Aris A. Syntetos, Service Parts Management - Demand Forecasting and Inventory Control-Springer-Verlag London (2011)
Aris A. Syntetos, Service Parts Management - Demand Forecasting and Inventory Control-Springer-Verlag London (2011)
Aris A. Syntetos, Service Parts Management - Demand Forecasting and Inventory Control-Springer-Verlag London (2011)
123
Editors
Assoc. Prof. Nezih Altay Assoc. Prof. Lewis A. Litteral
Department of Management Robins School of Business
DePaul University University of Richmond
1 E. Jackson Blvd. Westhampton Way 28
Depaul Center 7000 Richmond, VA 23173
Chicago, IL 60604 USA
USA e-mail: llittera@richmond.edu
e-mail: naltay@depaul.edu
DOI 10.1007/978-0-85729-039-7
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the
publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued
by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be
sent to the publishers.
The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of
a specific statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the
information contained in this book and cannot accept any legal responsibility or liability for any errors
or omissions that may be made.
There are 14 distinct contributions to this volume from authors who hail from more
than ten countries representing universities from six countries around the world.
Although their approaches to the management of spare parts are widely divergent,
everyone involved in the project agrees on two things: first, the management of
spare parts is important because of its prevailing nature and magnitude and second,
the problems associated with the management of spare parts are complex and
really very hard. The first point is the motivation for the volume and we think that
the second point is moderated somewhat by the talent, experience, and hard work
of the authors whose work is presented.
Many industries rely on the management of spare parts including military
weapon systems in naval and aircraft applications, commercial aviation, infor-
mation technology, telecom, automotive, and white goods such as fabrics and large
household appliances. Some of these applications involve providing service parts
to end users while others involve the maintenance of manufacturing facilities for
products like textiles and automobiles.
According to an article in the McKinsey Quarterly authored by Thomas Knecht,
Ralf Leszinski, and Felix A. Weber, the after-sales business accounts for 10–20%
of revenues and a much larger portion of total contribution margin in most
industrial companies.Equally important, during a given product’s life cycle, after-
sales can generate atleast three times the turnover of the original purchase,
especially for industrial equipment. A well-run after-sales business can also pro-
vide strategic benefits. Customers are usually less concerned about spare part
prices than about speed of delivery and availability of service know how, whether
on-site or via telephone. The reason is simple: down-time costs typically run at
anywhere from 100 to 10,000 times the price of spare parts or service. And that
means good performance can boost customer satisfaction, and thus, build repur-
chase loyalty in the original equipment business.
With specific regard to spare parts management policies used by the armed
services of the United States, the General Accounting Office reported in 1997 that
the inventory of service parts at non-major locations was valued at over $8.3 billion
and that the need for many of the items stored at non-major locations is
vii
viii Preface
This volume represents the work of many individuals. Our greatest debt is to Tricia
Fanney of the Robins School of Business at the University of Richmond who has
cheerfully and carefully shaped the manuscript into its current form. We are also
grateful to each of the authors, many of whom also served as reviewers. Jonathan
Whitaker and Steve Thompson of the Robins School served as reviewers and we
appreciate their service. Our thanks go to the editorial team at Springer for
encouraging and supporting the research in this area. Special thanks are due to
Claire Protherough who worked closely with Tricia in bringing this project to
publication.
Respectfully submitted:
Nezih Altay and Lewis A. Litteral
ix
Contents
xi
xii Contents
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
List of Contributors
xiii
xiv List of Contributors
1.1 Introduction
Intermittent demand patterns are very difficult to forecast and they are, most
commonly, associated with spare parts’ requirements. Croston (1972) proved the
inappropriateness of single exponential smoothing (SES) in an intermittent
demand context and he proposed a method that relies upon separate forecasts of
the inter-demand intervals and demand sizes, when demand occurs. His method for
forecasting intermittent demand series is increasing in popularity. The method is
incorporated in statistical forecasting software packages (e.g. Forecast Pro), and
demand planning modules of component based enterprise and manufacturing
solutions (e.g. Industrial and Financial Systems-IFS AB). It is also included in
integrated real-time sales and operations planning processes (e.g. SAP Advanced
Planning & Optimisation-APO 4.0).
An earlier paper (Syntetos and Boylan 2001) showed that there is scope for
improving the accuracy of Croston’s method. Since then two bias-corrected
Croston procedures have been proposed in the academic literature that aim at
advancing the practice of intermittent demand forecasting. These are the Syntetos–
Boylan Approximation (SBA, Syntetos and Boylan 2005) and the Syntetos’
method (SY, Syntetos 2001; Teunter and Sani 2009).1
A. A. Syntetos (&)
University of Salford, Salford, UK
e-mail: A.Syntetos@salford.ac.uk
J. E. Boylan
Buckinghamshire New University, Buckinghamshire UK
e-mail: John.Boylan@bucks.ac.uk
1
At this point it is important to note that one more modified Croston procedure has appeared in
the literature (Leven and Segerstedt 2004). However, this method was found to be even more
biased than the original Croston’s method (Boylan and Syntetos 2007; Teunter and Sani 2009)
and as such it is not further discussed in this chapter.
In this paper, these estimators as well as Croston’s method and SES are presented
and analysed in terms of the following statistical properties: (i) their bias (or the lack
of it); and (ii) the variance of the related estimates (i.e. the sampling error of the
mean). Detailed derivations are offered along with a thorough discussion of the
underlying assumptions and their plausibility. As such, we hope that our contribu-
tion may constitute a point of reference for further analytical work in this area as well
as facilitate a better understanding of issues related to modelling intermittent
demands.
Parametric approaches to intermittent demand forecasting rely upon a lead-time
demand distributional assumption and the employment of an appropriate forecasting
procedure for estimating the moments of the distribution. However, a number of
non-parametric procedures have also been suggested in the literature to forecast
intermittent demand requirements (e.g. Willemain et al. 2004; Porras and Dekker
2008). Such approaches typically rely upon bootstrapping procedures that permit a
re-construction of the empirical distribution of the data, thus making distributional
assumptions redundant. In addition, a number of parametric bootstrapping
approaches have been put forward (e.g. Snyder 2002; Teunter and Duncan 2009).
These approaches also rely upon bootstrapping but in conjunction with some
assumptions about the underlying demand characteristics. Although it has been
claimed that all the approaches discussed above may have an advantage over pure
parametric methodologies, more empirical research is needed to evaluate the con-
ditions under which one approach outperforms the other. In this chapter, we will be
focusing solely on parametric forecasting. In particular, we will be discussing issues
related to the estimation procedures that may be used. The issue of statistical dis-
tributions for parametric forecasting of intermittent demands is addressed in Chap. 2
of this book. A discussion on non-parametric alternatives may be found in Chap. 6.
The remainder of this chapter is structured around two main sections: in the
next section we discuss issues related to the bias of intermittent demand estimates,
followed by a discussion on the issue of variance. Some concluding remarks are
offered in the last section of the chapter and all the detailed derivations are pre-
sented in the Appendices.
l; t ¼ np þ 1
Yt ¼ ð1Þ
0; otherwise
Yt0 ¼ Yt1
0 0
þ aet ¼ aYt þ ð1 aÞYt1 ð2Þ
Croston, assuming the above stochastic model of arrival and size of demand,
introduced a new method for characterising the demand per period by modelling
demand from constituent elements. According to his method, separate exponential
smoothing estimates of the average size of the demand and the average interval
4 A. A. Syntetos and J. E. Boylan
between demand incidences are made after demand occurs. If no demand occurs,
the estimates remain the same. If we let:
p0t = the exponentially smoothed inter-demand interval, updated only if
demand occurs in period t so that Eðp0t Þ ¼ Eðpt Þ ¼ p, and
Z0t = the exponentially smoothed size of demand, updated only if demand
occurs in period t so that E(Z0t ) = E(Zt) = z
=
then following Croston’s estimation procedure, the forecast, Yt for the next
time period is given by:
=
= Zt
Yt ¼ =
ð6Þ
pt
and, according to Croston, the expected estimate of demand per period in that case
would be:
0
0 Zt E Zt0 l
E Yt ¼ E 0 ¼ 0 ¼ ð7Þ
pt E pt p
where
Rt is the replenishment level to which the stock is raised,
mt is the estimated mean absolute deviation of the demand size forecast errors
and
K is a safety factor.
Schultz (1987) proposed a slight modification to Croston’s method, suggesting
that a different smoothing constant value should be used in order to update the
inter-demand interval and the size of demand, when demand occurs. However, this
modification to Croston’s method has not been widely adopted (an exception may
1 Intermittent Demand: Estimation and Statistical Properties 5
be the study conducted by Syntetos et al. 2009) and it is not discussed further in
this chapter.
Croston advocated separating the demand into two components, the inter-demand
time and the size of demand, and analysing each component separately. He
assumed a stationary mean model for representing the underlying demand pattern,
normal distribution for the size of demand and a Bernoulli demand generation
process, resulting in geometrically distributed inter-demand intervals.
Three more assumptions implicitly made by Croston in developing his model
are the following: independence between demand sizes and inter-demand inter-
vals, independence of successive demand sizes and independence of successive
inter-demand intervals. As far as the last assumption is concerned it is important to
note that the geometric distribution is characterised by a ‘memory less’ process:
the probability of a demand occurring is independent of the time since the last
demand occurrence, so that this distributional assumption is consistent indepen-
dent inter-demand intervals.
The normality assumption is the least restrictive one for the analysis conducted by
Croston, since the demand sizes may be, theoretically, represented by any probability
distribution without affecting the mathematical properties of the demand estimates.
The remaining assumptions are retained for the analysis to be conducted in this
chapter. These assumptions have been challenged in respect of their realism (see, for
example, Willemain et al. 1994) and they have also been challenged in respect of
their theoretical consistency with Croston’s forecasting method. Snyder (2002)
pointed out that Croston’s model assumes stationarity of demand intervals and yet a
single exponential smoothing (SES) estimator is used, implying a non-stationary
demand process. The same comment applies to demand sizes. Snyder commented
that this renders the model and method inconsistent and he proposed some alternative
models, and suggested a new forecasting approach based on parametric bootstrap-
ping (see also Sect. 1.1). Shenstone and Hyndman (2005) developed this work by
examining Snyder’s models. In their paper they commented on the wide prediction
intervals that arise for non-stationary models and recommended that stationary
models should be reconsidered. However, they concluded, ‘‘…the possible models
underlying Croston’s and related methods must be non-stationary and defined on a
continuous sample space. For Croston’s original method, the sample space for the
underlying model included negative values. This is inconsistent with reality that
demand is always non-negative’’ (Shenstone and Hyndman, op. cit., pp. 389–390).
In summary, any potential non-stationary model assumed to be underlying
Croston’s method must have properties that do not match the demand data being
modelled. Obviously, this does not mean that Croston’s method and its variants,
to be subsequently discussed in this section, are not useful. Such methods do
constitute the current state of the art in intermittent demand parametric
6 A. A. Syntetos and J. E. Boylan
then the demand in any period is the sum of the orders in that period and both the
individual orders and the number of them in a given period are stochastic
variables:
X
N
W¼ Si ð10Þ
i¼1
Under the assumption that the order arrival process can be modelled as a
Poisson stream and combining Clark’s calculated mean and variance of the dis-
tribution of the summation of a number of stochastic random variables (1957):
W 1 ¼ N 1 S1 ð11Þ
W2 ¼ N1 S2 þ N2 ðS1 Þ2 ð12Þ
Using Cox’s asymptotic equations (1962) for relating the number of orders (N)
to the more easily measurable inter-demand interval ðIÞ counting from a random
point in time rather than an initial event (i.e. demand occurrence):
1
N1 ¼ ð13Þ
I1
I2 1 ðI2 Þ2 I3
N2 þ þ ð14Þ
ðI1 Þ3 6 ðI1 Þ4 3ðI1 Þ3
where I3 is the third moment about the mean for the inter-order interval.
