, 8(1), 55-59.
[5] Honold, T. (2001). Characterization of finite Frobenius rings. Archiv der
Mathematik, 76(6), 406-415.
[6] MacWilliams, F. J. (1962). Combinatorial problems of elementary abelian groups
(Doctoral dissertation, Radcliffe College).
[7] Niederreiter, H. (1991). A combinatorial problem for vector spaces over finite
fields. Discrete Mathematics, 96(3), 221-228.
[8] Ward, H. N., & Wood, J. A. (1996). Characters and the equivalence of codes.
Journal of Combinatorial Theory, Series A, 73(2), 348-352.
[9] Wood, J. A. (1999). Duality for modules over finite rings and applications to coding
theory. American journal of Mathematics, 121(3), 555-575.
[10] Wood, J. (2008). Code equivalence characterizes finite Frobenius rings. Proceedings
of the American Mathematical Society, 136(2), 699-706.
17
PARALEL SESSIONS
Algebra
ON FINITE MONOTHETIC DISCRETE TOPOLOGICAL

GROUPS OF PONTRYAGIN DUALITY
L.F.D. BALI1, TULUS2, MARDININGSIH3
1Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Sumatera Utara, Medan 20155, Indonesia, lukasfagolo@gmail.com

2Department of Mathematics, Faculty of Mathematics and Natural Sciences, University
of Sumatera Utara, Medan 20155, Indonesia, tulus@usu.ac.id

University of Sumatera Utara, Medan 20155, Indonesia, mardiningsih@usu.ac.id
Abstract. A group which has a dense cyclic subgroup plays fundamental role in
theory of topological groups. Van Dantzig introduced such group as monothetic
groups. Pontryagin duality is continuous homomorphism that relate some
topological groups with circle groups , that is Hom( , ). Set that contains all
the continuous homomorphism called by Pontryagin Dual Groups or just Dual
Groups, and denoted by i.e. | ∶ → , continuous
homomorphism }.
This paper study the dual groups structure when is monothetic and finite. It
will be shown that monothetic and its dual have same order and they’re
isomorphic.
Key words and Phrases: circle group, cyclic, dual group, monothetic, pontryagin
duality.
Abstrak. Dalam teori grup topologi, grup yang memiliki subgrup siklik yang
padat menjadi hal yang sangat mendasar. Van Dantzig memperkenalkan grup
dengan struktur tersebut sebagai grup monotetik. Dualitas Pontryagin merupakan
homomorfisma kontinu antara grup topologi dan grup lingkaran T, yakni
Hom( , ), dan himpunan semua homomorfisma tersebut membentuk grup dual,
disimbolkan , dengan | ∶ → , homomorfisma kontinu }.
Penelitian ini mengkaji struktur grup dual bilamana adalah grup monotetik dan
berhingga. Akan diperlihatkan dan grup dualnya memiliki orde yang sama dan
keduanya isomorfik.
Kata kunci: dualitas pontryagin, grup dual pontryagin, grup lingkaran,

monothetic, siklik.
18
19
1. Introduction
Cyclicity concept in Group Theory has developed by using number theory as

the foundation of its structure. Gauss’ Disquisitiones Arithmaticae in 1801, shows
that the set of all nonzero integers modulo prime p, each of its element is
representation of every element in the set. Later, the set is known by cyclic group
Zp. Gallian [1] define the cyclic group in general, a group G is said to be cyclic if
there exist an element a ∈ G so that G = {an | n ∈ Z}. The element a is called
generator and n is order of G if n is the least integer which satisfied an = e, that is an
identity element of G.
Topological groups is topological space which also has group structure. A
topological group G is said to be monothetic if there exist a dense cyclic subgroup
H in G. The generator element of cyclic subgroup H is also the generator of
monothetic group G. Falcone et al. [2] said that a group that has a dense cyclic
subgroup plays a fundamental role in theory of discrete topological groups.
The circle group of complex number with modulus 1 is a multiplicative group
with identity 1, denoted by T, that is T = {z | |z| = 1,z ∈ C}. Armacost [3] stated that
circle group T is a monothetic group, and every its proper subgroup is isomorphic
to finite cyclic group Zn of addition group modulo integers n. The circle group T is
also isomorphic to quotient group R/Z with addition operation.
The Pontryagin Duality was named after Lev Semenovich Pontryagin who
laid down the foundation of topological groups, specially locally compact abelian
groups and their duality. The Pontryagin duality relates some topological groups G
and circle groups T by continuous homomorphism, that is Hom(G,T). The set of all
continuous homomorphism which satisfy the duality is called Pontryagin Dual
Groups or Dual Groups, denoted by Gb.
This paper discuss about Pontryagin Duality and the dual groups of
monothetic groups and circle groups. The main structure of monothetic groups that
has a dense cyclic subgroup will affect on the duality to circle groups. Cyclic
subgroup which generated by single element will show that the images of all
elements from monothetic groups will completely determine the isomorphism
between the monothetic group and its dual.
2. Discrete Topological Groups and Continuous Homomorphism
All terms, definitions and theorems related with topology could be found in
Mendelson [4] and Aliprantis & Burkinshaw [5]. In general, topological space is a
set that contains subset from its underlying set with some properties.
Definition 2.1. Let X be a nonempty set. A collection τ of subsets of X is said

to be a topology on X if satisfies the following properties
(i) ∅ ∈ τ and X ∈ τ
(ii) For every U,V ∈ τ, then U ∩ V ∈ τ
(iii) If {Vi | i ∈ I} is a family members in τ, then ∪i∈IVi ∈ τ
If τ is a topology in X, then an ordered pair of (X,τ) is to be a topological

space, and the element of τ is called open subset. The topology τ is said to be
discrete if τ consists of all subsets from X, or τ = P(X), that is the power set of X. A
neighborhood of x, denoted by Nx is a open subset that contain x. A topological
space become Hausdorff Space if it satisfy the Hausdorff Separated Axiom, that is,
20
for each pair of distinct points x,y ∈ X, there are neighborhood Nx and Ny such that
Nx ∩ Ny = ∅. A point x in subset A of X is called a closure point if every of its
neighborhood contains at least one point from A, or Nx ∩ A 6= ∅. A set of all
closure points in is to be a closure of , symbolized by ̅. A subset A of is to

be dense if ̅ . It is obvious that closure is a closed set. Mendelson [4] assert
that the closedness of subset assured by its density.
Theorem 2.2. A is closed if and only if ̅.
Proof. We know that ̅, is closed, so if ̅ , then A is closed. Conversely,

suppose
is closed. In this event, A itself a closed set containing A, so, therefore ̅, ⊂ A.

On the other hand, for an arbitrary subset A we have ⊂ ̅, for if x ∈ A, then each
neighborhood Nx of x contains a point of A, namely x itself. Thus, if A is
closed, ̅.
Topological space that has a group structure is said to be a topological
groups. These structure were connected by algebra properties of the group structure
that affect on its topology structure, and vice versa. Next is definition of
topological groups written by Hewitt and Ross [6].
Definition 2.3.
Let G be a set that is a group and also a topological space. Suppose that:
(i) the mapping (x,y) 7→ xy of G × G → G is a continuous
(ii) the mapping x 7→ x−1 of G → G is a continuous. Then G
is called a topological group.
A discrete topological space which also has a group structure is called
discrete topological group. The discreteness of a topology closely related with
Hausdorff property. A discrete topological group that has two distinct points will
assure the availability of their disjoint neighborhood, since the discreteness will
render all possibilities of open subset in its topology.
Continuous Homomorphism
This part will started with definition of continuous function between two
topological spaces.
Definition 2.4. A function f : (X,τ) → (Y,τ1) is said to be continuous at a point a ∈

X if for each neighborhoood Nf(a) of f(a), f−1(Nf(a)) is neighborhood of a. The
function f is said to be continuous if f is continuous at each point of X.
An open subset that contained in topological space will mapped to open

subset too by continuous function.
Theorem 2.5. A function f : (X,τ) → (Y,τ1) is continuous if and only if for each open
subset O of Y , f−1(O) is an open subset of X.
Proof. First, suppose that f is continuous and that O is an open subset of Y . For
each a ∈ f−1(O), O is a neighborhood of f(a) therefore f−1(O) is neighborhood of a.
21
Since f−1(O) is neighborhood for each of its point, f−1(O) is an open subset of X.
Conversely, suppose that for each open subset O in Y , f−1(O) is an open subset of
X. Let a ∈ X and neighborhood Nf(a) of f(a). Nf(a) contains an open subset
O containing f(a), so f−1(Nf(a)) contains f−1(O) containing a. Thus, f−1(Nf(a)) is a
neighborhood of a and f is continuous at a. Since a was arbitrary in X, f is
continuous.
We could use theorem 2.5 to prove the continuous function in case of closed
set, since the complement of open subset O that is Oc is closed.
Homomorphism function shows the property of image, therefore we can
conclude property of the domain. Note that, the operation in both groups domain
and the image could be different. Next, the definition of homomorphism from
Gallian[1].
Definition 2.6. A homomorphism φ from group G to G0 is a mapping that preserves

the operation, that is φ(ab) = φ(a)φ(b) for all a,b ∈ G.
If there exist two groups hG,∗i and hG0,◦i, then homomorphism φ denoted by
φ(a ∗ b) = φ(a) ◦ φ(b). If φ is a bijection mapping, then φ is said to be an
isomorphism, and both groups are called isomorphic, symbolized by G ∼= G0. The
bijection mapping is a one to one and onto mapping, that is φ is said to be one to
one if whenever a 6= b then φ(a) 6= φ(b), and said to be onto if for each φ(a) ∈ G0
there is a φ−1(a) ∈ G. The continuous homomorphism is a continuous function that
preserves the operation in the group.
3. Image of Monothetic and The Dual Groups
Dikranjan [7] explained that discrete Hausdorff topological group is closed.

In this case, monothetic group will satisfy such properties. By continuous function,
the image of monothetic group will also closed.
Lemma 3.1. Let φ be a continous function and M be a monothetic group. If M

closed then φ(M) is closed.
Proof. Let consider a point z /∈ M. Since M is discrete topological group, then there
exist a neighborhood Nz thus Nz ∩ M = ∅. From definition 2.4, that if φ continuous
then there exist φ(Nz) that is a neighborhood of φ(z). For all arbitrary point z /∈ M,
we can conclude that ∪φ(z)∈/φ(M)Nφ(z) = φ(M)c is open. So, φ(M) is closed.
Every element in M will mapped onto T by continuous function. Every
subgroup in M will also become a subgroup in T by continuous mapping. As the
consequence from lemma 3.1 that explained the image of monothetic group was
closed, so that the image of subgroup of monothetic will be closed.
Corollary 3.2. Let M be a monothetic group and H its subgroup which H ⊆ M, if H

is closed then φ(H) is closed, by φ continuous.
Proof. Since M is closed, so does H. Consider a point z /∈ H, or z ∈ Hc, then there

exist neighborhood Nz ⊆ M but Nz ∩ H = ∅. So that for φ(z) ∈ T, there exist Nφ(z) ⊂
T. Because of Nφ(z) is open, and z is arbitrary point in Hc, so φ(H)c is union of all
neighborhood, that is also an open subset in T. Since φ(H)c open, so φ(H) is closed.
22
Suppose that M to be a monothetic group with generator m. By continuous

homomorphism φ, image of m, namely φ(m) become generator of φ(M).
Lemma 3.3. If M is a monothetic group with generator m, then φ(M) will

generated by φ(m).
Proof. Since φ is continuous, every distinct element in group will be mapped to

distinct element too. Consider that the generator should be singleton, so the image
will be singleton too. Note that H is cyclic subgroup in M, with generator m, that is
hmi = H, then H = hmi. Because of H is dense in M, or H = M, then lemma
3.1 assert that φ(H) is closed with φ(H) = hφ(m)i. Since φ(H) dense in φ(M), then
φ(m) generate φ(M).
Both lemmas and corollary implicitly explained the properties of
monothetic’s image. The following theorem will assert them.
Theorem 3.4. Let M be a monothetic discrete topological groups, φ(M) is

monothetic if and only if φ is continuous function.
Proof. If φ continuous, then φ(M) monothetic. Clearly that lemma 3.1 shows the
closedness of φ(M), and corollary 3.2 state that subgroup H of M mapped to
subgroup φ(H) of φ(M). Lemma 3.3 explain that φ(m) is generator if m is generator
in M. By continuous function φ, M is monothetic. Conversely, theorem 2.5’s proof
will stated that closed set will mapped to closed set by continuous function in two
topological spaces. Since φ(M) is closed then φ(M)c ⊂ T is open. This is similar
when M is closed, then Mc is open, with φ : Mc → φ(M)c. Because of φ(M)c is open
and contain neighborhood for every point inside, then we can conclude that φ(M)c =
∪Nφ(z),∀z /∈ M. For an arbitrary point z /∈ M, then φ−1(∪Nφ(z)) = Mc, that surely
open. According to definition 2.4 and theorem 2.5, then φ is continuous.
Pontryagin Dual Groups Structure

Suppose that M be a monothetic group with cyclic subgroup H. Then H become
closed subgroup which generated by an element m, and the order of H suppose w,
so that mw is the identity element in H. Consider that an element m in H will
mapped to z ∈ T.
φ(m) = z ∈ T
Since mw is identity in H, by trivial continuous mapping, φ(mw) = 1, with 1 is the
identity of circle group T. Lemma 4.2 state that image of generator will be
generator too,
m 7→ φ(m)
φ(m)w = 1 ∈ φ(H)
zw = 1
Note that the homomorphism φ determined by z ∈ φ(H) which satisfy zw = 1.
Previously, we know that T is isomorphic with quotient group of addition R/Z, that
is a representative of all real numbers in [0,1). Since each element z = eiα ∈ T be
representated by α ∈ [0,2π), then α should be convert to real numbers [0,1) by
continuous function f i.e. f : [0,1) → [0,2π).
f : [0,1) → [0,2π)
f(a) = a2π
with a ∈ [0,1) and α ∈ [0,2π). For each z = eia2π and w an order of H, we could form
23
∈ 0,1 with 0 ≤ k ≤ w. So that , and consequently,

, by Euler Identity that state eiπ = −1, then eiπ(2k) = 1. In
other word, for all z ∈ φ(H) ⊂ T representing all homomorphism between H and
φ(H) ⊂ T. So, the dual group H can be written as
Hb = {φ | φ : m 7→ z,hmi = H,∀z ∈ φ(H) ⊂ T,φ continuous homomorphism}
Beside that, the number of element in H is equal to Hb, the notation would be ||H|| =
||Hb||. Since H dense in M, it certainly satisfied that ||M|| = ||Mc||.
Theorem 3.5. If M is monothetic group, then Mc is monothetic group.
Proof. Firstly, it will show that Mc satisfy the definition of topological group.
Suppose that continuous homomorphism φa,φb ∈ Mc and m be the generator of M.
Note that φa dan φb is continuous homomorphism in Mc that bring m ∈ M to a and b
di T. The Ordered pair (φa,φb) that contained in Mc×Mc can be mapped by ψ to Mc,
that is ψ : (φa,φb) 7→ (φaφb)(m).
ψ : Mc× Mc → Mc
ψ : (φa,φb) 7→ (φaφb)(m) (φaφb)(m) =
φa(m)φb(m)
Since a ∈ T has an inverse a−1 ∈ T, then ψ bring φa to its inverse, that is a function
which mapped to a−1. It can written as follows, ψ : φa →7 φa−1 . So that Mc is
topological group. Next, Mc is monothetic group if it contain a dense cyclic
subgroup Hb. This can be proved by showing the generator of Mc. If m generator of
M and φ(m) generator of φ(M), then m and φ(m) also a generator of H and φ(H),
respectively. So there exist φ1 ∈ Mc which bring m to φ(m). The operation in φm will
depend on φm(m). If φm(m) is generator of φ(M), it can written
If w the order of M and φ(M), then φ1(m)w = 1 ∈ φ(M) that is the identity of φ(M).
Since theorem 2.4 assert that the homomorphism which mapped to identity is
homomorphism identity, then e. So, φm that bring m ∈ M to φ(m) ∈ φ(M) is
the generator of Mc.
Note that in this case, monothetic group has same number of element with its
dual group. Next, the bijection between M dan Mc will be shown, so that both of
them are isomorphic.
Theorem 3.6. If M is a monothetic group, then M ∼= Mc
Proof. To show the isomorphism between M and its dual, let ϕ be a continuous
function. Suppose that the order of M and Mc is w. The binary operation on M is ∗
that is for ma,mb ∈ M, 0 ≤ a,b ≤ w with ma∗mb ∈ M. The continuous function ϕ will
define as ϕ(ma ∗ mb) = (φaφb)(m). If we set mb = e as the identity, then
ϕ : M →Mc
ϕ(ma ∗ e) = (φaφe)(m)
ϕ(ma) = (φa)(m)
The similar case will occur if we set ma = e. So, for ma 6= mb obtained (φa)(m)
6= (φa)(m), that is ϕ is one to one function. Next, for every φa(m), there exist
24
ϕ−1(φa(m)) = ma, which shows ϕ is onto. Because ϕ is one to one and onto, then the
bijection has been satisfied. For every ma ∗ mb ∈ M occur
ϕ(ma ∗ mb) = (φaφb)(m)
= φa(m)φb(m)
= ϕ(ma)ϕ(mb)
So that ϕ is homomorphism, which means ϕ is isomorphism, or M ∼= Mc.
4. Concluding Remarks
Let M be a monothetic group, and Mc is Pontryagin Dual Groups, then the

conclusion as follows.
1. By continuous homomorphism φ, image of monothetic group in circle
group also monothetic. If m generate M, the φ(m) generate φ(M) ⊆ T. If H
⊆ M is cyclic subgroup, then φ(H) ⊆ φ(M) is also a cyclic subgroup.
2. The number of homomorphism φ ∈ Mc determined by mapping of generator
m. Cyclic subgroup H is isomorphic with Hb, it also occur to M ∼= Mc .
References
[1] Gallian, J. A., Contemporary Abstract Algebra 7th, Belmont: Brook/Cole, (2010).
[2] Falcone, G., Plaumann, P., Strambach, K., Monothetic Algebraic Groups, J. Aust.
Math. Soc. , Vol. 82, pp. 315-324, (2007).
[3] Armacost, D. L., The Structure of Locally Compact Abelian Groups, New York:
Marcel Dekker, Inc., (1981).
[4] Mendelson, B., Introduction to Topology 3rd, Boston: Allyn and Bacon, Inc.,
(1975).
[5] Aliprantis, C. D. & Burkinshaw, O., Principles of Real Analysis 2nd, London:
Academic Press, Inc , (1990).
[6] Hewitt, E. & Ross, K. A., Abstract Harmonic Analysis 2nd, New York: Springer-
Verlag,
[7] (1979).
[8] Dikranjan, D. Introduction to Topological Groups. Topologia 2. (2013).
Algebra
LINEAR INDEPENDENCE OVER THE SYMMETRIZED

MAX PLUS ALGEBRA
GREGORIA ARIYANTI1, ARI SUPARWANTO2, AND BUDI SURODJO 2
1 Departmentof Mathematics Education, University of Widya Mandala

Madiun, Indonesia, ariyanti gregoria@yahoo.com
2 Department of Mathematics, University of Gadjah Mada, Yogyakarta, Indonesia, ari
suparwanto@ugm.ac.id, surodjo b@ugm.ac.id
Abstract. The symmetrized max plus algebra is an algebraic structure which is a

commutative semiring, has a zero element , the identity element e = 0,
and an additively idempotent. Motivated by the the previous study as in
conventional linear algebra , in this paper will be described the necessary
condition of linear independence over the symmetrized max plus algebra.We
show that if a the columns of a matrix over the symmetrized max plus algebra
are linearly dependent, then the determinat of that matrix is
Key words and Phrases: the symmetrized max plus algebra, linear independence,
determinant
Abstrak. Aljabar maks plus tersimetris adalah suatu struktur aljabar yang
merupakan semiring komutatif, mempunyai elemen nol , elemen identitas
e = 0 dan idempoten penjumlahan. Termotivasi seperti dalam aljabar linier
konvensional, dalam makalah ini akan dikembangkan syarat perlu bebas linier
atas aljabar maks plus tersimetris. Akan ditunjukkan bahwa jika kolom dari
matriks atas aljabar maks plus tersimetris adalah tak bebas linier maka
determinan matriks tersebut adalah
.Kata kunci: aljabar maks plus tersimetris, bebas linier, determinan
1. Introduction
The system max plus algebra lacks an additive inverse. Therefore, some
equations do not have a solution. For example, the equation 3⊕x= 2 has no
solution since there is no x such that max(3,x) = 2. In Schutter (1997) and Singh
(2008) state that one way of trying to solve this problem is to extend the max plus
algebra to a larger system which will include and additive inverse in the same way
25
we have that the system (S, ⊕, ⊗) is called the symmetrized max plus algebra and
S = R2 /B with B is an equivalence relation. Motivated by linear dependence
dan independence over the conventional algebra, in this paper will be described
linear independence over the symmetrized max plus algebra. Akian et al. (2009)
state that a family m1 , m2 , ..., mk of elements of a semimodule M over a semiring
S is linearly dependent in the Gondran-Minoux sense if there exist two subset
I, J ⊆ K := 1, 2, ..., k,P
I ∩ J = ∅, I ∪PJ = K, and scalars α1 , α2 , ..., αk ∈ S, not all
equal to 0, such that i∈I αi mi = j∈J αj mj .
2. The Symmetrized Max Plus Algebra
For the set of all real numbers R = R ∪ {} with
:= −∞ and e := 0.
For all a, b ∈ R , the operations ⊕ and ⊗ are defined as follows:
a ⊕ b = max (a, b) and a ⊗ b = a + b
and then,(R , ⊕, ⊗ ) is called the max plus algebra.
Definition 2.1. (Schutter (1996)) Let u = (x, y), v = (w, z) ∈ R2 .
1. Two unary operators and (.)• are defined as follow :
u = (y, x) and u• = u ⊕ ( u)
2. An element u is called balances with v, denoted by u ∇ v, if
x ⊕ z = y ⊕ w.
3. A relation B is defined as follows :

(x, y)∇(w, z), if x 6= y and w 6= z
(x, y)B(w, z) if
(x, y) = (w, z), otherwise
Because B is an equivalence relation, we have the set of factor S = R2 /B

and the system (S, ⊕, ⊗) is called the symmetrized max plus algebra, with the
operations of addition and multiplication on S is defined as follows
(a, b) ⊕ (c, d) = (a ⊕ c, b ⊕ d)
(a, b) ⊗ (c, d) = (a ⊗ c ⊕ b ⊗ d, a ⊗ d ⊕ b ⊗ c), for (a, b), (c, d) ∈ S
The system (S, ⊕, ⊗) is a semiring, because:
(1) (S, ⊕) is associative
(2) (S, ⊗) is associative
(3) (S, ⊕, ⊗) satisfies both the left and right distributive
Lemma 2.2. (Schutter (1996)) Let (S, ⊕, ⊗) be the symmetrized max plus algebra.
Then the following statements holds.
(1) (S, ⊕, ⊗) is commutative
(2) An element (, ) is a zero element and an absorbent element.
(3) An element (e, ) is an identity element.
(4) (S, ⊕, ⊗) is an additively idempotent.
The system S is divided into three classes :

• S⊕ consists of all positive elements or
S⊕ = {(t, )|t ∈ R } with (t, ) = {(t, x) ∈ R2 |x < t}
• S consists of all negative elements or
S = {(, t)|t ∈ R } with (, t) = {(x, t) ∈ R2 |x < t}
• S• consists of all balanced elements or
S• = {(t, t)|t ∈ R } with (t, t) = {(t, t) ∈ R2 }
Because S⊕ isomorfic with R , so it will be shown that for a ∈ R , can be expressed
by (a, ) ∈ S⊕ .
Furthermore, it is easy to verify that for a ∈ R we have :
a = (a, ) with (a, ) ∈ S⊕
a = (a, ) = (a, ) = (, a) with (, a) ∈ S
a• = a a = (a, ) (a, ) = (a, ) ⊕ (, a) = (a, a) ∈ S•
Lemma 2.3. For a,b ∈ R ,a b = (a, b).
Proof.
a b = (a, ) (b, ) = (a, ) ⊕ (, b) = (a, b)
Lemma 2.4. For (a, b) ∈ S with a,b ∈ R , the following statements hold :
(1) If a>b then (a, b) = (a, )
(2) If a<b then (a, b) = (, b)
(3) If a=b then (a, b) = (a, a) or (a, b) = (b, b)
Proof.
(1) For a > b we have that a ⊕ b = a. In other words, a ⊕ = a ⊕ b . The result
that (a, b)∇(a, ). So it follows that (a, b)B(a, ). Therefore (a, b) = (a, ).
(2) For a < b we have that a ⊕ b = b. In other words, a ⊕ b = b ⊕ . The result
that (a, b)∇(, b). So it follows that (a, b)B(, b). Therefore (a, b) = (, b).
Corollary 2.5. For a, b ∈ R ,


 a, if a > b
a b = b, if a < b
 •
a , if a = b
3. Matrices over The Symmetrized Max Plus Algebra
Let S the symmetrized max plus algebra, n a positive integer greater than 1
and Mn (S) is the set of all nxn matrices over S. Operation ⊕ and ⊗ for matrices
over the symmetrized max plus algebra is defined :
C = A ⊕ B ⇒ cij = a Lij ⊕ bij
C = A ⊗ B ⇒ cij = l ail ⊗ blj
Zero matrix nxn over S is n with (n )ij = and identity matrix nxn over S is En
with
e, jika i = j
[En ]ij =
6 j
, jika i =
Definition 3.1. We say that the matrix A∈ Mn (S) is invertible over S if

A ⊗ B∇En dan B ⊗ A∇En
for any B ∈ Mn (S).
Definition 3.2. Let a matrix A ∈ Mn (S). The determinant of A is defined by
M O n
det A = sgn(σ) ⊗ ( Aiσ(i) )
σ∈Sn i=1
with Sn is the set of all permutations of {1, 2, ..., n}, and

0, if σ is even permutation
sgn(σ) =
0, if σ is odd permutation
Note that the operator ”(∇)” and the systems of max-linear balances hold :
Lemma 3.3. (1) ∀a, b, c ∈ S, a c∇b ⇔ a∇b ⊕ c
(2) ∀a, b ∈ S⊕ ∪ S , a∇b ⇒ a = b
(3) Let A ∈ Mn (S). The homogeneous linear balance A ⊗ x∇nx1 has a non
trivial solution in S⊕ or S if and only if det(A)∇.
4. Linear Independence over The Symmetrized Max Plus Algebra

The symmetrized max plus algebra is an idempotent semiring, so in order to
define rank, linear combination, linear dependence, and independence we need def-
inition of a semimodule. A semimodule is essentially a linear space over a semiring.
Let S be a semiring with as a zero.
A (left) semimodule Mn×1 (S) over S is a commutative monoid (S, ⊕) with zero
element ∈ Mn×1 (S), together with an S−multiplication
S × Mn×1 (S) −→ Mn×1 (S), (r, x) −→ r.x = r ⊗ x

such that, for all r, s ∈ S and x, y ∈ Mn×1 (S), we have
(1) r ⊗ (s ⊗ x) = (r ⊗ s) ⊗ x
(2) (r ⊕ s) ⊗ x = r ⊗ x ⊕ s ⊗ x
(3) e⊗x=x
(4) r⊗=
(5) r ⊗ (x ⊕ y) = r ⊗ x ⊕ r ⊗ y
In a similar way, a right semimodule can be defined.
The rank, linear combination, and linear independence are given in the next defi-
nitions.
Definition 4.1. (Schutter (1997)) Let A ∈ Mmxn (S). The max-algebraic minor
rank of A, rank⊕ (A), is the dimension of the largest square submatrix of A the
max-algebraic determinant of which is not balanced.
Definition 4.2.LLet a1 , a2 , ..., an ∈ Mnx1 (S) and α1 , α2 , ..., αm ∈ S.

m
The expression i=1 αi ⊗ ai is called a linear combination of {a1 , a2 , ..., an }.
Definition 4.3. A set of vectors ai ∈ Mnx1 (S)|i = 1, 2, ..., m is said

Lmto be a linearly
independent set whenever the only solution for the scalars αi in i=1 αi ⊗ ai ∇nx1
is the trivial solution αi = .
The relation between linear dependence and linear combination are given in
the following theorem.
Lemma 4.4. If the set of vectors {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is linearly dependent
then one of the vectors can be presented as a linear combination of the other vectors
in the set.
Proof.
Let α1 ⊗ a1 ⊕ α2 ⊗ a2 ⊕ ... ⊕ αn ⊗ an ∇.
Because {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is a linearly dependent set, without loss of
generality, we can take α1 6= .
−1
So, there is a scalar α1⊗ such that
−1 −1
α1⊗ ⊗ (α1 ⊗ a1 ⊕ α2 ⊗ a2 ⊕ ... ⊕ αn ⊗ an )∇α1⊗ ⊗
−1 −1
a1 ⊕ α1⊗ α2 ⊗ a2 ⊕ ... ⊕ α1⊗⊗ αn ⊗ an ∇
n
M
a1 ∇β2 ⊗ a2 ⊕ β3 ⊗ a3 ⊕ ... ⊕ βn ⊗ an or a1 ∇ βi ⊗ ai
i=2
This leads to a characterization of linear dependent in term of determinants.
Theorem 4.5. Let {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} be a vector

set.
Construct a matrix A such that A = a1 a2 ... an .
The set of vectors {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} are linearly dependent if and only
if det(A)∇.
Proof.
(⇒) Because {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is a linearly dependent set, this implies
that one of the vectors (without loss of generality, let’s say its the vector a1 ) can
be presented as a linear combination of the other vectors in the set. Or,
n
M
a1 ∇β2 ⊗ a2 ⊕ β3 ⊗ a3 ⊕ ... ⊕ βn ⊗ an or a1 ∇ βi ⊗ ai
i=2
Ln
Next, take matrix A and subtract a1 with i=2 βi ⊗ ai . This results in another
matrix (say A’) whose last column is a vector.
A = a• a2 ... an ∇ a2 ... an

Because det a2 ... an ∇ so, we now have that det(A0 )∇.

It follows from the fact that one of the columns of the matrix being , that det(A)∇.
(⇐) Let {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is a linearly independent.
We can show that det(A) not balanced with .
Construct
α1 ⊗ a1 ⊕ α2 ⊗ a2 ⊕ . . . ⊕ αn ⊗ an ∇
We have that
a1 ⊗ α1 ⊕ a2 ⊗ α2 ⊕ . . . ⊕ an ⊗ αn ∇
 
α1
  α2 

Consequently, a1 a2 ... an ⊗  .  ∇.
 .. 
αn
Because {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is a linearly independent, so we have
α1 = α2 = ... = αn =
We can see, that homogenous linear balance A ⊗ x∇ has a trivial solution x = .
Since it follows from lemma 3.3 that the homogeneous linear balance A ⊗ x∇ has
a non trivial signed solution if and only if detA∇, so we have det(A) not balanced
with .
Example
 1: Let {v1 , v2 , v
3 , v4 } ⊆ M
4×1 (S)
 with  
0 2 0
 7   9   3   7 
  , v2 =  5  , v3 =  6  , and v4 =  9 
v1 =        
2 4 0 3
We have, v2 and v4 are linear combinations from {v1 , v3 } such that
v2 = 2 ⊗ v1 ⊕ (−1) ⊗ v3
and
v4 = v1 ⊕ 3 ⊗ v3 .
 
0 2 0
 7 9 3 7 
We have det v1 v2 v3 v4 = det  

5 6 9 
2 4 0 3
= (18 ⊕ 16 ⊕ 12) ( 10 17 ⊕ 18 ⊕ ) = 18 18 = 18• ∇.
From theorem 4.5, we have that {v1 , v2 , v3 , v4 } is linearly dependent.
The following theorem shows that relation between non trivial solution and the
rank of matrix in the homogeneous linear balance.
Theorem 4.6. Let A ∈ Mmxn (S).

The homogeneous linear balance A ⊗ x∇ has a non trivial signed solution (i.e.
x " Mn×1 (S• )) if and only if rank⊕ (A) < n.
Proof.
(⇒) Let rank(A) = n = r.
We have Ar ⊗ x∇ can be represented as
Er
Ar ⊗ x = ⊗ x∇.

So, the only solution of Ar ⊗ x∇ is x∇.
(⇐) Let A ∈ Mmxn (S) and rank(A) = r < n.
Suppose
 Ar , the row echelon formof A can be represented as a form
e ... ∗ ... ∗
 e ... ∗ ... ∗ 
 .. .. .. .. .. 
 
 . . . . . 
  Er C
Ar =  ... e ∗ ... ∗  =
  .
 ... ... 
 
 . . .. .. .. 
 .. .. . . . 
... ...
Without loss of generality, the rank A is r and there is at least r + 1 columns.
Therefore,C has at least onecolumns and
Er C Er C z
Ar ⊗ x = ⊗x = ⊗ ∇. Therefore, Er ⊗ z ⊕ C ⊗ y∇
y
or z = C ⊗ y.
C ⊗ y
We have shown that x = is non trivial solution for y not balanced
y
with .
Example 2 :
2 1• 0
 
Let A ⊗ x∇ with A =  1 0 .

1 0• 1
 
(−2)
With row echelon form, we have rank(A) = 3 < 4 and C =  (−1) 
(−2)•
 
−1
 0 
We can show that for y = 1, we have x =   (−1)•  is nontrivial solution.

Theorem 4.7. Let the set of vectors S = {ai ∈ Mnx1 (S)|i

= 1, 2, ..., n}.
Construct a matrix A such that A = a1 a2 ... an with m × n.
If n > m then S is linearly dependent.
8 Gregoria Ariyanti, Ari Suparwanto, and Budi Surodjo
Proof.
Let {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} and construct α1 ⊗ a1 ⊕ α2 ⊗ a2 ⊕ . . . ⊕ αn ⊗ an ∇
We have that a1 ⊗ α1 ⊕ a2  ⊗ α2 ⊕. . . ⊕ an ⊗ αn ∇
α1
  α2 

So, a1 a2 ... an ⊗  .  ∇.
 .. 
αn
Because A = a1 a2 ... an with m × n, m < n, and rank(A) ≤ n.
This implies that rank(A) ≤ m < n.  
α1
 α2 

Since it follows from Theorem 4.7, so a1 a2 ... an ⊗  .  ∇ has a non
 .. 
αn
trivial solution.
Therefore, there is αi 6= .
This means that S = {ai ∈ Mnx1 (S)|i = 1, 2, ..., n} is linearly dependent.
References
[1] Akian, M., Gaubert, S., and Guterman, A., 2008. Linear Independence over Tropical Semir-
ings and Beyond.
[2] Farlow, Kasie G.,2009. Max Plus Algebra, Master’s Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University in partial fulfillment of the requirements
for the degree of Masters.
[3] Gaubert, S., 1992.T héoriedesSystèmesLinéairesdanslesDioïdes. PhD thesis, Ecole Na-
tionale SupérieuredesM inesdeP aris, F rance.
[4] Schutter, B. De., 1996. Max-Algebraic System Theory for Discrete Event Systems, PhD
thesis, Faculty of Applied Sciences, K.U. Leuven, Leuven, Belgium.
[5] Schutter, B. De and Moor, B. De, 1997. A Note on The Characteristic Equation in The
Max-Plus Algebra. Linear Algebra and Its Applications, vol. 261, no. 13, pp. 237250.
[6] Singh, D., Ibrahim, M., and Singh, J.N., 2008. A Note on Symmetrized Max-Plus Algebra.
Journal of Mathematical Sciences and Mathematics Education. Vol. 5. No.1.
Algebra
ON GRADED N- PRIME SUBMODULES
SUTOPO1, INDAH EMILIA WIJAYANTI2, SRI WAHYUNI3
1.Ph.D student of Depth. Math Gadjah Mada University, sutopo_mipa@ugm.ac.id

2.Depth.Math. Gadjah Mada University, ind_wijayanti@yahoo.com
3.Depth. Math .Gadjah Mada University, swahyuni5950@yahoo.com
Abstract. Let be any ring with identity, right modules and

. Submodule of is called fully invariant submodule of if for
any ∈ , we have ⊂ . Let be is a fully invariant proper submodule of
. Then is called a prime submodule of if for any ideal of and any fully
invariant submodul of , ⊂ implies ⊂ or ⊂ . In this paper
we will define consept submodule from Sanh in graded module and we
called graded prime submodule, Furthermore we investigate some
characterization. We found out that.
Key words and Phrases: prime submodule, graded prime submodule.
1. Introduction
Dauns[4] have introduced the prime submodule and investigated some

properties in 1978. Research about graded ring and graded module have introduced
by C. Natasescu and F.Van Oystaeyen in 1982. In 2006, Atani[1] have introduced
the graded prime submodule of graded module and investigated some properties .
Also , graded prime submodule have been introduced and stadied in [2],[5].
Before we state some result, let us introduce some notations and
terminologies . Let G be an abelian group with identity e and let R be any ring with
identity. The ring R is called graded ring if R ⨁ ∈ R where R is an additive
subgroup of R and R R ⊆ R for every g, h in G. The summands are called
homogeneous components. If ∈ then can be written uniquely as ∑ ∈
where is component of in . In this case , is a subring of and 1 ∈ .
Let be a graded ring and an module. We say that is a graded
module if there is exists a family of subgroups of such that
∈
⨁ ∈ and ⊆ , here denotes the additive subgroup of
consisting of all finite suma of elements with ∈ and ∈ . If is a
graded module, then is module for all ∈ . Submodule of
graded module is called graded submodule of if ⨁ ∈ where
∩ for ∈ .
Sanh [7] have introduced the prime submodule of fully invariant submodule of
module . Definition of prime submodule from Sanh is base on the properties
33
34
that every homomorphism from to is a left multiplication and that

is isomorphic to . In this paper, the prime submodule is defined by
Sahn denoted by prime submodule.
In this paper we will define consept submodule from Sanh in graded module
and we called graded prime submodule, Furthermore we will investigate some
characterization.
2. Main Results
Let be a graded ring, are graded module and : → is an

module homomorphism. Then said to be a graded module
homomorphism of degree if ⊆ for each ∈ . Graded
homomorphism without an indication of degree are understood to have degree
zero. We let be the set of graded module homomorphism from to
of degree and let ⨁ ∈ . Then is a
graded ring and is a subring of . The graded ring
⊕ ∈ is isomorphic to ⊕ ∈
Let be graded right module and , its endomophism
ring. Graded submodule X of is called a fully invariant graded submodule of
if for any ∈ , ⊂ . By the definition , the family of all fully invariant
graded submodule of graded module is non-empty. If then a ideal of is
fully invariant.
Let , graded ideals of dan a graded submodule of , then we define
∑ | ∈∪ ∈ , ∈∪ ∈ , ∈ , and ∑ ∈∪ ∈
. For any right graded module and any right graded ideal of graded ring ,
the set is a fully invariant graded submodule of .
Definition . . Let be a graded right module and a fully invariant proper

graded submodule of . Then is called a graded prime submodule of if
for any ideal of and any fully invariant graded submodule of , ⊂
implies ⊂ or ⊂ .
Especially, if we take module , a graded ideal of is a graded
prime ideal if for any graded ideals , of with ⊂ implies ⊂ or ⊂ .
If is a graded ring and is a graded module , then is subring of
and is module. From this fact, we have the following definition.
Definition 2.2. Let be a graded right module , a fully invariant proper

graded submodule of and ∈ . A fully invariant graded submodule of
module is called prime submodule if for any ideal of and for
any a fully invariant submodule of with ⊂ implies ⊂
or ⊂ .
Relation between graded prime submodule and its homogeneous component

is said the following lemma.
Proposisi 2.3. Let be a graded ring , a graded module and a fully

invariant graded submodule of . If is a graded prime submodule of then
is prime submodule of module for every ∈ .
35
Proof.
Let be a graded prime submodule of graded module . Take any ideal of
and any a fully invariant submodule of such that ⊂ . Because
of ⊂ , then ⊂ ⊂ . From fact that is a graded prime
submodule, we have ⊂ or ⊂ . If ⊂ , then ⊂
because is graded submodule. If ⊂ then ⊂ because ⊂ and
∩ . Proving that X is N prima submodule of R modul M for
every g ∈ G.
The following theorem gives some characterization of graded prime
submodule. The proof the following theorem similar with theorem 1.2 in [7].
Theorem 2.4. Let be a graded module and a proper fully invariant

graded submodul of . Then the following conditions are equivalent :
1 . is a graded prime submodule of
2 .For any a graded right ideal of , any graded submodule of if ⊂
then either ⊂ or ⊂
3 .For any ∈ and fully invariant graded submodule of , if ⊂
then either ⊂ or ⊂
Example 2.5. Let , and 0 for 0 ∈ . is graded

ring . Let be a module. module Module is graded module
with 0 , 0 and 0 for 0,1 ∈ . Let
2 0 be a graded submodule , This submodule is graded prima
submodule.
Lemma 2.6. Let be a graded a right module and . Suppose

that is a fully invariant graded submodule of . Then the set ∈
| ⊂ is a graded ideal of .
Proof.
Take any ∈ and ∈ . Then ⊂ ⊂ and ⊂ ⊂ .
So , in . It is a clear that , is an abelian group. Proving that is
a ideal of . Furthermore, we will prove that is graded ideal of , i.e
⨁ ∈ ∩ for every ∈ . For every ∈ , ∩ ⊂ , so we obtain
⨁ ∈ ∩ ⊂ . Take any ∈ , Then ∑ ∈ and
∑ ∈ ⊂ , we will prove that ∈ ⨁ ∈ ∩ . Clear that ∈ , so
we have to prove that ∈ for every ∈ . From is homogeneous
component of . Then ∈ and ⊂ ⊂ , so ∈ .
Theorem 2.7. Let be a graded right module, and a fully

invariant garded submodule of . If is a graded prime submodule then
is a graded prime ideal of
Proof.
Let , be graded ideal of such that ⊂ . Then ⊂ ⊂ .
If assume that ⊄ , than ⊄ and submodule is graded prime
submodule then ⊂ , so we obtain ⊂ . Proving that is a graded
36
prime ideal of .
Let be a graded ideal of and is a graded modul. Let
∑ ∈ and ∈ | ⊂ . We have The following proposition.
Proposition 2.8. Let be a graded module , X is a fully invariant graded

submodule of and is a graded ideal of . The set ∑ ∈ ⊂ if
only if ⊂ ∈ | ⊂ .
Proof.
Take any ∈ , ∈ and ⊂ , we obtain ∈ . From ⊂ ,
then ∈ , and we have ∈ . Conversely, let ⊂ and the set
∑ ∈ . From ⊂ , we have ∑ ∈ ⊂∑ ∈ ⊂
Teorema 2.9. Let be a graded modul , X is a fully invariant graded

submodule of .Then is a graded prime submodule if only if for any
graded ideal of and any fully invariant graded submodule of such that
⊂ implies ⊂ or ⊂ .
Proof.
Use definition 2.2 and proposition 2.8 .
Acknowledgement. This work was supported by Department of Mathematics,

Faculty of Mathematics and Natural Sciences .
References
[1] Atani,S.E, 2006, On Graded Prime Submodules, Chiang Mai J. Sci, 33(1),pp: 3-7.
[2] Atani S.E, 2011, Notes the graded primes submodule, International journal of
Algebra
[3] Bland,P.E, 2011, Ring and Their Module, Walter de Gruyter & Co.KG.Berlin/New
York.
[4] Dauns, J, 1978, Prime Modules, Journal fur die reine angewandte mathematic, 298,
pp:156-181.
[5] Natasescu,C and Oystaeyen V,F, 1982, Graded Ring Theory, North –Holland
Pubishing Company.
[6] Oral, K.H, Tekir U, and Agargun,A.G, 2011, On graded prime and primary
submodules, Turk J Math, 35,pp:159-167.
[7] Sanh, N.V, 2010, Primeness In Module Category, Asian-European Journal of
Mathematics.Vol 3, No 1, pp:145-154
Analisys and Geometry
REGRESSION MODEL FOR SURFACE ENERGY

MINIMIZATION BASED ON CHARACTERIZATION OF
FRACTIONAL DERIVATIVE ORDER
ENDANG RUSYAMAN1, EMA CARNIA2, KANKAN PARMIKANTI3,
1Department of Mathematics Unpad, erusyaman@yahoo.co.id

2Department of Mathematics Unpad, ema_carnia@yahoo.com
3Department of Mathematics Unpad, parmikanti@yahoo.co.id
Abstract. If the midpoint of a rectangular elastic surface is pressed from the

bottom to the top, then at all points on the surface will formed a potential
energy that leads down. This energy will depend on the surface elasticity and the
magnitude of the pressure. Assuming that the surface that occurred after pressed
will be a function of two variables form a double sine series, and the energy that
formed is an integral of the square of the Laplace operator that expanded into a
fractional order, this paper will present a discussion about how to make a model
of energy minimization problems that occurs, as well as characterizes the
fractional derivatives order based on the surface elasticity.
Key words and Phrases : energy, elasticity, fractional, modeling, double sine.
1. Introduction
Suppose given a surface in form of a rectangular-shaped elastic plate.

Furthermore we press under the object right in the middle of the surface. Obviously,
at any point on the surface will occur opposite energy downward. This energy will
depend on two things which are surface elasticity and the magnitude of surface
tension. Similarly, if the surface is pressed on two or three different points,
assuming the pressure exerted on the two or three points are equal, the energy will
still depend on two things above.
Mathematically, if the square-shaped elastic surface is placed on the lines
x = 0, x = a, y = 0, and y = b, and then pressed from under the surface at the center
point, then it will look like in Figure-1 below.
37
38
Figure-1
According Langhar [6], the pressed surface could be represented by the

following double sine series function
∞ ∞
that satisfies the boundary condition w = 0, wxx = 0, and wyy = 0 for x = 0 or x

= a, and for y = 0 or y = b.
The tension energy on the downward-pointing plate surface is as follows:
1 1
∆ 2
2 2
with D represents plate surface elasticity, that according to (1) obtained

∞ ∞
1
. 3
8
2. Previous Research
In a previous research, Rusyaman et al [2] and H.Gunawan et al [4], have

guaranteed the existence of function as interpolation results passing points (xi , yj ,
cij) shaped surface which is described as double sine series
 
u(x,y) :=   a
m 1 n 1
mn sin m  x sin n  y  (4)
with 0  y  1 and 0  x 1,and minimize the energy in form of
/
≔ ∆ . 5
2 2
where  =  is Laplace operator, and  is the fractional-order of
x 2 y 2
derivative.
39
With u(x,y) as in (4), then the energy Eβ (u) will be as follow:
 a m 
 
1 2 
 mn
2 2
 n2 (6)
4 m 1 n 1

where a function u(x,y) exists, unique and continuous if and only if  > 1. [4]
The existence and uniqueness of the solution of this problem follows from the best
approximation theory in Hilbert spaces [1] [3]. In this study, to obtain the solution
of this problem, particularly in determining the value of the minimum energy,
carried through the stages: determining initial function u0(x , y), determining and
orthogonalizing basis by use the Gram Schmidt method, and the last is determining
minimum energy value. Technically, this process uses iterative system with the
help of software until obtain minimum energy value.
3. The Main Problem
Since the function in (1) is equal to the function (4) for a = b = 1, and the
energy in (3) differ only in the constants in (6), the results in [1], [4] and [5] ie
surface shape interpolation that minimizes energy, can be adopted as a surface that
is pressed from below as in [6].
Due to determination of the minimum energy value require a long process,
which must use the software through hundreds and even thousands of iterations,
then in this paper it will be prepared some regression models which show the
minimum energy as a function of  that can be identified with the surface elasticity,
and C which states the amount of pressure on the surface.
By regression, the minimum energy value can be predicted by substituting the value
of  and C into the model [5], as was done by Roberto Z Freire et al [7], in
Development of Regression Equation for Predicting Energy and Hygrothermal
Performance of Buildings.
With the existence of this regression model, the software will no longer be needed
to determine the amount value of the energy, that we could simply use the model
for easier and faster way.
The discussion of the problems is divided into three sections as follows :
1. Pressing from under a surface at a single point, which is right in the center
(1/2 , 1/2 , 1 ) , as far as one unit , so that the energy only depends on .
2. Pressing from under a surface at two different points are (1/4 , 1/2 , C ) and (
3/4 , 1/2 , C ), as far as C units so that the energy depends on  and C.
3. Pressing from under a surface at three different points are ( 1/4 , 1/4 , C ), (
1/3 , 2/3 , C ) and ( 3/4 , 1/2 , C ) as far as C unit. Even in this case, the
energy will depends on  and C
40
4. Main Results
4.1 Problem-1: Surface pressed at one point

In the first problem, given a surface in form of a rectangular-shaped elastic
plate placed on lines x = 0, x = 1, y = 0, and y = 1. Furthermore midpoint of
the plate is pressed from the bottom to the top as far as C = 1 unit as seem in
Figure-1 above, so that the maximum point is in the middle of the field, with the
exact coordinates ( 1/2 , 1/2 , 1 ).
The Surface formed is assumed as a function as in (4) and the value of the
surface energy as in (5) and (6). The value of the energy E will depend on the
value of . Without losing the generality by eliminating constant 2 from the
equation (6), by using the calculations have been described in [2] and [4] and using
software until 1225th iteration, it obatined data of twenty minimum energy value
are generated from twenty values of  with the range between 1.00 and 2.00 .
Scatter plot diagram is shown in Figure-2 below, where the red line shows the least
squares line, while the green line shows the linear regression line.
Data analisys results showed that the correlation between  and E is positive
value, amounting to 0.9939006. This means that there is mutual influence which is
proportional because its value is close to 1 , ie, the greater the value of  , the
greater the value of E .
With a residual standard error of 0.02163,

and freedom degrees of 18, as well as multiple
R-squared amounting to 0.9878, then
resulting intercept value = –0.52547 and 
coefficient = 0.69240.
Thus, the linear regression model is:
 0.52547 0.69240  .
with error square of the model-1 is
0.00842482.
Figure-2
4.2 Problem-2: Surface pressed at two points

In the second problem, given a surface in form of a rectangular-shaped
elastic plate placed on the lines x = 0, x = 1, y = 0, and y = 1, as shown in Figure-
3. Then the plate is pressed from te bottom to the top on two distinct points, i.e
(1/4, ½) and (3/4, ½) as far as C unit as seem in Figure-4 below. So that the two
maximum points have coordinates (1/4 , ½ , C) and (3/4, 1/2, C).
Figure-3 Figure-4
41
The value of the energy E will depends on the value of  and C. Without
losing the generality by eliminating constant 2 from the equation (6), by using the
calculations have been described in [2] and [4] and using software until 1225th
iterations, it obatined data of twenty-five minimum energy value generated from
twenty-five value of  with the range between 1.0 until 2.2 and also twenty-five
value of C with distribution between 0.1 until 1.0.
Data analisys results showed that the correlation between  and E is
positive value amounting to 0.1778537. This means that proportionally less
influence because its value is relatively small, while correlation between C and E
is better since the value is close to 1, i.e 0.8507899. Since the correlation value is
positive, then both  and C are proportional to E . Therefore, the greater the value
of  and C, the greater the value of E .
With the residual standard error of 0.2518, and the freedom degrees of 22,
as well as multiple R-squared of 0.8528 resulting intercept value = –1.5651,
coefficient of  = 0.6978, and the coefficient of C = 1.8696 .
Thus the linear regression model is :

 1.5651 0.6978  1.8696 .
with error square of the model-2 is 1.39497136
4.3 Problem-3: Surface pressed at three points
On the third problem, given a surface in form of a rectangular-shaped elastic

plate placed on the lines x = 0, x = 1, y = 0, and y = 1, as shown in Figure-5.
Then the plate is pressed from the bottom to the top on three distinct points, i.e
(1/4 , 1/4), (1/3 , 2/3) and (3/4 , 1/2) as far as C-units as in Figure-6 below, so that
the three maximum points have coordinates (1/4 , 1/4 , C), (1/3 , 2/3 , C) and
(3/4 , 1/2 , C).
Figurer-5 Figure-6
Same with the two-points problem, the value of the energy E will depends
on the value of  and C. Without losing the generality by eliminating constant 2
from the equation (6) , by using the calculations have been described in [2] and [4]
and using software until 1225th iteration, it obatined data of twenty-five minimum
energy value generated from twenty-five value of  with the range between 1.0
until 2.2 and also twenty-five value of C with range between 0.1 until 1.0.
Data analisys results showed that the correlation between  and E is
positive numbers amounting to 0.2774373, means proportionally less influence
because its value is relatively small. While correlation between C and E is better
since the value is close to 1, i.e 0.8131555. Since the correlation value is positive,
42
then both  and C are proportional to E . Therefore, the greater the value of  and
C, the greater the value of E .
With a residual standard error of 0.2104 , and the freedom degrees of 22, as
well as multiple R - squared of 0.8578 resulting intercept value = –1.5348,
coefficient  = 0.7362, and the coefficient C = 1.6467 .
Thus the linear regression model is :

 1.5348 0.7362  1.6467 .
with error square of the model-3 is 0.9742403.
5. Conclusion
The data processing and analysis of model resulted in a conclusion that the
value of the fractional-order derivative , can be identified with surface elasticity
as denoted by D in equation (2) and (3). Furthermore, the value of  and C ie the
amount of pressure from buttom is directly proportional to the amount of the
minimum energy  which means the more rigid material surface, and the greater
the pressure exerted, then the greater the energy value.
Acknowledgement. This work is fully supported by Universitas Padjadjaran

under the Program of Penelitian Unggulan Perguruan Tinggi Program Hibah
Desentralisasi No. 2002/UN6.RKT/KU/2013. We thank also to Prof. Hendra
Gunawan who have inspiring me, and to Akmal S.Si, MT for his help in
computation field.
References
[1] Atkinson.K & Han.W, 2001, Theorical Numerical Analysis A Functional Analysis
Framework, Springer Verlag, New York,.
[2] E. Rusyaman, H. Gunawan, A.K. Supriatna, R.E. Siregar, 2009, A 2-D
Interpolation Method That Minimizes An Energy Integral, Presented at IICMA in
Yogyakarta,.
[3] H. Gunawan, F. Pranolo, E. Rusyaman, 2007 , An Interpolation Method That
Minimizes an Energy Integral of Fractional Order. Singapore: Proceeding of Asian
Symposium of Computer Mathematics.
[4] H. Gunawan, E. Rusyaman & L. Ambarwati, 2011, Surfaces with Prescribed Nodes
and Minimum Energy Integral of Fractional Order ITB J.Sci., Vol. 43 A, No. 3,
page 209-222.
[5] Hogg and Craig,1995 , Introduction to Mathematical Statistics, Prentice-Hall, New
Jersey. .
[6] Langhaar.H.L, 1962, Energy Methods in Applied Mechanics, John Wiley & Sons,
New York.
[7] Roberto Z Freire, Gustavo H. C Oliveira, and Nathan Mendes, 2008, Development
of regression Equation for Predicting Energy and Hygrothermal Performance of
Buildings, Energy and Building 40 , page 810-820, www.sciencedirect.com.
THE HENSTOCK-STIELTJES INTEGRA IN
LUH PUTU IDA HARINI1 AND CH. RINI INDRATI2
1Departement of Mathematics, Udayana Unyversity, ballidah@gmail.com

2Departement of Mathematics, Gadjah Mada Unyversity, rinii@ugm.ac.id
Abstract. Lim, et.al. [4] studied the Henstock-Stieltjes integral on the real line
and proved some of its properties. In this paper, we discuss the Henstock-
Stieltjes integral in the dimensional Euclidean space . The main
purpose of this article is to define the Henstock-Stieltjes integral in and
develop some of its characteristics. At first, we define a bounded variation
function on a cell ⊂ . We use the equivalency between the interval function
and point function on a cell ⊂ to construct the definition. Some
characterizations of bounded variation function defined on a cell ⊂ will be
developed. The results will be used to give the sufficient condition for existence
of the Henstock-Stieltjes integral in and to investigate the properties of the
constructed integral. Some convergence theorems will be stated at the end of
the paper
Keywords and Phrases: Henstock-Stieltjes integral, bounded variation function,

convergence theorems.
1. Introduction
Integral theory is a branch of mathematics that is deductive analysis and it is

still growing and developing both theoretical and applicative. In its development,
integral theory emerges through a descriptive and constructive definition [2]. At
first Henstock-Stieltjes integral applies only to functions with monotonous
integrator. Lim in 1998 studied the Henstock-Stieltjes integral on the real line
and gave some sufficience conditions for a function to be Henstock-Stieltjes on
[a, b] with respect to an increasing function on [a, b] [4]. Furthermore the
emergence of the concept bounded variation functions give an opportunities to
analyze the existence of the Henstock-Stieltjes integral in a larger class of
functions. A real function on [ a, b] has bounded variation if its total variation is
finite. A function of bounded variation on [ a , b ] can be expressed as a difference of
two increasing functions [3]. From this result, the Henstock-Stieltjes integral with
respect to the integrator of monotone function can be generalized with respect to
the integrator of bounded variation function.
43
44
A more generalized concept of space in dimensional Euclidean space

R  inspires researchers including [2] and [5] which give various definitions and
n
basic properties of multidimensional integrations. Indrati [1] generalizes the

Henstock-Kurzweil integral in the dimensional Euclidean space. The
opportunities for developing Henstock-Stieltjes integral with respect to the
bounded variation function integrator in R n are still widely open. This is inspired
by the success of [1] who developed the Henstock-Kurzweil integral in the
dimensional Euclidean space. The main purpose of this article is to define and
develop Henstock-Stieltjes integral in R n . First, we define a bounded variation
function on a cell E  R n . Next, we use the equivalency between the interval
function and point function on a cell E  R n to construct the definition. Some
characterizations of bounded variation function defined on a cell E  R n will be
developed. Based on those characteristics, we generalized the Henstock-Stieltjes
integral with respect to the integrator of bounded variation function on the cell
E  R n . Then, we investigate the properties of the constructed integral and give
the sufficient condition for existence of the Henstock-Stieltjes integral in R n .
Some convergence theorems will be stated at the end of the paper.
Before discussing the main results of the research, we begin with some
terminology notations. The scope of this paper is the dimensional Euclidean
n
 
space R , with R  x   x1 , x2 , x3 ,..., xn  ; xi  R, and 1  i  n . For
n
x  x1 , x2 ,..., xn   R n , x  max xi : 1  i  n and the distance between two

points x , y  R n is defined as follows:
d x , y   x  y 
 max xi  yi .
1i  n
Furthermore, we also have four types of interval in R n . For a , b  R n we have

(i). a , b   x  x , x ,..., x  : a  x  b , i  1,2,3,..., n
1 2 n i i i
(ii). a , b   x  x , x ,..., x  : a  x  b , i  1,2,3,..., n

1 2 n i i i
(iii). a , b   x  x , x ,..., x  : a  x  b , i  1,2,3,..., n

1 2 n i i i
(iv). a , b   x  x , x ,..., x  : a  x  b , i  1,2,3,..., n

1 2 n i i i
In this paper, a cell E  R n stands for a non-degenerate closed and bounded

 
interval a , b , where a  b . Its volume will be represented by E . Then if E a
cell, x  E , and  is a positive function on E , we defined an open ball centered
at x  E with radius  ( x ) , represented by B ( x ,  ( x )) , is defined as
B( x ,  ( x ))  y : x  y   ( x ).
Next, we will discuss about the   fine partition and the   fine interval. A
collection of cells I i : i  1,2,3,..., p is called non-overlapping if I i  I j   ,
o o

for i  j . A collection of a finite non-overlapping cell D  I   I 1 , I 2 ,..., I p , 
p
with  I   Ii  E is called a partition of E . By taking a positive function 
I D i 1
defined on a cell E , we get a collections of the pairs of a finite point-cell
D  I , x   I i , xi ; i  1,2,3,..., p with 
x1 , x2 , x3 ,..., x p  E and
45
I i  I j   for i  j , and its called a  -fine partition on a cell E if

o o
p
I i  B ( x i ,  ( x i ))  E , for i  1,2,..., p and E   I i . If x i  I i , for
i 1
i  1,2,..., p then D  I , x   I i , x i ; i  1,2,3,..., p is called Perron  -fine

partition on a cell E . The point x with I, x   D is called associated point and
I, x   D is called  -fine interval with associated point x .
2. Bounded Variation Function in The Dimensional Euclidean Space
We shall now discuss the concept of functions of bounded variation in the

dimensional Euclidean space. A function of bounded variation is a real-valued
function whose total variation is bounded. In 2000, Lee and Vyborny gave relation
between monotonic function and bounded variation function. A function of
bounded variation can be expressed as a difference of two increasing functions [3].
The generalization of the definition of bounded variation function in
multidimensional space is started by defining interval function and point function
on a cell E  R n .
A function F : I ( E )  R is called real valued interval function. A

function F is said to be an additive interval function if for every I 1 , I 2  I ( E )
with I 1  I 2   and I 1  I 2 interval then F ( I1  I 2 )  F ( I 1)  F ( I 2) . An
o o
interval function can be constructed by the point function and otherwise point
function can be constructed by the intervals function.
Definition 2.1. Let E  [a , b ]  R n be a cell, I (E ) be a collection of all

intervals subset E and F : I ( E )  R is an interval function. Based on the
interval function F , we may define point function f : E  R . If x  E , the
point function f at x , written by f  x  is defined as follows:
 0, if x with xi  ai for at least one i
f (x)  
 F a , x , for x otherwise .
Opposite case is given in the following definition. Given a point function f
defined on E , we may defined a corresponding interval function F as follows.
Definition 2.2. Let E  [a , b ]  R n be a cell andf : E  R is a point

function. Based on the interval function f , with the value of the function f at the
point x  E written by f  x  . We may define interval function F : I E   R as
 
follows. If I  c , d  E with c  c1 , c2 , c3 , c4 ,..., cn  , d  d1 , d 2 , d 3 , d 4 ,..., d n 
and    1 ,  2 ,  3 ,  4 ,... n   I where  i  ci , d i  for i  1,2,..., n , and n( )
denotes the number of terms in  for which  i  ci for any i with i  1,2,..., n ,
then the value function F at I, written by F (I ) , denotes
f   . Where the summation is over all vertices  and in this
n ( )
F ( I )   (1)
46
case  stated the corner points of the interval I .

Furthermore, we have monotonically properties between a corresponding
interval function and point function. The point function has monotonically
properties if only if the interval function has monotonically properties. From the
equivalency between the interval function and point function on a cell E  R n , we
construct the definition of bounded variation function in the Dimensional
Euclidean Space. Some characterizations of bounded variation function defined on
a cell E  R n will be developed.
Definition 2.3. Let E  [a , b ]  R

n
be a cell and F : I ( E )  R interval
function. The variation of F on a cell E , written by V F E  or V F ; E  , is
define as follows
 
V F E   V ( F ; E )  sup  F ( I )  ,
 ID 
with supremum is over all D partition in E . If V F E    then F is said to be
of bounded variation on E .
Let BV (E ) stand for a collection of all bounded variation function on a cell

E  R n to a real number R . If F is bounded variation function on a cell E
then we will be represented by F  BV ( E ) . Furthermore, if an additive interval
function F is bounded variation function on a cell E then F is bounded.
Theorem 2.4. Let E  [a , b ]  R be a cell, X  E , F , G : I ( E )  R are

n
additive interval function and c is a real number. If F , G  BV ( X ) then

F G, cF , FG , max F , G  , min F , G  BV ( X ) with
F  G I   F ( I )  G ( I ) , cF I   cF (I ) and FG I   F ( I )G ( I ) for
every I  I ( E ) with the starting and end point of I belong to X .
Based on Theorem 2.4. we have that BV ( E ) is a linear space, because if

F , G  BV ( X ) then F  G  BV ( X ) , and cF  BV ( X ) for every c  R .
Next we will discuss properties of bounded variation interval functions associated
with the domain.
Theorem 2.5. Let E  [a , b ]  R be a cell. An additive interval function

n
F : I ( E )  R and I1 , I 2  E two non-overlaping cell. For each X 1  I 1 dan

X 2  I 2 if F  BV ( X 1 ) and F  BV ( X 2 ) then F  BV ( X 1  X 2 ) .
From Theorem 2.5., we have the corollary 2.6.
Corollary 2.6. Let E  [a , b ]  R be a cell, interval function F : I ( E )  R

n
and X 1 , X 2  E . If F  BV ( X 2 ) and X 1  X 2 then F  BV ( X 1 ) and then

VF ( X 1 )  VF  X 2  .
47
That means V F (.) is increasing interval function. Furthermore, based on Corollary

2.6. we have the Corollary 2.7.
Corollary 2.7. Let E  [a , b ]  R be a cell and F is an additive interval

n
function. A function F  BV (E ) if and only if it is the difference of two

increasing functions on E .
In the other word, if a function F on a cell E has a bounded variation then it is

can defined with the difference of two increasing functions on E , and if F is
increasing function on a cell E then F  BV (E ) . Next, we will discuss the
bounded variation properties in the point function and the associated interval
function.
Definition 2.8. Let E  [a , b ]  R be a cell and f : E  R is a point function.

n
The oscillation function f , written by  ( f , ) , with ( f , ) : I( E)  R defined

*
by  ( f , I )  sup f ( x )  f ( y ) , for each I  I ( E ) .

x , yI
We defined the bounded variation of point function on a cell E as in Definition

2.9.
Definition 2.9. Let E  [a , b ]  R be a cell and f : E  R is a point function.

n
A point function f is said to be of bounded variation on E , if there exist a positif

number M  0 such as for every D  I i ; i  1,2,3,...m partition on a cell E we
have
m
D   ( f , I )    ( f , I i )  M .
i 1
The number of V f ( E )  V ( f , E )  supD   ( f , I ) is said to be a variation of

point function f on a cell E , with the supremum is over all
D  Ii ; i  1,2,3,...m partition in E .
Furthermore, if a point function f is bounded variation function on a cell

E then f is bounded. Beside that BV (E ) is a linear space. In addition the
following theorem applies.
Theorem 2.10. Let E  [a , b ]  R be a cell, f : E  R is a point function

n
and F : I E   R is an additive interval function that associated with point

function f . A function f  BV ( E ) if and only if F  BV ( E ) .
Based on the theorem is concluded that the bounded variation properties is
maintained for the point function and intervals function are interlinked with each
other. Because in the next chapter we will discuss the generalization of integral
48
with respect to the bounded variation function G then we need more general
definition of the continuity, its called the continuity with respect to G .
Definition 2.11. Let E  [a , b ]  R be a cell, F , G : I ( E )  R are additive

n
functions on E and G  BV (E ) . We have
(i). An interval function F is said to be continuous with respect to G at

a cell E if for every   0 there is   0 such that for any cells
I  E with GI    then F (I )   .
(ii). An interval function F is said to be continuous with respect to G at

an open interval U  E , if for every   0 there is   0 such that
for any cells I  U with GI    then F (I )   .
Based on those definition, in the next section, we generalized the Henstock-

Stieltjes integral with respect to the integrator of bounded variation function on cell
E  Rn .
3. The Henstock-Stieltjes Integral with Respect to Integrator of Bounded

Variation Function in The Dimensional Euclidean Space
The Henstock-Stieltjes integral in R n is generalization of the Henstock-Stieltjes

integral on the interval a, b in the real number system R . Bounded variation is
important to the existence of Henstock–Stieltjes integral in real line. Beside that the
Henstock-Stieltjes integral in R n is also generalization of the Henstock integral in
R n . The success of Indrati (2002), who developed the Henstock-Kurzweil integral
in the dimensional Euclidean space, is opening the opportunities for
developing Henstock-Stieltjes integral with respect to the bounded variation
function integrator in R n . The discussion will begin by establishing definitions
and basic properties of Henstock-Stieltjes integral with respect to the integrator G
on a cell E  R n .
Definisi 3.1. Let E  [ a , b ]  R n be a cell, a function G : E  R is a bounded

variation function on a cell E ,  is a positif function on E and let
D  I , x   I i , xi ; i  1,2,3,..., p be a Perron  -fine partition on a cell E .
 
For any function f on x1 , x 2 , x3 ,..., x p , we set
p
 ( f , D, G )  D  f ( x )G ( I )   f ( xi )G ( I i ) .
i 1
and call this number the G -Stieltjes Sum of f associated with Perron  -fine
49
partition D on a cell E .
Now, we present the definition of the Henstock-Stieltjes integral by
constructing the definition of the Henstock-Stieltjes integral with take on a
bounded variation functions G : E  R as an integrator of the integral.
Definition 3.2. Let E  [ a , b ]  R n be a cell and G : E  R is a bounded

variation function on a cell E . A function f : E  R is said to be Henstock-
Stieltjes integrable (HS- integrable) with respect to G on E , in short,
f  HS (G , E ) , if there is a real number A such that for every   0 there is a
positive function  on E such that for every Perron  -fine partition
D  I , x   I i , xi ; i  1,2,3,..., p of E , we have
D  f ( x )G ( I )  A ,
p
where D  f ( x )G ( I )   f ( xi )G ( I i ) .
i 1
The numbers A in Definition 3.2. is unique and called the value of
Henstock-Stieltjes integral and will be written as A  ( HS ) f dG . If E is a 
E
degenerate interval, we defined ( HS )  fdG  0 . The HS (G , E ) in Definition 3.2.
E
represent a collection of all Henstock-Stieltjes integrable function from a cell E to
R with respect to a bounded variation function G on a cell E  R n .
Furthermore, the function f is called integrand and the function G is called
integrator. If the integrator function G is an identity function G( I )  I for every
I  I(E ) then The Henstock-Stieltjes integral become the Henstock integral in
R n . Thus the Henstock integral in R n is one of the special cases of the Henstock-
Stieltjes integral in R n . From the Definition 3.2., we have some basic properties of
the Henstock-Stieltjes integral as follows.
Theorem 3.3. Let E  [ a , b ]  R n be a cell, f , f1 , f 2 : E  R are functions

on E and G : E  R is a bounded variation function on a cell E .
(i) If f  HS (G, E ) and   R then f  HS (G, E ) and f  HS (G, E ) .
Moreover, we have
( HS )  fdG   ( HS )  fdG  ( HS )  fd G  .
E E E
(ii) If f1 , f 2  HS (G, E ) then f1  f 2  HS (G, E ) .

Moreover, we have
( HS )   f1  f 2 dG  ( HS )  f1dG  ( HS )  f 2 dG .
E E E
(iii) If f  HS (G, E ) and f  HS ( H , E ) then f  HS G  H , E  .

Moreover, we have
( HS ) fd G  H   ( HS ) fdG  ( HS ) fdH .
  
