Geethanjali: Department of Mechanical Engineering

Department of Mechanical Engineering
Geethanjali
College of Engineering and Technology (Autonomous)
Cheeryal (V), Keesara (M), Medchal Dist 501301. Telangana State
(Affiliated to JNTUH, Approved by AICTE, New Delhi, Accredited by NAAC ‘A’ Grade, Accredited by NBA)
COURSE HAND-OUT
OPERATIONS RESEARCH
IV Year B.Tech I Semester
June 2019 – November 2020
Course Handout: IV Year I Sem Page 1

VISION OF THE INSTITUTE

Geethanjali visualizes dissemination of knowledge and skills to students, who would
eventually contribute to the well being of the people of the nation and global
community.
MISSION OF THE INSTITUTE

 To Impart adequate fundamental knowledge in all basic sciences and
engineering, technical and inter-personal skills to students.
 To bring out creativity in students that would promote innovation, research and
entrepreneurship.
 To preserve and promote cultural heritage, humanistic and spiritual values

promoting peace and harmony in society.
VISION OF THE DEPARTMENT

The Mechanical Engineering department strives to be recognized globally for
outstanding education and research, imparting quality education, churning well-
qualified engineers, who are creative, innovative, and entrepreneurial, solving
problems for societal development.
MISSION OF THE DEPARTMENT

M1: Imparting quality education to students to enhance their skills and make them
globally competitive.
M2: Prepare graduates to engage in life-long learning, possess intellectual
capabilities, serving society with a strong commitment to their profession,
meeting technical challenges and exhibiting ethical responsibility for societal
development.
M3: Conduct quality research, create opportunities for students and faculty to
showcase their talent, disseminate knowledge, and promote community
development, leading to peace and harmony in society.

PROGRAMME EDUCATIONAL OBJECTIVES (PEOs)

PEO1: To prepare students with strong fundamental knowledge in basic sciences
mathematics, engineering courses which would facilitate them find gainful
employment with a sense of appreciation to pursue life-long learning for
professional development.
PEO2: To inculcate problem solving skills in students, imbibing creativity and

innovation which would enable them to develop modern machinery involving
cutting edge technologies of multidisciplinary nature for societal
development.
PEO3: To develop critical thinking with an aptitude to conduct research and

development, instill professional ethics, develop effective communication
and interpersonal skills with positive attitude to contribute significantly
towards their chosen profession, thereby supporting community
development.
PROGRAMME OUTCOMES (POs)

PO1: Engineering knowledge: Apply the knowledge of mathematics, science,
engineering fundamentals, and an engineering specialization to the solution of
complex engineering problems.
PO2: Problem analysis: Identify, formulate, review research literature, and analyze
complex engineering problems reaching substantiated conclusions using first
principles of mathematics, natural sciences, and engineering sciences.
PO3: Design/development of solutions: Design solutions for complex engineering

problems and design system components or processes that meet the
specified needs with appropriate consideration for the public health and
safety, and the cultural, societal, and environmental considerations.
PO4: Conduct investigations of complex problems: Use research-based

knowledge and research methods including design of experiments, analysis
and interpretation of data, and synthesis of the information to provide valid
conclusions.
PO5: Modern tool usage: Create, select, and apply appropriate techniques,
resources, and modern engineering and IT tools including prediction and
modeling to complex engineering activities with an understanding of the

limitations
PO6: The engineer and society: Apply reasoning informed by the contextual
knowledge to assess societal, health, safety, legal and cultural issues and the
consequent responsibilities relevant to the professional engineering practice.
PO7: Environment and sustainability: Understand the impact of the professional

engineering solutions in societal and environmental contexts, and
demonstrate the knowledge of, and need for sustainable development.
PO8: Ethics: Apply ethical principles and commit to professional ethics and
responsibilities and norms of the engineering practice.
PO9: Individual and team work: Function effectively as an individual, and as a

member or leader in diverse teams, and in multidisciplinary settings.
PO10: Communication: Communicate effectively on complex engineering activities

with the engineering community and with society at large, such as, being able
to comprehend and write effective reports and design documentation, make
effective presentations, and give and receive clear instructions.
PO11: Project management and finance: Demonstrate knowledge and

understanding of the engineering and management principles and apply these
to one’s own work, as a member and leader in a team, to manage projects
and in multidisciplinary environments
PO12: Life-long learning: Recognize the need for, and have the preparation and
ability to engage in independent and life-long learning in the broadest context
of technological change.

PROGRAMME SPECIFIC OUTCOMES (PSOs)

PSO1: Apply Continuity, Energy and Momentum equations to mechanical systems,
design and perform experiments in all fields of mechanical engineering
PSO2: Able to analyze, design and develop/model mechanical and its allied
systems using software tools such as AUTOCAD, ANSYS, Creo etc
PSO3: Able to design layouts for process and manufacturing industry taking into
consideration optimization of resources for effective operation and
maintenance.
INDEX Page No
1 Semester Plan
2 Assignment Schedule
3 Scheme
4 ME411: Operations Research
4.1. Course Information Sheet
4.2. Course Plan
4.3 Sample Question unit wise
4.4 Tutorial Questions
4.5 Quiz Questions

GEETHAN JALI COLLEGE OF ENGINEERING AND TECHNOLOGY (AUTONOUMUS)

SEMESTER PLAN, IV YEAR B.TECH I SEMESTER
JUNE 2018 – NOVEMBER 2019
SEMESTER PLAN
JUNE JULY AUGUST SEPTEMBER OCTOBER NOVEMBER

3. SCHEME
FOURTH YEAR – SEMESTER – I
Scheme of
Category
No. of Periods Examination with No. of
S. Course per Week Credits
Course Maximum Marks
No Code
L T P/D CIE SEE Tot C
1 16ME4101 Heat Transfer PC 3 1 - 30 70 100 3
2 16ME4102 Operations Research PC 3 1 - 30 70 100 3
Soft Core – II
Industrial Engineering
16ME4103
3 and Management SC 3 1 - 30 70 100 3
Production Planning
16ME4104
and Control
Professional Elective – III
Non-destructive
16ME4105
Testing
4 16ME4106 Robotic Engineering PE 3 - - 30 70 100 3
Work Study and
16ME4107
Ergonomics
16ME4108 Mechatronics
Open Elective - II
Supply Chain
16MB4131
Management
Knowledge
16CS4132
Management
5 Energy Conservation OE 3 - - 30 70 100 3
16EE4133
and Management
Basics of
16EC4134 Communication
Systems (ECE)
16CE4136 Building Technology
Open Elective – III
16MB4141 Banking and Insurance
16CS4142 Database Systems
Micro-Electro-
16EE4143
Mechanical Systems
Principles of Wireless
16EC4144 Communication
6 Systems (ECE) OE 3 - - 30 70 100 3
16CE4146 Green Buildings
Foreign Language-
16EN4147
French
Foreign Language-
16EN4148
Spanish
Foreign Language-
16EN4149
German
7 Soft Core - II Lab. SC - - 3 30 70 100 2
16ME41L1 Work Study Lab

16ME41L2 Facility Design Lab
8 16ME41L3 Heat Transfer Lab PC - - 3 30 70 100 2
Industry Oriented Mini
9 16ME4111 CC - - - - 100 100 1
Project
10 16ME4112 Major Project Seminar CC - - 2 100 - 100 1
Total 18 3 08 340 660 1000 24
Total Periods per Week 31

4: 16ME4102; OPERATIONS RESEARCH
4.1. COURSE INFORMATION SHEET
Program: Mechanical Engineering Degree: B.Tech

Course: Operations Research Year & Semester: IV – I, Credits: 4
Course Code: 16ME4102, Regulations: AR16 Course Type: Core
Contact Hours: 3 (Lectures) Hour/Week
Course Area/Domain:
Tutorials = 1 Hour/Week
Corresponding Lab Course Code (If any): NIL Lab Course Name: NIL
Course Instructor: Dr. M. Devaiah Course Coordinator: Dr. M. Devaiah
Scope of the Course:
Engineers, scientists, analysts and managers are often faced with the challenge of making
tradeoffs between different factors in order to achieve desirable outcomes. Optimization is
the process of choosing these trade-offs in the best way. Optimization problems, having
reached a degree of maturity over the past several years, are encountered in physical sciences,
engineering, economics, industry, planning, and many other areas of human activity.
Objective of the course is to familiarize the students with standard methods of solving
optimization problems.
Operation Research is also called OR for short and it is a scientific approach to decision
making which seeks to determine how best to design and operate a system under conditions
requiring allocation of scarce resources. Operations research as a field, primarily has a set or
collection of algorithms which act as tools for problems solving in chosen application areas.
OR has extensive applications in engineering, business and public systems and is also used
by manufacturing and service industries to solve their day to day problems.
This course deals with details of various aspects associated with optimization These include
description of optimization techniques, namely, Linear Programming, and their applications
to various engineering and science disciplines including economics and finance. Multi-
objective optimization which handles optimization aspects of more than one objective is also
discussed.