1 Intermittent Demand: Estimation and Statistical Properties 7
S2 ðS1 Þ2
W2 ¼ þ ð16Þ
I1 I1
Thus, the forecasts can be generated from estimates of the mean and variance of
the order size and the average inter-demand interval.
The SI method was compared with SES on theoretically generated demand data
over a wide range of possible conditions. Many different average inter-demand
intervals (negative exponential distribution), smoothing constant values, lead times
and distributions of the size of demand (negative exponential, Erlang and rect-
angular), were considered. The comparison exercise was extended to cover not
only Poisson but also Erlang demand processes. The results were reported in the
form of the ratio of the mean squared error (MSE) of one method to that of
another. For the different factor combinations tried in this simulation experiment
the SI method was superior to SES for inter-demand intervals greater than 1.25
review periods and in that way the authors showed how intermittent demand needs
to be in order to benefit from the SI method (based on Croston’s concept) more
than SES.
At this stage it is important to note that the estimate of mean demand is
identical between Croston’s method and the SI method. Thus, later comments on
=
Zt
bias of the = (or SI11 ) estimator hold for both methods.
pt
We know (assuming that order sizes and intervals are independent) that
0
Z 1
E 0t ¼ E Zt0 E 0 ð17Þ
pt pt
but
1 1
E 0 6¼ 0 ð18Þ
pt E pt
We denote by Pt the inter demand interval that follows the geometric distri-
bution including the first success (i.e. demand occurring period) and by p1t the
probability of demand occurrence at period t. Now the case of a = 1 is analysed
since it is mathematically tractable; more realistic a values will be considered in
the next sub-section. Assuming that a = 1, so that p0t = pt we then have:
8 A. A. Syntetos and J. E. Boylan
X 1 x1
1 11 1
E ¼ 1
pt x¼1
xp p
1X 1
1 p 1 x1
¼
p x¼1 x p
½for p [ 1ði:e: demand does not occur in every single time periodÞ
x
p1
1X 1
1 p 1 1 X 1
1 p1 x
¼
1 ¼ p1
p x¼1 x p1 p p x¼1 x p
p
" #
1 p1 1 p1 2 1 p1 3
¼ þ þ þ
p1 p 2 p 3 p
1 1
¼ log
p1 p
Therefore:
0
Zt 0 1 1 1
E 0 ¼ E Zt E 0 ¼ l log ð19Þ
pt pt p1 p
So if, for example, the average size of demand when it occurs is l = 6, and the
average inter-demand interval is p = 3, the average estimated demand per time
period using Croston’s method (for a = 1) is not lp ¼ 63 ¼ 2 but it is
6 * 0.549 = 3.295 (i.e. 64.75% bias implicitly incorporated in Croston’s estimate).
The maximum bias over all possible smoothing parameters is given by:
1 1 l
Maximum bias ¼ l log ð20Þ
p1 p p
This is attained at a = 1. For realistic a values, the magnitude of the error is
smaller and it is quantified in the next sub-section.
For a values less than 1 the magnitude of the error obviously depends on the
smoothing constant value being used. We show, in this sub-section, that the bias
associated with Croston’s method in practice can be approximated, for all
a
smoothing constant values, by: 2a l ðp1
p2
Þ
ð22Þ
og 1
¼ ð23Þ
oh1 h2
og h1
¼ 2 ð24Þ
oh2 h2
o2 g
¼0 ð25Þ
oh21
o2 g 1
¼ 2 ð26Þ
oh1 oh2 h2
!
o2 g 2 2h1
2
¼ h1 3 ¼ ð27Þ
oh2 h2 h32
Therefore:
x1 h1 1 o2 g
E ¼ þ Var ðx2 Þ þ ð28Þ
x2 h2 2 oh22
Assuming that the inter-demand interval series is not auto-correlated and that
the inter-demand intervals ( pt) are geometrically distributed with a mean of p and
homogeneous variance2 of p( p - 1), it follows that:
a a
Varðx2 Þ ¼ Var p0t ¼ Varðpt Þ ¼ pðp 1Þ
2a 2a
Assuming that demand sizes (zt) are distributed with a mean, l, Eq. (28)
becomes:
x1 h1 1 a 2h1
E þ pðp 1Þ ð29Þ
x2 h2 2 2 a h32
0
z l a ð p 1Þ
E t0 þ l ð30Þ
pt p 2a p2
Subsequently, the bias implicitly incorporated in Croston’s estimates is
approximated by (31):
a ð p 1Þ
BiasCroston l ð31Þ
2a p2
Syntetos (2001) showed by means of experimentation on a wide range of
theoretically generated data that, for a B 0.2, the difference between the theo-
retical bias given by (31) and the simulated bias lies within a 99% confidence
interval of ±0.2% of the mean simulated demand.
2
The issue of variance in the geometric distribution is discussed in the next section.
1 Intermittent Demand: Estimation and Statistical Properties 11
0
z l a ð p 1Þ
E k t0 k þ k l
pt p 2a p2
We can then set an approximation to the bias equal to zero in order to specify
the value of parameter k:
l a ð p 1Þ
Bias ð1 kÞ k l ¼0
p 2a p2
a
k 2a l ðp1
p2
Þ
1k¼ l
p
a p1
1¼k 1þ ð33Þ
2a p
1 1 ð2 aÞp
k¼ a p1
¼ 2papþapa ¼
1þ 2a p ð2aÞp
2p a
1 a2
k¼ a
1 2p
We call this method, for the purpose of our research, the SY method (after
Syntetos 2001; for a further discussion see also Teunter and Sani 2009). The
expected estimate of mean demand per period for the SY method is given by
Eq. (35).
0 1 a2 z0t l
E Yt ¼ E ð35Þ
p0t a2 p
This approximation is not necessarily accurate when higher order terms are
taken into account.
a
But, as p ! 1; k ! 1 .
2
Therefore a possible estimation procedure, for intermittent demand data series
with a large inter-demand interval, is the following:
a
z t
=
Yt0 ¼ 1 ð36Þ
2 p=t
As in the case of the SY method, the smoothing constant value is considered for
generating demand estimates. The above heuristic seems to provide a reasonable
approximation of the actual demand per period especially for the cases of very low
a values and large p inter-demand intervals. This estimator is known in the lit-
erature as the SBA method (after Syntetos–Boylan Approximation, Syntetos and
Boylan 2005). The expected estimate of mean demand per period for the SBA
method is given by Eq. (37).
0 a
z0t l al
E Yt ¼ E 1 2 ð37Þ
2 p0t p 2p
This approximation is not necessarily accurate when higher order terms
are taken into account. For the detailed derivation of (37) see Appendix A.
The empirical validity and utility of the SBA have been independently established
in work conducted by Eaves and Kingsman (2004) and Gutierrez et al. (2008).
Before we close this section, we need to say that all the work presented above is
based upon the assumption of a Bernoulli demand arrival process and SES esti-
mates of sizes and intervals. Boylan and Syntetos (2003) and Shale et al. (2006)
presented correction factors to overcome the bias associated with Croston’s
approach under a Poisson demand arrival process and/or estimation of demand
sizes and intervals using a simple moving average (SMA). The correction factors
are summarized in the following table (where k is the moving average length and a
is the smoothing constant for SES).
At this point it is important to note that SMA and SES are often treated as
equivalent when the average age of the data in the estimates is the same
(Brown 1963). A relationship links the number of points in an arithmetic
average (k) with the smoothing parameter of SES (a) for stationary demand.
Hence it may be used to relate the correction factors presented in Table 1.1
for each of the two demand generation processes considered. The linking
equation is:
k ¼ ð2 aÞ=a
1 Intermittent Demand: Estimation and Statistical Properties 13
Equation (36)
According to Croston (1972), and under the stochastic demand model he assumed
for his study, the variance of demand per unit time period is:
p 1 2 r2
VarðYt Þ ¼ l þ ð38Þ
p2 p
and the variance of the exponentially smoothed estimates, updated every
period:
0 a a p 1 2 r2
Var Yt ¼ VarðYt Þ ¼ l þ ð39Þ
2a 2 a p2 p
where b = 1 - a.
14 A. A. Syntetos and J. E. Boylan
For x1 = Zt0 and x2 = p0t , considering Eqs. (42), (43), (44) and (45), the variance
of the estimates produced by using Croston’s method is calculated by Eq. (47)
0
0 Zt a ðp 1Þ 2 r2
Var Yt ¼ Var 0 ¼ l þ 2 ð47Þ
pt 2a p4 p
assuming that the same smoothing constant value is used for updating demand
sizes and inter-demand intervals and that both demand size and inter-demand
interval series are not auto-correlated and have homogeneous variances.
1 Intermittent Demand: Estimation and Statistical Properties 15
Rao (1973) pointed out that the right-hand side of Eq. (47) is only an
approximation to the variance. This follows since Eq. (46) is, in fact, an
approximation.
By taking (48) into consideration, the variance of the demand per period estimates,
using Croston’s method, would become:
0
Z a p 1 2 r2
Var t0 l þ ð49Þ
pt 2 a p3 p2
3
Equation (10) in the original paper.
16 A. A. Syntetos and J. E. Boylan
h1
where gðhÞ ¼ is just the first term in the Taylor series and not necessarily the
h2
population expected value.
For:
E½gðxÞ ¼ gðhÞ þ e ð51Þ
If we set:
x1 = Zt0 , the estimate of demand size, with E(Zt0 ) = l
and x2 = p0t , the estimate of the inter-demand interval, with E( p0t ) = p
so that g(x) = Yt0
it has been proven, in the previous section, that:
l
E Yt0 6¼ or E½gð xÞ 6¼ gðhÞ
p
Based on that, we argue that the error term in Eq. (51) cannot be neglected
and therefore approximation (52) cannot be used to represent the problem in
hand.
Our argument is discussed in greater detail in Appendices 2 and 3, where we
also derive a correct approximation (to the second order term) of the variance of
Croston’s estimates. That variance expression is given by (53).
0
r2
Zt a ð p 1 Þ 2 a 2
Var 0 l þ r þ 2
pt 2a p3 2a p
a4 l2 1 1 2
þ 1 9 1 p þ 1 ð53Þ
1 ð1 aÞ4 p4 p p
0
r2
Zt a ðp 1Þ 2 a 2
Var l þ r þ 2 ð54Þ
p0t 2a p3 2a p
The estimation equation for the SY method presented in the previous section is
given by:
0 1 a2 Zt0 1 a2 Zt0
Yt ¼
¼
1 2pa 0 p0t p0t a2
t
and the expected estimate produced by this method was shown to be as follows:
0 1 a2 Zt0 l
E Yt ¼ E
p0t a2 p
In Appendix D we perform a series of calculations in order to find the variance
of the estimates of mean demand produced by the SY method. The variance is
approximated by Eq. (55).
h i
a
0
p a 2 2
r þ pð p 1 Þl 2
þ a
p ð p 1 Þr 2
1 2 Zt að2 aÞ 2 2a
Var
p0t a2 4 p2 a 4
a 2 2
4 1 l
a 2 2 1 1 2
þ p 1 9 1 p þ 1 ð55Þ
1 ð1 aÞ4 p a2 6 p p
The estimation procedure for the SBA method discussed in the previous
section is:
a
Zt0
Yt0 ¼ 1
2 p0t
with
0 a
z0t l al
E Yt ¼ E 1 2
2 p0t p 2p
The variance of the estimates produced by the SBA is calculated as:
0
a
z0t a
2 zt
Var Yt0 ¼ Var 1 ¼ 1 Var ð57Þ
2 p0t 2 p0t
Considering approximation (54) we finally have:
r2
a
Zt0 að2 aÞ ðp 1Þ 2 a 2
Var 1 l þ r þ 2 ð58Þ
2 p0t 4 p3 2a p
Extensive analysis conducted by Syntetos (2001) justified the choice of
(58) instead of (57) for the purpose of approximating the variance of the SBA
method.