E E E
50
Based on the Theorem 3.3., we have that HS (G , E ) is a linear space. For

the next discussion, we define the Cauchy criterion that provides an alternative to
determine of Henstock-Stieltjes integrable function on a cell E  R n without
having to determine the value of the integral.
Theorem 3.4. (Cauchy’s Criterion Theorem) Let E  [ a , b ]  R n be a cell and

G : E  R is a bounded variation function on a cell E . A function
f  HS G, E  if and only if for every   0 there exists a positive function 
on E such that for every two Perron  - fine partitions D1  I, x  dan
D2  I, x  of E , we have
D1  f ( x )G ( I )  D2  f ( x )G ( I )   .
PROOF. () From f  HS (G , E ) , there is a real number A  HS  f dG such 
E
that for every   0 there is a positive function  on E such that for every
Perron  -fine partition D1  I, x  of E and Perron  -fine partition
D2  I, x  of E , we have
D1  f ( x )G( I )  A   and D2  f ( x )G( I )  A   .
2 2
Then we have
D1  f ( x)G(I )  D2  f ( x)G(I )  D1  f ( x )G(I )  A  A  D2  f ( x )G(I )
 D1  f (x)G(I )  A  A  D2  f (x)G(I )
 D1  f (x)G(I )  A  D2  f (x)G(I )  A

 
   .
2 2
1
, for n N , there is a positive function  n on E such
*
() For every  
n
that for any two Perron  n - fine partitions D1  {( I , x )} and D2  {( I , x )}
* * *
of E , we have
    1
D1  f ( x )G ( I )  D2  f ( x )G ( I )  .
* *
n
For every n N , we define a positive function  n on E , where
 
 n ( x )  min 1* ( x ), 2* ( x ), 3* ( x ),..., n* ( x ) for every x  E . From the
definition we have that  n   m , for every n  m , so we have, every Perron  n -
fine partition on E also Perron  m - fine partition on E , when n  m . For every
n , we can choose only one Perron  n -fine partition Dn  {(I , x )} on E then we
construct Sn  Dn  f ( x )G( I ) . Moreover, we have Sn  the sequence in R .
51
With Archimedian properties, that given any real number   0 there exists
1
natural number no  , then for every m, n  no , n  m and by taking n a

positive function on E , we have
1 1
S n  S m  Dn  f ( x )G ( I )  Dm  f ( x )G ( I )    .
m no
Thus obtained that Sn  is a Cauchy sequence in R , then Sn  is convergence
sequence. So, there exists a real number A as the limit of Sn  . That means for
2
every   0 as the above, there exist natural number n  , then for every


n  N , n  n we have S n  A  . Let    n is a positive function on E ,
2
then for every Perron  -fine Partition D  {( I , x )} , on E we have
D f ( x )G(I )  A  D f ( x )G(I )  Sn  Sn  A
 D  f ( x )G( I )  Sn  Sn  A
1   
    . 
n 2 2 2
So we can proved that A is the Henstock-Stieltjes integral value of f with
respect to G on E . That means f  HS (G , E ) .
Theorem 3.5. Let E  [ a , b ]  R n be a cell , G : E  R is a bounded

variation function on a cell E and E1 , E2 be non-overlapping cell in R n with
E  E1  E2 . if f  HS G, E1  and f  HS G, E2  then f  HS G, E  .
Moreover, we have
HS  fdG  HS   fdG  HS   fdG .
E E1 E2
By Cauchy’s Criterion and Theorem 3.5. we have Theorem 3.6. as follows.
Theorem 3.6. Let E  [ a , b ]  R n be a cell, B  E and G : E  R is a

bounded variation function on a cell E . If f  HS G, E  then f  HS G, B 
for every cell B  E .
PROOF. Since f  HS G , E  , By Cauchy’s Criterion, for every   0 , there
exists a positive function  on E such that for every two Perron  - fine
partitions D *  I , x  and D **  I , x  of E , we have
D  f ( x )G(I )  D  f ( x )G(I )   .
* **
If B  E then we have the trivial proof. If B  E then there exists collection of

non–overlapping finite subcells E , C  C1 , C 2 , C 3 ,..., C m , such that
52
m
cl ( E  B)   Ci . By Cousin’s lemma there exists Perron  -fine partition Di on
i 1
a cell Ci , for every Ci C , i  1,2,3,.., m . Furthermore, if DB and PB are
 -fine partition on a cell m 

Perron B then D   DB    Di  dan
 i 1 
m 
D   PB    Di  also Perron  -fine partition on E . Then we have
 i 1 
DB  f ( x )G( I )  PB  f ( x )G ( I )
m  m 
 DB  f ( x )G ( I )    Di  f ( x )G ( I )    Di  f ( x )G ( I )  PB  f ( x )G( I )
 i 1   i 1 
 m    m  
  DB  f ( x )G ( I )    Di  f ( x )G ( I )      Di  f ( x )G ( I )  PB  f ( x )G ( I ) 
  i 1     i 1  
  m    m 
  DB    Di   f ( x )G ( I )   PB    Di   f ( x )G ( I )
  i 1     i 1  
 D f ( x )G ( I )  D f ( x )G ( I )   .
By Cauchy’s Criterion, f  HS (G , B ) .
Theorem 3.7. Let E  [ a , b ]  R n be a cell, B  E and f , G : E  R are

functions on a cell E . If f is a continuous function on E and G is a bounded
variation function on a cell E then f is Henstock-Stieltjes integrable with
respect to G on a cell E .
PROOF. Since G is a bounded variation function on a cell E , then there exists a
constant M  2V (G; E )  1  0 such that G ( I )  M , for every I  I (E ) , with
V (G; E) is variation of G on a cell E . We choose  n  be an arbitrary
decreasing sequence of real number and converges to 0. Since f is a continuous
function on a closed and bounded cell E then f uniformly continuous on E . So
for every natural number n there exists positif constan  n  0 , such that for every
x , y  E , x  y   n , we have
n
f (x)  f ( y)  .
M
Without loose of the meaning, assume that  n1   n , for every n  N , then we
choose arbitrary Perron  n -fine partition D n  I , x  on a cell E . Construct
S n  such that
S n  Dn  f ( x )G ( I ) for every n  N .
Furthermore, we choose m, n  N with m  n . Then we have that  m   n for
every m  n . From the definition we have that  n   m , for every n  m , so we
have, every Perron  n -fine partition on E also Perron  m - fine partition on E ,
53
when n  m . We have for every Perron  m -fine partition Dm  I , x  on E

also Perron  n -fine partition on E . Therefore, if I  I    then there exists
point z  I  I  and we have
f ( x )  f ( x )  f ( x )  f ( z )  f ( z )  f ( x )
 f ( x )  f ( z )  f ( z )  f ( x )
m n n
  2 .
M M M
1
Without loose of the meaning, we choosed  n  , for n  1,2,3,... then  n  is a
n
decreasing sequence of real number and converges to 0. With Archimedian
1
properties, that given any real number   0 , there exists natural number K 

, then for every m, n  K , n  m , we have  m   n   K   . For every
m, n  K , n  m , put one positive constant  K on E , then we have
S m  S n  Dm  f ( x )G ( I )  Dn  f ( x )G ( I )
 Dm  f ( x )G( I )  Dm  f ( z )G( I ) Dm  f ( z )G( I ) Dn  f ( x )G( I )

 Dm  f ( x )G( I )  Dm  f ( z )G( I )  Dn  f ( z )G( I ) Dn  f ( x )G( I )
 Dm   f ( x )  f ( z ) G ( I )  Dn   f ( z )  f ( x ) G ( I )
m n n
 V (G; E )  V (G; E )  2V (G; E )  n   .
M M M
Thus obtained that Sn  is a Cauchy sequence in R then Sn  is convergence
sequence. So, there exists a real number A as the limit of Sn  . That means for
2
every   0 as the above, there exist natural number n  , then for every


n  N , n  n we have S n  A  . Let    n is a positive function on E ,
2
then for every Perron  -fine Partition D  {( I , x )} , on E we have
D f ( x )G(I )  A  D f ( x )G(I )  Sn  Sn  A
 D  f ( x )G( I )  Sn  Sn  A
1   
   . 
n 2 2 2
So we can proved that A is the Henstock-Stieltjes integral value of f with
respect to G on E . That means f  HS (G , E ) .
From the above theorem we have that the sufficient condition to guarantee
the existence of the Henstock-Stieltjes integral function with respect to a bounded
variation function G on a cell E  R if f a continuous function.
n
Some convergence theorem of the integral are still prevailed on the Henstock-
54
Stieltjes integral with respect to the integrator of bounded variation function G on

a cell E  R n . In the theory of integral, the convergence theorems give some
sufficience conditions for a sequence of Henstock-Stieltjes integrable functions
 f k  such that
lim( HS )  f k dG  ( HS )  lim f k dG .
k  k 
E E
In this section, we discuss some convergence theorem such as the Uniformly
Convergence Theorem and the Monotone Convergence Theorem on the Henstock-
Stieltjes integral with respect to the integrator G on a cell E  R n .
Theorem 3.8. (The Uniformly Convergence Theorem) Let E  [a , b ]  R be a

n
cell, G : E  R is a bounded variation function on a cell E and f k : E  R

with f k  HS(G, E) for every k  N . If  f k  uniformly converges to a function
f on a cell E then f  HS (G , E ) . Furthermore,
( HS ) fdG  lim( HS ) f k dG .
k 
E E
PROOF Since G is a bounded variation function on a cell E , then there exists a
constant M  0 such that for every P  I i ; i  1,2,3,...., m partition on a cell
m
E we have P  G ( I )   G ( I i )  M . Let   0 be given. Without loose of
i 1
the meaning, assume that  f k  uniformly converges to a function f on a cell E

then there is exists natural number k o  N , then for every x  E and k  k o , we

have f k (x)  f (x)  .
3 .2 k 1
M  1
Since f k  HS (G , E ) for every k  N , then there is exists real Ak then for every
  0 at the above, there is  positive function on acell E such as for every
Perron  - fine partitions D  I , x  on a cell E we have
D  f k ( x )G( I )  Ak   .
We defined Ak  is the sequence, where
Ak  ( HS )  f k dG
E
for every k .
So Ak  R for every k . Furthermore we must show that Ak  is a Couchy
sequence. merupakan barisan Cauchy. For every   0 at the above, we choose
k o  no . Then for every natural number m, n  no and for arbitrary Perron  -
fine partition D on a cell E , we have
An  Am  ( HS )  f n dG  ( HS )  f m dG
E E
55
 ( HS )  f n dG  D  f n ( x )G ( I )  D  f n ( x )G ( I )  D  f ( x )G ( I )
E
 D  f ( x )G ( I )  D  f m ( x )G ( I )  D  f m ( x )G ( I )  ( HS )  f m dG
E
   D   f n ( x )  f ( x ) G ( I )  D   f ( x )  f m ( x ) G ( I )  

 

  G( I )   G( I )  
3.2 ( M  1)
n 1
n
m 1 3.2 ( M  1)
m
 
       3 .
3 3
So, for every k , Ak  is a Cauchy sequence in R and convergence sequence.
So, there exists a real number A as the limit of Ak . Katakan konvergen ke
suatu A  R . That means for every   0 as the above, there exist natural
number n1 , then for every k  n1 we have
Ak  A   .
We put k *  maxno , n1  then we defined a positive function  ( x )   k * ( x ) for
every x  E and for every Perron  k * -fine partition D * on a cell E we have
D  f ( x )G ( I )  A  D  f ( x )G ( I )  D  f k ( x )G ( I )
 D  f k ( x )G ( I )  Ak  Ak  A
 D  f ( x )  f k ( x ) G ( I )    

 
 G ( I )  2   2  3 .
k 1 3.2 ( M  1)
k
3
That means
A  ( HS )  fdG .
E
Moreover, we have
( HS )  f dG  lim ( HS )  f k dG
k 
E E
Next, we will discuss the monotone convergence theorem in the Henstock-Stieltjes
integral on E  R n .
Theorem 3.9. ( Monotone Convergence Theorem) Let E  [a , b ]  R be a

n
cell, G : E  R is a bounded variation function on a cell E , and

f k , f : E  R for every k  N . If
(i). f k  f almost everywhere in E as k   with f k  HS(G, E) for
every k  N ,
(ii).  f k  is monotone in E , and
56
(iii). Fk (E) , the sequence of the value of Henstock-Stieltjes integral

f k with
respect to integrator of G on E , converges to A as k   , then
f  HS (G , E ) to A on E and
( HS ) fdG  lim( HS ) f k dG .
k 
E E
PROOF Since G is a bounded variation function on a cell E , then there exists a
constant M  0 such that for every P  I i ; i  1,2,3,...., m partition on a cell
E we have
m
P  G ( I )   G ( I i )  M .
i 1
For fk  is increasing monotone sequence in a cell E . Let   0 be given.

Based on (i) f k  HS (G , E ) for every k  N . Then for every k  N and   0
at the above, there is positive function  k -fine on E such as for every
Dk  I , x  Perron  k - fine partition on E we have

Dk  f k ( x )G ( I )  Fk ( I )  .
3.2 k 1
Furthermore we have

f k ( x )G ( I )  Fk ( I )  Dk  f k ( x )G ( I )  Fk ( I ) 
3.2 k 1
for every k  N .
Based on (ii),  fk  increasing monotone sequence in a cell E and
f k  HS (G, E ) . By Theorem 3.6.. we have f k  HS (G, I ) where I  E for
every k  N . Moreover, we have
Fk ( I )  ( HS )  f k dG  ( HS )  f k 1 dG  Fk 1 ( I ) (3.1)
I I
Then Fk (I ) is increasing monotone sequence and bounded. That means

Fk (I ) converges. Suppose Fk (I ) converges to F (I ) . Based on (iii) Fk (E )
converges to real nuber A . Moreover we have
lim Fk ( E )  lim ( HS )  f k dG  A .
k  k 
E
So there is positive number k o such that for every k  k o we have


Fk ( E )  A  .
3
Next, without loose of generality we suppose f k  f almost everywhere on E
for k   . Then for every x  E and   0 at the above, we can choose the
positive integer K ( x ,  )  k o , then for every k  K ( x ,  ) we have

f k (x)  f (x)  .
3.2 ( M  1)
k
Furthermore, we defined a positive function  on a cell E , with

 ( x )   K ( x , ) ( x ) , for every x  E . So, for any Perron  - fine partition
57
D  I, x  on a cell E we have

a) There is infinite associated point in D , so we can choose
p  minK ( x ,  ) ; x  I , x  D. By formula (3.1) we have
F p ( I )  FK ( x , ) ( I ) ,
for every I  E .
Furthermore, we have
F p ( E )  D  F p ( I )  D  FK ( x , ) ( I )  F ( E )  A .
Then we have
A  D  FK ( x , ) ( I )  A  D  F p ( I ) .
b) For every pairs of a finite point-cell I, x   D are Perron  - fine partition on
a cell I and union for all the pairs of a finite point-cell I , x   D are Perron
 - fine partition on a cell E .
Moreover, we have
D  f ( x )G ( I )  A  D  f ( x )G ( I )  f K ( x , ) ( x )G ( I )
 D  f K ( x , ) ( x )G ( I )  FK ( x , ) ( E )  D  FK ( x , ) ( E )  A
 D   f ( x )  f K ( x , ) ( x ) G ( I )  D  f K ( x , ) ( x )G ( I )  FK ( x , ) ( E )
 D  F p ( I )  A
   2 
  K ( x , )
G(I )   K ( x , ) 1

K ( x , ) 1 3.2 ( M  1) 3.2 K ( x , ) 1 3
  2  
  K ( x , )
M  K ( x , ) 1

K ( x , ) 1 3.2 ( M  1) K ( x , ) 1 3.2 3
  
  . 
3 3 3
That means f  HS (G, E ) to the number A on a cell E . Then by (iii), we have
lim Fk ( E )  lim ( HS )  f k dG  A  F ( E )  ( HS )  fdG .
k  k 
E E
In the other word, we have
( HS )  f dG  lim( HS )  f k dG .
k 
E E
By remaining this condition: “if fk  decreasing monotone on a cell E then

 f k  increasing monotone on a cell E ”, that the Theorema 3.9. also prevailed
for  f k  decreasing monotone on a cell E.
The properties of the Henstock-Stieltjest integral in the real line still hold in
the dimensional Euclidean space, with respect to the integrator of bounded
variation function. Some theorems include the Cauchy’s Criterion (Theorem 3.4.),
the existence of the Henstock-Stieltjes integral function with respect to a bounded
variation function G on a cell E  R (Theorem 3.7.), the Uniformly
n
58
Convergence Theorem (Theorem 3.8.) and the Monotone Convergence Theorem

(Theorem 3.9.) are still valid on the Henstock-Stieltjes integral with respect to the
integrator of bounded variation function G on a cell E  R n .
References
[1] Indrati, C.R., The Henstock-Kurzweil Integral in n - dimentional Euclidean Space,

Disertation, Gadjah Mada University, Yogyakarta, 2002.
[2] Lee P.Y., Lanzhou Lectures on Henstock Integration, World Scientific, Singapore,
1989.
[3] Lee P.Y. & Výborný, R., The Integral: An Easy Approach after Kurzweil and
Henstock, Cambridge University Press, 2000.
[4] Lim, J. H., Yoon, S., L., and Eun, G.S, On Henstock-Stieltjes Integral, Kangweon –
Kyungki Mathematics Journal 6, No:1, pp. 87-96, 1998.
[5] Pfeffer, W.F., The Riemann Approach to Integration, Cambridge University Press,
New-York, USA, 1993.
ON UNIFORM CONVERGENCE OF SINE INTEGRAL

WITH CLASS p-SUPREMUM BOUNDED VARIATION
FUNCTIONS
MOCH. ARUMAN IMRON1, CH. RINI INDRATI2AND WIDODO3
1Department of Mathematics , University of Brawijaya and Graduate School,

University of Gadjah Mada, maimr@ub.ac.id
2Department of Mathematics, University of Gadjah Mada, rinii@ugm.ac.id
3Department of Mathematics, University of Gadjah Mada,
widodo_mathugm@yahoo.com
Abstract. Recently, the monotone decreasing coefficients of sine series has

been generalized by classes of general monotone sequences, non-one sided
bounded variation sequences, supremum bounded variation sequences. Further
class of general monotone functions as counterpart of class of general monotone
sequences can be extended to classes of p-general mononotone functions and p-
non-one sided bounded variation functions. In this paper we construct class of p-
supremum bounded variation functions, a generalization of p-general monotone
functions and p-non-one sided bounded variation functions. The second we
study the uniform convergence of sine integral with this class.
Key words and Phrases: p-supremum bounded variation functions, sine integral,
uniform convergence.
1. Introduction
Chaundy and Jollife [1] proved the following classical theorem:
Theorem 1.1. Suppose that ⊂ 0,∞ is decreasingly tending to zero. A

necessary and sufficient conditions for the uniform convergence of the series
sin 1.1
is lim 0.
→
Denote by MS the monotone decreasing sequences of coefficients (1.1), the class

of MS has been generalized by many researchers such as Tikhonov [11], Zhou et.
al. [12], and Korus [7]. These classes are GMS (general monotone sequences),
NBVS ( non-one sided bounded variation sequences ), MVBVS (mean value
59
60
bounded variation sequences) and SBVS (supremum bounded variation sequences).

Zhou et al [12] proved that ⊊ ⊊ ⊊ and Korus [7]
showed that ⊊ .
Futhermore, Liflyand and Tikhonov [9] generalized GMS to (p-

general monotone sequences), 1 ∞. Let and be two
sequences of complex and non-negatif numbers, respectively, a pair , ∈
if there exists 0 such that
/
| | ,
44
for p, 1 ∞ . Theorem 1.2. was proved by Dyachenko and Tikhonov [2]
and it is generalization of Theorem 1.1.
Theorem 1.2. Let , ∈ , if for p=1 and 1 , then the series

(1.1) converges uniformly on 0, 2 .
Imron, et al [6] generalized MVBVS and SBVS to (p-mean value

bounded variation sequences) and (supremum bounded variation
sequences) respectively. A pair , ∈ if there exist 0 and
2 such that
/
| | ,
/
for p, 1 ∞and , ∈ if there exist positif constant and 1

such that
/
| | sup ,
/
for p, 1 ∞. Alittle modification of definition of gives a class
2 .A pair , is said to be p-supremum bounded variation sequences of
second type , written , ∈ 2 , if there exist positive constant and
⊂ 0, ∞ tending monotonically to infinity depending only on , such
that
/
| | sup ,
holds for p, 1 ∞. Imron, et al [ 5] have shown that ⊊

⊊ 2 .
The following theorem was proved by Imron, et. al. [4] and it is more general than
Theorem 1.2.
/
Theorem 1.3. Let , ∈ 2 , if sup ∑ 1 , for
1 ∞ , then series (1.1) converges uniformly on 0, 2 .
Recently certain efforts have been made in extending the class of monotone
sequences to class of monotone functions. Liflyand and Tikhonov [9] introduced
the class of GM (general monotone functions) as counterpart of GMS and
generalized GM to ,1 ∞. Furthermore Imron et. al. [3] generalized
to (p-non-one sided bounded variation) . The definition of and
are stated in Definition 1.4. For the rest of the discussion, we assumed f,
61
defined on , is locally absolute continuous on where 0, ∞ .
Definition 1.4. Let and be complex valued and non-negatif functions,

respectively and defined on ,
(i) A pair , is said to be p-general monotone, written , ∈ , if there
exists a positive constant C such that
/
| ′ | , ∈ ,
(ii) A pair , is said to be p-non-one sided bounded variation, written , ∈

, if there exists a positive constant C such that
/
| ′ | 2 , ∈ ,
for 1 ∞.
Furthemore Moricz [10] defined , and Korus [8] defined

and 2 in Definition 1.5.
Definition 1.5. A complex valued function belongs to the class of

(i) mean bounded variation functions if there exist constants 0,
0 and 2 depending only on f such that
| ′ | | |
for .
(ii) supremum bounded variation functions if there exist constants
0, 0 and 2 depending only on f such that
| ′ | sup | |
/
for .
(iii) supremum bounded variation functions of second type 2 if there exist
constants 0, 0 and a function : → 0, ∞ tending monotonically
to infinity, depending only on f such that
| ′ | sup | |
for .
The counterpart of SBVS called will be used to observe the
sufficient condition of the integral (1.2) where : → and ∈ to be
uniform convergence of the sine integral
62
sin . 1.2
The uniform convergence of (1.2) means the uniform of
sin , as → ∞.
Due to (1.2), Moricz [10] used a member of to proved Theorem 1.6.
Theorem 1.6. Assume ∈ ,

(i) if : → , and
| | ∞ as → ∞, 1.3
then the integral (1.2) is uniformly bounded as → ∞.
(ii) If : → 0, ∞ and integral (1.2) is uniformly bounded, then condition (1.3)
is satisfied.
In the present paper, we shall construct classes of p-mean value bounded

variation functions , p-supremum bounded variation functions
and p-supremum bounded variation functions of second type 2 .
Our goal is to extend the Theorem 1.3. from sine series (1.1) to sine integral (1.2)
by the results of the new construction and using Theorem 1.6.
2. Main Results
In this section, we construct class of p-mean value bounded variation

functions, p-supremum bounded variation functions and p-supremum bounded
variation functions of the second type. Furthemore we investigate the relations
among those classes and also study other properties of class of p-supremum
bounded variation functions of the second type. The Goal of this paper is to study
the uniformly corvergence of sine integral (1.2) in Theorem 2.11 and uniformly
bounded of integral (1.2) in Theorem 2.12. We always assume that each such
complex valued function f is defined on measurable and locally absolute
continuous on .

respectively and defined on . A pair , is said to be p-mean bounded
variation function, written , ∈ , if there exist constants
0, 0 and 2 depending only on f such that
/
| ′ | ,
for and 1 ∞.

63
respectively and defined on . A pair , is said to be p-supremum bounded

variation functions, written , ∈ if there exist constants 0,
0 and 2 depending only on f such that
/
| ′ | sup
for and 1 ∞.
Futhermore, we define a class is more general than class , that is class

of 2 (p-supremum bounded variation functions of second type), which
is stated in Definition 2.3.

respectively and defined on , for 1 , a pair , is said to be p-supremum
bounded variation functions of second type, written , ∈ 2 , if
there exist constants 0, 0 and a function : → 0, ∞ tending
monotonically to infinity, depending only on f such that
/
| ′ | sup ,
for and 1 ∞.
The following properties, we prove that classes of and
are subclass of 2 .
Theorem 2.4. If 1 ∞, then ⊆ .
PROOF. Let , ∈ , and be the integer such that 2

2 . By Definition 2.1 there exist constants 0, 0and Theorem 2.3 [8] by
replacing f with , we have
| ′ |
1
sup
/
Therefore , ∈ , with constant 1 , A and .□

64
Theorem 2.5. If 1 ∞, then ⊆ ⊆ .

PROOF. From definition 1.1, it is clear that
⊆ .
Second, we proof that ⊆ . By idea proof of Theorem
3 introduced by Moricz [10], if ∈ obtained
4
| ′ | | | | 2 | | |
for 4 and a positive constant 4C. Since , ∈

/
4
| ′ | 2
for non-negatif function defined on . Therefore , ∈
Theorem 2.6. If 1 ∞, then ⊆
PROOF. Let , ∈ , by Definition 2.2 there exist constants

0, 0 and for such that
/
| ′ | .
For 1 ∞ and applying Holder’s inequality to | | and 1,
| | | | .1
/ / /
/ /
| | 1 | | .
Therefore
/ /
/ /
′
| ′ | | ′ | .
Thus , ∈ .□
Theorem 2.7. If 1 ∞, then ⊆ 2 .

PROOF. Take / , it is easy to see that ⊆ 2 .
Theorem 2.8. If 1 ∞, then 2 ⊆ 2

PROOF.This proof is similar with the proof of Theorem 2.6 by substituting
65
/ with .□
Some properties of 2 ,1 ∞, are stated below

Theorem 2.9. If , ∈ 2 and
/
sup decreasing monotone for ∈ then
| ′ | /
sup
holds for p, 1 ∞.
PROOF. Let , ∈ 2 with constants C, A and a function B. By

definition 2.2, we have
| ′ | | ′ |
/
/
| ′ | 2
2
| |
2
/
2
′ sup
2
1
′ /
sup
2
/
sup
The proof is complete. □
/
Theorem 2.10. If , ∈ 2 and sup
tending to 0 for → ∞ , then
| ′ | ∞
66
for p, 1 ∞.
/
PROOF. Let sup tending to 0 and we denote
/
sup sup .
then → 0. Let , ∈ 2 and from proof of Theorem 2.9. we

obtained
/
2
| ′ | ′ sup
2
, 2
2
Therefore
| ′ | ∞.
In the properties bellow, we always assume that for : → ,

. locally integrable on (2.1)
/
Theorem 2.11. Let , ∈ 2 and sup is
tending to 0 for → ∞, 1 ∞,
(i) If condition
→ 0 → ∞, 2.2
is satisfied, then sine integral of (1.2) converges uniformly in t.
(ii) If : → 0, ∞ and sine integral of (1.2) converges uniformly in t, then
condition (2.2) is satisfied.
PROOF.
(i) We denote
/
sup sup .
/
Since sup → 0, then → 0. Let , ∈
2 , we may use proof of Theorem 2.10. for every 0 there

exists ∈ , such that
67
| ′ | 2.3
for all . Furthemore, first for 1/ and by condition (2.1)

and (2.2) we have
/
1
| | | |
1/
Second, for he case , integration by parts and (2.3) we have
1
| | | | | |
| | | | | | 3
From first and second case, the integral (1.2) converges uniformly in t.
(ii) Let integral (1.2) converges uniformly in t, this means that for every 0, there
exists ∈ such that
sin , for all and .
We start with the equality
, 0 2 . 2.4
Since , ∈ 2 and from (2.4) that

/
| | | | /
2.5
/
sup
We integrate (2.5) with respect to y over the interval ,2 to obtain
/
sup 2.6
68
/
Since sup tending to 0 for → ∞ , from (2.6) we
have
. 2.7
Set
, ∈ ,
4
then
, 2 .
4 2
From these inequality and (2.7)we have
sin sin sin ,

4 4
for all ∈ .
Since integral (1.2) is supposed to converge uniformly in t, therefore
→ 0 as → ∞.
/
Theorem 2.12. Let , ∈ 2 and sup is
tending to 0 for → ∞, 1 ∞,
(i) If condition
| | ∞ as → ∞, 2.8)
then the sine integral of (1.2) is uniformly bounded as → ∞.
(ii) If : → 0, ∞ and the sine integral of (1.2) is uniformly bounded, then
PROOF.
(i) We may use proof of Theorem 2.10, there exists ∈ , such that
| ′ | 2.9
for all . Furthemore, first for 1/ and by condition (2.1)

and (2.8) there exists constant 0 such that
69
1
| | | | .
Second, for case , integration by parts and (2.9) we have
1
| | | | | ′ |
| | | | | | 2
From first and second case, the partial integrals of (1.2) uniformly are
bounded.
(ii) Let partial integrals of (1.2) are buniformly bounded, this means that for every
constant 0, there exists ∈ such that
sin , for all and . 2.10
Similarly with proof of Theorem 2.11. (ii) wehave
sin sin sin ,

4 4
for all ∈ .
By inequality (2.9) we have
as → ∞.
/
Corollary 2.13. Let , ∈ and sup is
/
tending to 0 for → ∞, 1 ∞.
70
(i) If condition (2.8) satisfies, then the integral of (1.2) isuniformly bounded as
→ ∞.
(ii) If : → 0, ∞ andthe sine integral of (1.2) is uniformly bounded, then
/
Corollary 2.14. Let , ∈ ,1 ∞ and /
is tending to 0 for → ∞.
(i) If condition (2.8) satisfies, then the sine integral of (1.2) is uniformly
bounded as → ∞.
(ii) If : → 0, ∞ and the sine integral of (1.2) is uniformly bounded, then
In this paper we have introduced the class 2 . We have investigated

that
(i) The class of 2 is more general than class by Theorem
2.4, Theorem 2.5 and Theorem 2.7.
(ii) The sufficient condition of Uniformly convergence of (1.2) in Theorem 2.11
/
are , ∈ 2 for 1 ∞, sup is
tending to 0 for → ∞ and → 0 as → ∞.
(iii) The sufficient condition of Uniformly bounded of (1.2) in Theorem 2.12 are
/
, ∈ 2 for 1 ∞, sup is tending
to 0 for → ∞ and | | ∞ as → ∞.
Acknowledgement. The authors gratefully acknowledge the support of the

Department of Mathematics, Faculty of Mathematics and Sciences University of
Brawijaya and Graduate School Department of Mathematics, Faculty of
Mathematics and Sciences, University of Gadjah Mada.
References
[1] Chaundy, T. W. and Jollife, A.E., The Uniform Convergence of certain class
trigonometric series, Proc. London, Soc. 15, 214-116, 1916.
[2] Dyachenko, M. and Tikhonov, S., General monotone sequences and convergence of
trigonometric series, in: Topics in Classical Analysis and Applications in Honor of
Daniel Waterman ,World Scientific, Hakensack, NJ, pp. 88-101, 2008.
[3] Imron, M. A., Indrati, C, R. and Widodo, On p-non One Sided Bounded Variation
Sequences and Functions, Proceeding of 2nd Basic Science International
Conference, M46-M50,FMIPA, UB, Malang, 2012.
[4] Imron, M.A., Indrati, C. R. and Widodo, On uniform convergence of trigonometric
series under p-supremum bounded variation condition, Proceeding of the third
Basic Science International Conference, M041-M043,FMIPA, UB, Malang, 2013.
71
[5] Imron, M. A., Indrati, C. R. and Widodo, Some properties of class of p-Supremum
bounded Variation sequences, Int.Journal of Math. Analysis, Vol 7, no.35, 1703-
1713, 2013. online:http//dx.doi.org/10.12988/ijma.2013.3494.
[6] Imron, M.A., Indrati, C. R. and Widodo, Relasi Inklusi pada Klas Barisan p-
Supremum Bounded variation, Jurnal Natural A ,No 1, Vol 1, (1-6) FMIPA, UB,
Malang, 2013.
[7] Korus, P., Remark On the uniform And L1-Convergence of Trigonometric Series,
Acta Math. Hungar, 128(4), 2010.
[8] Korus, P., On the uniform Convergence Of special sine integrals, Acta Math.
Hungar, 133 (I), 82-91, 2011.
[9] Liflyand, E. and Tikhonov, S., A concept of general monotonicity an application,
Math. Nachr, 284, No.8-9, 1083-1098, 2011.
[10] Moricz, F., On the uniform convergence of sine integral, J.Math Anal Appl, 254,
213-219, 2009.
[11] Tikhonov, S., Best approximation and moduli of Smoothness computation and
Equivalence Theorems, Journal of Approximation Theory, 153, 19-39, 2008.
[12] Zhou, S.P., Zhou, P. and Yu, D. S., Ultimate generalization to monotonicity for
Uniform Convergence of Trigonometric Series,
online:http//arxiv.org/abs/math/0611805v1
Applied Mathematics
APPLICATION OF OPTIMAL CONTROL OF THE CO2

CYCLED MODEL IN THE ATMOSPHERE BASED ON
THE PRESERVATION OF FOREST AREA
AGUS INDRA JAYA1, RINA RATIANINGSIH2, INDRAWATI3
1MathematicsStudy Program of Tadulako University, ratianingsih@yahoo.com

2MathematicsStudy Program of Tadulako University, ratianingsih@yahoo.com
3Mathematics Study Program of Tadulako University, indrawati@yahoo.com
Abstract. The CO2 content in the atmosphere cycles related to the plant
assimilation process, respiration process and many other human activities.
There are two scenarios to be obse rved in th is paper. The first
scenario is c onside r the na ture in to a tmo sphere and biosphere zone,
while the sec ond one is conside r th e nature into the a tmosphere,
biosphere a nd sea-z one. The growth o f the CO2 content in those zones is
controlled by propose the forest area prese rvation ( ) and the succesess
ra te o f the reforesta tion program . The optimal control of it is also
designed to satisfy the stability of te mperature and pressure intera ction
in the atmo sp here. The H am ilton ia n functio n is gene rated in ord er to
find the va lu es of and using the Lagrange equa tio ns tha t must
satisfy the statione r cond ition and both state and co -state equa tions.
It could be see n fro m the research result that no t on ly the CO2 content
in the atmosphere and biosphere zones must be controlled but also the CO2
content in the sea zone. This effort will reduces the CO2 content in each zones
and keeps the stable temperature and pressure interaction in the atmosphere.
Key words and Phrases: Hamilton ia n function, Lag range e quations,
optimal control, state and c o-sta te eq uatio ns, sta tioner codition.
1. Introduction
The growth of CO2 emition rate is one of the most strategical issues topic in
climate changing research. Forest and green zone area, as a main biotic resources,
are able to change CO2 to be O2 such that could play a role to maintain a stable
interaction of temperature and pressure as main climate unsures. The climate
changing actually indicates the forest unability to reduce the CO2 emition rate such
that we come to the global warming phenomena.
Haneda [2] stated that Global warming describes the dynamic of
temperature that gives impact to the climate changing. The impacts could be seen
from the changing of wind and climate pattern, atmospheric hurricane and
hydrological cycle [11]. Tjasyono stated on [14] that green house effect, i.e CO2,
gives a significant contribution to the growth of CO2 content and pressure in the
atmosphere. Soedomo [11] supports [14] by stated that it already indicated the
72
73
growth of temperature because the CO2 content rise continuesly. Global warming
could be avoided by arrange the forest area to control the CO2 content and to resist
the growth of temperature in the asmosphere. The problem is how to control it
based on the preservation of forest area. Mathematically this problem is stated as
the CO2 content controlling problem. The related optimal control could be
got by building such performance index that satisfy Pontryagin
maximum principe in [6], [7] and [12].
It could be understand that the contribution of CO2 content in the atmosphere
will determine the temperature profile such that causes the dynamics of the other
main climate unsure, i.e. pressure. The dynamics of temperature and pressure that
caused by the growth of CO2 content in the atmosphere is very important to be
observed. Refering to [3], the interaction between both main climate unsures is
derived from the thermodynamics law and is analysed by consider the represented
mathematical model that describe the interaction between temperature and pressure
in the atmosphere. The most important result of [2] is the dependencies of the
stability of the system with respect to the parameter that represent the caloric
exchange rate to the atmosphere. This parameter must be bounded by the
difference between the the heat capacity and the specific volume of gas. This is the
main reason to consider the contribution of CO2 content in the atmosphere to the
system by extend the system in a dynamical system form of temperature, pressure
and CO2 content in the atmosphere.
The observation of the extended system in [10] shows that the caloric
exchange rate also appear in the requirement of its stability with the same criteria
of stability sistem in [3]. It means that the system keeps the parameter to be the
determined factor that bring the system into an stable or unstable condition.
Another condition must be investigated by consider the eigen value of the
linearization of the nonlinear system represented the extended system. The
behaviour of CO2 content in the atmosphere is evaluated to decide wether the
system must be revised or not.
In case that the system is needed to be revised, we must consider the cycle of
CO2 content in the universe that [1] devided it into atmosphere, biosphere and sea
zone. The cycle describes the CO2 transport that delivered from the sea, as the
place of respiration and assimilation of plant in the earth, to the atmosphere and
driven to the biosphere. If the interpretation of the solution of the system of [4] is
not represent the reduction of CO2 content, this paper propose to revised the third
equation of [4] by the derived equations that build in [1]. The new tentative system
is observed until the phenomena of the CO2 reduction in the atmosphere could be
found. This paper also revise the compartements that built in [1] by promote a
related parameter due to the CO2 reduction program in biosphere.
The CO 2 reduction program in biosphere, that being considered in this
paper, is the green zone preservation and the reforestation program. A cycled CO2
content model is proposed by [8]. To reduce the growth CO2 content the model
must be controlled by proposing the forest area preservation ( ) and the
succesess rate of the reforestation program . This program will revise
the changing of CO 2 content in the biosphere with respect to the time
changing. Mathematically it will cause the appearance of new term on the
related equation.
74
2. Main Results
2.1 The Initial Model
A dynamical system that represents the interaction of temperature, pressure

and CO2 content in the atmosphere is stated in [4] by
∝

∝
1 .... (1)
Where T : temperature, p : pressure, x : CO2 content in the atmosphere, Cp:

heat capacity in fixed pressure condition, α: specific volume of gas, Q: caloric
exchange rate and b, c and e: related parameters with respect to the production and
consumption of CO2.
An optimal control of the initial model is designed by generating the
Hamiltonian function to find the values of control parameters and
using the Lagrange equations that must satisfy the stationer
condition and both state and co-state equations. The parameters
respectively represents the growth of the CO2 content in the biosphere
zone and the rate of forest area preservation identified by the succesess
rate of the reforestation program. The optimal solution of the optimal
controlled problem is arranged such that the stability between
temperature and pressure interaction in the atmosphere could be
reached. The system stated on (1) is controlled by places parameter A in the
third equation of system (1) that interpreted as the atmospheric CO 2 absorption
program that is designed to minimized the CO2 production in the atmosphere
with performance index , ,
.. A
Hamiltonian function , , ∑ , , , , , ,
∝
, , , ∝
, , , ∝
, , , 1 is
generated to find the value of A using the Lagrange equations
, , ∑ , , , , 0,
0, 0. If the stationer condition 0 satisfy

both respectively state and co-state equations
1
, , , 2
, , , 3
, , and we got :

, jika

, jika

, jika
75
The simulation of the initial model, with Q = 35, Cp = 36,775, 0,017, K =

0,23, b =1, e =1,8 is refer to [8], [9] and [10] , gives the plots of temperature,
pressure and atmospheric CO2 content shown in Figure 1.
Figure 1. The plots of temperature, pressure and atmospheric CO2 content
Figure 1 shows that the CO2 content in the atmosphere already could be reduced
but the atmospheric CO2 is rised and reduced fluctuatively. It means that the
stability interaction between the temperature and pressure could be disturbed. This
result makes must be revised by redisign the following tentative model.
2.2 The Tentative Model
The tentative model is designed by consider the cycle of CO2 content in the
nature. The CO2 cycle scheme could be seen in Figure 2.
Figure 2. The cycle of CO2
A revised dynamical system that represents the interaction of temperature,

pressure, CO2 content in the atmosphere, biosphere and the died organic material
rate is stated in the following system :
∝

∝
76
.... (2)
Where Xl , Xa, Xd is respectively the CO2 content in the biosphere,

atmosphere and the died organic material rate, Q : caloric exchange rate, Cp : heat
capacity, α : gas specific volume, is positive constant, : respiration intensity,
: assimilation intensity, died organic material rate, : died organic caloric
exchange rate, : forest area preservation rate and : the successes of
reforestation program. The last three equations of system (2) is governed from the
cycle of CO2 in figure 2.
The system is controlled by places two parameters 1 and 1 to the system
(2) that interpreted as the atmospheric CO2 absorption programs that is
designed to minimized the CO2 content in the atmosphere and biosphere,
with performance index , , , 1, ,
.
A Hamiltonian function , , , , ∑ , , , ,
and , , , , ∑ , ,
, , , , , with , , ,
∝
∝
, , , ,
∝
, , , , ,
, , , , , , , , ,
, , , , 0, 0, 0,
0, 0. If the stationer condition 0,
1 , 2 satisfy both respectively state and co-state equations
1
, 1, 2, , 2
, 1, 2, , 3
, 1, 2, , 4
, 1, 2, , 5
, 1, 2,
and we got :
, jika
, jika ,
, jika

, jika

, jika

, jika
The simulation of the initial model, with Q = 35, Cp = 36,775, 0,017,

0,017, K = 0,23, 0,03 4,3. 10 , 2,0. 10 , 1,6. 10 , 0,21
are refer to [8], [9],10] and [1], gives the plots of respectively temperature,
pressure, atmospheric and biospheric CO2 content and the died organic material
rate shown in Figure 3.
77
Figure 3. The plots of temperature, pressure, atmospheric CO2 content, biospheric

CO2 content and the died organic material rate
Figure 3 shows that the CO2 content in the atmosphere and biosphere already
decreases with respect to time while the died organic material rate is still increase
that makes the increasing of temperature with a fluctuative pressure condition. It
means that a stable interaction of temperature ang pressure is not obtained yet. This
result makes must be revised by redesign an extended model that consider the CO2
content in the sea zone in the nature CO2 cycle.
2.3 The Extended Model
An extended model is designed by consider the role of CO2 content in the

sea zone in the nature. The extended model is designed by consider the
contribution of CO2 content in the sea zone to the CO2 cycle in universe. A revised
scheme of CO2 cycle is proposed in Figure 4.
Figure 4. The revised cycle of CO2
An extended dynamical system that represents the interaction of temperature,

pressure, CO2 content in the atmosphere, biosphere, sea zone and the died organic
material rate is stated in the following system :
∝

∝
78
…….. (3)
where Xs is the CO2 content in the seazone, : the sea zone respiration
intensity and : the sea zone assimilation intensity. The last four equations of
system (3) is governed from the cycle of CO2 in figure 4.
The system is controlled by places two parameters 1 and 1 to the system
(3) that interpreted as the atmospheric CO2 absorption programs that is
designed to minimized the CO2 content in the atmosphere and biosphere,
with performance index , , , , ,
.
A Hamiltonian function , , , , ∑ , , , ,
and , , , , ∑ , , ,
, , , , , , with , , ,
∝
∝
, , , ,
∝
, , , , ,
, , , , , , , , , , ,
+ , ,
, , , , 0, 0, 0,
1
, 1, 2, , 2
, 1, 2, , 3
, 1, 2, , 4
, 1, 2, , 5
, 1, 2, , 6
, 1, 2,
and we got :
, jika
, jika ,
, jika

, jika

, jika

, jika

0,017, K=0,23, 0,03 0,21 are
4,3. 10 , 2,0. 10 , 1,6. 10 ,
refer to [8], [9],10] and [1], gives the plots of respectively temperature, pressure,
atmospheric, biospheric and sea zone CO2 content and the died organic material
79
Figure 5. The plots of temperature, pressure, atmospheric, biospheric and sea zone
CO2 content and the died organic material rate of the extended model
Figure 5 shows that the temperature and biospheric CO2 content will deacreases
with respect to time while the decreasing of atmospheric CO2 content just occur in
the first 7 years and increases later on. The increasing of died organic material rate
just happen in the begining, after that it will decrease with respect to time. The
most important result is the sea zone CO2 content is still uncontrolled yet. In the
next session it will be consider a parameter control in the fifth equation of system
(3).
2.4 The Fixed Model
To control the growth of CO2 content in the atmosphere and sea zone shown
in preview section, the system (3) is controlled to minimized the CO 2 content in
the atmosphere, biosphere and sea zone, with performance index ,
, , , , , . A
Hamiltonian function , , , 1, , ∑ , , , and
, , , 1, , ∑ , , ,
, , , , , , , with , , ,
∝
∝
, , , , ∝
, , , , ,
, , , , , , , , , , ,
+ , ,
, , , , 0, 0, 0,
1
, 1, 2, , 2
, 1, 2, , 3
, 1, 2, , 4
, 1, 2, , 5
, 1, 2, , 6
, 1, 2,
and we got :
, jika
, jika ,
, jika
80

, jika

, jika

, jika

0,017, K=0,23, 0,03 4,3. 10 , 2,0. 10 , 1,6. 10 , 0,21 are
refer to [8], [9],10] and [1], gives the plots of respectively temperature, pressure,
atmospheric, biospheric and sea zone CO2 content and the died organic material
Figure 6. The plots of temperature, pressure, atmospheric CO2 content,

biospheric CO2 content, sea zone CO2 content and the died organic material rate
Figure 6 shows that the growth of temperature could be controled properly with
with small volatility. It could be concluded from the linear relation between
temperature and pressure and seen from the plot of pressure. The CO2 content in
the atmosphere and biospheric also already well controlled while the same
behaviour of died organic material rate of the extended system is still found. The
most important result is the sea zone CO2 content already could be controlled.
The optimal CO 2 content in the universe that makes the stable

interaction between the two main climate unsure, temperature and pressure,
is reached when the controller is not only placed in the atmosphere and
biosphere zone terms but also in the atmosphere zone term. It could be
interpretated that, to have a stable interaction between the temperature and pressure
in the atmosphere, it is not enough to control the atmospheric CO 2 content and
the biospheric CO2 content. We can’t eliminate the importance role of sea zone,
as a place of assimilation and respiration zone of marine biotic, because the area of
the sea zone is wider than the continent zone.
81
Acknowledgement
The authors would like to thank to the Directory General of Higher Education of
The Ministary Education and Culture for the funding of Fundamental Research
scheme 2013.
References
[1] Eriksson, E. and Welander, P., 1955, On a Mathematical Model of The Carbon
Cycle in The Nature, International Meteorological Institute, Stockholm.
[2] Haneda, accessed 11 Nopember 2008, Hubungan Efek Rumah Kaca, Pemanasan
Global dan Perubahan Iklim, http://climatechange.menlh.g.id.
[3] Jaya, A. I., 2010, Peran Penting Laju Perubahan Kalor pada Model Dinamik
Unsur-Unsur Utama Iklim, Jurnal Ilmiah Matematika dan Terapan Volume 7 no 2
ISSN 1829-8133, UniversitasTadulako, Palu.
[4] Jaya, A.I., 2011, Mencari Perluasan Model Dinamik Unsur-Unsur Utama Iklim,
Jurnal Ilmiah Matematika dan Terapan Volume 8 no 1 ISSN 1829-8133, Universitas
Tadulako, Palu.
[5] Kato, S., Tri W. H dan Joko W., 1998, Dinamika Atmosfer, ITB Publisher,
Bandung.
[6] Naidu, D.S., 2002, Optimal Control Systems, CRS Presses LLC, USA.
[7] Pontryagin, L.S.,et al, 1962, The Mathematical Theory of Optimal Process, Vol.4,
Interscience.
[8] Ratianingsih, Rina, Jaya, A.I. and Tarende, T.E, 2013, The Design of CO 2
Absorption Model with Respect to The Stability of Temperature and
Pressure Interaction in The Atmosphere, Procedia Environmental
Sciences CRISU-CUPT VIII International Conference 2013, Kendari.
[9] Ratianingsih, Rina, 2011, Dynamic Model of Pressure, Temperature and CO2
Concentration : Identification of Stability Parameters Potency, Presented on The
SEAMS-GMU Conference on Mathematics and Its Applications, Universitas
Gadjah Mada, Yogyakarta.
[10] Ratianingsih, Rina, 2011, Identifikasi Model Konsumsi Gas CO2 di Atmosfer untuk
Mendapatkan Interaksi Unsur – Unsur Utama Iklim yang Stabil, akan terbit,
Prosiding Seminar Nasional Sains IV, Institut Pertanian Bogor.
[11] Soedomo, M., 2001, Pencemaran udara, Kumpulan Karya Ilmiah, Penerbit ITB,
Bandung.
[12] Subchan, S and Zbikowski, R, 2009, Computational Optimal Control : Tools and
Practices, John Willey & Sons Ltd. Publishing,UK.
[13] Sutimin, 2010, Model Matematika Konsentrasi Gas Oksigen Terlarut pada
Ekosistem Perairan Danau, UniversitasDiponegoro, Semarang.
[14] Tjasyono, B., 2004, Klimatologi, ITB Publisher, Bandung.
Applied Mathematics
COMPARISON OF SENSITIVITY ANALYSIS ON

LINEAR OPTIMIZATION USING OPTIMAL
PARTITION AND OPTIMAL BASIS
(IN THE SIMPLEX METHOD) AT SOME CASES
1
BIB PARUHUM SILALAHI, 2 MIRNA SARI DEWI
1Lecturer at Bogor Agricultural University, bibparuhum1@yahoo.com

2Student at Bogor Agricultural University, mirnasaridewikara@gmail.com
Abstract. Sensitivity analysis describes the effects of coefficient changes of a

linear optimization problem to the optimal solution. Usually we use the optimal
basis approach as in the simplex method. This paper discussed the sensitivity
analysis with another approaches: analysis using an optimal partition based on the
interior point method to determine the range and shadow price. We then compare
the results obtained with those produced by the simplex method with the help of
software LINDO 6.1. The results of sensitivity analysis, obtained through the
optimal partition approach is more accurate than using the optimal basis approach
(the simplex method), especially for cases where the primal or the dual optimal
solution is not unique. But when the primal and the dual have a unique optimal
solution, the simplex method and the optimal partition approach produce same
information.
Key words and Phrases : sensitivity analysis, shadow price, range, optimal
partition, optimal basis.
1. Introduction
Linear Optimization (LO) is concerned with the minimization or

maximization of a linear function, subject to constraints described by linear
equations and/or linear inequalities.
Sensitivity analysis describes the effect of changing the parameters of the
linear optimization model, i.e. studying the effect of changing the coefficients of
objective function and right-hand side value constraints to the optimal solution.
Sensitivity analysis that is used in the classical approach (the simplex method) based
on the optimal basis. This paper will present briefly sensitivity analysis by using
another approach, the analysis using the unique partition (optimal partition) based
on the interior point method. This method is presented by Roos, Terlaky and Vial
[1]. By using the optimal partition approach, we determine shadow price and range.
For the same problem we also performed a sensitivity analysis using the simplex
82
83
method with the help of software LINDO 6.1. Then we compare the obtained results.
The structure of this paper is as follows. In section 2, we review shortly the
primal-dual problem, optimal partition and optimal sets, range and shadow price, and
sensitivity analysis with classical approach. In section 3, we present three cases of
LO problems to be analyzed and compared by using optimal partition and by using
LINDO 6.1. At the end we give concluding remarks.
2. Sensitivity Analysis
2.1. Primal - Dual
Every linear optimization problem can be modeled mathematically into a form

called the primal form (P) and the dual form (D).
The standard form of a primal and a dual form are as follows:
(P) min {cTx : Ax = b, x ≥ 0},
(D) max {bTy : ATy + s = c, s ≥ 0 },
where c, x, s ∈ n, b, y ∈ m and A ∈ m x n is matrix with rank m.
Suppose the optimal value of (P) and (D) symbolized by v(b) and v(c) :
v(b) = min {cTx : Ax = b, x ≥ 0},
v(c) = max {bTy : ATy + s = c, s ≥ 0}.
The feasible regions of (P) and (D) are denoted by P and D, respectively:
P := {x ∈ n : Ax = b, x ≥ 0},
D := {(y, s) ∈ m: ATy + s = c, s ≥ 0}.
If (P) and (D) are feasible then both problems have optimal solutions, and we
denote it by P* and D*,
P* := {x ∈ P: cTx = v(b)}
D* := {(y, s) ∈ D: bTy = v(c)}.
2.2. Optimal Partition and optimal sets
The followings are the theorems that used as base of forming an optimal partition.
Theorem 1. (Duality Theorem, cf. [1] Theorem II.2) If (P) and (D) are feasible then
both problems have optimal solutions. Then, if x ∈ P and (y, s) ∈ D, these are optimal
solutions if and only if xTs = 0. Otherwise neither of the two problems has optimal
solutions, either both (P) and (D) are infeasible or one of the two problems is
infeasible and the other one is unbounded.
Theorem 2. (Goldman-Tucker, cf. [1] Theorem II.3) If (P) and (D) are feasible
then there exists a strictly complementary pair of optimal solutions, that is an
optimal solution pair (x*, s*) satisfying x* + s* > 0.
The optimal partition of (P) and (D) are the partition that splits the index of x
(and s) into B and N, as follows:
B : = {i : xi > 0 for some x ∈ P*},

N : = {i : si > 0 for some (y, s) ∈ D*}.
We may check that the duality theorem implies B ∩ N = Ø, and Goldman-

84
Tucker theorem implies B ∪ N = {1, 2, ..., n}.

We use xB and xN as notations refer to the restriction of the vector x ∈ n to
the index set B and N respectively. Similarly, AB and AN represent the restriction of
A to the columns with indices of set B and N respectively. We then have the following
lemma.
Lemma 1. (cf. [1]) P* and D* can be expressed by the terms of the optimal
partition into
P*= {x : Ax = b, xB ≥ 0, xN = 0},
D* = {(y, s) : ATy + s = c, sB = 0, sN ≥ 0}.
2.3. Range and Shadow Price
Sensitivity analysis determines the shadow price and range of all the
coefficients b (the value of the right side of primal constraints) and c (the value of
the right side dual constraints). In one case, the value of coefficient b or c may be a
break point. If the coefficient is a break point, then we have two shadow prices: the
left shadow price and right shadow price. If the coefficient is not a break point, then
there is a shadow price at an open linearity interval and range of the coefficient is in
the linearity interval. Figure 1 shows an example of change in the optimal value for
the change in the value of cj (cj=1 and cj=2 are break points).
Figure 1. Optimal value function for cj
Suppose that (P) and (D) are feasible. According to optimal partition approach
[1], range of bi is obtained by minimizing and maximizing bi over the set
{bi : Ax = b, xB ≥ 0, xN = 0}. (1.1)
Left and right shadow price of bi are determined by minimizing and maximizing yi
over the set
{yi : AT y + s = c, sB = 0, sN ≥ 0}. (1.2)
Range of cj is obtained by minimizing and maximizing the value of cj over the set
{cj: AT y + s = c, sB = 0, sN ≥ 0}. (1.3)
Left and right shadow price of cj are determined by minimizing and maximizing xj
over the set
{xj: Ax = b, xB ≥ 0, xN = 0}. (1.4)
85
2.4. Sensitivity Analysis with the Classical Approach
Sensitivity analysis with the classical approach based on the simplex method
to solve linear optimization problems. The optimal solution of this classical approach
is determined by an optimal basis.
Assume that A is a matrix of size mxn and rank (A) = m. Indices of a basis
variable is denoted by B'. Then sub-matrix is a non-singular matrix of size mxm
with =b, =0 where N' is the set of non-basis variable index of A. A primal
basic solution x can be determined by
, (1.5)
′
and a dual basic solution can be determined by
, . (1.6)
′
Sensitivity analysis with the classical approach uses also formulas (1.5) - (1.8)
to determine the range and shadow price, but with the optimal basis partition (B ', N
') instead of (B, N). In fact, P and D may have more than one optimal basis, and
therefore this classical approach may also provides different shadow price and range
[2].
3. Cases
We consider three cases as follows:

1. Optimal solution of the primal problem is unique and optimal solution of the
dual problem is not unique.
2. Optimal solution of the primal problem is not unique and optimal solution of the
dual problem is unique.
3. Optimal solution of the primal and the dual problems are unique.
3.1. Case I
Suppose the primal problem (P) is defined as follows:

Min 4x1 - 5x2 + 11x3
s.t -x2 + 3x3 = 0
x1 - x2 - x3 = 1
x1, x2, x3 ≥ 0
The dual problem (D) is

Maxy2
s.t y2 ≤ 4
- y1 - y2 ≤ -5
3y1 - y2 ≤ 11
The feasible region of the dual problem is depicted in Figure 2. From Figure 2, it
can be seen that the set of optimal solutions of (D) is D* = {(y1, y2): 1 ≤ y1 ≤ 5, y2
= 4} and the optimal value is 4 . Slack variable of each of the dual constraints are
as follows:
86
y2 + s1 = 4  s1 = 4 - y2
-y1 - y2 + s2 = -5 s2 = -5 + y1 + y2
3y1 – y2 + s3 = 11 s3 = 11 - 3y1 + y2
It can be concluded that all the slack can be positive at an optimal solution unless the
slack value of the constraint y2 ≤ 4, i.e. s1 = 0. This means that the optimal partition
of set N is N={2, 3}. Hence B={1}.
By using Lemma 1, we get:
P* = {x ∈ P: x2 = x3 = 0} and (P) has a unique solution x = (1, 0, 0).
Figure 2. Feasible region of case I.
Next we show examples of finding range and shadow price of b1 = 0 and c1 = 4. The
other range and shadow price can be found in the same way.
Range and Shadow Price for b1 = 0

By using (1.1), range b1 can be determined by minimizing and maximizing b1 over
the set {b1: Ax = b, xB ≥ 0, xN = 0}.
We have Ax = b as follows
1
0 1 3 1
2 = .
1 1 1 3 1
From the above system we get
0 = b1,
x1 = 1.
Hence the range of b1 is the interval [0, 0]. Therefore b1 = 0 is a break point.
By using (1.2), the shadow price of b1 can be determined by minimizing and
maximizing y1 over the set {y1 : AT y + s = c, sB = 0, sN ≥ 0}.
Using that y ∈ D*, the minimum value of y1 is 1 and the maximum value is 5, so the
shadow price for b1 is [1, 5].
Range and Shadow Price for c1 = 4

Range of c1 determine by minimizing and maximizing c1 over the set {c1: AT y + s
= c, sB = 0, sN ≥ 0}, as in (1.3).
Matrix multiplication of AT y + s = c:
87
0 1 1 1 1
1 1 2
+ 2 = 5
3 1 3 11
Based on Figure 1, if we eliminate the first constraint, y2 will be in the interval [1,
∞). By substituting s1 = 0 and y2 to the first constraint, we get y2 = c1. This means
that 1 ≤ c1 ≤ ∞, hence the range for c1 is the interval [1, ∞).
By using (1.4) shadow price of c1 is determine by minimizing and maximizing x1
over the set {x1: Ax = b, xB ≥ 0, xN = 0}. Because of x1 = 1, then the shadow price of
c1 is 1.
In Table 1, we present range and shadow price of case I which are obtained by using
optimal partition approach. We also present range and shadow price obtained from
calculation by using LINDO (Table 2). We may see sensitivity analysis of the
simplex method (LINDO) did not detect that b1 = 0 is a break point.
Table 1. Range and shadow price obtained from optimal partition approach
(Case I)
Coefficient Range Shadow price
b1 = 0 0 [1, 5]
b2 = 1 [0, ∞ 4
c1 = 4 [1, ∞ 1
c2 = -5 [-9, ∞ 0
c3 = 11 [-1, ∞ 0
Table 2. Range and shadow price obtained from LINDO (Case I)

b1 = 0 (-∞, 0] 1
b2 = 1 [0, ∞ 4
c1 = 4 [1, ∞ 1
c2 = -5 [-9, ∞ 0
c3 = 11 [-1, ∞ 0
3.2. Case II

Min 4x1 + 31x2 - 5x3 + 11x4
s.t 3x2 - x3 + 3x4 = 0
x1 + 7x2 - x3 - x4 = 1
x1, x2, x3, x4 ≥ 0.
The dual problem (D) is

Maxy2
s.t y2 ≤ 4
3y1 + 7y2 ≤ 31
-y1 - y2 ≤ -5
3y1 - y2 ≤ 11
88
The feasible region of the dual problem is shown in Figure 3. From Figure 3, we
obtain that the optimal solution of (D) is D* = {(y1, y2): y1 = 1, y2 = 4} and the optimal
value is 4. Slack variable of each of the dual constraints are as follows:
y2 + s1 = 4 s1 = 4 - y2
3y1 + 7y2 + s2 = 31 s2 = 31 - 3y1 -7 y2
-y1 - y2 + s3 = -5 s3 = -5 + y1 + y2
3y1 - y2 + s4 = 11 s4 = 11 - 3y1 + y2
Figure 3. Feasible region of case II.
By substituting y1 = 1, y2 = 4, we obtain the values of each slack. Slack in these

constraints: y2 ≤ 4, 3y1 + 7y2 ≤ 31 and -y1 - y2 ≤ -5 are 0, i.e. s1 = s2 = s3 = 0. Hence
in the primal problem only x1, x2 and x3 can be positive. Therefore the optimal
partition (B, N) is obtained, i.e. N = {4} and B = {1, 2, 3}.
By using Lemma 1, we get:

P* = {x ∈ P: x4 = 0} and (P) has not unique solution : { (x1, x2, x3) : (a , ¼ - ¼ a, 3 (
¼ - ¼ a) ) } , 0 ≤ a ≤ 1.
By using the same calculation as in case I, we get ranges and shadow prices of case
II (Table 3). Table 4 shows ranges and shadow prices of case II obtained by using
LINDO.
(Case II).
b1 = 0 (-∞, 3/7] 1
b2 = 1 [0, ∞ 4
c1 = 4 4 [1, 0]
c2 = 31 31 [¼, 0]
c3 = -5 -5 [¾, 0]
c4 = 11 [-1, ∞ 0
89
Table 4. Range and shadow price obtained from LINDO (Case II).
b1 = 0 (-∞, 0] 1
b2 = 1 [0, ∞ 4
c1 = 4 [1, 4] 1
c2 = 31 [31, ∞ 0
c3 = -5 [-5, ∞ 0
c4 = 11 [-1, ∞ 0
From Table 3 and Table 4, there are differences in range and shadow price obtained
by using optimal partition and the simplex method. At the coefficient b1 = 0, for the
same shadow price, the optimal partition detect a greater range. Next, at the
coefficients c1 = 4, c2 = 31, and c3 = -5, analysis using the simplex method does not
detect any break points.
3.3. Case III

Min 31x1 - 5x2 + 11x3
s.t 3x1 - x2 + 3x3 = 0
7x1 - x2 - x3 = 1
x1, x2, x3 ≥ 0
Dual problem (D) is

Max y2
s.t 3y1 + 7y2 ≤ 31
-y1 - y2 ≤ -5
3y1 - y2 ≤ 11
The feasible region of the dual problem is shown in Figure 4. From Figure 4, it can
be determined that the optimal solution of (D) is D* = {(y1, y2): y1 = 1, y2 = 4} and
the optimal value is 4 . Slack variable of each of the dual constraints are as follows:
3y2 + 7y2 + s1 = 31 s1 = 31 - 3y1 - 7y2

-y1 - y2 + s2 = -5 s2 = -5 + y1 + y2
3y1 - y2 + s3 = 11 s3 = 11 - 3y1 + y2
We can check that at y1 = 1 and y2 = 4, all the slack can be positive except slack in
the constraint 3y1 + 7y2 ≤ 31 and -y1 - y2 ≤ -5, at the constraints mentioned we have
s1 = s2 = 0. Hence the optimal partition (B, N) is N = {3} and B = {1, 2}.
By using Lemma 1, we obtain:
P* = {x ∈ P: x3 = 0} and (P) has a unique solution x = (¼, ¾, 0).
By using the same calculation as before, we get ranges and shadow prices of case III
(Table 5). Table 6 shows ranges and shadow prices of case III obtained by using
LINDO.
90
Figure 4. Feasible region of case II
(Case III).
b1 = 0 (-∞, 3/7] 1
b2 = 1 [0, ∞ 4
c1 = 31 [19, ∞ ¼
c2 = -5 [-7, ∞ ¾
c3 = 11 [-1, ∞ 0
Table 6. Range and shadow price obtained from LINDO (Case III).
b1 = 0 (-∞, 3/7] 1
b2 = 1 [0, ∞ 4
c1 = 31 [19, ∞ ¼
c2 = -5 [-7, ∞ ¾
c3 = 11 [-1, ∞ 0
We may see that the range and shadow price using optimal partitioning and the
simplex method are same.
The results of sensitivity analysis by using the simplex method (using the optimal
basis approach) for cases where one of the primal or dual optimal solution is not
unique, is not as perfect as the results obtained by using optimal partition approach.
When the primal and the dual have a unique optimal solution, simplex method and
optimal partition approach give the same information.
References
[1] C. Roos, T. Terlaky and J.-P. Vial, Interior Point Methods for Linear Optimization,
New York: Springer, 2006.
[2] B. Jansen, J. de Jong, C. Roos and T. Terlaky, "Sensitivity Analysis in Linear
Programming: Just be Careful!," European Journal of Operations Research, vol. 101,
pp. 15-28, 1997.
Applied Mathematics
APPLICATION OF OPTIMAL CONTROL FOR A

BILINEAR STOCHASTIC MODEL IN CELL CYCLE
CANCER CHEMOTHERAPY
D. HANDAYANI1, R. SARAGIH 2, J. NAIBORHU 3, N. NURAINI 4
1,2,3,4Departmentof Mathematics, Faculty of Mathematics and Natural Sciences,

Institut Teknologi Bandung
Email : dhandayanimtk@students.itb.ac.id, roberd@math.itb.ac.id,
janson@math.itb.ac.id2,nuning@math.itb.ac.id
Abstract. A model of stochastic bilinear is introduced to study the effect of

cancer chemotherapy on the kinetic of the cell cycle disturbed by additive noise.
In the design of the chemotherapy or the diseases treatment, it is important to
quantify the effect of a drug and to accommodate the random disturbance during
treatment period. Optimal control is required to control the effect of
chemotherapy because the substance contained in the drug may not only kill the
cancer cells but also the normal cells. The normal cells which being exposed to
the drug and the other disturbances factor on cell cycles, we assume as the
additive noise. The additive noise disturbs the system at random. In this paper to
see the numbers of cancer cells during chemotherapy period, will be solved a
Riccati equation which the main problem on optimal control theory. Numerical
simulation will explain the effect of the drug to number of cancer cells during
treatment with random disturbance on each phase of the cell cycle. Particles
Swarm Optimization method guess the initial of adjoin equation to solve
stochastic optimal control. This methodology enables the authors quantify the
effects of chemotherapy and solve bilinear stochastic optimal control.
Key words: bilinear stochastic control, optimal control, white noise, cell-cycles,
cancer chemotherapy
1. Introduction
In the design of cancer chemotherapeutic agents it is important to quantify
the effect of the drugs to the cancer cells in cell cycles and to accommodate the
disturbances during treatment period. This information will enable to improve the
effectiveness of chemotherapy treatment. Optimal control is required to control the
effect of chemotherapy because the substance contained in the drug may not only
kill the cancer cells but also the normal cells.
The cell cycles is divided into four phases : Pre-synthesis phase (G1), DNA
synthesis phase (S), Post-synthesis phase (G2) and Mitotic Phase (M) in which cell
division occurs. Each cell passes through a sequence of phases from cell birth to
cell division. Starting point is a growth phase G1 after which the cell enters a phase
91
92
S where DNA synthesis occurs. Then a second growth phase G2 takes place in
which the cell prepares for mitosis or phase M. Here cell division occurs.
Depending on the medical aspects taken into account, the cell-cycle is divided into
compartments which describe the different cell phases or combine phases of the
cell cycle into clusters. The simplest and at the same time most natural models
divide the cell cycle into three compartments, respectively. In these models the
phases G2 and M are combined into one compartment as mitosis phase. On the
basis of DNA contents, the fractions of cells in the G1, S, and G2 + M can be
determined. Thus the effects of chemotherapeutic agents on cell cycle progression
be studied.
We will see the effect of 1 mg Melphalan on cell cycle chemotherapy.
Melphalan, known as Alkeran, is used to treat multiple myeloma, as well as
ovarian, breast, and prostate cancer. It is also used for other cancers and sometimes
for noncancerous conditions. Melphalan is a member of the general group of
chemotherapy drugs known as alkylating agents. It works by interfering with and
stopping the growth of cancer cells, which causes them to die.
The research on bilinear control systems applied to cancer chemotherapy
was be studied by [1], [4], and [5]. In this research, the bilinear model in [1] will be
disturb by additive noise. The additive noise disturbs the system at random and is
assumed as a Brownian Motion/Wiener Processes, known as white noise. The early
research [1] concluded that Melphalan was most effective in mitosis phase. But, it
is not accommodate the others factors we didn’t know that affect the cell cycles
such as (kidney disease, liver disease (including hepatitis), heart disease,
congestive heart failure, diabetes, gout, or infections). In practise, these conditions
may impact the medicine to give the more effect in cell cycles. Therefore, we think
its important to add the noises in this paper.
In this paper to see the number of cancer cells during chemotherapy period,
we use the complete state information. We will solve a Riccati differential equation
for bilinear systems which the main problem on optimal control theory. Numerical
simulation will explain the effect of the drug to number of cancer cells during
treatment with random disturbances on each phase of the cell cycle. Particles
Swarm Optimization method guess the initial of adjoin equation to solve stochastic
optimal control. This methodology enables the authors quantify the effects of
chemotherapy and solve bilinear stochastic optimal control
2. Formulation of The Problem

The dynamics of the cell cycles is represented by the compartmental model
shown in Figure 1. This model yields the following bilinear system of differential
equation is based on [1]. x1,x2,x3 represent the number of cells at time t in the
phases of the cell cycle pre-synthesis G1, synthesis S, and post-synthesis G2 + M,
respectively and k1 + u1,k2 + u2,k3 + u3 represent the flow rate parameter of the cell
cycles change in time when cells are exposed to drugs, while unperturbed cells
maintain constant parameters. From [1] we can define that the constant k1,k2,k3
represent the constant flow rate parameters of unperturbed cells and control input
u1,u2,u3 represent the effects of the drug on each phase of the cycle at time t.
We assume that the dynamic of x1,x2,x3 was affected by the random vectors
W1(t),W2(t),W3(t) because there are the others factors we didn’t know that affect the
dynamic of x1,x2,x3 besides the numbers of circulated cells in cell cycle. While,
W1(t),W2(t),W3(t) are the random disturbances on systems represented as stochastic
processes because on biological dynamic systems evolve under stochastic force.
93
The most important stochastic process in continuous time is the Wiener process
also called Brownian Motion.
x˙1(t) = 1.6(k3 + u3(t))x3(t) − (k1 + u1(t))x1(t) + BW1W1(t) (1)

x˙2(t) = (k1 + u1(t))x1(t) − (k2 + u2(t))x2(t) + BW2W2(t) (2)
x˙3(t) = (k2 + u2(t))x2(t) − (k3 + u3(t))x3(t) + BW3W3(t) (3)

Figure 1. Compartmental Model
Let the vector X(t) = (x1(t),x2(t),x3(t))T, u(t) = (u1(t),u2(t),u3(t))T, and W(t) =

(W1(t),W2(t),W3(t))T. So the models (1),(2), and (3) can be written in the form of
completely controllable, stochastic system for bilinear :
(4)
X(0) = X0 (5)
where,
, and BW is defined as random disturbances weight matrix will

be determined when we construct the stochastic optimal control in next section.
The chemotherapy effects can be seen from the numbers of the cells which
enter to each phase in cell cycles when cells are exposed to drugs. It is represented
by the flow rate parameters k1 + u1, k2 + u2, and k3 + u3. To define the effect of
chemotherapy we assume that the combined effects of the drug and the washing
cells (cells which were put in a new medium but were not exposed to the drug) use
the data for cells exposed to the drug and the values of the constant cell cycle
parameters. The constant parameters of the model k1,k2,k3 with ui for i = 1,2,3 was
be identified from the unperturbed cells data, i.e. cells which were without washing
or being exposed to the drug.
3. Stochastic Optimal Control

3.1. Wiener Processes.
Definition 3.1. (Brownian Motion) A stochastic process W(t) is called Brownian
94
Motion if it satisfies the following condition

(1) Independence : W(t + ∆t) − W(t) is independent of {W(τ)} for all τ ≤ t
(2) Stationarity : The distribution of W(t + ∆t) − W(t) does not depend on t
(3) Continuity : for all δ >0
This definition induces the distribution of the process W(t).
Theorem 3.2. If W(t) is a Brownian motion, then W(t) − W(0) is a normal random
variable with mean µt and variance σ2t, where µ and σ are constant real numbers.
The random disturbances is assumed as a standard Brownian Motion. We

have the definition is as following:
Definition 3.3. A Brownian Motion is called standard if it satisfies the following

condition
W(0) = 0
E[W(t)] = 0
E[W2(t)] = 1
3.2. The bilinear stochastic optimal control for identifying the drugs effects.
In this section, the following stochastic optimal control problem is
considered for a dynamic system with the state vector X(t) R3, the admissible
control vector u(t) U R3 (where U is a time-invariant, convex, and closed subset
of R3) and the standard vector Brownian Motion W(t) R3.
dX(t) = (AX(t) + Bu(t) + XNixiui)dt + BWdW(t) (6)
i=1
X(0) =X0(7) where A, Bu, Ni for i = 1,2,3 was defined in equation (4)
and (5), and Bw is defined as covariance error matrix,
with b1 = k1 + u1(t), b2 = k2 + u2(t), and b3 = k3 + u3(t).
In this paper, we consider the stochastic models for cancer chemotherapy

with objective function. The objective is to find the control input u(t) which yields
the optimal to fit X(0) = X0, X(t) = Xt, and X(tf) = Xtf fixed to the data from [1]. The
optimal control minimize the objective function is as following:
(8)
where Ptf,Q,R R with Ptf >0, Q ≥ 0, and R >0 for all t [0,tf]. Subject to the
constrained equation (6).
We use the following theorem to solve stochastic optimal control for bilinear
systems in (6).
95
Theorem 3.4. (Stochastic Hamiltonian Jacobi Bellman Theorem) If the partial

differential equation
with the boundary condition J(X,tf) = Ptf admits a unique solution, the globally
optimal state feedback control law is
Proof. A rigorous proof of this theorem can be found in [2].