SYLLABUS
GEETHANJALI COLLEGE OF ENGINEERING AND TECHNOLOGY

(Autonomous)
Cheeryal (V), Keesara (M), Medchal District–501 301 (T S)
16ME4102 – OPERATIONS RESEARCH

L T P/D C
IV Year B.Tech, ME - I Semester
3 1 - 3
Pre-requisites: None
COURSE OBJECTIVES:
1. Students will understand the significance of Operations Research concept and techniques
and formulation of LPP models.
2. Students will understand the Algorithms of Graphical and Simplex Methods.
3. Students will understand the Transportation and Assignment techniques.
4. Students will understand the concepts of sequencing and Replacement.
5. Students will understand the concepts of Game theory and Inventory Control.
6. Students will understand the concepts of queuing theory and DPP.
COURSE OUTCOMES:
At the end of the course, student would be able to
Bloom’s
CO No STATEMENT Taxonomy
Level
Identify the importance and value of Operations Research and Applying,
mathematical formulation in solving practical problems in industry Creating
CO1
and Formulate a managerial decision problem into a mathematical (Level 3 & 6)
model
Applying,
Formulate and solve engineering and managerial situations as
CO2 Creating
Transportation and Assignment problems.
(Level 3 & 6)
Applying,
CO3 Apply sequencing and replacement concepts in industry applications
(Level 3)
Applying,
CO4 Apply game theory and inventory concepts in industry applications
(Level 3)
Formulate multi-stage applications into a dynamic programming Applying,
CO5 framework and Apply queuing theory concepts in industry Creating
applications (Level 3 & 6)

Detailed Syllabus:
UNIT-I: DEVELOPMENT: Definition– Characteristics and Phases – Types of models –
Operations Research models – applications.
ALLOCATION: Linear Programming Problem - Formulation – Graphical solution –
Simplex method – Artificial variables techniques: Two–phase method, Big-M method;
Duality Principle.
UNIT-II: TRANSPORTATION PROBLEM: Formulation – Optimal solution, unbalanced
transportation problem – Degeneracy.
ASSIGNMENT PROBLEM: Formulation – Optimal solution - Variants of Assignment
Problem; Traveling Salesman problem.
UNIT-III: JOB SEQUENCING: Introduction – Flow Shop sequencing, n jobs through 2
machines, n jobs through 3 machines, Job shop sequencing, 2 jobs through ‘m’ machines -
graphical model.
REPLACEMENT MODEL: Introduction – Replacement of items that deteriorate with
time, when money value is not counted and counted, Replacement of items that fail
completely, Group Replacement.
UNIT-IV: THEORY OF GAMES: Introduction –Terminology – Solution of games with
saddle points and without saddle points, 2 x 2 games, m x 2 and 2 x n games - graphical
method, m x n games, dominance principle.
INVENTORY MODELS: Introduction – Concept of EOQ, Single item - Deterministic
models – Types - Purchase inventory models with one price break and multiple price breaks,
Stochastic models – demand discrete variable or continuous variable – Single Period model
with no setup cost.
UNIT-V: QUEUING THEORY: Introduction – Terminology - Single Channel – Poisson
arrivals and Exponential Service times – with infinite population and finite population
models– Multichannel – Poisson arrivals and exponential service times with infinite
population.
DYNAMIC PROGRAMMING: Introduction – Terminology - Bellman’s Principle of
Optimality – Applications of dynamic programming - shortest path problem – linear
programming problem.
References:
Text books:
1. J.K.Sharma, Operation Research, MacMilan India Ltd, 5th edition, 2012.
2. A.C.S.Kumar, Operations Research, Yesdee, 2015.
Reference books:
1. A.M.Natarajan, P.Balasubramaniam, A. Tamilarasi, Operations Research, Pearson
Education,2009
2. Harvey M.Wagner, Principles of Operations Research, Prentice Hall, 1975.
3. Frederick S. Hillier &Gerald J.Libermann, Introduction to Operation Research, TMH,
2001.
4. Hamdy A.Taha, Introduction to Operation Research, Prentice Hall, 1997.

CO-POs and CO-PSOs Mapping
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12 PSO1 PSO2 PSO3
CO1 3 2 - - 3 - - - - - - 2 - - -
CO2 3 2 2 - 3 - - - - - - 2 - - -
CO3 3 2 2 - 3 - - - - - - 2 - - -
CO4 3 2 2 - 3 - - - - - - 2 - - -
CO5 3 2 2 - 3 - - - - - - 2 - - -
C401 3.00 2.00 2.00 - 3.00 - - - - - - 2.00 - - -
1- Low correlation (Low), 2- Medium correlation (Medium), 3-High correlation (High)
Justifications for CO - PO Mapping

MAPPING LOW/MEDIUM/HIGH JUSTIFICATION
CO1 - PO1 H Acquire the knowledge of the logic behind operations research techniques
Knowledge in programming codes that are used in NC part programming and computer
CO1 - PO5 H
aided part programming will enable the students analyse different
CO1 - PO12 M Ability to understand and adopt the new innovative methods in simulation
Students will be able to analyze the practical application of operations research in
CO2 - PO1 H
Engineering
CO2 - PO2 M Students will get the ability to analyze the different types optimization problems
Knowledge in programming codes that are used in NC part programming and computer
CO2 - PO3 M
aided part programming will enable the students analyze different

Students will be able to analyze the different codes used in part programming so that they
CO2 - PO5 H
can develop effective methods
CO2 - PO12 M Ability to understand and adopt the new innovative methods in transportation and assignment
Students will be able to apply and solve different operations research methods such as job
CO3 - PO1 H
sequencing and replacement
Students will be able to analyze and solve different operations research methods such as job
CO3 - PO2 M sequencing and replacement models in different mechanical streams such as thermal,
manufacturing etc.,
Students will be able to formulate, design and solve different operations research methods
CO3 - PO3 M
such as job sequencing and replacement models for societal benefit.
Students will be able to use modern tools and solve different operations research methods
CO3 - PO5 H
such as job sequencing and replacement models.
Ability to understand and adopt the new innovative methods in such as job sequencing and
CO3 - PO12 M
replacement models
Students will be able to understand the importance of different operations research methods
CO4 - PO1 H
such as theory of games and inventory.
Students will be able to analyze and solve different operations research methods such as
CO4 - PO2 M theory of game and inventory models in different mechanical streams such as thermal,
manufacturing etc.,
CO4 - PO3 M
such as theory of games and inventory models for societal benefit.
CO4 - PO5 H
such as theory of games and inventory models.
Ability to understand and adopt the new innovative methods in such as theory of games and
CO4 - PO12 M
inventory models
CO5 - PO1 H Students will be able to apply and solve different operations research methods such as

queuing and dynamics programming problems.