1.5 Conclusions
Research in the area of forecasting and stock control for intermittent demand items
has developed rapidly in recent years with new results implemented into software
products because of their practical importance (Fildes et al. 2008). Simple expo-
nential smoothing (SES) is widely used in industry to deal with sales/demand data
but is inappropriate in an intermittent demand context. Rather, Croston’s method is
regarded as the standard estimator for such demand patterns. Recent studies have
shown that there is scope for improving Croston’s estimates by means of
accounting for the bias implicitly incorporated in them. Two such methods have
been discussed in this chapter and their statistical properties have been analysed in
detail along with those of both Croston’s method and SES. We hope that our
contribution may constitute a point of reference for further analytical work in this
area as well as facilitate a better understanding of issues related to modelling
intermittent demands.
1 Intermittent Demand: Estimation and Statistical Properties 19
0 a
z0t
E Yt ¼ E 1
2 p0t
a l 2a a p1
1 þ l
2 p 2 2 a p2
a l 2a a 1 1
¼ 1 þ l 2
2 p 2 2 a p p
a
l a l a l
¼ 1 þ
2 p 2 p 2 p2
l al al a l
¼ þ
p 2 p 2 p 2 p2
l al
¼ 2
p 2p
This proves the result given by Eq. (37).
og h1
¼ 2 ðB:3Þ
oh2 h2
o2 g
¼0 ðB:4Þ
oh21
o2 g 1
¼ 2 ðB:5Þ
oh1 oh2 h2
!
o2 g 2 2h1
¼ h1 3 ¼ ðB:6Þ
oh22 h2 h32
We set:
x1 = Zt0 , the estimate of demand size, with E(Zt0 ) = l
and x2 = p0t , the estimate of the inter-demand interval, with E( p0t ) = p
so that g(x) = Yt0 ,
It has been proven, in Sect. 1.2, that:
1 o2 g
E Yt0 ¼ E½gðxÞ gðhÞ þ Eðx2 h2 Þ2
2 oh22
considering the first three terms in the Taylor series.
Therefore:
Var Yt0 ¼Var½gð xÞ¼E½gð xÞE½gð xÞ2
8 2 2 92
og
<oh
1
ðx1 h1 Þþ ohog
2
ðx2 h2 Þþ oho1 oh
g
2
ðx1 h1 Þðx2 h2 Þþ 12 oohg2 ðx2 h2 Þ2 =
2
E
:1o2 g Eðx h Þ2 ;
2oh22 2 2
( 2 2 2 2
og 2 og 2 o g
¼E ðx1 h1 Þ þ ðx2 h2 Þ þ ðx1 h1 Þ2 ðx2 h2 Þ2
oh1 oh2 oh1 oh2
!2 2 2 h
1 o2 g i2
4 1 o g 2 og og
þ 2
ð x2 h 2 Þ þ E ð x 2 h2 Þ þ2 ðx1 h1 Þ ðx2 h2 Þ
4 oh2 4 oh2 oh1 oh2
1 Intermittent Demand: Estimation and Statistical Properties 21
og o2 g og o2 g
þ2 ðx1 h1 Þ ðx1 h1 Þðx2 h2 Þþ ðx1 h1 Þ 2 ðx2 h2 Þ2
oh1 oh1 oh2 oh1 oh2
og o2 g og o 2
g
ðx1 h1 Þ 2 Eðx2 h2 Þ2 þ2 ðx2 h2 Þ ðx1 h1 Þðx2 h2 Þ
oh1 oh2 oh2 oh1 oh2
og o2 g 3 og o g
2
þ ð x2 h 2 Þ ðx2 h2 ÞEðx2 h2 Þ2
oh2 oh22 oh2 oh22
o2 g o2 g
þ ðx1 h1 Þðx2 h2 Þ 2 ðx2 h2 Þ2
oh1 oh2 oh2
o2 g o2 g
ðx1 h1 Þðx2 h2 Þ 2 Eðx2 h2 Þ2
oh1 oh2 oh2
!2 )
2
1 og 2 2
ðx2 h2 Þ Eðx2 h2 Þ
2 oh22
ðB:8Þ
where:
xt represents the demand size (zt) or inter-demand interval (pt),
x0t is their exponentially smoothed estimate (z0t , p0t ) and
E(x) is the population expected value for either series.
Consideration of (B.11) and (B.12) necessitates the adoption of the following
assumptions:
no auto-correlation for the demand size and inter-demand interval series
homogeneous moments about the mean for both series
same smoothing constant value is used for both series
Taking also into account that:
Var(zt) = r2 and Var(pt) = p(p - 1)
(B.10) becomes:
0 a
2 r2 pðp 1Þ
z a r2 a 2 pðp 1Þ
Var t0 þ l þ
pt 2 a p2 2 a p4 2a p4
3 2 4
a 2l 3 a l2
3 p5
E ð pt p Þ þ 4 p6
Eðpt pÞ4
1 ð1 aÞ 1 ð1 aÞ
a4 l2 2 a
2 l2
2
þ
2 6 p ðp 1Þ p2 ðp 1Þ2 ðB:13Þ
2 p 2 a p6
1 ð1 aÞ
1
The third moment about the mean in the geometric distribution, where: is the
p
probability of success in each trial, is calculated as:
1 1 1 p1 1 1 p 1 1 2p 1
E ð p t pÞ 3 ¼ 1 1þ1 3
¼ 3
2 ¼
p p p p p p p p3 p
ðp 1Þð2p 1Þ
¼ ðB:14Þ
p5
and the fourth moment:
2 2
3
9 1 1p 9 11
4
1 1p 1 4 p 15
Eðpt pÞ ¼ 1
þ 1 ¼ 1 1
þ1
p4 p2
p p4 p2
1 1
¼ 1 9 1 p4 þ p2
p p
2 1 1 2
¼p 1 9 1 p þ1 ðB:15Þ
p p
1 Intermittent Demand: Estimation and Statistical Properties 23
a4 a
2
2 ¼ ;
2a
1 ð1 aÞ2
(B.13) becomes
0 a
2 r2 pðp 1Þ
z a r2 a 2 pðp 1Þ
Var t0 þ l þ
pt 2 a p2 2 a p4 2a p4
a3 2l2 ðp 1Þð2p 1Þ
1 ð1 aÞ3 p10
a4 l2 1 1 2
þ 1 9 1 p þ 1 ðB:16Þ
1 ð1 aÞ4 p4 p p
Since the fourth part of approximation (B.16) becomes almost zero even for
quite low average inter-demand intervals, finally the variance is approximated by
(B.17):
0
r2
z a ðp 1Þ 2 a 2
Var t0 l þ r þ 2
pt 2a p3 2a p
a4 l2 1 1 2
þ 1 9 1 p þ 1 ðB:17Þ
1 ð1 aÞ4 p4 p p
Appendix C: The 2nd, 3rd and 4th Moment About the Mean
for Exponentially Smoothed Estimates
If we define:
X
1
x0 ¼ að1 aÞxjtj ði:e: the EWMA estimateÞ ðC:1Þ
j¼0
X
1
Eðx0 Þ ¼ að1 aÞ j E xtj
j¼0
X
1
x 0 E ð xÞ ¼ a ð1 aÞ j xtj Eð xÞ
j¼0
( )n
n
X1
j
0
½ x Eð x Þ ¼ a ð1 aÞ xtj Eð xÞ
j¼0
and
( )n
n
X
1
j
0
E ½ x Eð x Þ ¼ E a ð1 aÞ xtj Eð xÞ ðC:3Þ
j¼0
n
X
1 n
E ½x0 Eð xÞ ¼ an ð1 aÞnj E xtj Eð xÞ ðC:4Þ
j¼0
and assuming E[xt-j - E(x)]n = E[x - E(x)]n for all j C 0, i.e. homogeneous
moments of order n
n an
E ½x0 Eð xÞ ¼ E ½x Eð xÞn ðC:5Þ
1 ð1 aÞn
For n = 2
a2
Varðx0 Þ ¼ Varð xÞ
1 ð1 aÞ2
and for n = 3
3 a3
E ½x0 Eð xÞ ¼ E ½x Eð xÞ3
1 ð1 aÞ3
For n = 4, Eq. (C.3) becomes:
( )4
4
X
1
j
0
E ½ x Eð x Þ ¼ E a ð1 aÞ xtj Eð xÞ
j¼0
assuming no auto-correlation
X
1 4
¼ a4 ð1 aÞ4j E xtj Eð xÞ
j¼0
X
1 X
1 2
þ a2 ð1 aÞ2j E xtj Eð xÞ a2 ð1 aÞ2i E ½xti Eð xÞ2
j¼0 i¼0
1 Intermittent Demand: Estimation and Statistical Properties 25
4 X
1 X
1 X
1
¼ a4 E xtj Eð xÞ ð1 aÞ4j þa4 ½Varð xÞ2 ð1 aÞ2j ð1 aÞ2i
j¼0 j¼0 i¼0
ðC:6Þ
X
1
1
ð1 aÞ4j ¼ ðC:7Þ
j¼0 1 ð1 aÞ4
X
1 X
1
ð1 aÞ2j ð1 aÞ2i ¼ 1 þ ð1 aÞ2 þ ð1 aÞ4 þ ð1 aÞ6 þ
j¼0 i¼0
ðC:9Þ
If we multiply the first and the second part of Eq. (C.9) with (1 - a)2, we then
have:
X
1 X
1
ð1 aÞ2 ð1 aÞ2j ð1 aÞ2i ¼ ð1 aÞ2 þ2 ð1 aÞ4 þ3 ð1 aÞ6 þ
j¼0 i¼0
ðC:10Þ
Subtracting (C.9)-(C.10)
h iX
1 X
1
1 ð1 aÞ2 ð1 aÞ2j ð1 aÞ2i
j¼0 i¼0
1
¼ 1 þ ð1 aÞ2 þ ð1 aÞ4 þ ð1 aÞ6 þ ¼
1 ð1 aÞ2
Therefore:
X
1 X
1
1
ð1 aÞ2j ð1 aÞ2i ¼
2 ðC:11Þ
j¼0 i¼0 1 ð1 aÞ2
4 a4 a4
E ½ x 0 Eð x Þ ¼ 4
E ½x Eð xÞ4 þ
2 ½Varð xÞ
2
ðC:12Þ
1 ð1 aÞ 1 ð1 aÞ 2
x1 ¼ 1 z0t
2
with expected value:
h a
i a
a
a a
4 4
E ðx2 h2 Þ4 ¼ E p0t p þ ¼ E p0t p
2 2
a4 a4
¼ 4
E ðpt pÞ4 þ
2 ½Varðpt Þ
2
1 ð1 aÞ 1 ð1 a Þ2
a4 1 1 2 a
2 2
¼ p 1 2
9 1 p þ 1 þ p ðp 1Þ2
1 ð1 aÞ4 p p 2a
ðD:6Þ
(assuming that the same smoothing constant value is used for both x1 and x2 series
and that both series are not auto-correlated and have homogeneous moments about
the mean).