We have that optimality conditions satisfies:
(9)
Where, i = 1,2,3 and Jx(X,t) = P(t)X(t). With this optimal control law, the
Hamiltonian Jacobi Bellman partial differential equation in Theorem (3.4) has the
following from
To solve the stochastic optimal control, we use a adjoint equation known as Riccati
differential equation. It is as following:
[ATP + PA − PBuR−1BuTP + Q]−

R−1[BuT + XT(P3i=1 Ni)T]PX(P3i=1 Ni)P−
P˙ = (11)
PBuR−1XT(P3i=1 Ni)TP−
PR−1[BuT + XT(P3i=1 Ni)T]PX(P3i=1 Ni)

P(tf) = 0 (12)
with P is the symmetric and positive definite matrix.
4. Numerical simulation
In the early research by [1] with deterministic optimal control approach, they
got that Melphalan given the effective effect in mitosis phase. It was consistent
with known data. In this research, we will see the effects of 1 mg Melphalan on cell
cycle with stochastic optimal control approach. The optimal control is accounted
through the Riccati differential equation (11). The chemotherapy effects is
determined from the cells traverse rate k1 + u1(t), k2 + u2(t), and k3 + u3(t).
We have to solve stochastic optimal control in equation (9) which minimize
the objective function J subject to the systems (6) and (11). The procedure to solve
the optimization problem is based on the Particle Swarm Optimization algorithm.
PSO algorithm is used to solve optimal control so that the objective function (8)
minimized. The constant parameters k1,k2,k3 was determined from unperturbed data
where k1 = 0.050/hour, k2 = 0.087/hour, and k3 = 0.159/hour. In next explanation,
we will quantify the effects of the chemotherapy as the flow rate k1 + u1(t), k2 +
u2(t), and k3 + u3(t) with ui(t) is dependent time.
For this problem, PSO algorithm to solve optimal control is as following
96
(1) Create an agent population (particles) as solution of initial adjoin

equation.
Define ˜x0j and vj0 with j is the numbers of agents population.
(2) Solve the systems with these result.
(3) Evaluate the objective function J(x,u˜).
(4) Define the best particle position, i.e. particle that has an optimal value.
(5) Update particle’s velocities
(pbest (gbest
where vjt is particle’s inertia, (pbest ) is personal influences, and
(gbest ) is social influences.
(6) Update particle’s
(7) Go to 2nd step unless some termination criteria are met.
Figure 2 shows the result for relative numbers of cells on unperturbed cells.
The results of the determination of the constant parameters k1,k2,k3 and the graph
shows the fit of the data [1]. We use PSO algorithm to fit simultaneously the three
set of data for the contents of G1, S, and G2 + M. For stochastic model (the blue
one) we see that he numbers of cells in G1, S, and G2 +M increase allow the time (in
hour). It’s mean that, the normal cells still grow and die in controlled (normal) way.
Laboratory data in Figure 2 are shown by the green dot one. Objective
function for this problem has Q = 0, but we expect the final point of state close to
the data. Thus, we can see the original dynamics of bilinear models. The effects of
either washing the cells or the effects of the drugs the sum of the value for the
control functions and the parameters, i.e. k1+u1,k2+u2,k3+u3 must be considered.
Figure 3 shows the average result of cells traverse rate as effect of the drugs
in 5 hours for k1 + u1,k2 + u2,k3 + u3 in the case of washing the cells. k1 + u1 represent
the cells traverse from G1 (pre-synthesis) phase to S (synthesis) phase, k2 + u2
represent the cells traverse from S (synthesis) phase to G2 + M (mitosis) phase, and
k3 + u3 represent the cells traverse from G2 + M (mitosis) phase back to G1 (pre-
synthesis) phase. In Figure 3 we can see that k1 + u1 for stochastic model is greater
enough than k1 + u1 for deterministic model. Both of them toward to 0.05/hour in 40
hours. Then, k2 +u2 both of stochastic and deterministic model closer together. Its
mean that deterministic and stochastic model have same control effect in synthesis
phase. And k3 + u3 both of them stable and toward to 0.1597/hour in 40 hours.
Figure 4 shows the numerical simulation result for cell numbers of washed
cells. The numbers of normal cells in washed cells for stochastic model relatively
increase. This results are consistent enough with the known experimental data.
Figure 5 is the result of the drug effects for exposed cells. the sum of the value for
the control functions and the parameters, i.e. k1 +u1,k2 +u2, and k3 +u3 must be
considered for exposed cells. The effects was be accounted on average of 10 hours
due to the period effects of exposed cells. Figure 6 shows cells relative numbers for
cells exposed to the drug 1 mg Melphalan.
Comparing the figures for the different experiments, we observe the following:
97
(1) In the first period of the washed cells experiment, following the
exposure to the drug Melphalan, there is a enough increasing in the
parameter k1 +u1, which corresponds to cell traverse from G1 to S phase.
Then, there is decreasing in the parameter k2 +u2, corresponds to cell
traverse from S to
Figure 2. Relative numbers of unperturb cells

98
Figure 3. The effects of the drugs for washed cells

99
Figure 4. Relative numbers of washed cells

100
Figure 5. The effects of the drugs for exposed cells

101
Figure 6. Relative numbers of exposed cells G2 + M. This effect was

prolonged as the drug dosage was same in 1mg
Melphalan (Figure 3a and Figure 5a). In same doses tested, this parameter
returned to the original condition of the unperturbed cells within
40 hours from the start of the experiment.
102
(2) In all experiments, exposure to the drug resulted is stable and closer
together between deterministic and stochastic model in the parameter
k2+u2 which means cell traverse from S phase to G2 + M phase (can be
seen at Figure 3b and Figure 5b).
(3) Cell traverse from synthesis S to the mitosis of the cycle G2 + M which
is reflected by k3 +u3 values, is affected by the drug in a dose related
fashion. The early research [1] concluded that Melphalan was the most
effective on cells on mitosis where the parameter k3 + u3 dropped to a
low constant level. While in this research, the control in which cells
were sustained in a medium being exposed to the drug, the parameter k3
+ u3 is random enough. In the case where cells were exposed to the
smallest dosage (1 mg Melphalan) there is an initial transient sometimes
increase and decrease in the parameter rapidly. Therefore, the random
disturbances
This simulation result show that with stochastic optimal control we get that 1
mg Melphalan give random effect to parameter k3 +u3(t) on mitosis phase G2 + M of
exposed cells, whereas the early research by [1] concluded that Melphalan given
the most effective on mitosis phase because the rate of cell traverse k3 + u3
decreased along treatment period. With stochastic optimal control we can capture
the phenomena effect of the disturbances factor on cell cycles chemotherapy. On
the mitosis phase, stochastic result conclude that Melphalan is not effective enough
on mitosis phase. According to washed and exposed cells stochastic model
simulation, the results are consistent enough with the experimental data. There is
no significant different between the effects in synthesis phase for washed cells and
exposed cells. Thus, stochastic result conclude that 1mg Melphalan is effective on
cells at synthesis phase and suggest that the optimal control approach provides a
quantitative method for determining drug effects on cell cycles.
References
[1] Yael Biran and Bayliss McInnis, Optimal Control of Bilinear Systems: Time
Varying Effects of Cancer Drugs, Automatic vol.15 pp. 325-329, 1979
[2] A. Chinchuluun et al. (eds.), Optimization and Optimal Control, Springer
Optimization and Its Applications 39, DOI 10.1007/978-0-387-89496-6 18, Springer
Science+Business Media, LLC, 2010
[3] Suzanne Lenhart and John T. Workman, Optimal Control Applied to Biological
Models, Chapman and Hall/CRC, 2007
[4] Urzula Ledzewicz and Heinz Schattler, Analysis of a cell cycle specific model for
cancer chemotherapy, J. of Biological Systems, 10, pp. 183-206, 2005
[5] Roberd Saragih, Ednawati Rainarli, Aplikasi kontrol bilinier pada sumsum tulang
dengan kemoterapi cell-cycle spesifik, Prosiding KNM XIII 793-800, 2006
Applied Mathematics
OUTPUT TRACKING OF SOME CLASS

NON-MINIMUM PHASE NONLINEAR SYSTEMS
FIRMAN1, JANSON NAIBORHU2, ROBERD SARAGIH 3
1,2,3Industrial & Financial Mathematics Group, ITB,
fina@students.itb.ac.id, janson@math.itb.ac.id, roberd@math.itb.ac.id
Abstract. In this paper, we will design an input control to track the output of a
non-minimum phase nonlinear system. The design of the input control is based
on an input-output linearization method and gradient descent control. To
perform the design of the input control, the other output should be selected such
that the systems becomes minimum phase systems. Furthermore, the desired
output of the output which has been selected will be set based on the desired
output of the original system.
Key words and Phrases: input-output linearization, steepest descent control,

minimum phase system,non-minimum phase system.
1. Introduction
A system called minimum phase systems if the origin of the zero dynamics
is asymptotically stable, but if zero dynamics is unstable, then the system called
non-minimum phase [6] . In [1], Isidori has shown that for the minimum phase
systems, then it can be choose a static control law such that the output of the system
goes to zero while keeping the state of the system bounded locally. While that in
[2] has introduced a dynamic feedback control which is a modification of the
steepest descent control. The modified steepest descent control success to handle
the problem that arise if relative degree of the nonlinear system not well defined.
Recently, output tracking problems on nonlinear non-minimum phase systems have
been investigated intensively. The stable inversion proposed in [3], [4] is an
iterative solution to the tracking problem with the unstable zero dynamics. This
method requires the system to have well defined relative degree and hyperbolic
dynamics, i.e. no eigenvalues on imaginary axis. In [5], proposed the control design
procedure for the output tracking. The design procedure consists of two steps. In
the first step, the standard input output linearization is applied. In the second step,
we group a subset of the output with the internal dynamics as one subsystems,
which is usually nonlinear, and the rest of output as the other subsystem which is
linear, the nonlinear subsystems is linearized about its equilibrium. In [7], Riccardo
Marino and Patrizio Tomei have shown how to design a globally stabilizing
dynamic output feedback controller of order n + 2(ρ − 1)(n is the system order, ρ is
the relative degree) for a class of nonlinear nonminimum phase systems. The
103
104
system are required to be minimum phase with respect to a linear combination of

the state variables. In [8], the asymptotic output tracking which is a class of causal
nonminimum phase uncertain nonlinear systems is achieved by using higher order
sliding modes (HOSM) without reduction of the input-output dynamics order. In
[9], a new nonlinear dynamic controller is described based on the gradient descent
control. Performance index is generated by error of output system from output
desired value and internal state of the system. adding of an internal state to
maintain the stability of internal dynamic of the system.
In this paper, we will design the input control which ensures that the nonlinear
system is stable asymptotically. If the system has relative degree well defined, we
used the input output linearization method [1] to design input control. Then if the
relative degree of the system is not well defined, to design of the input control
based on the modification of the steepest descent control. Modification is the
addition of an input artivisial of the steepest descent control. Furthermore, the
desired output of the output which has been selected will be set based on the
desired output of the original system.
2. Output Tracking
We will investigate the output tracking for a non-minimum phase nonlinear
system. The non-minimum phase system in the following form :
x˙ = Ax + bu + φ(y), x Rn, u R (1)

y = x1 (2)
in which φ(y) is a smooth vector field in Rn with φ(0) = 0, b = [0,...,0,br,...,bn]T with
br 6= 0,
A .
Our objective is to make the output system (1)-(2) tracks the desired output
while keeping the state bounded. To keep the state bounded is difficult for non-
minimum phase system. In this paper, we design a controller such that the output
system (1)-(2) tracks the desired output while keeping the state bounded. To
perform the design of the input control, the other output should be selected such
that the system (1) becomes minimum phase with respect to a new output. Thus, in
this paper we assume that
Assumption 1. There exists a linear combination of the state variables µ = t1x such
that the zero-dynamics (1)-(2) is asymptotically stable.
We consider the system (1)-(2). Based on assumption 1, consider now a new
output
z1 = µ = t1x = (t11,...,t1n)x
with the relative degree of system (1) with respect to µ is still equal to r. The
system (1) can be transformed to
105
z˙1 =z2 z˙2=z3
.. . (3)
z˙r = b(z) + a(z)u, η˙ =
q(z) y = z1.
Before applying the control law, we have to set up the output desired for z1, i.e. z1d.
In this paper we consider the system which satisfies the following assumption
Assumption 2. λl(x) = xl, l {2,...,n} then x˙l = fl(xl,x1) can

be solved by substituting x1 = yd(t)
Thus, λld = xl(t).

Based on assumption 2, we obtain
z1d = t11yd + t12λ2d + ··· + t1nλnd
let . Thus
e˙k = ek+1, 1,...,r − 1
e˙r = , (4)
η˙ = q(e + zd,η,η).
By choosing
)) (5)
where c0,c1,...,cr−1 are real numbers,
Then system (4) can be written as

e˙ = Ae, (6)
η˙ = q(e + zd,η), (7)
where the matriks A has a characteristic polynomial : p(s) = c0 + c1s + ··· +
cr−1sr−1 + sr.
By choosing the value of ci;i = 0,...,r such that all the roots oh the polynomial p(s)
have negative real part, and applying the control law (5), then the equilibrium point
(e,η) = (0,0) of the system (4) is asymptotically stable. (see Proposition 4.5.1 in
[1]).
Thus e1 tend to zero if time t goes to infinity. Then z1 tend to z1d if time t goes to
infinity. Thus x1 tracks to the desired output yd(t).
Next, if the relative degree of the system (1)-(2) is not well defined.
We construct the performance index as a descent function as follow :
. (8)
By”Trajectory Following Method” [10], the control u is determined from the
differential equation
106
, (9)
where the control law in (9) is called the steepest descent control [2]. Furthermore
calculate the time derivative of descent function (8) along the trajectory of the
extended system
(10)
(11)
(12)
. (13)
Now, we have
. (14)
From equation (14), we see that the value of time derivative of the descent function
along the trajectory of the extended system can not be guaranteed to be less than
zero for t ≥ 0.
Now we modify the steepest descent control (9) by adding an artificial input v.
Then the extended system (1) becomes
x˙ = Ax + bu + φ(y), x Rn, u R (15)
(16)
From equation (14), we have
. (17)
From Sontag formula, we get
. (18)
The control law in equation (16) is called as modified steepest descent control.
Based on the modified steepest descent control, then F˙(y,y,...,y˙ (r)
) <0, if
= 0. Thus, if we choose aj such that the polynomial
p(s) = a0 + a1s + ··· + ar−1sr−1 + sr is Hurwitz, z1 tend to z1d if time t goes to infinity.
Thus x1 tracks to the desired output yd(t).
Example 1. Consider the nonlinear system (SISO)

x˙1 =
x˙2 = (19)
107
x˙3 =
y = x1, yd = sin(t).
The nonlinear system (19) has relative degree 2 at any point x0 (relative degree of
the system is well defined). In normal form, the nonlinear system (19) becomes
z˙1 = z2
z˙2 = (20)
η˙ = η − z2.
Because the stability of zero dynamics is unstable, the nonlinear system (19) is the
non-minimum phase. Now, redefining output z1 = µ = x1 + 2x2 + 2x3. By considering
the new output, the relative of the system (19) is 2 at any point x0
and normal form
z˙1 = z2
z˙2 = b(z) + a(z)u (21)
η˙ = −η + z2,
where b(z) = x3 − 6x21 − 4x1x2 − 8x31, a(z) = 1. The zero-dynamics of the system
(19) are asymptotical stable. Thus the system (19) is the minimum phase with
Figure 1 : Output tracking z1 to z1d
Figure 2: Output tracking (original system) y to yd

108
respect to a new output.

Let yd(t) = sin(t) = x1d(t). Next, we chose z1d(t) such that if z1(t) tracks z1d(t), then
y(t) tracks to the desired output yd(t). By replacing x1 with x1d(t) = sin(t), then we
have x2d = cos(t) − 2sin2t. By replacing x2 with x2d(t), we have a differential
equation ˙x3 − x3 = sin(t) + 2sin(2t) − 2sin2t.
Thus x3d = −1/2cos(t) − 1/2sin(t) − 2cos2t + 2. Next, z1d = x1d + 2x2d + 2x3d = cos(t).
According to (5), the input control is
u = −b(z) + z¨1d − c0(z1 − z1d) − c1(z˙1 − z1˙d)
Simulation results are shown in Figure 1 and in Figure 2 for constants: c0 = 6, c1 =
10. Initial value: x1(0) = 0, x2(0) = 1, x3(0) = 1.
In Figure 1, the output which has been selected such that the system become
minimum phase track the desired output z1d.
In Figure 2, the output of the original system track the desired output yd.
Figure 3: Output tracking z1 to z1d
Example 2.
x˙1 =
x˙2 = (22)
x˙3 = x2u
y = x1, yd = sin(t).
The nonlinear system (22) has relative degree 2 at any point x0 6= 0 (relative degree
of the system is not well defined). The system (22) is the non-minimum phase. By
considering the new output z1 = µ = x1 +2x2 +2x3, the zero dynamic are ˙η = −η.
Therefore the system (22) is minimum phase with respect to the new output. By
the same method as in example 1, obtained z1d = x1d + 2x2d + 2x3d = cos(t).
According to (9), the modified steepest descent control is u˙ = −2x2a2(a0(z1 − z1d) +
a1(z˙1 − z1˙d) + a2(z¨1 − z¨1d)) + v, (23) with v as in (18). Simulation results are
109
shown in Figure 3 and in Figure 4 for constants: a0 = 35, a1 = 12, a2 = 1. Initial

value: x1(0) = 0, x2(0) = 1, x3(0) = 1, u(0) = 1.
In Figure 3, By modified steepest descent control, the the output which has been
selected such that the system become minimum phase track the desired output z1d.
In Figure 4, the output of the original system track the desired output yd.
Figure 4: Output tracking (original system) y to yd
3. Conclusions
In this paper, we have investigated the output tracking for a class of
nonlinear non-minimum phase system (1)-(2). The input control has been designed
for the output tracking. To perform the design of the input control, the systems (1)
are required to be minimum phase with respect to a new output, where the new
output is a linear combination of the state variables. Furthermore, the desired new
output will be set based on the desired output of the original. If the system (1) has
relative degree well defined with respect to the new output, we used the input
output linearization method to design the input control. Then if the relative degree
of the system (1) is not well defined with respect to the new output , to design of
the input control based on the modification of steepest descent control.
References
[1] A. Isidori.Nonlinear Control Systems: An Introduction. Secon Edition Springer-

Verlag Berlin, Heidelberg 1989.
[2] J Naiborhu.Trajectory Following Method on Output Regulation of Affine Nonlinear
Control Systems with Relative Degree not Well Defined. ITB J.Sci.,Vol.
43A,No.1,2011,73-84.
[3] D.Chen and B.Paden (1996), Stable inversion of nonlinear non-minimum Phase
systems, Int.J.Control , Vol.64,No.1, 45-54.
[4] D.Chen (1993), Iterative solution to stable inversion of nonlinear non-minimum
Phase systems, Proc. American Control Conference, June, 2960-2964.
110
[5] Dong Li. (2005), Output Tracking of Nonlinear Nonminimum Phase Systems : an
Enginering Solution, Proceedings of the 44th IEEE Conference on Decision and
control, and the European Control Conference, Sevilla,Spain,Dec 12-15, 3462-
3467.
[6] Hassan K.Khalil. (2002), Nonlinear Systems, Prentice Hall, New Jersey , Third
Edition.
[7] Riccardo Marino, Patrizio Tomei. (2004), A class of Globally Output Feedback
Stabilizable Nonlinear Non-minimum Phase Systems, 43rd IEEE Conference on
Decision and Control, Atlantis, Paradise Island,Bahamas Des 14-17, 4909-4914.
[8] S. Baev, Y.Shtessel, I. Shkolnikov. (2007), HOSM driven outputvtracking in the
nonminimum-phase causal nonlinear Systems, Proceeding of the 46th IEEE
Conference on Decision and Control, New Orleands,LA,USA, Des 12-14, 3715-
3720.
[9] J. Naiborhu.Output Tracking of Nonlinear Non-minimum Phase Systems by
Gradient Descent Control. proceeding of the IASTED International Conference
Identification, Control, and Applications (ICA 2009),August 17-19, 2009, Honolulu
Hawaii, USA, 110-115
[10] Vincent,T.L. and W.J. Grantham,Nonlinear and Optimal Control Systems, John
Wiley and Sons, Inc., New York, 1997.
Applied Mathematics
AN ANALYSIS OF A DUAL RECIROCITY BOUNDARY

ELEMENT METHOD
IMAM SOLEKHUDIN1, KENG-CHENG ANG2

Gadjah Mada University, Yogyakarta 55281, imamsolahuddin@yahoo.com
2Mathematics and Mathematics Education, National Institute of Education,
Nanyang Technological University, 1 Nanyang Walk, Singapore 637616,

kengcheng.ang@nie.edu.sg
Abstract. Infiltration from irrigation channels is governed by Richard’s

equation. This equation may not be solved analytically or numerically. To study
the infiltration more conveniently, the governing equation is transformed to a
Helmholtz equation, which may be solved numerically. A numerical method that
may be employed to solve the Helmholtz equation is the dual reciprocity
boundary element method (DRBEM). In this study, we employ the DRBEM, to
solve a problem involving infiltration from periodic flat channels. The DRBEM
is implemented on MATLAB. Using the analytic solution of this problem
obtained by Batu, an analysis of the method is made.
Key words and Phrases:Dual reciprocity boundary element method, infiltration,

periodic channels, homogeneous soil
1. Introduction
In the last few decades, the problem of water infiltration has been studied by
a number of researchers such as Gardner [12], Philip [14, 15], Batu [7], and Basha
[6]. Gardner proposed an exponential relationship between the hydraulic
conductivity and the suction potential. Philip wrote an excellent review of soil
water physics, he also investigated infiltration from spherical cavities, and obtained
the exact solution using the method of separation variables. A problem involving
steady infiltration from periodic flat channels was studied by Batu. Batu obtained
the analytic solution of the problem using the same method as Philip. Basha
employed Green’s function to study infiltration toward a shallow water table.
The studies carried out by some of these researchers were limited, since the
methods used may not be applied to solve infiltration problems with arbitrary
geometry of the boundary or boundary conditions. A possible way to deal with
such problems is to use numerical methods. A numerical method used to solve such
111
112
problems was the boundary element method (BEM). This was used by Pullan and
Collins [16], Pullan [17], Azis et al [5], Lobo et al [13], Clements et al [10], and
Clements and Lobo [11].
The method has successfully handled various infiltration problems.
However, this method is only used for special governing equations that can be
transformed to a type of Helmholtz equation. This is due to the requirement of the
fundamental solution of the Helmhotz equation in the BEM formulation. If the
governing equation cannot be transformed to the type of Helmholtz equation, the
BEM may not be used.
A method that may be employed to solve more general Helmholtz equations
is the dual reciprocity boundary element method (DRBEM). This method requires
only the fundamental solution of the Laplace equation. However, to apply this
method a number of constant line segments and a number of collocation points
must be constructed. These two numbers must be chosen in such a way, so that
some degree of accuracy and optimum computational time are achieved.
In this paper, a problem involving infiltration from periodic flat channels is
solved numerically using the DRBEM. Different sets of constant line segments and
collocation points are tested to obtain numerical solutions. The corresponding
analytic solution is used to analyze error obtained from these different sets.
2. Problem Formulation
We consider an array of equally spaced identical flat channels, each of width
of 2L. The distance between two consecutive channels is 2D. We assume that the
channels are sufficiently long and that there is a large number of such channels.
Hence the flow pattern is two-dimensional and the influence of outer channels is
negligible. The fluxes on the channels and on the soil surface outside the channels
are v0 and 0 respectively. The geometries described are as shown in Figure 1.
flat channel
0
X
v0
2L 2D
Z
Figure 1. Cross-section of periodic flat channels.
Due to the symmetry of the problem, it is sufficient to consider the

semiinfinite region defined by 0 ≤ X ≤ L + D and 0 ≤ Z ≤ ∞. This region is denoted
by R with boundary C. Along the surface of the channel, the boundary is
symbolized by C1 and along the surface of the soil outside the channel by C2. The
fluxes across C1 and C2 are v0 and 0 respectively. The boundary along X = L+D and
X = 0, denoted by C3 and C4, have zero fluxes.
113
3. Basic equation
Steady infiltration in porous media is governed by the following Richard’s
equation
, (1)
where K is the hydraulic conductivity, Z is the vertical physical space coordinate,
pointing positively downward [17, 18, 19].
The Kirchhoff transformation
where Θ is the matric flux potential, and the exponential relation

K(ψ) = Kseαψ, α >0, ψ ≤ 0,
where Ks is the saturated hydraulic conductivity and α is an empirical parameter [7,

17, 18, 19], transform equation (1) to the linear equation,
. (2)
The horizontal and vertical components of the flux, which are functions of the
matric flux potential, are
and
respectively [7]. The flux normal to the surface with outward pointing normal, n =
(n1,n2), is given by
.
Using the dimensionless variables
and
and the substitution
Φ = ϕez, (3)
in equation (2), we obtain

(4)
which is a type of Helmholtz equation. The dimensionless flux is
which yields
114
(5)
Fluxes at the surface of the channels are v0, and therefore dimensionless
fluxes are 2π/αL. This implies
, for 0 and z = 0. (6)

The condition of zero flux across the soil surface outside the channels, and along
X = 0 and X = L + D implies
for ) and z = 0, (7)
, for x = 0 and z ≥ 0, (8)
and
. (9)
It is assumed that as X2 + Z2 → ∞, ∂Θ/∂X = 0 and ∂Θ/∂Z = 0 [7]. Thus
f = 2ϕ ez. (10)
Hence, using equation (5), we have
) and z = ∞. (11)
A DRBEM is employed to obtain numerical solutions to equation (4) subject

to boundary conditions (6) to (11). The DRBEM was initially proposed by Brebbia
and Nardini [8]. Other researchers have been using this method to solve various
boundary value problems, such as Zhu et al. [20], Ang [3], and Ang and Ang [4].
An integral equation for solving equation (4) is
where
is the fundamental solution of the two-dimensional Laplace’s equation, and lies on

smooth part of (C) ∈R
Integral equation (12) may be solved numerically. Boundary C is discretized

by constant line elements, and a number of interior points are chosen as collocation
points. The midpoints of every segments are chosen as collocation points besides
the interior collocation points. Let N and M be the numbers of the line segments
and the interior points respectively. C(1), C(2), ···, C(N) be the line segments, and
115
(a(i),b(i)) is the midpoint of C(i), i = 1,2,··· ,N. Points (a(N+1),b(N+1)), (a(N+2),b(N+2)), ···,
(a(N+M),b(N+M)) be the interior collocation points. The value of ϕ can be approximated
by
Where
(13)
and
(14)
and
,
where
and
From the numerical values of ϕ obtained, The dimensionless MFP, Φ, is

computed using equation (3).
116
4. Results and Discussion

The method described in the preceding section is tested on a problem
involving steady infiltration from periodic flat channels in homogeneous pima clay
loam (PCL). The value of both L and D are set to be 50 cm. The value of α for PCL
is 0.014 cm−1 [2, 9].
One way to analyse the numerical solutions is to compare them with the
analytic solution of the problem. For the values of L, D, and α set as mentioned
above, the analytic solution is
, (15)
where
and
The DRBEM is used to obtain numerical solutions of Φ, and implemented

using MATLAB. In the implementation of the DRBEM, a numerical integration
package in MATLAB, quadl, is used to compute the integrals in equations (13) and
(14). In the DRBEM formulation, the domain must be bounded by a simple closed
curve. Therefore, the value of z in the domain must satisfy 0 ≤ z ≤ c, for a positive
real number c. In this study, the value of c is set to be 4. In other words, z = 4 is an
imposed boundary.
We first set the number of constant line elements, N, to be 200. Using this
value of N, six different values of M are chosen. For easy reference, this set of M
and N is called set A. We then fix the value of M, to be 625. With this value of M,
different values of N are chosen, and this set of M and N is denoted by set B. Sets A
and B are summarized in Table 1.
Table 1. Sets of constant line segments and interior collocation points.

Set A Set B
N M N M
200 100 200 625
200 225 300 625
200 400 400 625
200 625 500 625
200 900 600 625
200 1225 700 625
Numerical solutions of Φ at six interior points (0.1,0.8), (0.4,0.8), (0.1,2.0),

(0.4,2.0), (0.1,3.2) and (0.4,3.2) are obtained using the values of N and M in Table
1 above. These points are chosen, such that two points are at z = 0.8 (near the
surface of soil), two are at z = 2.0 (middle of the domain in z-direction), and the
two other are at z = 3.2 (near the imposed boundary). Using the analytic solution
117
(15), errors of the numerical solutions are computed, and shown graphically in
Figures 2 and 3.
0.04 0.04
(0.4,3.2)
(0.1,3.2)
0.03 0.03
Er
ro 0.02 0.02
(0.1,2.0) (0.4,2.0)
0.01 0.01
(0.1,0.8) (0.4,0.8)
0 0 100
100 225 400 625 900 1225 225 400 625 900 1225
M M
(a) Errors at three points along x = 0.1. (b) Errors at three points along x = 0.4.
Figure 2. Plot of dimensionless errors at selected points

for N = 200 and six different values of M.
Figure 2 shows errors of Φ obtained at the six interior points, using N and M
in Set A. It can be seen from Figure 2 that the errors increase as z increases. In
particular, the errors at z = 0.8 are below 0.004, at z = 2.0 are about 0.01, and at z =
3.2 vary between 0.03 and 0.04. The larger errors could be due to the imposed
boundary at z = 4.
It can also be seen that at z = 0.8 and z = 2.0, there are no significant
differences in the errors obtained from the six different sets of N and M. In contrast,
there are significant differences in the errors obtained at z = 3.2. The error
decreases as M increases, and remains approximately constant when M ≥ 625.
These results indicate that a larger value of M tends to yield more accurate values
of Φ than smaller values of M in the region beyond a certain depth. In addition,
there is a threshold value of M, at which the accuracy may not be affected by any
increase in the value of M. In this particular case, for N = 200, this threshold value
for M is 625.
0.04 0.04
(0.1,0.8)
(0.4,0.8)
(0.1,2.0)
(0.4,2.0)
0.03 (0.1,3.2)
0.03 (0.4,3.2)
E 0.02
0.02
0.01
0.01
0 0
200 300 400 500 600 700
200 300 400 500 600 700
N
N
a) Errors at three points along x = 0.1. (b) Errors at three points along x = 0.4.
Figure 3. Plot of dimensionless errors at selected points for

M = 625 and six different values of N.
118
Figure 3 shows errors of numerical solutions obtained using Set B. As

before, the errors obtained increase as z increases. However, unlike the results in
Figure 2, the increase in N affects the errors at the selected values of z. In
particular, the errors decrease as N increases. The decrease in the error at some
value of z is steeper than that at smaller values of z. This means that a value of N
yields more accurate solutions than smaller values of N, especially when z
approaches 4.
In the DRBEM construction, for n constant elements and m interior
collocation points, the number of the line integrals (13) and (14) evaluated is
(n+m)×2n. Thus, if for instance we have two pairs of N and M, which produce the
same number of collocation points, the pair with larger value of N produces bigger
number of such integrals.
As mentioned earlier, a built in numerical integration function quadl was
used to compute the line integrals. It is found that quadl tends to take a
considerable amount of time to compute these integrals. Hence, it is expected that a
greater proportion of computational time is spent on evaluating the line integrals.
Hence, a bigger number of such line integrals takes longer computational times
than smaller one.
From the results presented, more accurate solutions are obtained by
increasing the number of constant elements. Despite the accuracy of the solutions
obtained, a number of constant line segments makes the computational time taken
much longer than smaller numbers of constant line segments. In the study of
infiltration from irrigation channels, researchers are normally interested in the
solutions at shallow level of soil. A good accuracy at this level may be achieved by
using small numbers of constant elements. Hence, small numbers of constant
elements give more advantages in infiltration studies, especially in computational
time to obtain solutions.
A problem involving steady infiltration from periodic flat channels have
been solved numerically using a DRBEM. Two sets of constant line elements, M,
and interior collocation points, N, are considered to obtain numerical solutions. The
first set contains six pairs of M and N, with fixed M, and the other set contains six
other pairs with fixed N.
Using the solutions obtained, errors are computed using the analytic solution
of the problem. From the errors obtained, the number of interior points may not
affect the accuracy of the solutions at small values of z. However, when z
approaches 4, the accuracy is affected by the number of interior collocation points.
In contrast, the number of constant line elements gives impact on the accuracy,
especially at z close to 4.
Using a higher number of constant elements tends to result in longer
computational time. This is likely to be due to the fact that more line integrals need
to be evaluated and compounded by the use of a built in function in MATLAB,
quadl. Such problems may be overcome by using other mean of implementation,
and more efficient schemes of numerical integration.
Acknowledgement Imam Solekhudin wishes to thanks the Directorate General of

the Higher Education of the Republic of Indonesia (DIKTI) for providing financial
support for this research.
119
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equation for a class of problems concerning infiltration from periodic channels”,
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equation with application to evaporation from a water table”, Soil Sci. 85 (1957) 228
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10.
Applied Mathematics
AN INTEGRATED INVENTORY MODEL WITH

IMPERFECT-QUALITY ITEMS IN THE PRESENCE OF
A SERVICE LEVEL CONSTRAINT
NUGHTHOH ARFAWI KURDHI1 AND SITI AMINAH2
1
Department of Mathematics, Faculty of Mathematics and Natural Science,
Sebelas Maret University, math_anomali@yahoo.co.id
2Department of Mathematics, Faculty of Mathematics and Natural Science,
Sebelas Maret University, sithiamin@yahoo.co.id
Abstract.This article develops an integrated production-inventory model with

imperfect quality items. The production process is imperfect and produces a
certain number of defective items with a known probability density function.
Our work is based on the paper of Lin [11]. We extend the model by considering
a service level constraint and lead time and ordering cost reductions act
dependently. It is assumed that shortages are allowed and partially backlogged
on the buyer’s side, and that the lead time demand is assumed to be normally
distributed. The objective is to determine the optimal order quantity, reorder
point, lead time and the number of deliveries simultaneously so that the
expected total system cost is minimized. Furthermore, numerical example and
sensitivity analysis are carried out to demonstrate the benefits of the model and
to illustrate the effect of parameters on the decision and the total system cost.
Keywords and Phrases:Integrated model, imperfect quality, service level cons-

traint, controllable lead time.
1. Introduction
The vendor-buyer integrated production inventory system received a lot of
attention in recent years. Strategic vendor-buyer alliance is a formalized type of
collaborative relationship between a vendor and a buyer in a supply chain. It
involves commitment to long-term cooperation, shared benefits and cost, joint
problem solving and information sharing. This close partnership will ultimately
improve product quality and reduce inventory cost and lead time of the supply
chain. Therefore, several authors (e.g., Ben-Daya and Hariga [2], Lin [12], Rad and
Khoshalhan [16], Shaha and Shahb [17]) have presented the integrated inventory
management system. Goyal [4] is among the first who analyzed an integrated
inventory model for a single-vendor single-buyer system, in which he assumed that
the vendor’s production rate is infinite. Banerjee [1] modified Goyal’s model and
presented a joint economic-lot-size model where a vendor produces for a buyer to
121
122
order on a lot-for-lot basis. Goyal [5] further generalized Banerjee’s [1] model by
relaxing the assumption of the lot-for-lot policy of the vendor and suggested that
the vendor’s economic production quantity should be a positive integer multiple of
the buyer’s purchase quantity.
The classical economic order quantity (EOQ) model assumes that items
produced are of perfect quality, which is usually not the case in real production. In
practice, it can often be observed that there are defective items being produced due
to imperfect production process. Porteus [15] first incorporated the effect of
defective items in the basic EOQ model. Recently, several authors (e.g., Hsu1 and
Hsu2 [7], Hsua and Hsub [8],Lin [11]) consider imperfect quality item in integrated
vendor-buyer inventory model. Lin [11] explored the lead time reductions and
imperfect quality items problems on the integrated production-inventory system,
where shortages are allowed with partial backorder. We notice that the possible
relationship between ordering cost and lead time is ignored. In practices, the lead
time and ordering cost reductions may be related closely; the reduction of lead time
may accompany the reduction of ordering cost, and vice versa. For example, the
implementation of electronic data interchange (EDI) can reduce both the lead time
and ordering cost simultaneously (see, Chen et al. [3], Ouyang, et al. [14]). In
integrated vendor-buyer models described above, the stock out cost is one of the
components in the objective function. However, the stock out cost often includes
intangible components such as loss of goodwill and potential delay to the other
parts of the inventory system, so it is difficult to explicitly express the stock out
cost. Instead of having a stock out cost term in the objective function, a service
level constraint, which implies that the stock out level per cycle is bounded, is
considered. Moreover, a service level criterion is generally easy to interpret and
establish. Thus service level constraint models are more popular in real-life
inventory systems than full-cost models (see, Lin [10], Ma and Qiu [13]).
Based on the survey above, in this paper, we extend Lin’s [11] model by
considering the service level constraint and the reduction of lead time accompanies
a decrease of ordering cost. The objective is to determine the optimal order
quantity, reorder point, lead time and the number of deliveries simultaneously so
that the expected total system cost is minimized. By constructing Lagrange
function, the analysis regarding the solution procedure is conducted, and a solution
algorithm is then provided. Moreover, numerical examples and sensitivity analysis
are carried out to demonstrate the benefits of the model and to illustrate the effects
of parameters on the decision and the total system cost. This paper is organized as
follows. In Section 2, the notation and assumptions used in this paper are
introduced. In Section 3, we develop a mathematical model that integrates the
vendor’s and the buyer’s expected total cost and takes into consideration imperfect-
quality items and service level constraint. In Section 4, a Lagrange function and an
efficient algorithm are constructed to find the optimal solutions. Section 5 provides
a numerical example and discussion of the results. Finally, in Section 6 we draw
some concluding and give suggestions for the future research.
2. Notations and Assumptions

In this paper, we propose an integrated vendor-buyer inventory model with
imperfect-quality items and service level constraint, in which shortages are allowed
with partial backorders. The model is developed using the following notations and
assumptions.
123
2.1 Notation
Expected demand per unit time on the buyer (for non-defective items).
Production rate on the vendor.
Transportation cost per order.
Unit treatment cost of defective items on the vendor.
Fixed penalty cost per unit short on the buyer.
Marginal profit (i.e., cost of lost demand) per unit on the buyer.
Original length of lead time (before any crashing of lead time is
made).
The length of lead time (decision variable).
Reorder point of the buyer for non-defective items (decision variable).
Setup cost per set-up for the vendor.
Inventory holding cost per non-defective item per unit time for the
buyer.
Inventory holding cost per defective item per unit time for the buyer
.
Inventory holding cost per item per unit time for the vendor.
Original ordering cost (before any crashing of lead time is made).
Ordering cost per order for the buyer.
Proportion of demands which are not met from stock, that is, 1 is
the service level.
Fraction of the demand during the stock-out period that will be
backordered,0 1.
Screening rate on the buyer.
Screening cost per unit for the buyer.
Order quantity of the buyer for non-defective items.
Order quantity of the buyer per order including defective items, i.e.,
shipping quantityfrom the vendor to the buyer per shipment (decision
variable).
The number of lots in which the items are delivered from the vendor
to the buyer in one production run, a positive integer (decision
variable).
The period during which the vendor produces.
The period duringwhich the vendor supplies from inventory.
The lead time demand with finite mean and standard deviation
√ ,
where denotes the standard deviation of the demand per unit time.
The number of defective items in a lot size , a random variable.
∙ Mathematical expectation.
2.2 Assumptions
1. There is single-buyer and single-vendor for a single-product in this model.

2. Inventory is continuously reviewed. The buyer places an order or requests
for successive shipments when on hand inventory level (based on the
number of non-defective items) falls to the reorder point . The reorder point
expected demand during lead time + safety stock , and
124
(standard deviation of lead time demand), i.e., √ , where

is known as the safety factor.
3. The lead time has mutually independent components. The
component has a minimum duration and normal duration , and a
crashing cost per unit time . The components can be rearranged such that
⋯ . The components are crashed starting from the least
crashing cost per unit time.
4. Let be the length of the lead time with component 1,2, … , crashed to
their minimum duration and Σ Σ , 1,2, … , .
Also, we let Σ and Σ , the
lead time crashing cost per cyclefor a given ∈ , .
5. The extra costs incurred by the vendor will be fully transferred to the buyer
if shortened lead time is requested.
6. The buyer orders a lot of size (for non-defective items) and will receive
the batch quantity in equally-sized shipments of size , where is a
positive integer.
7. An arriving lot may contain some defective items. It is assumed that the
number of defective items , in an arriving order of size is a random
variable which has a binomial distribution with parameters and where
0 1 represents the defective rate in the order lot. Upon the arrival
of the order, all the items in the lot are inspected with the screening rate by
the buyer, and defective items in each lot are discovered and returned to the
vendor at the time of delivery of the next lot.
8. Vendor’s production rate for the non-defective items is greater than buyer’s
demand rate, i.e., 1 .
9. The screening process and the demand proceed simultaneously, but the
screening rate is greater than the demand rate.
3. Mathematical Model
In this section, a mathematical model is formulated to minimize the joint
total expected cost per unit time with service level constraint by finding the optimal
order quantity, lead time and safety factor of the buyers and the number of
shipments in a production cycle between the vendor and the buyer. The integrated
inventory model is designed as follows. If the buyer orders quantity , then the
vendor produces at one set-up, with a finite production rate , in order to reduce
its set-up, where is a positive integer. The production process of the vendor is
assumed to deteriorate resulting in a random number of defective items, .
Therefore, the expected length of each ordering cycle for the buyer is
and the expected length of each production cycle for the vendor
is . The buyer adopts 100% screening process to inspect all
the items upon arrival and all the defective items in each lot are returned to the
vendor at the time of delivery of the next lot.
3.1. Buyer’s expected average total cost per unit time

Based on the above notations and assumptions, the inventory system of the
buyer can be depicted in Figure 1. The first step in formulation of the total
expected cost per unit time for the buyer is to derive the expressions of the separate
cost components for the buyer. For the model without interaction between lead
time and ordering cost and without service level constraint, we will closely follow
125
the model in Lin [11]. Such that, holding cost for non-defective itemis
1 , holding cost for defective
items is , stock-out cost is 1 where
is the expected number of shortages at the end of the cycle, screening cost is
, and lead time crashing cost is . The buyer’s total cost per cycle, given that
there are defective items in an arriving shipment of size , is the sum of the
ordering cost, transportation cost, holding cost for non-defective item, holding cost
for defective items, stock-out cost, screening cost, and lead time crashing cost.
Symbolically, the buyer’s total cost per cycle can be represented by:
, , ;
1
2 2
1 , (1)
Figure 1. The inventory system for the buyer
The number of defective items in a

lot follows a binomial random variable with parameters and , where 0
1 represents the defective rate in an order lot. That is,
1 , for 0,1,2, … , .
Therefore, one has
and 1 .
The expected length of the cycle time is and therefore
the expected average total cost per unit time for the buyer is
, , ;
, ,
1
1
1
2 1 2
1 . (2)
According to the opinion of Chen et al. [3] and Ouyang et al. [14], the
reduction of lead time may accompany the reduction of ordering cost, and vice
126
versa. In this subsection, we consider that the lead time and ordering cost
reductions have the following logarithmic functional relationship:
ln , (3)
where 0 is constant scaling parameter for the logarithmic relationship between
percentages of reductions in lead time and ordering cost. In this case, the ordering
cost can be written as
ln , (4)
where ln and 0. Substituting (4) into (2), the model (2)
can be rewritten as
, , ln 1
1
1
2 1 2
1 . (5)
For a given safety factor which satisfies the probability that lead time
demand at the buyer exceed the reorder point, the actual proportion of demands not
met from stock should not exceed the desired value of . Therefore, the service
level constraint can be established as

.

Symbolically, the service level constraint can be formulated as

. Using this definition and replacing the stock out cost by a definition
on the service level in (5), the problem is transformed to
Minimize , , ln
1 1
, (6)
subject to .
4.2. Vendor’s expected average total cost per unit time

The production rate of vendor’s non-defective items is greater than the
buyer’s demand rate and then the vendor’s inventory level will increase gradually.
When the total required amount is fulfilled, the vendor stops producing
immediately. Therefore, the vendor’s inventory per production cycle can be
obtained by subtracting the accumulated buyer inventory level from the
accumulated vendor inventory level. Figure 2 shows the vendor’s holding cost per
cycle can be obtained as (see, for example, Hsua and Hsub [7] and Lin [12]):
Holding costper cycle
[bold area – shaded area]
1 2 ⋯ 1
2
1 1 2 ⋯ 1
2
. (7)
127
The total cost of the vendor per production cycle includes the set-up cost, defective
item treatment cost,and holding cost. One has
, . (8)
Therefore, the expected average total cost per unit time for the vendor can be
obtained as
,
,
1 1
2 2 1
. (9)
Figure 2. Time-weighted inventory for the vendor

128
4.3. The joint total expected average cost per unit time
The joint expected average total cost per unit time consists of the expected
total cost of the buyer and the expected total cost of the vendor. Therefore, we have
the following problem:
Minimize , , , , , ,
1
1 1 1
1 2 2 2
1
2 1 2
1 , (10)
subject to .
Note that when the lead time and ordering cost reductions are assumed to act
independently and there is no service level constraint, model (10) can be reduced to
Lin’s [11] model.
As mentioned earlier, it is assumed that the lead time demand follows a
normal distribution with finite mean and standard deviation √ . We note that
√ , and hence, the expected shortage quantity, , where
max , 0 ,can be written as
√ √ Ψ 0, 11
whereΨ 1 Φ . Notations andΦdenote the standard normal
probability density function and cumulative distribution function, respectively.
Substituting (11) into (10) and using the safety factor as a decision variable instead
of , the model (10) can be transformed to
Minimize , , , ln
1 1 1
1 2 2 2
√ 1 √ Ψ
2 1 2
1 , (12)
√
subject to .
4. Optimal Solution
The problems are to find the optimal values of , , ,and such
that , , , in (12) is minimum and satisfying the service level constraint.
The inequality constraint in (12) can be converted into equality by adding a slack
variable 0. One has
Minimize , , , ln
1 1 1
1 2 2 2
129
√ 1 √ Ψ
2 1 2
1 , (13)
subject to √ Ψ 1 .
The Lagrange function of (13) is

, , , , , ln
1
1 1 1
1 2 2 2
√ 1 √ Ψ
2 1 2
1 √ Ψ 1 , (14)
where is the Lagrange multiplier.
For any given , , , , , , , , , , is a concave down function in
variable for ∈ , , because
, , , , ,
1 Ψ Ψ 0. (15)
√
Hence, for fixed , , , , the minimum expected joint total cost will occur at the
boundary points and .
Taking the first partial derivatives of (14) with respect
to , , ,and respectively and equalizing the results to zero, we have
, , , , ,
ln
1
1 1 1
1 2 2 2
1 0, (16)
, , , , ,
√ 1 1 Φ 1 0,(17)
, , , , ,
√ Ψ 1 0, (18)
and
, , , , ,
2 0. (19)
From (19),if 0, then ought to be equal to zero. Substituting 0into (17), it is
reduce to
√ 1 1 Φ 1 0. (20)
An unreasonable solution, Φ 0(since the value of a CDF cannot be
negative) is obtained when (20) is solved for .Hence 0and 0can be
√
expected and the constraint is active when the optimal solution is
obtained.
Solving equations (16) to (18) respectively, we have the following results:
, (21)
∗
,(22)
and
∗
Ψ
√
. (23)
130
Substituting (22) into (21) yields
∗
.
(24)
The following proposition shows that, for fixed and ∈ , , the
point ∗ , ∗ is the local optimal solution, which satisfies the active
constraint √ and minimizes the expected average total cost , , , .
Proposition 1. For given and ∈ , , the point ∗ , ∗ satisfies the second

order sufficient condition (SOSC) for the minimizing problem with a single
constraint.
√
PROOF.For fixed and ∈ , , the constraint is active when the
optimal solution is obtained (i.e. slack variable 0). Therefore, in what follows,
∗ ∗
we show that , satisfies the second order sufficient condition for the
minimizing problem with a single equality constraint. We first obtain the bordered
Hessian matrix H as follows:
, , , , , , , , , ,
0
, , , , , , , , , , , , , , ,
.
, , , , , , , , , , , , , , ,
where
, , , , ,
ln ,
, , , , , , , , , ,
0,
, , , , , , , , , ,
1 ,
, , , , ,
√ 1 ,
, , , , , , , , , ,
√ 1 Φ .
For a given value of and , since there are two variables , and one
constraint, therefore, we need to check the sign of the last one principal minor
determinant of at point ∗ , ∗ . If the sign of it is negative, then this point
satisfies the second order sufficient condition for the minimizing problem (see,
Taha [19]). Now we proceed by checking the sign of the last one principal minor
determinant of at point ∗ , ∗ .
| | | |
2
√ 1 Φ ln
1
√ 1 1 0.
Since the sign of | |is negative, hence, it can be concluded that the optimal value
∗ ∗
, satisfies the sufficient condition for the minimizing problem with a
constraint.□
131
Note that there is a negative term in the denominator of ∗ . Recall that we

assume 1 / , that is, 1 / . We have
1 1 0.
∗
Thus, in order to have a feasible solution (the denominator of is positive), the
maximum allowable value of in (30) is
. (25)
The problem is to determine the value of that minimizes , , , . If

we take the second derivatives of (14) with respect to , we have
, , , , ,
0. (26)
Therefore, , , , , , is a convex function of , for fixed , , , ,and .

Because the number of shipments per batch production run, , is a discrete variable,
the optimal value of (denoted by ∗ ) can be obtained when
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
, , , , , , 1 ,
and
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
, , , , , , 1 .
∗
Once we obtain the value, the optimal order quantity of the buyer for non-
defective items is given by ∗ ∗ ∗ ∗ ∗
1 .
We note that an explicit general solution for ∗ , ∗ is not easy to obtain
because the evaluation for equations (23) and (24) requires knowledge of the value
of the other. The optimal value ∗ , ∗ can be obtained by using a similar graphical
technique used in Hadley and Whitin [6]. The same numerical search technique
also has been used in Lin [11], Lin [12], and others. Therefore, we now develop an
algorithm to find the optimal values for lot size, reorder point, lead time, and the
number of shipments per production run from the vendor to the buyer.
Algorithm.
Step 1.Set 1.
Step 2. For each , 0,1,2, … , , perform (i) to (v).
(i) Start with 0and getΦ 0.5.
(ii) Substituting Φ into (24) to evaluate .
(iii) Using determine Ψ from (23).
(iv) Check Ψ from Silver and Peterson [18] to find , and then Φ .
(v) Repeat (ii) to (iv) until no changes occurs in the values of and .
(vi) Compute the corresponding , , , , 0,1,2, … , .
Step 3. Find min , , , , 1,2, … , .
∗
If , ∗, ∗ , min , , , , 1,2, … , , then the
point ∗ , ∗ , ∗ , is the optimal solution for fixed .
∗
Step 4. Set 1, and repeat Step 2 to Step 3 to get , ∗, ∗ , .
∗
Step 5.If ∗
, ∗, ∗ , , ∗ , ∗ , 1 , then go to Step 4,
otherwise go to Step 6.
∗
Step 6. Set ∗
, ∗, ∗ , , ∗ , ∗ , 1 . Then ∗ , ∗ , ∗ , ∗ is
∗ ∗ ∗
the optimal solution, the optimal reorder point is √ ∗ and the
optimal effective order quantity (i.e., the optimal quantity of non-defective
132
∗ ∗ ∗
items) is 1 .
If the vendor and the buyer do not work together in a cooperative manner
towards minimizing their mutual total costs, both the vendor and the buyer
determine their inventory policy independently. The buyer use model (6) to
compute his economic order quantity, reorder point and lead time. Substituting (11)
and √ into (6), model (6) can be transformed to
Minimize , , ln
√ 1 √ Ψ 1
, (27)
√
subject to .
Applying the above solution processes developed in this section, it can be shown
that the minimum expected average total cost for buyer will occur at the end points
of the interval , , and for fixed ∈ , , the buyer’s independent optimal
solution is
∗
(28)
∗
Ψ
√
(29)
∗
. (30)
For a given buyer’s optimal order quantity, the vendor will determine the
optimal number of deliveries, , so that his expected average total cost in (8) is
minimum. Since the number of deliveries is a positive integer, we compute
equation (8) by setting 1,2,3, …. The optimal value of for vendor and
∗ ∗
, can be obtained.
5. Numerical Example
In this section, a numerical example is utilized to demonstrate the feasibility
of the proposed solution procedure. Consider an integrated vendor-buyer
cooperative inventory model with the following data: 600units/year, 2000
units/year, $200/order, $1500 /setup, $2/unit/year, $4
/unit/year, $3 /unit/year, $25/shipment, $4/unit, $0.5/unit,
175200unit/year, 7units/week, 0.8, 0.025.We assume that the maximum
allowable proportion of demands which are not met from stock is 0.015(i.e. the
worst service level is1 0.985). It is assumed that the lead time demand is
normally distributed and the lead time is divided into three components as shown
in Table 1.
Table 1. Lead time data

Lead time Normal duration Minimum duration Unit fixed
Component (days) (days) crashing cost ($ per day)
1 20 6 0.4
2 20 6 1.2
3 16 9 5.0
133
The constant scaling parameter of the logarithmic relationship between lead

time and ordering cost reductions has five different values. There
are0, 0.2, 0.5, 0.8and 1respectively. Algorithm procedure is applied to yield
the results for various values as shown in Table 2. The optimal policy for
∗
each value can be determined by comparing , ∗, ∗ , , 1,2, …,and the
results are summarized in Table 3. Moreover, we also list the results of fixed
ordering cost model (i.e., take 0 in the same table in order to observe the
relationships between lead time and ordering cost. From the results in Table 3, as
the value of decreases, the integrated policy has the higher frequency of
deliveries, smaller lot size, smaller ordering cost, and lower expected average total
cost (the larger savings of the total cost).
If the inventory policies of the vendor and the buyer are determined independently,
in the case of 0.5, we obtain ∗ 244.74 units, ∗ 0.23 i. e. ∗ 40units) and
∗
3 weeks for the buyer and ∗ 5 for the vendor. The expected average total cost for
∗ ∗ ∗
the buyer is , , $1171.62 and the expected average total cost for the vendor
∗ ∗
is , $1568.92. The joint total cost of the non-integrated model is $2740.44.
For the same value, the expected average total cost in an integrated model is $2205.98. It
is lower than the joint total cost in non-integrated model.
∗
Table2.The results for various using the solution procedures ( in weeks)
∗ ∗ ∗ ∗ ∗ ∗ ∗
, ∗, ∗ ,
1 3 57.4 200.00 649.687 0.62 27 2258.45
0
2 3 57.4 200.00 432.924 0.22 32 2258.68
1 3 57.4 160.77 642.273 0.61 27 2237.62
0.2 2 3 57.4 160.77 424.409 0.21 32 2221.70
3 3 57.4 160.77 331.214 0.00 35 2289.76
1 3 57.4 101.92 630.890 0.59 29 2232.32
0.5 2 3 57.4 101.92 411.020 0.18 34 2205.98
3 3 57.4 101.92 317.284 0.03 36 2262.34
1 3 57.4 43.070 619.317 0.57 28 2200.95
0.8 2 3 57.4 43.070 397.235 0.15 33 2149.83
3 3 57.4 43.070 302.603 0.07 36 2182.87
1 3 57.4 3.8300 611.568 0.56 28 2179.36
1.0 2 3 57.4 3.8300 387.807 0.13 33 2111.27
3 3 57.4 3.8300 292.401 0.10 36 2127.99
∗
Table 3. Summary of the optimal integrated policy for various ( in weeks)
∗ ∗ ∗ ∗ ∗ ∗ ∗
, ∗, ∗, ∗
Saving(%)
0 1 3 200.00 649.687 0.62 27 2258.45
0.2 2 3 160.77 424.409 0.00 32 2221.70 1.63
0.5 2 3 101.92 411.020 0.18 34 2205.98 2.32
0.8 2 3 43.070 302.603 0.07 36 2149.83 4.81
1.0 2 3 3.8300 387.807 0.13 33 2111.27 6.52
Note: Saving is based on the fixed ordering cost model (i.e., 0 .
A fair and acceptable profit sharing mechanism is the key to the success of an
integrated model. A profit sharing mechanism is first suggested by Goyal [4]. The total
∗
annual cost , ∗ , ∗ , ∗ should be allocated to the vendor and the buyer as
134
follows:
∗, ∗, ∗
∗, ∗, ∗ ∗, ∗ ,
∗ ∗ ∗ ∗
cost to the buyer , , , ,
∗ ∗ ∗ ∗
cost to the vendor 1 , , , .
.
For example, when 0.5,
. .
0.4275. The allocated buyer’s
total cost is$943.09 and the allocated vendor’s total cost is$1262.89. The non-
integrated policy and integrated policy are shown in Table 4. The allocations for
various values of are summarized in Table 5. With profit sharing, it is shown that
the integrated policy reduces the joint total cost.
Table 4. Non-integrated policy and the total cost shaving for the case of 0.5
Non-integrated model Integrated model
Buyer’s order quantity 244.74 Buyer’s order quantity 411.020
Lead time 3 Lead time 3
Safety factor (reorder point) 40 Safety factor (reorder point) 34
Vendor’s production 2000 Vendor’s production quantity 2000
quantity
Vendor’s setup cost 1500 Vendor’s setup cost 1500
Buyer’s total cost 1171.62 Buyer’s total cost 1412.04
Allocated buyer’s total cost 943.09
Vendor’s total cost 1568.92 Vendor’s total cost 1595.47
Allocated vendor’s total cost 1262.89
Joint cost 2740.44 Joint cost 2205.98
Table 5. Allocation of the total cost for each case of

Integrated model
Non-integrated model
Buyer Vendor

Allocated Allocated

total cost total cost
0 1409.8 1557.4 2967.2 1775.4 1073.0 1682.2 1185.3 2258.4
0.2 1323.3 1553.3 2876.7 1429.1 1022.0 1573.4 1199.6 2221.7
0.5 1171.6 1568.9 2740.4 1412.0 943.00 1595.4 1262.8 2205.9
0.8 987.30 1584.5 2571.9 1395.0 825.30 1620.6 1324.5 2149.8
1.0 831.40 1595.7 2427.2 1383.7 723.20 1639.4 1388.0 2111.2
Next, the value of lower bound of service level (1- is varied from 0.96 to 0.99
with equal interval 0.1 to perform sensitivity analysis and give some observations and
managerial implications. From the results in Table 6, it is seen that the increasing 1-
value results in smaller lot size, higher safety factor which implies higher safety stock and
reorder point, and higher expected average annual total cost. It seems that a lower service
level benefits manager in profit. However, the lower service will produce negative
influences on brand and customer loyalty which are crucial to building competitive
advantage in market. From this perspective, the manager should determine a proper lower
bound of service level which can balance short-term income and long-term development.
135
Table 6. Effect of change in parameter

∗ ∗ ∗ ∗ ∗ ∗
1‐ , ∗, ∗, ∗

0.96 1 3 662.820 2.13 9 2170.37
0.97 1 3 661.082 1.57 16 2195.94
0.98 1 3 655.856 0.97 23 2230.82
0.99 1 3 637.278 0.21 32 2303.34
This paper extends the Lin’s [11] model by considering that the reduction of
lead time accompanies a decrease of ordering cost. Additionally, the stock-out cost
component in the objective function is replaced by a condition on a service level to
formulate the model. The production process is imperfect and produces a random
number of defective items in buyer’s arrival order lot. We consider the policy in
which the delivery quantity to the buyer is identical at each shipment. Once the
buyer receives the lot, a 100% screening process of the lot is conducted with a
fixed screening rate. It is assumed that the lead time demand and the number of
defective item in an order lot follow the normal distribution and binomial
distribution, respectively. A Lagrange function and an efficient algorithmic
approach are proposed to find the optimal order quantity, safety factor, lead time of
the buyer and number of lot delivered from the vendor to the buyer in a production
cycle while minimizing the joint total expected cost of the vendor-buyer integrated
system and satisfying the service level constraint on the buyer. Moreover, the
results contained in this research are illustrated and verified by a numerical
example. In the future research, we may consider a capital investment in the
reduction of setup cost. Another feasible extension of the present research is to
follow Khan et al. [9] by considering the possibility of incorrectly classifying non-
defective item as defective (a type I error), or incorrectly classifying a defective
item as defective (a type II error). Thus, some defective items may be sold to
customers, who in turn will detect the quality problem and return them to the
buyer.
136
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[4] Goyal, S. K., 1976, An Integrated Inventory Model for a Single Supplier-Single
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Imperfect Items and Planned Back Orders, International Journal Adv Manuf
Technol, DOI 10.1007/s00170-013-4810-7.
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Quality and Inspection Errors, International Journal of Production Economics, vol.
133, no.1, pp. 113-118.
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Model Involving Controllable Backorder rate and Variable Lead Time with a
Service Level Constraint, Mathematical Probles in Engineering, Volume 2012,
Article ID 689061, doi: 10.1155/2012/689061.
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Quality Items, Controllable Lead Time and Distribution-Free Demand, Yugoslav
Journal of Operations Research, 23 (2013), Number 1, 87 - 109.
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Applied Mathematics
HYBRID MODEL OF IRRIGATION CANAL AND ITS

CONTROLLER USING MODEL PREDICTIVE
CONTROL
SUTRISNO
Sanata Dharma University, Yogyakarta, Indonesia,

e-mail: tresnowijoyo@gmail.com,
Abstract. In this paper, we formulate a hybrid model of irrigation canal that

contains four reaches where each of reach has two state events. These state
events of this hybrid model are triggered by the height of the water level which
has two different surfaces. We formulate this hybrid model in piecewise affine
(PWA) form, transform it into mixed logical dynamic (MLD) form using hybrid
system description language (HYSDEL) that was embedded in hybrid toolbox
for MATLAB and control this MLD using model predictive control (MPC). We
control this irrigation canal so that the water level on each reach will be located
at the desired level. Finally, we simulate this system and its controller to desire
given desired level or set point. From the simulation results, the water level of
all reaches are located at the desired level.
Key words and Phrases: Hybrid system, piecewise-affine (PWA) system, mixed
logical dynamic (MLD) system, Model predictive control, irrigation canal
1. INTRODUCTION
Following [4], hybrid system is a mathematical model that consists of
dynamics of real valued variables, discrete variables and their interaction. A
particular case of discrete hybrid systems is piecewise-affine (PWA) systems
whose mode depends on the current location (or event) of the state vector. Hybrid
system in PWA form can be transformed into equivalent mixed logical dynamical
(MLD) form [2] and it can be done using HYSDEL [14] that was embedded in
hybrid control toolbox for MATLAB [2]. This MLD form is more suitable for
solving optimization problem corresponding to the optimal control problem [4].
The state of an MLD model contains state and input of the PWA model and some
auxiliary variables i.e. binary variable and real variable. Furthermore, besides the
original constraint of PWA, MLD has auxiliary constraint in the form of matrix
inequality. The controllability and observability of this hybrid model was given by
[3] in order to be controllable and observable by a control design.
Hybrid predictive control is a famous control method that was applied in
138
139
some problems like optimal semi-active suspension [5], direct injection stratified
charge engines [6] and the stability of MPC for hybrid system was guaranteed by
[7]. Predictive control can be applied to control an MLD system with some
modifications such as the corresponding optimization problem. Predictive control
for MLD system for finite time horizon can be done by forming the vector
prediction of all state variables of MLD over time horizon, substituting them into
objective function and solving the corresponding optimization problem. In case
where the objective function has quadratic form, the corresponding optimization
can be solved using mixed integer quadratic programming (MIQP) [2].
Irrigation canal is needed to be controlled to ensure that the water level of
each reach is located at the desired level. To control this system, we need the
mathematical model of the system. For flat irrigation canal (there is no different
surface for any water level), the mathematical model was given by [13] in the linear
discrete dynamics (non-hybrid model) and the control designs for this system were
given by some researches like coordinated DMPC [11], distributed model
predictive control [10, 12] and its performance analysis was given by[8]. For non-
flat irrigation canal (there is different surfaces for some water levels), we need a
new mathematical model formulation, that is a new hybrid model in order to
control this system.
In this paper, we formulate the hybrid model of irrigation canal that have
two different surfaces for each reach. This hybrid model will be presented as PWA
form and it will be transformed equivalently into MLD form in order to be more
suitable for designing its controller. We will apply MPC to control this MLD
model. MPC for MLD contains a mixed integer quadratic optimization and we will
solve it using MIQP. To observe how this hybrid model and its controller are
working, we simulate this system so that the water level of each reach will be
located at the desired level and show the output that is the water level of each
reach.