Students will be able to analyze and solve different operations research methods such as
CO5 - PO2 M queuing and dynamics programming problems in different mechanical streams such as
thermal, manufacturing etc.,
CO5 - PO3 M
such as queuing and dynamics programming problems for societal benefit.
CO5 - PO5 H
such as queuing and dynamics programming problems
Ability to understand the new innovative methods in such as queuing and dynamics
CO5 - PO12 M
programming problems
Justifications for CO - PSO Mapping

MAPPING LOW/MEDIUM/HIGH JUSTIFICATION
- - -

GAPS in the Syllabus - to meet Industry/Professional Requirements:
RELEVANCE PROPOSED
S.No DESCRIPTION
TO PO/PSO ACTIONS
Dual and Primal simplex problems and PO2, PO5, NPTEL +

1
Simulation Models PO12 Reading Books
Topics beyond Syllabus/Advanced Topics/Design: NIL
Proposed actions: topics beyond syllabus/assignment/industry visit/guest lecturer/NPTEL etc
Delivery/Instructional Methodologies:
Chalk & Talk  Student Assignment  Web Resources
LCD/Smart Board  Student Seminar ADD-ON Courses
Assessment Methodologies - Direct
Assignments  Mid-Term Exams  Semester End Exams 
Student lab practice Student Seminar Student Vive
Mini/Major Projects Certifications Add-On Courses
Tutorials  Others
Assessment Methodologies – Indirect
Assessment of Course Outcomes by Course End Survey (End of the Semester) 
Student Feedback on Teaching-Learning Process (Twice in a Semester) 
Assessment of Mini/Major Projects
Other
IV Year B.Tech I Semester, Course Handout Page 15

EVALUATION SCHEME
Continuous Internal Evaluation (CIE) (Theory 30 Marks)
Evaluation Method Course with Assignment
Mid Examinations 25
Assignments 5
Total 30
Semester End Evaluation Theory (70)
Part- –A
20
Objective type questions
Part –B
There should be five questions from five units. Each question should be for maximum
of 10 Marks
50
Each unit will have an internal choice.
Both the questions should be of the same complexity in terms of COs and Bloom’s
taxonomy level.
Total 70

COURSE ASSESSMENT METHODS
Assessment To Max Contribution to

What Frequency of conduction Evidence
Method whom Marks Course Outcome
Mid Examinations Two 25 Answer Scripts

Assessment
Methods
CIE 40%
Direct
Assignment One from each unit 5 Reports

Students 100% 75%
End of every semester Consisting
SEE Semester End Examination 70 Answer Scripts 60%
of Part-A and Part-B
Assessment
methods
Course End Survey End of course 10%

Indirect
Questionnaire
Students -
Based on COs
Once before I- Mid and once
Online feedback 15%
before II -mid

4.2. COURSE PLAN
S. Lecture Reference to
UNIT Learning Objectives Topics to be Covered
No. Nos Text book
To understand the meaning of operations
Evolution and development of operations
1 I research and its evolution and 1 T1 and T2
research
development
To Explain characteristics, OR phases and
its applications in different area such as Characteristics and Phases, Operation Research
2 I 2-3 T1 and T2
finance, physical sciences, engineering, Models and its Applications
economics, industry, planning etc.
3 I To understand the Formulation of LPP Formulation of LPP and some Problems 4-6 T1 and T2
Discussion on how to solve two variables
4 I LP models by the graphical solution Graphical Solutions and some Problems 7-8 T1 and T2
procedure
To obtain an understanding of why and
how the simplex calculations are made
5 I Simplex Method, 9 - 10 T1 and T2
and know how to recognize the special
situations
To solve the problems using special Artificial Variable Technique- Big M Method
6 I 11 - 13 T1 and T2
solution algorithms and Two Phase Method and some Problems
Introduction to Transportation Problems, Finding
Initial Basic Feasible Solution (IBFS) by using
To formulate transportation as LPP and different methods namely, NWCM, LCEM,
7 II 14 - 20 T1 and T2
how to solve these problems RMM CMM, VAM, Optimality Test,
Variations in TP Unbalanced TP and some
Problems
To formulate assignment problems as LPP Introduction to Assignment Problems, finding

8 II optimal solution by Hungarian Algorithms, 21 - 25 T1 and T2
and how to solve these problems
Variation in Assignment Problems and some

Problems, Unbalanced Assignment problems,

Restrictions in Assignment Problems and some
Problems
 The ability to sequence requires Introduction to Sequencing Problems, Processing
9 III higher-order thinking skills, from of n jobs on 2 machines (Johnson’s Algorithm) 26 - 28 T1 and T2
recognizing patterns to determining and some Problems
cause and effect and more.
 Sequencing helps students understand Processing of n jobs on 3 machines and
10 III and organize material they've learned Processing of n jobs on m machines and some 29 - 31 T1 and T2
as well as helps them solve problems Problems
11 III To describe replacement models Introduction to Replacement Problems, 32 - 33 T1 and T2
 To design Replacement of items that
deteriorate with time when money Replacement of items when fails with constant
value is constant money and some Problems
 To design Replacement of items that
12 III Replacement of items when fails with money 34 - 35 T1 and T2
deteriorate with time when money
value increases with time and some Problems.
value is change with time
 To derive and Analyze replacement of Group Replacement and some Problems
items that fail completely
Introduction to Theory of games, Two person
13 IV To propose the best strategy using zero sum games with saddle point and some 36 - 37 T1 and T2
decision making methods under game Problems
theory and to solve the zero-sum two- Games without saddle point, Dominance
14 IV 38 - 41 T1 and T2
person games, algebraic method and Principle and some Problems
graphical method Algebraic method and Graphical method and
15 IV 42 - 46 T1 and T2
some Problems
 To understand concept of inventory
management and doctrine of inventory Introduction to Inventory models, Concept of
16 IV 47 - 52 T1 and T2
 To describe the basic concept of EOQ Inventory, derivation of Economic Order
model

 To describe the economic production Quantity (EOQ)

quantity and its assumptions and solve
typical problems model Model –I and II Problems and some Problems
 To describe the quantity discount and
Model –III & IV Problems and some Problems
solve typical problems model
 To describe reorder point and solve
typical problems models and solve
typical problems.
To understand the meaning of Queuing Introduction to Queuing Models (Waiting line
17 V model or waiting line models. 53 - 55 T1 and T2
theory), single channel problems
To find out the optimum service rate and
the number of servers so that the average
18 V Multi Server Problems 56 - 59 T1 and T2
cost of being in queuing system and the
cost of service are minimized.
To explain fundamentals of dynamic Dynamic Programming Problems, Bellman’s
19 V programming 60 - 62 T1 and T2
principle of Optimality
To solve multi-level decision problems Short path problems, LP problems, Applications
20 V using dynamic programming method 63 - 64 T1 and T2
of DPP

4.3. UNIT WISE QUESTIONS AND ANSWERS
UNIT –I: Evolution, development of OR, Graphical and Simplex Method

1. What is operations research?
2. Give various definitions of operations research?
3. What is meant by redundant constraint?
4. What are the applications of OR in industry?
5. What are the shadow prices? What is its significance in simplex method of solving LPP?
6. What is a model? List the various classification schemes of Operations Research models.
7. What are the advantages of using linear programming?
8. Discuss the advantages and limitation of good Operations Research model.
9. Define linear programming problem. Give an example.
10. Define iso- profit and iso-cost lines. How do these help to obtain a solution to an LP
problem?
11. Classify OR models according to problem for which they are developed and explain
12. Describe the Phases in O.R and Write the characteristics of O.R.
13. What are the advantages of using linear programming?
14. Define O.R? With examples, describe the models of O.R
15. Define the following. Give example for each
(a) Alternate optimum solution
(b) Unbounded solution
(c) Infeasible solution
(d) Criterion value
(e) basic variable
16. Old hens can be brought at Rs. 20 each and young ones at Rs. 50 each. The old hens lay
3 eggs per week and the young ones lay 5 eggs per week, each egg being worth of Rs.
1.50 ps. A hen (young or old) costs Rs. 1.50 per week to feed; I have only Rs.800 to
spend for hens. How many of each kind should I buy to give a profit of at least Rs. 60/-
per week, assuming that I cannot house more than 20 hens?
17. A firm manufactures headache pills in two sizes A and B. Size A contains 2 grains of
aspirin, 5 grains of bicarbonate and 1 grain of codeine. Size B contains 1 grain of aspirin,
8 grains of bicarbonate and 6 grains of codeine. It is found by users that it requires at
least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing
immediate effect. It is required to determine the least number of pills a patient should
take to get immediate relief. Formulate the problem as a standard LPP.
18. A farmer has 100 acre farm. He can sell all tomatoes, lettuce, or radishes he can raise.
The price he can obtain is Rs 1.00 per kg for tomatoes, Rs 0.75 a head for lettuce and Rs
2.00 per kg for radishes. The average yield per acre is 2000 kg of tomatoes, 3000 heads
of lettuce and 1000 kgs of radishes. Fertilizer is available at Rs 0.50 per kg and the

amount required per acre is 100 kgs each for tomatoes and lettuce, and 50 kgs for
radishes. Labour required for sowing and harvesting per acre is 5 man-days for tomatoes
and radishes, and 6 man-days for lettuce. A total of 400 man-days of labour are available
at Rs 20.00 per man-day. Formulate this as a Linear-Programming model to maximize
the farmers total profit.
19. Solve the following LPP using graphical method.
Maximize, Z = 10X1 + 8X2
Subject to constraints
X1 + 2X2 ≤ 1000
X1 ≤ 300
X2 ≤ 500
X1 and X2 ≥ 0
20. Use the graphical method to solve the following LP Problem
Minimize = 20x1 + 10x2
Subject To
x1 +2x2 40
3x1+x2 30
4x1+3x2 60
x1, x2, 0
21. Use the graphical method to solve the problem:
Maximize Z = 2x1 +x2,
Subject to
x2 ≤ 10
2x1 + 5x2 ≤ 60
x1 + x2 ≤ 18
3x1 + x2 ≤ 44
and x1 ≥ 0, x2 ≥ 0
22. Solve graphically:
Maximize z = 3x1  5x2
Subjected to constraints
2 x1  6x2  50
3 x1  2x2  35
5x1  3x2  10
3 x2  20 where x1 , x2 , x3  0
23. Solve the following LPP using graphical method and verify by simplex method
24. Solve the following problem by simplex method