Consequently we apply Taylor’s theorem to a function of two variables,
gð xÞ ¼ xx12 with:
og 1
¼ ðD:7Þ
oh1 h2
og h1
¼ 2 ðD:8Þ
oh2 h2
o2 g
¼0 ðD:9Þ
o h1 2
o2 g 1
¼ 2 ðD:10Þ
oh1 oh2 h2
!
o2 g 2 2h1
2
¼ h1 3 ¼ ðD:11Þ
oh2 h2 h32
with:
x1 1 a2 z0t l
E½gð xÞ ¼ E ¼E ðD:13Þ
x2 p0t a2 p
h1 1 a2 l l
gð hÞ ¼ ¼ 6¼
h2 p a2 p
28 A. A. Syntetos and J. E. Boylan
1 o2 g
E½ g ð x Þ gð hÞ þ E ðx2 h2 Þ2
2 oh22
Since the fourth part of approximation (D.16) becomes almost zero even for
quite low average inter-demand intervals, and the last two terms cancel each other,
finally the variance is approximated by (D.17):
! h i
a 0
p a 2 2
r þp ð p 1 Þl 2
þ a
pð p 1 Þr 2
1 2 zt a a 2 2 2a
Var 1
p0t a2 2a 2 p2 a 4
a
2 2
a4 12 l 2 1 1 2
þ p 1 9 1 p þ 1
1 ð1 aÞ4 p a 6 2
p p
ðD:17Þ
This proves the result given by Eq. (55).
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Chapter 2
Distributional Assumptions
for Parametric Forecasting
of Intermittent Demand
2.1 Introduction
further discussed in Sect. 2.5 of the chapter. Intermittent demand items may be
engineering spares (e.g. Mitchell 1962; Hollier 1980; Strijbosch et al. 2000), spare
parts kept at the wholesaling/retailing level (e.g. Sani 1995), or any SKU within
the range of products offered by all organisations at any level of the supply chain
(e.g. Croston 1972; Willemain et al. 1994). Such items may collectively account
for up to 60% of the total stock value (Johnston et al. 2003) and are particularly
prevalent in the aerospace, automotive and IT sectors. They are often the items at
greatest risk of obsolescence.
Research in the area of forecasting and stock control for intermittent demand
items has developed rapidly in recent years with new results implemented into
software products because of their practical importance (Fildes et al. 2008). Key
issues remaining in this area relate to (i) the further development of robust
operational definitions of intermittent demand for forecasting and stock control
purposes and (ii) a better modelling of the underlying demand characteristics for
the purpose of proposing more powerful estimators useful in stock control. Both
issues link directly to the hypothesised distribution used for representing the rel-
evant demand patterns. Surprisingly though, not much has been contributed in this
area in the academic literature.
Classification for forecasting and stock control entails decisions with respect to
an appropriate estimation procedure, an appropriate stock control policy and an
appropriate demand distributional assumption. The subtle linkages between op-
erationalized SKU classification procedures and distributional assumptions have
not been adequately explored. In addition, the compound nature of intermittent
demand necessitates, conceptually at least, the employment of compound distri-
butions, such as the negative binomial distribution (NBD). Although this area has
attracted some academic attention (please refer also to the second section of this
chapter) there is still more empirical evidence needed on the goodness-of-fit of
these distributions to real data.
The objective of this work is three-fold: first, we conduct an empirical inves-
tigation that enables the analysis of the goodness-of-fit of various continuous and
discrete, compound and non-compound, two-parameter statistical distributions
used in the literature in the context of intermittent demand; second, we critically
link the results to theoretical expectations and the issue of classification for
forecasting and stock control; third, we provide an agenda for further research in
this area. We use three empirical datasets for the purposes of our analysis that
collectively constitute the individual demand histories of approximately 13,000
SKUs. Two datasets come from the military sector (Royal Air Force, RAF UK and
US Defense Logistics Agency, DLA) and one from the Electronics industry. In all
cases the SKUs are spare/service parts.
At this point it is important to note that some non-parametric procedures have
also been suggested in the literature to forecast intermittent demand requirements
(e.g. Willemain et al. 2004; Porras and Dekker 2008). Such approaches typically
rely upon bootstrapping procedures that permit a re-construction of the empirical
distribution of the data, thus making distributional assumptions redundant.
Although it has been claimed that such approaches have an advantage over
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 33
Intermittent demand
In this section, we first describe the datasets used for the purposes of this empirical
investigation, followed by a discussion of the statistical goodness-of-fit tests
conducted and the empirical results.
The empirical databases available for the purposes of our research come from the
US Defense Logistics Agency (DLA), Royal Air Force (RAF) and Electronics
Industry and they consist of the individual monthly demand histories of 4,588,
5,000 and 3,055 SKUs, respectively. Some information regarding these datasets is
presented in Table 2.1, followed by detailed descriptive statistics on the demand
data series characteristics for each of the datasets presented in Tables 2.2, 2.3, and
2.4. At this point it should be noted that the time series considered have not been
tested for stationarity.
Two tests have been mainly used and discussed in the literature for checking
statistically significant fit, namely: the Chi-Square test and the Kolmogorov–
Smirnov (K–S) test (see, for example, Harnett and Soni 1991). These tests measure
the degree of fit between observed and expected frequencies. Problems often arise
with the standard Chi-Square test through the requirement that data needs to be
grouped together in categories to ensure that each category has an expected
frequency of at least a minimum of a certain number of observations. Some
modifications of this test have also been considered in the literature. A modified
Chi-Square test has been developed for the purpose of testing the goodness-of-fit
for intermittent demands (Eaves 2002). This test differs in that boundaries are
specified by forming a certain number of categories with similar expected fre-
quencies throughout, rather than combining groups just at the margins. However,
the implementation of this test requires the specification of the number of cate-
gories to be used. We encountered a difficulty in using the standard or modified
Chi-Square test in our research, namely that of deciding how to specify the cat-
egories’ intervals or the number of categories. On the other hand, the K–S test does
not require grouping of the data in any way, so no information is lost; this elim-
inates the troublesome problem of categories’ intervals specification.
In an inventory context one could argue that measures based on the entire
distribution can be misleading (Boylan and Syntetos 2006). A good overall
goodness-of-fit statistic may relate to the chances of low demand values, which
can mask poor forecasts of the chances of high-demand values. However, for
inventory calculations, attention should be restricted to the upper end of the dis-
tribution (say the 90th or 95th percentiles). The development of modified good-
ness-of-fit tests for application in inventory control, and even more specifically in
an intermittent demand context, is a very important area but not one considered as
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 39
part of this research. Consequently, we have selected the K–S test for the purpose
of assessing goodness-of-fit.
The K–S test assumes that the data is continuous and the standard critical values
are exact only if this assumption holds. Several researchers (e.g. Noether 1963,
1967; Walsh 1963; Slakter 1965) have found that the standard K–S test is con-
servative when applied to data that is discrete. The standard exact critical values
provided for the continuous data are larger than the true exact critical values for
discrete data. Consequently, the test is less powerful if the data is discrete as in the
case of this research; it could result in accepting the null hypothesis at a given
significance level while the correct decision would have been to reject the null
hypothesis. Conover (1972) proposed a method for determining the exact critical
levels for discrete data.
As discussed in the previous section, we are considering five distributions the fit
of which is tested on the demand data related to 12,643 SKUs. The distribution of the
demand per period has been considered rather than the distribution of the lead-time
demand; this is due to the lack of information on the actual lead times associated with
the dataset 1. (Although this may be very restrictive regarding the performance of
the normal distribution, this would still be expected to perform well on the time
series that are associated with a small coefficient of variation of demand per period.)
Critical values have been computed based on K–S statistical tables for 1 and 5%
significance levels. We consider that:
• There is a ‘Strong Fit’ if the P-value is less than both critical values;
• There is ‘Good Fit’ if the P-value is less than the critical value for 1% but larger
than the one for 5%;
• There is ‘No Fit’ if the P-value is larger than both critical values.
In Table 2.5 we present the percentage of SKUs that satisfy the various degrees of
goodness-of-fit taken into account in our research, for each of the datasets and
statistical distributions considered.
As shown in Table 2.5, the discrete distributions, i.e. Poisson, NBD and stut-
tering Poisson provide, overall, a better fit than the continuous ones, i.e. Normal
and Gamma. More precisely, and with regards to ‘Strong Fit, the stuttering Poisson
distribution performs best in all three datasets considered in our research. This is
followed by the NBD and then by the Poisson distribution. On the other hand, the
normal distribution is judged to be far from appropriate for intermittent demand
items; this is partly due to the experimental structure employed for the purposes of
our investigation that relied upon the distribution of demand per time period rather
than the distribution of the lead time demand.
Contrary to our expectations, the gamma distribution has also been found to
perform poorly. This may be explained in terms of the inconsistency between the
distribution under concern, which is continuous in nature, and the discreteness of the
40 A. A. Syntetos et al.
(demand) data employed in our goodness-of-fit tests. We return to this issue in the
last section of the chapter where the next steps of our research are discussed in detail.
Johnston and Boylan (1996) offered for the first time an operationalised definition
of intermittent demand for forecasting purposes (demand patterns associated with
an average inter-demand interval ( p) greater than 1.25 forecast revision periods).
The contribution of their work lies on the identification of the average inter-
demand interval as a demand classification parameter rather than the specification
of an exact cut-off value. Syntetos et al. (2005) took this work forward by
developing a demand classification scheme that it relies upon both p and the
squared coefficient of variation of demand sizes (CV2), i.e. the contribution of
their work lies in the identification of an additional categorisation parameter for
demand forecasting purposes. Nevertheless, inventory control issues and demand
distributional assumptions were not addressed. Boylan et al. (2007) assessed the
stock control implications of the work discussed above by means of experimen-
tation on an inventory system developed by a UK-based software manufacturer.
The researchers demonstrated, empirically, the insensitivity of the p cut-off value,
for demand classification purposes, in the approximate range 1.18–1.86 periods.
In this section, we attempt to explore the potential linkages between demand
distributional assumptions and the classification scheme developed by Syntetos
et al. (2005). In the following figures we present for dataset #1 and each of the
distributions considered, the SKUs associated with a ‘Strong Fit’ as a function of
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 41
0
0 5 10 15 20
p
4
3
2
1
0
0 5 10 15 20
p
42 A. A. Syntetos et al.
CV^2
4
3
2
1
0
0 5 10 15 20
p
0,6
0,4
0,2
0
0 5 10 15 20
p
3
2
1
0
0 5 10 15 20
p
gamma distribution provides also a strong fit to the SKUs with very high values of
p (i.e. SKUs with an inter-demand interval going up to 12 periods in dataset #1 and
24 periods in datasets #2 and #3) and high CV2 values (i.e. SKUs with CV2 up to 6
in dataset #1, CV2 = 10 in the dataset #2 and CV2 = 8 in the dataset #3). This is
also expected since the gamma distribution is known to be very flexible in terms of
its mean and variance, so it can take high values for its p and CV2 and can be
reduced to the normal distribution for certain parameters of the mean and the
variance.
Based on the goodness-of-fit results presented in this section, we have
attempted to derive inductively an empirical rule that suggests which distribution
should be used under particular values of the inter-demand interval and squared
coefficient of variation of the demand sizes. That is to say, we have explored the
possibility of extending the classification scheme discussed by Syntetos et al.