(a) Irrigation canal with their surface areas four reaches
(b) A reach with two state events triggered by
Figure 1. Irrigation canal and a reach
2. MODELING OF HYBRID SYSTEM OF IRRIGATION CANAL

A hybrid model presents the dynamic of a system that constituted by parts
described by logical parameter such as on/off switches, event states, if-then-else
rules, etc [4]. Firstly, we will formulate the hybrid model of irrigation canal in the
PWA form. Consider an irrigation canal system illustrated by Figure (1a) that has
four reaches. Let xi(k) be the water level in meter unit (m) for reach- be the
140
water surface area (m2) for reach -and qiout be the in-flow (m3/s) and out-flow
3
(m /s) for reach-i which are measured on upstream and downstream respectively
and p4 be the flow that passing the pump on reach-4. Let Ts be the sampling time,
then the dynamic for reach-i can be written as [13].
. (1)
For reach- be the water surface area if xi > li (mode-0) and be the water
surface area if xi ≤ li (mode-1) as illustrated by Figure (1b). Then the dynamic of
reach i can be written as the following PWA model
) if xi > li,(mode-0) )
if xi ≤ li,(mode-1)
yi(k) = xi(k),
for i = 1,2,3 and especially for is replaced by p4.
3. MODEL PREDICTIVE CONTROL FOR MLD SYSTEM

Model predictive control is one of a modern control design that has been
applied by many researches to solve an optimal control problem including optimal
control problem for a hybrid system. Let s be the number of the model parts or
events and k denote the time step, then PWA model for discrete time in general
form can be described as follows [4].
A1xc(k) + B1uc(k) if δ1 = 1,
) if δ2 = 1,
Asxc(k) + Bsuc(k) if δs = 1.
yc(k) = Cxc(k) + Duc(k) (2)
nc mc pc
where xc(k) R ,uc(k) R and y(k) R denote the state, input and output
vectors respectively at time step k, i = 1,2,...,s, δi {0,1}, and matrices Ai,Bi,C and
D are real matrices with appropriate dimensions. This PWA model can be
transformed into equivalent MLD and vice versa as follows [1]. Let δ {0,1} be
the binary variable, nc >0 be the number of the state of the system then the MLD
model can be described as follows.
x(k + 1) = Ax(k) + B1u(k) + B2δ(k) + B3z(k) (3)
y(k) = Cx(k) + D1u(k) + D2δ(k) + D3z(k) (4)
E2δ(k) + E3z(k) ≤ E1u(k) + E4x(k) + E5 (5)
where is the new state vector for MLD model, xc(k) Rnc,
is the new output vector for
MLD model, is the new input vector for MLD

141
model, z(k) Rrc and δ(k) {0,1}rl are auxiliary variables. A,Bi,C,Di and Ei are real
constant matrices and E5 is a real vector.
To control this MLD model, that is determining some input values such that
the output track some desired set point or reference trajectories, we will use MPC
controller by assuming that this MLD model is controllable and reachable. The
objectives of the MPC can be described as the state gain to the set point or
references. MPC minimizes this objective function that can be described as the
following optimization [4].
(6)
[u,δ,z]T t=0
(7)

subject to :
x(k + 1) = Ax(k) + B1u(k) + B2δ(k) + B3z(k) (8)
− E4x(k) − E1u(k) + E2δ(k) + E3z(k) ≤ E5 (9)
where T be the prediction horizon length, Q1, Q2, Q3, Q4, and Q5 are the symmetric
and positive definite weighting matrices, xr, ur, δr, and zr are the references and
k
kv 2Q = vTQv.
The optimization (6) can be transformed into mixed integer quadratic (MIQ)
optimization by forming the vector predictions of state, input u, δ, and z over
horizon prediction T and substituting them into the objective function. This
transformation gives the following MIQ optimization as follows.
R argmin R0S1R + 2(S2 + x00S3)R (11)
R
subject to :
F1R≤ F2 + F3x0 (12)
ABR = xf − ATx0, (13)
R = [u(0),...,u(T − 1),...,δ(0),..., (14)
T
δ(T − 1),z(0),...,z(T − 1)] (15)
where S1,S2, and S3 are the real constant matrices with appropriate dimension,
A B2 B3, F1 = E−E4G4, F2 = E5,
E2 E ,
Ei Bi
E
Ei = i .. ,i = 1,2,3,4, and Bi = ... ,i = 1,2,3.
. Bi
Ei
142
Optimization (11) can be solved using MIQP that embedded in hybrid toolbox [2].
The optimal value R = [u ,δ ,z ] obtained from (11) contains the optimal value
for u , δ , and z over time horizon. The control action that will be applied to the
system is u at current time step.
4. SIMULATION RESULTS
We simulate the hybrid model of irrigation canal that contains four reaches
associates to (3)-(4) by applying MPC controller that the optimal input values are
obtained by optimization (11) with two initial state values. For all i = 1,2,3,4, the
parameter values for this system are li = 2, As,ai = 600, As,bi = 800, and 0 ≤ ui(k) ≤ 4.
The first simulation is using initial state or initial water level x(0) = [1.3,2,2.5,1.8]T.
The simulation results including desired water levels (set points) for all reaches are
appeared on Figure (3-4).
Figure (2) shows the optimal control actions for all reaches generated by MPC.
These control actions was determined by MIQP programming (11). At time step
k(second), the value u1(k) presents the inflow (m3/s) for reach-1 that passed through
the gate-1 and analogously for ui(k) for i = 2,3,4,5. These control actions are
applied to the system and give the outputs (water level for all reaches) that given by
Figure (3).
Figure 2. Optimal control inputs where x(0) = [1.3,2,2.5,1.7]
Figure 3. Water levels for all reaches where x(0) = [1.3,2,2.5,1.7]

143
Figure (3) shows the water level for all reaches. For reach-1, the initial water
level is 1.3 m. From this figure, it can be seen that the water level is located at the
desired level after 100 s and analogously for reach-i for i = 2,3,4. From this figure,
it can be conclude that the controller brings the water levels of all reaches to the
desired levels very well.
Figure (4) shows the modes for all reaches along simulation. For reach-1, the
initial water level is 1.3 m, lower than l1 = 2m, it means that this state is located at
mode-1. At time step 600 s, the water level of reach-1 is 2.1 m, upper than
Figure 4. Modes for all reaches where x(0) = [1.3,2,2.5,1.7]
l1 = 2m, it means that this level is located at mode-0. We can observe analogously for other
reaches. These mode values are dependent to the water level showed by Figure (3).
Figure 5. Optimal control inputs where x(0) = [1.8,2.5,1.6,2.2]
The second simulation is using initial state x(0) = [1.8,2.5,1.6,2.2]. Analogously to

the first simulation, Figure (5) shows the optimal input values for all reaches, Figure (6)
shows the water level for all reaches and Figure (7) shows the modes for all reaches. From
Figure (6), it can be observed that the output of this system i.e. the water level for each
reach followed the given desired level or set point. The mode for each reach on Figure (7) is
following the output values on
144
Figure 6. Water levels for all reaches where x(0) = [1.8,2.5,1.6,2.2]
Figure 7. Modes for all reaches where x(0) = [1.8,2.5,1.6,2.2]
Figure (6), that is if the water level is lower or equal to 2 m, then the mode is 1 and
if the water level is upper than 2 m, then the mode is 0. These mode values are
dependent to the water level showed by Figure (6).
5. CONCLUSIONS
In this work, the modeling and control of irrigation canal with two state
events were considered. The mathematical model of this system was presented as
hybrid model in the PWA form and it can be transformed equivalently into MLD
form using HYSDEL and mld function on hybrid toolbox for MATLAB. The
controller for this MLD model was designed using MPC for MLD. From the
simulation results, the water level for all reaches were followed the given desired
levels very well.
145
REFERENCES
[1] Bemporad, A., 2004, Efficient Conversion of Mixed Logical Dynamical Systems
Into an Equivalent Piecewise Affine Form, IEEE Transactions On Automatic
Control, Vol. 49, No. 5, May 2004., pp. 832838.
[2] Bemporad, A., Hybrid Toolbox - Users Guide, April 20, 2012.
[3] Bemporad, A., Ferrari-Trecate, G., and Morari, M., ”Observability and
Controllability of Piecewise Affine and Hybrid Systems”, IEEE Transactions on
Automatic Control, vol. 45, no. 10, October 2000.
[4] Borrelli, F., Bemporad, A., Morari, M., 2011, Predictive Control for linear and
hybrid systems, June 16, 2011.
[5] Giorgetti,N. , Bemporad,A., Tseng,H.E. , and Hrovat,D. , Hybrid model predictive
control application towards optimal semi-active suspension, Int. J. Control, vol. 79,
no. 5, pp. 521533, 2006.
[6] Giorgetti, N., Ripaccioli,G., Bemporad,A., Kolmanovsky, I.V., and Hrovat,D.,
Hybrid model predictive control of direct injection stratified charge engines,
IEEE/ASME Transactions on Mechatronics, vol. 11, no. 5, pp. 499506, Aug. 2006.
[7] Lazar, M., Heemels,W. P. M. H., Weiland, S., and Bemporad, A.., ”Stabilizing
Model Predictive Control of Hybrid Systems”, IEEE Transactions on Automatic
Control, vol. 51, no. 11, November 2006.
[8] Li, Y., and De Schutter, B., Performance analysis of irrigation channels with
distributed control, Proceedings of the 2010 IEEE International Conference on
Control Applications, Yokohama, Japan, pp. 21482153, Sept. 2010.
[9] Maciejowski, J.M., 2002, Predictive Control with Constraints, Prentice Hall Inc.,
England.
[10] Negenborn, R.R., and De Schutter, B., A distributed model predictive control
approach for the control of irrigation canals, Proceedings of the International
Conference on Infrastructure Systems 2008: Building Networks for a Brighter
Future, Rotterdam, The Netherlands, 6 pp., Nov. 2008. Paper 152.
[11] Negenborn, R.R., van Overloop, P.J., and De Schutter, B., Coordinated distributed
model predictive reach control of irrigation canals, Proceedings of the European
Control Conference 2009, Budapest, Hungary, pp. 14201425, Aug. 2009.
[12] Negenborn, R.R., van Overloop, P.-J., Keviczky, T., and De Schutter, B.,
Distributed model predictive control of irrigation canals, Networks and
Heterogeneous Media, vol. 4, no. 2, pp. 359380, June 2009.
[13] Overloop, V., 2006, Model Predictive Control on Open Water Systems, Ph.D Thesis,
Delft University Press, The Netherlands.
[14] Torrisi, F.D., and Bemporad, A., HYSDEL A tool for generating computational
hybrid models, IEEE Trans. Contr. Systems Technology, vol. 12, no. 2, pp. 235249,
Mar. 2004.
Applied Mathematics
FUZZY EOQ MO DEL WITHTRAPEZOIDAL AND

TRIANGULAR FUNCTIONS USING PARTIAL
BACKORDER
ELIS RATNA WULAN1, VENESA ANDYAN2
1
Mathematics Department Islamic State University, elisrwulan@yahoo.com
2
Mathematics Department Islamic State University, venessa.andyann@gmail.com
Abstract. EOQ fuzzy model is EOQ model that can estimate the cost from
existing information. Using trapezoid fuzzy functions can estimate the costs of
existing and trapezoid membership functions has some points that have a value
of membership . value results of trapezoid fuzzy will be higher than usual
TRC value results of EOQ model . This paper aims to determine the optimal
amount of inventory in the company, namely optimal Q and optimal V, using the
model of partial backorder will be known optimal Q and V for the optimal
number of units each time a message . EOQ model effect on inventory very
closely by using EOQ fuzzy model with triangular and trapezoid membership
functions with partial backorder. Optimal Q and optimal V values for the
optimal fuzzy models will have an increase due to the use of trapezoid and
triangular membership functions that have a different value depending on the
requirements of each membership function value. Therefore, by using a fuzzy
model can solve the company's problems in estimating the costs for the next
term.
Key words and Phrases: Inventory, EOQ, Fuzzy Logic, Partial Backorder,
Membership function.
1. Introduction
Inventory control can be defined as activities and measures that are used to
determine the right amount to meet the supply of an item. For the effective
management of inventory greatly affect a company's profits. Common problems in
the inventory system is an important issue for a company. Therefore in order to
control inventory can be resolved properly used the "economic order quantity".
Economic order quantity is the volume or number of purchases made on the most
economical for each purchase. With the EOQ method , the company will reduce
the inventory that has been improperly used or damaged and minimize the
constraints of a production company to produce, or work delays and missed
146
147
opportunities in the company sells its products due to run out of stock, then it must
be determined from the value of EOQ (economic order quantity) to minimize the
cost of the annual inventory. The basis of the EOQ formula introduced by Haris
1993. [2]
where: R = annual demand, C = order cost, H = holding cost per year.
In the EOQ model there are two EOQ model that is deterministic inventory
model is characterized by the demand characteristics and arrival time before orders
can be known with certainty, it is a basic assumption for deterministic models and
probabilistic models are characterized by the demand and arrival time orders can
not be previously known with certainty. However, in the real case many things to
consider and many EOQ modeling studies using probabilistic approach to deal with
uncertainty. Assumptions for the probabilistic model is characterized by demand
and lead time that can not be known in advance with certainty so that needs to be
approached with a probability distribution. But a lot of inventory parameter
uncertainty and lack of cost information, to solve this problem it will be used fuzzy
model because it can estimate the cost of the existing lack of cost information that
already exists.
Fuzzy logic is a logic that has a value of vagueness or ambiguity between
right and wrong, the fuzzy logic a value can be true and false at the same time, but
how much truth and error a value depends on the weight of its membership.
Membership function is a curve that shows the mapping of points of data input into
its membership, which has a value between 0 to 1 interval.
General EOQ model can not estimate the cost of existing ones, while using a
fuzzy model can estimate the cost of lack of information that already exists . EOQ
to be simplified modeling the EOQ model using fuzzy partial backorder and the
principle of fuzzy membership functions. Partial backorder EOQ model is part
deterministic EOQ model is a model which every ordering and inventory shortages
can be known in advance. This paper will discuss the determination of the optimal
amount of inventory model with partial backorder, backorder partial fuzzy EOQ
models using the principle of triangle and trapezoid membership functions.
2. Main Results
2.1. Inventory Model with Backorder Using Partial
Assumptions used in the partial backorder EOQ model are goods ordered and kept
only one kind, inventory immediately comes, and partial backorder allowed[1].
Notation used in this model are:
TRC = Total Relevant Cost
t₁ = Inventory depletion time period
t₂ = Inventory backorder time period
R = Total annual requirement
V = Maximum Inventory
Q = Number of units each order
C = Order cost
148
H = Holding cost
K = Backorder cost
L = Loss cost
δ = Number of backorder units
In this discussion the total relevant costs for this model comes from order
cost, holding cost, backorder costs, and losses sales cost.
For backorder unit is denoted by δ. In the model there are two phases of the
inventory cycle namely: when the inventory depletion (t ₁) and when inventory
backorder (t ₂).
Value of t ₁ and t ₂ defined[1]:
₁ (1)
₂ (2)
For order costs, holding costs, backorder costs and the loss cost is defined
as[1]:
1. Order Cost:
(3)
If assumed C = order cost, R = total annual requirement, Q = number of unit each
order.
2. Holding cost:
(4)
for H = holding cost, V = maximum inventory
3. Backorder cost:
(5)
assumed K = backorder cost
4. Loss cost:

(6)
assumed L = loss cost
thus the total relevance cost is[5]:

(7)

In (7) still contained the parameter t ₁ and t ₂ which has a different dimension to the
planning period requirement R, therefore the dimensions should be synchronized
first. The authors obtained the total relevance cost as follows:
² ² ²
(8)
The purpose of this model formulation is finding the optimal Q and V, therefore
the total relevant cost become:
149
² ²
(9)
Requirement to minimize (9) are:
0 0
then theoptimal Q and V for δ = 1 are:
(10)
(11)
and for δ = 0, as follows:
(12)
(13)
2.2. Partial Backorder Inventory Models for The Function of Trapezoidal

Fuzzy
Notation for the use of trapezoidal fuzzy model are:
T C : Total relevant costs under fuzzy models
: Fuzzy number for holding cost
: Fuzzy number for order cost
: Fuzzy number for the number of annual demand
: Fuzzy number backorder cost
: Fuzzy number for loss cost
⊗ : Fuzzy number for multiplication
⊘ : Fuzzy number for division
⊕ : Fuzzy number for sum
: Unit number of once orderby fuzzy model
: The maximum number of inventory by fuzzy model
ᴀ : The level of a member function value for x
: The value of the membership function
By using (7), TRC for fuzzy model is described as follows [3]:
⊗ ⊗ Ṽ² ⊗ Ṽ ² ⊗ 1 Ṽ ²

2 2 2
Principle use of trapezoidal functions denoted by (a ₁, a ₂, a ₃, a ₄; w) where

₁ ₂ ₃ ₄is the principle function. Fuzzy calculation operation for two
trapezoidal member functions and are [3]:
⊕ ₁ ₁. ₂ ₂, ₃ ₃, ₄ ₄ ; (14)
150
⊗ ₁ ₁. ₂ ₂. ₃ ₃. ₄ ₄ ; (15)
Optimal solutions obtained using the condition:
0 0
thus obtained:
∑ ∑ ∑ ∑
∑ ∑ ∑
(16)
∑ ∑ ∑
∑ ∑ ∑ ∑
(17)
The authors used fuzzy membership function consists of a crisp value, thus for a
trapezoidal function, ₁ ₂ ₃ ₄ . The results of the , and
T C are:
2 1
. . 1
2 1
Ṽ
1

(18)

2.3. Partial Backorder Inventory Models for Triangular Fuzzy Function
Notation for the use of triangular fuzzy model are:

T C : Total relevance cost according to the fuzzy model
: Fuzzy number for holding cost
: Fuzzy number for order cost
: Fuzzy number for annual demand
: Fuzzy number untuk backorder cost
: Fuzzy number for loss cost
⊗ : Fuzzy number for multiplication
⊘ : Fuzzy number for division
⊕ : Fuzzy number for the sum
: Unit number of once order according to fuzzy models
Ṽ : The maximum amount of inventory on the fuzzy model
ᴀ : Level of member function value for x
: Value of the membership function
151
By using (7), TRC for fuzzy model is described as follows [3]:

⊗ ⊗ Ṽ² ⊗ Ṽ ² ⊗ 1 Ṽ ²

2 2 2
Principle use of the triangle function denoted by (e₁, e₂, e₃, ; w) where ₁ ₂
₃is the principle function. Operation calculation for two fuzzy triangular member
function of and are [3]:
⊕ ₁ ₁. ₂ ₂, ₃ ₃ ; (19)
⊗ ₁ ₁. ₂ ₂. ₃ ₃ ; (20)
.
Optimal solutions obtained using the condition:
0 0
thus obtained:
∑ ∑ ∑ ∑
∑ ∑ ∑
(21)
∑ ∑ ∑
∑ ∑ ∑ ∑
(22)
The authors used fuzzy membership function consists of a crisp value, thus for the
triangular function, ₁ ₂ ₃ . The results of the , and T C are:
2 1
. . 1
2 1

1

(23)

2.4. Analysis
Aspects of the costs associated with inventory Super Star mattress:
a. Super Star Mattress = Rp.700.000
b. Order Cost (C) = Rp. 59.000/ order
c. Holding Cost (H) = Rp. 52.000/unit
d. Backorder Cost (K) = Rp. 38.000/unit
e. Loss Cost (L) = Rp. 35.000/ unit
f. Number Requirement Usage / year (R) = 10272 pcs
g. Unit number of backorder / Least backlog ( δ ) = 50 %
h. Average usage/ week = 214 pcs / week
152
Here is a product of consumer demand data Super Star mattress from March
to May 2011 are presented in Table 1[4]:
Table 1 Consumer Demand Data
Mattress Type Consumer Demand

1 216
2 106
3 193
4 338
5 246
6 185
7 133
8 161
9 213
10 282
11 219
12 275
Total 2566
Average 213,83
2.4.1. Partial Backorder Inventory Model

By using data that has been obtained from the company to find the total
relevant cost (TRC) and the optimal amount of inventory to be able to calculate the
partial backorder models with reference to the previous formula, to calculate the
TRC:
a. Step 1: calculate the Q and V
The results of = 237,734
The results of V = 98,04
b. Step 2: Substitute Q and V to (7) to produce a number of TRC:
The results obtained for TRC is 5,131,110.63
2.4.2. Partial Backorder Inventory Model Using Trapezoidal Fuzzy Function

Notation and cost using fuzzy model is:
a. Super Star Mattress = Rp.700.000
b. Order cost ( ) = Rp. 59.000 – 72000 order
c. Holding cost ( ) = Rp. 52.000 – 74000 / unit
d. Backorder cost ( ) = Rp. 38.000 – 59000 / unit
e. Loss cost ( ) = Rp. 35.000 – 65000 / unit
f. Number Requirement Usage / year ( )= 10272 – 12240 pcs
g. Unit number of backorder / Least backlog ( δ ) = 50 %
h. Average usage/ week = 214 pcs – 255 /week
To find value , and of partial backorder model backorder using the

following steps:
a. Step 1: calculate the and
The resultsof = 231,60
The results of Ṽ = 101,84
153
b. Step 2: Substitute andṼto (18) for find value

The resultsobtained for T C = 6.381.506,20
At the time variable has a crisp value = (c₁ = c₂ = c₃ = c₄ = C ), it is a

crisp value determination. For example the annual demand, ordering cost, holding
cost, backorder cost and the cost of lost sales, is determined from the average
value, then the value would appear to R = 11203, C = 65 750, H = 62 750, K =
48500, L = 50000 . Trapezoidal fuzzy membership functions will be = (c₁ = c₂ =
c₃ = c₄ = 65750), = (r₁ = r₂ = r₃= r₄ = 11203), =(h₁ = h₂ = h₃= h₄= 62750),
= ( k₁ = k₂ = k₃ = k₄ = 48500), = ( l₁ = l₂ = l₃ =l₄= 50000).
with an average value are generated using fuzzy trapezoidal membership
functions as follows:
a. Step 1: calculate andṼusing equation (16) and (17).
Then value ofṼ = 101,606
b. Step 2: Determine and to (18) for obtained T C:
The resultsof 6.375.773, 585
To prove that the results of crisp values for trapezoidal fuzzy models
will be equal to the model without backorder partial fuzzy function can be
calculated using equation (7). TRC with the following information:
a. Step 1: determine Q and V valueusing equation(16) and (17).
The results of Q = 231,062
The results of V = 101,606
b. Step 2: Substitute and Ṽ to (7) for obtained T C:
Then the results of 6.375.773, 585
2.4.3. Partial Backorder Inventory Model Using Triangular Fuzzy Functions

Partial backorder inventory models using triangular fuzzy membership
function is another model that can be used to consider the lack of value of
information costs required by the company for the next period.
a. Step 1: Calculation and valueusing equation (21) and (22)
Then results of = 101,61
b. Step 2: Substitute and to (23) for obtained
The resultsof T C = 6.402.157,80
For crisp value = (c₁ = c₂ = c₃ = c₄ = C ), which is an irreducible value.

For example the annual demand, ordering cost, holding cost, backorder cost and
the cost of lost sales for the model of triangular fuzzy membership function is
determined from the average value, then the value would appear to R = 11256, C =
65500, H = 63000, K = 48500, L = 50000. Fuzzy membership functions will be =
(c₁ = c₂ = c₃ = 65500), = (r₁ = r₂ = r₃= 11256), = (h₁ = h₂ = h₃= 63000), =
(k₁ = k₂ = k₃ = 48500), = (l₁ = l₂ = l₃ = 50000).
with an average value generated using triangular fuzzy membership functions
as of the previous formula is:
a. Step 1: Calculate and using equation (21) and (22).
The results ofṼ = 101,336
154
b. Step 2: Substitute andṼto (23) for obtained T C:

Then the results of 6.384.220,93
To prove that the results of crisp values for trapezoidal fuzzy models
will be equal to the model without backorder partial fuzzy function can be
calculated using equation (7). TRC with the following information:
a. Step 1: determine Q and V using equation (21) and (22)
The results of Ṽ = 101,336
b. Step 2: Substitute andṼto (23) to obtained T C :
Then the results of 6.384.220,93
This shows that when the lack of information is to be considered to find out
how much it costs to run the next time. can be proved also for partial
backorder models using fuzzy models will be higher than the TRC models without
fuzzy partial backorder due to the use value of a trapezoidal or triangular
membership functions are determined to be greater or equal to the specified cost,
this is a realistic to get the fuzzy model in order to consider the uncertainty
due to the lack of information about the costs of existing ones. Fuzzy EOQ models
can then be used to identify and estimate the likely costs that will occur for the next
period.
EOQ influence on inventory very closely then by using EOQ fuzzy model
with triangular and trapezoidal membership functions using partial backorder will
be different with EOQ model. Optimal Q and V values for the optimal fuzzy
models will have an increase due to the use of trapezoidal and triangular
membership functions have a different value depending on the requirements of
each membership function value can be an increase or equal. Whereas without the
fuzzy it only has one fixed value of the costs. Therefore, by using a fuzzy model
can solve the company's problems in estimating the costs for the next period.
References
[1] Purnomo, H.D,Wee, Hui-Meng, and Chiu,Yu Fang, Fuzzy Economic Order Quantity
Model With Partial Backorder,Penang,Malaysia ICMBSE(2012).
[2] Heilpern, S., “The Expected Value of A Fuzzy Number,” Fuzzy Sets and Systems,
47, 81-86 (1992).
[3] Kusumadewi, S.,Teknik dan Aplikasinya: Logika Fuzzy,Artificial Intelligence
Publisher (2003).
[4] UKRIDA Data Collection and Prosessing, Available:
http://www.ukrida.ac.id/.../jkunukr-ns-s1-2009-222005011-1923-karya_prima-
chapter3 - Ukrida [August 8, 2012]
[5] Yamit, Z.,Inventory Management, Ekonisia Publisher, Economic Faculty
UII,Yogyakarta(1999).
Applied Mathematics
A GOAL PROGRAMMING APPROACH TO SOLVE

VEHICLE ROUTING PROBLEM USING LINGO
ATMINI DHORURI1, EMINUGROHO RATNA SARI2, AND DWI

LESTARI3
1Department of Mathematics, Yogyakarta State University, atmini_uny@yahoo.co.id

2Department
of Mathematics, Yogyakarta State University,
eminugrohosari@gmail.com
3Department of Mathematics, Yogyakarta State University, dwilestari@uny.ac.id
Abstract. Vehicle routing problem (VRP) which discusses a set of routes for some
vehicles, starting and ending at a depot, serving a set of customers such that each
customer must be visited once by exactly one vehicle, and having a time constrain is
called vehicle routing problem with time window (VRPTW). This paper presents a
goal programming approach to solve VRP, especially VRPTW. We have considered
an objective function with four main goals: to maximize utilization of vehicle
capacity, to minimize the total waiting time, to minimize the total cost to serve the
customers and to maximize the number of served customers. The proposed model was
implemented and has been solved numerically using LINGO software and the optimal
solution is presented.
Key words and Phrases: goal programming, vehicle routing problem, LINGO.
1. Introduction
Vehicle Routing Problem (VRP) is a set of routes which formed to serve a set
of customers using vehicles, starting and ending at a depot. In VRP, each customer
must be visited once by exactly one vehicle. The route has to be designed such that
the total demands of all customers must not exceed the capacity of the vehicle.
Based on Jolai and Aghdaghi [1], Sousa et al [2], Azi et al [3], if VRP has a
time constrains on the periods of the day in which each customer must be visited, it
is called Vehicle Routing Problem with Time Window (VRPTW). VRPTW is one
of an important problem occurring in distribution systems. So, it has received many
attentions not only on the development of the theory, but also on its application.
For example, postal deliveries, delivery service of food business, Liquefied
Petroleum Gas (LPG) deliveries. Routes which designed should be in a short
duration and must be satisfy time constrain. Larsen [4], Cook and Rich [5],
Cordeau et al [6] proposed an exact method for the VRPTW. As development of
research, there are many other methods to solve VRPTW.
On the other side, because of its wide application to real-life situations,
155
156
distribution problems have other objectives than minimizing the total travel time or
distance. Multiple objectives of VRP can be found in Hong [7], Calvete [8], and
Hashimoto [9]. The goal programming approach is a great method to solve the
multiple objectives. The aim of goal programming is to minimize the deviations
between the goals. Jolai and Aghdaghi [1], Hong [7], and Calvete [8] used goal
programming to model the VRPTW.
In this paper we use goal programming approach to solve VRP, especially
VRPTW. The remainder of this paper is organized as follow. In Section 2, we
present the mathematical formulation of VRPTW which is described as a goal
programming approach. In Section 3, we implemented the model using a set of
data and solve it using LINGO. Finally, Section 4 concludes the paper.
2. Model Formulation

We formulate the problem as follow. Let G N , E be a directed graph  
associated with the problem. We define N  1, 2,..., n be a set of nodes such that
each representing a customer location. Notation N   0,1, 2,..., n, n  1 is the set
of nodes, where node 0 represents the depot, and node n  1 refers to a copy of
depot. We have notation E   i, j  : i, j  N  as 
the set of directed arcs
(potential route between the customers and the depot).
There is a known demand di for every customer i  N . The sum of the
demands of the customers served by a designed route cannot exceed q, where q is
the capacity of vehicle that used for delivering goods from the depot to the
customers. We also define si as a known service time for each customer i.
Each arc  i, j   E has an associated travel cost cij and a travel time tij . It
represents cost and time of going from node i to node j through arc  i , j  . Node 0
and node n  1 are only incident to outgoing and incoming arcs, respectively. The
set of route denoted by R  1, 2,..., r , and it should be noted that the route taken
by a vehicle have to depart from node 0 and terminate at node n  1. For each
k  R , we define Ck as the travel cost to serve the customers at route k, and we
use notation Tk as a distribution time of all customers in route k. So, we can
assume each customer in a route has to be served before Tmax with total cost less
than Cmax .
To formulate the model, we define the following variables:
Decision variables:
xijk  1,0, ifotherwise
there is a vehicle travel from node i to node j in a route k
, (1)
yik  1,0, ifotherwise

node i is visited in route k
, (2)
wik is the time of beginning of service at node i in route k
In this paper we use goal programming approach for the problem which has
four main goals. First goal, maximize utilization of vehicle capacity. Second,
minimize total distribution time of all customers. Third, minimize the total travel
157
cost to serve the customers, and forth goal, maximize the number of served
customers. So, we define deviational variables of each goal.
Deviational variables:

kR

1k is negative deviational variable of first goal,
 
2 is positive deviational variable of second goal,
 
3 is positive deviational variable of third goal,
 
4i  1,0, otherwise
if customer i can be served
, it is negative deviational variable of forth goal.
i A
The purpose of goal programming is to minimize the given deviations
between the goals. From the goals that has presented before and variables we
define, if 1 ,..., 4 as weight of each deviational variable, respectively, so the
objective function of the problem can be written as follow:
min Z  1  1k  22  33  4  4i ,

kR iA
(3)
The problem has some constrains. We will explain one by one as follow:
Because the sum of the demands of the customers served at route k cannot exceed
q, and represent from the first goal by using negative deviational variable, so we
have
d y
iN
k
i i  1k  q, k  R . (4)
The distribution time in route k is derived from sum of total service time and total
travel time in route k. Using notation Tk as distribution time in route k, so
Tk   si yik    tij xijk , k  R .

iN i0 N jN n 1
(5)
We should be noted that total distribution time have to less than the given time,
Tmax , so
T
kR
k  Tmax . (6)
Related to the first goal, in order to guarantee that total distribution time is
minimize by using positive deviational variable, so

kR
Tk  2  0 . (7)
Travel cost to serve the customers at route k is sum of travel cost such that there is
a vehicle going from node i to node j through arc  i , j  in route k. Using notation Ck as
travel cost to serve the customers at route k, then
Ck   cij xijk , k  R . (8)
 i , j E
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But, total travel cost should less than Cmax
C
kR
k  Cmax . (9)
According to the third goal, we define Eq. (10) to guarantee that total travel cost is
minimize by using positive deviational variable, that is
C
kR
k  3  0 .
(10)
If customer i can be served at route k, then should be there is a vehicle travel from
customer i to the other customer in route k. It means, we have to guarantee each
customer can only visited once
x k
ij  yik , i  N  , k  R . (11)
jN 
Refer to the forth-goal, each customer can only be visited once, and from the
objective function, sum of the negative deviational variable 

iA

4i which is
weighted by 4 computes the number of customers who have not been served, then
in order we can maximize the number of served customers
y
kR
k
i  4i  1, i  N . (12)
To ensure that all routes leave and return to the depot, so we write
x
jN
k
0j  1, k  R , (13)
x
iN
k
i n 1
 1, k  R . (14)
For every vehicle that has visited customer i, it means the vehicle will leave
customer i,

i0 N
xik  
jN n 1
xkj  0,    N , k  R . (15)
If wik is a notation for the start time of service at customer i of route k, then we can
guarantee feasibility of the time schedule by
wik  si  tij  M 1  xijk   wkj , i, j  N  , k  R , (16)
where M an arbitrary large constant. Each customer i has a time window  i ,  i 

meaning the vehicle must arrive during the interval, so the start time of service at
customer i should be in the interval
 i  wik   i , i  N , k  R , (17)
Eq. (3) subject to (4) – (17) is called goal programming model of the VRPTW. In
159
the next section, we will discuss how to solve this problem using LINGO.
3. Computational Result
Now we present some tests and results to a real problem of the Liquefied
Petroleum Gas (LPG) agent. The LPG will be distributed to the customers. In this
paper we only take the first 5 customers as an example. We denote each customer
by N1, N2,…, N5. For depot and copy of depot will be denoted by N0 and N6,
respectively. The travel time, travel cost and the demand of the customers are
presented in Table 1, Table 2, and Table 3.
Table 1. Travel time between nodes (minutes)

Depot N1 N2 N3 N4 N5
Depot 0 2 3 3 5 3
N1 2 0 1 4 5 4
N2 3 1 0 5 6 5
N3 5 4 5 0 5 5
N4 5 6 7 3 0 5
N5 3 7 8 6 7 0
Table 2. Travel cost between nodes (x Rp 1000,00)

Depot N1 N2 N3 N4 N5
Depot 0 0.27 0.3 0.41 0.54 0.38
N1 0.25 0 0.06 0.48 0.67 0.48
N2 0.28 0.06 0 0.45 0.70 0.51
N3 0.45 0.41 0.45 0 0.54 0.41
N4 0.54 0.67 0.70 0.35 0 0.51
N5 0.67 0.77 0.80 0.64 0.57 0
Table 3. Demand and service time for each customer

Customer N1 N2 N3 N4 N5
Demand 70 420 80 120 240
(units)
Service 23 140 27 40 80
time
(minutes)
160
Complete script of the problem solved in LINGO 11.0 shown bellow
MODEL:
SETS:
N/N0,N1,N2,N3,N4,N5,N6/:q,s,v,h,d2;
R/R1..R2/:B,TR,d4;
E(N,R):Y,W;
D(N,N,R):X;
A(N,N):c,t;
ENDSETS
DATA:
U = 56;
UC = 3;
UT = 6;
q = 0 7 42 8 12 24 0;
s = 0 23 140 27 40 80 0;
!0; !1; !2; !3; !4; !5; !10;
c = !0; 0.0.27 0.30 0.41 0.54 0.38 0.
!1; 0.25 0. 0.06 0.48 0.67 0.48 0.27
!2; 0.28 0.06 0.0.45 0.70 0.51 0.30
!3; 0.45 0.41 0.45 0.0.54 0.41 0.41
!4; 0.54 0.67 0.70 0.35 0. 0.51 0.54
!5; 0.67 0.77 0.80 0.64 0.57 0.0.38
!10; 0. 0.27 0.30 0.41 0.54 0.38 0. ;
!0; !1; !2; !3; !4; !5; !10;

t = !0; 0233530
!1; 2014542
!2; 3105653
!3; 5450555
!4; 5673055
!5; 3786703
!10; 0233530 ;
v = 0 1 1 2 1 1 0;
h = 0 1 1 2 1 1 0;
ENDDATA
min = d1 + @sum(N(I)|I#NE#1 #AND# I#NE#7:d2) + d3 + @SUM(R(K):d4);
@FOR(R(K):@SUM(N(I):q(I)*Y(I,K))+d4(K)=U);
@FOR(R(K):(@SUM(A(I,J):t(I,J)*X(I,J,K))+@SUM(N(I):s(I)*Y(I,K)))/60=TR);
@SUM(R(K):TR)<=UT;
@SUM(R(K):TR)-d3=0;
@FOR (R(K):B=@SUM(A(I,J):c(I,J)*X(I,J,K)));
@SUM(R(K):B)<=UC;
@SUM(R(K):B)-d1=0;
@FOR(R(K):@FOR(N(I):@SUM(N(J):X(I,J,K))=Y(I,K)));
@FOR(N(I)|I#NE#1#AND# I#NE#8:@SUM(R(K):Y(I,K))+d2=1);
@FOR(R(K):@SUM(A(I,J)|J#EQ#7:X(I,J,K))=1);
@FOR(R(K):@SUM(A(I,J)|I#EQ#1:X(I,J,K))=1);
@FOR(R(K):@FOR(N(I):@SUM(N(J)|I#NE#1 #AND# I#NE#7:X(I,J,K))-
@SUM(N(J)|I#NE#7 #AND# I#NE#1:X(J,I,K))=0));
@FOR(R(K):@FOR(N(I):@FOR(N(J):(W(I,K)+S(I)+t(I,J))-100000*(1-
X(I,J,K))<=W(J,K))));
@FOR(N(I):@FOR(R(K): v*Y(I,K) <= W(I,K)));
@FOR(N(I):@FOR(R(K): W(I,K) <= h*Y(I,K)));
@FOR(D(I,J,K):@BIN(X));
@FOR(R(K):@BIN(Y));
@FOR(N(I):@BIN(d2));
END
Fig. 2. Complete Script Using LINGO 11.0
By inputting vehicle capacity 560 units, maximum distribution time 6 hours a

week, and maximum distribution cost Rp 3.000,00 a week, we get two route which
summarize in Table 4.
161
Table 4. Summarize output of LINGO

Route Total Distribution Total distribution Total unit
Cost time (hour) LPG
N0‐N2‐N1‐N6 Rp 610,00 2.82 490
N0‐N4‐N3‐N5‐N6 Rp 1.970,00 2.77 440
Sum Rp 2.580,00 5.59 930
From Table 4, all customers can be served by the agent. If we decrease the
maximum distribution time to 5 hours a week, then only 4 customers can be served.
It is also logic, if we decrease to 4 hours a week, then only 2 customers can be
served. Decreasing the number of customers which can be served also happened
whenever we decrease maximum distribution cost to Rp 2.000,00. It is only 4
customers can be served.
4. Conclusion
In this paper, we proposed a goal programming approach to solve Vehicle
Routing Problem with Time Windows (VRPTW). Eq. (3) subject to (4) – (17) is
called goal programming model of the VRPTW. The model was implemented and
the results had been obtained using LINGO. From the simulation, by increasing the
value of Tmax , it is followed the increasing customers can be served. As a future
research, we suggest improving our model to solve larger nodes.
References
[1] Jolai, F., & Aghdaghi, M. (2008). A Goal Programming Model for SIngle Vehicle
Routing Problem with Multiple Routes. Journal of Industrial and Systems
Engineering , 154-163.
[2] Sousa, J. C., Biswas, H. A., Brito, R., & Silveira, A. (2011). A Multi Objective
Approach to Solve Capacitated Vehicle Routing Problems with Time Windows Using
Mixed Integer Linear Programming. International Journal of Advanced Science and
Technology , 1-8.
[3] Azi, N., Gendreau, M., & Potvin, J.-Y. (2007). An exact algorithm for a single-vehicle
routing problem. European Journal of Operational Research , 755-766.
[4] Larsen, J. (1999). Parallelization of the Vehicle Routing Problem with Time Windows.
Institute of Mathematical Modelling. Denmark: Technical University of Denmark.
[5] Cook, W., & Rich, J. (1999). A Parallel Cutting-Plane Algorithm for the Vehicle
Routing Problems with Time. Department of Computational and Applied
Mathematics. Houston: Rice University.
[6] Cordeau, J.-F., Gendreau, M., Laporte, G., Potvin, J.-Y., & Semet, F. (2002). A guide
to vehicle routing heuristics. Journal of the Operational Research Society , 512-522.
[7] Hong, S., & Park, Y. (1999). A heuristic for bi-objective vehicle routing with time
window constraints. International Journal of Production Economics , 249-258.
[8] Calvete H.I., G. C. (2007). A Goal Programming Approach to Vehicle Routing
Problems With Soft Time Windows. European Journal of Operational Research ,
1720-1733.
[9] Hashimoto H., I. T. (2006). The Vehicle Routing Problem With Flexible Time
Windows and Travelling Times. Discrete Applied Mathematics , 1364-1383.
Computational Mathematics
CLUSTERING SPATIAL DATA USING AGRID+
ARIEF FATCHUL HUDA1, ADIB PRATAMA2
1
Mathematic Dept., UIN Sunan Gunung Djati Bandung, afhuda@gmail.com
2
Mathematic Dept. of UIN Sunan Gunung Djati Bandung, pratama.adib@gmail.com
Abstract. Clustering is a branch of data mining which focus to form clusters or

groups from data. Many methods has been developed to accomplish this task,
one of them is AGRID+. In this paper, we proposed a new approach of
clustering that is using simplify AGRID+ to cluster spatial data. The idea is to
separate the location attribute from the main attribute. We form the cell based on
location attribute and calculate the proximity of two objects based on the main
attribute. Experimental result that are using simulated data and criminal data
from POLDA METRO JAYA area are reported.
Key words and Phrases:Clustering, AGRID+, Spatial Data, Grid Clustering,

Criminal Data
1. Introduction
Clustering is one of the main techniques in data mining that focus to form
clusters (or groups) from data where objects that belong to same cluster are similar
or close to each other, and objects that not in same cluster are not similar. Many
methods and algorithms has been developed to solve this problem. Most clustering
methods fall into four categories : hierarchy based, partition based, grid based, and
density based[4]. Grid based algorithms apply a grid structure into data which then
quantize the data into finite number of cells. The benefit of this technique is it’s
fast processing time so this approach is suitable for handling a large data. One of
the grid based clustering algorithm is AGRID+[13]. AGRID+ combines grid based
and density based approach to clustering large data with complex shape, although
the experiment was done only using normal data that need no special treatment.
In clustering, there are some kind of data that can be clustered. One of them
is spatial data. Spatial data is a data that have not only the main attribute (normal
attribute) but also contain the location attribute of the data. This attribute differ
from main attribute because its function is not to measure the similarity between
objects. With spatial attribute, we can say whether an object located in north of
another object or not. There is not much algorithms than can handle this kind of
data because it cannot be clustered directly and need special treatment. In this
paper, we formulate a new approach to clustering spatial data using AGRID+.
162
163
2. Related Work
Some clustering algorithms of spatial data are the results of an extension
from previous clustering algorithms[15][18]. Therefore, the categorization of
spatial data clustering algorithms follows the categorization of common clustering
algorithms. The conversion of common clustering algorithms into spatial clustering
algorithms is done by involving spatial elements in the clustering algorithms. Each
clustering algorithm has its own unique way of involving spatial elements. The
following paragraph presents some spatial clustering algorithms which are the
extensions of usual clustering algorithms.
Partition method divides data into subsets or partitions by evaluating the
distance between data and cluster representation. Some proposed and developed
algorithms which are based on partition method are k-means, k-medoid, fuzzy k-
means, PAM, CLARA, and CLARANS[19][20][23]. CLARANS algorithm
enhances PAM and CLARA algorithms in terms of their computation efficiency.
Two methods of clustering which adopt CLARANS algorithm for spatial data are
termed spatial dominant or SD approach (CLARANS) and non-spatial dominant or
NSD approach (CLARANS)[20][22]. CLARANS algorithm has also been
implemented for spatial data in the form of Raymond’s polygon[20].
Some algorithms that perform clustering process based on density are
DBSCAN, DENCLUE, and OPTICS[23]. Sander extended DBSCAN by involving
spatial elements and non-spatial elements simultaneously to cluster spatial data in 2
to 5 dimensional format. DBCLASD is another extension of DBSCAN in spatial
clustering algorithm. Likewise, the density-based clustering algorithm developed
by Yang et.al. (2010) and Liu et.al. (2012) was extended by using Delaunay
triangulation[20][17].
3. AGRID+ for Spatial Data

There are 4 main features of AGRID+. First, instead of cells, objects are
taken as the smallest units. Second is the concept of ith-order neighbors, where
neighbor cells are organized into a couple of groups. Third is density compensation
to improve accuracy. The last feature is new distance measure, minimal subspace
distance for subspace clustering.
In grid based clustering algorithms, the concept of neighbor cell is very
important. There two concept of neighbor cell i.e. full neighbor and immediate
neighbor. The accuracy of full neighbor is high because its consider all cell that
around the main cell. But the computation is quite high too, especially with the
increase of dimensionality of the data. To solve this problem, new neighbor
concept is introduced, that is the ith-order neighbors, where cells are grouped based
on its contribution. With this concept, we don’t need to consider all cells, but only
those who have high contribution.
With the concept of ith-order neighbors, not all cells are considered. Then
the result is not as good as when we use the full neighbor. Density compensation is
introduced to handle this problem. With density compensation, the ratio between
volume of used cell and volume all cell are considered. With this technique, the
accuracy of AGRID+ will increase even if we didnt use the full neighbor.
With that features, AGRID+ can found clusters with different shape and with
its grid technique, it can clustering large data or data with high dimension more
efficiently. The procedure of AGRID+ consist 7 main steps.
1. Partitioning. Data space is partitioned into cells. Each object then are labeled
to its cell according to its attribute.
164
2. Computing distance threshold. The distance threshold is computed based on

interval each dimension.
3. Calculate density. For every object, count the number of object in its
neighbor cells that the distance between both is less than distance threshold.
4. Density compensation. For each object, count the ratio between all cells and
the number of cells used. This ratio multiply with the object density and save
as the new density for the object.
5. Compute Density Threshold (DT). DT is the average of all compensated
density.
6. Clustering. First, each object whose density greater than DT is taken as a
cluster. Then, for every object in its neighbor whose its density greater than
DT too, if the distance between those two less than distance threshold, then
merge those two cluster. Continue the merging process until all object have
been checked.
7. Removing outliers. Some cluster might only have few objects in it. They too
small to be considered as a meaningful clusters, so they are removed and
labelled as noises.
In this paper, there are some modified of AGRID+ algorithm. There are two
modification, i.e first, density compensation, instead of volume, we used the
number of cells as the ratio. Second, the dissimilarity measured is used euclidean
distance as the distance measure instead of minimum subspace distance of
AGRID+. That’s way we called simplified AGRID+.
Our proposed is to cluster spatial data using AGRID+. Using AGRID+
directly to cluster spatial data, it will treat the spatial attribute of an object same as
the main attribute. Therefore the clusters that exist in main attribute are distorted.
So, the proposed method is using spatial attribute to determine the cells (grid).
Clustering method done in grid that developt from spatial attributes of the data set.
3. Experiment and Analysis

To get the real result from spatial clustering, we have to treat the spatial
attribute differently. In our method, we separate the spatial attribute from the main
attribute. We used the spatial attribute only to form cells, and we used the main
attribute to measure similarity between two objects.
We have done 4 experiments using this method. In the first two experiments,
we used data that we generate randomly. We set this data to have cluster in spatial
and main attribute so we can see whether the method can found the cluster based
on main attribute only and not affected by spatial aspect of the object. For the last
two experiments, we used crime data. The crime data used is crime data of under
POLDA METRO JAYA jurisdiction area (Jakarta, Depok, Bekasi, Tangerang)
from year 2009 to 2012. In each experiment, the result were taken based on its
evaluation. We combined some parameters to get the best result.
The first experiment was done using simulation data with the number of
main attribute is two and the number of objects are 1000. The data were set so
there only two clusters exist in one spatial area. The clustering result can be seen in
figure 1. The data clustered into two clusters although in spatial space, it only has
one cluster. From this experiment, we can see that our method can detect the
cluster very well without being affected by spatial aspect of the object. We did the
second experiment using 2000 data that we set into three clusters where spatially
165
form two spatial areas. We also set one of the cluster to exist in both spatial area.
This method can found all the clusters, including cluster that exist in both spatial
area (see figure 1).
From the figure, we can see that using our approach, spatial attribute from
the object is only used for determine the neighbor of an object and therefore not
related to the measure of similarity of the object.
Figure 1:Clustering result using 2D generated data
The next two experiments were done using crime data from Jakarta, Depok,
Tanggerang, Bekasi, Kabupaten Tanggerang and Kabupaten Bekasi from 2009 to
2012. There are 16 kinds of crime, that is murder, mayhem, thievery, hold up
crime, raiding, hijacking, 2 wheels vehicle, 3 wheels vehicle, 4 wheels vehicle, fire,
gambling, extortion, rape, narcotics, and Juvenile Delinquency. But not all kind
crime were used, we only used a few kind (as attribute) that look more
“interesting”. Every object is a representation of each district in each region. The
spatial component were taken approximately using grid technique.
The third experiment was done using only the narcotics attribute. To get the
result, we combine some parameters and then we take the best result based on its
evaluation index. In our experiment, we used silhouette evaluation method (SH) to
evaluate the clustering result[10]. In SH, the higher the index is (get close to 1)
means the more good the result is. The clustering result and its evaluation index
can be seen in table 1.
Table 1. Clustering Result using narcotics attribute
Year SH Ratio (%) Clusters

2009 0.31005 22.33 2
2010 0.3609 35.922 4
2011 0.822 13.59 2
2012 0.529 12.417 2
From the table above, we can see that the evaluation result using our method
is not quite high. Only in 2011 that SH shows value above 0.7, the rest falls mainly
between 0.3 – 0.5. And also as we can see, the number of object that are clustered
is quite small too, where the ratio of clustered object still under 35%. The result of
clustering in 2010 can be seen in figure 2.
166
Figure 2:Clustering result in 2010 using narcotics attribute
The last experiment was done using 3 attributes. In this experiment we

combine some attribute into one. The first we combine mayhem, hold up crime,
raiding, and hijacking attribute into one attribute, theft with violence. And also all
vehicle crime (2 wheels, 3 wheels and 4 wheels) into one vehicle crime. The last
attribute is thievery. The experiment scenario still same, that is we combine several
value of parameter and find the best result. The best result still based on SH
evaluation index. Table 2 shows the clustering result from 2009 to 2012.
Table 2: Clustering result of thievery, theft with violence, and vehicle attributes
Year SH Ratio (%) Clusters

2009 0.139 22.33 2
2010 0.602 18.446 2
2011 0.378 17.475 2
2012 0.460 19.417 2
Clustering result in 2010 can be seen in figure 3.

167
Figure 3:Clustering result in 2010 using 3 attributes
In our research, we propose a new approach to clustering spatial data. The
result of our simulation data shows that our method can found the two clusters
which in spatial space only have one cluster. For real data, the result of our
experiment shows in some year, data are well clustered. This can be shown with
the value of evaluation index that greater than 0.7. But since we used silhouette
(SH) index for the evaluation and the SH is not designed to evaluate spatial data,
the result might be not optimal. This is our future work to provide an evaluation
technique that can evaluate spatial data.
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CHAOS-BASED ENCRYPTION
ALGORITHM FOR DIGITAL IMAGE
EVA NURPETI1, SURYADI MT2
1
Departement of Mathematics, Universitas Indonesia, eva.nurpeti@sci.ui.ac.id
2
Departement of Mathematics, Universitas Indonesia, yadi.mt@sci.ui.ac.id
Abstract. Today, storage and transmissions of data or information is

unproblematic by supports of information and communication technology. Data
or information presented in digital form is highly vulnerable by attack of data or
information abusing. Digital image is one of digital data or information which is
frequently becomes in target crime. Therefore, reliable, secure, and fast security
technique is required in data or information digital image. In this study,
encryption algorithm is built using logistic map as a random number generator.
This algorithm applied in digital image. Designing in chaos-based encryption
algorithm is improved endurance from brute force and known plaintext attack.
Performance on algorithm endurance was based on key space analysis,
sensitivity analysis to the initial values, and histogram analysis using a
goodness-of-fit test. According to testing and analysis, this algorithm has a key
space of 10 and sensitivity to initial values is very sensitive, up to 10 . It
can be concluded that, the algorithm is very difficult to be cracked by brute
force attack.
Key words and Phrases: Chaos, Logistic map, Encryption algorithm, Digital
image.
1. Introduction
Until the 1990s the encryption algorithm that is often used is DES, but
because of the advanced technology, these algorithms are not considered safe
anymore. Therefore, an international competition held by the National Institute of
Standards and Technology (NIST)and Rijndael algorithm was selected as the
winner after passing through various stages of the selection in 2001, which was
later renamed AES algorithm [4].The performance of an algorithm can be seen
from the resistance against attacks and security algorithms computational time.
Traditional ciphers such as Data Encryption Standard (DES), Advanced Encryption
Standard (AES), and Rivest-Shamir-Adleman Algorithm (RSA) etc. to encrypt
image requires a large computational time and high computing power. But, only
169
170
thoseciphers are preferable which take lesser amount of time and atthe same time
without compromising security [2].
To provide a better solution from the image security issues, a number of
image encryption techniques have been proposed, one of which is a chaos -based
image encryption. This method gives a good combination of speed, high security,
complexity , and computational power [3].
Chaos is the type of behavior of a system or function that is random,
sensitive to initial values, and ergodicity. Function that has chaos propertieswas
called chaos function. Chaos function have been proved very suitable to design
facilities for data protection [1]. With these properties, chaos function can be used
as a random number generator. One of the simple function that shows the chaos
properties is the logistic equation or commonly called the logistic map. Other
functions that have chaos properties are Henon chaotic map, Arnold 's cat map, and
the tent map. Logistic map function is defined as a function :
∶ → , 1
which is a function of two variables and . λ variable value in the interval 0,4
and are in the interval 0,1 . Meanwhile, the presentation of logistic map
function is in the form of iterative. It is :
1 (1)
with 0 , 1,2,3 . . .. and is the initial value of iteration .
For the last decade , a lot of chaos -based cryptographic techniques have
been studied, such as secret communication based on chaos, chaos based block /
stream cipher,chaos -based random number generator, etc. As for the chaos -based
security applications such as image encryption and chaos based copyright
protection of multimedia [1]. Chaos-based encryption also been extensively studied
by researchers because of its superior in safety and complexity [6].Therefore, in
this paper. we will discuss about security of digital imageusing chaos -based
encryption method , by using the logistic map as a chaos function.
Testing of algorithm was done based on key space analysis, key sensitivity
analysis, the encryption average time. The analysis was carried out to see the
resistance againtsbrute force attack and known plaintext attack .
2. Main Results
2.1 Algorithm
Digital image encryption algorithm in this paper uses logistic map as a chaos
function (equation 1). The sequence of the process of securing the digital image
can be seen in Figure 2.1 below :
A.
Original Bit stream Image

Image Encyption encrypted encrypted
Result
Key stream
, Logistic map
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B.
Image Bit stream
decrypted Encyption Image
encrypted
Result encrypted
Key stream
, Logistic map
Figure 2.1. The sequence of the process of securing the digital image
A.) Encryption process B.) Decryption process
Figure 1 shows the flow in securing digital images. On A, first input the
original image and then encrypt it using logistic map as a key stream generator,
where logistic map needa key , and from that we obtain the bit stream
encrypted, that is the image result. On B , the reverse process from encryption
process on A, with using the same key.
Encryption algorithm is described in the flowchart shown in Figure 2.2:
no
yes
no
yes
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Figure 2.2 Flowchart of digital image encryption algorithm using the logistic map
Flowchart explanation is shown in Figure 3.4 :
Step 1 : Insert thekey , andoriginal imagewith size
Step 2 : Do 200 times iteration the logistic map equation and we will get
decimal fractions.
Step 3 : Check condition.
Step 3a : If yes , then do 3 times the logistic map iteration and we will obtain the
results are decimal fractions , such as .
Step 3b : if not, so the encyption process is done for all part of image and we will
obtain encrypted image.
Step 4 : Check whether the pixel block is the last or not.
Step 4a : If yes,Select the first 15 number behind the decimal from decimal
fraction that has been placed before ( for example ) , that are the
result of 3 iterations logistic map. Then divide 15 number to 5 integer
with each integernya has 3 points . Then take as much . , 5
integer . Do operation mod 256 to each integer , so we get a Chaos
integer number . , 5 byte integer chaos . 1 byte chaos
number is called key stream .
Step 4a.1 : Take the pixel value information at each grayscale as much as
. , 5 . Each 1 byte information of the image is called P.
Step 4a.2 : Do step 5 . , 5 times.
Step 4.b : If not , then do transformation from real to integer, like in the step 4a,
but take by 5 integer. Then take 5 integer. Do Operation mod 256 to
each integer, so we get 5 bytes chaos integer number.
Step 4b.1 : Take the pixel information at each pixel grayscale by 5 . Each 1 byte
information of the image is called P.
Step 4b.2 : Do the step 5 by 5 times.
Step 5 : Do bitwiseXOR operation on each byte chaos integer numberwithevery
byte image data. Otherwise, do: ⊕
Step 6 : Back to Step 3.
Information so that the original image data can be accessed again, it must be
done the decryption process. Than the decryption process is the reverse process of
encryption. Then, input the decryption process is an image that has been encrypted
and the result is the original image. Key used in the decryption process must also
be the same as the key used during the encryption process.
2.2 Results
Tests performed to see the computational time and the resistance of chaos-
based encryption algorithm against brute force attack and known plaintext attack.
Key space analysis and key sensitivity analysis were done to the resistance againts
brute force atttack, while analysis of uniform distribution of pixel values
(histogram analysis) were done to the resistance againts known plaintext attack.
A. Analysis of Encryption Time

Tests toward all digital image test data, performed using the same key value
for both encryption and decryption process. Parameter values that used are
0.1 and 4 . The test results of the cat grayscale digital image (1-5) and lena (6-
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10) with variety of sizes shown in Table 2.1 :
Table 2.1
Test Average Encryption Average Decryption

Pixel Size Image Name
Data Time (second) time (Second)
1. 2560 x 1920 99.8870000839 98.3209999648
2. 1280 x 960 25.0730001926 23.10300073012
3. 640 x 480 Cat 6.14599990845 7.44099995434
4. 320 x 240 1.46299982071 1.48399996758
5. 80 x 60 0.0940001010895 0.109000205994
6. 2048 x 2048 87.3000001907 88.1790001392
7. 1024 x 1024 21.6100001335 21.6159999371
8. 512 x 512 Lena 5.30499982834 5.35699987411
9. 256 x 256 1.29999995232 1.27800011635
10. 128 x 128 0.353000164032 0.371999979019
While the test results of the cat colored digital image (1-5) and lena (6-10) with
variety of sizes shown in Table 2.2 :
Table2.2
Average
Test Average Decryption
Pixel Size Image Name Encryption Time
Data time (Second)
(second)
1. 2560 x 1920 182.141000032 183.129999923
2. 1280 x 960 45.3589999676 43.8740000725
3. 640 x 480 Cat 11.4220001698 11.0429999828
4. 320 x 240 3.10699987411 2.74099993706
5. 80 x 60 0.138999938965 0.19200000876
6. 2048 x 2048 154.983999968 156.345999956
7. 1024 x 1024 38.5360000134 38.9079999924
8. 512 x 512 Lena 9.66899991035 9.90400004387
9. 256 x 256 2.45600008965 2.59000000954
10. 128 x 128 0.605999946594 0.79069999925
B. Key Space Analysis

The random number generator which was used to generate key stream is
logistic map. Keys that are used on logistic map are and , where and are
real number. If we use a higher level of precision, for example 64-bit double
precision IEEE standard, the precision level will reach 10 . So, the total
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possibilities of keys are 10 10 10 .

C. Key Sensitivity Analysis
The value of the key that is used is always same for each digital image test
data in this paper . While the decryption process will be tested with many
differentkey value. The results are presented in Tables 2.3, 2.4 and 2.5 .
.
Table 2.3 The result of key sensitivity (case-1)
Encryption Result Decryption Result

Oginal Image
( . , ) ( . , )
Same
In Table 4.3 are shown the results of the encryption and decryption process
simulation using catimage with the same key that is 0.1 , 4 . Thus seen
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that the decryption process succeeded in opening the original data . Histogram
display for each column in a row that the components R , G , and B shows the
distribution of pixel values . Look at the histogram of original image and decrypt
result image are very similar and have been proved also with the help of python
that its values are the same. Table 4.4 shows the decryption attempt to use a
different key to the encryption key.
Decryption Result
Encryption Result (
Original Image
( . , ) . ,
)

Difference of Valueis10
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Original Image Encryption Result Decryption Result

( . , ) (
. ,
)

`

Difference of Valueis10
The test has tested with difference value of when the decryption
proccesto when the encryption proccess.Shown in Table 4.4 that the attempt to
decrypt using a key difference between the value by 10 did not succed to get
the original image . But when the difference reaches 10 the decryption proccess
got the information of original image . It shows that the numbers 0.1 and
0.10000000000000001 is considered to be the same number that is 0.1. So we get
the sensitivity of this algorithm is up to 10 .
This algorithm is so hard to be cracked by brute force attack. Because the
key space reach to 10 , so it needs long time to find the key. Beside that, there are
high key sensitivity factor that up to 10 and also because of random unpattern
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key streamfactor makes this algorithm is hard to be cracked to get the

original data.
References
[1] Kocarev, L., & Lian, S, 2011, Chaos-based cyrptography. Berlin Heidelberg :
Springer-Verlag.
[2] Pareek, N.K., Patidar, V., Sud, K.K., 2006, Image encryption using chaotic logistic
map. Journal of Image and Vision Computing, 24, 926-934.
[3] Patidar, V., Pareek, N.K., Sud, K.K., 2009, A new subtitution-diffusion based
image cipher using chaotic standard and logistic maps.Journal of Communications
in Nonlinear Science and Numerical Simulation, 14, 3056-3075.
[4] Stallings, W, 2011,Computer and Network Security : Principle and Practice (5
ed.). New York: Prentice hall.
[5] Suryadi MT, 2013, New Chaotic Algorithm for Video Encryption, 4th The
International Symposium on Chaos Revolution in Science, Technology and
Society 2013, Jakarta, August, 28-29.
[6] Zhang, W., Wong, K.W., Yu, H., Zhu, Z.L, 2013, An image encryption scheme
using reverse 2-dimensional chaotic map and dependent diffusion. Journal of
Communications in Nonlinear Science and Numerical Simulation, 18, 2066-2080.
APPLICATION OFIF-THEN MULTI SOFT SET
RB. FAJRIYA HAKIM
Department of Statistics, Universitas Islam Indonesia,

hakimf@fmipa.uii.ac.id
Abstract. A soft set as a relatively new mathematical tool for dealing with
uncertainties was first introduced by Molodtsovhas experiencedrapid growth.
Variousapplicationsof softsetfor the purpose ofdecision-makinghave beenshown
by several researchers. From various studies presen tedmostly shows the role of
soft sets as a tool in the collection o fthe various attributes needed by a person
to determine which decisions will be taken. The development ofthe use of soft
setmay actually be more than that, this paper will show how soft set can play a
rolein the decisionmade bya personbased onahistory ofdecisionsthathave been
madebysomepeopleearlierandused asa referenceforthe nextdecision. Therefore,
this paper introduce an (if-then) multi soft sets as a developments of application
of soft set which is stated in the form if (antecedent) then (consequence) with
antecedent and consequence are derived from previously several decisions that
have been made by people when using a soft set as a tool to help them for
making a decision.
Key words and Phrases: Soft Set, Multi Soft Set, If – then, Decision
1. Introduction
The buyer-oriented industry puts the buyer at the highlight of everything
business does. The ability of industrial management to understand the buyer could
make a significant increase in its sales. Understanding the buyer or any kind of
terms that refer to deep recognizing the buyer such as buyer persona could help the
industry understand what the buyer really want. Even though many people who do
the marketing knowing the big impact of understanding the buyer but very few
companies do a research of buyer (even a simple research), they just regarding this
problem as an sense or intuition that the buyer engage with their business
product’s. Most corporations expect that collecting the information/insights about
the buyer just wasting time effort that make an ineffective work and spending a lot
of money that will lessen their sales.
Today, people are very stingy to give their personal information even when
the salespersons do the very warm welcome and humble conversation. It is difficult
to satisfying in the field of buyer persona to obtain reference of the buyer such as
buyer demographics, buyer’s role, buyer psychology or deeper personal
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179
information about the buyer. This could be understood that buyer keep their
individual information for security reason. Therefore, salesperson should be
exerting all effort to observe the buyer even though only using a simple
observation. Buyer just asking what they need and they do not want to be asked
outside of the things they asked. Buyers get annoyed when the shop assistant
started asking personal things. They will only provide personal information that
they feel is important to be known by the shop owner, such as their address in
which the company must deliver the goods they had bought.
Once the company has gathered a sufficient amount of simple but deep
observation then all company’s team participate inevaluating the buyers. Rough
observation of buyers could be put in the form of soft set. Discussion should be
made as an effort to approximate accurately reflect who the buyer is and their
decision that have been made that could be made as a basis for recommendation to
the next buyer in choosing company’s product.
In this article will show the simple steps of marketing and sales based on soft
set theory to quickly understand the buyer via simple observation of the
buyer.Decision that has been made by buyer to purchase is a result of complicated
policy from point of view of buyer. Choosing one product to be purchased could be
started by describing the product they want using some simple characteristics,
attributes, information or knowledge they have about those product. Any
parameters which had been regarded as an important characteristic that might be
owned by the product to be bought could be collected in the structure of
mathematical notion. Collection of those parameters can be laid on the form of soft
set theory. Soft set theory first introduced by Molodtsov [15] in 1999 and has been
applied in many fields by researchers. Hakim et al. [7] had proposed a
recommendation analysis as a buyer tool to assist their decision in purchasing a
product. Many researchers (Chen et al. [2], Feng et al. [5][6], Herawan and Mat
Deris [8], Jiang et al. [9], Kong et al. [11], Maji et al. [13][14], Roy and Maji [18])
mostly show the role ofsoftsetsas a tool inthe collection ofvariousattributes or
parameters of objects neededbyapersonand thendetermineusing some calculations
whichdecisionswill betaken. The development ofthe use of softsetmay actually be
morethanthat, this paperwill showhowsoftsetcanplay a rolein the decisionmade bya
personbased onahistory ofdecisionsthathave been made by some people earlier and
used asa reference for the nextdecision.
Deciding a product to be purchased is a difficult matter for a buyer. Hakim et
al. [7] has introduced a recommendation system based on soft set theory to
purchase a product from buyer side. This recommendation analysis is an advantage
for buyers in helping them to determine the product they need. This paper also
trying to use a soft set theory from a view of store team to observe the personality
of buyer by means of the ability of the store owner and store assistant to evaluate
their buyer when purchase goods. Since store team observation also will be put in
the form of soft set theorythen we have a structure of dual usage of soft set theory.
First is recommendation analysis for buyer in choosing a store’s product and
secondly to assist buyer in finding out the products based on observations of the
store team to its buyers. Both of them are based on the parameters which are
collected from the information of buyer and store team. Therefore, it can be seen as
two parts, which are condition and decision model problem. The ‘condition’ and
the ‘decision’ parts will be determined after remarking two usage of soft set which
first useis application of soft set in buyer’s view when purchasing product and
second use is application of soft set in store team observation to their customers.
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We will develop this new application of soft set in the form if (antecedent) then
(consequence) with antecedent as a condition attribute and consequence as a
decision attribute that are derived from previously several decisions which had
been made by other buyers. Because it involves a condition and decision attribute,
we need a language of ‘decision rules’. A decision rule is an implication in the
form if A then B, where A is called the ‘condition’ and B the ‘decision’ of the rule.
Decision rules state relationship between conditions and decisions. In this paper,
we are trying to combine the decision rules and dual soft sets that will produce a
new application of soft set which can be known as if-then multi soft-set. This
application not only helping buyer in deciding the product to be chosen but also
help the store to map their buyer when determining the product needed.
2. Literature Reviews and Main Results

2.1. Soft Set Theory
Molodtsov [15] first defined a soft set which is a family of objects whose definitions
depend on a set of parameter. Let U be an initial universe of objects, E be the set of
adequate parameters in relation to objects in U. Adequate parametrization is desired to
avoid some difficulties when using probability theory, fuzzy sets theory and interval
mathematics which are in common used as mathematical tool for dealing with uncertainties.
The definition of soft set is given as follows.
Definition 2.1. (Molodtsov [15]).A pair (F, E) is called a soft set over U if and
only if F is a mapping of E into the set of all subsets of the set U.
From definition, a soft set (F, E) over the universeU is a parameterized family that
gives an approximate description of the objects inU.Let e any parameter in E, eE,
the subsetF(e) Umay be considered as the set ofe-approximate elements in the
soft set (F, E).
Example 2.1. Let us consider a soft set (F, E)which describes the “attractiveness
of houses” that Mr. X is considering to purchase.
U – is the set of houses under Mr. X consideration
E – is the set of parameters. Each parameter is a word or a sentence
E = {expensive, beatiful, wooden, cheap, in the green surroundings,
modern, in good repair, in bad repair}
In this example, to define a soft set means to point out expensive houses,
that shows which houses are expensive due to the dominating parameter is
‘expensive’ compared to other parameters that are possessed by the house, in the
green surrounding houses, which shows houses that their surrounding are greener
than other, and so on. Molodtsov [15] also stated that soft set theory has an
opposite approach which is usually done in classical mathematics that should
construct a mathematical model of an object and define the notion of the exact
solution of this model. Soft set theory uses an approximate nature as an initial
description of the objects under consideration and does not need to define the
notion of exact solution. Common mathematical tools to solve complicated
decision problems with uncertainties are probability theory, fuzzy theory and
interval mathematics, but there will get difficulties on using them. Probability
theory must perform a large number of trials, fuzzy theory must set the
membership function in each particular case and the nature of the membership
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function is extremely individual, interval mathematics should construct an interval

estimate for exact solution of a problem but not sufficiently adaptable for problem
with different uncertainties. To avoid these difficulties, in the soft set theory, when
someone is faced to the decision problems with many uncertainties, problem or
object is determined by the ability of the person to explain various things related to
that object. Various things that might be related are referred to as the object
parameter in the soft set. He or she could express the parameters use any
information that might be possessed by the objects. This relevant information
refers to the necessary parameter of the objects. The necessary parameters could be
a particular interest that he or she can express their preference, knowledge,
perception or common words in a simple way to the objects under consideration.
Parameters attached to the objects are said to indispensable if he or she considers
that the information involved to identifying a problem is sufficient to elucidate the
objects.Since object parameters have been determined and then give a fair
valuation to each object based on those parameterwould yield a solution that gives
a suggestion to make a decision. Setting the objects and their necessary
information using words and sentences, real numbers, function, mappings, etc., is a
parametrization process that makes soft set theory applicable in practice.
Maji et al. [13] has extended example 2.1 to decision making problem of
choosing six houses based on the attractiveness of houses as a houses parameters.
Some parameters are absolutely belonging to some houses and some parameters
are absolutely not belonging by some houses. Their example of choosing of houses
problem based on soft set has initiated many important applied and theoretical
researches that have been achieved in soft set decision making problem. However,
soft set theory has not been yet find out the right format to the solution of soft set
theory due to many research using binary, fuzzy membership or interval valued for
parameters of objects valuation which actually should have been avoided as
notified by Molodtsov [15].
2.2. Soft Solution of Soft Set Theory.