Maximize z = 3x1  4 x2  x3

Subjected to constraints
x1  2 x2  5x3  90
2 x1  x 2  x3  60
3x1  x2  2 x3  80 where x1 , x2 , x3  0
25. Solve the following LPP

Maximize Z = 15x1+6x2+9x3+2x4
Subject to constraints 2x1 + x2 + 5x3 + 6x4 ≤ 20
3x1 + x2 + 3x3 + 25x4 ≤ 20
7x1 + x4 ≤ 70
x1, x2, x3 and x4 ≥ 0
26. Use simplex method to
Minimize Z = x2 - 3x3+2x5
3x2 - x3 + 2x5 ≤ 7
-2 x2 + 4x3 ≤ 12
-4 x2 + 3x3 + 8x5 ≤ 10
x2, x3 and x5 ≥ 0
27. Solve the following problem by simplex method adding artificial variable
Maximize Z = 2x1+5x2+7x3
3x1 + 2x2 + 4x3 ≤ 100
x1 + 4x2 + 2x3 ≤ 100
x1 + x2 + 3x3 ≤ 100
x1, x2, x3 ≥ 0
28. What do you mean by LPP? What are its limitations? Use penalty (or Big-M) method to
Maximize Z = 3x1- x2
2x1 + x2 ≥ 2
x1 + 3x2 ≤ 3
x2 ≤ 4
x1, x2 ≥ 0
29. Use Big-M Method to solve the following
Maximize = 6x1 + 4x2
Subject To
2x1 +3x2 ≤30
3x1+2x2 ≤24
x1+x2 ≥3
x1, x2,≥0
30. Solve the following LPP problem by two-phase method
Max = 2x1 + 3x2 + 5x3

3x1 +10x2 + 5x3 15
33x1-10x2 +9x3 33
x1+2x2+3x3 4
x1, x2, x3 0
UNIT –II: Transportation problems and Assignment Problems
1. What is meant by optimal solution
2. Explain the steps in transportation algorithm.
3. When does degeneracy occur in Transportation Problem?
1. Distinguish between assignment and allocation problem. Write the situation where an
assignment problem can arise?
4. Give the mathematical formulation of transportation problem.
5. What is meant by balanced transportation problem? Give the method of solving
transportation problem with capacities.
6. With the example of your choice, explain the algorithm of North West corner method to
obtain IBFS in a transportation problem.
7. What is IBFS. Give step by step approach to find IBFS in a transportation problem using
Vogel’s Approximation method.
8. Write the algorithm to find IBFS in a transportation problem using least cost entry
method. Give suitable examples.
9. What is assignment problem?
10. How the assignment problem can be viewed as a linear programming problem?
11. Formulate the travelling –Salesman problem as an assignment problem.
12. What is unbalanced assignment problem? How is it solved by the Hungarian method?
13. What is the difference between transportation problem and assignment problem?
14. Explain the Hungarian method to solve an assignment problem.
15. What is the unbalanced Assignment problem? How is it solved by the Hungarian method?
16. How can you maximize an objective function in the Assignment problem?
17. Find the total cost using North-west corner method. Also find the optimal assignment.
W1 W2 W3 W4 Capacity
F1 95 105 80 15 12
F2 115 180 40 30 7
F3 195 180 95 70 5
Requirement 5 4 4 11
18. Solve the following transportation problem, by findings; find the IBFS by North West
corner rule and OBFS by stepping stone method, where the entries are cost coefficients.
To Destination Availability
1 2 3 4
From 1 15 0 20 10 50
Origins 2 12 8 11 20 50
3 0 16 14 18 100
Requirement 30 40 60 70 200

19. Solve the following transportation problem for the optimal solutions. Use North-West
corner method to generate initial BFS.
Factory\Warehouse W X Y Z Availability
A 19 30 50 10 7
B 70 30 40 60 9
C 40 8 70 20 18
Requirement 5 8 7 14
20. A company has three plants A, B, C which supplies to ware houses 1, 2, 3, 4, 5. Monthly
plant capacities are 800, 800, 900 units respectively while the monthly requirements at
the warehouses are 400, 400, 500, 400, 800 units respectively. The unit transportation
costs are shown in table below. Determine optimum distribution for the company in order
to minimize the total transportation cost.
1 2 3 4 5
A 5 8 6 6 3
Plants
B 4 7 7 6 6
C 8 4 6 6 3
21. Use Vogel’s’ approximate method to obtain an initial basic feasible solution of the
transportation problem and find optimal solution
Warehouse
W X Y Z Supply
Factory
A 11 13 17 14 250
B 16 18 14 10 300
C 21 24 13 10 400
Demand 200 225 275 250
22. An oil corporation has got three refineries P, Q and R and it has to send petrol to four
different depots A, B, C and D. The cost of shipping 1 gallon of petrol and the available
petrol at the refineries are given in the table. The requirement of the depots and the
available petrol at the refineries are also given. Find the minimum cost of shipping after
obtaining an initial solution by VAM.
Depots
Available
A B C D
P 10 12 15 8 130
Refineries
Q 11 11 9 10 150
R 20 9 7 18 170
Required 90 100 140 120
23. Find the initial basic feasible solution for the following transportation problem by VAM
method and test the solution for optimality by MODI method.
A B C D E Supply
F 2 1 4 2 5 50
G 4 3 6 4 5 60
H 1 7 7 5 8 75
I 8 8 7 6 9 80
Demand 75 60 50 80 100

24. There are three sources or origins which store a given product. These sources supply
these products to four dealers. The capacities of the sources (S1) and the demands at
dealers (D1) are as given below
S1 = 150, S2 = 40, S3 = 80, D1 = 90, D2 = 70, D3 = 50, D4 = 60
The cost of transporting the product from various sources to various dealers is shown
in the table below
D1 D2 D3 D4
S1 27 23 31 69
S2 10 45 40 32
S3 30 54 35 57
Find out the optimal solution for transporting the products at a minimum cost.
25. Determine the initial basic feasible solution to the following transportation problem
using northwest corner cell method.
26. A Military equipment is to be transported from origins 1, 2, 3 to destinations 1, 2, 3, 4.

The supply at the origins, the demand at the destinations and time to shipment is given
below. Work out a transportation plan that the required for shipment is the minimum.
Destination Supply
10 22 0 22 8
Origin
15 20 12 8 13
20 12 10 15 11
Demand 5 11 8 8
27. Find the initial basic feasible solution for the following transportation problem by VAM
method and test the solution for optimality by MODI method.
A B C D Supply
F 5 2 4 3 22
G 4 8 1 6 15
H 4 6 7 5 8
Demand 7 12 17 9
28. Raju and Co. has four lathe machines on which four workers operate. Any worker can
operate any machine but due to the difference in skill and machine complexity the time
of operation varies. The average times in hours when same job dome on each machine by
each worker is given below.
L1 L2 L3 L4
W1 7 6 4 9
W2 5 5 8 8
W3 4 5 4 6
W4 7 8 5 8

(a) Find optimal allocation

(b) The company wants to replace the less efficient lathe with a new machine. The
probable times (in hrs) that each worker can operate is estimated as 4, 5, 6 and 6
respectively. Verify whether the company has to replace any machine. If so, which
machine is to be replaced?
29. Find the optimal solution for the assignment problem with the following cost matrix.
I II III IV V
A 11 17 8 16 20
B 9 7 12 6 15
C 13 16 15 12 16
D 21 24 17 28 26
E 14 10 12 11 15
30. Suggest optimum assignment of 4 workers A, B, C and D to 4 jobs, I, II, III and IV. The
time taken by different workers in completing the different jobs is given in table below.
Jobs
I II III IV
A 5 5 - 2
Workers
B 7 4 2 3
C 9 3 5 -
D 7 2 6 7
31. A hospital wants to purchase three different types of medical equipments and five
manufacturers have come forward to supply one or all the three machines. However, the
hospital’s policy is not to accept more than one machine from any one of the
manufactures. The data relating to the price (in thousands of rupees) quoted by the
different manufactures are as given in table. Find the optimum assignment policy.
Machines
1 2 3
A 30 31 27
Sub
B 28 29 26
Ordinates
C 29 30 28
D 28 31 27
E 31 29 26
32. Find the optimal solution for the assignment problem with the following cost Matrix
I II III IV
A 12 30 21 15
B 18 32 09 31
C 44 25 24 21
D 23 30 28 14