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 43
2
CV
Gamma
NBD/SP
1 Poisson
Normal
0 p
1 2
Lengu and Syntetos (2009) proposed a demand classification scheme based on the
underlying demand characteristics of the SKUs (please refer to Fig. 2.8). SKUs are
first categorised as non-qualifying if the variance of the demand per period is less
than the mean or qualifying if the variance is at least equal to the mean. Compound
Poisson distributions can be used to model the demand series of qualifying SKUs
but they are regarded as not appropriate for modelling the demand of non-quali-
fying SKUs. Let us assume that demand is generated from a compound Poisson
44 A. A. Syntetos et al.
SKUs
Qualifying SKUs
Non-qualifying SKUs
model (i.e. demand ‘arrives’ according to a Poisson process and, when demand
occurs, it follows a specified distribution). If we let l denote the mean demand per
unit time period and r2 denote the variance of demand per unit period of time, then
l ¼ klz ð1Þ
r2 ¼ k l2z þ rz 2
ð2Þ
where k is the rate of demand occurrence, and lz and r2z the mean and variance,
respectively, of the transaction size when demand occurs. Note that
r2 kðl2z þ r2z Þ
¼ 1 ð3Þ
l klz
since l2z C lz (the transaction size is at least of 1 unit) and r2z is non-negative. The
compound Poisson demand model is therefore not appropriate for SKUs associated
with r2/l \ 1 (non-qualifying). Note that the actual rate of demand occurrence k
does not affect the classification of SKUs as to whether they are qualifying or not.
usually equal to one but can also take higher values. The Poisson-Geometric
compound distribution also accommodates the case of clumped demand since the
Poisson distribution is a special case of the Poisson-Geometric distribution. Spe-
cifically, if the parameter of the Geometric distribution Ge(pG) is 1, then the
transaction size can only take one value (transaction size 1). With the transaction
size being clumped, the demand model is now reduced to a standard Poisson
distribution. In the empirical goodness-of-fit tests, the Poisson-Geometric distri-
bution provided the most frequent fit of all the distributions considered (see
Table 2.5).
model to distinguish between the demand occurrence process and the transaction
size distribution. Such a model could however be useful for modeling slow-
moving non-qualifying SKUs and we will consider it in the next steps of our
research.
The normal distribution and the gamma distribution seem to be the least promising
of all the distributions considered in the empirical part of this chapter. For either
distribution, the variance can be less than, equal to or larger than the mean. The
two distributions can therefore be used to model both qualifying and non-quali-
fying SKUs. Furthermore, the normal distribution and the gamma distributions
have been studied extensively and tables of the critical values for both distribu-
tions are widely available. However, in the empirical study, the two distributions
provided the least frequent fit and there is no clear pattern associated with the
SKUs for which the distributions provided a good fit. The normal distribution and
the gamma distribution might be convenient to use but that should be contrasted to
their rather poor empirical performance.
As we have mentioned in Sect. 2.2, that the K–S test assumes that the data is
continuous and the test is less powerful if the data is discrete as in the case of this
research. The standard exact critical values provided for the continuous data are
larger than the true exact critical values for discrete data. Conover (1972) and
Pettitt and Stephens (1977) proposed a method for determining the exact critical
levels for the K–S test for discrete data. Choulakian et al. (1994) proposed a
method of calculating the critical values of the Cramér–von Mises test and the
Anderson–Darling test for discrete data. These tests have one significant drawback
because of their sensitivity: their critical values are very much dependent upon the
model being tested. Different tables of the critical values are therefore required for
each demand model being tested. Steele and Chaselling (2006) have compared the
power of these different goodness-of-fit tests for discrete data but their study was
not extensive enough to indicate which test is the most powerful for our purposes.
studies are required in order to develop our understanding on the adequacy of these
distributions under differing underlying intermittent demand structures; (ii) there is
some scope for linking demand distributional assumptions to classification for
forecasting and stock control purposes. Both these issues are explored as part of
the research work presented in this chapter. The empirical databases available for
the purposes of our investigation come from the US DLA, RAF and Electronics
Industry and they consist of the individual monthly demand histories of 4,588,
5,000 and 3,055 SKUs, respectively.
The empirical goodness-of-fit of five distributions (of demand per period) has
been assessed by means of employing the Kolmogorov–Smirnov (K–S) test. These
distributions are: Poisson, Negative Binomial Distribution (NBD), stuttering
Poisson, Normal and Gamma. The results indicate that both the NBD and stut-
tering Poisson provide the most frequent fit. Both these distributions are compound
in nature, meaning that they account explicitly for a demand arrival process
(Poisson) and a different distribution for the transaction sizes (Log series and
Geometric for the NBD and stuttering Poisson, respectively). Despite previous
claims, the gamma distribution does not perform very well and the same is true for
the normal distribution. This may be attributed to the continuous nature of these
distributions (since their fit is tested on discrete observations) but also to the fact
that we model demand per unit time period as opposed to lead time demand. Upon
reflection, this is viewed as a limitation of our work since lead time demand could
have been considered for two of the three datasets available to us (in those cases
the actual lead time was available). If that was the case, both the Normal and
gamma distribution would be associated potentially with a better performance. The
Poisson distribution provides a ‘reasonable’ fit and this is theoretically expected
for slow moving items.
Some recent work on the issue of demand classification (Syntetos et al.
2005) has focused on both the demand arrival pattern and distribution of the
demand sizes. In this chapter, we have attempted empirically to link the
goodness-of-fit of the above discussed distributions to the classification scheme
proposed by Syntetos et al. (2005). Although some of the results were matched
indeed by relevant theoretical expectations this was not the case when the
inventory implications of the proposed scheme were considered. Goodness-of-fit
tests focus on the entire demand distribution whereas stock control performance
is explicitly dependant upon the fit on the right-hand tail of a distribution. This
is an important issue in Inventory Management and one that has not received
adequate attention in the academic literature. The empirical results discussed
above have also been contrasted to some theoretical expectations offered by a
conceptual demand classification framework presented by Lengu and Syntetos
(2009). The framework links demand classification to some underlying char-
acteristics of intermittent demand patterns and although it seems capable of
explaining a number of empirical results it may not be utilized in an opera-
tionalised fashion yet.
The work presented in this chapter has revealed a number of interesting
themes for further research. Distributional assumptions play a critical role in
48 A. A. Syntetos et al.
Acknowledgements The research described in this chapter has been partly supported by the
Engineering and Physical Sciences Research Council (EPSRC, UK) grants no. EP/D062942/1
and EP/G006075/1. More information on the former project may be obtained at http://www.
business.salford.ac.uk/research/ommss/projects/Forecasting/. In addition, we acknowledge the
financial support received from the Royal Society, UK: 2007/Round 1 Inter. Incoming Short
Visits—North America.
Appendix
Goodness-of-Fit Results
2
1
0
0 5 10 15 20 25 30
p
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 49
CV^2
4
3
2
1
0
0 5 10 15 20 25 30
p
4
3
2
1
0
0 5 10 15 20 25 30
p
0,15
CV^2
0,10
0,05
0,00
0 5 10 15 20 25 30
p
6
4
2
0
0 5 10 15 20 25 30
p
50 A. A. Syntetos et al.
CV^2
2
1
1
0
0 5 10 15 20 25 30
p
3
2
1
0
0 5 10 15 20 25 30
p
3
2
1
0
0 5 10 15 20 25 30
p
0,8
0,6
0,4
0,2
0,0
0 5 10 15 20
p
2 Distributional Assumptions for Parametric Forecasting of Intermittent Demand 51
CV^2
5
4
3
2
1
0
0 5 10 15 20 25 30
p
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Chapter 3
Decision Trees for Forecasting
Trended Demand
3.1 Introduction
Service parts, by their very nature, are often subject to trends in demand. The
trends may be short-term, as a consequence of changing market conditions, or
long-term, related to the life-cycle of the part. Fortuin (1980) suggested that there
are three phases of a service part’s life history: initial, normal and final. The initial
phase is often a time of growth in demand, as more original equipment begins to
fail. The normal phase may be more stable, but still subject to shorter-term trends,
perhaps reflecting wider market trends. The final phase is generally one of long-
term decline in demand, as the original equipment is replaced by newer models
and the service parts are required less frequently.
The nature of the trend in demand is not always clear, particularly during the
normal phase of demand, when noise in demand may mask trends. Therefore,
forecasting may be enhanced by improving the selection process between trended
and non-trended series, and between different types of trended series.
In many organisations, service parts are voluminous, amounting to thousands or
tens of thousands of items. This has led to the implementation of automatic
methods in forecasting and inventory management software. One approach to the
selection of methods is to use a ,pick best’ method, whereby a competition of
methods is conducted for each series. However, this approach is not suitable for
forecasting trended demand when the length of demand history is short, making it
difficult to discern growth or decline. Another approach is to simply allow users to
choose a forecasting technique from a menu. This presupposes that users are aware
N. N. Atanackov (&)
Belgrade University, Belgrade, Serbia
e-mail: Natasha_atanackov@yahoo.co.uk
J. E. Boylan
Buckinghamshire New University, High Wycombe, UK
e-mail: John.Boylan@bucks.ac.uk
of the strengths and weaknesses of the methods on the menu, and know how to
conduct appropriate accuracy comparisons. This is not always a reasonable
assumption, particularly for those new to demand forecasting, or insufficiently
well-trained.
An alternative approach is to develop decision rules for method selection based
on the analysis of the models that underpin forecasting methods. Such selection
methods, known as protocols, can be implemented quite straightforwardly in
forecasting software. Analysing the problem from this perspective may bring new
insights and offer the opportunity of more accurate forecasts.
The aim of this chapter is to assess protocols for method selection, based on
rules derived from the time series features. The pool of methods consists of
well known non-seasonal exponential smoothing methods, namely Single
Exponential Smoothing (SES), Holt’s linear method, and damped Holt’s
method. The damped Holt’s method, introduced by Gardner and McKenzie
(1985), includes a damping parameter in Holt’s method to give more control
over trend extrapolation. It is included in many service parts applications and
has been found to perform well in forecasting competitions on real data. In this
chapter, a distinction will be maintained between a forecasting method, which
is a computational procedure to calculate a forecast, and a forecasting model,
which is a stochastic demand process. A model-based method takes the sto-
chastic process as given, and is based on an appropriate estimation procedure
for the parameters specified in the model. A schematic representation is shown
in Fig. 3.1.
The models considered in this chapter are restricted to non-intermittent demand,
for which the standard smoothing methods are appropriate. Intermittent demand
occurs when there is no demand at all in some time periods and requires different
smoothing approaches, such as Croston’s method. The selection of forecasting
methods for intermittent items is an important problem and has been a subject of
growing interest in the academic literature. Boylan and Syntetos (2008) review
recent developments and offer a framework for classification of service parts,
including consideration of intermittence and erraticness of demand. The approa-
ches suggested for the selection of methods for intermittent demand are quite
different to those that have been developed for non-intermittent demand. Both
categories of demand are vital for service parts management but, in this chapter,
we shall concentrate on non-intermittent demand.
The reminder of this chapter will develop an approach to select from the models
that underpin SES, Holt’s and damped Holt’s methods. Then, the approach will be
Identification Estimation
Set of Procedure Procedure
Forecasting Forecasting Forecasting
Models Model Method
examined and tested on real-world data. Three datasets will be employed. The first
two are ,standard data sets’, having been used in previous forecasting competi-
tions, while the third has not been analysed previously. The third dataset consists
of weekly demand histories; forecasts of weekly demand are required for inventory
management purposes.
where dt represents the observation for the current time period t; ^lt represents an
estimate of the underlying mean level of the series; d^tþn denotes a forecast made at
time t for n-periods ahead; and a represents a smoothing parameter. Equation (2.1)
shows how the smoothed level is updated when a new observation becomes
available. Equation (2.2) represents the n-step ahead forecast using observations
up to and including time t.
The choice of values of smoothing parameters in ES methods can be made in a
number of ways. Firstly, the parameter values can be set arbitrarily to values
suggested in the academic literature. Secondly, the estimation period of the time
series (discussed in Sect. 3.6.2) can be used to establish the parameters by
selecting values that minimise Mean Squared Error (or other error measure). The
values will be kept constant regardless of the new observations that become
available. Finally, the smoothing parameters can be optimised at each period,
before the forecasts are generated, meaning that all the available time series
information is used. In this study, the second approach is used, allowing for
optimisation of parameters, but without the computational burden of re-optimi-
sation every period. For SES, the smoothing constant is optimised from the range
0.05 to 0.95, with a step value of 0.05.