Maji et al.[13][14] applied the theory of soft set to solve a decision making
problem using example which was described by Molodtsov. They used tabular
representation with the entry is 1 if an object has a particular parameter, and zero if
it does not has then decision is based on the maximum of cumulative number of
objects that have parameters of the soft set.To handle the binary valuation, Maji et
al. [12] tried to introduce the W-soft set or weighted soft sets, but their effortwould
not give a new approach to decision analysis caused by the weightsare multiplied
to each parameter (as attribute) and hence, this method would not change the
result.Later, Herawan and Mat Deris [8] and Zou and Xiao [20] have proven that
soft set could be transformed to binary information system. Maji et al. [13] also
used rough set theory to reduce the parameters that have been hold to every object
in the universe. Unfortunately, rather than optimize the worth of parameter as
necessaryinformation; they preferred to reduce the parameter. The information
involved in the parameter will be loosed. Molodtsov has insisted the adequacy of
parameterization to objects of universe rather than reducing the parameter that has
been belonged to every object. Due to binary value of the entries, their decision
result gives an exact solution that might contradicted to the philosophy of the initial
Molodtsov’s soft set that insisted the approximation to the result which is caused
by soft information accepted in a parametrization family of soft set.Chen et al.[2]
and Kong et al.[11] also wanted to reduce the parameter of the objects, but
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Molodtsov already pointed out that the expansion of the set of parameters may be
useful due to the expansion of parameters will give more detailed description of the
objects. Since the adequacies of parameter arecrucial in soft set theory to describe
the houses, reduction of parameters may bring out a set of indispensable parameter
from a set of parameters. This reduction parameter expected to deduct valuable
information from a set of indispensable parameter of soft set.
Roy and Maji [18] combined the fuzzy set and soft set, fuzzy numbers is
used to evaluate the value of the parameter’s judgment for each object. This idea
develops to the hybrid theory of fuzzy soft set. This fuzzy soft set also initiated by
Yang et al. [19]. They also said that rather than using binary value, it will be better
to use the degree of membership to represent the objects which hold the parameter.
Since the parameter’s value of each object filled by fuzzy numbers, there must be
an expert to determine the membership values that represent the matching number
for each house. It may become more difficult since the valuation of parameters of
the objects are on the interval-valued fuzzy number (Feng et al. [5], Jiang et al.
[9]). An expert should give not only matching number of parameter but should
determine the lowest and the highest number as the value of the objects parameters.
Molodtsov had been stated that this is the nature difficulties when dealing with
fuzzy numbers and should be avoided. The idea to substitute the value of each
object parameters using binary, fuzzy number or interval-valued fuzzy number may
still be used as a reference because it gives results that can be used as a benchmark
for a person in making decision.
All researchers on soft set in decision making could be grouped into two
groups. First, researchers that treat the soft set as an attribute of information system
(Herawan and Mat Deris [8], Zou and Xiao [20]) then using Rough Set to handle
the vagueness for making a decision (Feng et al. [6]). Second, researchers that use
fuzzy theory to soft set (Jun et al. [10], Feng et al. [5] and Jiang et al. [9]). Both of
them gave techniques which produce best decision based on binary or fuzzy
number rather than recommendation that may be little bit more satisfying
Molodtsov’s soft set philosophy. Of the entire study could be seen that the whole
objects under consideration was assessed through the parameters by the decision
makers and will get the solution in the form of a subset of the objects itself that
each of them has a dominating parameters. According to this understanding we will
give the definition for soft solution of soft sets.
From that definition 2.1., a soft set (F, E) over the universeU is a
parameterized family that gives an approximate description of the objects inU.Let e
any parameter in E,eE, the subsetF(e) U may be considered as the set ofe-
approximate elements in the soft set (F, E). It is worth noting that the sets F(e) may
be arbitrary. Some of them may be empty, some may have nonempty intersection.
That is, the solution of the soft set is a set which are a subset of object and a subset
of parameters that shows the objects and its parameters.
Definition 2.2. (soft solution). A pair (F’, E’) over U’ is said to be a soft solution of soft set
(F, E) over U if and only if
i) U’  U
ii) {e|U’ | e E} = E’ where e|U’ is the restriction parameter of e to U’
iii) F’ is a mapping of E’ into the set of all subsets of the set U’
We shall use the notion of restriction parameter of eE’ to U’ in order to obtain the
parameters which dominate an object compared to other parameters that may be possessed
by those objects.
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A soft set (F, E) over U might be considered as an information system (U, AT)
(Demri and Orlowska [3]) such that AT = {F} and value of a mapping function of F = eE
make available the same information about objects from U. It is a common thing to identify
a wide range of matters (parameters) relating to the object and then create a collection of
objects that possess this parameters. To compose this intuition, for a given soft set S = (F,
E) over U, we define a soft set formal context S = (U, E, F) where U and E are non-empty
sets whose elements are interpreted as objects and parameters (features), respectively, and
FU x E is a binary relation. If xU and eE and (x, e) F, then the object x is said to
have the feature e.In this concept, the soft set formal context provide the following
mappings ext: P(E)P(U), that shows extensional information for objects under
consideration. This means an object parameters may be able to be expanded on someone
views as the set of those objects that possess the parameters.
2.3. Multi Soft Set
In the situation where a lot of things involved in decision-making, these situation
can be separated and described in some soft sets. Here are some thought that underlying the
application of multi soft sets.
def
Definition 2.3. For all X  U and e  E we define ext(E)  {xU | (x,e)F, for every
eE};ext(E) is referred to as the extent of E.
A soft set formal context S = (U, E, F) is a urnfor a collection of soft sets. Not
necessarily soft set formal context will only give one soft set. S = (U. E, F) could
be viewed as Multi Soft Set, say dual soft set S1 and S2 where S1, S2S and S1 is
soft set (F1, E1) over U1, S2 soft set (F2, E2) over U2, and U1, U2U and E1, E2E
and E1 ∩ E2 = Ø
Lemma2.1. For Soft set formal context S = (U. E, F), S1, S2 S and S1 is soft set (F1, E1)
over U1, S2 soft set (F2, E2) over U2, for all U1, U2 U and E1, E2 E if E1 ∩ E2 = Ø, then
ext(E1) ∩ ext(E2) = Ø
def
P ROOF .Let E1, E2E. Assume that xU1U and for every eE1Ethen ext(E1) 
{xU1| (x, e) F1, for every eE1 E} since E1, E2Eand E1 ∩ E2= Ø,xU1U and for
every eE1Ethen (x,e)F1 that is ext(E1) and never ext(E2).□
Lemma 2.1 shows that a soft formal context can be divided into a number of
soft sets (Multi Soft Sets) with each object and its parameters are different
but still in the same context. It is in line with Alkhazaleh and Salleh [1] that
has introduced the definition of Fuzzy Soft Multiset with a collection of
universes and reminds that any change in the order of universes will produce
a different Fuzzy Soft Multiset. But they did not explicitly state the
relationship between each soft set inside Multi Soft Set with a different
multiset. This notion can be exemplified as follows, in a furniture store, the
buyer establish a soft set to choose which one to buy based on desired
parameters, while the shop owner build a soft set to observe the behavior of
the buyer to make his choice. There will be relation between furniture
chosen with the buyer performance. In this paper we will use the definition
of Multi Soft Set in the formal context and clearly state the relationship
between soft sets with if-then decision rules.
3. The Proposes Application Of if-then Multi Soft Sets.

In this paper we will develop again the examples given by Hakim et al [7]
184
which illustrate a user interface of soft set recommendation analysis for purchasing
furniture products in some furniture store. System (figure 3.2) displays three
columns, first columns consists of customer identification and buyers are offered to
get assistant from furniture expert for choosing and question of some specific
purpose in intending buying the furniture. All collections of furniture items are
shown in second column and buyer was asked to choose one of collections. In this
example, buyers choose dining chairs then the third column display all collection
of dining chairs. Four selected chairs (figure 3.3) are chosen by customer and
buyers could determine their own requirements for their dining chairs. Buyer has
several things that he thought as a dining chairs precondition, he could type any
perspective inside the form, for example, ‘match with my dining room decoration’,
‘fit the space of my dining room’, ‘cheap’, ‘comfort’, ‘classic’ and ‘wood color’.
This could be expressed in the form of soft set. A soft set (F1, E1) of this example
could be described as the preconditions of the chairs which buyer is going to buy.
U1 – is the set of chairs under consideration {CH1, CH2, CH3, CH4}
E1– is the set of parameters. Each parameter is a word or a sentence.
E1= {match with my dining room decoration, fit the space of my dining room,
cheap, comfort, classic and wood color}
Those preconditions could be regarded as parameters of each chair. He thought

that, those information/ knowledge/parameters are necessary parameters for him to
choose a chair that he need for inviting a special guest for dinner. Soft set has
applied here, that someone could use any parametrization he wants for purchasing
chairs. It might he only knows what he need and conditions that he must consider
putting the chairs then. Meanwhile, in this cases we offer a judgment from expert
based on buyer’s precondition which is available in the form below the ‘customer
request’, this form shows what Expert Says with the valuation of each chair below
of its pictures. In the second column, it also displays the form of customer
evaluation that he could determine his own judgment for each chair. The simple act
to evaluate the selected items is to compare them in a fairly flexible way by giving
a mark to the chairs that meet his requirements. More asterisks more meet
parameters (table 3.1).
Table 3.1. Evaluation of chairs’ parameter by buyer
Fit the Wood

Chair Match cheap Comfort Classic
space color
Chair1 *** ** * ** ** ***
Chair2 * * * *** *** ***
Chair3 ** ** *** * ** *
Chair4 * ** ** * * *
To better utilize information from the tables and provide added value to our
recommendation for Mr. X, we will use multidimensional scaling techniques. Non-metric
multidimensional scaling techniques are common techniques which based on ordinal or
qualitative rankings of similarities data (Kruskal [12]). Therefore, table 3.1 needs to be
transformed via the numbers into an ordinal table.Using the software R (R Development
Core Team [17]) withveganpackage and metaMDSprocedure (Dixon and Palmer [4]), we
get the mapping of houses and its parameters ranking on figure 3.1.
185
match
+
0.15
0.10
CH1
CH3
NMDS2
0.05
fit the +space

0.00
cheap
+
wood+ color
classic
+
-0.05
CH2
comfort
+
CH4
-0.2 -0.1 0.0 0.1 0.2 0.3
NMDS1
Figure 3.1. Multidimensional scaling plot of chairs and its parameters
Based on the figure 3.1 the soft solution of soft set for this problem is
Soft solution (F1’,E1’) = {(Match the dining room decoration) = CH1,

(Woodcolor, Comfort, Classic) = CH2,
(Fit the space of dining room, Cheap, Match the
dining room decoration) = CH3,
(Cheap) = CH4}
This set of soft solution is used as a recommendation forbuyer to purchase

the chairs. This soft solution is a result of soft set using hierarchical clustering and
multidimensional scaling techniques. Outcome of this solution is a
recommendation based on buyer’s evaluation, for example, the first picture show
that the chair tends to match with the dining room decoration while second picture
meet a lot of customer requests, i.e., wood color, comfort and classic style. The
final decision is verified by customer to buy that chair. The last row could be
utilized as an offer to buyer for buying another product which is usually bought by
others while buying that chair.
186
Figure 3.2. Soft set recommendation analysis first page.
Figure 3.3. Soft set recommendation analysis second page.
Nowadays buyers are miserly to give personal information due to security

reason and get annoyed when shop assistant started asking personal things. Shop
assistant also cannot force the buyers ask for personal information, but much better
187
if observing the behavior of buyers when selecting products and make their choice.
Simple research will be carried out if the shop owner does not assume the arrival of
buyer to their shop only as a destiny. Buyer that has already coming to their shops
should be noticed use any kind of characteristics or parameters which the owner or
shop assistant could do. Some simple parameters that might could be used to
differentiating one person to another such as, appearance of buyers, style of buyers
when asking something, gesture of buyers, speaking style of buyer and so on.
In this example, the furniture store owner and their team trying to observe
their buyer using several parameter which could be put in the form of set {tidy
appearance, looks wealthy, age-old, too much questions, modern lifestyle,
complicated requests, busy and in hurry, too much bargain} that they think
sufficient to evaluate their buyer. Other owners could add or reduce the parameter
used in this set, depend on the observation to their own buyers. A soft set (F2, E2)
of this example could be described as the behavior of buyers as the result of
observation of the owner
U2 – is the set of buyers under consideration {A, B, C, D}

E2– is the set of parameters. Each parameter is a word or a sentence.
E2= {{tidy appearance, looks wealthy, age-old, too much questions, modern
lifestyle, complicated requests, busy and in hurry, too much bargain}.
The owners could ask his/her employee to help him in judging their buyers. Of
course, the owner and other employee does not need to become an expert in advance at the
field of human personality evaluation. They just give simple evalution in accordance to
their ability to observe and what they feel about their buyer. In this example suppose the
owner choose 4 buyer that had been made a transaction with him. Those buyer will be
evaluated based on observation in which the owner and its team remembering again
behavior of the buyer when they was in their store. The way of dealing with evaluation
usually using ranking or rating to the objects under consideration and express their
perception in an easiest and fairly flexible way. The simplest expression is give ranking by
using an asterisk that show more asteriks in the parameter means more adjacent the
parameter belonging to the object. The owner will give more asteriks if one buyer meet the
parameters than other buyer. For instance, in parameter ‘tidy appearance’ Mr. B seems the
most tidy than others. Mr. C seem as tidy as Mr. D, even though both of them not really
equal tidy in appearance. Mr. A is the most not tidy compared to other three buyers. Mr. C
looks the wealthiest than others, Mr. A and Mr. D looks same wealthy and Mr. B looks not
so wealthy. Evaluation could be continued to the next parameters. Tabular representation of
the buyers and parameters would be useful to describe the response of shop owner and his
team in evaluating their buyers (table 3.2).
Table 3.2. Evaluation of shop owner to his/her buyers.
Too Busy Too

Tidy Looks Age- Modern Complicated
much and in much
appearance wealthy old lifestyle requests
question hurry bargain
A * ** *** ** * ** * **
B *** * *** *** * *** * ***
C ** *** ** * *** * *** **
D ** ** * *** *** * ** *
Doing the same thing to the first soft set, after evaluation and bring table 3.2 to numbers
and used multidimensional scaling technique to mapping those table, we get the soft
solution for the second soft set. Soft solution of soft set for the behavior of buyers is
188
Soft solution (F2’,E2’) = {(Age-old, Too much bargain) = A,

(Tidy appearance, Too much questions,
Complicated requests) = B,
(Wealthy, Busy and hurry) = C,
(Modern lifestyle) = D}
Figure 3.4. Interface of application if-then multi soft set
From this two soft set (F1, E1) and (F2, E2) give a result of two soft solution which
are (F1’, E1’) and (F2’, E2’) and due to high relationship between two soft set, the
owner could get the decision rules of two soft solution which are
If (F1’, E1’) then (F2’, E2’) or if (F2’, E2’) then (F1’, E1’)
Say, the owner would like to take one of those buyer to see the decision rules of
multi soft set, say Mr. B, then he will get the rules,
if Mr. B bought chair then Mr. B is {Tidy appearance, Too

{Classic, Comfort, Wood much questions, Complicated
color} = (Ch2) requests}
or
189
if Mr. B is {Tidy then Mr. B bought chair {Classic,

appearance, Too much Comfort, Wood color} = (Ch2)
questions, Complicated
requests}
Second rule seems reasonable for recommendation which will be used by the
owner and his sales person to a buyer who have behavior looks like Mr. B. Of
course this rule will not be disclosed to the buyer, because the rules are based on
the assessment of shopkeeper to their buyers quietly. Even though this
recommendation is not exact decision but this rules could help the owner and the
sales person to assist the buyers while they are determining to choose one chair
from several chairs of dining room. Figure3.4. shows the interface of
recommendation analysis of the owner to their buyers. This work has already
shown the applied of soft set when it is implemented in the rules if-then. The usage
of Multi Soft Set, could help not only buyer when he/she need to choose the object
he wants but also help the shop owner to give recommendation to his/her buyer
based on buyer behavior.
4. Conclusion
Soft set as a new mathematical tool in decision making already give lack of
restrictions to the one who using them to get the final decision. Anyone could use
any parameters in deciding which objects will be choosen. Frommany previous
studies mostly shows the role ofsoftsetsas a tool inthe collection ofthe
variousattributes neededbyapersontodeterminewhichdecisionswill betaken
however in this paper we have already shown the development ofthe use of
softsetto multi soft set and its lemma.We show howif-then multi softsetcanplay a
rolein the decisionmade bya personbased onahistory ofdecisionsthathave been
madebysomepeopleearlierandused asa referenceforthe nextdecision.
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ITERATIVE UPWIND FINITE DIFFERENCE METHOD

WITH COMPLETED RICHARDSON EXTRAPOLATION
FOR STATE-CONSTRAINED OPTIMAL CONTROL
PROBLEM
HARTONO1, L.S. JENNINGS2, S. WANG3
1Mathematics Department, The University of Sanata Dharma, Yogyakarta, Indonesia,

yghartono@usd.ac.id
2,3Mathematics Department, The University of Western Australia, Western Australia.
Abstract. This paper proposes a numerical algorithm for solving state-

constrained optimal feedback control problems. In this work, we use the Upwind
Finite Difference Method introduced in Wang [24] in an iterative fashion to
improve the speed of the method. In addition, with the combination of
Completed Richardson Extrapolation (see Richards [15]) the accuracy of the
method can be increased.
Key words and Phrases: Optimal Control, Hamilton-Jacobi-Bellman Equation,

Finite Difference Method, Penalty Function, Richardson Extrapolation.
1. Introduction
In this article we present a numerical method for solving state-constrained
optimal feedback control problems. Consider the following optimal problem,
subject to x˙ = f(x(t),u(t),t), for all t ∈ (s,tf],

x(s) = y,
where x ∈ Rn and u ∈ U ⊂ Rm are the state and control, tf > 0 is a constant, (s,y) ∈
[0,tf) × Rn, , y ∈ Rn is a given point. L : Rn+m+1 → R, φ : Rn → R and f : Rn+m+1 → Rn
are known functions.
191
192
If s = 0 and y = x0 is fixed then the formulation is reduced to an open-loop optimal

control problem. The solution to the open-loop problem constitutes an optimal
control u along the optimal trajectory x from the initial state x0. Unluckily, if
perturbations exist in the state x such that the state is off the optimal trajectory then
the corresponding optimal control solution is no longer available. On the other
hand, the solution to the closed-loop problem is robust or stable because it is
defined over a time-space region that contains the optimal trajectory. Hence, the
corresponding optimal control still can be found if the disturbances are present in
the state x.
For that purpose, we firstly need to translate the problem into first order
partial differential equation called the Hamilton-Jacobi-Bellman(HJB) equation.
By defining the value function V (s,y) = inf J(s,y,u) u

and using the Dynamic Programming approach, this problem can be converted to
HJB,
)) = 0 (1)
with terminal condition
V (tf,x) = φ(x(tf)).
In this equation there are two unknowns, the value function V and the optimal
control u.
Generally, the solution to HJB equation is continuous but nonsmooth. To deal
with this nonsmooth solution, the concept of viscosity solution was introduced by
P.L. Lions etc.[3], [4] and [10]. In the presence of constraints, H.M. Soner [18]
broadened the definition of the viscosity solution to the constrained viscosity
solution. More detail about viscosity solutions of HJB equation can be found in
Bardi [1] and Fleming [6].
Although HJB equation theoretically has been solved, unfortunately, it is in
general difficult to find the analytic solution. Hence, in practice numerical
approximation is necessary. There are many numerical methods available in the
literature for solving unconstrained HJB equation, such as in Bardi [1], Guo [7],
[8], Huang [9], Wang [24], [25] to name but a few.
Among these methods, we are interested in the Upwind Finite Difference
Method introduced in Wang [24] due to its simplicity. For one dimensional
problems, the discretization of equation (1) is as follows:
arg inf , , , ,
∆
for k = 1,...,N and i = k,...,M − k, where M,N are the number of partitions of spatial
and time intervals respectively, sign denotes the sign of and
.
This method is the Upwind Finite-Difference because it takes into account upwind
directions from which characteristic information propagates. If 0 the scheme
switches to the forward-difference scheme and to backward-difference scheme for
the opposite sign.
193
However, there is a drawback, namely trapezoidal propagation of the spatial

domain with each time step. This propagation causes a large initial region so that it
leads to expensive computation due to a greater number of computed grid points.
To address this problem and to improve the speed and accuracy of the method, we
introduce an Iterative Finite Upwind Difference Method (inspired by Luus’s work
in Dynamic Programming [11]) in combination with Completed Richardson
Extrapolation (Richards [15]).
As is commonly known, Richardson extrapolation is a technique for
improving the order of accuracy of numerical results. The main idea behind this
technique is as follows. If the rate of convergence of a discretization method with
grid refinement is known and if discrete solutions on two systematically refined
grids (coarse and fine grids) are available, then this information can be used to
provide higher-order solution on the coarse grid. As a result, it is easily
implemented as a postprocessor to solutions regardless of the methods or equations
producing them.
However, in order to work well, Richardson Extrapolation requires some
assumptions such as smoothness and asymptotic range of the solution. Smoothness
of the solution is mainly important because the analysis of Richardson
Extrapolation is based on Taylor series expansion. The asymptotic range of
solution means that the sequence of systematically refined grid points over which
discretization error reduces at the formal order of accuracy of the discretization
scheme. For example, if the order of accuracy of scheme is p, then the asymptotic
range is reached as h is small (enough) such that the hp term dominates any
remaining higher-order terms. Further details can be read in Oberkampf [14].
2. Iterative Upwind Finite Difference Method

In this section we present an Iterative Upwind Finite Difference Method to
solve state-constrained viscosity solution to Hamilton-Jacobi-Bellman equations.
The iteration part of the method applies to the doubling of the number of discrete
xvalues in order to gain better accuracy. Adjustments from iteration to iteration are
designed to create efficiencies. In order to iterate without a trapezoidal propagation
of the spatial domain in each time step, we impose artificial boundary conditions
explained later. Firstly, we convert the state-constrained problem,
Problem 2.1.
∈ , , ϕ x t
subject to x˙ = f(x(t),u(t),t), for all t ∈ (s,tf],

x(s) = y,
where
Ω = {u(t) ∈ Rq | g(x,u,t) ≤ 0}, for all t ∈ (0,tf], tf > 0 is a constant,
(s,y) ∈ [0,tf) × Rn, x ∈ Rn, y ∈ Rn is a given point and ∶ → , ∶ →
, ∶ → and ∶ → are known functions.
194
to an unconstrained optimal control problem by incorporating linear penalty terms

in the objective function,
Problem 2.2.
subject to x˙ = f(x(t),u(t),t) for all t ∈ (s,tf],

x(s) = y,
where r = (r1(t),r2(t),...,rm(t))T is a vector satisfying ri(t) > 0 for i=1,2,...,m and ∶
T
→ is the smoothed version of the constraints g = (g1,g2,...,gm) (see Teo
[21] and [23] for more detail) defined as
0 if g < −ρ;
if − ρ ≤ g ≤ ρ;
g if g > ρ.
Remark 2.1. Note that in this case smoothing the sharp corner of the function g at
zero is necessary with the aim of applying standard optimization routines.
The corresponding HJB equation of this unconstrained problem is as follows:
)) = 0 (2)
Remark 2.2. For the convergence analysis of the linear penalty on constrained
viscosity solution of HJB equations, we refer to Bardi [1], Bokanowski [2] and
Loreti
[12].
In order to have a smooth solution as required by Richardson Extrapolation

method, we need to change equation (2) to the singularly perturbed
convectiondiffusion equation
=0 (3)
The difference between equation (2) and (3) is only the diffusion term −ε 2V , ε >
0, which represents a small perturbation parameter. As ε → 0 the solution of (3)
converges to the solution (2)(see Bardi [1]).
2.1. Discretization of HJB. To simplify notation, let us consider an optimal

control problem with one control u and one state variable x ∈ [a,b]. Extension to a
multivariable optimal control problem can be easily done with some adjustments to
notation. In addition, without loss of generality and for the intention of numerical
195
tests later, we will set s = 0, y = x0 and tf = 1.

We start with constructions of spatial discretization and time stages. We
select a positive integer M and divide the space interval [a,b] into M equal
partitions so that
.
Therefore, the discretization for space interval becomes
xi = a + (i − 1)∆x
where i = 1,...,M + 1.
In order that this spatial discretization always contains the initial point, an
appropriate shifting might be necessary. Let arg min | | and make the
adjustments
xi ← xi + (x0 − x(j)), i = 1,...,M
a ← a + (x0 − x(j))
b ← b + (x0 − x(j))
Next, we impose a limit on control u(t) where t ∈ [0,1], specifically, the lower
bound ul and the upper bound uu. This constraint is usually determined by physical
limits of the system control values.
The time interval [0,1] is then divided into N equal partitions with ∆t = −N1 so
that
tk = 1 + (k − 1)∆t, k = 1,...,N + 1
is the backward partition. This means that t1 and tN+1 correspond to t = 1 and t = 0
respectively.
With notation ) and ) for the value function and
control variable at point xi and time tk, we split the equation (3) into 2 equations and
discretize it for i = 1,...,M + 1 as follows
(4)
arg inf , , , , , ,
∆
where ) and sign denotes

the sign of f at point xi and time tk.
Using and , equation (4) can be rewritten as follows:
for i = 2,...,M. (5)

196
Because the Upwind Finite-Difference Method is an explicit method, we need

to take account of the stability of the scheme under some conditions on the step
length ∆t and ∆x.
Theorem 2.1. Under the condition
. (6)
scheme (5) is stable.
Proof. It is known from Strikwerda [19], that the scheme (5) is stable if and only if
with L+rgρ = 0 it is also stable. The scheme (5) with L+rgρ = 0 is equivalent
to
Denote the discrete maximum norm
and
then, under the condition 0 < α + β < 1, we prove that

1
≤ 1 | | | | | |
Taking the maximum for both sides with respect to i and using the induction
method, we get
1 ∞ ∞ . . . 1 ,
in other words, the scheme is stable under the condition 0 < α + β < 1.
Furthermore, in terms of N and ∆x, the above stability condition becomes
Remark 2.3. Note that the stability condition (6) is the extension of the stability
condition in Wang [24] in the case for ε 6= 0. Next, we set the initial value function
according to
197
and initial control value for i = 2,...,M + 1
u1i = argmin .
The effect of a trapezoidal propagation on the spatial domain for each time
stage can be avoided by setting up some artificial boundary conditions for control
and value function based on linear extrapolation of the closest known points. The
linear extrapolation to boundaries is chosen because it is simple to apply in
computation and the HJB equation is a first order PDE. Moreover, it gives freedom
for the edge points to flip following the line directed by the values of two closest
points. Thereby, for i = 1
.
Next, we update the value function for k = 1,...,N and i = 2,...,M according to
(5) and do extrapolations for both boundaries
2 .
2 .
Moreover, to update control we set for 1, . . . , and 2, . . . ,

1,
!
= argmin
u
and for the left boundary

.
So far we have obtained and for i = 1,...,M + 1 and k = 1,...,N + 1.

These constitute the first iteration of the method.
2.2. Finding Optimal Trajectory and Control. To iterate, we first need to
determine the optimal trajectory from the first iteration. Starting with the initial
value, we integrate forward the state equation ˙x = f(t,x,u) using the following
predictor-corrector method. Let us name the resultant trajectory and control yp and
up for predictor, yc and uc for corrector with yp(1) = yc(1) = x0 and uc(1) = u(x0,tN+1)
respectively.
The control value used during the integration is the optimal control value
corresponding to the closest grid point to the resultant state as suggested in
Tremblay [22]. Thus, for l = 2,...,N + 1 yp(l) = yc(l − 1) − ∆tf(t−l+N+3,yp(l),uc(l − 1))
up(l) = u(xi∗(l),t−l+N+3)
where i*=arg mini |yp(l) − xi|, i = 1,...,M + 1
where i∗ = arg mini |yc(l) − xi|, i = 1,...,M + 1.

The resultant pair (yc(l),uc(l)) for l = 1,...,N + 1 makes an optimal trajectory and
control for all time steps from the first iteration of the HJB. In addition, the value
function along the optimal trajectory can be determined by the value function of the
corresponding closest grid points. The penalty value and objective function value
198
can also be evaluated by forward integration along the optimal trajectory of

corresponding terms.
2.3. Region Size Reduction. Now, we determine a procedure for region reduction
based on the optimal trajectory and control from previous iteration. This new
region is applied to the next iteration in order to improve computational speed and
accuracy. What we need, first, is the maximum and the minimum value of resultant
control and trajectory. Thus, for l = 1,...,N + 1
xmax = maxyc(l)
xmin = minyc(l)
umax = maxuc(l)
umin = minuc(l)
In view of the fact that for the next iteration the number of interval partitions M
will be doubled, we set the region for the next iteration as follows:
a = xmin − c ∆x
b = xmax + c ∆x
M = 2M
and the lower and upper bound for the control
ul = buminc
uu = dumaxe
where ul and uu are consecutively the lower bound and upper bound for the control
and bzc means rounding the elements of z to the nearest integer less than or equal
to z, dze rounding the elements of z to the nearest integer greater than or equal to z
and c some given positive integer. c is used to make the region larger so as to
improve stability.
We repeat the above steps, i.e. discretization of the computation of the HJB
equation, forward integration of optimal trajectory and region reduction until
iteration has nearly reached convergence. For that purpose, we may prescribe a
lower bound for space interval shrinkage factor %x, i.e. the ratio of latter space
interval length to former. The smaller the shrinkage factor is, the larger the
reduction of the space interval length for the next iteration. The shrinkage factor
close to 1 indicates that the length of space interval for the next iteration does not
change much. By setting a lower bound for space interval shrinkage factor high, for
instance 95%, it will ensure that the asymptotic range requirement for applying
Richardson Extrapolation is satisfied. Afterwards, we run additional iteration with
Completed Richardson Extrapolation on the region reduction from the last
iteration. This improves the result accuracy from first-order to second-order.
3. Completed Richardson Extrapolation

The Completed Richardson Extrapolation proposed by Roache and Knupp in
[16] is an extension of the original Richardson Extrapolation. They completed the
method by giving higher-order solution not only on the coarse grid but on the entire
fine grid. In particular, they presented application of the extrapolation on numerical
solution of time-independent partial differential equations as examples.
Furthermore, Richard [15] modified it in order to be used on time-dependent partial
differential equation problems.
199
In short, the formulas for Completed Richardson Extrapolation are as follows.

Let ϕc,i and ϕf,j denote respectively the first order approximate solution at node i on
the coarse and j on the fine grid. The fine grid here is formed by bisecting the
coarse grid such that the fine grid coincide with the coarse grid only when the
indices are odd (j = 2i−1) where i = 1,2,...,N +1. Then the extrapolated second order
approximate solution by the Completed Richardson Extrapolation, ϕRE,j, is
determined by
ϕRE,j = 2ϕf,j − ϕc,i for j = 2i − 1 (7)
ϕRE,j+1 = ϕf,j+1 + 0.5(ϕRE,j − ϕf,j + ϕRE,j+2 − ϕf,j+2) for j + 1 even (8)
From the last iteration, we choose M as the number of coarse grid so that
Analoguous with coarse grid, we can determine ∆xf;∆tf;η1,f;η2,f using Mf = 2Mc and
Nf = 2Nc for fine grid.
Remark 3.1. At this stage, an appropriate shifting in the spatial discretization

might be necessary to include the starting point x0.
Theorem 3.1. The condition Nf = 2Nc fulfills the stability condition in (6).
Proof.
Then, we run an extra iteration as before and update the value function (VRE) and
control (uRE) for the fine grids according to (7) and (8). The resultant pair of
matrices (VRE,uRE) is the solution of HJB equation which has a second order of
accuracy.
Remark 3.2. The use of artificial boundary conditions, i.e. linear extrapolations to
the boundaries, in the algorithm does not create boundary layers. Hence,
nonuniform mesh layer-adapted meshes in our case are unnecessary. For further
information related to boundary layers, we refer to [5], [13], [17], [20] and the
200
references cited therein.
4. Numerical Experiment
To test the effectiveness of this algorithm, we take an example containing 1
state, 1 control and 1 mixed (state-control) inequality constraint from [26]. The
problem is to minimize Example 4.1.
subject to
x˙ = u x(0) = 0
g(x,u,t) = −(x + u − t2 − 1) ≤ 0
2 2
The analytic optimal solution for this problem is x∗(t) = t and u∗(t) = 1, so that
the constraint is active for all t ∈ [0,1]. The value function for this solution is
The corresponding HJB initial-value problem for this example is
The numerical simulation is done using MATLAB R2010A and MATLAB

Optimization Toolbox. We start with region −1 ≤ x ≤ 2 and −2 ≤ u ≤ 2 for the first
iteration and then reduce it progressively according to our proposed method. The
problem has been resolved for ε = 10−10 and various values of M. The first four
iterations are purely computed with Iterative Upwind Finite-Difference Method
while the last iteration for M = 256 is the result of implementation of the
Completed Richardson Extrapolation. The summary of our computations are given
in Table 1.
it. M N pen. obj. value [a,b] [ul,uu] %x

1 16 11 0.2799 - - [-1.000, [-2, 0.56
0.2472 0.1039 2.000] 2]
2 32 38 0.0429 - - [-0.375, [0, 2] 0.71
0.1792 0.1466 1.319]
3 64 107 0.0039 - - [-0.106, [0, 2] 0.89
0.1646 0.1570 1.098]
4 128 238 0.0107 - - [-0.038, [0, 2] 0.96
0.1688 0.1600 1.039]
5 256 497 0.0040 - - [-0.017, [0, 2] –
0.1659 0.1623 1.015]
Table 1. Computational result for e = r = 2,ε = 10−10.
The first, second and third column are respectively the number of iterations,
the number of spatial and time partitions. The penalty value and objective function
value along the optimal trajectory for each iteration are shown in fourth and fifth
columns. These values are evaluated by forward integration of corresponding terms
201
along the optimal trajectory whereas the value function in sixth column is the value
function at the initial point obtained from the Upwind Finite Difference Method.
The discrepancy between the objective function value and the value function in
each iteration is caused by the use of state and control values of the corresponding
closest grid points along the optimal trajectory in the evaluation of the objective
function value. However, from iteration to iteration this discrepancy becomes
smaller. This indicates that the use of the state and control values of the
corresponding closest grid points is a reasonable choice. It can be seen also that in
general the penalty and value function decrease as the number of iterations and M
increase. Additional information related to the space and control interval used
during the iteration are in seventh and eighth column. Last column contains the
space interval shrinkage factor.
Tables 1, 2 and 3 indicate that the computed optimal control and state
converge to the analytic solution as the error decreases significantly.
M
Error 16 32 64 128 256

‖. ‖ 0.1406 0.0296 0.0059 0.0068 0.0015
‖. ‖ 0.2346 0.0776 0.0332 0.0420 0.0194
Table 2. Computed error for u in the maximum and L2 norm.
Error 16 32 64 128 256

‖. ‖ 0.0562 0.0083 0.0013 0.0016 0.0004
‖. ‖ 0.1049 0.0250 0.0069 0.0094 0.0072
Table 3. Computed error for x in the maximum and L2 norm.
The computational results for the last iteration are plotted in the following
figures. It can be seen that the value function and control shown in Figures 1 and 3
are smooth in the solution domain. This shows the success of the linear
extrapolation used.
In this article we present the Iterative Upwind Finite-Difference Method for
the approximation of constrained viscosity solutions to Hamilton-Jacobi-Bellman
Equations. As has been seen from the example, this method is very effective not
only to improve the accuracy but also reduce computational time compared to the
available Upwind Finite-Difference Method. In this method the trapezoidal
propagation does not occur so that it reduces the number of computed grid points.
Hence, the computational time is reduced.
202
FIGURE 1. Value function
FIGURE 2. Value function
value function
0.6
0.5
0.4
0.3
V 0.2
0.1
−0.1
−0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
along optimal trajectory
203
Figure 3. Optimal control
Figure 4. Optimal control along optimal trajectory

optimal control
1.002
computed optimal control u
analytical optimal control u
1.0015
1.001
1.0005
0.9995
0.999
0.9985
0 0.2 0.4 0.6 0.8 1
t
204
Figure 5. Optimal state along optimal trajectory

optimal state
1.4
computed optimal state x
analytical optimal state x
1.2
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
t
Figure 6. Constraint along optimal trajectory

−3
x 10 constraints
2
g
−1
−2
−3
−4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
Figure 7. Penalty along optimal trajectory for Example 4.1

−3 penalty term
x 10
8
P 5
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
205
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Biomathematics
STABILITY ANALYZE OF EQUILIBRIUM

POINTS OF DELAYED SEIR MODEL WITH
VITAL DYNAMICS
RUBONO SETIAWAN
Mathematics Education
Teacher Training and Education Sciences Faculty (F.KIP)
Sebelas Maret University (UNS)
Ir. Sutami Street No. 36A, Kentingan, Surakarta City, Indonesia
email : rubono.matematika@gmail.com
Abstract An SEIR (Susceptible Exposed Invected Recovered) epidemic model

with vital dynamics, incubation time length, and constant recruitment is
formulated. We take incubation time length as discrete time delay parameter into
bilinear incidence term of our model. With mathematical algebraic
manipulation, we derive the basic reproduction number as threshold value to
determine whether the disease dies out found. Then, it also determine the
existence and stability of our two kind of equilibrium point, that is disease and
disease free equilibrium. We also analyze the effect of our time delay as latent
delay parameter to the stability of those two equilibrium point and determine
whether this time delay lead to Hopf bifurcation of not.
Key words and Phrases: SEIR Model, Delay Time, Vital Dynamics, Constant
Recruitment
1. Introduction
In SEIR epidemiological models with vital dynamics, class of population is
divided into four subclasses that is susceptible, exposed, infectious and removed
classes. where S denotes the number of individuals that are susceptible to
infection,E denotes the number of exposed individual, I denotes the number of
individuals that are infectious, and R denotes the number of individuals that have
been removed with immunity. This model use the standard bilinear incidence SI (
where   0 is the average number of contacts per infective per unit time) rate.
For example about complete assumption of this model is given by Cappasso 1 .
We assume the latent period as discrete time delay parameter is constant,
denoted by  and we take it into bilinear incidence term S t I t    . The number
of the susceptible individuals that become exposed at time t    . Zhang, et.al
has investigated the dynamics of SIR epidemic model with nonlinear incidence.
207
208
Setiawan in [2] has investigated the dynamics of SIR epidemic model with vital
dynamic and also use the bilinear incidence term S t I t    . Yan and Liu, 2006
in 3 investigated a class of SEIR epidemiological models under assumption that
there is a probability value that individuals survives the latent period t    is
e   , where  is per capita death rate due to causes other than the disease. The
different of model in 3 with model in this paper is assumption that all of the
individuals will be survives in the latent period and becoming infective at time t.
We arrange our paper as follows. In Section 2, we give the model
formulation of our problem and its initial condition. By mathematical analysis, we
prove that the existence of our equilibrium are completely determined by the values
of the threshold value R0 . The stability of disease free equilibrium is also
determined by R0 and we can conclude that R0  1 implies that E0 is always
locallyasymptotically stable stable, and there is no effect of presence of time delay
to the stability of disease free equilibrium, in other words there is no bifurcation of
E0 . Finally we also study the stability of endemic equilibrium.
2. Main Results
2.1. Mathematical Formulation
We consider the common SEIR model with bilinear incidence or we can say
SEIR model with vital dynamics. The detail of those model can be found in many
references by many authors, as example by Capasso, in 1 . We assume the latent
period as discrete time delay parameter is constant, denoted by  and we take it
into bilinear incidence term S t I t    .The number of the susceptible
individuals that become exposed at time t   . We assume that all of the
individual will be survive in the latent period and becoming infective at time t.
Finally, we can get SEIR epidemiological model with vital dynamic and discrete
time delay as follows :

S t      .S (t )  S t I t   

E t   S t I t     E (t )  E (t ) 2.1

I t   E (t )  I t    .I (t )

Rt    .I t    .R (t )
N (t )  S (t )  E (t )  I (t )  R (t ),
where  ,  ,  ,  ,  are real constants parameter, where  is the parameter that

measures the natural mortality rate,  is average number of contacts per infective
1
per unit time,  is latent delay time, is average latent period and  is the

natural recovery rate that of the infective individuals.
The initial conditions    1 ,  2 ,  3 ,  4  of (2.1) are defined in the
Banach space,
209

C    C(  ,0, R 4 ) : 1    S  ,2    E ,3  I ( ),4  R( ),    ,0 
2.2
 
where R 3  S , E , I , R   R 4 : S  0, E  0, I  0, R  0 and by a biological
meaning, we assume that i  0 , i = 1,2,3. Because total population size N(t) is
constant, then for convenience, we may assume that N(t)= S(t)+E(t)+I(t)= 1.
2.2 Equilibrium States and The Basic Reproduction Number

In this section we study the existence of the equilibrium points of system
(2.1) by the following theorem
Theorem 2.1.System (2.1) always disease free equilibrium Eˆ 0  1,0,0,0  . Let


R0  .We can get the other equilibrium based on the value of R0 .
      
(1) If R0  1 there is one additional equilibrium point, that is the other disease
        
free equilibrium E0   ,0,0,0 
  
(2) If R0  1 there is one additional equilibrium point, that is disease equilibrium,

 
E1  S * , E * , I * , R * , where
S* 
       ,     1  1  ,  
1 
1 
,
E* I* 
         R0          R0 
  1 
R*  1  
        R0 
PROOF.
Equilibrium points is the states of a system that are satisfied the following
  
conditions S t   0 , I t   0 and Rt   0 . From system (2.1) we get
   .S (t )  S t I t     0
S t I t     E (t )  E (t )  0 2.3
E (t )  I t    .I (t )  0
 .I t    .R(t )  0
It’s easy to prove that from equation (2.3), we can get the first disease free
equilibrium Eˆ 0  1,0,0,0  . Based on third and fourth equation of (2.3) we get
R* 
 *
I and E * 
    I *
 
Then, if we substitute E * to first equation of (2.3) we also get
210
S* 
       I *

* * *
Finally, if we substitute S , E and R back to first equation of (2.3) and then by
algebraic manipulation we get
  1 
I*  1  
        Ro 

where R0  . If R0  1 then we have an endemic equilibrium
      
 
E1  S * , E * , I * , R * with
On the otherhand, if R0  1 then we have an additional disease free equilibrium
Q.E.D.
2.3 Linearized System

 
Let Eˆ  Sˆ , Eˆ , Iˆ, Rˆ be any equilibrium of system (2.1). Then, we will find
linearized system of system (2.1) at Eˆ  Sˆ , Eˆ , Iˆ, Rˆ   by linearizing about
 
Eˆ  Sˆ , Eˆ , Iˆ, Rˆ with Taylor Series Method. We can write system (2.1) in matrix
form as follows :
2.4
let
,
211
, 2.5
Then, we will determine Taylor series of

, and
at as follows
Then, by same method we can get

212
analogue,
and So, we get
linearized system of system ( 2.1)as follows :
2.6
Finally, we can find the roots of system (2.6) by solving the following equation
2.7 
2.4 Stability of Disease Free Equilibrium

In this section, we study the local stability of disease free equilibrium of system
(2.1) and the effect of time delay to the stability of disease free equilibrium.If we
evaluate equation (2.7) at E0 then we have the following equation
213
2.8
one of root of equation (2.8) is always    which is independent of any
parameters. We can get the others roots of equation (2.8) from the following
equation
For τ = 0, we get
Based from the above equation, we can conclude that τ = 0 then all of the value of
roots of equation (2.8) are negative, therefore E0 is asymptotically stable. In this
paper we do not analyze E0 for τ ≠ 0 and we left this problem to further research.
2.5 Stability of DiseaseEquilibrium

In this section, we study the local stability of disease equilibrium E1 . If we
evaluate equation (2.7) at E0 then we have the following equation
   1  
   1                    2       e   0

         R0  
2.9
Let A     and B     , then we rewrite equation 2.9  into
  1  
       1       A  B     ABe   0 2.10
 
 AB  Ro   
or equivalent with
3  C  1    1   1 
 2  1  2  1  2C1    C1       e ;  0
AB  AB  R0    R0   R0 
2.11

where C  1 
AB
Theorem 2.2.
2C
Let Rc  , which is Rc  1 . If R0  Rc , then E1 is
2C  1
locallyasymptotically stable stable for   0
214
PROOF.
Based on equation 2.11 for   0 , we will get the following equation
3  C  1  2   1   1 
 2  1     2C 1     C 1      0 2.12
AB  AB  R0    R 0   R 0 
If we use Routh-Hurwitz stability criterion, especially for a fourth-order

polynomial, to determine the sign of all the roots of equation (2.12). Let
It’s easy to prove that equation (2.12) is satisfy a n  0, n  0,1,2,3,4 and also
a 2 a1  a 3 a 0 , so all the roots of equation (2.12) have negative sign. Q.E.D.
Then, we will study the effect of time delay to the stability of disease equilibrium
(case   0 ). Without loss of generality, if   i ,   R then, if we substitute to
the equation (2.9) we get
2.13
By separating the real dan imaginary parts, and then squaring and adding those
both equations, then we have the following equation :
2.14
It’s difficult to get direct conditions to get real solutions of equation (2.14) and
we left this problem to further research.
From system (2.1) we get two equilibrium, that is disease free equilibrium
and endemic equilibrium. If R0  1 implies that E0 is always locallyasymptotically
stable stable, and there is no effect of presence of time delay to the stability of
disease free equilibrium, in other words there is no bifurcation of E0 . Then, if
215
2C
Rc  , which is Rc  1 , E1 is locallyasymptotically stable stable for
2C  1
  0 if and only if R0  Rc .
References
[1] Cappaso, V,1993,Mathematical Structures of Epidemic Systems ,Vol 97 Lecture-

Notes in Biomathematics, Springer – Verlag Berlin – Heidelberg.
[2] Setiawan, R.,2009, Stability of Delayed S I R Model With Vital Dynamics, Proc.
IndoMS International Conference on Mathematics and Its Application (IICMA)(
2009), 471-478.
[3] Yan, P. and Liu, S.,2006, SEIR Epidemic Model with Delay, ANZIAM Journal 48
(2006), pp. 119-134.
[4] Zhang, J Z, Jin, Z, Liu,QX, Zhang, ZY, Analysis of a Delayed SIR Model with
Nonlinear Incidence, Discrete Dynamics in Nature and Society, Vol.2008, Article
ID 636153(2008), Hindawi Publishing Company.
Graph and Combinatorics
THE TOTAL VERTEX IRREGULARITY STRENGTH OF

A CANONICAL DECOMPOSABLE GRAPH,
G = S(A,B)◦ tK1
D. FITRIANI1, A.N.M. SALMAN2
1,2Combinatorial Mathematics Research Group

Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung
Jl. Ganesa 10 Bandung 40132 Indonesia
dinnyfitriani@students.itb.ac.id, msalman@math.itb.ac.id
Abstract. A vertex-irregular total k-labeling λ : V (G) ∪ E(G) → {1,2,··· ,k} of a

graph G is a labeling of the vertices and the edges of G in such a way that for
any different vertices v and w, their weights wt(v) and wt(w) are distinct. The
weight of a vertex v is the sum of the label of v and the labels of all edges
incident with v. The total vertex irregularity strength of G, denoted by tvs(G), is
defined as the minimum k for which a graph G has a vertex-irregular total k-
labeling. In this paper, we consider a canonical decomposable graph. A
canonical decomposable graph G = S(A,B) ◦ tK1 is a graph formed by taking the
disjoint union of two graphs, the first is a split graph S whose an independent set
A and a clique B, and the second is t copies of K1, and adding to it all edges
having an endpoint in B and the other endpoint in V (tK1). In this paper we
determine the total vertex irregularity strength of G.
Keywords and Phrases: canonical decomposable graph, total vertex irregularity

strength, vertex-irregular total k-labeling
1. Introduction
All graphs in this paper are simple, finite and undirected. Baˇca et al. [2]
introduced a vertex-irregular total k-labeling λ : V (G)∪ E(G) → {1,2,··· ,k} of a
graph G as a labeling of the vertices and edges of G such that for every pair distinct
vertices v and w, wt(v) = λ(v) + P λ(vx) =6 λ(w) + P λ(wy) = wt(w).vx∈E(G) wy∈E(G).
The total vertex irregularity strength of G, denoted by tvs(G), is the minimum
positive integer k for which G has a vertex-irregular total k-labeling.
Baˇca et al. [2] have determined the total vertex irregularity strength for
some classes of graphs, namely cycles, stars, and prisms. Whereas the total vertex
irregularity strength of trees and disjoint union of t copies of path had been
determined by Nurdin et al. in [3] and [5]. Furthermore, Nurdin et al. [4]
determined the total vertex irregularity strength for several types of trees
containing vertices of degree 2, namely a subdivision of a star and a subdivision of
216
217
a particular caterpillar. They also derived the total vertex irregularity strength for a
complete k − ary tree. Wijaya et al. [8] determined the total vertex irregularity
strength of complete bipartite graphs. In [6], Tong et al. determined the total vertex
irregularity strength of toroidal grid . Besides that, Al-Mushayt et al. [1]
determined the total vertex irregularity strength for convex polytope graphs. In this
paper, we determine the total vertex irregularity strength of a canonical
decomposable graph, G = S(A,B) ◦ tK1.
2. Main Results
A clique in a graph is a set of pairwise adjacent vertices; an independent set
is a set of pairwise non-adjacent vertices. A split graph S(A,B) is a graph whose
vertex set can be partioned into an independent set A and a clique B.
Let a,b in V (G). The graph G + ab is a graph obtained from G by adding a
new edge ab. Let Σ and Γ denote the sets of split graphs and of simple graphs,
respectively. Define the composition ◦ : Σ × Γ → Γ as follows:
if σ ∈ Σ, σ = S(A,B), H ∈ Γ then
σ ◦ H = (S(A,B) ∪ H) + {bc : b ∈B,c ∈V (H)}.
(The edge set of the complete bipartite graph with parts B and V (H) is added to the
disjoint union S(A,B) ∪ H). A graph G is called decomposable, if G = σ ◦ H for
some σ ∈ Σ and for some H ∈ Γ. Otherwise, G is indecomposable [7].
In this paper, we consider t copies of K1 as H.
Figure 1. G = S(A,B) ◦ tK1 where deg(a) = k with 1 ≤ k ≤ n − 1
We use the concept of multisets. Define a multiset as

a number family containing ki numbers of ai for i ∈ {1,2,...,n}, n ∈N. Let X,Y be
two multisets, then the multiset X UY = {α : α ∈X;α ∈Y }. Notice that if α appears r
times in X and s times in Y, r,s ∈ {0,1,2,...}, then α appears r+s times in X UY.
Let X be a multiset containing positive integer. X is said to be anti balanced if
218
there exist submultisets X1,X2,...,Xk for some k ∈N such that |Xi| = |Xj|,
K
ΣXi 6= ΣXj, i 6= j, i,j ∈ {1,2,...,k}, and U Xi = X.

i=1
Lemma 2.1. Let n,s, and t be integers at least 2, z be an integer at least 1, and k ∈
{1,2,...,n − 1}. Let m and z + o be two integers at most
;
and ;
and ,
and two multisets
k+1
X = ] {1k1,2k2,...,mkm}
i=1
k+n+2
Y = ] {zk1,(z + 1)k2,...,(z + o)ko+1}.

j=k+2
Then there exist submultisets X1,X2,...,Xs and Ys+1,Ys+2,...,Ys+t such that

• for each i ∈ {1,2,...,s}, |Xi| = k + 1;
• for each j ∈ {s + 1,s + 2,...,s + t}, |Yj| = n + 1;
P
• Xk 6= PXl, k 6= l, k,l ∈ {1,2,...,s}, PYp 6= PYq, p 6= q, p,q ∈ {s + 1,s + 2,...,s
+ t}; and
So X and Y is anti-balanced.
Proof. For each i ∈ {1,2,...,s} define multiset Xi = {aib(1)(i),aib(2)(i),...,aib(k)(i) ,ai}

where for 1 ≤ l ≤ k :
if 1 ≤ i ≤ l;
i (l)(i) =
m,if (m − 2)k + m + l − 1 ≤ i ≤ (m − 1)k + m
+l−1
and
if 1 ≤ i ≤ k + 1; i = m, if (m −
1)k + m ≤ i ≤ mk + m.
For each j ∈ {s + 1,s + 2,...,s + t} define multiset Yj = {cjb10,cjb20,...,cjbn0

0
,ci} where for 1 ≤ u ≤ n and n ≥ k + s :
219
0
if 1 ≤ j ≤ u ;
0 0 if (m − 2)n + m + u − 1 ≤ j ≤ (m − 1)n +
m+u−1
and
if 1 ≤ j ≤ n + 1; if (m − 1)n + m ≤
j ≤ mn + m.
0 0
For n < k+s, let k+s−n = p +z(n+1)−y−x where p ∈ {1,2,...,n+1}, z ≥
1, y ∈ {0,1,...,n + 1}, x ∈ {1,2,...,n} and
z, if 1 ≤ j − s ≤ x;
z + 1, if x + 1 ≤ j − s ≤ x + y; cjbp0 = z + 2,
if x + y + 1 ≤ j − s ≤ x + y + n + 1;
z + o, if x + y + (o − 2)n + o − 1 ≤ j − s ≤ x + y + (o − 1)n + o − 1
where cjbn+10 defined as cj.

Next, define
Al ={aib(l)(i)|1 ≤ i ≤ s}
Ak+1 ={ai|1 ≤ i ≤ s}
Ak+u0+1 ={cjbu0|1 ≤ j ≤ t} Ak+n+2
={cj|1 ≤ j ≤ t}.
k+1 k+n+2
0 where 1 ≤ l ≤ k and 1 ≤ u ≤ n. We have U Ai = X, U
Ai = Y, and
i=1 i=k+2
s s+t
U
Xi = X, U Yi = Y. Besides that, for each i ∈ {1,2,...,s}, |Yi| = k + 1 and
i=1 i=s+1 for each j ∈ {s + 1,s + 2,...,s + t},
|Yj| = n + 1.
Case 1: n ≥ k + s
Consider ai and ai+1 where i ∈ {1,2,...,s − 1}. Let aib(l)(i) = m, then
(m − 2)k + m + l − 1 ≤ i ≤ (m − 1)k + m + l − 1
(m − 2)k + m + l ≤ i + 1 ≤ (m − 1)k + m + l
(m − 2)k + m + (l + 1) − 1 ≤ i + 1 ≤ (m − 1)k + m + (l + 1) − 1.
So, aib(l)(i) = m = ai+1b(l+1)(i+1) and
ΣXi+1 = ai+1 + ai+1b(k)(i+1) + ... + ai+1b(1)(i+1)
= aib(k)(i) + aib(k−1)(i) + ... + aib(1)(i)) + ai+1b(1)(i+1)
= ΣXi − ai + ai+1b(1)(i+1). (1)
Let ai = m, then
220
(m − 1)k + m ≤ i ≤ mk + m
((m + 1) − 2)k + m ≤ i ≤ ((m + 1) − 1)k + m
((m + 1) − 2)k + m + 1 ≤ i + 1 ≤ ((m + 1) − 1)k + m + 1.
So ai+1b(1)(i+1) = m + 1. From (2) we have
ΣXi+1 = ΣXi + 1.
By using a similar way, we have ΣYj+1 = ΣYj +1, j ∈ {s+1,s+2,...,s+ t − 1}.
Hence, if n ≥ k + s, we obtained ΣXk 6= ΣXl for k 6= l, k,l ∈ {1,2,...,s} and
ΣYp 6= ΣYq for p 6= q, p,q ∈ {s + 1,s + 2,...,s + t}.
Case 2: n < k + s
Consider cj and cj+1 where j ∈ {s+1,s+2,...,s+t−1}. Let cjbu0 = z+o, then
x + y + (o − 2)n + o − 1 ≤ j ≤ x + y + (o − 1)n + o − 1 x +
y + (o − 2)n + o ≤ j + 1 ≤ x + y + (o − 1)n + o
0
Since x + y = u + z(n + 1) − k − s,
0 x + y + (o − 2)n + o = (u + 1) + z(n + 1) −
k − s + (o − 2)n + o − 1.
So, cjbu0 = z + o = cj+1bu+10 and
ΣYj+1 = cj+1 + cj+1bn0 + ... + cj+1b10
= cjbn0 + cjbn−10 + ... + cjb10 + cj+1b10
= ΣYj − cj + cj+1b10. (2)
Let cj = z + o, then x + y + (o − 2)n + o − 1 ≤ j ≤ x + y + (o − 1)n

+o−1
x + y + (o − 2)n + o ≤ j + 1 ≤ x + y + (o − 1)n + o.
Since cj = cjbn+10,
x + y + (o − 2)n + o = n + 1 + z(n + 1) − k − s + (o − 2)n + o
= 1 + z(n + 1) − k − s + ((o + 1) − 2)n + (o + 1) − 1.
So cj+1b10 = z + o + 1. From (2) we have
ΣYj+1 = ΣYj + 1.
Hence, if n < k + s, we obtained ΣXk 6= ΣXl for k 6= l, k,l ∈ {1,2,...,s} and
ΣYp 6= ΣYq for p 6= q, p,q ∈ {s + 1,s + 2,...,s + t}.
So X and Y are anti-balanced.
Theorem 2.2. Let n,s, and t be integers at least 2 and k ∈ {1,2,...,n − 1}. Let G =
S(A,B) ◦ H with H = tK1, |A| = s, and |B| =
n. If deg(a) = k for all a ∈A, then
;
and ;
and .
Proof. Let φ be an optimal total k-labeling with respect to the tvs(G). To get the
221
smallest k such that the weight of every vertices v ∈V (G) are distinct then the
weight of the smallest sequence should started from vertices a ∈A whose degree
are k, 1 ≤ k ≤ n−1 with all edges joining that vertices to the vertices b ∈B. Next,
give label for the vertices c ∈V (H) whose degree are n with all edges joining that
vertices to the vertices b ∈B.
• Since the degree of every vertex in A is k, the smallest weight of all
vertices in A is at least k + 1 and the largest weight of all vertices in A is
at least k + s.
• Since the degree of every vertex in C is n, the smallest weight of all
vertices in H is at least
;
and the largest weight of all vertices in
H is at least
; s.
Thereby, the largest label of G is at least
;
where
;
and
;
+1)
.
Therefore,
;
and ;
and
.
Next, we will show that
;
and ;
and
.
For each vertex ai, i ∈ {1,2,...,s}, define {b(1)(i),b(2)(i),...,b(k)(i)} as its neighbor set
and indexing them based on index in bj ∈B, j ∈ {1,2,...,n}. Give label edges
aib(l)(i), l ∈ {1,2,...,k}, and vertices ai:
222
+m+−1 i
(m − 1) + m + − 1,
and wt(ap) = λ(ap) + λ(apb(k)(p)) + λ(apb(k−1)(p)) + ... + λ(apb(1)(p)).

For vertices in B, define f(bj) to be the sum of the labels of all edges joining
bj to the vertex ai ∈A. Denote the vertices of B as b10,b20,...,bn0, indexing them in
non increasing order their values under f. Next, give label the vertices in H and
edges cobj0, o ∈ {s+1,s+2,...,s+t}, by using the similar way as label that given for
the vertices in A and edges aib(l)(i) with the next smallest number. Then give label
vertices bj0 ∈B with all edges joining that vertices to bq0 ∈B \{bj0} by using the
similar way as label that given to vertices A dan V (H) with all edges incident
with that vertices. Since f(bn0) is the smallest one, we give label vertices bj0 ∈B
start from bn0.
From Lemma 2.1 we have
• for each ai,ai+1 ∈A, wt(ai+1) = wt(ai) + 1; • for each
co,co+1 ∈V (H), wt(co+1) = wt(co) + 1; and
• wt(ai) < wt(co).
Let bi0 and bj0 be vertices of B with i < j; by definition, f(bi0) ≥ f(bj0). Since
deg(co) = n < n−1+t ≤ deg(bn0) and the way to give label of each vertex in B is
similar as in A and V (H) then we obtained wt(ct) < wt(bn0) and wt(bj0) < wt(bi0).
Furthermore, we have wt(ai) < wt(co) < wt(bj0).
Therefore,
;
and ;
and .
Hence,
;
and ;
and .
223
Figure 2. A vertex-irregular total 2-labeling for G = S(A,B) ◦
6K1 with |A| = 3, |B| = 5 and deg(a) = 2 for all a ∈A

From Figure 2 above we can see that two multisets X = {16,23} and Y =
{121,215} are anti-balanced since we have following multisets from X and Y :
• X1 = {1,1,1}; • X2 =
{1,1,2};
• X3 = {1,2,2};
• Y4 = {1,1,1,1,1,1}; •
Y5 = {1,1,1,1,1,2}; •
Y6 = {1,1,1,1,2,2}; •
Y7 = {1,1,1,2,2,2}; •
Y8 = {1,1,2,2,2,2}; •
Y9 = {1,2,2,2,2,2}.
References
[1] O. Al-Mushayt, A. Arshad, M.K. Siddiqui, Total vertex irregularity strength of

convex polytope graphs, Acta Math. Univ. Cominianae 82 (2013) 29-37.
[2] M. Baˇca, S. Jendro´l, M. Miller, J. Ryan, On irregular total labellings, Discrete
Math., 307 (2007) 1378-1388.
[3] Nurdin, E.T. Baskoro, A.N.M. Salman, N.N. Gaos, On the total vertex-irregularity
strength of trees, Discrete Math. 310 (2010) 3043-3048.
[4] Nurdin, E.T. Baskoro, A.N.M. Salman, N.N. Gaos, On total vertex-irregular
labellings for several types of trees, Util. Math. 83 (2010), 277-290.
[5] Nurdin, E.T. Baskoro, A.N.M. Salman, N.N. Gaos, On the total vertex-irregular
strength of a disjoint union of t copies of a path, J. Combin. Math. Combin. Comput.
71 (2009) 227-233.
224
[6] C. Tong, X. Lin, Y. Yang, L. Wang, Irregular total labellings of , Util. Math.
81 (2010) 3-13.
[7] R. Tyshkevich, Decomposition of graphical sequences and unigraphs, Discrete
Math., 220 (2000) 201-238.
[8] K. Wijaya, Slamin, Surahmat, S. Jendro´l, Total vertex-irregular labellings of
complete bipartite graphs, J. Combin. Math. Combin. Comput. 55 (2005) 129-136.
THE ODD HARMONIOUS LABELING OF kCn-

SNAKE GRAPHS FOR SPESIFIC VALUES OF , THAT
IS, FOR AND
FITRI ALYANI, FERY FIRMANSAH, WED GIYARTI,KIKI A.SUGENG
Magister Program, Department of Mathematics

University of Indonesia, Depok 16424, Indonesia
Email: fitri.alyani@sci.ui.ac.id, fery.firmasah@sci.ui.ac.id
wed.giyarti@sci.ui.ac.id, kiki@sci.ui.ac.id
Abstract. A graph , is called , -graph if it has vertices and

edges. A , -graph is said to be an odd harmonious if there exists an
injection : → 0, 2 1 such that the induced ∗ is a
bijection from onto 1, 2 1 1,3,5, … ,2 1 and is said to be an odd
harmonious labeling of . A graph is called kC -snake graphs with k 1 is a
connected graph with k blocks whose block-cutpoint graph is a path and each of
the k blocks is isomorphic to C .In this paper, we constructthe odd harmonious
labeling on kCn-snake graph for several values of n. Moreover, we also give
odd harmonious labeling construction for a class of graph which is variation of
kCn-snake graph.
Key words and Phrases: Odd harmonious labeling,kCn-snake graph, Snake graph.
1. Introduction
In this paper we consider simple, finite and undirected graph. A graph
, is called , -graph if it has vertices and edges. There are
several types of graph labeling, some of them have been motivated by practical
problems, but most of its come from the coriousities of the beautiful of the art of
graph labelings. We refer to Gallian [1] for a dynamic survey of various graph
labeling problems along with extensive bibliography. Let be the ring of integer,
and be the residue ring of integers modulo , the simbol , is defined by
| ∈ , , and , is defined by
| ∈ , , ≡ mod . If is a set, denotes the set
| ∈ [3]. Graham and Sloane [2] intoduced harmonious labeling and
defined as follows :
Definition 1.1 [2] A , -graph is said to be harmonious if there exists an

injection : → such that the induced mapping ∗
225
226
is a bijection from onto and is said to be harmonious

labeling of .
Liang and Bai [3] intoduced odd harmonious labeling and defined as follows :
Definition 1.2 [3] A , -graph is said to be odd harmonious if there exists an

injection : → 0, 2 1 , such that the induced mapping ∗
is a bijection from onto 1, 2 1 1,3,5, … ,2 1 and is said to
be a odd harmonious labeling of .
Liang and Bai [3] have obtained the necessary conditions for the existence of
odd harmonious labeling of graph. Futhermore, they proved if is an odd
harmonious graph, then is a bipartite graph and if a , -graph is odd
harmonious, then is the number of vertices is bound by 2 1. The
maximal label of all vertices in an odd harmonious graph is at most 2 ,
where is the minimum degree of the vertices of . Let , ,…, be a
degree sequence of , -graph , if the graph is odd harmonious, then the
equation ∑ has non-negative integer solutions , ,…,
satisfying if and 2 for ∈ 1, . Let , ,…, be
a degree sequence of , -graph then the necessary conditions for a graph to
be odd harmonious is , ,…, | . If all trees are odd harmonious, then
the equation ∑ 1 has distinct non-negative integer solutions for
any integer p satisfying ∑ 2 1 and 0, ∈ 1, .
In the same paper Liang and Bai [3] proved that cycle is odd harmonious
if and only if ≡ 0 mod 4 , a complete graph is odd harmonious if and only if
2, a complete k-partite graph , ,…, is odd harmonious if and only
if 2, a windmill graph is odd harmonious if and only if 2. They also
proved that the graph ⋁ , the graph , ,…, , the graph
, the graph ∪ ⋃ , some trees and the product graph
etc are odd harmonious graph.
Vaidya and Shah [6] proved that the shadow graphs of path and star ,
are odd harmonious. Futhermore, they proved that the split graphs of path and
star , admited odd harmonious labeling.
Saputri [5] proved that ladder graphs for 2, dumbbell graphs , ,
for ≡ 0 mod 4 , 4, palm graphs , for ≡ 0 mod 4 , 4,
3, 1, generalized prism graphs for ≡ 0 mod 4 , 2 and sun
graphs ⊚ , ≡ 0 mod 4 , 2 admited odd harmonious labeling.
Rismayanti [4] proved that corona graphs ⊚ for ≡ 0 mod 4 ,
2, sun graphs ⊚ for ≡ 0 mod 4 , hairy cycle graphs ; for ≡
0 mod 4 , 1 1, 2, 1, cycle shadow graphs for ≡
0 mod 4 and generalized of cycle shadow graphs for ≡ 0 mod 4 ,
3 admited odd harmonious labeling.
2. Main Results
Definition 2.1 [1] -snake graphs with 1 is a connected graph with
blocks whose block-cutpoint graph is a path and each of the blocks is
isomorphic to .
227
The vertex notation and construction of -snake graphs is shown in Figure 2.

u11 u12 ui1 u1k
v0 v1 v2 vi vk
u12 u22 ui2 uk2
Figure 1 : -snake graphs
Definition 2.2 [1] -snake graphs with 1 is a connected graph with

blocks whose block-cutpoint graph is a path and each of the blocks is isomorphic
to .(Gallian,2012).
The vertex notation and construction of -snake graphs is shown in Figure 3.

w 1k
1 1
w w
1 2
u 12 k 1 u12k
1 1 1
u1 u 1
2 u 3 u 4
v0 v1 vk1 vk
u12 u 22 u 32 u42
2
u 22k 1 u 2k
w12 w22
w k2
Figure 2 : -snake graphs
,
Definition 2.3[1]Bracelet of graph is a connected graph which consists of k
block whose block-cutpoint graph is a path and each of the blocks is isomorphic
,
to .
,
The vertex notation and construction of Bracelet of C graph is shown Figure
3.
,
Figure 3 : Bracelet of graphs
Theorem 2.1 -snake graphs with 1 is an odd harmonious graph.