33. Find the optimal solution for the assignment problem with the following cost Matrix
I II III IV
A 5 40 20 5
B 25 35 30 25
C 15 25 20 10
D 15 5 30 15
UNIT –III: Sequencing Problems and Replacement Problems

1. What are the assumptions made in the sequencing problem
2. What is priority sequencing and what are the priority sequencing rules.
3. Explain the possibility and working rules of a maximization case in sequencing
4. Define Total elapsed time.
5. Define the problem of sequencing.
6. Explain the problem of processing n jobs through three machines.
7. Explain the Jackson’s conditions in working with ‘n’ jobs through 2 machine sequencing
problems.
8. Give 3 examples of sequencing problem from your daily life?
9. Explain briefly the importance of replacement analysis.
10. What are the situations which make the replacement of items necessary?
11. What are the conditions recommended for the replacement of a machine with a new one
when you already have an old one?
12. What do you mean by ‘money value is not counted and counted’ in replacement
analysis?
13. Define present worth factor.
14. Explain group replacement concept and its applications.
15. Find the sequence that minimizes the total elapsed time required to complete the
following jobs.
Processing times in hours
No. of Jobs 1 2 3 4 5 6
Machine A 4 8 3 6 7 5
Machine B 6 3 7 2 8 4
16. Find the sequencing that minimizes the total time required in performing the following
jobs on three machines in the order A-B-C as shown in the below table. Also find the
total elapsed time
Machine/Job 1 2 3 4 5 6
A 8 3 7 2 5 1
B 3 4 5 2 1 6
C 8 7 6 9 10 9
17. There are six jobs, each of which must go through machines A, B and C. processing time
(in hours) are given in the following table.
Job 1 2 3 4 5 6
Machine A 12 10 9 14 7 9
Machine B 7 6 6 5 4 4
Machine C 6 5 6 4 2 4
Order of the processing of each job is ACB. Sequence suggested is 5-3-6-2-1-4. Find the total
time elapsed for the sequence suggested.
18. Consider following 3 machines (A, B, C) and 7 jobs problem. Find the optimal sequence
if the processing order is ABC and also determine makes pan time for the optimal
sequence.
Job 1 2 3 4 5 6 7
A 5 7 3 4 6 7 12
B 2 6 7 5 9 5 8
C 10 12 11 13 12 10 11
19. We have five jobs each of which must go through the machine A, B and C in the order
ABC:
Processing Times (in Hours)
Job No 1 2 3 4 5
Machine A 5 7 6 9 5
Machine B 2 1 4 5 3
Machine C 3 7 5 6 7
Determine a sequence for the jobs that will minimize the total elapsed time
20. The maintenance cost and resale value per year of a machine whose purchase price is Rs.
7000 is given below
Year 1 2 3 4 5 6 7 8
Maintenance cost in Rs 900 1200 1600 2100 2800 3700 4700 5900
Resale value in Rs 4000 2000 1200 600 500 400 400 400
When the machine should be replaced.
21. An individual is planning to purchase a car will cost Rs. 1, 20, 000. The resale value of
the car at the end of the year is 85% of the previous year value. Maintenance and
operation costs during the first year are Rs. 20, 000 and they increase by 15% every year.
The minimum resale value of car can be Rs. 40, 000. (a) When should the car be replaced
to minimize average annual cost (ignore initial) (b) If interest of 12% is assumed, when
should the car be replaced?
22. A truck owner finds from his pas records that the maintenance cost per year of a truck
whose purchase price is Rs. 8000, are given below:
Year 1 2 3 4 5 6 7 8
Maintenance cost in Rs. 1000 1300 1700 2200 2900 3800 4800 6000
Resale Price 4000 2000 1200 600 500 400 400 400
Determine at what time it is profitable to replace the truck?

23. A machine owner finds from his past records that the costs per year of maintaining a
machine whose purchase price is Rs.6000 are as given below:
Year 1 2 3 4 5 6 7 8
Maintenance
1000 1200 1400 1800 2300 2800 3400 4000
cost (Rs)
Resale Price 3000 1500 750 375 200 200 200 200
24. Equipment A costs Rs.9000. Annual operating costs are Rs.200 for the first year and then
increases by Rs.2,000 every year. Determine the best age at which to replace the
equipment. Equipment B costs Rs.10, 000.Annual operating costs are Rs.400 for the first
year and then increases by Rs.800 every year. Now you have a equipment of type A
which is one year old. Should you replace it with B, if so when?
25. A scooter cost Rs 6,000 when new. The running cost and salvage value at the end of
different years are as follows: (in Rs):
Year 1 2 3 4 5 6 7
Running cost 1200 1400 1600 1800 2000 2000 2400
Scrap Values 4000 2666 2000 1500 1000 600 600
26. A truck is priced for Rs.60000 and running costs are at Rs.6000 for each of first 4 years
increasing by Rs.2000 per year in the fifth and subsequent years. If money is worth 10%,
when the truck should be replaced. Assume scrap value = 0.
27. The yearly cost of two machines A and B, when money value is neglected is shown
below. Find their cost patterns if money is worth 10 per cent per year and hence find
which machine is more economical.
Year: 1 2 3
Machine A (Rs): 1,800 1,200 1,400
Machine B (Rs): 2,800 1,200 1,400
28. A manufacturer is overfed two machines A and B. A is priced at Rs.5000 and running
costs are estimated at Rs.800 for each of the first five years, creasing by Rs.200per year
in the sixth and subsequent years. Machine B, which has the same capacity as A, costs
Rs.2500 but will have running costs of Rs.1200 per year for six years, increasing by
Rs.2000 per year thereafter. If the money is worth 10% per year, which machine should
be purchased assuming that both machines will eventually be sold for a scrap at a
negligible value?
29. If the interest rate is 10 per cent per year and running costs are assumed to have occurred
mid year, find when the scooter should be replaced. The following failure rates have
been observed for a certain type of light bulb.
End of Week: 1 2 3 4 5 6 7 8
Probability of failure: 0.05 0.13 0.25 0.43 0.68 0.88 0.96 1.00
The cost of replacing an individual bulb is Rs. 2.25. The decision is made to replace all
bulbs simultaneously at fixed intervals, and also to replace individual bulbs as they fail in
service. If the cost of group replacement is 60 paise per bulb and the total number of
bulbs is 1,000, what is the best interval between group replacements?
30. A newspaper boy buys papers for 50 paise each and sells them for 90 paise each. He

cannot return unsold newspapers. Daily demand R for news papers follows the
distribution.
R 10 11 12 13 14 15 16
probability 0.05 0.16 0.42 0.18 0.10 0.05 0.05
If each day’s demand is independent of the previous day’s, how many papers should be
ordered each day?
31. A computer contains 10000 resistors. When any resistor fails, it is replaced. The cost of
replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same
time, cost per resistor would be reduced to 35 paisa. The % of surviving resistors say S(t)
at the end of month t and the P(t) the probability of failure during the month t are
t 0 1 2 3 4 5 6
S(t) 100 97 90 70 30 15 0
P(t) -- 0.03 0.07 0.2 0.4 0.15 0.15
What is the optimum replacement policy?
32. A factory has large number of bulbs, all of which must be in working condition. The
mortality of Bulbs is given in the following table:
Week : 1 2 3 4 5 6
Proportion of bulbs failing : 0.10 0.15 0.25 0.35 0.12 0.03
If a bulb fails in service, it costs Rs 3.50 to be replaced; but if all the bulbs are replaced at
a time it costs Rs 1.20 each, nd the optimum group replacement policy
UNIT –IV: Theory of Games and Inventory Models