56 N. N. Atanackov and J. E. Boylan
When trend is present in the time series, the SES method needs to be generalised.
Holt’s linear trend method (Holt 1957, 2004a, b) produces estimates of the local
level and of the local growth rate. The method may be written as follows:
^lt ¼ adt þ ð1 aÞ ^lt1 þ ^bt1 ð2:3Þ
^
bt ¼ b ^lt ^lt1 þ ð1 bÞ^bt1 ð2:4Þ
d^tþn ¼ ^lt þ n^
bt ð2:5Þ
where ^bt represents an estimate of the underlying trend of the series; a and b
represent smoothing parameters; and the remaining notation is unchanged.
Equation (2.3) shows how the smoothed level of the series is updated when a new
observation becomes available, while (2.4) serves the same purpose for the trend
estimate. Equation (2.5) gives a straight line trend-projection for the n-step ahead
forecast.
As for single exponential smoothing, the parameters can be chosen arbitrarily,
or they can be estimated from the time series. In this chapter, the two parameters
are optimised once, with both parameters being required to lie in the interval 0.05–
0.95 (with step value of 0.05).
It is not clear when the idea of damped trend was introduced for the first time in the
academic literature. Roberts (1982) introduced a predictor for sales forecasting
with an incremental growth estimate and a damping parameter, and proved its
optimality for an ARIMA (1, 1, 2) process. Gardner and McKenzie (1985) dis-
cussed Holt’s linear method, and suggested a generalised version of the method by
adding an autoregressive-damping parameter, /, to give more control over trend
extrapolation. The damped Holt’s method may be written as follows:
^lt ¼ adt þ ð1 aÞ ^lt1 þ /^bt1 ð2:6Þ
bt ¼ b ^lt ^lt1 þ ð1 bÞ/^bt1
^ ð2:7Þ
X
n
d^tþn ¼ ^lt þ /i ^
bt ð2:8Þ
i¼1
In the above equations, / is a damping parameter between zero and one, and the
remaining notation is unchanged. Depending on the value of the damping
3 Decision Trees for Forecasting Trended Demand 57
Extrapolating trends is risky because if the trend forecast is in the wrong direction,
the resulting forecast will be less accurate than the random walk. This may lead to
significant over-stocking or under-stocking of service parts, particularly if lead-
times are long. Therefore, it is desirable to examine methods which can discern
when trend is present and when it is absent.
Numerous approaches to model and method selection have been discussed over
the last 40 years. Box and Jenkins (1970) proposed the analysis of the auto-
correlation function (ACF) and the partial auto-correlation function (PACF), after
appropriate differencing of series, to select between ARIMA models. Originally,
this procedure depended on careful analysis of the ACF and PACF of each indi-
vidual series, and was not suited to automatic forecasting systems. Since then,
automatic selection of models has been incorporated in forecasting packages such
as Autobox.
A less sophisticated approach is prediction validation (see, for example,
Makridakis et al. 1998). A subset of the dataset is withheld and various methods
are compared on this ,out-of-sample’ subset, using an error criterion such as Mean
Square Error or Mean Absolute Percentage Error. Billah et al. (2005) analysed
simulated and empirical data, and showed that prediction validation can be
improved upon by approaches based on ,encompassing’ and information criteria.
An ,encompassing approach’ relies on the most general method being applied to
all series, where all other methods under consideration are special cases of the
general method. Billah et al. (2005) compared SES, Holt’s method and Holt–
Winters’ method, using Holt’s as the encompassing approach for non-seasonal
data and Holt–Winters’ for seasonal data.
Information criteria are often recommended for the selection of an appropriate
forecasting method. These criteria penalise the likelihood by a function of the
number of parameters in the model. They have stimulated much academic interest
and are used in service parts computer applications such as Forecast Pro. The
following information criteria are well-established in the literature: Akaike’s
Information Criterion (Akaike 1974), the bias-corrected AIC (Hurvich and Tsai
1989), the Bayesian Information Criterion (Schwarz 1978), and the Hannan–Quinn
Information Criterion (Hannan and Quinn 1979). More recently, linear empirical
information criteria have been proposed by Billah et al. (2005). Gardner (2006)
comments that studies comparing the performance of different information criteria
should also include a comparison with universal application of damped Holt’s
method, as this offers a benchmark which is difficult to beat. This offers a more
general encompassing benchmark approach than Holt’s method (for non-seasonal
data) and is adopted in this study.
An alternative approach to model selection is to employ expert systems. Such
systems are based on sets of rules recommended by experts in the field. Collopy and
Armstrong (1992) gave rules to select between a random walk, time-series regres-
sion, Brown’s double exponential smoothing method and Holt’s linear method.
3 Decision Trees for Forecasting Trended Demand 59
Vokurka et al. (1996) gave a fully automatic expert system to distinguish between
SES, the damped trend method, classical decomposition and a combination of all
methods. Gardner (1999) tested the expert systems of Collopy and Armstrong
(1992) and Vokurka et al. (1996) and found them to be less accurate than universal
application of the damped trend method. A subsequent study by Adya et al. (2001)
enhanced the rules of Collopy and Armstrong (1992) and found the new rules to
perform better than universal damped trend for annual data and about the same for
seasonally adjusted quarterly and monthly data. In the light of these studies, and the
comments by Gardner (2006) on the analysis of methods based on information
criteria, discussed previously, universal application of damped Holt’s method will
be used as a benchmark for comparison with the performance of the protocols and
decision trees tested in this study.
Another approach to method selection is based on time-series characteristics.
Shah (1997) developed a rule for selecting the best forecasting method for each
individual series, using discriminant analysis. He demonstrated that a choice of
forecasting method using summary statistics for an individual series was more
accurate than using any single method for all series considered. Meade (2000)
designed an experiment for testing the properties of 25 summary statistics. Nine
forecasting methods were used, divided into three groups, and tested on two data
sets, the M1-competition data and Fildes’ telecommunications data. The author
concluded that the summary statistics can be used to select a good forecasting
method or set of methods, but not necessarily the best.
Gardner and McKenzie (1988) proposed an approach based on comparison of
the variances of the original series, the once-differenced and the twice-differenced
series. Tashman and Kruk (1996) analysed three protocols for method selection on
real time series: Gardner and McKenzie’s (1988) variance procedure, the set of
rules from Rule-Based Forecasting (Collopy and Armstrong 1992), and a method-
switching procedure developed by Goodrich (1990). Tashman and Kruk found that
the protocols are effective in selecting the appropriate applications of strong trend
methods but do not effectively distinguish applications of non-trended and weak-
trended methods. This issue will be addressed in this chapter. A modification to the
Gardner and McKenzie protocol will be suggested, and tested on simulated and
real data.
Under the Steady State Model (SSM) assumption, the mean level of the time series
fluctuates stochastically over time. The SSM model is given in the following form:
dt ¼ lt þ et ð4:1Þ
lt ¼ lt1 þ ct ð4:2Þ
where dt represents the observation for the current time period t; lt represents the
unobserved series mean level at time t; et and ct are uncorrelated normally dis-
tributed random variables each with zero mean E(et) = E(ct) = 0 and constant
variance V(e) = const. and V(c) = const.; et and ct are also serially uncorrelated.
The above model is often referred to as the local level model (eg Commandeur and
Koopman 2007). It corresponds to an ARIMA (0, 1, 1) process.
Theil and Wage (1964) analysed generating processes for trended time series.
Harrison (1967) extended their work and formulated a probabilistic Linear Growth
Model (LGM) of the following form:
dt ¼ lt þ et ð4:3Þ
lt ¼ lt1 þ bt þ ct ð4:4Þ
bt ¼ bt1 þ dt ð4:5Þ
where bt denotes the underlying trend; et, ct, and dt are uncorrelated normally
distributed random variables each with zero mean E(et) = E(ct) = E(dt) = 0 and
constant variance V(et) = const., V(ct) = const. and V(dt) = const; et, ct, and dt are
also serially uncorrelated. The above model is often referred to as the local linear
trend model (e.g. Commandeur and Koopman 2007) and it corresponds to an
ARIMA (0, 2, 2) process.
3 Decision Trees for Forecasting Trended Demand 61
In the first Damped Trend Model (DTM 1), it is assumed that the dampening starts
in the second time period after the end of the historical data:
dt ¼ lt þ et ð4:6Þ
lt ¼ lt1 þ bt þ ct ð4:7Þ
bt ¼ /bt1 þ dt ð4:8Þ
where dt represents the observation for the current time period t; lt represents the
unobserved series level at time t; bt denotes the underlying trend; et, ct, and dt are
uncorrelated normally distributed random variables each with zero mean
E(et) = E(ct) = E(dt) = 0 and constant variance V(et) = const., V(ct) = const.
and V(dt) = const; et, ct, and dt are also serially uncorrelated; / represents a
damping parameter.
In the second Damped Trend Model (DTM 2), it is assumed that the dampening
starts in the first time period after the end of the historical data:
dt ¼ lt þ et ð4:9Þ
bt ¼ /bt1 þ dt ð4:11Þ
where dt, lt, bt, et, ct, dt and / are as defined in the previous sub-section. In this
model, the damping parameter is applied one period earlier than in DTM 1.
Atanackov (2004) analysed the Serial Variation Curves, discussed in the next
section, for both model forms. She showed that the Serial Variation Curve for
DTM 1, for the special case of / = 0, is not consistent with the Serial Variation
Curve for the Steady State Model (SSM). On the other hand, the Serial Variation
Curve for DTM 2 is consistent with SSM for / = 0 and is consistent with the
Linear Growth Model for / = 1. Therefore, for the remainder of this chapter,
DTM 2 will be adopted.
Protocols and decision trees for method selection purposes will be analysed in this
section. The aim is to propose a more coherent approach that exhibits good per-
formance in practical applications. The research process will be based on the
selection of mathematical models, discussed in Sect. 3.4, that underpin the
methods discussed in Sect. 3.2, namely SES, linear Holt’s and damped Holt’s
methods.
62 N. N. Atanackov and J. E. Boylan
Harrison (1967) suggested testing the lagged second differences of the data in
order to assess the suitability of a data generating process. The idea was originally
applied to the Linear Growth Model (LGM) only, and has not been extended since.
In this chapter, the SVC procedure will be applied for the first time to the Damped
Trend Model (DTM). It will be shown that the SVC approach applied to the LGM
and DTM models can form an independent diagnostic, from now on called the
SVC protocol, to select between the two trended models.
Defining D1(t) = dt - dt-1 as the first difference of the observations at time t,
and defining Dn(t) = D1(t) - D1(t - n) as the second difference of the obser-
vations lagged by n periods, and taking the squared expectations of the above
differences, Harrison (1967) showed that, for n C 2:
2/2 ð1 /n Þ
E½Dn ðtÞ2 ¼ 4VðeÞ þ 2VðcÞ þ VðdÞ ð5:2Þ
1 /2
Equation (5.1) is a special case of (5.2) when / = 1 (by application of
L’Hôpital’s rule to (5.2)). Equation (5.2) shows that the Serial Variation Curve of
the lagged second differences of the data for the DTM 2 (0 \ / \ 1) is expected to
be a curve, as illustrated in Fig. 3.2.
Because the Serial Variation Curve (SVC) of the LGM model is linear, and the
SVC of the DTM model is non-linear, a protocol can be designed to distinguish
between series that follow a weak trended model (DTM) and series that follow a
strong trended model (LGM). The new protocol, called the SVC protocol, is based
3 Decision Trees for Forecasting Trended Demand 63
Fig. 3.2 The SVC of the two The SVC for the LGM and DTM models
trended models: LGM and 16
LGM model DTM model
DTM (/ = 0.8) 14
12
Variance
10
8
6
4
2
0
1 2 3 4 5 6 7 8 9 10
Lag length
on fitting a straight line, using least squares regression, through the SVC points
(that is through the variances of the lagged second differences) and on testing the
significance of the autocorrelation (AC) of the residuals.