228
Proof. Let be the graph with 1 . The vertex set and edge set of are
defined as follows ∪ 1 , 1,2 ∪
|1 and
0 1, 1,2 ∪ 1 , 1,2
then | | 3 1 and | | 4
Define the vertex labels : → 0,1,2, … ,8 1 as follows
0
4 2 5, 1 , 1,2
4 ,1
The labeling will induced mapping define ∗ : → 1,3,5,7, … ,8 1
which is defined by ∗ . Thus we have the edge labels as
follows
∗
8 2 1, 0 1,
∗
1,2 8 2 5, 1 , 1,2
It is not difficults to show that the mapping is an injective mapping and the
mapping ∗ admits a bijective labeling. Hence -snake graphs with 1 is an
odd harmonious graph.
Since the labeling for odd and even are different. In Figure 6 and 7, we
give examples of odd harmonious labeling for 3 and 4 -snake graphs.
Theorem 2.2 -snake graphs with 1 is an odd harmonious graph.
Proof. Let G be the graph with 1. The vertex set and edge set of are
defined as follows ∪ 1 2 , 1,2 ∪
1 , 1,2 ∪ |1 and
0 1, 1,2 ∪ 1 , 1,2 ∪
1 , 1,2 ∪ |1 , 1,2
then | | 16 1 and | | 8
Define the vertex labels : → 0,1,2, … ,8 1 as follows
0
4 2 5, 1 2 , 1,2
8 4 2, 1 , 1,2
8 , 1
The labeling will induced mapping define ∗ : → 1,3,5,7, … ,8 1
which is defined by ∗ . Thus we have the edge labels as
follows
∗
16 2 1, 0 1,
1,2
∗
16 2 5, 1 , 1,2
∗
16 2 3, 1 , 1,2
∗
16 2 7, 1 1,2 , 1,2
mapping ∗ admits a bijective labeling. Hence -snake graphs with 1 is an
229
odd harmonious graph.
,
Theorem 2.3bracelet of graphs with 1 is an odd harmonious graph.
,
Proof. Let G be the bracelet of C graphs with 1. The vertex set and edge
, ,
set of bracelet of C graphsare defined as follows C |0
2 , ∪ |1 2 1, ∪ 1,2,3, … , , 1,2 and
,
C | ,0 2
∪ ,1 2 , 1,2
∪ ,1 2 1, 1,2
, ,
then C 5 1 and C 5
,
Define the vertex labels : C → 0,1,2, … ,10 1 as follows
, 0 2 ,
, 1 2 1,
5 2 5 , 1,2,3, … , j=1,2
,
The labeling will induced mapping define ∗ : → 1,3,5,7, … ,10
∗
1 which is defined by . Thus we have the edge labels as
follows
∗
5 1, ,0 2
∗
5 2 5, ,0 2 ,
∗
1,2 5 2 4, ,1 2
1 , 1,2
,
mapping ∗ admits a bijective labeling. Hence bracelet of C graphswith
1 is an odd harmonious graph.
Example 2.4 Odd harmonious labeling for 4 -snake graph is shown in Figure 5.
Figure 4 : 4 -snake graphs and its odd harmonious labeling
Example 2.5 Odd harmonious labeling for 3 -snake graph is shown in Figure 6
230
Figure 5 : 3 -snake graphs and its odd harmonious labeling
,
Example 2.6 Odd harmonious labeling for bracelet of C graphsis shown in
Figure 6
,
Figure 6 : bracelet of C graphsand its odd harmonious labeling
Acknowledgements. The authors are thankful to the anonymous referee for
valuable comments and kind suggestions.
References
[1] Gallian, J. A., 2012, Dynamic Survey of Graph Labeling,Electronic Journal of

Combinatorics, 26.
[2] Graham, R. L., Sloane, N. J. A., 1980, On Additive Bases and Harmonious Graphs,
SIAM J.Algebra. Disc.Math., Vol 1, No 4, 382-404.
[3] Liang, Z., Bai, Z., 2009, On The Odd Harmonious Graphs with Applications,J.
Appl. Math. Comput.,29, 105-116.
[4] Rismayati., 2012, Pelabelan Harmonis Ganjil pada Graf Hairy Cycle, Graf Shadow
Lingkaran dan Graf Generalisasi Graf Shadow Lingkaran, Tesis S2, Departemen
Matematika., Universitas Indonesia., Depok.
[5] Saputri, G. A., 2013, The Odd Harmonious Labeling of Dumbbell and Generalized
Prism Graphs, AKCE Int. J. Graphs Comb., 10, No. 2(2013), pp. 221-228.
[6] Vaidya, S. K., Shah, N.H., 2011, Some New Odd Harmonious Graphs. IJMSC, Vol
1, No 1, 9-16.
Proceedings of IICMA 2013
CONSTRUCTION OF , ‐VERTEX-ANTIMAGIC
TOTAL LABELINGS OF UNION OF TADPOLE GRAPHS
PUSPITA TYAS AGNESTI, DENNY RIAMA SILABAN, KIKI

ARIYANTI SUGENG
Department of Mathematics
University of Indonesia Depok 16424
puspita.tyas, denny, kiki@sci.ui.ac.id
Abstract. Let be a graph with vertex set and edge set , where | | and | |
be the number of vertices and edges of . A bijection from ∪ to the set
1, 2, … , | | | | is an , -vertex-antimagic total labeling of G if the
vertex weight set form an arithmetic progression with the initial term 0 and
the common difference 0. In this paper, we give the construction of , -
vertex-antimagic total labeling on disjoint union of tadpole graphs, for 1.
Key words and Phrases : , -vertex-antimagic total labeling, tadpole graph.
1. Introduction
In this paper all graphs are finite, simple, and undirected. The graph has vertex
set and edge set . The number of vertices and edges on are | | and | |,
respectively.
A total labeling of a graph is a mapping from the set of vertices and
edges of to a set of numbers (usually positive integers). Bača et al. introduced the
notion of an , -vertex-antimagic total labeling [2]. The , -vertex-
antimagic total labeling (VTAL) of a graph G is a bijective mapping from ∪
to the set 1, 2, … , | | | | such that the set of weight of vertices, that is
∑ ∈ | ∈ with is the set of all vertices adjacent to , form an
arithmetic progression , , 2 ,…, | | 1 where the initial term
0 and the different 0 are two fixed integers. A graph is called (a, d)-
vertex-antimagic total if it admits an , -vertex-antimagic total labeling. If
0 then we call as a vertex-magic total labeling. The concept of the vertex-magic
total labeling was introduced by MacDougall et al. [5].
A tadpole graph , is formed by joining an end point of a path with
vertices to a vertex of a cycle with vertices, where the end point of path is one of
the vertices of cycle. The union of tadpole graphs, ⋃ ,
, consists of
231
232
vertex set | 1 ,1 ∪ 1 ,1
where ( the end point of th path is one of the vertices of th
cycle) for every and the edge set 1 1, 1 ∪
1 1, 1 ∪ 1 . The number of
vertices and edges on ⋃ ,
is equal, that is | | | | ∑ 1 .
There are several results on , -vertex-antimagic total labeling of

particular classes of graphs. For examples, Bača et al. [2] gave some basic
properties of an , -vertex-antimagic total labeling. They also showed that paths
and cycles have , -vertex-antimagic total labelings for a wide variety of and
. Gray and MacDougall [4] showed that a tadpole has a vertex-magic total
labeling (or , 0 -vertex-antimagic total labeling). Agnesti, Silaban, and Sugeng
[1] showed that a tadpole has , -vertex-antimagic total labeling for 1, 2, 3
and particular values of . For the case of disjoint union of graphs, Parestu,
Silaban, and Sugeng [6] proved that ∪ ∪ …∪ is , -vertex-antimagic
total for 1, 2, 3, 4, and 6 and for a particular value of . Further results on
vertex-magic total labeling and , -vertex-antimagic total labeling can be found
in Gallian [3].
For a graph , if is the smallest degree of , then the minimum possible
weight of vertices is at least 1 2 ⋯ 1 , consequently
. (1)
If ∆ is the largest degree of , then the maximum weight of vertices is less than the
sum of the ∆ 1 largest labels, that is | | | | ∆ | | | | ∆ 1
⋯ | | | | . Thus,
∆ | | | | ∆
| | 1 . (2)
Then we obtain the restriction of as follows
∆ | | | | ∆
| |
. (3)
Since tadpole has 1 and ∆ 3, then we have
7. (4)
In this paper we give an , -vertex-antimagic total labeling of , and
⋃ ,
for 1.
2. Main Results
Theorem 1 gives the , 1 -vertex-antimagic total labeling of isomorphic

copies tadpole , .
Theorem 1. For 3 and 2, the tadpole , has a 2 2 2

2, 1 -vertex-antimagic total labeling, where 2.
233
PROOF. Label the vertices and the edges of the tadpole , as follows
 For 2 :
2 , 1, 1
2 1 1
, 1, 2
.
2 1 1 ,2 1, 1 (5)
2 , , 1
2 2 , , 1
. (6)
2 1 1 , , 2
, 1, 1
.
1 ,2 1, 1 (7)
2 1 , 1.
(8)
3 , 1, 1
.
2 1 , 1, 2 (9)
 For 3 :
2 2 3 2 1 , 1, 1
2 2 3 1 ,2 1, 1.
(10)
2 2 3 2 2 , , 1
2 1 ,2 1, 1
. (11)
2 2 3 , , 1
, 1, 1
. (12)
1 ,2 1, 1
3 1 , 1. (13)
, 1, 1
. (14)
,2 1, 1
Under the labeling we have

1 1, 1 2, … , 2 2 2 (15)
and
1, 2, 3, … , 1 (16)
where 3 and 2. It means that the labeling is a bijection from the set ∪
onto the set 1, 2, … , 2 2 2 .
Also, we have
, 1
,2 1. (17)
,
,
. (18)
,2 1
By substituting equations (5)-(9) and (10)-(14) to (17)-(18) we get vertex weight
set of , as follows.
|1 ∪ | 2
2 2 2 2, 2 2 2 3, … , 3 3 1 (19)
for 2 and
|1 ∪ | 2
234
2 2 2 2, 2 2 2 3, … , 3 3 3 1 (20)
for 2.
Since the vertex weight consists of consecutive integers starting from 2 2
2 with different 1, thus is a 2 2 2 , 1 -vertex-antimagic total
labeling on tadpole , . □
Figure 1. 74, 1 -VATL on 3 ,
An example of a 74, 1 -vertex-antimagic total labeling of 3 , can

be seen at Figure 1.
For the union of non-isomorphic tadpoles, we get only for a special

case, when 2 for all 1, 2, … , , as given in Theorem 2.
Theorem 2. For 3, 1, 2, … , the tadpole ⋃ ,

has a 2 ∑ 1
1 2, 1 -vertex-antimagic total labeling, where 2.
P ROOF . Let 1 1, 1, … , and define
,
, .
0 , otherwise
For 1, 2, … , , label the vertices and the edges of tadpole ⋃ ,

as
follows (21)
2∑ 2 1 1
2∑ 1 1 ∑ , 2, … , 1
(22)
2∑ 1
2∑
, 1 (23)
. (24)
1 , ,2 1
(25)
235
.
2 .
It can be checked that

∑ 1 1, ∑ 1 2, … ,2 ∑ 1 (26)
and
1, 2, 3, … , ∑ 1 . (27)
It means that the labeling is a bijection from the set ∪ onto the set
1, 2, … , 2 ∑ 1 .
By substituting equations (5)-(9) and (10)-(14) to (17)-(18) we get vertex weight

set of ⋃ ,
as follows.
|1 ∪ 2∑ 1 2, 2 ∑ 1 3, … , 3 ∑ 1 1.
(28)
Since the vertex weight consists of consecutive integers starting from 2 ∑ 1
2 2 ∑ 1 1 2 with different 1, thus is a 2 ∑ 1 1 2, 1 -
vertex-antimagic total labeling on tadpole ⋃ ,
.□
1 9 1
1 1
5 1
6 2
3 2
3 2 3 2 2 3
7
1 1 4 1
2 1
8 2
1 1
2 2 3 2 2 3
Figure 2. 70, 1 -VATL on , ∪ , ∪ , ∪ ,
An example of a 70, 1 -vertex-antimagic total labeling on , ∪ , ∪ , ∪ ,

can be seen at Figure 2.
236
We conclude this paper with an open problem for further research as

follows.
Open Problem. Find if there exists an , -vertex-antimagic total labeling on

union of tadpole graphs for other values of .
References
[1] Agnesti, P.T., Silaban, D.R., Sugeng, K.A., 2013, Construction of , -Vertex-
Antimagic Total Labeling on Tadpole Graphs, Proceeding The 1st Indonesian
Student Conference on Science and Mathematics, to appear.
[2] Bača, M. et al., 2003, Vertex-Antimagic Total Labelings of Graphs, Discuss. Math.
Graph Theory, 23, 67–83.
[3] Gallian, J.A., 2013, A Dynamic Survey of Graph Labeling, The Electronic Journal
of Combinatorics 19, #DS6.
[4] Gray, I.D., and MacDougall, J.A., 2008, Vertex-Magic Labelings: Mutations,
Australas. J. Combin, v45, 189-206.
[5] MacDougall, A. et al., 2002, Vertex-Magic Total Labelings of Graphs, Utilitas
Mathematics, 61, 68-76.
[6] Parestu, A., Silaban, D.R., Sugeng, K.A., 2008, Vertex Antimagic Total Labelings
of Union of Suns, JCMCC, 71, 2009, 179-188.
Proceedings of IICMA 2013
SUPER ANTIMAGICNESS OF TRIANGULAR BOOK

AND DIAMON LADDER GRAPHS
DAFIK1, SLAMIN2, FITRIANA EKA R3, LAELATUS SYA’DIYAH4
1 MathEdu. Depart., FKIP-University of Jember, d.dafik@gmail.com

2SystemInformation Depart, University of Jember, slamin@gmail.com
3Math Edu. Depart., FKIP-University of Jember, fitriana.eka@gmail.com
4Math Edu. Depart., FKIP-University of Jember, laelatus@gmail.com
Abstract. A graph G of order p and size q is called an (a,d)-edge-antimagic

total if there exists a bijection f : V (G) ∪ E(G) →{1,2,...,p + q} such that the
edgeweights, w(uv) = f(u)+f(v)+f(uv),uv ∈ E(G), form an arithmetic sequence
with first term a and common difference d. Such a graph G is called super if the
smallest possible labels appear on the vertices. In this paper we study super
(a,d)-edgeantimagic total properties of Triangular Book and Diamond Ladder
graphs. The result shows that there are a super (a,d)-edge-antimagic total
labeling of graph Bt n and Dln , if n ≥ 1 with d ∈ 0,1,2}.
Key Words: (a,d)-edge-antimagic total labeling, super (a,d)-edge-antimagic

total labeling, Triangular Book, Diamond Ladder.
1. Introduction
Mathematics consists of several branches of science. Branch of current
mathematics associated with a computer science is a graph theory. One of the
interesting topics in graph theory is a graph labeling. Several applications of graph
labeling can be found in [3]. There are various types of graph labeling, one is a
super (a,d)edge antimagic total labeling (SEATL for short). Assigning a label on
each vertex and each edge such that the edge-weights form an arithmetic sequence
is considered to be an NP-complete problem.
By a labeling we mean any mapping that carries a set of graph elements
onto a set of numbers, called labels. In this paper, we deal with labelings with
domain the set of all vertices and edges. This type of labeling belongs to the class
of total labelings. We define the edge-weight of an edge uv ∈ E(G) under a total
labeling to be the sum of the vertex labels corresponding to vertices u, v and edge
label corresponding to edge uv.
These labelings, introduced by Simanjuntak at al. in [16], are natural extensions of
the concept of magic valuation, studied by Kotzig and Rosa [14] (see also
[2],[11],[12],[15],[18]), and the concept of super edge-magic labeling, defined by
237
238
Enomoto et al. in [10]. Many other researchers investigated different forms of

antimagic graphs. For example, see Bodendiek and Walther [4] and [5], and
Hartsfield and Ringel [13].
In this paper we investigate the existence of super (a,d)-edge-antimagic total
labelings for connected graphs. Some constructions of super (a,d)-edge-antimagic
total labelings for m£n and m£i,j,k have been shown by Dafik, Slamin, Fuad and
Rahmad in [6] and super (a,d)-edge-antimagic total labelings for disjoint union of
caterpillars have been described by Baˇca in [1]. Dafik et al also found some
families of graph which admits super (a,d)-edge-antimagic total labelings, namely
and m caterpilars in [7, 8, 9].

All graphs in this paper are finite, undirected, and simple. For a graph G, V (G) and
E(G) denote the vertex-set and the edge-set, respectively. A (p,q)-graph G is a
graph such that |V (G)| = p and |E(G)| = q. We will now concentrate on Triangular
Book and Diamond Ladder graphs, denoted by Btn and Dln.
2. Three useful Lemmas

We start this section with a necessary condition for a graph to be a super (a,d)-
edge-antimagic total, which will provide a least upper bound, for a feasible value
d.
Lemma 1. [17] If a (p,q)-graph is super (a,d)-edge-antimagic total then d ≤
.
Proof. Assume that a (p,q)-graph has a super (a,d)-edge-antimagic total labeling f :
V (G) ∪ E(G) → {1,2,...,p + q} with the edge-weight set W = {w(uv) : uv ∈ E(G)}
= {a,a + 1,a + 2,...,a + (q − 1)d} . The minimum possible edge weight in the
labeling f is at least 1+2+p+1 = p+4. Thus, a ≥ p+4. On the other hand, the
maximum possible edge weight is at most (p − 1) + p + (p + q) = 3p + q − 1. Hence
a + (q − 1)d ≤ 3p + q − 1. From the last inequality, we obtain the desired upper
bound for the difference d.2
The following lemma, proved by Figueroa-Centeno et al. in [11], proves a
necessary and sufficient condition for a graph to be super edge-magic (super
(a,0)edge-antimagic total).
Lemma 2. [11] A (p,q)-graph G is super edge-magic if and only if there exists a
bijective function f : V (G) → {1,2,...,p}, such that the set S = {f(u) + f(v) : uv ∈
E(G)} consists of q consecutive integers. In such a case, f extends to a super edge-
magic labeling of G with magic constant a = p+q +s, where s = min(S) and S = {a
− (p + 1),a − (p + 2),...,a − (p + q)}.
A similar lemma with Lemma 2, Baˇca, Lin, Miller and Simanjuntak, see
[2], stated that a (p,q)-graph G is super (a,0)-edge-antimagic total if and only if
there exists (a − p − q,1)-edge-antimagic vertex labeling. They extended the study
with the following lemma.
Lemma 3. [2] If (p,q)-graph G has an (a,d)-edge antimagic vertex labeling then G

has a super(a+p+q,d−1)-edge antimagic total labeling and a
super(a+p+1,d+1)edge antimagic total labeling.
239
In this paper, we will use the last lemma to prove the existence of a super (a,0)-
edge-antimagic total labeling and super (a,2)-edge-antimagic total labeling for
triangular book Btn and diamond ladder Dln.
3. Triangular Book Btn

Triangular Book graph denoted byS Btn is a connected graph with vertex setS V
(Btn) = {xi;i = 1,2} {yj,1 ≤ j ≤ n} and edge set E(Btn) = {x1x2} {xiyj;i = 1,2,1 ≤ j ≤
n}. Thus |V (Btn)| = p = n + 2 and |E(Btn)| = q = 2n + 1.
If Triangular Book graph has a super (a,d)-edge-antimagic total labeling then it
follows from Lemma 1 that the upper bound of d is d ≤ 2 or d ∈ {0,1,2}. The
following theorem describes an (a,1)-edge-antimagic vertex labeling for Triangular
Book graph.
5
y4
4 6
y3 6 14 y5
13 7
3 5 7
15
y2 y6
12 8
4 16
2 8
y1 11 9 y7
3 17
x1 x2
10
1 9
Figure 1. Example of (3,1)-edge antimagic vertex labeling Bt7 with its

edge weight
Theorem 1. A triangular book Btn has an (a,1)-edge-antimagic vertex labeling if n

≥ 1.
Proof. Define the vertex labeling α1 : V (Btn) → {1,2,...,n + 2} in the following
way:
α1(xi) = (i − 1)n + i, for i = 1, 2
α1(yj) = j + 1, for 1 ≤ j ≤ n
The vertex labeling α1 is a bijective function.
The edge-weights of Btn, under the labeling α1, constitute the following sets
wα1(x1x2) = n + 3
wα1(xiyj) = n(i − 1) + j + (i + 1), for i = 1, 2 dan 1 ≤ j ≤ n
It is not difficult to see that the union of the set wα1 equals to {3,4,5,...,n +
3,...,2n+3} and consists of consecutive integers. Thus α1 is a (3,1)-edge antimagic
vertex labeling.2
240
Figure 1 gives an example of (a,1)-edge-antimagic vertex labeling of Btn.

With Theorem 1 in hand and by using Lemma 3, we obtain the following result.
Theorem 2. A triangular book Btn has a super (3n + 6,0)-edge-antimagic total
labeling and a super (n + 6,2)-edge-antimagic total labeling for n ≥ 1.
Proof.
We have proved that the vertex labeling α1 is a (3,1)-edge antimagic vertex
labeling. With respect to Lemma 2, by completing the edge labels p+1,p+2,...,p+ q,
we are able to extend labeling α1 to a super (a1,0)-edge-antimagic total labeling and
a super (a2,2)-edge-antimagic total labeling, where, for p = n+2 and q = 2n+1, the
value a1 = 3n + 6 and the value a2 = n + 6. 2
Theorem 3. A triangular book Btn has a super (2n + 6,1)-edge-antimagic total

labeling.
Proof. Label the vertices of Btn with α2(xi) = α1(xi) and α2(yj) = α1(yj), for i = 1,2
and 1 ≤ j ≤ n; and label the edges with the following
way.
; if n is odd
; if n is even
For n odd, any j, and i = 1,2
For n even, any j and i = 1,2
The total labeling α2 is a bijective function from V (Btn) ∪ E(Btn) onto the set
{1,2,3,...,3n+3}. The edge-weights of Btn, under the labeling α2, constitute the
following sets:
; if n is odd
; if n is even
For n odd, any j, and i = 1,2, the edge-weights of xi
;
For n even, any j, and i = 1,2,
j iji;
It is not difficult to see that the union of the set wα2 equals to {2n + 6, 2n+7,2n+8,
...,4n+6} and contains an arithmetic sequence with the first term 2n + 6 and
common difference 1. Thus α2 is a super (2n + 6,1)-edge-antimagic total labeling.
This concludes the proof. 2
241
4. Diamond Ladder Dln
Diamond ladder graph denoted by Dln is a connected graph with a vertex set V
(Dln) = {xi,yi,zj;1 ≤ i ≤ n,1 ≤ j ≤ 2n} and an edge set E(Dln) = {xixi+1,yiyi+1;1 ≤ i ≤ n
− 1} ∪ {xiyi;1 ≤ i ≤ n} ∪ {zjzj+1;2 ≤ j ≤ 2n − 2 for j even} ∪ {xiz2i−1,xiz2i,yiz2i−1,yiz2i;1
≤ i ≤ n}. Thus |V (Dln)| = p = 4n and |E(Dln)| = q = 8n − 3.
If diamond ladder graph has a super (a,d)-edge-antimagic total labeling then it
follows from Lemma 1 that the upper bound of d is d ≤ 2 or d ∈ {0,1,2}. The
following lemma describes an (a,1)-edge-antimagic vertex labeling for diamond
ladder.
2 6 10 14
x1 x2 3
24
x4
8 16
3 6 11 14 19 22 27 30
4 5 8 9 12 13
9 z4 17 25 z 7 z 8 16
1 z1 5 z2 z3 13 z5 21 z6 29
4 7 12 15 20 23 28 31
y1 10 y2 18 y3 26 y4
3 7 11 15 x
Figure 2. A (3,1)-edge antimagic vertex labeling of Dl4
Theorem 4. If n ≥ 2 then the diamond ladder graph Dln has an (a,1)-edge-

antimagic vertex labeling.
Proof. Define the vertex labeling β1 : V (Dln) → {1,2,...,4n} in the following way:
β1(xi) = 4i − 2, for 1 ≤ i ≤ n β1(yi) = 4i − 1, for 1 ≤ i ≤ n
, for 1 ≤ j ≤ 2n
The vertex labeling β1 is a bijective function. The edge-weights of Dln, under the
labeling β1, constitute the following sets
wβ1(xixi+1) = 8i; for 1 ≤ i ≤ n − 1;
wβ1(yiyi+1) = 8i + 2; for 1 ≤ i ≤ n − 1;
wβ1(xiyi) = 8i − 3; for 1 ≤ i ≤ n;
wβ1(zjzj+1) = 4j + 1; for 2 ≤ j ≤ 2n − 2 j even;
wβ1(xiz2i−1) = 8i − 5; for 1 ≤ i ≤ n;
wβ1(xiz2i) = 8i − 2; for 1 ≤ i ≤ n;
wβ1(yiz2i−1) = 8i − 4; for 1 ≤ i ≤ n;
wβ1(yiz2i) = 8i − 1; for 1 ≤ i ≤ n;
It is not difficult to see that the union of wβ1 = {3,4,...,8n − 1} and consists of
consecutive integers. Thus β1 is a (3,1)-edge antimagic vertex labeling. 2 Figure 2
gives an example of a (3,1)-edge antimagic vertex labeling of Dl4.
In similar way, with Theorem 4 in hand and by using Lemma 3, we obtain the
following result.
242
Theorem 5. If n ≥ 2 then the graph Dln has a super (12n,0)-edge-antimagic total

labeling and a super (4n + 4,2)-edge-antimagic total labeling.
Theorem 6. If n ≥ 2, then the graph Dln has a super (8n + 2,1)-edge-antimagic
total labeling.
Proof. Label the vertices of Dln with β2(xi) = β1(xi), β2(yi) = β1(yi) and β2(zj) = β1(zj),
for 1 ≤ i ≤ n and 1 ≤ j ≤ 2n; and label the edges with the following way.
β2(xixi+1) = 12n − 4i − 1; for 1 ≤ i ≤ n − 1
β2(yiyi+1) = 12n − 4i − 2; for 1 ≤ i ≤ n − 1
β2(xiyi) = 8n − 4i + 2; for 1 ≤ i ≤ n
β2(zjzj+1) = 8n − 2j; for 2 ≤ j ≤ 2n − 2 even
β2(xiz2i−1) = 8n − 4i + 3; for 1 ≤ i ≤ n
β2(xiz2i) = 12n − 4i; for 1 ≤ i ≤ n
β2(yiz2i−1) = 12n − 4i + 1; for 1 ≤ i ≤ n
β2(yiz2i) = 8n − 4i + 1; for 1 ≤ i ≤ n
The total labeling β2 is a bijective function from V (Dln) ∪ E(Dln) onto the set
{1,2,3,...,12n−3}. The edge-weights of Dln, under the labeling β2, constitute the
sets
Wβ2(xixi+1) = 12n + 4i − 1; for 1 ≤ i ≤ n − 1
Wβ2(yiyi+1) = 12n + 4i; for 1 ≤ i ≤ n − 1
Wβ2(xiyi) = 8n + 4i − 1; for 1 ≤ i ≤ n
Wβ2(zjzj+1) = 8n + 2j + 1; for 2 ≤ j ≤ 2n − 2, for j even
Wβ2(xiz2i−1) = 8n + 4i − 2; for 1 ≤ i ≤ n
Wβ2(xiz2i) = 12n + 4i − 2; for 1 ≤ i ≤ n
Wβ2(yiz2i−1) = 12n + 4i − 3; for 1 ≤ i ≤ n
Wβ2(yiz2i) = 8n + 4i; for 1 ≤ i ≤ n
It is not difficult to see that the union of the the set Wβ2 = {8n + 2,8n + 3,...,16n −
2} contains an arithmetic sequence with the first term 8n + 2 and common
difference 1. Thus β2 is a super (8n + 2,1)-edge-antimagic total labeling.
This concludes the proof. 2
Figure 3 gives an example of super (a,d)-edge antimagic total labeling of Dl4 for d
= 1.
243
2 6 10 14
x1 x2 3 x4
43 39 35
31 44 27 40 8 23 36 13 19 32
4 5 9 12
28 z z4 24 22 20 z z 8 16
1 z1 30 z2 3 26 z5 z6 7 18
45 29 41 25 37 21 33 17
y1 42 y2 38 y3 34 y4
3 7 11 15 x
Figure 3. Super (34,1)-edge antimagic total labeling of Dl4
5. Conclusion
In this paper, we have studied the existence of super antimagicness of two special
families of graphs, namely triangular book and diamond ladder. The research
shows the following results:
(1) The upper bound d of a super (a,d)-edge-antimagic total labeling at Btn and
Dln is d ≤ 2
(2) There are a super (a,d)-edge-antimagic total labeling of graph Btn and Dln,
if n ≥ 1 with d {0,1,2}.
Further interested research is then to answer following problem: If a graph Btn and
Dln are super (a,d)-edge-antimagic total, are the disjoint union of multiple copies of
the graphs Btn and Dln super (a,d)-edge-antimagic total as well? Therefore, we
propose the following open problem.
Open Problem 1. For the graph mBtn, n ≥ 1 and m ≥ 2, determine if there exists a
super (a,d)-edge-antimagic total labeling with any feasible upper bound d.
Open Problem 2. For the graph mDln, n ≥ 1 and m ≥ 1, determine if there exists a
super (a,d)-edge-antimagic total labeling with any feasible upper bound d.
References
[1] M. Baˇca, Dafik, M. Miller and J. Ryan, On super (a,d)-edge antimagic total
labeling of caterpillars, J. Combin. Math. Combin. Comput., 65 (2008), 61–
70.
[2] M. Baˇca, Y. Lin, M. Miller and R. Simanjuntak, New constructions of
magic and antimagic graph labelings, Utilitas Math. 60 (2001), 229–239.
[3] G.S. Bloom and S.W. Golomb, Applications of numbered undirected
graphs, Proc. IEEE, 65 (1977) 562–570.
[4] R. Bodendiek and G. Walther, On (a,d)-antimagic parachutes, Ars Combin.
42 (1996), 129– 149.
[5] R. Bodendiek and G. Walther, (a,d)-antimagic parachutes II, Ars Combin.
46 (1997), 33–63.
244
[6] Dafik, Slamin, Fuad and Riris. 2009. Super Edge-antimagic Total Labeling
of Disjoint Union of Triangular Ladder and Lobster Graphs.Yogyakarta:
Proceeding of IndoMS International Conference of Mathematics and
Applications (IICMA) 2009.
[7] Dafik, M. Miller, J. Ryan and M. Baˇca, Antimagic total labeling of disjoint
union of complete s-partite graphs, J. Combin. Math. Combin. Comput., 65
(2008), 41–49.
[8] Dafik, M. Miller, J. Ryan and M. Baˇca, On super (a,d)-edge antimagic total
labeling of disconnected graphs, Discrete Math., 309 (2009), 4909-4915.
[9] Dafik, M. Miller, J. Ryan and M. Baˇca, Super edge-antimagic total
labelings of mKn,n,n, Ars Combinatoria , 101 (2011), 97-107
[10] H. Enomoto, A.S. Lladó, T. Nakamigawa and G. Ringel, Super edge-magic
graphs, SUT J. Math. 34 (1998), 105–109.
[11] R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, The place
of super edgemagic labelings among other classes of labelings, Discrete
Math. 231 (2001), 153–168.
[12] R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, On super
edge-magic graph, Ars Combin. 64 (2002), 81–95.
[13] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press,
Boston - San Diego New York - London, 1990.
[14] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math.
Bull. 13 (1970), 451–461.
[15] G. Ringel and A.S. Lladó, Another tree conjecture, Bull. Inst. Combin.
Appl. 18 (1996), 83–85.
[16] R. Simanjuntak, F. Bertault and M. Miller, Two new (a,d)-antimagic graph
labelings, Proc. of Eleventh Australasian Workshop on Combinatorial
Algorithms (2000), 179–189.
[17] K.A. Sugeng, M. Miller and M. Baˇca, Super edge-antimagic total labelings,
Utilitas Math., 71 (2006) 131-141.
[18] W. D. Wallis, E. T. Baskoro, M. Miller and Slamin , Edge-magic total labelings,
Austral. J. Combin. 22 (2000), 177–190.
STUDENT ENGAGEMENT MODEL OF MATHEMATICS

DEPARTMENT’S STUDENTS OF UNIVERSITY OF
INDONESIA
1STRIANTI SETIADI1
1Departemen Matematika FMIPA-UI, ririnie@yahoo.com.sg
Abstract. Every department in a University should strive to create students who

feel engaged to their Department. An engaged student is a student who is
satisfied, loyal, and attached to his/her Department. The model of the
relationships among Student Engagement Components should be observed to
obtain a better policy for evaluating and improving the department. Student
Satisfaction can be used as a measuring instrument for a department’s self-
evaluation. High student Loyalty level will result in reliable resources for the
department. While high student Attachment level will result in alumni who will
still support the Department towards its improvement in the future. Hence, if a
department possesses engaged students, the continuity of the department’s
progress can be achieved. Based on logical reasoning, a satisfied student usually
is a loyal student, and a loyal student usually is an attached student. However
this is not the case in reality. The model of the relationship is heavily dependent
on students’ characteristics, such as Batch, GPA, Parents’ educations, Parents’
occupations, and the Students’ Moral Foundations. Students can be categorized
into groups based on those characteristics. Each group of student characteristic
will provide different relationship model of engagement components. This
article will present a general conclusion of engagement component relationship
model by taking into account the existence of student characteristic groups. The
methods applied are Two Step Clustering and MetaSem. This article is written
as a result of study conducted in Mathematics Department, University of
Indonesia 2013.
Key words and Phrases:Student Engagement, MetaSem.
1. Introduction
A. Background
A College Department’s success is very dependent on the quality of its

students and alumni. High quality students will create a passionate learning
processand creating valuable works. The same principle applies for the alumni.
High quality alumni of a department will bring pride to the department. Especially
if the alumni have emotional relation with the Department, they will be more than
willing to support the developments of the Department. Hence the developments
245
246
can be attained continuously.

In the banking sector, there is this term called Customer Engagement which
refers to customers who are willing to defend and make every effort for the sake of
certain company or product. There are three main components in Customer
Engagement; Satisfied Customer, Loyal Customer, and Attached Customer
(Customer with emotional bond with the company or product). Engaged customer
is a very valuable asset to the company, especially as a free-of-charge marketing
agent.
Borrowing the expression, the writer would like to introduce what is dubbed
as Student Engagement. It refers to students who are willing to defend their
Department in college both when they are still registered students and also when
they have been alumni of the Department in the future. Student Engagement cannot
be measured immediately because usually someone’s engagement to Department is
shown only after that person becomes an alumnus. Even so,an engaged student can
be mold through the creation of essential components of Student Engagement.
There are three components of Student Engagement, namely Satisfied Student,
Loyal Student, and Attached Student (student with emotional bond with the
Department where he/she is studying)
Satisfied student is student who feels that his/her expectation of the
Department is in accordance with the reality. Students’ satisfaction comprises of
satisfaction towards learning materials, lecturers, teaching/educational system,
facilities, and campus environment. Students’ satisfaction towards the Department
can be utilized as a measurement tool of Department’s service quality.
Loyal student is student who is willing to participate and support activities initiated
by the Department and is willing to do the best for in order to improve the
Department. A loyal student will be an asset and a very much needed human
resource to the Department.
Attached student is student who has emotional bond with his/her
Department. Attachment is something, such as a tie, band, or fastener, which
attaches one thing to another. A student with high attachment level to the
Department will have a sense of belonging towards the Department. He/she will be
proud when the Department is deemed well by other parties or is obtaining a
certain accomplishment. He/she will also be saddened over the fact that the
Department fails to win certain achievement, etc.
Due to the fact that Student Engagement can be developed through the
shaping of its components, namely Satisfaction, Loyalty, and Attachment, thus the
relationship model of the Student Engagement components ought to be evaluated
in order to ensure the Department’s ability to determine which component(s)
should be first in line to be fixed in attempt to increase the Students’ Engagement
level.
Based on logical understanding, the three components of Student
Engagement will form a certain relationship model. A satisfied student will usually
be a loyal student, and a loyal student will usually be an attached student to his/her
Department. If the relationship model is assumed as so, thus in order to obtain an
Engaged student, the Department needs to put increasing student satisfaction level
on the top list. If the students are satisfied, he will be loyal and followed by being
attached to his/her Department. If the three components can be improved, there will
be a higher probability that the Department get an Engaged Student.
However, in reality the relationship among the Student Engagement
components are not always in that model. There are students who are satisfied but
247
not loyal to the Department, and there are also ones who are loyal but not attached
to the Department. If the relationship model is not as so, the Department have to
find ways in order to obtain Engaged Student. The difference in this relationship
model is very likely to be influenced by the students’ characteristics, such as batch,
GPA, parents’ occupation, parents’ educational background, and the moral
foundation of the students. Based on those characteristics, the writer attempted to
classify the students into several different categories in which students with the
same characteristics are classified in the same categories, and those with different
characteristics are classified into different categories. This relationship model of
Student Engagement components needs to be examined for each of the student
characteristic groups to finally attain a general model that will be used as a
reference in generating regulations that will help increasing Student Engagement
level towards the Department.
B. Research Purpose
To attain a general relationship model of the Student Engagement
components by taking into account the existence of student characteristics groups.
C. Research Method
a. Sample : Students of the Departement of Mathematics University of
Indonesia, batch 2010 (28 students), 2011 (40 students), 2012 (45
students) which are sampled by purposive sampling method. Data
collection technique used is questionnaire distribution.
b. Measurement instrument used is measurement tool with Likert scale to
measure satisfaction, loyalty, attachment and moral foundation of the
students. This particularmeasurement tool is designed based on the
theoretical elements used as its foundation.
The reliability of the measurement tool is 0.759 and every single item is
valid item.
c. Data analysis methods used are Clustering and MetaSem (combination of
Meta Analysis and Structural Equation Model)
2. Main Results
A. Descriptive Statistics
 Batch: 2010 (28 students), 2011 (40 students), 2012 (45 students)
 GPA: ≤ 2.75 (8 students), >2.75 (104 students), missing (1 students)
 Parent’s education: ≤ high school (42 students), > high school (67
students), missing (4 students)
 Parent’s job: Government Employee (26 students), Private Employee
(32 students), Entrepreneur (31 students), others (19 students), missing
(5 students)
 Average Satisfaction score/item : 3.08
 Average Loyalty score/item : 3.54
 Average Attachment score/item : 3.82
 Average Moral Foundation score/item : 3.32
Since a respondent’s answer is considered high when the score is > 3.5, it
can be concluded that the respondents’ have an average of low Satisfaction, high
Loyalty, high Attachment, and low Moral Foundation.
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B. Student Characteristic Groups Profile based on Batch, GPA,

Occupation, Eduction, and Moral Foundation
By using Two Step Clustering method, the writer attempted to classify the
respondents based on Batch, GPA, Parent’s education, Parent’s job and Moral
Foundation variables but unfortunately there was only one variable that was able to
separate the respondents into groups with the same characteristics within the same
categories and different characteristics across categories. The variable that was able
to create the aforementioned student characteristic groups is the Batch variable.
Subsequently to find the relationship model of Student Engagement components,
Batch variable will be taken into account for the analysis.
C. Relationship Model of Student Engagement Components

The relationship of Student Engagement components by taking batch into account
will be attained by using Meta SEM method. This method is a combination of Meta
Analysis method and Structural Equation Model method.
Structural Equation Model

Student Engagement components (Satisfaction, Loyalty, Attachment) are latent
variables measured by indicators in the form of items containing all of the
theoretical concepts. The items have been verified to be having Alpha Chronbach
Reliability = 0.759 and every single one is a valid item.
Structural Equation Model method is set up by modeling the concept in the form of
path diagram. Parameters are estimated by matching the correlation model matrix
with the correlation matrix obtained from the samples of indicator-variables
forming latent variables. In the working process, statistics software, will
automatically transform raw data to covariance/correlation matrix form.
Consequently in Structural Equation Model analysis, data can be input in the form
of raw data, covariance matrix or correlation matrix of the indicator-variables. In
this paper, correlation matrix will be used as input data.
The correlations among indicator-variables in each batch are as follows:
Batch 2010:
 Q7 Q 22 Q 25 Q2 Q15
Q11 Q12
 Q7 1 
 
Q 22 0.089 1 
 
Q 25 0.315 0.379 1 
 Q2 0.302 0.246 0.06 1 
 
 Q11 0.307 0.415 0.268 0.036 1 
 Q12 0.317 0.215 0.442 0.087 0.369 1 
 
 Q15 0.477 0.197 0.508 0.438 0.29 0.472 1 
249
Batch 2011:
 Q7 Q 22 Q 25 Q2 Q11 Q15
Q12
 Q7 1 
 
Q 22 0.276 1 
 
Q 25 0.386 0.402 1 
 Q2 0.527 0.072 0.011 1 
 
 Q11 0.002 0.017 0.005 0.123 1 
 Q12 0.211 0.016 0.492 0.304 0.271 1 
 
 Q15 0.368 0.140 0.445 0.521 0.300 0.549 1 
Batch 2012:
 Q7 Q 22 Q 25 Q2 Q11 Q15
Q12
 Q7 1 
 
Q 22 0.579 1 
 
Q 25 0.378 0.414 1 
 Q2 0.095 0.119 0.153 1 
 
 Q11 0.009 0.181 0.110 0.163 1 
 Q12 0.419 0.370 0.511 0.154 0.095 1 
 
 Q15 0.100 0.141 0.234 0.061 0.047 0.301 1 
Subsequently, to find the relationship of the Student Engagement components, the

overall correlation that represents the existing batch correlation should first be
obtained. Overall correlation is obtained using MetaAnalysis method. The
coefficient of the overall correlation will be used as an entry for input matrix in
Structural Equation Model.
Meta Analysis Method

As previously mentioned, the respondents of this research are batch 2010, 2011,
and 2012 students of the Mathematics Department, University of Indonesia. By
taking batch into account, overall correlation of the indicator-variables will be
obtained based on the correlation among the indicator- variables of each batch.
Overall correlation will be calculated using Meta Analysis method.
Assumed is correlation coefficient between two indicator variables in batch-i.
Define 0.5 ln
it can be shown that has variance ≈ 3.
Before finding overall correlation, the equality of , is evaluated by testing
hypothesis ∶ .
The conclusion is “ is not rejected”.
Define 0.5 ln ; is correlation estimation of .
∗ ∑
Compute ∑
where
250
∗
1
1
∗
is the overall correlation we want to obtain.
Values of ∗ for every pair of indicator-variabeles will be used as entry matrix of
input matrix in Structural equation Model.
By using Meta Analysis, the following result of overall correlations of each pair of
the indicator-variables is obtained
 Q7 Q 22 Q 25 Q2 Q11 Q15
Q12
 Q7 1 
 
Q 22 0.2884 1 
 
Q 25 0.3705 0.3244 1 
 Q2 0.1416 0.3143 0.3540 1 
 
 Q11 0.3952 0.1620 0.2426 0.2000 1 
 Q12 0.1656 0.070 0.1503 0.2000 0.1657 1 
 
 Q15 0.1096 0.1708 0.4231 0.2712 0.2316 0.4548 1 
The above mentioned overall correlation matrix is then used as input matrix in
Structural Equation Model.
Meta SEM Method

Meta SEM method is a combination of Meta Analysis method and SEM method.
Meta Analysis method is used to find the overall correlation and SEM is used to
find the relationship model of the Student Engagement components by utilizing
overall correlation matrix as input matrix.
The best model of Student Engagement Model obtained is as follow:
SATISFACTION
0.890
LOYALTY
ATTACHMENT
0.812
Student Engagement Model
The numbers in the above diagram presents standardized effect, thus the values can
be compared. Goodness of fit of Structural Equation Model can be examined using
several different methods. In this paper, Goodness of fit of the obtained Model will
be examined using
 Chi Square test, model is fit if p-value > 0.05
 CFI : model is fit if CFI > 0.9
251
 CMIN/DF : model is deemed fit if CMIN/DF lies between 1 – 3

 RMSEA : model is fit if RMSEA = 0.05 – 0.08
From the result of data processing, Goodness of fit of the model is obtained:
P - Chi Square = 0.056 > 0.05
CFI = 0.941 (standard > 0.9)
C MIN/DF = 1.718 (standard 1 – 3)
RMSEA = 0.08 (standard 0.05 – 0.08)
Conclusion: Model is fit
Note: When using Structural Equation Model method, the use of overall correlation
matrix as input matrix did not allow the writer to examine multivariate normality
assumption.
Based on the data analysis above, it can be concluded that the assumption
about how satisfied student will result in loyal student, loyal student will result in
attached student is true for student of the Mathematics Department, University of
Indonesia. While satisfied students does not automatically become attached
student. Therefore the Department must pay a good attention to both satisfaction
and loyalty to achieve attached students.
The numbers in the above mentioned diagram presents standardized effect,
thus the values can be compared. It can be seen that the influence of Satisfaction
towards Loyalty is higher than the influence of Loyalty towards Attachment.
Therefore it is essential for the Department to prioritize its effort in improving
students’ satisfaction. For that reason, the Department is required to understand the
needs of the students and look into the currently available services. A separate
survey is needed for to learn about the students’ needs and to compare it with the
currently available services. The purpose is so that the Department can evaluate
what segments of the Department to be improved and what segments to be
preserved. As a result, students’ satisfaction level can be optimized.
Aside from that, we can see from the model that Loyalty is another
influencing factor to Attachment, while Satisfaction does not directly influence
Attachment. Thus, to get students who are attached to the Department, an effort to
enhance students’ loyalty in order that the students are willing to do their best for
the sake of the Department is needed. By the existence of loyal students, attached
students will exist accordingly.
The existence of satisfaction, loyalty and attachment in the students create
engaged students. Engaged students are extremely valuable assets for the
Department in order to maintain the incessant improvement of the Department’s
quality.
Acknowledge:
Thank you for my students who have helped me to collect the data.
References
[1] Cheung, M. L. W., Chan, W. (2005).Meta-Analytic Structural Equation Modeling: A

Two-Batch Approach. Psychological Methods2005, Vol. 10, \No. 1 : 40 – 64
[2] Bollen Kenneth A. (1989)Structural Equation with Latent Variables; John Wiley &
Sons
252
[3] Rencher AC.(2002).Methods of Multivariate Analysis, John Wiley & Sons

[4] Virginia Gunawan.(2013). Hubungan Pola Asuh Orang tua dan Fondasi Moral
pada Remaja (thesis)
[5] Zhilin Young, Robin T Pererson. (2004).Customer Perceieved Value,
Satisfaction and Loyalty: the Role of Switching cost Psychology &
Marketing, Vol. 21(10): 799-822 (October 2004), Wiley
InterScience(www.interscience.com)
DESIGNING ADDITION OPERATION LEARNING IN

THE MATHEMATICS OF GASING FOR RURAL AREA
STUDENT IN INDONESIA
RULLY CHARITAS INDRA PRAHMANA1 AND SAMSUL

ARIFIN2
1Program Studi Pendidikan Matematika, STKIP Surya, Jl. Scientia Boulevard Blok U/7,
Gading Serpong-Tangerang, Indonesia, rully.charitas@stkipsurya.ac.id
2 Program Studi Pendidikan Matematika, STKIP Surya, Jl. Scientia Boulevard Blok U/7,
Gading Serpong-Tangerang, Indonesia, samsul.arifin@stkipsurya.ac.id
Abstract. Learning number operations at the primary school is important for

learning other subjects. It’s because learning number operations tends to an
understanding of notation, symbols, and other forms to represent (reference
number), so it can support the students’ thinking and understanding, to solve
their problems [6]. Several studies on mathematics learning for Papua's student
indicate that students have difficulty in understanding the concept of number
operations [13]. It’s supported by the results of rural area's student, namely
Ambon, Serui, and Sorong Selatan, classroom observations toward to learning
number operations conducted by researchers in pre-test. Students are more likely
to be introduced by the use of the formula without involving the concept itself
and learning number operations separate the concrete situation of learning [8,9].
Addition operation is number operation which first must be mastered before
student learns another number operations starting from introduction of number
[10]. This underlies the researcher to try designing addition operation learning in
the mathematics of GASING (Math GASING) for them, which always starts
from the concrete (informal level) to the abstract (formal level) [11]. Concrete
means real, can be touched, seen, and explored. Once students are able to relate
the concrete forms to abstract (mathematics symbols), they are required to do
much exercise (drill) with mental arithmetic namely mencongak [8]. The
purpose of this study is to look at the role of learning addition operation in Math
GASING in helping students' understanding and mastering of the addition
concept from the informal (concrete) into formal level. The research method
used is a design research with preliminary design, teaching experiments, and
retrospective analysis stages [1,3,5]. This study describes how the Math
GASING make a real contribution of students understanding in the concept of
addition operation. The whole strategy and model that students discover,
describe, and discuss the construction or contribution shows how students can
use to help their initial understanding of the addition concept. The stages in the
learning trajectory has important role in understanding the addition concept from
informal to formal level [2,4].
Key words and Phrases: Design Research, Math GASING, Addition Operation,
Rural Area’s Student.
1. Introduction
Professional teacher as the product of reform in education must have higher
253
254
education and be able to innovate in teaching and learning [7]. So, every
prospective teacher should be able to prepare themselves to become professional
teachers to equip themselves with a high education and knowledge of the learning
and teaching process. In the other hands, prospective teachers who come from rural
areas have a few access to get decent education and information as requirements to
become a professional teacher. Surya College of Education has responsibility for it.
Here, they get a great education to become a professional teacher including
mathematics teacher. In addition, Surya [13] have made and apply a learning
innovation in mathematics education, named Math GASING. This learning has
been applied to student from Papua, which began with the introduction of number
and number operations, and produce many Olympic champions both nationally and
internationally.
Furthermore, learning number operations at the primary school is important for
learning other subjects. It’s because learning number operations tends to an
understanding of notation, symbols, and other forms to represent (reference
number), so it can support the students’ thinking and understanding, to solve their
problems [6]. Based on the results of several previous studies show students have
difficulty to understand the number operation concept [8,9]. It’s supported by the
results of rural area's student, namely Ambon, Serui, and Sorong Selatan,
classroom observations toward to learning number operations conducted by
researchers in pre-test. Students are more likely to be introduced by the use of the
formula without involving the concept itself and learning number operations
separate the concrete situation of learning. Addition operation is number operation
which first must be mastered before student learns another number operations
starting from introduction of number [10]. This underlies the researcher to try
designing addition operation learning in the mathematics of GASING (Math
GASING) which always starts from the concrete (informal level) to the abstract
(formal level) for matriculation prospective teachers students at Surya College of
Education Tangerang derived from Ambon, Serui, Yapen, and South Sorong,
Papua.
2. Theoretical Framework
In this study, the literature on Math GASING and addition operation was learn to
see the typical learning processes used by real situations (concrete) to abstract with
the steps that has been in the design.
Math GASING
Surya and Moss [13] stated that GASING has several basic premises. First is that
there is no such thing as a child that cannot learn mathematics, only children that
have not had the opportunity to learn mathematics in a fun and meaningful way.
Second is that mathematics is based on patterns and these patterns make math
understandable. Third is that a visual context to mathematical concepts should
come before the symbolic notation. Lastly is that mathematics is not memorization,
but knowing basic facts comes easily with a conceptual and visual understanding.
Memorization of basic mathematics facts is easy if it is based on conceptual
learning and visual representations. Additionally, Shanty and Wijaya [11]
describes that in Math GASING, the learning process make students learning easy,
fun, and enjoyable. Easy means the students are introduced to mathematical logic
that is easy to learn and to remember. Exciting means the students have motivation
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which comes from by them to learn mathematics (intrinsic factor). Fun is more in
the direction of outside influences such as visual aids and games (extrinsic factor).
In the other hand, Prahmana [8] had been conducted research for division topic in
Math GASING, where the learning process begins with the activities share sweets
fairly, then move into the process of how each student gets distributed sweets after
a fair amount of candy (concrete), ranging from division without remainder to
division with remainder, and ends with the completion of division operation in
Math GASING (abstract). Math GASING shows
how to change a concrete sample into an abstract symbol so the students will be
able to read a mathematical pattern, thus gain the conclusion by themselves
Addition Operations in Math GASING

Math GASING as one of innovations in learning mathematics offers critical point
in its learning process. When studying a topic in Math GASING, there is a critical
point that we must pass that is called GASING’s critical point. After reaching this
critical point, students will not be difficult anymore to work on the problems in that
topic [13]. The critical point in learning addition is addition of two numbers
between 1 and 10 with a sum less than 20. In the other words, when a student has
mastered addition of two numbers between 1 and 10 with a sum less than 20, the
student can learn a variety of addition operation problems more easy.
The Hypothetical Learning Trajectory (HLT) in this study had several learning
goals expected to be reached by the students. To reach the goals formulated,
researcher designs a sequence of instructional learning for learning addition in
Math GASING on the following diagram.
Recognizing Addition of Recognizing Addition of

numbers two numbers numbers two numbers
from 1 to 10 whose sum from 11 to whose sum is
is 10 or less 19 11-19
Figure 2.1 the HLT of Learning Addition in Math GASING

The explanation of Figure 2.1 is as follows:
1. Students are recognizing numbers from 1 to 10 by using their fingers. For
example, teacher introduced the notation of number “1” by showing 1 index
finger or 1 thumb or 1 middle finger or 1 ring finger or 1 little finger; the
notation of number “2” by showing 1 index finger and 1 thumb or 2 index
fingers or 2 thumbs, and various other variations; the notation of number “3”,
“4”, until “10” by using various other variations from their finger.
2. Students learn the addition of two numbers whose sum is 10 or less process by
using their fingers. For example, teacher showed 3 fingers on the right hand
and said “these are three fingers”, and then showed 2 fingers on the left hand
and said “these are two fingers”. After that, she combined the fingers on both
hands together and said “these are five fingers”, so “three plus two is equal to
five”. Lastly, students learned how to write in abstract symbols: 3 + 2 = 5.
Teacher showed all various combination of addition from 2 to 10.
3. Students are recognizing numbers from 11 to 19 by using “number card”. For
example, “black card” as tens and “white card” as unit. The teacher showed 1
black card and said “this is ten”, and then showed 1 white card and said “this
is one”. After that, she combined the two cards that consists of 1 black card
and white card, and said “this is eleven”. This way is able to make the students
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imagine that eleven consists of ten and one. Teacher showed all numbers from
1119 by using black card and white card.
4. Students learn the addition of two numbers whose sum is 11-19 by using
“number card”. The learning process consists of some addition types namely
10+, 9+, 8+, 7+, and 6+ and starts from 10+ type where student add the
number 10 with numbers from 1 to 9, to 6+ type (see table 2.1). They also
learn about the commutative of addition according to table 2.1.
Table 2.1 Addition of two numbers whose sum is between 11 and 19

10 + 9 + 8 + 7 + 6 +
10 + 1 =
10 + 2 = 9 + 2 =
10 + 3 = 9 + 3 = 8 + 3 =
10 + 4 = 9 + 4 = 8 + 4 = 7 + 4 =
10 + 5 = 9 + 5 = 8 + 5 = 7 + 5 = 6 + 5 =
10 + 6 = 9 + 6 = 8 + 6 = 7 + 6 = 6 + 6 =
10 + 7 = 9 + 7 = 8 + 7 = 7 + 7 =
10 + 8 = 9 + 8 = 8 + 8 =
10 + 9 = 9 + 9 =
Research Question
Based on a few things mentioned in the introduction above, then researcher
formulates a research question in this study, as follows:
"How is student learning trajectory of learning addition in Math GASING, which
evolved from informal to formal level for rural area's student at Surya College of
Education?
3. Methods
This study uses a design research approach, which is an appropriate way to answer
the research questions and achieve the research objectives that start from
preliminary design, teaching experiments, and retrospective analysis [9]. Design
research is methodology that has five characteristic, which is interventionist nature,
process oriented, reflective component, cyclic character, and theory oriented [1].
To implementation, design research is a cyclical process of thought experiment and
instruction experiments [3]. There are two important aspect related to design
research. There are the Hypothetical Learning Trajectory (HLT) and Local
Instruction Theory (LIT). Both will be on learning activities as learning paths that
may be taken by students in their learning activities.
According to Freudenthal in Gravemeijer & Eerde [5], students are given the
opportunity to build and develop their ideas and thoughts when constructing the
mathematics. Teachers can select appropriate learning activities as a basis to
stimulate students to think and act when constructing the mathematics.
Gravemeijer [4] states that the HLT consists of three components, namely (1) the
purpose of mathematics teaching for students, (2) learning activity and devices or
media are used in the learning process, and (3) a conjecture of understanding the
process of learning how to learn and strategies students that arise and thrive when
learning activities done in class.
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For the data, researchers have collected research data is derived from multiple
sources of data, to get a visualization of the students' mastery of basic concepts of
addition operations, namely video recording, documentation (learning activities
photo), and the data is written (the results of students' answers and observation
sheet). Furthermore, the data were analyzed retrospectively with HLT as a guide.
In addition, these studies have been completed in 2 days in the first semester of
academic year 2013/2014 with the subjects are 11 matriculation prospective
teachers students at Surya College of Education Tangerang derived from Ambon,
Serui, Yapen, and South Sorong, Papua, and also a teaching assistant who acted as
a model teacher.
This study consists of three steps done repeatedly until the discovery of a new
theory that a revision of the theory of learning is tested. Overall, the stages that will
be passed on this research can conclude in the form of the following diagram in
Figure 3.1. [9]:
Figure 3.1 Phase of the design research
4. Results and Analysis
The learning activities start from recognizes number between 1 and 10 using
students’
fingers to introduce the concept of number in concrete level as sum of their fingers.
Furthermore, Students learn the addition of two numbers whose sum is 10 or less
process by using their fingers. Lastly, students are recognizing numbers from 11 to
19 by using “number card” and learning the addition of two numbers whose sum is
11-19 by using “number card” that consist of black card as tens and white card as
units. At the end of the second meeting, students do mental arithmetic activity
namely mencongak as one of assessment process in this learning activities and
exercise by using student evaluation sheet. As a result, students was able to master
the addition operation in Math GASING seen from the results of the final
evaluation and was pleased to learn Math GASING can be seen from the
comments of students who wish to abandon the old way of learning mathematics.
The results of this study indicate that learning design of addition operation in Math
GASING have a very important role as the starting point and improve students'
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motivation in learning addition operation.

For more details, researchers will discuss the results of this study, which is divided
into three stages that are called preliminary design, teaching experiments, and
retrospective analysis.
Preliminary Design
At this stage, researcher is beginning to implement the idea of addition operation in
Math GASING by reviewing the literature, conducting observations in
matriculation class, and ends with designing hypothetical learning trajectory
(HLT), as shown in Figure 2.1. A set of activities for learning addition operation in
Math GASING has been designed based learning trajectory and thinking process of
students who hypothesized. The instruction set of activities has been divided into
four activities that have been completed in 2 meetings, start from recognize
numbers between 1 and 10 using students’ fingers as the concrete form, learn the
addition of two numbers whose sum is 10 or less process by using their fingers,
recognize numbers from 11 to 19 by using “number card”, learn the addition of
two numbers whose sum is 11-19 by using “number card” that consist of black
card as tens and white card as units, do a variety of fun activities that make
students happy in the learning process, and end with the evaluation process.
Teaching Experiment
In teaching experiment, researcher tests the learning activities have been designed
in the preliminary design stage. When the teacher models have started to see
students do not get excited, then the teacher models provide educational games that
make fun learning activities, because it is becoming one of characteristics in Math
GASING learning process. There are four activities in this stage. First, teacher
introduced the notation of number “1” until “10” by showing her finger and student
recognize it by using their fingers as the concrete form using various other
variations from their fingers (see in Figure 4.1).
Figure 4.1 students recognize numbers between 1 and 10 using their fingers
Second, students learn the addition of two numbers whose sum is 10 or less
process by using their fingers. Student showed 4 fingers on the right hand and said
“this is four”, and then showed 2 fingers on the left hand and said “this is two”.
After that, she combined the fingers on both hands together and said “this is six”,
so “four plus two is equal to six”. Lastly, students write in abstract symbols: 4 + 2
= 6. Teacher showed all various combinations of addition from 2 to 10 (see in
Figure 4.2).
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Figure 4.2 students learn addition of two numbers (left) and all various
combinations (right)
Third, students are recognizing numbers from 11 to 19 by using “number card”

that consists of “black card” as tens and “white card” as unit. Student showed 1
black card and said, “This is ten”, and then showed 1 white card and said, “This is
one”. After that, she combined the two cards of 1 black card and 1 white card, and
said, “This is eleven”. Teacher showed all numbers from 11-19 by using black card
and white card (see in Figure 4.3).
Figure 4.3 the concrete form of 10+ type (left) and students presented the number
card (right)
Lastly, students learn the addition of two numbers whose sum is 11-19 by using
“number card”. The learning process consists of some addition types namely 10+,
9+, 8+, 7+, and 6+ and starts from 10+ type where student add the number 10 with
numbers from 1 to 9, to 6+ type (see table 2.1). They also learn about the
commutative of addition according to table 2.1. Student count nine white card plus
three white card, and then count all white card that she gets. After that, she switch
10 white card with 1 black card and now she gets 1 black card and 2 white card,
and said, “This is twelve” (see in Figure 4.4).
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Figure 4.4 students learn addition of two numbers (left) and all various
combinations 9+ (right)
The learning process in this study ends with 2 forms of evaluation. First, students
are given about spontaneously problem in front of class and direct answer that
problem on whiteboard. Second, students are given a worksheet that consists of
many questions about addition and should be able to finish it within a few minutes
(see in Figure 4.5).
The results of the evaluation process are quite amazing that all students get
satisfactory results and are able to explain it either concretely or abstractly.
Figure 4.5 the first evaluation form (left) and the second evaluation form (right)
Retrospective Analysis
Addition process in Math GASING is different with addition process in
mathematics in general. As a result, all activities which have been designed can be
used to answer the research question above. The activities are as follows:
1. Learning trajectory which has been modeled in Figure 2.1 are the activities
undertaken in this study to guide students mastered addition operation. So
that, researcher designed an activity using students’ fingers to recognize
numbers from 1 to 10 and to learn by using their fingers too. The goal is that
students are able to imagine the concrete form of number notation between 1
and 10 and the addition of two numbers whose sum is 10 or less process.
Fingers are the greatest learning tool to introduce the concrete form of number
between 1 and 10 and that addition. Next, to introduce a number greater than
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10 and its sum, researcher uses a number card starting from units, tens,
hundreds, etc. by using different colors such as white card for the unit, the
black card to tens, the orange card to hundreds, etc.
2. Furthermore, from these activities, teachers guide students toward the concept
of number notation and addition using their fingers and number card. As a
result, when students have mastered the addition process of two numbers
whose sum is between 11 and 19, they were able to complete the various
forms of addition operations more easily, using the addition process in Math
GASING. So that, they can do mencongak to solve all addition problems
given. Addition process in Math GASING is starting from “front addition”
where addition process start from left to right and “scratch system” for the
addition of two numbers whose sum is between 10 and 19. For more details,
see in Figure 4.6 below.
Figure 4.6 Yobel’s answer sheet using “scratch system”
3. In Figure 4.6, Yobel’s answer sheet shows addition process in Math GASING
where every addition of two numbers whose sum is more than or equal to 10
262
then he did ”scratch” on that number and so on until finished and then count
the number of scratch made and write it followed by the last number that he
count. For example, based on problem 4, 8 + 8 is equal to 16. It means he
must do scratch on “8”, and then 6 + 6 is equal to 12. It means he must do
scratch again on “6”, and then 2 + 9 is equal to 11. It means he must do
scratch on “9”, and then 1 + 8 is equal to 9. It means he don’t do scratch on
“8”, and continue until finished. Lastly, he get 6 scratch and “6” as the last
number that he count. So, he can write “66” as the result of that problem. In
the other hands, Figure 4.7 shows that rosita can solve 100 addition problems
only 7 minutes and gets 1 mistake using “front addition”. It is apparent that
addition process in Math GASING is much more effective than the usual
process of addition, when students have mastered the addition of two numbers
whose sum is between 11 and 19.
Figure 4.7 Rosita’s answer sheet using “front addition”
4. Based on all the activities above, it can be seen that the students have gone
through the process of activity based on experience using their fingers and
“number card”, moving toward a more formal, the understanding of formal
level from the critical point, and then reached into the formal level desired as
the ultimate goal of this learning activities.
5. In the design of this study, researcher used the learning steps of addition
operation in Math GASING as shown in Figure 2.1. When the activity takes
place, the dialogue is very good in the process of introducing the basic
concepts of addition operations. In the dialogue, it seems that students feel
learning addition operation in Math GASING looks so easy and so much fun.
As a result, the learning process can guide students in understanding addition
operations. It can also be seen from the student evaluation of learning addition
process given by the teacher to evaluate student understanding. As a result,
students seemed to be able to apply addition operation process in solving each
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problem is given in terms of evaluation (see in Figure 4.8). Therefore, it can

be seen that learning addition operation in Math GASING can use to raise
students' understanding in integer addition operations or in other words, the
design of this study can be used as the starting point of learning addition
operations.
Figure 4.8 some student’s answer sheets
Based on the result and analysis of this research that has been described above,
researcher can conclude that the learning of addition operation in Math GASING
have a very important role as the starting point and improve students' motivation in
learning addition operation. In addition, the activities that have been designed in
such way those students find the concept of addition operation starting from
recognizing number between 1 and 20 to calculating addition operation of two
numbers whose sum is between 11 and 19 which is the critical point of addition
operation in Math GASING. This process begins with the activities recognize
number between 1 and 19 using their finger and number card (black and white
card), and then move into the process of how to calculate two numbers addition
operation of two numbers whose sum is between 11 and 19. Lastly, each student
can do mencongak for any given addition problem and resolve many addition
questions very quickly and precisely where is both of this are one of assessment
forms in Math GASING.
Acknowledgement. Researchers would like to thank Petra Suwasti as a research
assistant for their contribution in order to collect data and to be a model teacher in
264
this research.
References
[1] Akker, J.V.D., Gravemeijer, K., McKenney, S., and Nieveen, N. (2006). Education
Design Research. London: Routledge Taylor and Francis Group.
[2] Carpenter, T. P., & Fennema, E. (1992). Cognitively guided instruction: Building on
the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform:
The case of mathematics in the United States. Special issue of the International
Journal of Educational Research (pp. 457–470). Elmswood, NY: Pergamon Press,
Inc.
[3] Gravemeijer, K. (1994). Developing Realistic Mathematics Education. Utrecht:
Technipress, Culemborg.
[4] Gravemeijer, K. (2004). Local Instructional Theories as Means of Support for
Teacher in Reform Mathematics Education. Mathematical Thinking and Learning,
6(2), 105-128, Lawrence Erlbaum Association, Inc.
[5] Gravemeijer, K., Eerde, D.V. (2009). Design Research as a Means for Building a
Knowledge Base for Teaching in Mathematics Education. The Elementary School
Journal Volume 109 Number 5.
[6] National Council of Teachers of Mathematics (NCTM). (2000). Principles and
Standards for School Mathematics. Reston, VA: National Council of Teachers of
Mathematics.
[7] Ornstein, A.C., Levine, D.U., Gutek, G.L. (2011). Foundations of Education. (11th
ed.). Belmont: Wadsworth.
[8] Prahmana, R.C.I. (2013). Designing Division Operation Learning in The
Mathematics of Gasing. Proceeding in The First South East Asia
Design/Development Research (SEA-DR) Conference 2013, pp. 391-398.
Palembang: Sriwijaya University.
[9] Prahmana, R.C.I., Zulkardi, Hartono, Y. (2012). Learning Multiplication Using
Indonesian Traditional Game in Third Grade. Journal on Mathematics Education
(IndoMS-JME) Vol. 3 No. 2, pp. 115-132. Palembang: IndoMs.
[10] Reys, R. E., Suydam, M. N., Lindquist, M. M., & Smith, N. L. (1984). Helping
Children Learn Mathematics. (5th ed.). Boston: Allyn and Bacon.
[11] Shanty, N.O., Wijaya, S. (2012). Rectangular Array Model Supporting Students’
Spatial Structuring in Learning Multiplication. Journal on Mathematics Education
(IndoMS-JME) Vol. 3 No. 2, pp. 175-186. Palembang: IndoMs
[12] Surya, Y. (2011). Petunjuk Guru: Dasar-Dasar Pintar Berhitung GASING.
Tangerang: PT. Kandel.
[13] Surya, Y., Moss, M. (2012). Mathematics Education in Rural Indonesia. Proceeding
in the 12th International Congress on Mathematics Education: Topic Study Group
30, pp. 62236229. Seoul: Korea National University of Education.
Mathematics Finance
MEASURING AND OPTIMIZING MARKET RISK USING

VINE COPULA SIMULATION
KOMANG DHARMAWAN 1
1Department of Mathematics, Udayana University

E-mail: dhramawan.komang@gmail.com
Abstract. Copula models have increasingly become popular for the modeling of
the dependence structure of financial risks. The most recent copula topic is vine
copula or pair copula construction (PCC). This paper is concerned with the
application of pair-copula construction for measuring and optimizing the market
risk of a portfolio. The dependence among the assets is modeled using a copula
based on pair-copula constructions or known as vine copula, C-vine and D-vine.
Furthermore, a pairwise copula model is also discussed in this paper. In order to
achieve this aim, the return of each stock price is characterized individually. The
main objective of this paper is firstly to show how PCC may be useful
measuring and modeling market risk. The paper are also combining important
results published recently. Secondly, to show PCC can be used to model the tail
dependence of log-return distribution. Thirdly, to show the uses of stepwise
semiparametric estimators for the estimations of market risk in multivariate log-
return data. Fourthly, to show how PCC may be used on a daily basis in finance,
in particular for constructing efficient frontiers and computing Conditional VaR.
The paper is also to verify whether and how the optimal portfolio composition
discussed may change utilizing various types of copula families and their
construction, such as pairwise or vine. To do this, four Indonesian Blue Chip
stocks: ASSI=Astra International Tbk, BMRI=Bank Mandiri (Persero) Tbk.,
PTBA=Tambang Batubara Bukit Asam Tbk., INTP=Indocement Tunggal
Perkasa Tbk recorded during the period of 6-11-2006 to 2-08-2013 are used to
compose the stock portfolio. The VaR and Conditional VaR of the stock
portfolio is minimized by various types of copula and their pair constructions.
The use of stepwise semiparametric estimation is also demonstrated in this
paper.
Key words and Phrases: Value at Risk, Conditional Valur at Risk, Vine Copula,
multivariate copula
1. Introduction
In the last 10 years copula modeling has become a frequently used tool in
financial market analysis. Copula modeling in multivariate distribution was
initiated by Joe [10] in 1997. The copula theory available in [10] is based on
hierarchical model and has been used in some new developments in multivariate
modeling. The idea of Joe [10] originally proposed the pair-copula construction
265
266
(PCC) is further developed by Bedford and Cooke [4, 5], [8] and Kurowicka and
Cooke [11]. Aas et al. [1] have given the inferential methodology of how to infer
the parameters available in PCC. This inferential method has stimulated the use of
the PCC in various applications (see, for example, Schepsmeier et al. [20],
Nikoloulopoulos et al. [18], Min and Czado [14], and Mendes et al. [12]. There
have also been some recent applications of copulas in the context of time series
models (see the survey by Patton [19] and the recently developed COPAR model of
Brechmann and Czado [7], which provides a vector autoregressive VAR model for
analyzing the non-linear and asymmetric co-dependencies between two series).
This paper emphasizes the uses of vine copulae as they are useful tools to be
implemented efficiently in simulating the multivariate distribution. In fact, copula
functions can be used to model the dependence structure independently of the
marginal distributions. By this method, a multi-variate distribution with different
margins and a dependence structure can be constructed. This paper is intended to
provide some applications of pair copula constructions in portfolio risk
computations and efficient frontiers. The estimation of the parameters is based on
the maximum likelihood method given by Vogiatzoglou [23].
Furthermore, this paper also applies the supplement (or alternative) to VaR,
that is the Conditional Value-at-Risk. The CVaR risk measure is closely related to
VaR. For continuous distributions, CVaR is defined as the conditional expected
loss under the condition that it exceeds VaR, see Rockafellar and Uryasev [21, 22].
For continuous distributions, this risk measure also is known as Mean Excess Loss,
Mean Shortfall, or Tail Value-at- Risk. However, for general distributions,
including discrete distributions, CVaR is defined as the weighted average of VaR
and losses strictly exceeding VaR. Recently, the redefinition of expected shortfall
similarly to CVaR can be found in [2]. For general distributions, CVaR, which is a
quite similar to VaR measure of risk has more attractive properties than VaR.
CVaR is sub-additive and convex explained ([21] and [22]. Moreover, CVaR is a
coherent measure of risk discussed in [3] and proved in [15].
The main objective of this paper is firstly to show how PCC may be useful
measuring and modeling market risk. The paper are also combining important
results published recently such as [1],[4, 5], [8], [11], [12], [14], and [18].
Secondly, to show PCC can be used to model the tail dependence of log-return
distribution. Thirdly, to show the uses of stepwise semiparametric estimators for
the estimations of market risk in multivariate log-return data. Fourthly, to show
how PCC may be used on a daily basis in finance, in particular for constructing
efficient frontiers and computing Conditional VaR.
2. Pair-Copulas Constructions: A brief review

Consider a stationary d dimensional process X = (X1,··· ,Xd). If X is a
continuous random vector with joint cumulative distribution function (c.d.f.) F with
density function f, and marginal c.d.f.s Fi with density functions fi, for i = 1,··· ,d,
then there exists a unique copula C defined on [0,1]d such that
C(F1(x1),··· ,Fd(xd)) = F(x1,··· ,xd) (1)

holds for any (x1,...,xd)  R (Sklar’s theorem, Sklar (1959), Nelsen [17]). Therefore
d
a copula is a multivariate distribution with standard uniform margins. Multivariate

267
modeling through copulas allows for factoring the joint distribution into its
marginal univariate distributions and a dependence structure, its copula. By taking
partial derivatives of (1) one obtains
,…, ; , … ,…, ; 2
for some d-dimensional copula density c1···d with parameter θ.