1. Explain the rules to determine a saddle point.
2. List out the assumptions made in the game theory
3. Briefly explain the importance of game theory.
4. Define: (i) Competitive game (ii) Pure strategies (iii) Mixed strategies (iv) Two – person
–zero sum game (v) Payoff matrix (vi) saddle point.
5. Which competitive situation is called a game?
6. What are the characteristics of game theory?
7. State and explain the column dominance property.
8. Define Maximin-Minimax principle
9. How is concept of dominance principle used in simplifying rectangular games?
10. Differentiate between strictly determinable games and non-determinable games.
11. What is inventory build up? Explain the reasons for build up.
12. What is inventory management? Write the major decisions concerning inventory?
13. What are the consequences of over-inventory and under-inventory situations?
14. List and explain different types of costs incurred in inventory system. with suitable
examples. How they are inter-related

15. Explain the concept of EOQ.

16. Define (i) Set-Up cost, (ii) Carrying cost, (iii) Ordering Cost
17. Derive the expression for EOQ of Wilson-Harris Inventory model
18. Two companies A and B are competing for the same product. Their different strategies
are given in the following payoff matrix.
Company A
A1 A2 A3
Company B B1 2 -2 3
B2 -3 5 -1
Determine the best strategies for both the players
19. Obtain the optimal strategies for both persons and the value of the game for zero-sum
two-person game whose payoff matrix is given below:
3 2 4 0
2 4 4 2
4 2 4 0
0 4 0 8
20. Solve the following game using dominance property and also algebraic method
Player B
B1 B2 B3
Player A A1 1 7 2
A2 6 2 7
A3 5 2 6
21. The payoff matrix of a game is given. Find the solution of the game to the player A and B
B
I II III IV V
I -2 0 0 5 3
II 3 2 1 2 2
A
III -4 -3 0 -2 6
IV 5 3 -4 2 -6
22. A and B play game in which each has three coins 5p, 10p and a 20p. Each selects a coin
without the knowledge of the others choice. If the sum of the coins is an odd amount, A
wins B’s coin. If the sum is even B wins A’s coin. Find the best strategy for the each
player and the value of the game
23. Find the saddle point (or points) and hence solve the following game :
B
I II III IV
A I -5 2 1 20
II 5 5 4 6
III 4 -2 0 -5

24. Solve the following game graphically
Player B
B1 B2 B3
A1 1 3 11
Player A
A2 8 5 2
25. Write the assumptions made in game theory. Solve the following game graphically
1 -3
3 5
-1 6
4 1
2 2
-5 0
26. Solve the game by reducing to 2 x 2 game by graphical method.
27. Solve the game given in the table by reducing to 2 X 2 game by graphical method.
PLAYER B
I II III
PLAYER A I 4 -1 0
II -1 4 -2
28. A stockiest has to supply 400 units of a product every Monday to his customers. He gets
the product at Rs.50 per unit from the manufacturer. The cost of ordering and
transportation from the manufacturer is Rs.75 per order. The cost of carrying inventory is
7.5% per year of the cost of the product. Find (a) the economic lot size (b) the total
optimal cost.
29. A company uses 24000 units of a raw material which costs Rs. 12.50 per unit. Placing
each order costs Rs 22.50 and the carrying cost is 5.4% per year of the average
inventory. Find the economic order quantity and the total inventory cost ( including the
cost of the material)
30. Find the most economic batch quantity of a product on a machine if the production rate
of the item on the machine is 300 pieces/day and the demand is uniform at the rate of
150 pieces/day. The set-up cost is Rs.300 per batch and the cost of holding one item in
inventory is Rs.0.81 per day. How will the batch quantity vary if the machine production
rate was infinite?
31. Find the most economic batch quantity of a product on a machine of the production rate
of the item on the machine is 300 pieces/day and the demand is uniform at the rate of 150
pieces/day. The set-up cost is Rs.300 per batch and the cost of holding one item in
inventory is Rs.0.81 per day. How will the batch quantity vary if the machine production
rate was infinite?
32. The annual demand for an items is 3200 units, the unit cost is Rs. 6/- and inventory
charges 25% per annum. If the cost of one procurement is Rs. 150/-, Determine (i).
Economic Order Quantity (ii). Number of orders per year (iii). Time between two
consecutive orders (iv). The optimal cost.
33. The demand for an item in a company is 15000 units per year and the company can
produce the items at a rate of 300 per month. The cost of one set-up is Rs. 500 and
holding cost of 1 unit per month is 15 paise. The shortage cost of one unit is Rs. 20 per
month. Determine: (a) Optimum production batch quantity and number of shortages (b)
Optimum cycle time and production time. (c) Maximum inventory level in the cycle. (d)
Total associated cost per year if the cost of the items is Rs. 20 per unit.
34. A company requires 10000 units of an item per annum. The cost of ordering is Rs 100 per
order. The inventory carrying cost is 20%.The unit price of the item is Rs 10.Calculate (a)
Economic order quantity (b) Optimum total annual cost
35. In a two wheeler manufacturing company pistons are being fed into the main assembly
line from a product line situated in the next bay. The annual demand for the pistons is
8000 units and the annual production capacity of the product line manufacturing the
pistons is 12000 units. The set up cost is Rs.125 per setup and carrying cost is Rs.4 per
piston per year. Find (a) Economic order quantity (b) Maximum inventory.
36. An engineering firm has determined from the analysis of past accounting and production
data that part number 607 has ordering cost of Rs 350 per order and it costs Rs 22 per
part. The inventory carrying cost is 15% of the cost of the item per year. The firm
currently purchases Rs 2, 20,000 worth of this part every year. Determine the economic
order quantity. What is the time between two orders? What is optimum numbers of
orders per year to minimize the cost for the firm?
37. If a product is to be manufactured within the company the details are as follows:
Annual demand rate, r = 24000 units
Production rate, k = 48000 units
Setup cost, C0 = Rs. 200 per setup
Carrying cost, Cc = Rs. 20/unit/year. Find the (i) EOQ and (ii) Cycle time.
38. A product ‘X’ is purchased by a company from outside suppliers. The consumption is
10,000 units per year. The cost of the item is Rs.5 per unit and the ordering cost is
estimated to be Rs.100 per order. The cost of carrying inventory is 25%. If the
consumption rate is uniform, determine the economic purchasing quantity.
39. Find the optimal order quantity for a product for which the price breaks are as follows:
Quantity Unit cost (Rs.)
0 ≤ q1 ≤ 500 10.00
500 ≤ q2 ≤ 750 9.25
750 ≤ q3 8.75
The monthly demand for a product is 200 units, the cost of storage is 2% of the unit cost
and the cost of ordering is Rs. 350.

UNIT – V: Queuing models and Dynamic programming problems

1. Describe the various elements of the queue.
2. Explain briefly the main characteristics of queuing system.
3. What do you understand by (M/M/I): (α/FCFS). Explain the terms
4. What do you understand by a queue? Give some important applications of queuing
theory?
5. Explain Kendall’s notations fpr representing queuing models.
6. State the Bellman’s Principle of Optimality
7. Define Bellman’s principle of optimality with examples.
8. What are the applications of dynamic programming?
9. State the Bellman’s principle of optimality in dynamic programming and give a
mathematical formulation of a dynamics programming problem?
10. What are the applications of dynamic programming?
11. What are the characteristics of dynamic programming?
12. A TV repair man finds that the time spent on repairing has an exponential distribution
with mean 30 min per unit. The arrival of TV sets is Poisson with an average of 10 sets
per day of 8 hours. What is his expected idle time per day? How many sets are there on
the average?
13. In a store with one server, 9 customers arrive on an average of 5 minutes. Service is done
for 10 customers in 5 minutes. Find (i) average number of customers in the system. (ii)
Average queue length. (iii) Average time a customer spends in the store. (iv) Average
time a customer waits before being served
14. Jobs arrival at a workstation in a manufacturing plant is in a Poisson fashion at an average
rate of five per hour. The time to machine one job is an exponential distribution with a
mean time of 20 minutes. What is the expected time a job has to wait at the workstation?
What will be the average number of jobs waiting at the workstation at any time? What is
the probability that there will be more than four jobs?
15. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day.
Assuming that the inter-arrival time follows an exponential distribution and service
time(the time taken to hump a train) distribution is also exponential with an average of
36 min. Calculate the following: (a) The average number of trains in the system. (b) The
probability of the queue size exceeds 10. (c) If the input of the trains increases to an
average of 33 per day, what will be the change in (a) and (b).
16. A person repairing radios, finds that the time spent on the radio sets has an exponential
distribution with mean 20 minutes. If the radios are repaired in the order in which they
come in and their arrival is approximately Poisson with an average rate of 15 for 8 –hour
day, what is the repairman’s expected idle time in each day? How many jobs are ahead of
the average set just brought in?
17. A supermarket has two girls ringing up sales at the counters. If the service time for each
customer is exponential with mean 4 minutes, and if people arrive in a Poisson fashion at
the counter at the rate of 10 per hour, then calculate: (a) The Probability of having to wait