From Fig. 3.2 it follows that there are two cases: (i) for the LGM model, there
is no significant AC of the residuals; (ii) for the DTM model, there is significant
positive AC of the residuals. The one-sided Durbin–Watson statistic (Durbin and
Watson 1951) is appropriate for testing the autocorrelation of the residuals.
In order to use the SVC protocol for real life applications it is necessary to
establish its operational rules. The lag length to be used for the SVC fitting is not
known in advance. Therefore, it will be tested as part of the simulation analysis.
Six different lag lengths of 15, 20, 25, 30, 35, and 40 are examined under two
significance levels: 1 and 5%. Their performance is compared in terms of the
percentage of correctly identified time series models and the best operational rules
for the SVC protocol are established. The findings are summarised in Sect. 3.6.5.
VðeÞ
Vð0Þ\Vð2Þ if and only if n\ þ2 ð5:3Þ
VðcÞ
where n is the length of the time series. Further analysis of the above expression
shows that the V(0)V(2) protocol can detect the SSM time series for low values of
the smoothing parameter a and for short time series. This result is confirmed by
simulation experiments.
For the LGM model, comparison of the variances V(0) and V(2) gives the
following inequality (Atanackov 2004):
where the notation is unchanged. Further analysis of the above expression shows
that the V(0)V(2) protocol can detect the LGM time series in a high percentage of
cases for the higher values of the a parameter, and for time series containing more
than 20 observations. For low a values and for short time series the V(0)V(2)
protocol detects non-trended series.
The performance of the V(0)V(2) protocol will be analysed further using sim-
ulation experiments on theoretically generated data, and tested on real data sets.
Two new decision trees are presented in this section. The first, Decision Tree A,
incorporates an extension of Harrison’s Serial Variation Curves. The second,
Decision Tree B, is based solely on the idea of comparing the variances of the
original series and the once- and twice-differenced series.
66 N. N. Atanackov and J. E. Boylan
The simulation experiment serves two purposes. The first purpose is to identify
suitable parameters for the SVC protocol and, subsequently, for the Decision Tree
A, namely the lag length and the relevant significance level, that cannot be
established otherwise. The second purpose is to assess the classification and
forecasting performance of the protocols under controlled conditions, where the
underlying models are known. The protocols and decision trees to be tested are as
follows: (i) The SVC protocol—extended Harrison’s Serial Variation Curves; (ii)
Original GM protocol—Gardner and McKenzie’s (1988) variance procedure; (iii)
V(0)V(2) protocol; (iv) Decision Tree A; and (v) Decision Tree B.
The three mathematical models outlined in Sect. 3.4 (SSM, LGM and DTM2)
will be employed in the first part of the experiment to simulate time series
exhibiting known characteristics. Choosing different combinations of smoothing
3 Decision Trees for Forecasting Trended Demand 67
parameters, and making connections between the method and model parameters, it
will be possible to generate time series data that follow a given model for which
the optimal forecasting method is known in advance. Eight different lengths of
time series, namely 10, 20, 30, 40, 50, 75, 100, and 150 observations per series,
will be generated for each of the three models. This covers all the series lengths in
the empirical study, to be presented in the following section. For every length,
10,000 replications will be made.
The protocols are tested for classification performance. Protocols will be
applied to the relevant data sets (for example, the SVC is relevant for the LGM and
DTM models) and the percentage of series for which the model is correctly
identified will be recorded. The aim of this classification is to establish the
operational rules necessary for the protocols’ practical application, and to support
the analysis of the forecasting performance.
Finally, the simulation is designed to test the forecasting performance of the
protocols. Forecasting performance will be evaluated over a six-period horizon.
Since the methods are MSE-optimal for the corresponding models, the Mean
Square Error (MSE) is used as an accuracy measure, and the Geometric Root Mean
Square Error (GRMSE) will be used in the simulation experiment to summarise the
accuracy results. In addition, the Mean Absolute Error (MAE) will be employed in
the simulation exercise because it is intended to use this error measure in the
empirical analysis. This error measure is less sensitive to outlying or extreme
observations. The error measures are discussed in more detail in Sect. 3.6.4.
The objective here is to establish the relationships between the method and the
model parameters for the three methods, to generate data sets that follow specific
models and exhibit specific forecasting methods as optimal. In order to cover many
different combinations of the smoothing parameter values, while keeping the
simulation experiment manageable, five different a values will be used for gen-
erating the SSM model, twelve pairs of a and b parameters will be chosen for
generating the time series that follow the LGM model, and eighteen triplets of
smoothing parameters and the damping parameter / will be used for the DTM2
model generation. Altogether, thirty five data sets, each containing 80,000 time
series will be generated for experimental purposes.
In order to simulate the time series that follow the SSM model and genuinely
have the SES as the optimal forecasting method, the connection between the
method and the model parameters needs to be established. In this particular case,
there is only one method parameter, the smoothing parameter a, and two model
parameters, the variances of the error terms et and ct, i.e. V(c) and V(e). Following
Harrison (1967), the connection between the optimal smoothing parameter for the
SES method and the variance parameters for the SSM model can be expressed as
follows:
68 N. N. Atanackov and J. E. Boylan
1a
V ðeÞ ¼ V ð cÞ ð6:1Þ
a2
The above formula will be used in the experiment to generate time series data
that follow the SSM model. Five different smoothing parameter values (0.1, 0.3,
0.5, 0.7, and 0.9) will be used, to cover the range between 0 and 1. Finally, in order
to generate the SSM process, ct and et will be chosen from two sets of indepen-
dently normally distributed random numbers, with zero means and constant
variances V(c) and V(e), respectively. The initial level will be set at 100. The run-
in procedure (also known as ,warming up’) will be used, to allow the time series to
become stable.
To simulate the LGM process, four parameters will be taken into account: two
smoothing parameters a and b for Holt’s linear exponential smoothing method,
and two variance ratios characterising the LGM model. The relationship between
the model and optimal method parameters, obtained by Harrison (1967) and
further discussed by Harvey (1984), can be expressed as follows:
a2 þ a2 b 2ab
VðcÞ ¼ VðdÞ ð6:2Þ
a 2 b2
1a
VðeÞ ¼ VðdÞ ð6:3Þ
a2 b 2
Twelve pairs of the smoothing parameter values, a and b, are selected, to assess
the influence of the parameters on the forecasting performance of Holt’s linear
method for the series following the LGM model, while covering a wide region of
possible combinations. The random components et, ct, and dt will be chosen from
three different sets of independently normally distributed random numbers, with
zero means and constant variances V(e), V(c) and V(d), respectively. The initial
level will be set at 100 and the run-in period will be applied.
The relationships between method and model parameters for damped Holt’s
method, given the DTM 2 model, can be expressed as follows (Atanackov
2004):
VðeÞ ð1 aÞ/2
¼ 2 ð6:5Þ
VðdÞ / a2 b2 þ abð1 /Þ /2 abð1 /Þ þ /a2 b 1 /2
The value of the damping parameter / has been assumed to be in the interval
(0, 1). The smoothing parameter values, a and b, will be chosen in such a way as to
test the influence of the parameters on the forecasting performance of damped
Holt’s method for time series that follow the DTM model, while encompassing a
wide region of possible combinations. The random components et, ct, and dt will
be chosen from three different sets of independently normally distributed
3 Decision Trees for Forecasting Trended Demand 69
random numbers, with zero means and constant variances V(e), V(c) and V(d),
respectively. The initial level will be set at 100 and the run-in period will be
applied (Figs. 3.3, 3.4).
Y N
V(0) < V(2)
Y SVC N
AC significant?
Y N
V(0) < V(2)
Y N
V(1) < V(2)
N
Estimation period Out-of-sample period
every combination of smoothing parameters relevant for the method under concern
and, finally, the errors are compared. The parameters that produce the minimum
error will be chosen as the optimal ones.
As argued by many researchers (e.g. Fildes 1992; Tashman 2000; Armstrong
2001) forecasting methods should be assessed for accuracy using the out-of-
sample period rather than goodness of fit to the past data. Therefore, in this study
the estimation period will not be used for comparison of the forecasting perfor-
mances of relevant protocols. During this research, the series will be split as shown
in Table 3.2.
After selecting a method and optimising its parameters, the actual observations
from the out-of-sample period will not be taken into account when selecting and
optimising a method. The last observation in the estimation period (Fig. 3.5) is
called the forecasting origin. There are two ways of performing an out-of-sample
test regarding the forecasting origins: (i) Single (or fixed-origin), and (ii) Multiple
(or rolling-origin).
Given the length of the performance period (Fig. 3.5) the single forecasting
origin will produce only one forecast for each of the relevant forecasting horizons
Table 3.2 The time series Time series length Initialisation Calibration Performance
splitting rules
14–15 2 6 6–7
16–18 3 6 7–9
19–21 4 6 9–11
22–24 5 7 10–12
25–27 6 12 7–9
28–30 8 12 8–10
31–33 8 13 10–12
34–36 8 16 10–12
37–39 8 19 10–12
[40 8 n – 8 – 12 12
At the bottom of the table, n represents the series length
3 Decision Trees for Forecasting Trended Demand 71
ððP þ 1Þ; ðP ¼ 2Þ; . . . ; ðP ¼ NÞÞ: Therefore, the forecast error value and,
consequently, the performance of a given forecasting method, or the protocol in this
particular case, will heavily depend on the choice of the starting point. If the selected
point is an irregular observation (i.e. outlier) then the errors for every subsequent
forecasting horizon might not reflect reality and completely distort the picture of the
pattern in the time series, and finally offer a misleading evaluation of the method
performance. Therefore, throughout the simulation experiment either with the
theoretically generated data or the real data, multiple time origins will be utilised.1
In order to report on the accuracy for a given method, the errors are firstly aver-
aged over the length of the performance period and, subsequently, averaged across
all series for a given length. Two error measures, the Geometric Root Mean Square
Error (GRMSE) and the Mean Absolute Error (MAE), will be employed to express
the accuracy of the protocols and universal application of forecasting methods
using theoretically generated time series data.
The Geometric Root Mean Squared Error (GRMSE) was first used in fore-
casting competitions by Newbold and Granger (1974) and is defined as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Y
n
ðYt Ft Þ2
2n
GRMSE ¼
t¼1
where Yt represents the observation at period t, and Ft represents the forecast made
for the period t, n is the number of data points in the performance period.
The Mean Square Error (MSE) is scale dependent and sensitive to outliers
(Chatfield 1988, 1992). This difficulty is mitigated by the Geometric Root Mean
Square Error (GRMSE) and eliminated when used as a relative measure (i.e.
calculating the GRMSE of one method to that of another) under an assumption of
multiplicative random error terms (Fildes 1992). However, relative measures will
not be used in simulation, since the purpose of the experiment is to test the
protocols using theoretically generated time series rather than real life series. This
error measure is well-behaved and has a straightforward interpretation, (Fildes
1992; Sanders 1997)
Another error measure, considered to be suitable for the simulation experiment,
will be the Mean Absolute Error (MAE). An accuracy measure must be robust
from one data set to another, and not be unduly influenced by outliers. From a
practical point of view, it must make sense, be easily understood, and convey as
much information about accuracy as possible. The MAE satisfies all the above
conditions. Therefore, the Mean Absolute Error will be employed in order to allow
the results achieved in the simulation experiment, concerning the comparison of
1
A consequence of this choice is that a direct comparison with M3-competition results will not
be possible.