This decomposition allows for estimating the marginal distributions fi
separated from the dependence structure given by the d-variate copula. In practice,
this fact simplifies both the specification of the multivariate distribution and its
estimation.
The decomposition of a multivariate distribution in a cascade of pair-copulas
was originally proposed by Joe [10], and later discussed in detail by Bedford and
Cooke [4],[5], Kurowicka and Cooke [11] and Aas et al.[1]. Refers to their results,
Equation (2) can be represented as
f(x1,··· ,xd) = fd(xd) · f(xd−1|xd) · f(xd−2|xd−1,xd)···f(x1|x2,··· ,xd). (3)
The conditional densities in (3) can be written as functions of the corresponding

copula densities. That is, for every j
f(x|v1,v2,··· ,vd) = cxvj|vj(F(x|vj),F(x|vj)),··· ,f(x|vj), (4)
where vj the d-dimensional vector v excluding the j-th component. For example
when d = 4, then the four-dimensional canonical vine (C-vine) structure is
generally expressed as
f(x1,x2,x3,x4;α,θ) = f(x1;α1) · f(x2;α2) · f(x3;α3) · f(x4;α4)
·c12(F(x1),F(x2);θ12)·c13(F(x1),F(x3);θ13)
·c14(F(x1),F(x4);θ14)·c23|1(F(x2|x1),F(x3|x1);θ23|1)
.c34|1(F(x2|x1),F(x4|x1);θ34|1)
· c34|12(F(x3|x1,x2),F(x4|x1,x2);θ34|12), (5)
and the D-vine structure as

f(x1,x2,x3,x4;α,θ) = f(x1;α1) · f(x2;α2) · f(x3;α1) · f(x4;α1)
· c12(F(x1),F(x2);θ12)·c23(F(x2),F(x3);θ13)
·c34(F(x3),F(x4);θ34)·c13|2(F(x1|x2),F(x3|x2);θ13|2)
· c24|3(F(x2|x3),F(x4|x3);θ24|3)
· c14|23(F(x1|x2,x3),F(x4|x2,x3);θ14|23). (6)
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where α and θ are the parameters of the margins and copula, respectively. The
general form of the joint probability density function (d-dimensional pdf), f() is
Figure 1. 4-dimensional D-vine copula
Figure 2. 4-dimensional C-vine copula
equal to
∏ , ∏ ∏ , | ,…,
(F(xi|xi+1,··· ,xi+j−1),F(xi+1|xi+1,··· ,xi+j−1);θ).
In a D-vine there d−1 hierarchical tress with increasing conditioning sets, and there
are d(d − 1)/2 bivariate copulas. For a detailed description, see Asa [1]. Figure 1
shows the D-vine decomposition for d = 4. It consists of 3 nested trees, where Tj
possess 5 − j nodes and 4 − j edges corresponding to pair copula. Figure 2 shows
the C-vine decomposition for d = 4. It consists of 3 nested trees, where Tj possess 5
− j nodes and 4 − j edges corresponding to pair copula.
As is mention in Berg and Aas [6] and Bedford and Cooke [4], the use of
vine copula is very flexible, in this case, one may choose different copula in the
bivariate.
For example, one may combine the following types of (bivariate) copulas:
Gaussian pair copula (no tail dependence, elliptical); t-student pair copula, Clayton
269
(lower tail) pair copula. See Joe [10] for a copula catalogue. Here below is the
example D-vine consisting of Gaussian pair-copulae and margins. Parameters,
θ12,θ23,θ34, are estimated using maximum likelihood estimation method as in
Vogiatzoglou [23]. One may refer to Haff [9] for stepwise semiparametric
estimations (SSP).
(7)
In this case, θ13,θ14, and θ24 are estimated using Kendall’s τ formula, that is
(8)
(9)
(10)
The following is the four dimensional C-vine consisting student’s-t margins and
Student’s t pair-copulae. Another method to calculate c23,c24 and c34 is to employ
the stepwise semiparametric (SSP) estimators developed by Haff [9]. The
following example is C-Vine copula for which the parameters are estimated using
SSP method.
(11)
where
(12)
270
where ui = tν(xi), and tν is the c.d.f. of the Students t-distribution with ν degrees of
freedom. See Haff [9] for d-dimensional C-vine formulation. As reported by
Vogiatzoglou in [23] that the log likelihood function of the canonical vine t-copula
model (tCVine) did not converge to a solution, hence SSP reported by Haff in [9] is
alternative estimation methods. Mention in by Haff [9] that SSP estimator is
computationally tractable even in high dimensions, as opposed to other methods
such as MLE or IFM method.
3. Optimization of Value at Risk

Consider a portfolio consist of d stocks and denote by wi the weight of stock i
allocated to the portfolio at time t. Let f(w,r) be the loss function of the portfolio,
with w ∈ Rd is the vector of the portfolio and r is the vector of asset returns. Let ξ
be a certain threshold, the Value-at-Risk of a portfolio at level α is defined as the
lower α−quantile of the distribution of the portfolio return
VaR(α) = inf{ξ ∈ R : P(f(w,r) ≤ ξ) ≥ α} (15)
The uses of VaR to measure financial risks have been increasingly popular since
1990s. VaR now becomes the standard risk measure used by financial analysts to
quantify the market risk of an asset or a portfolio.
CVaR is a supplement or an alternative to VaR. CVaR is another percentile
risk measure which is called Conditional Value-at-Risk. For continuous
distributions, CVaR is defined as the conditional expected loss under the condition
that it exceeds VaR. The following approach to CVaR is summarized from [21] and
[22]. Let rp be the return of the portfolio, then rp is defined as a random variable
satisfying rp = w1r1 + w2r2 + ··· + wdrd = wTr and the weight constrain condition is
imposed to be = 1 If the short position is not allowed then wi ≥ 0 for all t =
1,··· ,d. Let p(r) be the joint distribution of the uncertain return of the assets, the
probability of rp exceeding a certain amount r* is given by
271
∗ then is equivalent to Ψ , ,
16
where Ψ(w,) represents the cumulative distribution function for the associated
loss w. Assuming Ψ(w,ξ) is continuous with respect to ξ, VaR(α) and CVaR(α) for
the loss f(w,r) associated with w any probability level α  (0,1) can be defined by
VaR , min ∈ : Ψ , (17)
CVaR , ∈
max , ,0 (18)

Equation (17) is read as the smallest value ξ such that the probability P[f(w,r) > ξ]
of a loss exceeding ξ is not larger that 1 − α. Therefore, VaRα presents (1 − α)-
quantile of the loss distribution Ψ(w,ξ) and CVaRα presents the conditional
expected loss associated with w if VaRα is exceeded. Following [22], the CVaR(α)
of the loss associated with any w, it is found
CVaR(α) = minFα(w,ξ), (19)
ξ∈R
with
r (20)
Now, the conditional value-at-risk, CVaR, is defined as the solution of an
optimization problem
r (21)
where α is the probability level such that 0 < α < 1. CVaR also is known as Mean
Excess Loss, Mean Shortfall (Expected Shortfall), or Tail Value-at-Risk, see [2].
However, for general distributions, including discrete distributions, CVaR is
defined as the weighted average of VaR and losses strictly exceeding VaR, see
[21]. Some properties of CVaR and VaR and their relations are studied in [2] and
[15]. For general distributions, CVaR, which is a quite similar to VaR measure of
risk has more attractive properties than VaR. CVaR is sub-additive and convex see
[21]. Moreover, CVaR is a coherent measure of risk, proved first in [15], see also
[2] and
[21].
Therefore, the problem of minimizing the Conditional Value at Risk can thus
be formulated as the following:
minimize r (22)
subject to (23)
T *
−w E(r) ≤ −r (24)
272
Table 1. Descriptive Statistics

Statistics ASII BMRI PTBA INTP
Mean 0.0009 0.0006 0.0006 0.0009
Standard deviation 0.0276 0.0271 0.0317 0.0307
Skewness 0.1306 0.3690 ‐0.2404 0.370
Kurtosis 10.0123 8.2807 13.0502 45.0897
Jarque‐Bera Test (5%) 1 1 1 1
1 = reject H0
Jarque‐Bera Statistics 35700 20612 73398 12848
Jarque‐Bera p‐Values 0.001 0.001 0.001 0.001
Jarque‐Bera Crit‐Value 5.9586 5.9586 5.9586 5.9586
In this case the feasibility set is X defined on region satisfying (23) and (24). Set X
is convex (it is polyhedral, due to linearity in constraint (23) and (24). The
optimization problem (22)-(24) are a convex programming. This problem is solved
using PortfolioCVaR class of MATLAB R2013a.
4. Empirical studies
In this section, the models of stock portfolio risk measurement and
management (described in the previous sections) are implemented to a hypothetical
portfolio composed by 4 Indonesian Blue Chip stocks: ASSI=Astra International
Tbk, BMRI=Bank Mandiri (Persero) Tbk., PTBA=Tambang Batubara Bukit Asam
Tbk., INTP=Indocement Tunggal Perkasa Tbk. The historical data are recorded
during the period of 6 November 2006 to 2 August 2013. Precisely, the statistics of
the 4 stocks are described in Table 1.
Table 1 reflects the mean, standard deviation, skewness, and kurtosis of the
daily returns of four stocks over the period from 6 November 2006 to 2 August
2013. The kurtosis of all stock returns exceeds the kurtosis of normal distribution
(3.0) substantially. The Jarque-Bera tests of the null hypothesis that the return
distribution follow a normal distribution against the alternative that the return do
not come from a normal distribution. As seen from Table 1. that the test statistics
exceed the critical value at the 5% level of significant, this means that all stock
returns are not normally distributed, it shows a fat tailed distribution. This means
that the probability of extreme events is higher than the probability of extreme
events under the normal distribution. Therefore, capturing the stock returns using
normal distribution could be underestimated. When the skewness of the return data
are considered, it shows that all returns are right skewed except for PTBA. This
suggests that the returns are not symmetry. Again, capturing by normal distribution
may result in misleading conclusions.
Using results in Table 2 and Table 3, the value of c13,c14, and c24 can be
calculated using Equation (8)-(10), giving the correlation matrix ρ. Using ρ, the
273
Table 2. Estimated parameters and the (standard errors) from D-vine models
D‐vine copula
Clayton SJC
c12 0.3057 (0.012) 0.3374 (0.036) 0.2270 (0.038)
c23 0.2636 (0.014) 0.1480 (0.018) 0.0541 (0.038)
c34 0.1810 (0.015) 0.0992 (0.032) 0.0749 (0.033)
c13|2 0.1059 (0.015) 0.4167 (0.017) 0.3650 (0.027)
c24|3 0.1793 (0.015) 0.2269 (0.027) 0.0863 (0.028)
c14|23 0.0982 (0.014) 0.2212 (0.024) 0.0538 (0.029)
AIC ‐1483.7095 ‐1871.350
BIC ‐1450.9396 ‐1805.8104
LL 747.855 948.753
Table 3. Estimated parameters and the (standard errors) from C-vine

models
D‐vine copula
Clayton SJC
c12 0.3057 (0.012) 0.3374 (0.036) 0.2270 (0.038)
C13 0.2337 (0.014) 0.2274 (0.031) 0.0704 (0.031)
C14 0.2401 (0.014) 0.0415 (0.027) 0.0266 (0.023)
C23|1 0.1542 (0.015) 0.4127 (0.024) 0.3112 (0.030)
c24|1 0.1391 (0.016) 0.3133 (0.029) 0.1864 (0.033)
c14|2 0.0464 (0.014) 0.1604 (0.034) 0.0005 (0.003)
AIC ‐1487.152 ‐1873.5069
BIC ‐1454.382 ‐1807.9672
LL 749.576 947.675
return of the portfolio (rp) consisting of 4 assets are simulated by Gaussian

copularnd() of MatLab function. The vector weight w = [0.25 0.25 0.25 0.25] is
used to compose the portfolio. Then apply PortfolioCVaR class [16] to rp = wTr.
The results are summarized in Table 4. Note that, for t C-Vine model, we use
stepwise semiparametric estimator (SSP) proposed by Haff [9] (Eqn.(11)-(14)), to
construct correlation matrix ρ.
Table 4 reflects the value of VaR and CVaR simulated by Clayton D-Vine, t
C-Vine, Symmetrized Joe-Clayton (SJC) D-Vine, Gaussian pairwise, t student
pairwise, and multivariate normal at 1% and 5% significant levels. The multivariate
normal calculated using the standard Markowitz mean-variance (MV) framework.
As it was shown in [22] that when the loss functions come from normal distribution
then VaR and CVaR are equivalent in the sense that they generate the same
efficient frontier. However, in the case of nonnormal distribution, and especially
skewed distributions, CVaR and MV portfolio optimization approaches result in
significant
274
Table 4. Simulated VaR and CVaR
α = 0.01 α = 0.05
Methods
VaR (%) CVaR (%) VaR (%) CVaR (%)
Clayton D‐Vine 4.99 5.99 3.07 4.32
t C‐Vine 6.69 9.59 3.50 5.57
SJC D‐Vine 4.75 6.23 2.83 4.08
Gaussian pairwise 5.68 7.75 3.33 4.81
t student pairwise 5.76 8.07 3.19 5.05
Multivariate 5.01 5.77 3.71 4.63
Normal

differences. The main idea of using the CVaR optimization technique is that it can
reshape one tail of the loss distribution, which corresponds to high losses, and it
can not account for the opposite tail representing high profits. On the other hand,
the MV approach consider the risk as the variance of the loss distribution, and since
the variances (risk) can come from both tails (upper and lower), hence it is affected
by high gains as well as by high losses.
5. Concluding remark
In this paper we have explored the potentials of PCC to model the
dependence among financial data. A fully flexible multivariate distribution was
obtained by constructing marginals with different distribution into C-vine and D-
vines model. The marginals of the data show the asymmetric, high kurtosis, and
skewed right except for PTBA. The simulation results show that t-student C-vine
gives highest estimation compare with the other model (see Table 4). This is due to
the fact that the parameters of t-student C-vine model is estimated by stepwise
semiparametric estimation, which is according to Haff [9] giving higher estimates
for correlations.
The pair copulas construction still needs further research on tests for
choosing among copula families and among decompositions, and more powerful
goodness-offit tests. Further research topics include time-varying pair-copulas may
be one of interest in multivariate modeling.
Figure 3 shows that for a given risk (volatility), one would like to choose a
portfolio that gives you the greatest possible rate of return, in this case one may
choose Clayton CVine. Gaussian Pairwise gives the lowest rate of return for a
given risk.
275
Figure 3. Efficient frontiers simulated by pairwise and vine cop-

ula with α = 1%
References
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multiple dependence. Insurance: Mathematics Economics. (2009) 44:128-198
[2] Acerbi, C., Nordio, C., Sirtori, C. Expected Shortfall as a Tool for Financial Risk
Management. Working Paper, can be downloaded from:
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Risk. Mathematical Finance, 9, 203-228.
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dependent random variables modeled by vines. Annals of Mathematical and
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[5] Bedford, T. and Cooke, R. M. Vines: a new graphical model for dependent random
variables. Ann. Statist. (2002). 30(4): 1031.1068.
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comparison study. European Journal of Finance (2009), 15, 639-659.
[7] Brechmann, E.C., and C. Czado. COPAR - multivariate time-series modelling using
the COPula AutoRegressive model, Working Paper, Faculty of Mathematics,
Technical University of Munich. (2012)
[8] Cooke, R.M., H. Joe and K. Aas, Vines Arise, chapter 3 in DEPENDENCE
MODELING Vine Copula Handbook, Ed. D. Kurowicka and H. Joe, World Scienti c
Publishing Co, Singapore. (2011)
[9] Haff, I. H. Parameter estimation for pair-copula constructions. Bernoulli 19(2),
(2013), 462491
[10] Joe, H. Multivariate Models and Dependence Concepts. (1997). London: Chapman
& Hall
[11] Kurowicka, M. and Cooke, R. M. Uncertainty Analysis with High Dimensional
Dependence Modelling (2006), John Wiley & Sons, New York.
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finance. Financial Markets and Portfolio Management (2010), 24 (2), 193-213.
[13] Min, A., Czado, C., Bayesian inference for multivariate copulas using pair-copula
constructions. Journal of Financial Econometrics,(2010), 8(4).
[14] Min, A., Czado, C., Bayesian model selection for multivariate copulas using pair-
copula constructions. Canadian Journal of Statistics (2011), 39(2), 239-258.
[15] Pflug G. Ch. Some Remarks on the Val-ue-At-Risk and the Conditional Value-At-
Risk In: S. Uryasev, Ed., Pro- babilistic Constrained Optimization: Methodology and
Applications, Kluwer Academic Publishers, Dordrecht, (2000)
[16] MATLAB version 8.01.0 (R2013a). Natick, Massachusetts: The MathWorks Inc.,
(2013).
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Science+Business Media, Inc. (2006)
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[19] Patton A. J. A review of copula models for economic time series. Journal of
Multivariate Analysis 110 (2012) 418
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copulas: Vine Copula Package Version February 15, 2013.
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[21] Rockafellar R. T. and S. Uryasev. Optimization of Conditional Value-at-Risk.
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Distributions, Journal of Banking and Finance (2002), Vol. 26, pp. 1443-1471.
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156, Thessaloniki, Greece. Version: November 07, (2010)
Mathematics Finance
MONTE CARLO AND MOMENT ESTIMATION FOR

PARAMETERS OF A BLACK SCHOLES MODEL FROM
AN INFORMATION-BASED PERSPECTIVE (THE BS-
BHM MODEL):A COMPARISON WITH THE BS-BHM
UPDATED MODEL
MUTIJAH1, SURYO GURITNO2, GUNARDI3
1STAIN Purwokerto, mutipurwokerto@yahoo.com

Ph.D Student of Department of Mathematics Gadjah Mada University
2Department of Mathematics Gadjah Mada University, suryoguritno@ugm.ac.id
3Department of Mathematics Gadjah Mada University, gunardi@ugm.ac.id
Abstract. This paper presents estimation of parameters on the BS-BHM model by

using Monte Carlo and Moments estimate as they have been done in BS-BHM
Updated model. BS-BHM Updated model is BS-BHM model that it is improved the
result of Gaussian integral, especially in completing square. Estimation of parameters
use Monte Carlo and moments estimate under BS-BHM model results the equation of
polynomial of four degree . While estimation of parameters under BS-BHM Updated
model results the quadratic equation. Application for real data of Microsoft shares
(MSFT), under BS-BHM model results four different estimates values, while under
BS-BHM Updated model results one estimate value.
Key words and Phrases : BS-BHM model, Monte Carlo estimate, Moment
estimate,and Comparison.
1. Introduction
Black Scholes asset pricing model from an information-based perspective has been
developed by Brody Hughston Macrina (BHM). It is developed from a special
condition of asset pricing model from an information-based approaches by Brody
Hughston Macrina ( BHM model or BHM approach ). Further, Black Scholes
model from an information-based perspective is called BS-BHM model in this
paper. Explicitly, BS-BHM model is presented, see Macrina, A.[6]
ν√T ν√T
St PtT S exp rT - ν T

(1.1)
or
ν√T √T
St S exp rt - ν T (1.2)

277
278
or
St ν√T √T
= exp rt - ν T (1.3)
S
The BS-BHM model in equation (1.2) which it is equal to equation (1.3) has two
parameters i.e. the asset price volatility parameter ν and the information flow rate
S
parameter σ. It can be showed that the random variable of t has lognormal
S
distribution with density function
S
St log t A
S
f exp t(T-t)
(1.4)
S St t(T-t) b σ t
√ π σ t T
S T
or
S
St log t A
S
f St exp (1.5)
S √ π
S
ν√T
where A = rt - ν T and

t(T-t) √T t(T-t)
B =b σ t σ t
T T
St
In other words, random variable of log is normally distributed with mean is
S
ν√T √T t(T-t)
A=rt - ν T and variance is B σ t . Further,
T
St
it is written by log ∼N(A, B2).
S
The BS-BHM model as in equation (1.2) is derived from BHM model by a
special condition. BHM model is built for case of cash flow is payout of the
associated dividend of equity. Further, the process of deriving of BS-BHM model
as in Macrina,A [6] and Mutijah, Guritno, S. and Gunardi [7].
Explicitly, the asset pricing model is presented as follows,
St = PtTEℚ[DT| t] (1.6)
St is the value of cash flows at time t, 0 ≤ t < T from asset that payout single
dividend DT at time T . In equation (1.6), PtT represents the discount factors that it
is to be equal to e-r(T-t) with r is the interest rates. Then ℚ is the risk neutral
probability, and t is the market information filtration.
Modeling the infornation flows is based on an assumption that the
information about dividends which is available in market is contained by the
process { t}0≤t≤T defined by :
t = tDT + tT (1.7)
{ t} is a market information process. The market information process is composed
from two parts, they are tDT which refers to the true information about dividends
and {βtT}0≤t≤T which refers to a standard Brownian Bridge on interval [0, T]. In the
formula of asset pricing model by Brody Hughston Macrina in equation (1.6)
above, if random variable DT is equal to x which it has continuous distribution
then ,
Eℚ[DT| t] = Eℚ[DT| t] = x t x) dx (1.8)
where
d
t x) = ℚ DT x|ξ (1.9)
dx
By using Bayes formula in Box-Tiao [2], πt x) is presented in Brody, D.C,

279
Hughston, L.P, and Macrina, A [1], Caliskan, N [3], Macrina,A [6] and Mutijah,
Guritno, S. and Gunardi [7] as follows
x)ρ t |DT x
πt x) = (1.10)
t
and the final result of the BHM model or the BHM approach,
T 1 2
x p(x) exp σxξt - σ x2 t dx
T-t 2
St PtT 2
(1.11)
∞ T 1
p x exp σxξt - σ x2 t dx
T-t 2
Brody Hughston Macrina also built the other concept for the asset pricing
model that it is derived from the formula of equation (1.6) for a special condition
where it is a limited-liability asset which pays no interim dividends and at time T it
is sold off for the value ST. ST is log-normally distributed and has the form of
ST = S0 exp( T - T √T X ) (1.12)
where S0, , are given constants and XT is a standard normally distributed
random variable. The corresponding information process is given by
t = σtXT + tT (1.13)
The price proses {St}0≤t≤T is obtained from :
St = PtT Eℚ(∆T(XT)| t) (1.14)
Then for t < T , the equation St results :
St PtT ∆T x) πtT x) dx (1.15)
And by the Bayes formula, it is obtained πtT x) as follows
T
p(x) exp x x t
T-t
πtT x) = T
p(x) exp x x t
T-t
(1.16)
In this case, ST plays the role of single cash flow ∆T x) for XT = x.
So, it is obtained the equation St as follows
St PtT S exp rT - ν T
T
p(x) exp x x t
T-t
ν√T x T
dx (1.17)
p(x) exp x x t
T-t
Because XT is assumed to be standard normally distributed then
p(x) = exp(- x (1.18)
√
To follow the Gaussian integrals in Macrina, A [6] and Straub, W.O. [12] then St
becomes
T T
exp(- x exp √T x
√ T-t T-t
St PtT S exp rT - ν T T T
exp(- x exp x
√ T-t T-t
(1.19)
By using Gaussian integrals, the equation of asset pricing model St is given below
√T
St S exp rt - ν T

(1.20)
tT
where τ . Successive steps to obtain the model in equation (1.20) can be
T-t
seen in Mutijah, Guritno, S. and Gunardi [7]. Furthermore, the model in equation
(1.20) is called the BS-BHM Updated model.
In BS-BHM model, there are also the asset price volatility parameter ν and
280
true information flow rate parameter σ which they can not be observed directly.
This paper will discuss the two parameter estimation for the previous behaviour of
asset price. The estimation value of parameter ν and σ arisen from this general
procedure is called a historical volatility and information flow rate estimation.
The deriving of BHM model with a special condition which is built by Brody
Hughston Macrina , it must be as in equation (1.20).Therefore, this paper will
compare the results of parameter estimation between estimation of parameters and
its application for real data of Microsoft shares (MSFT) under the BS-BHM model
with under the BS-BHM Updated model. Hence, estimation of parameters and its
application for real data of Microsoft shares (MSFT) under the BS-BHM Updated
model can be seen in Mutijah, Guritno, S. and Gunardi [8].
2. Main Results
2.1. Monte Carlo Estimate

Estimation procedure of Monte Carlo under the BS-BHM the same as
estimation procedure of Monte Carlo under the BS-BHM Updated model. They are
based on procedure by Higham,D.J. [5] as follows
Suppose that historical asset price data is available at equally spaced time
St
values ti = i Δt ,so St is the asset price at time ti . Defined Ui log and
St
Ui are independent as in Higham, D.J. [5]. To estimate the asset price volatility ν
and the information flow rate σ on the BS-BHM model by using Monte Carlo
approach that it is written Higham, D.J. [5] as follows :
Suppose that t = t is the current time and that the M+1 is most current asset
prices. Stn-M , Stn-M+1, … Stn-1 , Stn is also available and by using the corresponding
M
log rasio data which is Un+1-i , then the sample mean dan variance estimation
i=1
are
aM ∑M
i=1 Un+1-i (2.1)
M
and
bM ∑M
i=1 Un+1-i aM (2.2)
M -1
Monte Carlo estimate is done by comparing the sample mean with the mean of BS-
BHM model or by comparing the sample variance with the variance of BS-BHM
model as below
ν√T
aM rt - ν T

or
2 (aM rt
ν ν 0 (2.3)
√T
and
2
√T t(T-t)
b2M σ 2 t2
T
2
√T t(T-t)
σ 2 t2 ν
T
or
b2M
ν 2 (2.4)
√T t(T-t)
σ2 t2
T
281
Substitution equation (2.4) to equation (2.3) is obtained by algebra tricks

successively as below
b2M b2M 2 (aM rt
2 . 0
√T t(T-t) √T √T t(T-t)
σ2 t2 σ2 t2
T T
Making the same in denumerator to the equation is obtained
t(T-t) t(T-t)
bM t t bM σ t 2 (aM √T σ t
T T
t(T-t)
0
√T σ t
T
By dividing to multiplying then
t(T-t) t(T-t)
bM t σ τ 1 στt bM σ t 2 (aM rt στ √T σ t 0
T T
Changing in form of equation the same as
t(T-t) 2
στt b2M σ2 t2 bM t σ τ 1 2 aM rt στ √T σ 2 t2
T
t T t
T
And by multiplying to dividing is obtained
2 t T t
t(T-t) b2M 2 aM rt √ σ2 t2
σ 2 t2 + T
T
t b2M
The left and right hands are squared
t(T-t) b2M t t T t
σ2 t2 + 4T aM rt σ 2 t2 σ τ 1
T σ τ T
t T t 2
4 aM rt 2 σ τ T σ2 t2
T
t b2M
Thus it is made in equation is to be equal to zero
b t σ τ 1 t T t
2
4T a rt σ t σ τ 1
σ τ T
4 a t(T-t) rt σ τ T σ t
0 - σ 2 t2
t bM T
Successively, by algebra tricks then it is held the final result is the polynomial
equation as below
Aσ8 Bσ6 Cσ4 Dσ2 E 0
where
4 aM rt 2 τ T t2
A = b2M t τ + 4T aM rt t τ +
b2M
2
t (T – t) τ2 8 aM rt τ t T(T-t)
B = b2M t τ + 4T aM rt 2t τ +
T b2M
4 aM rt 2 τ (T-t)
t (T – t)
C = 6 b2M t + 4T aM rt t +
b2M
4 b2M t t (T – t)
D= + 4 aM rt t T t
τ T
b2M
E=
τ
Let σ2 = x, then it is obtained the polynomial of four degree
Ax4 Bx3 Cx2 Dx E 0
The equation of polynomial of four degree has four solution in x = σ2 .
282
Further, to determine estimation of parameter ν then the solutions in x = σ2

are substituted to equation (2.3) that is
1 2 (aM rt
ν - 2 ν 0
√T (σ τ+ 1)
Because equation (2.3) has solution
1 2 (aM rt
ν - 2 4.1.
√T √T (σ τ+ 1)
or
1 2 (aM rt
ν - 2 4.1.
√T √T (σ τ+ 1)
so to determine estimation of parameter ν then the solutions in x = σ2 also can be

substituted to two solution above. Finally, the estimate of parameters ν and σ are
held.
2.2. Moment Estimate

Analogous to Monte Carlo estimate, suppose that historical asset price data
is available at equally spaced time values ti = i Δt ,so St is the asset price at time ti.
St
Defined Ui log and Ui are independent. Parameters estimation of asset
St
price volatility ν and the information flow rate σ of BS-BHM model using the
method of moment as follows
Suppose that t = t is the current time and that the M+1 is most current asset
prices. Stn-M , Stn-M+1, … Stn-1 , Stn is also available and by using the corresponding
M
log rasio data which is Un+1-i then the first sample moment (m mean) and
i=1
the second sample moment (m , see Higham, D.J.[5], Mutijah, Guritno, S., and
Gunardi [8], Shao, J. [11], and Subanar [13] are
m ∑M
i=1 Un+1-i (2.5)
M
and
m ∑M i=1 Un+1-i (2.6)
M
Estimation of parameters in moment estimate can be obtained by making the k-th
moment of the sample to be equal to the k-th moment of the model.
Suppose μ dan μ are the first moment and the second moment for BS-
BHM model, then
ν√T
μ E U rt - ν T (2.7)

and
2
μ2 E U2i Var Ui E Ui
√T t(T-t) ν√T
σ t rt - ν T (2.8)
T
From equation (2.5) and equation (2.7) are obtained equation belows
ν√T
m rt - ν T

or
m rt
ν ν+ 0 (2.9)
√T
283
and from equation (2.6) and equation (2.8) are also obtained equation
√T t(T-t) ν√T
m σ t rt - ν T
T
√T t(T-t)
σ t ν m
T
or
m m m m m m
ν σ τ σ τ (T-t)

√T t(T-t) σ τ σ τ (T-t)
σ t
T

m2 m1

σ τ tT σ τ (T-t)
(2.10)
Substitution of equation (2.10) to equation (2.9) is obtained
m2 m1 m2 m1 2(m1 rt
2 . 0
√T t(T-t) √T √T t(T-t)
σ2 t2 σ2 t2
T T
Making the same in denumerator to the equation is obtained
t(T-t) t(T-t)
m m t t m m σ t √T σ t
T T
t(T-t)
0
√T σ t
T
By dividing to multiplying then m2 m1 t σ τ 1 στt
t(T-t)
m2 m1 σ2 t2
T
2 t(T-t)
2(m1 rt στ √T σ 2 t2 0
T
Changing in form of equation the same as
t(T-t)
στt m2 m1 σ2 t2 m2 m1 t σ τ 1
T
2 t T t
2 m1 rt στ √T σ 2 t2
T
and by multiplying to dividing is obtained
2 t T t
t(T-t) m2 m1 2 m1 rt √ σ2 t2
σ 2 t2 + T
T t m2 m1
The left and right hands are squared
t(T-t) m2 m1 t t T t
σ 2 t2 + 4T m1 rt σ 2 t2 σ τ 1
T σ τ T
t T t 2
4 m1 rt 2 σ τ T σ2 t2
T
t m2 m1
Thus, it is made in equation is to be equal to zero
m m 1
4 1
σ τ2
4
t(T-t) σ τ T
- σ2 t2 0
t m m T
Successively, by algebra tricks then it is held the final result is the polynomial
equation as below
Aσ8 Bσ6 Cσ4 Dσ2 E 0
where
4 m1 rt 2 τ T t2
A = m2 m1 t τ + 4T m1 rt t τ +
m2 m1
284
t (T – t) τ2 8 m1 rt 2 τ t T(T-t)
B = m2 m1 t τ + 4T m1 rt 2t τ +
m2 m1
t (T – t) 4 m1 rt 2 τ (T-t)
C = 6 m2 m1 t + 4T m1 rt t +
m2 m1
4 m2 m1 t (T – t)
D= + 4 m1 rt t T t
m2 m1
E=
τ
Let σ2 = x, then it is obtained the polynomial of four degree
Ax4 Bx3 Cx2 Dx E 0
The equation of polynomial of four degree has four solution in x = σ2 .
Further, to determine estimation of parameter ν then the solutions in x = σ2
are substituted to equation (2.9) that is
1 2 (aM rt
ν - 2 ν 0
√T (σ τ+ 1)
Because equation (2.9) has solution
1 2 (aM rt
ν - 2 4.1.
√T √T (σ τ+ 1)
or
1 2 (aM rt
ν - 2 4.1.
√T √T (σ τ+ 1)
so to determine estimation of parameter ν then the solutions in x = σ2 also can be

substituted to the two solution above. Finally, estimation of parameters ν and σ are
held too.
2.3. Application for Example of Real Data

Under BS-BHM and BS-BHM Updated model, estimation of historical
volatility and the information flow rate of Microsoft (MSFT) shares for Monthly
data are done by using Monte Carlo and moment estimates. For two parameters
and two methods of estimates, it is assumed that the data corresponds to equally
spaced points in time as in Higham, D.J [5], and Mutijah, Guritno, S., and
Gunardi [8].
In Monte Carlo estimate, the monthly data runs over 5 years (T = 5 years)
and has 60 asset prices (M = 59), so it has dt = T/M = 5/59 ≈ 0,084746. For the
monthly data result in aM 1,47 x 10 and bM 1,056 x 10 . Under BS-
BHM model, estimation based on Monte Carlo produces estimation of parameters
ν and σ , they are
ν 3.1822 , σ 3.9882,
ν 0.0796 0.6060i, σ 1.0545 3.7081i,
ν 0.0796 0.6060i , σ 1.0545 3.7081i,
ν 1.3451 , σ 0.1084
ν 2.1200 , σ 3.9882,
ν 0.1205 0.9014i, σ 0.1205 0.9014i,
ν 0.1205 0.9014i, σ 1.0545 3.7081i,
ν 0.8933 , σ 0.1084
There are eight estimation of parameters ν and σ. They include four real
number and four imaginary number. Therefore, estimation of parameters ν and σ
are chosen four real number.
285
In moment estimate, the monthly data runs over 5 years (T = 5 years) and
has 60 asset prices (M = 59), so it has dt = T/M = 5/59 ≈ 0,084746. For the
monthly data result in m 1,47 x 10 and m 1,1 x 10 . Under BS-BHM
model, estimation based on the moment estimate produces estimation of
parameters ν and σ, they are
ν 3.1849 , σ 3.9911,
ν 0.0803 0.6057i, σ 1.0537 3.7091i,
ν 0.0803 0.6057i , σ 0.0803 0.6057i,
ν 1.3451 , σ 0.1085
ν 2.1218 , σ 3.9911,
ν 0.1215 0.9010i, σ 1.0537 3.7091i,
ν 0.1215 0.9010i , σ 0.0803 0.6057i,
ν 0.8933 , σ 0.1085
The same as Monte Carlo estimate, there are eight estimation of parameters ν and σ
too. They also include four real number and four imaginary number. Therefore,
estimation of parameters ν and σ are chosen four real number too.
Whereas, to compare the results of estimation parameters under BS-BHM model
with under BS-BHM Updated model, it can be seen in Mutijah, Guritno,S, and
Gunardi [8].
BHM model or BHM approach is modelled by Brody Hughston Macrina

(BHM) under the assumption that market participants do not have acces to the
information about the actual value of the relevant market variable. Brody Hughston
Macrina defined an asset by its cash flow structure and then the associated market
factor is the upcoming cash flow that is the upcoming dividend. Brody Hughston
Macrina also built BHM model by a special condition which it is called Black
Scholes model from an information-based perspective or it is called BS-BHM
model in this paper. Further, the BS-BHM model that it is improved the result of
Gaussian integral, especially in completing square is named BS-BHM Updated
model. Both BS-BHM model and BS-BHM Updated model have lognormal
distribution. BS-BHM model and BS-BHM Updated model also have the same as
two parameter that is the volatility parameter ν and the information flow rate
parameter σ. Estimation of the two parameters result an equation of polynomial of
four degree under BS-BHM model and rerult a quadratic equation under BS-BHM
Updated model. Application for real data of Microsoft shares results four value of
estimation of parameters, while under BS-BHM Updated model result one value of
estimation of parameter. All about BS-BHM Updated model can be seen
Mutijah,Guritno, S, and Gunardi [7, 8].
Acknowledgement. Authors are very grateful for Department of Mathematics of

Gadjah Mada University for the facilities that they are given us to our works.
Authors would like to thank for committee of The IndoMS International
Conference on Mathematics and Its Applications 2013 (IICMA2013) which they
have given opportunity to present our topic in IICMA2013. Especially, Authors are
grateful to anonymous reviewrs for comments and suggestions.
286
References
[1] Brody, D.C, Hughston, L.P, and Macrina, A.,2006,Information-Based Asset Pricing,
Working Paper.
[2] Box-Tiao,1973, Bayesian Inference in Statistical Analysis,Addison-Wesley Publising
Company,Department of Statistics University of Wisconsin.
[3] Caliskan, N.,2007,Asset Pricing Models:Stochastic Volatility and Information-
Based Approaches,Thesis,Department of Financial Mathematics.
[4] Cochrane, J.H.,2000,Asset Pricing,Graduate Scholes of Business University of
Chicago.
[5] Higham, D.J.,2004,An Introduction to Financial Option Valuation
Mathematics,Stochastics and Computation, Cambridge University Press,Department
of Mathematics University of Strathclyde.
[6] Macrina, A.,2006,An Information-Based Framework for Asset Pricing:X-Factor
Theory and its Applications, Dissertation, King’s College London.
[7] Mutijah, Guritno, S, and Gunardi,2012,A Black Scholes Model from an
Information-Based Perspective by Brody Hughston Macrina,International
Conference on Statistics in Science, Business and
Engineering,Ready_ICSSBE2012-109
[8] Mutijah, Guritno,S, and Gunardi,2013,Estimation of Parameters on the BS-BHM
Updated Model. Journal Applied Mathematical Sciences, Vol. 7, 2013, No. 72,
3555-3568.
[9] Rupert, D.,2004,Statistics and Finance An Introduction,Springer-Verlag New
York,LLC.
[10] Shreve, S.E.,2004,Stochastic Calculus for Finance II.Springer Science+Business
Media,Inc.
[11] Shao,J,2003,Mathematical Statistics Second Edition,Springer Science+Business
Media,LLC,New York.
[12] Straub, W.O., 2009,A Brief Look at Gaussian Integrals,Article,Pasadena California.
[13] Subanar,2013,Statistika Matematika,Graha Ilmu,Yogyakarta.
Mathematics Finance
EARLY DRUGS DETECTION TENDENCY FACTOR’S

MODEL OF FRESH STUDENTS IN MATHEMATICS
DEPARTMENT UI
DIAN NURLITA1, RIANTI SETIADI2
1Department Matematika FMIPA-UI, mathui2010@yahoo.com

2Department Matematika FMIPA-UI, ririnie@yahoo.com.sg
Abstract. In the last decade, Indonesia faced the increasing number of drug abuse
very serious especially among young people. One of the people that can be attached to
any drugs is a university student. The tendency of an attachment to drugs can be
detected early. One of the tools that can be used to measure the tendency of an
attachment to drugs is a psychological test. This tool can detect whether a person will
be tied to drugs or not. The purpose of this paper is looking for a significant factors
that affect this tendency and find the model based on those factors to determine
whether a person will have a tendency to be tied on drug or not. The factors that are
considered are gender, age, ethnic group, education of student’s parent, employment
status of student’s parent, with whom respondents resided, score OAT (Obvious
Attributes), score DEF (Defensiveness), score SAM (Supplemental Addiction
Measure), and score Cor (Correctional). The methods that will be used here are
clustering, decision tree, and discriminant analysis. The study is applied to fresh
students in Mathematics Department, University of Indonesia, 2013.
Key words and Phrases : Narcotics, Clustering, Tree, Discriminant Analysis.
1. Introduction
A. Background
In the last decade, drugs continues to threaten the lives of the people in
Indonesia. Based on data from Direktorat Tindak Pidana Narkoba Bareskrim Polri
& Badan Narkotika Nasional (BNN), Maret 2012, in 2010 , there are 26.677 cases
which is 17.897 include narcotics cases 1.181 include psikotropika cases, and
7.599 cases for other additive materials. In 2011, there is an increase of 11.69%
which is there are 29.796 cases which is 19.128 include narcotics cases, 1.601
include psikotropika cases dan 9.067 cases for other additive materials.The
increase continued to occur every year, where most of the victims of drugs abuse
are students.
Drugs abuse provides a variety of negative impacts very seriously for your
self, others, society, nation and the country. A drugs addict can spread HIV through
287
288
free sex and also might experience death from an overdose. On the other hand, an
addict often do criminal acts for get money to buy drugs, such as, stealing, robbing,
even comitted murder. This caused drug abuse very dangerous.
Pollich stated that the most effective efforts to prevent drug abuse is early
detection. He said that, “ ... the majority of well-controlled studies suggest that
early detection is associated with more positive and succesful treatment
outcomes.”. This detection can be used to identify a person’s tendency to do drugs.
This can lead to effects and more severe damage can be prevented immediately [4].
During this time, early detection is done with a urine toxicology test on
every person suspected of involved drug abuse. However, a urine toxicology test
was only able to detect any additive substances that last 48-72 hours in the human
body, more than that of the urine test was not able to detect it again texicologi. This
weakness can be anticipated with the use of a psychological test which this tool can
distinguish someone who will be tends to be dependent on a drug or not [4].
Therefore, the factors that affect the tendency will be tied to the drug needs
to be sought so that will be retrieved on a model for early detection of whether
someone is likely to be tied to the drug or not. The factors that are considered are
gender, age, ethnic group, education of student’s parent, employment status of
student’s parent, with whom respondents resided, score OAT (Obvious Attributes),
score DEF (Defensiveness), score SAM (Supplemental Addiction Measure), and
score Cor (Correctional). Furthermore, the tendency of dependence on a drug
called dependentnesss.
B. Research Problem
How to find a model for early detection of whether someone is likely to be

tied to the drug or not as well as searching for factors that can affect the
tendency will be tied to drugs?
C. Research Purpose
Looking for a significant factors that affect this tendency and find the model
based on those factors to determine whether a person will have a tendency
to be tied on drug or not.
D. Research Methods
a. Respondents : fresh students in Mathematics Department,

University of Indonesia, 2013 (108 students)
b. Measuring instruments is the Likert scale with the measurement tool to
detect whether someone is likely to be tied to a drug or not. Realibility
of these tool is 0.839 and all items are valid items.
c. The technique of data collection by spreading a questionnaire.
d. The sample technique used is purposive sampling.
e. A method of analysis data are clustering, tree, and diskriminant
analysis.
f. Operational Definition of The Variables
289
Some important terms used in this paper are:

1. Dependentness
The level of person’s who is dependence in drugs. High scores on this
score indicates a higher tendency to have himself against drug abuse.
2. OAT (Obvious Attributes)
OAT scale scores reflect an individual’s tendency to indorse statements
of personal limitation. High score may be relatively able to recognize in
themselves what are sometimes terms “character defects”, and they tend
to endorses statements suggesting personal limitation.
3. SAM (Supplemental Addiction Measure)
This scale can measure the level of dependency to something other than
drugs.
4. Cor (Correctional)
The COR scale can be used to assess the client’s level of risk for legal
problems. Clients who have elevated COR scale scores show response
patterns similar to adolescents who have been refered to correctional
program.
2. Main Results
a. Descriptive Statistics
 Gender : Male (39 students), female (69 students)
Table 1
Gender
Cumulative
Frequency Percent Valid Percent Percent
Valid Male 39 36.1 36.1 36.1
Female 69 63.9 63.9 100.0
Total 108 100.0 100.0
 Ethnic Group : Jawa, Sunda, Betawi (59 students), others (32 students)
Table 2
Ethnic Group
Cumulative
Valid Jawa/Sunda/Betawi 59 54.6 64.8 64.8
Others 32 29.6 35.2 100.0
Total 91 84.3 100.0

Missing System 17 15.7
Total 108 100.0
 Education of student’s parents : < = high school (37 students), D3 (3

290
students), S1 (50 students), > = S2 (11 students), others (4 students),

missing (3 students)
Table 3
Education of Student’s Parents
Cumulative
Valid <=high school 37 34.3 35.2 35.2

D3 3 2.8 2.9 38.1
S1 50 46.3 47.6 85.7
>=S2 11 10.2 10.5 96.2
Others 4 3.7 3.8 100.0

Total 105 97.2 100.0
Missing System 3 2.8
Total 108 100.0
 Employment’s status of students’s parents : no job (3 students),

goverment employee (23 students), private employee (41 students),
enterpreuner (35 students), others ( 6 students)
Table 4
Employment’s status of Student’s Parents
Cumulative
Valid No job 3 2.8 2.8 2.8

Goverment 23 21.3 21.3 24.1
Employee
Private Employee 41 38.0 38.0 62.0
Entrepreneur 35 32.4 32.4 94.4
Others 6 4.6 4.6 100.0

Total 108 100.0 100.0
 With whom respondents resided : parents / family (51 students),

homestay (44 students), dorm (13 students)
291
Table 5
With whom Respondents Resided
Cumulative
Valid Parents/family 51 47.2 47.2 47.2
Homestay 44 40.7 40.7 88.0
Dorm 13 12.0 12.0 100.0

Total 108 100.0 100.0
 Marital’s status of students’s parents: married (97 students), divorced

dead (4 students), divorced life (7 students)
Table 6
Marital’s Status of Students’s Parents
Cumulative
Valid Married 97 89.8 89.8 89.8
Divorced Dead 4 3.7 3.7 93.5
Divorced Life 7 6.5 6.5 100.0

Total 108 100.0 100.0
b. Drugs dependence based on dependentness

By using two step clustering method, the author will classify the
respondents based on variable dependentness. It turns out, with Two Step
Clustering obtained 2 categories of students who will have a low
dependency on drugs and high dependency of drugs, namely :
1. Low (49 students), score 23-31, mean score 29.37 (valid item=16)
2. High (56 students), score 32-49, mean score 35.9643 (valid item=16)
c. Factors that distinguish dependentness variable
The factors that distinguish the variable dependentness is sought by using a
tree diagram which the result obtained in Figure 1. So from the tree can be
conclude that factors that distinguishes the dependentness is OAT and
SAM. The main factor that most distinguishes is OAT which is outlined as
follows :
 OAT 22 (low OAT), valid item =10
There are 26 students (78.8%) which have a low dependency of drugs
and there are 7 students (21.2%) which have a high dependency of
drugs.
 OAT 22 (high OAT), valid item = 10
There are 23 students (31,9%) which have a low dependency of drugs
drugs.
Other factors besides OAT is SAM which is outlined as follows :

292
 SAM 10 (low SAM), valid item = 5

drugs.
 SAM 10 (high SAM), valid item =5
drugs.
d. Group profile characteristic students based on variables OAT and SAM
To classify students who will have low and high dependency on drugs, it
can be seen with variable OAT and SAM. The result analysis can
elaborated as follows :
1. If a students who have score OAT 22 and score SAM 10 then
there are 19 students (46.3%) have low dependentness and 22 students
(53.7%) have high dependentness.
2. If a students have score OAT 22 and score SAM 10 then there are
4 students (12.9%) have low dependentness and 27 students (87.1%)
have high dependentness
Figure 1. Tree
293
e. Discriminant
Using discriminant analysis, the authors make a function that distinguishes
student tendency on drugs. The result analysis can elaborated as follows :
1. Tests of Equality of Group Means
 OAT : Wilks’ Lambda = 0.854, F = 17.274, Sig. = 0.000 < 0.15,
conclusion : variabel OAT significant
 SAM : Wilks’ Lambda = 0.791, F = 26.631, Sig. = 0.000 < 0.15,
conclusion : variabel SAM significant
2. Box’s M test of equality of covariance matrics : Sig. = 0.157 > 0.15 so
assuming a covariance matrix are fulfilled.
3. Wilk’s Lambda : Sig. = 0.000 < 0.15 so it can be concluded that the
discriminant functions distinguishes the group dependentness.
So, the unstandardized estimate of discriminant function is :
D = -5.553 + 0.087 OAT + 0.344 SAM (1)
And Functions of Group Centroid is
1. Low = - 0.556
2. High = 0.504
Discriminat function can be used to determine whether one new students

has a high or low dependentness of drugs.
Cutoff value = (Nhigh. Centroidlow + Nlow. Centroidhigh)/ Nhigh + Nlow
From data, it is found tahat cutoff value = -0.06.
When new student has D score > -0.06, it can be concluded that the
students will has high score depentdentness to drugs.
Tendency of dependence on the drugs can be detected early by using a

measuring instrument psychology. The result of test states that there are 56 fresh
students in Mathematics Departement,University of Indonesia, 2013 who will have
high score dependentness. The factors that distinguishes dependentness on drugs
are OAT and SAM. Students who have high score of dependentness to drgus are
OAT > 2.2/item & SAM > 2,2/item (87.1%) and OAT > 2.2/item & SAM <=
2,2/item (53.7%). Suggestions for Departement Mathematics UI is it has to
concern about this results.
References
[1] Quinlan, J.R., 1986, Induction of Decision Trees, Kluwer Academic Publishers,
Boston.
[2] Rencher, A.C., 2002, Methods of Multivariate Analysis, John Wiley & Sons.
[3] Sharma, Subhash, 1996, Applied Multivariat Techniques, John Wiley & Sons.
[4] Vincent, Stephen, 2003, Adaptasi Subtances Abuse Subtle Screening Inventory
(SASSI-A2) : Sebagai Alat Bantu Deteksi Dini Level Penyalahgunaan Narkoba
Anak Remaja (thesis)
Statistic and Probability
COMPARISON OF LOGIT MODEL AND PROBIT

MODEL ON MULTIVARIATE BINARY RESPONSE
JAKA NUGRAHA
Dept. of Statistics, Islamic University of Indonesia, Kampus Terpadu UII, Jl. Kaliurang
Km.14, Yogyakarta, Indonesia. email : jk.nugraha@gmail.com
Abstract. On univariate binary response, Logit model is better interpretation

compared to Probit model. Logit model and Probit model may be used to
analyze same data sets for the same purpose but which model can perform better
analysis on multivariate binary data is an interesting topics to be studied. In this
study, a comparison of multivariate binary probit and logit models via a
simulation study was performed under different correlations between dependent
variables. We assume that each of n individual observed T response. Yit is tnd
response on ind individual/subject and each response is binary. Each subject has
covariate Xi (individual characteristic) and covariate Zijt (characteristic of
alternative j). Individual response i can be represented by Y i=(Y i1,....,Y iT), Y it is
tnd response on ind individual/subject and each response is binary. In order to
simplify, we choose one of individual characteristics and alternative
characteristics. We studied effects of correlations using data simulation. General
Estimating Equations (GEE) was used to estimate the parameters in this study.
Data were generated by using software R.2.8.1 as well as the estimation on the
parameters. Based on the result, it can be concluded that estimator in the logit
model is equivalent to 1.63 on the probit model. Estimator of the correlation
base on Chaganty-Joe is more accurate compared to GEE base on Liang-Zeger.
Key words and Phrases : Random Utility Model, GEE, Simulated maximum
likelihood estimator, Newton-Raphson, GHK.
1. Introduction
Investigators often encounter a situation in which plausible statistical models

for observed data require an assumption of correlation between successive
measurements on the same subjects (longitudinal data) or related subjects
(clustered data) enrolled in clinical studies. Statistical models that fail to account
for correlation between repeated measures are likely to produce invalid inferences
since parameter estimates may not be consistent and standard error estimates may
be wrong[1]. Statistical methods that appropriate for analyzing repeated measures
include generalized estimating equations (GEE) and multi-level/mixed-linear
294
295
models[2]. GEE involves specifying a marginal mean model relating the response to
the covariates and a plausible correlation structure between responses at different
time periods (or within each cluster). Estimated Parameter thus obtained are
consistent irrespective of the underlying true correlation structure, but may be
inefficient when the correlation structure is misspecified[2]. GEE parameter
estimates are also sensitive to outliers[2,3]. Summary statistics derived from the
likelihood ratio test can be used to check model adequacy in cross-sectional data
analyses[1,4,5]. For mixed linear models, the process is often not straightforward due
to the complexities involved[6]. Model selection is difficult in GEE due to lack of
an absolute goodness-of-fit test to help in choosing the "best" model among several
plausible models [4,5,7]. For repeated binary responses, Barnhart and Williamson[5]
and Horton et al.[4] proposed ad-hoc goodness-of-fit statistics which are extensions
of the Hosmer and Lemeshow method for cross-sectional logistic regression
models[4,5,8].
Frequently, some dependent variables are observed in each individual. This

observation results the multivariate data. Research of multivariate binary response
models still gets a little attention, however the applications of multivariate binary
response model are mostly extensive. GEE can be implemented on multivariate
binary response. Variable Yit with i=1,..n and t=1,..T in panel data (longitudinal)
refer to the same variable. In multivariate binary response, Yit refer to the different
variable (T variables)[2]. Nugraha et al. [9,10] has tested properties of estimator of
bivariate logistic regression using MLE and GEE. Both of MLE and GEE are
consistent. Logistic model on multivariate binary data using GEE are more
efficient compared to the univariate approximation. From simulation data, it was
concluded that GEE was better than MLE, specifically GEE able to accommodate
the correlations and the GEE’s estimator was more precise than MLE.
Furthermore, Nugraha et al.[11,12] discussed estimating parameter of Probit

model on multivariate binary response using simulated maximum likelihood estimator
(SMLE) methods to estimate the parameter based on Geweke-Hajivassiliou-Keane (GHK)
simulator. From computational side, simulation method applicable for Probit model is need
to be developed to overcome the limitation of GHK method. For this limitation,
Nugraha[13] have proposed mixed logit model. From simulation data, he conclude that
mixed logit model is better than logit model.
Based on the fact that GEE was acceptable than MLE and is widely
available in many statistical software applications, in this study, we compare the
probit model and logit model on multivariate binary response using simulation
data. In the R.2.8.1 program, the logit and probit models can be obtained by using
library(geepack) and library(mprobit). The geepack is the GEE that is based on
Liang-Zeger and the mprobit is the GEE that is based on Chaganty-Joe. Generating
data and estimating paremater using R 2.8.1 software[14].
2. Utility Model
We assume that each of n individual is observed for T response in that Yit is

tnd response on ind individual/subject and each response is binary. Response for ind
individual can be represented by following statement:
Yi = (Yi1,....,YiT)
296
that is a vector of 1xT. Yit = 1 if ind subject and tnd response choose the first
alternative and Yit =0 if the subject choose the second alternative. Each subject has
covariate of Xi (individual characteristic) and covariate of Zijt (characteristic of
alternative j=0,1). In order to simplify, one of individual characteristic and one of
characteristic of alternative were chosen.
Utility of subject of i choose alternative of j on response t is
Uijt = Vijt + ijt for t=1,2,...,T ; i=1,2,...,n ; j=0,1 (1)
where
Vijt = jt +jtXi + tZijt
Uijt is utility that it is latent variable and Vijt is named representative utility. In
Random Utility Model (RUM), assumption that decision maker (subject) choosing
alternative based on maximize utility, so equation (1) can represented in different
of utility,
Uit = Vit + it (2)
where Vit = (1t-0t) + (1t -0t)Xi + t(Zi1t - Zi0t) and it = i1t - i0t.
Association between Yit and Uit is
yit = 1 <=> Uit > 0 <=> -Vit < it and yit = 0 <=> Uit <0 <=> -Vit > it .
Probability of subject i choose (yi1 = 1,...., yiT = 1) is
= P(0 < Ui1,..., 0 < UiT) = P(-Vi1 < i1,..., -ViT < iT)
P(yi1 = 1,...., yiT = 1) =  I (V

i
it   it ). f ( i )d i t (3)
where ’i = (i1,..., iT). The value of probability is multiple integral T and depend
on parameters , ,  distribution .
Logit model derived by asumption that ijt have extreme value distribution
and independence each other. Density of extreme value (Gumbel) is
  ijt
 ijt
f ( ijt )     (4)
Marginal probability (for some t and i) is
exp(Vi1t )
P( yit  1)   it  (5)
[exp(Vi 0t )  exp(Vi1t )]
Probit model derived by assuming that vector  i' has a multivariate normal
distribution with the mean of null and covarians of . Density of i is
1 1
f ( i )   ( i )  exp[  i'  1 i ] (6)
(2 ) T /2
|| 1/ 2
2
297
Marginal probability (for some t and i) is
it = P(yit=1|Xi,Zi) = P(-Vit < it ) = (Vit) (7)
Vit 1 1
where (Vit )  
 (2 )t
2 1/ 2
exp[
2 t2
 it2 ]d it
3. Overview of GEE
Marginal models are often fitted using the GEE methodology, whereby the
relationship between the response and covariates is modeled separately from the
correlation between repeated measurements on the same individual[2]. The
correlation between successive measurements is modeled explicitly by assuming a
"correlation structure" or "working correlation matrix". The assumption of a
correlation structure facilitates the estimation of model parameters[2]. Examples of
working correlation matrices include: exchangeable, auto-regressive of order 1
(AR(1)), unstructured, and independent correlation structures[2]. For binary data,
correlation is often measured in terms of odds ratios[15]. Details of the correlation
structure and response-covariate relationship are included in an expression known
as the quasi-likelihood function[2], which is iteratively solved to obtain parameter
estimates. Estimates obtained from the quasi-likelihood function are efficient when
the true correlation matrix is closely approximated. In other words, the large-
sample variance of the estimator reaches a Cramer-Rao type lower bound[3].
GEE for  can present in form

n
G() = W  S
i 1
i i
1
i (Y'i  π'i )  0 (8)
 1   1 
   
Wi  diag   Xi ,...,  Xi   and
  (Z  Z )   (Z  Z )  
  i11 i 01   i1T i 0T  

 i  diag  i1 (1   i1 ) ...  iT (1   iT ) 
1/2 1/2
Yi = (Yi1,...,YiT); i = (i1,..., iT); Si = A i R i A i ;

A1i / 2  diag Var(Yi1 ) ... Var(YiT ) 
Ri is working correlation matrix Yi and Wi is an observation matrix.
To estimate Ri, Liang and Zeger[16] use vector of empirical corelation ri

with
(Yis   is )(Yit   it )
rist  (9)
[ is (1   is ) it (1   it )]1 / 2
298
rist is usbias estimator for ist with i = 1,2,..n and s,r=1,2,…T.
In probit model, Chaganti and Joe[17] use
(Vis ,Vit ;  st )  (Vis )(Vit ))

Kor (Yis , Yit )  (10)
[(Vis )(1  (Vis ))(Vit )(1  (Vit ))]1 / 2
to estimate Ri. If ist= st for all i then
1 n
̂ st   rist
n i 1 (11)
Equation (8) and (11) can be solved simultaneously for  and .
4. Generating Simulation Data
We will generate simulation data with T=3. Then, the equations of utility are
Ui0t = 0t + 0tXi + tZi0t + i01 and Ui1t = 1t + 1tXi + tZi1t + i11 (12)
for i=1,...,N; j=0,1 and t=1,...,3; ijt ~Extreme Value Type I for logit model and
ijt~N(0,1) for probit model. Equation (12) can be presented in difference of utility
Uit = Ui1t–Ui0t. On logit model, equations of utility difference are
Uit = t + tXi + tZit + it (13)
where Zit = (Zi1t – Zi0t); t= 0t- 1t ;t =0t - 1t.
We generate data on 1=-1, 2=1, 3=-1; 1 = 0.5, 1 = -0.5, t = 0.5, 1=0.3,

2=-0.3, 3=0.3 and some of correlations  =0;0.1;0.2;...; 0.9 using program
R.2.8.1. Utility 1 (Ui1) was correlated with utility 2 (Ui2) and both utility is not
correlated with utility 3 (Ui3). Data 1 are obtained from it~extreme value and Data
2 are obtained from it~N(0,1). For each of the data simulation, we estimate
parameter using logit model and probit model. GEE-1 are estimator of logit model
based on Liang-Zeger. GEE-2 are estimator of probit model based on Liang-Zeger.
GEE-3 are estimator of probit model based on Chaganty-Joe.
Those data can be further analyzed by using the program geepack and
mprobit, so the utility must be transformated in the form of:
T
Ui =   D
t 1
t it  t X i Dit   t ( Z i1t  Z i1t ) Dit  (14)
where Dir is dummy variable. Dir =1 for r=t and Dir =0 for rt, r=1,2,3. So
If t=1 then Di1 = 1 and Di2 = Di3 = 0. If t=2 then Di2 = 1 and Di1 = Di3 = 0. If t=3
then Di3 = 1 and Di1 = Di2 = 0
Ui = t + tXi + t(Zi1t- Zi0t)..

299
5. Main Result
Discrete Choice Model (DCM) was prepared based on the of the error
distribution. So far there is no method to test the assumption of truth because the
utilities(Ui) is also a latent variable can not be known by researchers in value.
5.1. Efek of  variation to the Estimator

Logit model is constructed based on the assumption that variance it is
valued with  2 / 3 . The value of it variance to the estimator is presented bellow.
Based on the simulation data, it can be remarked that the value of variance it give
influence to the estimator. The bigger deviation of variace it (from  2 / 3 ) can
affect to the resulted more bias estimator (bigger deviation). Suppose that Var(it)
= 2, where the utility model is
Uit = Vit + it
Logit model use the assumption that value of error variance is  2 / 3 , so the utility
model that will be estimated is
  
U it  Vit   it
 3  3  3
~ ~   2
U it  Vit  ~it where Var (~it ) 
 3 3

So, the estimator resulted will be deviated for (1  ) .
 3
5.2. EFFECT OF CORRELATION TO THE ESTIMATOR

In advance simulation, data were generated on n=1000 with 5 replications on
each correlations of utility (0 to 0.9). From the simulation (Figure 1 to Figure 10),
it can be concluded that :
a. Estimator GEE-1 (in the logit model) is equivalent to 1.63 GEE-2 on the
probit model. From Data 2, On the Probit model using the assumption that
error (it) having normal standard distribution, the value of correlation
between utilities give no effect to the estimator properties. Estimator in the
logit model is equivalent to 1.64 on the probit model. This is caused by
differences in the size of the variansi error on logit model (2=2/61.645)
and variansi error in probit models (2=1) (Table 1, Table 2 and Table 3).
b. Estimator of the correlation using GEE-3 is the most accurate (small bias)
compared to GEE-1 and GEE-2.. The estimator is not affected by the faulty
of error distribution assumptions. On GEE-1 and GEE-2, the estimator of
correlation tends to underestimate.
c. Estimator of the parameter  using GEE-3 is not accurate compared to
GEE-1 and GEE-2. But, estimator of the parameter  and  using the third
method the results are relatively the same.
d. Utility 1 (Ui1) was correlated with utility 2 (Ui2) and both utility is not
correlated with utility 3 (Ui3). Therefore the value of correlation was only
give effect to the parameters whitin Ui1 and Ui2. On both utilities,
parameter estimation on the model have big deviation comparable to the
correlation value. Therefore the value of correlation was only give effect to
300
the parameters whitin Ui1 and Ui2 on Logit and GEE-1. Estimator of the
parameter  and  using GEE-2 and GEE-3 are more accurate than Logit
and GEE-1
Figure 1. Bias of 1 Figure 2. Bias of 2
Figure 3. Bias of 3 Figure 4. Bias of 1
Figure 5. Bias of 2 Figure 6. Bias of 3
Figure 7. Bias of 1 Figure 8. Bias of 2

301
Figure 9. Bias of 3 Figure 10. Bias of 
Based on the results, it can be concluded that estimator in the logit model is
equivalent to 1.63 on the probit model. Estimator of the correlation base on
Chaganty-Joe is more accurate compared to GEE base on Liang-Zeger. So, we
recommended to estimate correlation using GEE base on Chaganty-Joe and then
using GEE base on Liang-Zeger to estimate coefisien regression.
References
[1] Dobson A, 2002, An Introduction to Generalized Linear Models Florida: Chapman

& Hall/CRC.
[2] Diggle PJ, Heagerty P, Liang K, Zeger SL, 2002, Analysis of Longitudinal Data,
Second edition. Oxford: Oxford University Press.
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Fitzmaurice GM., 1999, Goodness of fit for GEE: An example with mental health
service utilization, Stat Med, 18(2):213-222.
[5] Barnhart H. X.,; Williamson J. M.., 1998, Goodness-of-fit tests for GEE modeling
with binary responses. Biometrics 54(2):720–729.
[6] Schabenberger O., 2004, Proceedings of the twenty-Ninth Annual SAS Users Group
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inference. Stat Sci. ;15: 1–26.
[8] Hosmer DW, Lemeshow S., 1980, Goodness-of-fit tests for the multiple logistic
regression model. Communications in Statistics. A9:1043–1069.
[9] Nugraha J., Haryatmi S., Guritno S, 2009, Estimating Parameter of Logit Model on
Multivariate Binary Response Using MLE and GEE. Jurnal Ilmu Dasar, Vol. 10.
No. 1 , FMIPA Universitas Jember.
[10] Nugraha J., Guritno S., Haryatmi S. 2009, A Comparison of MLE and GEE on
Modeling Binary Panel Response, ICoMS 3th IPB, 2008.
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Respons Using SMLE, Jurnal Ilmu Dasar, FMIPA Univ. Jember,.
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[12] Nugraha J., 2011, Simulation Study Of MLE on Multivariate Probit Models,
Proceedings of The 6th SEAMS-UGM Conference 2011, FMIPA UGM.
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Maximum Likelihood Estimator and Generalized Estimating Equations. Asian
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Computing. R Foundation for Statistical Computing, Vienna, Austria.
[15] Fitzmaurice G.M., Laird N.M., Ratnitzky, A.G.,1993, Regression Models for
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Methodology), Vol. 66, No. 4, pp. 851-860.
MULTISTATE HIDDEN MARKOV MODEL FOR

HEALTH INSURANCE PREMIUM CALCULATION
RIANTI SISWI UTAMI1 AND ADHITYA RONNIE EFFENDIE2
1Department of Mathematics Gadjah Mada University, riantisu@gmail.com

2Department of Mathematics Gadjah Mada University, adhityaronnie@ugm.ac.id
Abstract. Health conditions over time can be modelled using multistate Markov
model. However, the information about health conditions is not always
available, but there is another information related to this condition. This paper
presents hidden Markov model to estimate transition intensity and observation
probability for multistate model where the true state is not observed. The
estimation of transition intensity and transition probability will be used to
calculate health insurance premium. By using this method, it is expected to get
an appropriate premium value. This method will be applied to patient’s visit in a
clinic in West Java.
Key words and Phrases : Multistate model, hidden Markov model, health
insurance premium.
1. Introduction
A multi-state model is a model for a stochastic process which occupies one

of a set of discrete states at any time. The inference in multi-state models is
traditionally performed under a Markov assumption for which past and future are
independent given its present, Meira-Machado [9]. Sometimes, states of the
process are not observed, but there there is a finite set of signals, and that a signal
from the set is emitted each time the Markov chain enters a state. A model in which
the sequence of signals is observed, while the sequence of underlying Markov
chain states is unobserved, is called a hidden Markov model, Ross [11].
Hidden Markov model is commonly used in areas such as speech and

signal processing, Juang and Rabiner [8], and the analysis of biological sequence
data, Durbin et al. [4]. In health, Chen et al. [3], described a hidden Markov model
for breast cancer screening, Satten and Longini [12] used hidden Markov models
for the progression between stages of HIV infection, and Jackson et al. [5] used
hidden Markov models in the study of abdominal aortic aneurysms by
ultrasonography.
In health insurance, the premium value can be calculated based on the
303
304
multistate model. In the process of calculation required data of the patients’ health
status over time. However, this data is not always available but it is possible that
there are other variables that can describe the health status of the patients.
Therefore, in this study utilized the results of estimation of hidden Markov models
to calculate health insurance premiums.
2. Multistate Model
Individual's health condition from time to time subject to change. Changes

in these conditions generate a data consists of the time of the event and the type of
event that occurred. A model, called the multistate model, is suitable for this kind
of data. For example, a simple multistate model that consists of 3 state, "1:
healthy", "2: diseased", and "3: dead", is illustrated in Figure 1.
1 2
Healthy Disease
Dead
Figure 1 The three-state model
In the model ilustrated above, a healthy person is possible to make a

transition into diseased or dead. A diseased person is possible to make a transition
into dead, but a dead person is impossible to life again, so dead state is called
terminal event or absorbing state, where once entered they are never left.
In general multistate model, the state of the individual at time t is denoted

by X(t), t ≥ 0. Transition rate from state i to state j at time t is indicated by the
transition intensity qij  t , Ft  ,
qij  t , Ft   lim

P X  t  h   j X  t   i, Ft 
h 0 h
where Ft is the observation history of the process up to the time preceding t. If the
process X(t) is assumed to be time homogeneous and Markov, then qij  t , Ft  is
independent of t and Ft. For simplification, qij  t , Ft  will then be written as qij  t 
. Transition intensity for all possible transitions between states can be formed into a
matrix Q(t), whose rows sum to 0, so that the diagonal entries are

qii  t    qij  t  . For multistate model in Figure 1, the transition intensity
j i
matrix is as follows.
305
 q11  t  q12  t  q13  t  

 
Q  t    q21  t  q22  t  q23  t  
 0 0 0 
Transition probability Pij  t  is the probability of being in state j at a time t

+ s in the future, given the state at time s is i. If the process is assumed to be
Markov then the conditional distribution of the future X(t + s) given the present
X(s) and the past X(u), 0 ≤ u < s, depends only on the present and is independent of
the past.

P X  t  s   j X  s   i, X  u   x  u  , 0  u  s 

 P X t  s   j X  s   i 
Transition probability for all possible transitions between states can be formed into
a matrix P(t) whose rows sum to 1. In general, the matrix P(t) is as follows.
 P11  t  P12  t   P1n  t  

 
P  t  P22  t   P2 n  t  
P  t    21
     
 
 Pn1  t  Pn 2  t   Pnn  t  
The transition probability matrix can be obtained from the Kolmogorov equation as
P  t  eQt .
In Jackson [7] the one-step transition probabilities in a discrete-time Markov chain

can therefore be parametrized via
P  eQ .
3. Hidden Markov Model
In a hidden Markov model (HMM) the states of the Markov chain are not
observed. The observed data are governed by some probability distribution
conditionally on the unobserved state. The evolution of the underlying Markov
chain is governed by a transition intensity matrix Q. Hidden Markov models are
mixture models, where observations are generated from a certain number of
unknown distributions. However the distribution changes through time according
to states of a hidden Markov chain, Jackson [6]. This figure gives an illustration for
hidden Markov model.
306
Observed yr1 yr2 yrMr
....
Underlying Xr1 Xr2 XrMr
Q
Figure 2 A hidden Markov model
Observation yrs is the observed data for the rth individual at the sth
observation, where r = 1, 2, ..., N, and s = 1, 2, ..., Mr. It can be either continuous or
categorical data. The underlying state is denoted by X rs  i, i  1, 2, , n .
Probability distributions of yrs conditional on the underlying state X rs  i are,
f1  y 1  , f 2  y  2  ,, f n  y n  where k is a vector of parameters for the state
i distribution. For example, if yrs is Normally distributed for each state in the
model, then
y rs  X rs  i 
N  i ,  i2 
1   yrs  i 2 

fi yrs i ,  i
2
  exp  
2 i2 
.
2 i2  
The most efficient, and by far the most popular, approach for estimating
the parameters of HMM is based on the Expectation-Maximization (EM) principle,
Cappé et al. [2]. The EM algorithm is an iterative optimization method for
maximum likelihood estimation of statistical models that involve unobserved (or
latent) data. The application of EM to hidden Markov model is not straightforward
and requires an additional inductive algorithm known as the forward-backward,
introduced by Baum et al. [1]. In particular, EM is known to converge very slowly
in some models. As an alternative, we used quasi-Newton optimization methods as
well as on an efficient purely recursive algorithm for computing the log-likelihood.
Jackson et al [5] describe, individual r’s contribution to the likelihood is

Lr  P yr1 , yr 2 , , yrM r 
(1)
  
  P yr1 , yr 2 , , yrM r X r1 , X r 2 , , X rM r P X r1 , X r 2 , , X rM r 
where the sum is taken over all possible paths of underlying states
X r1 , X r 2 ,, X rM r . Assume that the observed data are conditionally independent
given the values of the underlying states.
307
 
P yr1 , yr 2 , , yrM r X r1 , X r 2 , , X rM r  P  yr1 X r1  P  yr 2 X r 2  P yrM r X rM r . 
Also assume the Markov property,
P  X rs X r , s 1 , , X r1   P  X rs X r , s 1 
then
  
P X r1 , X r 2 , , X rM r  P  X r1  P  X r 2 X r1  P X rM r X r , M r 1 . 
Decompose the overall sum in equation (1) into sums over each underlying state.
The sum is accumulated over the unknown first state, the unknown second state
and so on until the unknown final state.
Lr   P  yr1 X r1  P  X r1  P  yr 2 X r 2  P  X r 2 X r1 
X r1 Xr2
(2)
  
 P yrM r X rM r P X rM r X r ,M r 1
X rM r

From the above likelihood, there are several parameters that must be
estimated, vector of parameters  i in the observation distribution,
fi  y i   P  yrs X rs  i  , transition probability,
Pij  t   P  X rs  j X r , s 1  i  for t  trs  tr , s 1 , and initial state distribution

 i  P  X r1  i  . Furthermore, the set of the parameters that must be estimated
denoted by    P, ,   where P is transition probability matrix,
 = 1 , 2 ,, n  , and  = 1 ,  2 ,,  n  .