for service (b) The expected percentage of idle time for each girl (c) If a customer has to
wait, find the expected length of his waiting time.
18. Cars arrive at a petrol pump, having one petrol unit, in Poisson fashion with an average of
10 cars per hour. The service time is distributed exponentially with a mean of 3 minutes.
Find (a) average number of cars in the system (b) average waiting time in the queue. (c)
average queue length (d) the probability that the number of cars in the system is 2.
19. Patients arrive at a clinic to a Poisson distribution at the rate of 30 patients per hour. The
waiting room does not accommodate more than 14 patients. The examination time per
patient is exponential with mean rate of 20 per hour. (a) Find the effective arrival rate at
the clinic. (b) What is the probability that an arriving patient will not wait (c) What is the
expected waiting time until a patient is discharged from the clinic.
20. A manufacturing firm producing refrigerators has given a contract to supply 50 units at
the end of the first month, 50 at the end of the second month, and 50 at the end of the
third. The cost of producing ′X′ number of refrigerators in any month is given by X2. The
firm can produce more members of refrigerators in any month and carry them to the next
month. However, a holding cost of Rs. 20 per unit is charged for any refrigerator carried
over from month to the next. Assuming here is no initial inventory, determine the
number of refrigerators to be produced in each month so as to minimize the total cost,
using dynamic programming approach.
21. A medical representative located at city 1 has to travel to city 10. He knows the distance
of alternative routes from city 1 to city 10 and has drawn the network map based on the
distance between the cities as in the following table. Draw the network and find the
shortest possible route. Also, find the shortest routes from any city to city 10.
From To city Corresponding distance in
city km
1 2, 3, 4 4, 6, 3
2 5, 6, 7 7, 10, 5
3 5, 6, 7 3, 8, 4
4 5, 6, 7 6, 10, 5
5 8, 9 4, 8
6 8, 9 3, 7
7 8, 9 8, 4
8 10 7
9 10 9
22. Use dynamic programming to find the value of
Maximize z = y1y2y3
Subject to the constraints:
Y1 + y2 + y3 = 5
Y1, y2, y3 ≥0

Quiz Questions
1. If the value of the game is zero, then that game/strategy is known as [ ]

(a) Pure strategy (b) mixed strategy (c) fair game (d) pure game
2. When minimax and maximin criterion match, then [ ]
(a) Saddle point exist (b) fair game is resulted (c) mixed strategies are adopted (d)
game will be unfair
3. Games with saddle points are _______________ in nature [ ]
(a) deterministic (b) probabilistic (c) stochastic (d) normative
4. A game is said to be fair if [ ]
(a) saddle point exists (b) pure strategies are used (c) value of the game is zero (d) it is a
zero sum game
5. At EOQ [ ]
(a) APC = AOC (b) AOC = ACC (c) ACC = ASC (d) ASC = APC
6. The stock maintained to withstand unknown demand changes is known as [ ]
(a) De-coupling (b) pipeline inventory (c) fluctuatory inventory (d) anticipatory inventory
7. Which of the following increases with quantity ordered per order [ ]
(a) Order cost (b) carrying cost (c) demand (d) purchase cost
8. The time difference between ordering point and replenishment point is called [ ]
(a) Re-order period (b) lead time (c) inventory carrying cycle time (d) inventory horizon
9. The optimality principal to solve dynamic programming problems is given by [ ]
(a) Dantzig (b) Richard Brown (c) Bellman (d) Kimball
10. The sub problem of a main problem in a DPP is said to be [ ]
(a) State (b) function (c) recursion (d) stage
11. Which of the following is considered as recursive optimization [ ]
(a) Linear programming problem (b) dynamic programming problem (c) assignment
problem (d) Transportation problem
12. Characteristics LPP and DPP approaches differ in [ ]
(a) Objective function (b) constraints (c) recursive function (d) transformation function
13. A machine is to be replaced if the average running cost [ ]
(a) Is not equal to current running cost (b) of current period is greater than that of next
period (c) till current period is greater than that of next period (d) of current period is
less than that of next period
14. When money value changes with time at 10% then PWF for first year is [ ]
(a) 0.909 (b) 1 (c) 0.826 (d) 0.9
15. Find the oddman out [ ]
(a) Present worth factor (PWF) (b) discounted rate (DR) (c) depreciation value (DV) (d)
mortality tables
16. Group replacement policy is applicable for [ ]
(a) Repairable items (b) items that fail partially (c)items that fail completely (d)
dissimilar items
17. Group replacement policy is most suitable for [ ]
(a) Blood group (b) street lights (c) truck (d) any of the above

18. Customer behaviour in which be moves from one queue to another in a multiple channel
situation is [ ]
(a) Balking (b) reneging (c) jockeying (d) alternating
19. Which of the following characteristics apply to queue system? [ ]
(a) Customer population 9b) Arrival process (c) both (a) and (b) (d) neither (a) and (b)
20. In (M/M/S) : (N/FIFO), which of the following is wrongly stated [ ]
(a) Poisson arrival (b) exponential service (c) single server (d) limited service
21. Which of the following is not considered as the negative behaviour of customer according
to queue discipline [ ]
(a) Reneging (b) jockeying (c) balking (d) boarding
22. At steady state in a queue [ ]
(a) λ > μ (b) λ < μ (c) λ ≤ μ (d) λ ≥ μ
23. SIRO discipline is found generally in __ [ ]
(a) Office filing (b) trains arriving to platform (c) lottery (d) loading and unloading
24. A Poisson arrival, exponential service by single server to limited queue selected randomly
is represented by ______________
(a) (M/E/S) : (α/SIRO) (b) (M/M/S) : (N/SIRO) (c) (M/M/1) : (N/SIRO) (d) (M/M/i) :
(α/SIRO)
25. A doctor rushing to an emergency case leaving his regular service is said to be
(a) Pre-emptive Q discipline (b) non-pre-emptive Discipline (c) reneging (d) Balking
26. A game with more than two players is called ______________________ player game.
27. The value of the game shoes pay off is a 2 × 2 identity matrix is _____________.
28. The game whose pay off is null matrix is ________________ game.
29. If minimax value is same as maximin value the game said to have ______________.
30. The list of all possible actions that will take for every pay off that might results is called
_____________________.
31. If gain of a player is loss of the other then the game is called___________.
32. The EOQ formula was first proposed by _________________

33. Formula for total cost in simple EOQ model is _____________
34. The gap between the ordering point and its immediate actual stock completion point is
called _______________________
35. First stage decision is solved last in ___________ recursive approach.
36. The algebraic equation that represents the benefits of a decision in DPP is called______
37. The optimality principle is given by_________
38. According optimality principle, if an optimal solution is obtained to a systems it must be
optimal to _______________ of the system.
39. The probabilistic model among replacement policies is _________________
40. Pre paid taxi at an airport follow ___________________ queue discipline
41. The unit of utilization factor is _______________________________
42. If the value of the game is zero, then that game/strategy is known as [ ]
A) Pure strategy B) mixed strategy C) fair game D) pure game

43. A game is said to be fair if [ ]

A) saddle point exists B) pure strategies are used C) value of the game is zero D) it is a
zero sum game
44. Games with saddle points are _______________ in nature [ ]
A) Deterministic B) probabilistic C) stochastic D) normative
45. At EOQ [ ]
A) APC = AOC B) AOC = ACC C) ACC = ASC D) ASC = APC
46. The optimality principal to solve dynamic programming problems is given by [ ]
A) Dantzig B) Richard Brown C) Bellman D) Kimball
47. The sub problem of a main problem in a DPP is said to be [ ]
A) State B) function C) recursion D) stage
48. When money value changes with time at 10% then PWF for first year is [ ]
A) 0.909 B) 1 C) 0.826 D) 0.9
49. Group replacement policy is most suitable for [ ]
A) Blood group B) street lights C) truck D) any of the above
50. Customer behavior in which be moves from one queue to another in a multiple channel
situation is [ ]
A) Balking B) reneging C) jockeying D) alternating
51. At steady state in a queue [ ]
A) λ > μ B) λ < μ C) λ ≤ μ D) λ ≥ μ
52. The EOQ formula was first proposed by _________________
53. A game with more than two players is called ______________________ player game
54. The game whose pay off is null matrix is ________________ game
55. The unit of utilization factor is _______________________________
56. According optimality principle, if an optimal solution is obtained to a systems it must be

optimal to _______________ of the system
57. Pre paid taxi at an airport follow ___________________ queue discipline
58. First stage decision is solved last in ___________ recursive approach
59. The gap between the ordering point and its immediate actual stock completion point is
called _______________________
60. The probabilistic model among replacement policies is _________________
61. Formula for total cost in simple EOQ model is _____________