T T
The full likelihood function is,

N
L   Lr
r 1
then log-likelihood function is

N
log L   log Lr . (3)
r 1

The first derivative of the log L, which in turn is written log L , will be

searched by using the forward and backward variables. Instead of using the
continuous transition probability, the transitions of a Markov chain are often
characterized by a discrete transition probability matrix. Here's the definition of a
forward and backward variables, Qin et al. [10]. For each r = 1, 2, ..., N, s = 1, 2,
308
..., Mr, and i, j = 1, 2, ..., n, forward variable is defined as follows.
 rs  i   P  yr1 , yr 2 , , yrs , X rs  i 
n
 r , s 1  j     rs  i Pij f j  y  j 
i 1
In matrix form
  rs 1 
 
  rs  2  
 rs  .
  
 
 rs  n  
For each r = 1, 2, ..., N, s = 1, 2, ..., Mr, and i, j = 1, 2, ..., n, backward variable is

defined as follows.

 rs  j   P yr , s 1 , yr , s  2 , , yrM X rs  j
r

n
 rs  i    Pij f j  y  j   rs  j 
j 1
In matrix form
  rs 1 
 
  rs  2  
 rs  .
  
 
  rs  n  
Matrix  rs is defined as follows.
 P  yrs X rs  1 0  0 
 
 0 P  yrs X rs  2   0 
 rs   
     

 0 0  P  yrs X rs  n  
 f1  y 1  0  0 
 
 0 f2  y 2   0 
 
     
 0
 0  f n  y  n  
so that the forward and backward variables can be expressed as a matrix

multiplication.
309
 rT, s 1   rsT PBr ,s 1

 rs  PBr ,s 1 r ,s 1
Note that the likelihood in equation (2) can be written as the following matrix
multiplication,
Lr  T B r1  PB r 2  PB r 3  PB rM r  1
where 1 be a column vector consisting of 1s. The derivatives of the log likelihood
can be formulated as
 log Lr
= fi  y i  ˆr1  i 
 i
 log Lr M r 1
=  ˆ rs  i ˆr , s 1  j  f j  y  j 
Pij s 1
M r 1
 log Lr n
=  j ˆr1  j    ˆr , s 1  j  ˆ rs  i  Pij
 j s 1 i 1
Where ˆ rs  i  and ˆr , s  j  are the scaled forward and backward variables.
Derivatives of the full likelihood is,

 N

log L   log Lr .
 r 1 
To find estimators ̂ that maximize log L, we used quasi-Newton method.
4. Health Insurance
Health insurance is an insurance product that provides certain benefits if

the insured suffered illness, accident, or receive medical care. Health insurance
product consist of social and commercial health insurance, Thabrany et al. [13].
Individual health insurance premium is the amount of money that must be

paid by the person as an insured or policyholder in return for a guarantee of cost as
a result of the onset of a disease risk as stated in the policy. Meanwhile, health
insurance premium are generally set at the sum of the individual premium in one
agency.
This paper discuss about an insurance product that can be organized by

small-scale medical institutions such as outpatient clinics. This insurance covers
the members when he fell ill and went to the medical institutions concerned.
Members only need to pay insurance premium once at the beginning of the policy.
Subsequently the patient does not have to pay again if he gets a medical
examination within the policy period. If the policy period had expired, members
310
can extend their health insurance by paying premium again, and so on.
Suppose the stochastic process  X s , s  0,1,, M  is a discrete time

Markov chain. At time s, the risk is in state h, X s  h . Health insurance premium
for the occurrence of transition from state i to state j with the benefit of 1 is as
follows.
t
Ahij  t    Phit qij vt (4)
s 0
Where,
Phit is t-step transition probability from state h to state i,
qij is transition intensity from state i to state j,
v t is present value of of a payment of 1 that will be expired in the coming period t,
with v  1  i  , and i is the interest rate.
1
5. Application
The data used is the patient visit data in Polyclinics Cihideung Garut West
Java between January 2012 to April 2012. The data is taken from a sample of 200
patients. The information used is the patient ID number, time of visit, and cost.
The model that will be used is the multistate model with 4 states. The
states show the patient's health condition,
1: healthy
2: light disease
3: moderate disease
4: severe disease.
Time unit used is the week. Informations regarding the time of recovery are not
available in the data because patients only visit when sick so assumed if the
patients do not visit again within one week then the patient is considered cured
(healthy). Status of the patient journey from a state to another state is assumed to
follow a Markov assumption. Direct transitions are possible from healthy state to
diseased state and vice versa as illustrated in the following figure.
1 3
4
Figure 3 Multistate model for patient’s visit data
Based on the picture above the transition intensity matrix can be arranged as
follows,
311
 q11 q12 q13 q14 

q q22 0 0 
Q   21
 q31 0 q33 0
 
 q41 0 0 q44 
where qii   q j i
ij , i, j = 1,2,3,4.
On patient visits data, there is no information about the type of the

diseased, but it is assumed there is a relationship between the type of diseased with
cost. Thus, in this model it is assumed the state of patients was not observed the
cost can be observed.
Patient’s cost is assumed Normally distributed. If yrs shows the cost to the
r patient on the sth visit, then the observation probability distribution can be
th
written as follows,

a. y rs  X rs  1 N 1 ,  12 , 
b. yrs  X rs  2 N   ,   , 2
2
2
c. yrs  X rs  3 N   ,   , 3
2
3
d. yrs  X rs  4 N   ,   . 4
2
4
When the patient is health then the costs equal to zero, so that 1  0 and 1  0 .
Parameter estimation is done with the help of library msm version 0.7 on R
software. In the model described earlier, there are 12 parameters to be estimated,
q12, q13, q14, q21, q31, q41, μ2, μ3, μ4, 2, 3, and 4. Estimation process is an iterative
process requiring initialization values for each parameter. The initialization value
can be determined based on the literature, previous research, or determined by the
researcher. In here the initialization of transition intensity is as follows.
 0, 4 0,1 0,1 0, 2 

 0,5 0,5 0 0 
Q0  
 0, 6 0 0, 6 0 
 
 0,8 0 0 0,8
Initialization values for the parameters in the probability distribution of the

observations is μ2 = Rp. 33.000,00, μ3 = Rp. 48.000,00, μ4 = Rp.87.000,00, 2 = Rp.
8.000,00, 3 = Rp. 5.000,00, dan 4 = Rp. 17.000,00.
By using initial values above, obtained estimates for the transition

intensities as follows.
312
 0, 4453 0, 03283 0,1444 0, 2681 

 0,5742 0,5742 0 0 
ˆ 
Q
 0, 6135 0 0, 6135 0 
 
 0,8222 0 0 0,8222 
Estimated values for the parameters in the probability distribution of the

observations are,
ˆ 2  Rp.33.000, 01, ˆ 2  Rp.17.010, 76,

ˆ3  Rp.48.000, 00, ˆ 3  Rp.11.030,10,
ˆ 4  Rp.86.999,94, ˆ 4  Rp.28.719,32.
Based on estimates of the transition intensity, in the first row can be seen
that the risk of a healthy patient will suffer severe disease is greater than light and
moderate diseases. The second, third, and fourth row show that the cure rate of
patients suffering from severe disease is greater than light and moderate.
Parameter estimation results in the probability distribution of the

observations show that the average cost of patients suffering from light disease is
Rp. 33000.01 with a standard deviation of Rp. 17010.76. The average cost of
patients who suffer moderate disease is Rp. 48000.00 with a standard deviation of
Rp. 11030.10. The average cost of patients suffering from severe disease is Rp.
86999.94 with a standard deviation of Rp. 28719.32.
Estimation of transition probabilities in a one week period are as follows.
 0, 7383 0, 0208 0, 0897 0,1512 

 0,3635 0,5688 0, 0247 0, 0430 
Pˆ  
 0,3812 0, 0060 0,5675 0, 0453
 
0, 4636 0, 0075 0, 0327 0, 4962 
In the above transition probability matrix, the first row shows that the probability
of healthy patients staying healthy in one week is greater than falling ill. Row 2, 3,
and 4 show the probability of a patient will recover within one week is less than
the probability of stay in the same diseased.
Premium value is determined by equation (4). The required cost when

patients suffer light, moderate, and severe diseases are assumed constant and
determined as ˆ 2  Rp.33.000, ˆ3  Rp.48.000, dan ˆ 4  Rp.87.000 .
Multistate model used assumes that the direct transitions can occur between states
1↔2, 1↔3, dan 1↔4, so that the premium calculation is done based on the values
of Pˆ11 , qˆ12 , qˆ13 , and qˆ14 with discrete time approach,
t
t t t
33.000 vt Pˆ11t qˆ12  48.000 vt Pˆ11t qˆ13  87.000 vt Pˆ11t qˆ14
s 0 s 0 s 0
313
where Pˆ11 is the elemet of Pˆ t in first row and first column, and qˆ12 , qˆ13 , qˆ14
t
respectively are the element in first row column 2, 3, and 4 of the matrix Q̂ , also t
in week.
If it is assumed that the interest rate is 0%, the health insurance premiums
is as follows.
a. Premium value for half-year period is Rp.480.897,00.
b. Premium value for one year period is Rp.895.467,40.
c. Premium value for one half-years period is Rp.1.256.972,00.
d. Premium value for two-years period is 1.572.204,00.
References
[1] Baum, L. E. dan Petrie, T., 1966, Statistical Inference for Probabilistic Functions of
Finite State Markov Chains, Ann. Math. Statist., 37, 1554–1563.
[2] Cappé, O., Buchoux, V., and Moulines, E., 1998, Quasi-Newton Method for
Maximum Likelihood Estimation of Hidden Markov Models, IEEE on Acoustics,
Speech, and Signal Processing, ICASSP98, 4, 2265–2268.
[3] Chen, H. H., Duffy, S. W., and Tabar, L., 1996, A Markov Chain Method to
Estimate the Tumour Progression Rate from Preclinical to Clinical Phase,
Sensitivity and Positive Predictive Value for Mammography in Breast Cancer
Screening, The Statiscian, 45, 3, 307-317.
[4] Durbin, R., Eddy, S., Krogh, A. dan Mitchison, G., 1998, Biological Sequence
Analysis: Probabilistic Models of Proteins and Nucleic Acids, Cambridge University
Press, Cambridge.
[5] Jackson, C. H., Sharples, L. D., Thompson, S. G., Duffy, S. W., and Couto, E.,
2003, Multistate Markov Models for Disease Progression with Classification Error,
The Statistcian, 52, 193–209.
[6] Jackson, C. H., 2006, Multistate Modelling with R: the msm package,
http://rss.acs.unt.edu/Rdoc/library/msm/doc/msm-manual.pdf, accessed 6th March
2013.
[7] Jackson, C. H., 2011, Multi-State Models for Panel Data: The msm Package for R,
Jornal of Statistical Software, 38.
[8] Juang, B. H. dan Rabiner, L. R., 1991, Hidden Markov Models for Speech
Recognition, Technometrics, 33, 251–272.
[9] Meira-Machado, L., 2011, Inference for Non-Markov Multi-state Models: An
Overview, REVSTAT, 9, 1, 83-98.
[10] Qin, F., Auerbach, A., and Sachs, F., 2000, A Direct Optimization Approach to
Hidden Markov Modeling for Single Channel Kinetics, Biophysical Jornal, 79,
1915–1927.
[11] Ross, S. M., 2010, Introduction to Probability Models, 10th edition, Elsevier,
Oxford.
[12] Satten, G. A. dan Longini Jr, I. M., 1996, Markov Chains with Measurement Error:
Estimating the ‘True’ Course of a Marker of the Progression of Human
Immunodeficiency Virus Disease, Applied Statistics, 45, 3, 275-309.
[13] Thabrany, H., Surachmad, S., Iskandar, K., Handayani, Nurhayati, Hidayat, B.,
2005, Dasar-dasar Asuransi Kesehatan Bagian A, PAMJAKI, Jakarta.
Extended Abstract of IICMA 2013
THEORETICAL METODOLOGY STUDY BETWEEN

MSPC VARIABLE REDUCTION AND
AXIOMATIC DESIGN
SRI ENNY TRIWIDIASTUTI
FMIPA, Universitas Terbuka, srienny@ut.ac.id
Abstract. This paper proposes a variable ratio reduction methodology

Multivariate Statistical Process Control (MSPC) and Axiomatic Design (AD).
The benefit of the proposed method is to reduce the number of variables that
must be measured, thus reducing the time and costs associated with the
inspection. MSPC method of dimension reduction approach is to choose a
variable process by maintaining as much information about the full set of
variables wherever possible, while the dimension reduction AD approach is
based on a series of work processes and customer needs. This paper examines
the concept of MSPC by Gonzales and Sanchez (2010) and AD theory of Suh
(2001) developed by Brown (2005). The findings yield advantages and
disadvantages from the point of view of Statistical Quality Control, the two
methods are complementary approaches can be used in solving the case of a
business process. Preliminary selection of variables is performed using MSPC
due to large data to be screened without sacrificing the quality of the
information. AD is then used in the next screening process to come up with most
significant variables for service quality business process analysis in air
transportation.
Key words: variables reduction, variables weight, MSPC, Axiomatic Design.
1. Introduction
The main objective of statistical process control (SPC) is to maintain a stable

process that will produces minimum variability. Monitoring is not difficult in case
only single variable is observed. On the contrary, problem may arise if the
observations were made in the multi-variable process. To further facilitate the
process monitoring, it is necessary to conduct selection of significant variables (as
representative of whole variables) without impair the existing/quality of
information. Disposing or reducing number of variables could imply risk of
reducing the effectiveness of process monitoring in detecting out-of-control
(OOC) situations. In cases where there are doubts about eliminating some
variables, the decision could be delayed until a set of OOC situations is available,
314
315
the analysis of which can help to reveal whether variables can be disposed off
without losing critical information. Variable selection has been an important issue
in many areas of both theoretical, and applied statistics. The selection of variables
for SPC should always follow some engineering criteria according to the
functionality of the part.
Based on some previous research, MSPC approaches consists of two groups. The
first grup which is procedures that rely on the multiple correlation coefficients
between the selected variables and the discarded ones. For instance, Beale,et
al.(1967) select the p variables using the interdependence analysis. This method,
selects the p variables that maximize the minimum multiple correlation
coefficient between the p variables and each of the K-p discarded ones (minimax),
analyzed were B = K! where K is initial variables and p is variables

p! K  p !
selected. Beale's opinion has the disadvantage that when K is large the cost of
observation will be high.
Jolliffe (1972) introduced the stepwise method to overcome these weaknesses. In

each step, the variable with the largest multiple correlation coefficients (compare
with the remaining ones) is discarded because it is the variable with the more
redundant information. Because these selection methods are based on correlation,
the solution does not depend on the scalling of the variables. A more efficient
approach is proposed in Mc.Cabe (1984). He introduced a concept of “principal
variables” as the p variables that contain as much sample infomation as possible,
measured by the conditional covariance matrix of the K-p variables given the
selected p, denoted as  K  p . p . This method could also be applied using the
correlation matrix of data instead of the covariance matrix. By doing so, the
selection does not depend on the scaling of the variables. One common difficulty
of all these methods is that the number p of selected variables must be chosen.
Beale, et al. (1967) proposes obtaining the value p as a function of the multiple
correlation coefficients. For Jollife’s methods (1972, 1973) proposes setting p
equal to the number of principle component (PCs) needed to account for some
proportion of the total variability. McCabe (1984) proposes choosing the value p
according the percentage of variation explained for the selected subset of p
variables.
The latest findings on variable selection proposed by Gonzalez and Sanchez

(2010) produced some pluses and minuses. The plus is statistical quality control
procedures that rely on the parameter estimates and the level of importance of
each variable. Whereas the only drawback is that the variables selection is only
made from the viewpoint of statistical techniques.
On the other hand it is necessary that the selection of these variables be done by
taking view from both standpoint of statistical and engineering process
respectively. This paper proposes an approaches, of which the weakness of their
findings can be reduced with the Axiomatic Design (AD) method which is based
upon the approach of process engineering. By using experienced technicians who
can provide information about potential error of the available data, we will avoid
the risk of reducing the effectiveness of the monitoring process by removing the
316
useful variables. In this paper, the criteria of the variable selection in SPC is
following variables selection in process engineering functions, that is the
observed quality characteristic is the most influential variable in the process. The
proposed methodology is the selection of variables for measuring customer
satisfaction on service quality of airlines industry, refering to the papers from
Gonzales and Sanchez (2010) as well as the processed data from Triwidiastuti
(2006).
2. Main Results
This article proposes a methodology for the selection of variables that have several
advantages compared with existing theories, as presented in Figure 1.
Initial
Variab
Variable Selection
Evaluatio Stop
continue
Index= level of Evaluate the ability of a subset of var that indicate

efficiency of variable SPC performance on simulated multivariate alarm
To compare the performance of CC that uses K

variables with CC using selected variables.
Performance of the CC is considered adequate

enough
Filtering
to filter out the remaining variables from the
selected variables
Dependent variable Independent variables

non-selected variables selected variables
Regression equation
y (non-selected variables) = f (x = selected variables)
Figure 1. The proposed methodology for the selection of variables by Gonzales and Sanchez (2010)
This proposed methodology Gonzales and Sanchez (2010) in Figure 1 consists of

three steps: (a) variable selection, (b) evaluation, and (c) filtering. These steps are
run in iterations until it is completed at a predetermined stage. The first round, step
(a) selecting a single variable that contains the greatest amount of information from
the initial variable K. They proposed procedure, based on an oblique rotation of the
independent factor obtained by PCA (Principal Component Analysis). The next
317
step, (b), is to evaluate the existing information in the variable that has been
selected in the previous stage, with aims to determine whether the selection could
stops or be continued. Two proposed criteria for step (b) is:
- The first criteria, a subset of selected variables were evaluated based on the
total amount of information contained. They specify the index that
measures the information, which can indicate the level of efficiency of
these variables. As a complementary tool, they propose the index of
benefits along with a general index that measures the information in the
selected variables.
- The second evaluation criteria, based on Principal Alarm Analysis

(Sanches, 2006), used to evaluate the ability of a subset of variables that
indicate SPC performance on simulated multivariate alarm. The simulated
alarm is based on a shift in direction towards the principal component of a
controlled data set. The second criterion is to compare the performance of
control chart (CC) that uses all the variables K with control chart (CC)
using selected variables. If the performance of the control chart is
considered adequate enough in the evaluation stage, the selection is
stopped.
The next step (c) is to filter out the remaining variables from the selected variables.
At this stage, they form a regression equation with the non-selected variables as the
dependent variable and the selected variables as independent variables so y (non-
selected variables) = f (x = selected variables). In the first iteration, the results of
step (c) is a K-1 data set that has been filtered. When they repeat the iterative
procedure will yield in a new variable selected, so that eventually create more and
more number of selected variables. At the end of the procedure, they will have a
subset of the variables p  K.
Variable-Selection Methodology
The goal of this selection step is to determine, at each iteration t, the variable that
contains the largest amount of information. In order to decide how to measure the
information in a set of variables, we will constraint to the linear relationships
among them. Then, the in-control covariance  K (or correlation) matrix will be
used as a measure of their joint information. In the first iteration of the method, the
selection step is performed with all the K variables. Once a variable is selected
using the statistical technique explained below, the next selection is performed with
the remaining K -1 nonselected variables, but filtered out from the information of
the selected one. This filtered dataset of K-1 variables obtained at the end of the
(1)
first iteration will be denoted as X . At the beginning of the t iteration t-1
S(t 1) X1 X 2 X 3 X t 1
variables have already ben selected, so that =( , , ... ) where
X i is the variable seleced in i iteration; X t 1 is the selection step of this iteration;
with the k = K — t + 1 remaining variables, but filtered out from the information of
S (kt 1)
the variable subset (t 1) . The covariance matrix of X is denoted as
t 1
which
is the covariance of the k variables conditioned on the selected ones.
318
Gonzales dan Sanches (2010) proposed to select the variable at each iteration is
based on PCA and be labelled as the OR method because it be based on oblique
rotations. The main goal of PCA is to project a set of dependent variables
X1 , X 2 ,
X 3 ,.... X k into a set of independent PCs Y1 , Y2 , Y3 ,... YK . PCA is based on the

singular value decomposition (SVD) of k and holds
 K = CK LK CK'
C  L
where K is the matrix with the eigenvectors j of K and K a diagonal matrix
with the corresponding eigenvalues , j=1,2,...K and SVD still exists in the
presence of multicollinearity. Jollife’s methods (1972, 1973) use the matrix c j c1of
CK
eigenvectors
c2 j cKj to perform the selection of variables, each eigenvector = ( j,
ci . ,... )’ is associated to the variable with the highest absolute
C
One drawback of using the matrix K , each eigenvector
c j value in column
is not strongly
c
related with only one variable, but all the elements in j might have similar values
without a dominant one. Also, a variable can have high loads in several
eigenvectors, which may lead to a poor selection capacity. To avoid this drawback,
the proposed OR method uses some rotation of these eigenvectors, by using
promax oblique rotation. The rotation is performed so that each new vector is more
clearly associated with a single variable or a subset of highly related variables. The
promax solution is achieved by first rotating to an orthogonal varimax solution, and
then by extending this rotation to an oblique solution to better fit a simple structure.
V
The varimax rotation method,1/2performsvan orthogonal rotation of the matrix K
VK CK LK ij VK
that is equal to = , with being vij a generic element of denoted
as loadings. It is well
Y known that the loadings are proportional to the covariance
X
between varii , and j . The goal
VK v var i of the varimax method is to obtain a newv var rotated
i
matrix with vectors j , which have a large number of loadings ij with
high absolute values and also a large number of loadings Y var i with low absolute values.
After varimax rotation, each new rotated component j , j = 1, 2, ...K, tends to be
associated with one or a few original variables X. The promax method aims
improve this varimax solution byvari searching a nonorthogonal rotated matrix K in
V pro to
V
which the higher loadings of K arepromade even higher and the lower loadings
V
smaller. As a result, this matrix Kpro determines the dominant variable (or a
V
clusters of variables). The matrix K is usually denoted in the factor-analysis
literature as a pattern-loadings matrix. The pattern loadings can be interpreted as
measures of the unique contribution that each rotated component makes to the
variance of the original variables. Consequently, the promax solution is a better
tool for variable selection than using simple PCA.
At each iteration, the variable that contains the largest amount of information of the
variables (initials or filtered). Because variables might be groupedvin pro clusters, the
selection should not only take into account the individual loadings ij , but should
also be based on all the loadings
v pro in each component. If the jth component has many
large absolute values of ij , that component will contain a large amount of
information due to a cluster of variables. Hence, weq can sort all 2the rotated
pro
j i ( vij )
components by the sum of the q squares of their loadings = . Once the
component with maximum j is determined, the selected variable will be the one
with the largest absolute loading in that component. It is interesting to note that the
proposed OR algorithm is not affected by multicollinearity. Note
t 1 too that, at each
iteration t > 1, this OR selection criteria is performed using K .
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Definition 2.1.
Evaluation Step
Two general approaches proposed by Gonzalez and Sanchez (2010) is to

evaluate the subset of selected variables. In the first approach, the selected
variables are evaluated as a function of the total amount of information contained
in them. This evaluation approach is not target oriented in the sense that it only
evaluates the ability of the selected subset to reproduce the general features of the
complete data, irrespective of the specific use of the variables by the analyst. In this
evaluation approach, we define two R-like indices that measure the amount of
information contained in the selected subset of variables. Conversely, the second
evaluation approach principal alarms analysis (PAA) is target oriented. The
subset of selected variables is evaluated according to their ability to perform an
efficient statistical monitoring of a multivariate process. This second approach,
which is more time consuming, is based on the simulation of some out-of-control
situations using the method of PAA. Evaluation step in this paper use first
approach, two R-like indices because the selected variables are evaluated as a func-
tion of the total amount of information contained in them.
This paper uses the Axiomatic Design (AD) approach to evaluate the subset of
selected variables, this design method was originally introduced by Suh (2001).
AD is a structured approach to performance improvement in complex systems at
the stages of concept, at preliminary or basic design and at detailed design phase of
systems engineering models. This analytical and structured method links functional
requirements for the product system with process design parameters. If there is a
change in the design parameters, AD quantitative methods can be used to assess the
sensitivity of the functional requirements in a particular process. AD techniques
reduce the risk incidences, reduce costs and speed up the process for the products
to each market / customer, with multiple stages (Suh, 2001). Variable reduction
process using AD has been done by Triwidiastuti (2010) to determine the factors
critical to quality which are used then to measure the service quality of Higher
Education.
Original AD concept developed by Suh (2001) is aim to reduce the incidence of

risk, reduce costs and speed up processes of product to market / customers by:
 Develop and incorporate concepts of process design into a continuous and
measurable activities tailored to specific needs
 Communicate the plan to all stakeholders before technical drawings
(Computer Aided Design) are prepared and documented.
 Improve the design quality by analyzing and optimizing the design
architecture.
 Track and define the customer wants accurately and incorporate such
requirement into detailed design specifications
 Document and communicate the design logics(how and why) clearly
 Identify design problems at early stage and complete the cycle: design-build-
test-redesign with minimal costs.
 Allow management to observe the interrelationship/dependencies among
design structures, to optimize the scheduling, identification and reduction of
risks
Axiomatic process can result in:

320
 detailed description of the functions of an object (usually derived from

customer needs desires).
 description of the object that will fulfill that functions.
 description of how the function will be fulfilled.
Axiomatic approach
This approach has been widely used for the design (both at initial design and at
redesign) of products, its processes and organizations starting from describing or
defining the problem or customer needs. Problem or customer needs definitions are
often subjective because of differences in interpretation by different groups or
individuals. Difficulties will arise towards the end of the design because of
differences in interpretation, therefore the problem or customer requirements
should be described as detailed and as accurate as possible at the earliest stage, the
details of which should have a physical operational features. AD concept helps the
designer to specify and incorporate such design into each phases of activity to suit
the customer's needs. Designers can use the design tool to produce a more efficient
and better design than before (Triwidiastuti, 2010).
There are four main concepts in AD, that are: Domain, Hierarchies, Zigzagging
and Design axioms (Brown, 2005).
a. Domain
Contains specific requirements that are mapped at design phase in the form of
characteristic parameters.
Tabel 3. Axiomatic Design Domain
Design Stage Design Domain Design Element

Design customer - Customer needs (CNs), benefit/values customer
Concept seeks
- Customer needs , identified and formulated into
functional forms
Product design functions - Functional description (FRs) of the solution
- Constraints (Cs)
physical - Able to cater functional requirement
Process design - Design parameter (DPs) solution alternatives
- Plans were formulated into a draft
process Variables/process atributes
The basic concept of this domain consists of 4 stages:

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For each pair of domains, left domain is "what we want to achieve" while right
domain is a "how to achieve it". Each domain means:
Customer : Advantages / benefits desired by customers
Functional : Functional requirements of the design
Features/Physical : Design parameter
Process : Variables / process attributes
Tabel 4. Axiomatic Design Notation
Functional is an independent set of minimum requirements that can

Requirements (FRs) complement the characteristics of the functional
requirements of the design at a functional domain
Constraint (Cs) constraint is limit of resolution / acceptable design
Design Parameters is an element of the design of the physical domain chosen for
(DPs) a particular FRS
Process Variables is an element in the process domain that shows
(PVs) characteristics of design parameters
Tabel 5. Examples of Axiomatic Design Application
Type of Domain
activities customer functions features Process
durable food refrigerate refrigerator how to make a
Food food refrigerator (in detail)
preservation
what how
what how
what how
Relationship mapping each domain can be in a form of a matrix (between FRS and
DPs, between DPs and PVs).
DP1 DP2 DP3 DP4
FR1 X O X O
FR2 X X X O
FR3 X O X O
FRn X X O X
Where the notation X indicates a correlation and notation O as uncorrelation, and
each cell shows the mathematical relationship between FR and DP.
b. Hierarchies (hirarki)
The 2nd axiomatic design concept is HIRARCHIES that are represented by a
hierarchical design (design architecture). Starting from the highest level designer
will choose the most suitable design and then decompose the FRs starting from the
highest to the lowest FR. Highest FRs is paired with the highest DPs. This
decomposition produces the appropriate level layer between FRS and DPs,
performed at each level one by one. The same decomposition also conducted on a
pair DPs and PVs, and for each level as well.
c. Zigzagging
The 3rd of axiomatic design concept is zigzagging that describes the design
decomposition processes into hierarchies and pairing inter-domain levels. The
322
designer follows the pattern domain "what = what" and "how = how" for each level
of the hierarchy.
d. Design axioms
Axioma 1: Independence Axioma
Part of the design that can be separated (separable) so that changes in one of such
separate part will not or minimize the effect to the other part as minimum as
possible in other words maintain independence between FRS and DPs DPs
received on the draft. Each DP is set by the FRS FRS without involving others.
Axioma 2 : Information Axiom

Minimize information: among design alternatives that fit with Axioma 1, the best is
to keep information to minimum and which will give maximum probability of
success. Cotoia and Johnson (2001) divides this type into three (3), ie uncoupled,
coupled and decoupled. Design that is not comply with the independence Axioma
is called coupled. This design occurs in two conditions, namely when there are
fewer DPs than the FRS ( DPs<FRs), so several FRS affects 1 DPs. Otherwise 1
DPs affect some FRS. Examples of this design is the water faucet, because each DP
control/influence 2 FRs (regulate temperature and water flow). This can be
explained as follows: FR1 = DP1 and DP2 ; FR2 = DP1 and DP2 The opposite is
also DP1 = FR1 and FR2 ; DP2 = FR1 and FR2 so that
 A11 A12 0 
 A   0 A22 A23 

 0 0 A33 

The design which comply with independence Axioma are called uncoupled or
decoupled. The design is called uncoupled if each DP is independent, only affects
the 1 (one) FRs, does not affect other FRs. For example FR1 affect DP1, DP2
affects FR2, FR3 affect DP3. Uncoupled characteristics are all non-diagonal
elements equal to zero.  FR  A DP with m = n = 3 where FR1  A11DP1 ,
FR2  A22 DP2 , and FR3  A33 DP3
 A11 0 0 
A   0 A22 0 
 0 0 A33 
As for the design of decoupled occurred on 2 conditions:

 first condition if 1 DPs affecting 2 or more FRS or FRs > DPs. This problem
can be solved by increasing the qualified DPs.
 the second condition if DPs is enough but 1 or more FRS depends on one or
more DPs. This is expressed in matrix in the form of m x n
 A32 0 0 
 A   A22 A23 0 

A 0 A11 
 12
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The data processing
The limitation of this model is data with finite positive matrices, minimum interval
scale, not in mass service industry, data in discreet and base on small subgroup
observation. Assumption in model development: variables/attributes as result of
reduction are able to represent the whole quality characteristics being observed, as
they represent most important variables/attributes from customer side of view
(Triwidiastuti, 2006).
The data from research made by Gonzales and Sanchez (2010) is a tangible data,
that is manufacturing data of production process of the window frame for the
vehicle door. While the proposed research data is an intangible customer
satisfaction data in form of ordinal scale. This data is the measurement of customer
perception of service quality of airline company. because of the perception
variables are difficult to measure quantitatively measured, then the solution is the
perception variables measured by ordinal scale later in the transformation.
In Triwidiastuti (2006), the determination of the variable refers to some literature

on airlines services (Banfe, 2001), quality of services (Ramaswamy, 1996;
Holloway, 2002) and a standard operating procedure (SOP) PT. Garuda Indonesia.
Questionaires to the respondent are based on the criteria: a frequent flyer
passengers, traveling by plane at least 5 times a year, and short-haul flights (Banfe,
2001). Following the theory of expert system then the questionnaires were given to
limited number of respondents that were carefully chosen and selected.
Quantitative measurements were performed for customer satisfaction reveal
requirement, since reveal requirements are more easily measured. Questions are
divided into 3 parts, namely pre-flight, in-flight and post-flight, and generate 41
variables. These variables are grouped into 2 categories: product quality variables
(11 variables) and quality of service (30 variables). Study variables are presented in
Table 6.
Table 6. Research Variable
Product safety quality of Reservation officer serving with friendly and courteous
quality security service Location reservations can be reached / contacted easily
insurance Speed the reservation process, to obtain confirmation
frequency Condition check-in counter (location, equipment,
clarity signage)
timing Cleanliness waiting room
flight Submission of information waiting room with clear
connection sound
punctuallity Explanation officer for cancellation, delay and
withdrawal time flight delivered immediately
airport Skilled personnel to handle disruptive or abusive
location passengers
accesability
seat The ability and professionalism of officers (handling an
accesability efficient check-in): service time, flexible on certain
requests, special requests confirmation of passengers
carried on reservationHow to accept the passenger
cabin crew during boarding (smile and friendly)
Knowing passenger profiles (type of diet, preferred
seating, etc.)
How officers handle the lost ticket (left behind ticket)
Explanation for the special circumstances attendant
baggage: excess baggage, baggage size is too big, too
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heavy baggage
Availability of special counters for check-in (without
luggage, faster time, re-check-in, check-in group,
family check-in, check-in counters for long-distance,
baggage counter to check-in remotely)
Special handling check-in: VIP, CIP, F / C Class
Executive card holders, transit without visa, passengers
with less limitations / should be guided (incapacitated
pax), passengers with extra body shape (tall, fat, large),
with a passenger seat wheel, passengers who are
pregnant, oxigen bottles, prisoners, passengers in
conditions deported
Explanation officer for passengers who would go on to
change planes (procedures and advanced flight
information)
Handling passenger name that does not exist in the list
of attendees
benefit for Explanation officer for cancellation, flight delay and
loyal withdrawal time
customer
airport quality Cabin crew welcome greeted politely and ask the
service passenger seat
greet and welcome politely cabin crew and ask the
passenger seat
Cabin crew always help passengers find their seats and
gave instructions that the luggage must be stored so as
not to disturb other passengers and the safety of flight
How to dress neat, clean, well maintained and in
harmony with body size
Well-timed in handling; 5 mintes before the time
departure, the plane’s door is locked, so that the engine
can be turned on and take off as scheduled.
Announcement: delay of the flight has to be informed
to the passanger as soon as possible.
Announcer has a clear intonation and profesional.
Safety demostrastion is done with actual clean kit;
demonstrated in profesional manner.
Safety and security kit check (seat and seat belts)
before take off and during the flight.
Polite and professional communication
Professional, efficient, curteous and friendly manner.
Helpful and aware of those who might need help.
Seat and seat belt checking n professional manner
60 pieces of questionnaire were distributed, 56 pieces were returened and as

many as 51 pieces can be processed. Ordinal scale is used (discrete) transformed
into interval scale (continuous) with Multidimentional Scaling and using
MATLAB software. Cronbach α is used for Data processing for reliability test of
respondents, done for one measurement (one shoot) only. Quality of products
variable produced α of 0,849, while for service quality α of 0.967. So it can be
concluded that the measuring instrument is reliable. Validity test conducted by
Principle Component Analysis. The data variables were selected according to the
stages proposed by Gonzales and Sanchez (2010), yielding 30 variables. This
variables then selected again using AD methods.
Field findings indicate that the proposed model is able to measure customer
satisfaction and measuring quality caracteristic of services (gap 3). Problems arisen
325
during field application is in getting interval data to gauge customer perception.

This is solved by statistic manipulation, which is data transformation.
Table 7. State of The Art SPC variable reduction metodology of Gonzales dan
Sanches (2010) and Brown (2001).
Procedure MSPC AD The Data Advantages Disadvantages
/ by by Brown Processing
Step Gonzales
and
Sanches
Evaluate PAA Qualified Measureme MSPC based on MSPC not
the subset technician nt variables OOC, while AD consider
of selected experience are considering the characteristics of
variables. and brain characteristi characteristic the process. AD
storming cs of quality process does not consider
engineering the process
variability
Significant Eigen Brain Matrix Eigen value  i to
level of value and storming,
data describe the
variable eigen based on
vector customer variability
requireme component i,
nts and which is the most
process important
quality component is
standard indicated by the
maximum
variability. AD
based on customer
requirement point
of view
Variable Correlati zigzaging Matrix - -
correlation on in
data
matrix
Variabilita Covarian - Matrix MSPC, the MSPC, the
s inter dan ce matrix covariance matrix covariance matrix
data
antar focuses on focuses on
variabel precision and precision and
accuracy of data accuracy of data
Reducing Performa Variables CtQ matrix AD: business
variables nce SPC, that can be processes can be
based on accepted mapped more
process as critical optimal.
capabilit to quality MSPC: fewer
y (CtQ) observations
MSPC and AD: a

more efficient
time and cost
The conclusion of this paper is

326
1. Gonzales and Sanches (2010) method approach results in large variabiity

of simulated CC conditions in the in controll data, this situation may
happen in some actual processes but may not occur in other processes,
which mean that the selected variables may not result in proper detection of
OOC condition.
2. The approach of Gonzales ad Sanchez (2010) is that the proposed method

is based on statistical procedures that rely on the parameter estimates.
Therefore, it is advisable to take into account the uncertainty due to
sampling estimates using re-sampling techniques. This paper however do
not use resampling techniques
3. Another drawback of their approach is the possibility of a variable that

contains useful information is lost in the process of selection variables.
This drawback can be reduced with AD approaches, namely the selection
of variables based on engineering processes. It is suggested that these
results should be reviewed by a qualified techniciann who can provide
more in-depth information as well as potential error in the available data.
So that we maybe able to avoid the risk of reduced effectiveness of the
monitoring process for wasting useful variables.
4. The strength of MSPC method is that the resulting matrix is a

representation of a covariance matrix, where the covariance matrix is a
matrix that describes the variability of each and between variables /
attributes. This matrix can also be used to determine the weight of the
components observed variables (characteristics of quality, QC) which are
considered some of the most important QC aspect.
Acknowledgement.
My knowledge in the field of Process Control is enriched with knowledge gained
from Gonzales and Sanchez (2010) and Brown (2005) that complement each other.
References
[1] Brown, C.A., 2005, Teaching Axiomatic Design to Engineers-Theory, Applications,

and Software, Journal of Manufacturing Systems; 24, 3; ProQuest Research Library.
[2] Cotoia, M., Johnson, S., (2001). “Applying the axiomatic approach to business
process redesign”, Business Process Management Journal, 7,4, ABI/INFORM
Global, 304
[3] Gonzales,I., dan Sanchez,I., 2010, Variable selection for Multivariate Statistical
Process Control, Journal of Quality Tehnology, Juli 2010, Vol. 42, Iss 3, 242.
[4] Suh, N.P., 2001, Axiomatic Design, Advances and Application, New York, Oxford
University Press.
[5] Triwidiastuti (2006), Disertasi, Data of service quality in air transportation.
AN APPLICATION OF ARIMA TECHNIQUE IN

DETERMINING THE RAINFALL PREDICTION
MODELS OVER SEVERAL REGIONS IN INDONESIA
EDDY HERMAWAN1 AND RENDRA EDWUARD2
1The Atmospheric Modeling Division of Atmospheric Science and Technology

Center of National Institute of Aeronautics and Space (LAPAN),
Jln. Dr. Djundjunan No. 133, Bandung 40173, Indonesia
E-mail: eddy_lapan@yahoo.com
2Geophysics and Meteorology Department of Bogor Agriculture University (IPB),
Jln. Raya Darmaga Kampus IPB Darmaga Bogor, Bogor 16680

E-mail: rendra_edwuard@rocketmail.com
Abstract. In this present study, we mainly concerned an application of ARIMA

(Auto-Regressive Integrated Moving Average) technique in determining the
rainfall prediction models over several regions in Indonesia. They are Lampung
(South Sumatera), Pandeglang (West Java), Indramayu (West Java), Banjarbaru
(South Kalimantan), and Sumbawa Besar (Nusa Tenggara Barat). We used the
monthly of surface rainfall data for period of January 1970 to December 2000.
This study is motivated by the importance of understanding the mechanism of
air and sea interaction between Monsoon and El-Niño event when they come
simultaneously as already recommended by the IPCC (Intergovernmental Panel
on Climate Change (IPCC) AR (Assessment Report) 4 and GEOSS (Global
Earth Observation System to System). By applying the Power Spectral Density
(PSD) and Wavelet analysis on those rainfall anomalies data, and also the
Monsoon global index data, represented by the AUSMI (Australian Monsoon
Index) and WNPMI (Western North Pacific Monsoon Index), we found a pre-
dominant peak oscillation of them is about 12 month period that we call as the
Annual Oscillation (AO). While, for the El-Niño event, represented by SST (Sea
Surface Temperature) Niño 3.4, we found 60 month period. Furthermore
analysis, we found more significant relationship between rainfall anomalies and
AUSMI, WNPMI, and SST Niño 3.4. Please note here, this study was undertaken
with assumption that Monsoon and El-Niño is interacted each other, and the
model is developed by using multivariate regression method that formulated by
Rainfall = a + b [AUSMI] + c [WNPMI] + d [SST Niño3.4]. By applying the
ARIMA technique, we found the rainfall prediction models. They ARIMA
(1,0,1)12, with model equation Zt = 0.9989Zt-12 - 0.9338at-12 + at (for AUSMI),
ARIMA (1,1,1)12, with model equation Zt = -0.0674Zt-12 + Zt-12 - Zt-24 -
0.9347a t-12 + a t, (for WNPMI), and ARIMA (2,0,2), with model equation Zt =
3.594Zt-1 - 0.8362Zt-2 – 1.634a t-1 - 0.1053a t-2 + a t (for SST Niño 3.4). By
applying these techniques, we can predict the rainfall behavior over those area
until the end of 2013.
Keywords: ARIMA Technique and Rainfall Prediction Models
327
328
1. Introduction
As a national research institute with one of the basic tasks and functions it
handles weather and climate issues, the National Institute of Aeronautics and Space
(LAPAN), especially of the Atmospheric Science and Technology Center of
LAPAN in Bandung which does have a vision and mission field of the climate
change do everything possible to be able to contribute ideas or real contribution,
particularly related to issues of climate predictions, especially predictions of
rainfall anomalies that occurs in several regions in Indonesia.
There is one serious problem issues facing the Government (in this case the
National Food Security Council) recently, ie, the impact of global climate change,
especially the arrival of the dry season or wet season length increasingly difficult to
predict properly, let alone on time and on target.
This problem arises because the climate prediction models, especially the
model prediction of rainfall anomalies that exist today, are generally not fully
consider the interconnection or teleconnections or interactions that occur between
the various global climatic phenomena. There are only a few have already started
applying it, as did the Harijono1. The prediction model developed at this time,
generally still single column, individual, local, and rough resolution, but that
sometimes happens interconnection mutually reinforcing, but sometimes mutual
weaken.
Experience we have, when a drought occurs (more than six months) in 1997,
followed by the long wet season (also more than six months) one year later, is the
result of merging two natural phenomena, namely El-Niño and Dipole Mode in the
same time period (known as simultaneous).
These events teach us to understand more deeply the mechanism of merging
of the two phenomena in the well and correctly. If only the El-Niño or La Niña are
coming, then it is not going to impact as severe if not followed by the presence of
Dipole Dipole Mode Mode Positive or Negative.
The basic idea of this research is based on the national needs of the
importance of monitoring the early indications (as a precursor) of impending
extreme climate (especially rainfall extremes) in the western part of Indonesia
region, which is relatively wet throughout the year. Then, prediction becomes very
important, when the resulting model is not timely or not well targeted. Many
models have been produced, based either dynamic or statistical. One of the
statistical model that commontly used is ARIMA (Auto-Regressive Integrated
Moving Average) or called as the Box-Jenkins method. However, there is one
problem arises, when the model was no longer accurately applied to a particular
region, although the data analysis used is long relatively.
Among the climatic parameters, rainfall is the most dominant elements of
the climate in Indonesia. In general, rainfall in Indonesia is dominated by two types
of Monsoon which is characterized by the rainy and dry season. Refer to the annual
Monsoon cycle that set explicitly state of the atmosphere during the dry phase (dry)
and wet phases (rain). This annual cycle split phase wet and dry phases into two
periods.
Although Monsoons occur periodically, but the beginning of rainy season
and the dry season is not always the same throughout the year. This is due to the
season in Indonesia is influenced by several phenomena not only by Monsoon
itself. They are the El-Niño, Indian Ocean Dipole (IOD), and also the local
influence. If Monsoon related variations in annual precipitation, the El-Niño and
IOD associated with variation in inter-annual rainfall.
329
Climate phenomenon affects the agricultural sector in Indonesia. The

incident shows the increasingly important role with the emergence of extreme
climatic conditions that has serious impacts on agricultural production such a shift
in rainfall patterns and changes in surface air temperature (IPCC)2. Some concrete
examples when the long dry season followed by a long wet season is from 1997 to
1998. At that period in Indonesia experienced a long dry season is almost 9 months
old and is also a long wet season with nearly the same period.
Relationship between Monsoon and El Niño related needs to be studied more
deeply their impact on global climate. Basically the process is gradual climatic
aberrations. Therefore, an attempt to anticipate the effects of climate deviations
should be understood as a whole start of the process until the proper handling and
true impact. Therefore, it is necessary the development of science that combines
the atmosphere and oceans, including interactions to account for the next several
years.
The accuracy and reliable models that will greatly assist in the development
of climate forecast information. However, research that examines the physical and
dynamical interactions between the El Niño and Monsoon in Indonesia are still
rare. This is presumably due to the complexity of the Monsoon dynamics in
random parts of Indonesia affected by some type of Monsoon, like the Monsoon
Asia, India, the Pacific, and Australia.
On this basis, the research was conducted with the primary objective to
determine the effect of the interaction than the Monsoon and the El Niño when the
two come together (simultaneously) to fluctuations in rainfall in some areas in
Indonesia, as well as to determine the predictive model of rainfall in some areas.
2. A Brief Review of Box-Jenkins Models
The Box-Jenkins ARMA model is a combination of the AR and MA

models. They are
where the terms in the equation have the same meaning as given for the AR and
MA model.
A couple of notes on this model.
1. The Box-Jenkins model assumes that the time series is stationary. Box and
Jenkins recommend differencing non-stationary series one or more times to
achieve stationarity. Doing so produces an ARIMA model, with the "I"
standing for "Integrated".
2. Some formulations transform the series by subtracting the mean of the
series from each data point. This yields a series with a mean of zero.
Whether you need to do this or not is dependent on the software you use to
estimate the model.
3. Box-Jenkins models can be extended to include seasonal autoregressive and
seasonal moving average terms. Although this complicates the notation and
mathematics of the model, the underlying concepts for seasonal
autoregressive and seasonal moving average terms are similar to the non-
seasonal autoregressive and moving average terms.
330
4. The most general Box-Jenkins model includes difference operators,

autoregressive terms, moving average terms, seasonal difference operators,
seasonal autoregressive terms, and seasonal moving average terms. As with
modeling in general, however, only necessary terms should be included in
the model. Those interested in the mathematical details can consult Box,
Jenkins and Reisel3, Chatfield4, or Brockwell and Davis6.
There are three primary stages in building a Box-Jenkins time series model. They
are (1). Model Identification, (2). Model Estimation, and (3). Model Validation
Box-Jenkins Model Identification

The first step in developing a Box-Jenkins model is to determine if the series
is stationary and if there is any significant seasonality that needs to be modeled.
Stationarity can be assessed from a run sequence plot. The run sequence plot
should show constant location and scale. It can also be detected from an
autocorrelation plot. Specifically, non-stationarity is often indicated by an
autocorrelation plot with very slow decay.
Seasonality (or periodicity) can usually be assessed from an autocorrelation
plot, a seasonal subseries plot, or a spectral plot. Box and Jenkins recommend the
differencing approach to achieve stationarity. However, fitting a curve and
subtracting the fitted values from the original data can also be used in the context
of Box-Jenkins models.
At the model identification stage, our goal is to detect seasonality, if it exists,
and to identify the order for the seasonal autoregressive and seasonal moving
average terms. For many series, the period is known and a single seasonality term
is sufficient. For example, for monthly data we would typically include either a
seasonal AR 12 term or a seasonal MA 12 term.
For Box-Jenkins models, we do not explicitly remove seasonality before
fitting the model. Instead, we include the order of the seasonal terms in the model
specification to the ARIMA estimation software. However, it may be helpful to
apply a seasonal difference to the data and regenerate the autocorrelation and
partial autocorrelation plots. This may help in the model idenfitication of the non-
seasonal component of the model. In some cases, the seasonal differencing may
remove most or all of the seasonality effect.
Once stationarity and seasonality have been addressed, the next step is to
identify the order (i.e., the p and q) of the autoregressive and moving average
terms. The primary tools for doing this are the autocorrelation plot and the partial
autocorrelation plot. The sample autocorrelation plot and the sample partial
autocorrelation plot are compared to the theoretical behavior of these plots when
the order is known.
Specifically, for an AR(1) process, the sample autocorrelation function
should have an exponentially decreasing appearance. However, higher-order AR
processes are often a mixture of exponentially decreasing and damped sinusoidal
components.
For higher-order autoregressive processes, the sample autocorrelation needs
to be supplemented with a partial autocorrelation plot. The partial autocorrelation
of an AR (p) process becomes zero at lag p+1 and greater, so we examine the
sample partial autocorrelation function to see if there is evidence of a departure
from zero. This is usually determined by placing a 95% confidence interval on the
sample partial autocorrelation plot (most software programs that generate sample
331
autocorrelation plots will also plot this confidence interval). If the software
program does not generate the confidence band, it is approximately +/-2/SQRT(N),
with N denoting the sample size.
The autocorrelation function of a MA(q) process becomes zero at lag q+1
and greater, so we examine the sample autocorrelation function to see where it
essentially becomes zero. We do this by placing the 95% confidence interval for
the sample autocorrelation function on the sample autocorrelation plot. Most
software that can generate the autocorrelation plot can also generate this
confidence interval.
The sample partial autocorrelation function is generally not helpful for
identifying the order of the moving average process. The following table
summarizes how we use the sample autocorrelation function for model
identification.
In practice, the sample autocorrelation and partial autocorrelation functions
are random variables and will not give the same picture as the theoretical functions.
This makes the model identification more difficult. In particular, mixed models can
be particularly difficult to identify.
Although experience is helpful, developing good models using these sample
plots can involve much trial and error. For this reason, in recent years information-
based criteria such as FPE (Final Prediction Error) and AIC (Aikake Information
Criterion) and others have been preferred and used. These techniques can help
automate the model identification process. These techniques require computer
software to use. Fortunately, these techniques are available in many commerical
statistical software programs that provide ARIMA modeling capabilities. For
additional information on these techniques, see Brockwell and Davis5,6.
Box-Jenkins Model Estimation

Estimating the parameters for the Box-Jenkins models is a quite complicated
non-linear estimation problem. For this reason, the parameter estimation should be
left to a high quality software program that fits Box-Jenkins models. Fortunately,
many commerical statistical software programs now fit Box-Jenkins models.
The main approaches to fitting Box-Jenkins models are non-linear least
squares and maximum likelihood estimation. Maximum likelihood estimation is
generally the preferred technique. The likelihood equations for the full Box-Jenkins
model are complicated and are not included here. See Brockwell and Davis5,6 for
the mathematical details. The Negiz case study shows an example of the Box-
Jenkins model-fitting.
Box-Jenkins Model Validation

Model diagnostics for Box-Jenkins models is similar to model validation for
non-linear least squares fitting. That is, the error term At is assumed to follow the
assumptions for a stationary univariate process. The residuals should be white
noise (or independent when their distributions are normal) drawings from a fixed
distribution with a constant mean and variance. If the Box-Jenkins model is a good
model for the data, the residuals should satisfy these assumptions.
If these assumptions are not satisfied, we need to fit a more appropriate
model. That is, we go back to the model identification step and try to develop a
better model. Hopefully the analysis of the residuals can provide some clues as to a
more appropriate model.
As discussed in the EDA chapter, one way to assess if the residuals from the
332
Box-Jenkins model follow the assumptions is to generate a 4-plot of the residuals

and an autocorrelation plot of the residuals. One could also look at the value of the
Ljung-Box statistic7.
An example of analyzing the residuals from a Box-Jenkins model is given in
the Negiz data case study.
3. Data and Method of Analysis
The main data used in this study is the monthly of rainfall data at Lampung,
Sumbawa Besar, Indramayu, Banjarbaru, and Pandeglang for period of 1976 to
2000 as described in Figure 1. The Sea Surface Temperature (SST) Niño 3.4 data
taken from (http://www.cpc.noaa.gov/data/indices/ nino34.mth.ascii.txt) for period
of 1950 to 2009, the Monsoon Index data for the same period time, represented by
the Australian Monsoon Index (AUSMI), and Western North Pacific Monsoon
Index (WNPMI) taken from
(http://iprc.soest.hawaii.edu/users/ykaji/monsoon/realtime-monidx.html).
Figure 1: The five an investigated regions
While the method of analysis used is divided in two phases, each spectral
analysis (FFT and WL) and statistical analysis (cross-correlation, multivariate, and
Box-Jenkins approach). Special for Box-Jenkins approach, is divided into several
stages, each model identification, parameter estimation models, testing or
validation of the model, the determination of ARIMA models are relatively most
suitable, and prediction 2013.
4. Results and Discussions
4.1. Multivariate Regression Analysis
We are showing firstly interconnection between Monsoon and El-Niño when

they come simultaneously near real time as shown in Figure 2 below. We can see
an a good pattern between rainfall anomalies behavior and all investigated
parameters, represented by AUSMI, WNPMI, SST Niño 3.4, including their
interconnections.
333
Figure 2: The time-series of Global Monsoon Index, SST Niño 3.4, and Rainfall
anomalies for period of January 1996 to December 1999
By assuming the rainfall anomalies id depend on the global climate indices,

represented by the AUSMI, WNPMI, and SST Niño 3.4, we can make a simple
formula, that is (Hermawan, 2010)5:
R = f (AUSMI, WNPMI, SST Nino 3.4)
where R, and f is showing rainfall and function, respectively.
Fron this assumption, we got a multivariate regression as shown at Table 1
below. Please note, since the rainfall data is started from January 1976, we present
here comparison between those data started from January 1976 to December 1999.
Table 1. The multivariate regressions between rainfall anomalies and AUSMI,

WNPMI, and SST Niño 3.4 for period of January 1976 to December 1999
Lag
Location CCF time STD r2 Multivariate Regression Formula
(month)
Sumbawa
0.69 0 78.23313 0.532 Y = 16.267X1 - 3.940X2 - 6.790X3 + 6.819
Besar
Indramayu 0.64 0 124.86441 0.445 Y = 20.549X1 - 5.976X2 - 15.042X3 + 10.883
Banjarbaru 0.76 0 84.99061 0.591 Y = 8.195X1 - 12.388X2 - 25.964X3 + 7.408
Pandeglang 0.70 0 74.67462 0.535 Y = 15.923X1 - 3.445X2 - 5.264X3 + 6.509
Lampung 0.69 0 75.64186 0.486 Y = 9.954X1 - 6.721X2 - 2.913X3 + 6.593
Please note : X1=AUSMI, X2=WNPMI, and X3=SST Niño 3.4
Multiple regression equation above describes the respective roles of climate

phenomena, the Monsoon and the El-Niño in influencing the behavior of rainfall in
the study area. It is used to create a rainfall model that can eventually be used to
predict rainfall in the study area. Relationship or correlation between rainfall and
rainfall models with observations is explained by the size of the correlation
coefficient (r2) as described by Hasan8.
334
Based on the analysis CCF (Cross Correlation Function), Banjarbaru have

seen that the relative value of the CCF at most, approximately 0.76 compared to
other regions. This suggests that Banjarbaru showed a relatively greater response
time of the interconect various indices of global climate, especially when the
phenomenon of Monsoon and the El-Niño united.
CCF analysis was also used to determine the time delay or time lag
between phenomena Monsoon and El-Niño on rainfall. Positive sign (+) and
negative (-) on the value of the CCF to orientate the relationship of two variables.
If either has sign (+), meaning that the two variables have a proportional
relationship, otherwise if the value of the CCF has a negative value (-), then the
two variables are inversely linked.
Of Table 1 shows that the entire study area (Lampung, Sumbawa Besar,
Indramayu, Banjarbaru and Pandeglang) has a value of CCF is positive (+). This
suggests that the phenomenon of interaction between the Monsoon and the El-Niño
on rainfall has a directly proportional relationship. That means they strengthened
the interaction phenomenon when rainfall in the study area will increase, and vice
versa weakening the interaction phenomena of rainfall in the study area will
decrease.
In addition to knowing the value of the CCF, in Table 1dapat seen also how
long lag time or delay time in some areas of study. Lag time or delay time is the
time taken by the phenomenon of interaction of Monsoon and El-Niño to affect
rainfall in the study area. In Table 1 above it can be seen that the entire study area
(Lampung, Sumbawa Besar, Indramayu, Banjarbaru, and Pandeglang) has a time
lag of 0 months. It means that the interaction between the incident and the El-Niño
Monsun has no delay time to influence rainfall in the region with the original data.
4.2. Verification of the Multivariate Regression
In the Figure 2 it can be seen that the regions of the five studies had a large
correlation value. This suggests that the model is good and can be used to explain
the rainfall events in the study area that have been affected by the interaction
between the Monsoon with SST Niño 3.4. Validation of multivariate model of
rainfall period January 2000 - December 2000 one year after 1999 as stated in
Figure 3 below. Please note here, we for all validation or verification results, we
got almost a good value, it’s almost 0.9 for all an investigated regions.
335
Figure 3. Validation of multivariate model of rainfall period January to

December 2000.
4.3. Determining of the ARIMA Model Prediction
Before determined the ARIMA-based prediction model, the first step that
needs to be done is to test the stationarity of data, they are AUSMI, WNPMI, and
SST Niño 3.4. Afterwards, the identification and assessment models, testing or
validation of the model, before finally forecasting or prediction.
4.3.1. Test of the Stationarity Data
Tests performed on the data stationary rainfall anomalies period January

1949 to December 1997 as calibration results of AUSMI, WNPMI, and SST Niño
3.4 with rainfall data for period of January 1976 to December 1999. This is done
because it is a requirement modeling of time series data, because the data are not
stationary hard to predict which cause the resulting model is not maximal. It should
be noted that most of the time series data are nonstationary and aspects of AR
(Auto-Regressive) and MA (Moving Average) of ARIMA models only pleased with
the time series data are stationary.
Stationary data means there is no growth or decline in the data. Data seen by
naked eye along the horizontal time axis. In other words, the data are fluctuations
around a mean value that is constant, not depending on the time and the variance
and thus appears to remain constant at all times.
To check stationarity can be viewed via the autocorrelation function, partial
autocorrelation function, and plot time series of data to be examined
stasioneritasnya. The data are not stationary data must be converted into stationary
by differencing, ie, calculate the change or difference in value of observation.
Obtained difference value, double-checked whether the data is stationary or not. If
such data are not stationary then performed again differencing. If the variance is
not stationary, then the logarithmic transformation.
336
Figure 4 shows an examination of the data for stationarity. The results of the
examination showed that the data used are not stationary. Therefore the data is
processed again by differencing technique, just one time differencing. After
differencing seen that the data become more stationary and predictable than before
differencing. ACF (Auto Correlation Function) and PACF (Partial Auto
Correlation Function) in the entire study area has a seasonal pattern of rainfall so
that the data is the data on the seasonal lag 1.
Figure 4. Plotting data, differencing, ACF, PACF for earch parameters, AUSMI,
WNPMI, dan SST Niño 3.4 for period of January to 1976 to December
1999.
4.3.2. Identification and Estimated Model
Through the process of identification and assessment, while the obtained

models of global climate index data plot is ARIMA (1,0,1)12, ARIMA (0,1,1)12,
ARIMA (1,1,1)12, and ARIMA (2,0,2). Obtained from all models while a suitable
model for global climate data is ARIMA (1,0,1)12 for AUSMI, ARIMA (1,1,1)12
for WNMPI, and ARIMA (2,0,2) for SST Niño 3.4. ARIMA model on top of an
equation obtained at each global climate (Table 2) where Zt is the data at month t
and at an error in month-t. Equation is then validated with observational data..
Table 2. Determining of the ARIMA Prediction Model

Parameter ARIMA Formula R2
AUSMI (1,0,1)12 Zt = 0.9989Zt-12 - 0.9338at-12 + at 0.94
WNPMI (1,1,1)12 Zt = -0.0674Zt-12 + Zt-12 - Zt-24 - 0.9347at-12 + at 0.94
Nino 3.4 (2,0,2) Zt = 3.594Zt-1 - 0.8362Zt-2 – 1.634at-1 - 0.1053at-2 + at 0.95
337
4.3.3. Testing or Validation Model
Testing or validation is done by comparing the results of model calculations

with observational data with the time period used to validate the model is January
2000 to December 2000 (Figure 5). Plot the data for the entire data ikim ARIMA
models globally have a large correlation value is above 0.9. This shows that the
ARIMA model is very good and can be used to explain events with El-NNO
Monsun. Based on the ARIMA model equations get global climate time series data
prediction in January 2013hingga December 2013 (Figure 5). The data is then
developed again with multivariate equations, to obtain predictions of rainfall for
the month of January 2013 - December 2013.
Figure 5. Plotting of validation and prediction of AUSMI, WNPMI, and SST Niño
3.4 for period of January to December 2000 (for validation), and
January 2013 to December 2013, respectively.
4.3.4. Prediction Rainfall Anomalies Based on the ARIMA Model
After passing through the stages of ARIMA, the predictive value of the data
obtained global climate phenomena (AUSMI, WNMPI, and SST Niño 3.4) in the
period January 2013 to December 2013. Global climate data are then substituted
into the multivariate equation in Table 1. Data obtained from the substitution of
rainfall anomalies for each area of study period January 2013 - December 2013
(Figure 6). Thus, it can be interpreted that the multivariate equation can be used to
explain the rainfall events that have been affected by the interaction that occurs
between the El-Niño Monsoon.
338
Figure 6. Prediction of rainfall anomalies over Lampung, Pandeglang, Indramayu,

Sumbawa Besar, and Banjarbaru respectively for period of January to
December 2013.
By looking at Figure 6 above, we can see the minimum peak of rainfall

anomalies, mostly occur in around July. The pattern of those rainfall looks similar
each other. It indicates that most region is mostly effected by the Monsoon system.
We have another phenomena that called as El-Niño, but we suspect until the end of
this year, the Monsoon system is still as the pre-dominant peak oscillation.
To get more information about our prediction, we present here the rainfall
pattern over those area taken from the in-situ and satellite data observation as
shown in Figure 7 below.
Figure 7. The rainfall pattern over Lampung, Pandeglang (Banten), Indramayu

(West Java), and Sumbawa Besar (NTB)
Figure 8. As the as Figure 7, but for Banjarbaru (South Kalimantan)

339
In the end our analysis, we are showing here, another technique that we call
as the composite technique analysis that very useful for describing the rainfall
behaviour over those area more than 30 years observation that commontly is
mentioned as climatologist. We start analysis from Lampung, then following by
Pandeglang, Indramayu, Sumbawa Besar, and Banjarbaru, respectively as shown in
Figure 9 below.
Figure 9(a). The rainfall pattern behaviour over Lampung from January to June for
period of January 1982 to December 2009.
Figure 9(b). As the as Figure 9(a), but from July to December.

340
Figure 10(a). The rainfall pattern behaviour over Pandeglang from January to June
for period of January 1982 to December 2009.
Figure 11(a). The rainfall pattern behaviour over Indramayu from January to June
for period of January 1982 to December 2009.
341
Figure 12(a). The rainfall pattern behaviour over Sumbawa Besar from January to
June for period of January 1982 to December 2009.

342
Since we are interest to predict the rainfall behaviour over those investigated area,
we try to focus on investigating in November and December, respectively as shown
in Figure 13 below.
343
Figure 13. The rainfall pattern behaviour over Lampung, Pandeglang, Indramayu,
Sumbawa Besar, and banjarbaru from July to December for period of
January 1982 to December 2009.
Looking at the Figure 11 carefully, we obtain the general information that

almost those area have the same rainfall pattern anomalies. We can see clearly that
transition season from dry to wet season is occurred in November. While, the wet
season will be started in December. We can suspect also that almost no extreme
rainfall will be occurred, at least until the end of this year. They are almost the
normal condition. This is will be occurred perfectly by assuming that El-Niño event
still in normal condition, although we need to always investigate the behaviour of
the Monsoon system, especially the AUSMI parameter that we known before this
parameter has a strong correlation with the rainfall behaviour over Indonesia,
including this our investigated area.
Based on the analysis of the model using the Box-Jenkins method and
through the process of identification, assessment and testing, the obtained
prediction model ARIMA (1,0,1)12 for AUSMI, ARIMA (1,1,1)12 for WNPMI, and
ARIMA (2,0,2) for SST Niño 3.4. The ARIMA equation for three indexes is Zt =
0.9989Zt-12 - 0.9338at-12 + at (AUSMI), Zt = -0.0674Zt-12 + Zt-12 - Zt-24 - 0.9347at-
12 + at, (WNPMI), and Zt = 3.594Zt-1 - 0.8362Zt-2 – 1.634at-1 - 0.1053at-2 + at (SST
Niño 3.4). The ARIMA equation shows, for forecasting future depends on the data
to the previous month and a-t-t error to the previous month.
The model obtained is the result of multiple regression analysis between the
data anomalies of monthly rainfall in Lampung, Pandeglang, Indramayu, Sumbawa
Besar, and Banjarbaru with global monsoon index data and also the data Niño 3.4
SST for the period January 1976 to December 1999. As for the model validation or
test performed using the data one year later, in the period January to December
2000 which showed a good correlation between the data ARIMA models with
original dat with a correlation coefficient close to 0.9. This shows that the ARIMA
models developed can be used to predict rainfall anomalies throughout the study
area.
Results predicted by the end of 2013 showed that the study area has the fifth
general rainfall patterns the same approach, ie dominated by monsoonal pattern
AUSMI parameters. Peak of the rainy season generally occurs during DJF
(December-January-February), while the peak of the dry season, generally occurs
during JAS (July-August-September). This happens in almost every area of study
are examined.
344
Specially for the month of November and December 2013, the beginning of
a season of transition or the transition from the dry season to the rainy season. In
addition to the results of the analysis indicated that diapatj ARIMA model, also
evidenced by the data of rainfall anomalies in each month using Hovmoller
technique, which clearly visible November is a month of transition. While
December is the beginning of the dry season. This prediction will run perfectly
well and, of course, assuming that factors Niño 3.4 SST was normal until the end
of December 2013. Likewise Monsoon index global parameters, especially AUSMI
is in normal position..
For further research, to obtain an accurate prediction model using time series
data should be longer and more complete. Moreover, to explain the distribution of
rainfall areas with better spatial analysis using Hovmoller analysis. Future studies
are expected to also be able to create a statistical model using other statistical
methods such as CCA (Canonical Correlation Analysis).
Acknowledgement. Thanks a lot are given to the Organizing Commettee of the

IICMA2013 who gives a good chance for us to present this study. Thanks also to
Ms. Naziah Madani who is suppoting the data analysis.
References
[1] Harijono, S.W.B. 2008. Analisis Dinamika Atmosfer di Bagian Utara

Ekuator Sumatera Pada Saat Peristiwa El-Niño dan Dipole Mode Positif
Terjadi Bersamaan, Jurnal Sains Dirgantara (JSD), 5(2), 130 – 148.
[2] IPCC. 2001 .Climate Change 2001 : Impact, Adaptation and Vulnerabi1ity.
Cambridge: Cambridge University Press.
[3] Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series
Analysis, Forecasting and Control, 3rd ed. Prentice Hall, Englewood Clifs,
NJ.
[4] Chatfield, C. (1996). The Analysis of Time Series, 5th ed., Chapman & Hall,
New York, NY.
[5] Brockwell, Peter J. and Davis, Richard A. (1987). Time Series: Theory and
Methods, Springer-Verlang..
[6] Brockwell, Peter J. and Davis, Richard A. (2002). Introduction to Time
Series and Forecasting, 2nd. ed., Springer-Verlang.
[7] Ljung, G. and Box, G. (1978). "On a Measure of Lack of Fit in Time Series
Models", Biometrika, 65, 297-303.
[8] Hasan, M. Iqbal. 2003. Pokok-Pokok Materi Statistik 2 (Statistik Iterensif)
Edisi Kedua. Jakarta: Bumi Aksara.

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Published by Indonesian Mathematical Society

Assalamu’alaikum Warahmatullahi Wabarakatuh

Good morning and best wishes for all of us

It is my pleasure to say that the proceedings of the Second IndoMS

IICMA 2013 is one of the activities of IndoMS period 2012-2014.

Budi Nurani Ruchjana

On behalf of the Organizing Committee of IndoMS International

IndoMS conveys high appreciation for the Directorate General of Higher

It remains to thank all members of Organizing Committee spread across 3

Yogyakarta, January 5th, 2014

a. Directorate General of Higher Education (DGHE)

1Department National Centre for Advance Research in Discrete Mathematics (n-

CARDMATH)), Kasalingam University, Anand Nagar, Krishnankoil-626 126, India

Abstract. Let , be a graph and let 0 be an integer. A function

Key words and Phrases: adjacent, defended, safe function.

[1] J. Alber, H. Fernau, and R. Niedermeier, Parametrized complexity:

[9] E. J. Cockayne, Irredundance, secure domination and maximum degree intrees,

COUNTING QUICKLY THE INTEGER VECTORS WITH

Mathematisch Instituut, Universiteit Leiden

INVESTIGATION OF CONSTRAINED BAYESIAN

KARTLOS JOSEPH KACHIASHVILI

Abstract. The article focuses on the discussion of basic approaches to

Key words and Phrases: hypotheses testing, likelihood ratio, frequentist

[1] Bauer P. at all., 1988, Multiple Hypothesenprüfung. (Multiple Hypotheses Testing.)

MATHEMATICS EDUCATION IN SINGAPORE – AN

1National Institute of Education, Nanyang Technological University, Singapore

Key words and Phrases: mathematics education, Singapore, differentiated

MODELING, SIMULATION ANDOPTIMIZATION:

ZuseInstitute, Matheon, and Technische Universitaet Berlin,

Abstract. Unlike engineering, physics, and chemistry, mathematics isalmost

Key words and Phrases: Applied mathematics, mathematics in industry and

STRONG AND WEAK TYPE INEQUALITIES FOR

Department of Mathematics, Institut Teknologi Bandung,

THE ENDLESS LONG-TERM PROGRAMS OF

Department of Mathematics Education, Uniersitas Pendidikan Indonesia (Indonesia

Abstract. To be proficient in teaching mathematics, mathematics teachers need

Key words and Phrases: Mathematics teacher professional development,

DIGRAPH CONSTRUCTION TECHNIQUES AND THEIR

Study Program of Information System

Abstract. A communication network can be modelled as a graphor a directed

Key words and Phrases: construction technique, directed graph.

AN APPLICATION OF ARIMA TECHNIQUE IN

EDDY HERMAWAN1 AND RENDRA EDWUARD2

1The Atmospheric Modeling Division of Atmospheric Science and Technology

Jln. Raya Darmaga Kampus IPB Darmaga Bogor, Bogor 16680

Abstract. In this present study, we mainly concerned an application of ARIMA

Keywords: ARIMA Technique and Rainfall Prediction Models

MAC WILLIAMS THEOREM FOR POSET WEIGHTS

ALEAMS BARRA1, HEIDE GLUESING-LUERSSEN2

1Department of Mathematics, Institut Teknologi Bandung,

Abstract. In this paper we introduce the notion of local-global property. We

Key words and Phrases: MacWilliams extension theorem, partitions, poset

ON FINITE MONOTHETIC DISCRETE TOPOLOGICAL

L.F.D. BALI1, TULUS2, MARDININGSIH3

1Department of Mathematics, Faculty of Mathematics and Natural Sciences,

University of Sumatera Utara, Medan 20155, Indonesia, lukasfagolo@gmail.com

of Sumatera Utara, Medan 20155, Indonesia, tulus@usu.ac.id

University of Sumatera Utara, Medan 20155, Indonesia, mardiningsih@usu.ac.id

Kata kunci: dualitas pontryagin, grup dual pontryagin, grup lingkaran,

For the set of all real numbers R = R ∪ {} with

Because B is an equivalence relation, we have the set of factor S = R2 /B

a = (a, ) with (a, ) ∈ S⊕

a = (a, ) = (a, ) = (, a) with (, a) ∈ S

a• = a a = (a, ) (a, ) = (a, ) ⊕ (, a) = (a, a) ∈ S•

Lemma 2.3. For a,b ∈ R ,a b = (a, b).

Corollary 2.5. For a, b ∈ R ,

Because det a2 ... an ∇ so, we now have that det(A0 )∇.

Let A ⊗ x∇ with A =  1 0 .