Tutorial Problems
Tutorial Sheet No. 1 (Formation of LPP)

1. A ship has three cargo holds, forward, aft and center. The capacity limits are: Forward
2000 tons, 100,000 cubic meters C e n t e r 3 0 0 0 t o n s , 1 3 5 , 0 0 0 c u b i c m e t e r s Aft
1500 tons, 30,000 cubic meters. The following cargoes are offered, the ship owners may
accept all or any part of each commodity:
Tutorial Sheet No. 2 (Simplex Method)

1. Solve the following LPP problem by two-phase method
Tutorial Sheet No. 3 (Artificial Variable Technique-Big M Method)

1. Use Big-M Method to solve the following
Tutorial Sheet No. 4 (Unbalanced and variations in TP)

Tutorial Sheet No. 5 (Unbalanced and variations in AP)

1. A company has five jobs V, W, X, Y and Z and five machines A, B, C, D and E. The
given matrix shows the return in Rs. of assigning a job to a machine. Assign the jobs to
machines so as to maximize the total returns.
Tutorial Sheet No. 6 (Job Sequencing Model)

1. Consider following 3 machines (A, B, C) and 7 jobs problem. Find the optimal sequence
if the processing order is ABC and also determine makes pan time for the optimal
sequence.
Job 1 2 3 4 5 6 7
A 5 7 3 4 6 7 12
B 2 6 7 5 9 5 8
C 10 12 11 13 12 10 11
Tutorial Sheet No. 7 (Individual Replacement Problem)
2. A manufacturer is offered two machines A and B. A has the cost price of Rs. 2,500/- its
running cost is Rs. 400 for each of the first 5 years and increase by Rs.100/- every
subsequent year. Machine B having the same capacity as A. and costs Rs. 1250/-, has
running cost of Rs.600/- for first 6 years, increasing thereby Rs. 100/- per year. Which
machine should be purchased? Scrap value of both machines is negligible. Money value
is 10% per year.
Tutorial Sheet No. 8 (Group Replacement Problem)
1. The following failure rates have been observed for a certain type of light bulb.
End of Week: 1 2 3 4 5 6 7 8
Probability of failure: 0.05 0.13 0.25 0.43 0.68 0.88 0.96 1.00
The cost of replacing an individual bulb is Rs. 2.25. The decision is made to replace all bulbs
simultaneously at fixed intervals, and also to replace individual bulbs as they fail in service.
If the cost of group replacement is 60 paise per bulb and the total number of bulbs is 1,000,
what is the best interval between group replacements?

Tutorial Sheet No. 9 (Theory of Games)

1. In a certain game player has three possible courses of action L, M and N, while B has two
possible choices P and Q. Payments to be made according to the choice made.
What are the best strategies for players A and B in this game? What is the value of the game
for A and B?
Tutorial Sheet No. 10 (Theory of Games)
1. Two players P and Q play the game. Each of them has to choose one of the three colours:
White (W), Black (B) and Red (R) independently of the other. Thereafter the colours are
compared. If both P and Q has chosen white (W, W), neither wins anything If player P
selects white and Player Q black (W, B), player P loses Rs.2/- or player Q wins the same
amount and so on. The complete payoff table is shown below. Find the optimum
strategies for P and Q and the value of the game.
Tutorial Sheet No. 11 (Inventory Models)

1. A shopkeeper has a uniform demand of an item at the rate of 50 items per month. He
buys from a supplier at a cost of Rs.6/- per item and the cost of ordering is Rs. 10/- per
order. If the stock holding costs are 20% of stock value, how frequently should he
replenish his stock? Suppose the supplier offers 5% discount on orders between 200 and
999 items and a 10% discount on orders exceeding or equal to 1000 units. Can the
shopkeeper reduce his costs by taking advantage of either of these discounts?
Tutorial Sheet No. 12 (Queuing Model)
1. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day.
Assuming that the inter-arrival time follows an exponential distribution and service time
(the time taken to hump a train) distribution is also exponential with an average of 36
min. Calculate the following: (a) The average number of trains in the system. (b) The
probability of the queue size exceeds 10. (c) If the input of the trains increases to an
average of 33 per day, what will be the change in (a) & (b).

Tutorial Sheet No. 13 (Dynamic Programming Problems)

1. In a cargo-loading problem, there are four items of different unit weight and value. The maximum
cargo load is 6 units. How many units of each item are loaded to maximize the value?
Problems: Students are strongly advised to work out all the above problems and problems in
the text-book and do similar problems from the reference books. It is also strongly
recommended that the students should tryout the algorithms on computers to get a better
understanding of the subject.
Chamber Consultation Hours: To be announced in the class by the respective Instructors.

Notice: All notices regarding the course will be put up on Notice board of Mechanical
Engineering Department.
Note: Lecture notes and solutions to above problems are provided separate document.
Prepared by
Dr. M. Devaiah, Course Instructor
Approved by
Dr. T. Siva Prasad
HoD, ME

Geethanjali: Department of Mechanical Engineering

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Department of Mechanical Engineering

Department of Mechanical Engineering

Course Handout: IV Year I Sem Page 1

VISION OF THE INSTITUTE

MISSION OF THE INSTITUTE

 To preserve and promote cultural heritage, humanistic and spiritual values

VISION OF THE DEPARTMENT

MISSION OF THE DEPARTMENT

Course Handout: IV Year I Sem Page 2

PROGRAMME EDUCATIONAL OBJECTIVES (PEOs)

PEO2: To inculcate problem solving skills in students, imbibing creativity and

PEO3: To develop critical thinking with an aptitude to conduct research and

PROGRAMME OUTCOMES (POs)

PO3: Design/development of solutions: Design solutions for complex engineering

PO4: Conduct investigations of complex problems: Use research-based

modeling to complex engineering activities with an understanding of the

PO7: Environment and sustainability: Understand the impact of the professional

PO9: Individual and team work: Function effectively as an individual, and as a

PO10: Communication: Communicate effectively on complex engineering activities

PO11: Project management and finance: Demonstrate knowledge and

Course Handout: IV Year I Sem Page 4

PROGRAMME SPECIFIC OUTCOMES (PSOs)

4.1. Course Information Sheet

4.2. Course Plan

4.3 Sample Question unit wise

4.4 Tutorial Questions

4.5 Quiz Questions

Course Handout: IV Year I Sem Page 5

GEETHAN JALI COLLEGE OF ENGINEERING AND TECHNOLOGY (AUTONOUMUS)

JUNE JULY AUGUST SEPTEMBER OCTOBER NOVEMBER

Course Handout: IV Year I Sem Page 6

16ME41L1 Work Study Lab

Course Handout: IV Year I Sem Page 8

4: 16ME4102; OPERATIONS RESEARCH

4.1. COURSE INFORMATION SHEET

Program: Mechanical Engineering Degree: B.Tech

Scope of the Course:

Course Handout: IV Year I Sem Page 9

GEETHANJALI COLLEGE OF ENGINEERING AND TECHNOLOGY

16ME4102 – OPERATIONS RESEARCH

Course Handout: IV Year I Sem Page 10

Course Handout: IV Year I Sem Page 11

CO-POs and CO-PSOs Mapping

Justifications for CO - PO Mapping

Course Handout: IV Year I Sem Page 12

Course Handout: IV Year I Sem Page 13

queuing and dynamics programming problems.

Justifications for CO - PSO Mapping

Course Handout: IV Year I Sem Page 14

GAPS in the Syllabus - to meet Industry/Professional Requirements:

Dual and Primal simplex problems and PO2, PO5, NPTEL +

Topics beyond Syllabus/Advanced Topics/Design: NIL

Proposed actions: topics beyond syllabus/assignment/industry visit/guest lecturer/NPTEL etc

Chalk & Talk  Student Assignment  Web Resources

LCD/Smart Board  Student Seminar ADD-ON Courses

Assessment Methodologies - Direct

Assignments  Mid-Term Exams  Semester End Exams 

Student lab practice Student Seminar Student Vive