72 N. N. Atanackov and J. E. Boylan
the protocols, to be tested on real life time series. The MAE will provide a ground
for comparison between the performance of the protocols in the theoretically made
environment and real forecasting applications. In the empirical analysis, the MAE
will be complemented by the Mean Absolute Percentage Error (MAPE).
The first stage in the simulation experiment is to examine the protocol classifi-
cation performance and to establish the SVC operational rules. Durbin and Watson
(1951) statistics are used to test the significance of the autocorrelation of the
residuals. Six different lengths 15, 20, 25, 30, 35, and 40 were tested, under 5 and
1% significance levels.2 The best simulation results for both LGM and DTM
models are presented in Figs. 3.6 and 3.7.
70%
60%
50%
40%
30%
20%
10%
0%
10 20 30 40 50 75 100 150
Time series length
alpha=0.4; beta=0.125 alpha=0.65; beta=0.231 alpha=0.85; beta=0.412
Fig. 3.6 Percentage of the LGM series recognised by SVC protocol for different smoothing
parameters, as a function of time series length (lag length 15, 5% significance level)
2
A drawback of the DW statistic has been taken into account. The Durbin-Watson statistic has a
gap between the significant positive autocorrelation, representing the DTM model, and not
significant autocorrelation, representing the LGM model. Therefore, if the result belongs to that
gap it follows that the DW test is inconclusive. Therefore, an operational rule had to be adopted in
order to overcome the problem. There were two possibilities: either to allocate the inconclusive
time series to the LGM model or to the DTM model. Having analysed the above issue in both
cases during the simulation experiment, it was concluded that the penalty in terms of forecast
accuracy is lower, if the inconclusive time series are allocated to the DTM model. Since the LGM
model is a special case of the DTM model (for / = 1), it follows that the LGM could be detected,
but not vice versa.
3 Decision Trees for Forecasting Trended Demand 73
80%
70%
60%
50%
40%
30%
20%
10%
0%
10 20 30 40 50 75 100 150
Time series length
phi=0.1;alpha=0.15;beta=0.05 phi=0.3;alpha=0.35;beta=0.2 phi=0.5;alpha=0.25;beta=0.15
phi=0.7;alpha=0.9;beta=0.4 phi=0.9;alpha=0.65;beta=0.1
Fig. 3.7 Percentage of the DTM series recognised by SVC protocol for different smoothing
parameters, as a function of time series length (lag length 15, 5% significance level)
Five sets of smoothing parameters for the LGM model were tested and the lag
length 15 was the best performing one. A significance level of 5% was selected.
Although a significance level of 1% for testing the difference between the curve
(DTM) and a straight line (LGM) performed well on the LGM time series, rec-
ognising a high percentage of series, it was not chosen because it did not perform
well on the DTM time series.
From Fig. 3.6, it follows that the SVC protocol detects the LGM model in a
higher percentage of cases for higher values of the smoothing parameters,
regardless of the length of the time series. According to Fildes et al. (1998) this
result is not surprising. On the other hand, for low values of the smoothing
parameters the SVC detects the LGM model only for the short time series. For
longer series, the LGM model is missed and the DTM model is detected in a high
percentage of cases.
From Fig. 3.7, it follows that the SVC protocol with lag length 15 and sig-
nificance level 5% performs better, in terms of percentage recognition of the DTM
time series, for the middle values of the damping parameter / (between 0.3 and 0.7
inclusive) and the series containing more than 30 observations. When using the
same combinations of the smoothing parameters and the damping parameter / but
using 1% significance level instead of 5%, the DTM time series percentage rec-
ognition by the SVC protocol drops heavily for every combination tested in the
experiment. Therefore, the 5% significance level will be adopted as a rule for the
SVC practical applications.
Based on the above analysis, the operational rules for the SVC protocol derived
from the theoretically generated time series can be summarised as follows: (i) The
lag length for the SVC fitting should be 15; (ii) The significance level to be
employed should be 5%; (iii) The number of observations in the time series should
be 30 or more.
74 N. N. Atanackov and J. E. Boylan
The Decision Tree A, designed to use the SVC protocol, will use these oper-
ational rules. The original Gardner and McKenzie variance procedure and its
modified version, called the V(0)V(2) protocol, as well as the Decision Tree B do
not require operational rules for their practical application.
Having generated 35 data sets and established the operational rules for the relevant
protocols, the next stage involves testing the protocols’ classification performance.
Each protocol will be applied on the time series following the models relevant for
the protocol of interest. The percentage recognition of the specific model, and the
forecast accuracy of the recommended method will be recorded for comparison
purposes.
Single exponential smoothing (SES) needs one parameter to be estimated from
the time series in order to generate forecasts. The smoothing parameter a will be
selected based on the outcome of the comparison of the MSE errors for the
estimation period. The method will be initialised using an average of up to 8
observations from the beginning of data history, and values of the smoothing
parameter will be chosen from a range of 0.05 up to 0.95 with a step of 0.05. In this
study, Holt’s linear and damped Holt’s methods will be initialised using simple
linear regression, using up to the first eight observations. In testing the perfor-
mance of the universal application of forecasting methods, the values of the
smoothing parameters a and b will be chosen from a range of 0.05 up to 0.95 with
a step of 0.05.
Forecasts from 1-, up to 6-periods ahead will be computed and the performance
of the forecasting methods and, consequently, selection protocols will be com-
pared for every separate length of the horizons. Finally, throughout this study,
minimum n-step ahead MSE will be used during the calibration period in order to
select the smoothing parameters for the length of forecasting horizon equals n.
This section reports on the protocols’ performance when time series follow the
Steady State Model (SSM). Five different values of smoothing parameter a were
used in order to generate five sets of the SSM time series. The relevant protocols to
be applied for the series following the SSM model are the original GM protocol
and the V(0)V(2) protocol. Both trees perform the same as the V(0)V(2) protocol,
while the SVC protocol is not applicable since it deals with trended series only.
Table 3.3 shows that both protocols perform less well for higher smoothing
parameter values. Such series exhibit behaviour that may appear to be trended in
the short-term. The declining performance of both protocols as a function of series
3 Decision Trees for Forecasting Trended Demand 75
length is due to the greater tendency for long-term drifts in the mean level to
emerge, which are misdiagnosed as trend.
The original GM protocol exhibits poor classification characteristics when
detecting series with no trend (Table 3.3) as it chooses the DTM in a high per-
centage of cases for longer time series (Table 3.4).
Both protocols and the two decision trees were tested for forecasting accuracy
(see Table 3.5). Protocols’ performance are compared with universal application
of the SES method, as it is the best predictor for the series following the SSM
model (a = 0.1).
Even though the percentage recognition of the adequate mathematical model is
identical, Tree B produces more accurate forecasts than Tree A, for all forecasting
horizons tested in the experiment and for lengths of data history of 20 observations
or more. The main difference is in the models that are identified. According to
Table 3.4 Percentage of SSM series misidentified by the protocols (SSM model, a = 0.1)
Protocols Time series length
10 20 30 40 50 75 100 150
Percentage DTM identified
Tree A 2.7 6.7 13.7 20.6 28.3 41.6 51.6 62.6
Tree B 8.0 17.6 28.3 38.2 47.7 66.7 78.5 91.6
Original GM 33.5 62.7 79.7 88.6 93.6 98.7 99.7 100.0
Percentage LGM identified
Tree A 5.9 10.9 14.6 17.6 19.4 25.0 26.9 29.0
Tree B 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Original GM 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0
76 N. N. Atanackov and J. E. Boylan
Table 3.5 Mean Absolute Errors for the universal application of the SES method, V(0)V(2)
protocol, Decision Trees and the original GM protocol (SSM model, a = 0.1)
Protocols Time series length
10 20 30 40 50 75 100 150
MAE 1-step ahead
SES 13.74 14.18 13.74 13.21 13.02 12.79 12.76 12.76
V(0)V(2) 14.16 14.56 14.13 13.53 13.32 13.09 13.02 13.05
Tree A 13.97 14.49 14.05 13.43 13.22 12.96 12.89 12.88
Tree B 14.01 14.36 13.92 13.34 13.14 12.88 12.81 12.79
Original GM 14.33 14.77 14.33 13.57 13.26 12.91 12.81 12.80
MAE 3-step ahead
SES 17.38 18.37 18.42 18.03 17.80 17.49 17.27 17.21
V(0)V(2) 18.55 19.47 19.58 19.20 19.00 18.61 18.44 18.37
Tree A 17.68 19.18 19.28 18.82 18.58 18.09 17.84 17.69
Tree B 17.94 18.74 18.77 18.39 18.15 17.75 17.48 17.35
Original GM 18.44 19.38 19.24 18.69 18.34 17.82 17.49 17.35
MAE 6-step ahead
SES – 25.53 22.92 23.12 23.02 22.73 22.51 22.38
V(0)V(2) – 28.46 25.19 25.57 25.64 25.42 25.41 25.34
Tree A – 27.50 24.58 24.72 24.57 24.07 23.77 23.50
Tree B – 25.70 23.45 23.73 23.62 23.20 22.89 22.62
Original GM – 25.41 24.29 24.03 23.75 23.22 22.91 22.62
Table 3.4, Tree A selects the LGM model more often, while Tree B selects the
DTM model more often. The better performance of Tree B may be explained by
the lower penalty of misdiagnosing an SSM series as DTM instead of LGM. Also,
Tree B produces more accurate forecasts than the original GM protocol for the
series containing up to 50 observations. For longer time series, the accuracy of
these two protocols is very nearly the same, regardless of the length of forecasting
horizon.
Twelve sets of smoothing parameters a and b relevant for Holt’s linear method
were used to generate time series that exhibit trended behaviour according to the
relationships between the method parameters and the LGM model parameters
shown earlier in the chapter.
When testing the protocols on the LGM time series with low values of the a
parameter, both decision trees, the V(0)V(2) protocol and the original GM protocol
detect the SSM model for the shorter time series (Table 3.6). For longer time
series, the trees and the original GM protocol detect the DTM model. The results
presented in Tables 3.6 and 3.7 confirm the theoretical expectation that the
V(0)V(2) protocol will identify series that exhibit strong trend; this is particularly
evident for series containing more than 50 observations. The SVC protocol
3 Decision Trees for Forecasting Trended Demand 77
Table 3.6 Percentage of LGM series correctly identified and misidentified by the protocols
(LGM model, a = 0.4 and b = 0.125)
Protocols Time series length
10 20 30 40 50 75 100 150
Percentage SSM identified
V(0)V(2), both trees 96.9 88.0 68.9 52.2 40.3 24.5 17.5 11.4
Original GM 78.7 47.9 29.3 21.4 16.6 9.5 6.9 4.2
SVC – – – – – – – –
Percentage DTM identified
Tree A 1.0 5.3 20.1 34.4 45.3 60.5 67.3 74.3
Tree B 3.1 12.0 31.1 47.8 59.7 75.6 82.6 88.6
V(0)V(2) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Original GM 21.2 52.1 70.7 78.6 83.4 90.5 93.1 95.8
SVC 46.8 54.8 69.8 74.6 77.6 80.6 82.0 83.8
Percentage LGM recognised
Tree A 2.1 6.7 11.0 13.4 14.4 15.1 15.2 16.3
Tree B 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0
V(0)V(2) 3.1 12.0 31.1 47.8 59.7 75.6 82.6 88.6
Original GM 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0
SVC 53.2 45.2 30.2 25.4 22.4 19.4 18.0 16.2
recognises the LGM model for short time series only while for longer series the
DTM model is identified.
For higher values of the smoothing parameter a and the same value of b
parameter as in the previous table (see Table 3.7), the SVC protocol detects the
LGM time series in approximately 60% of cases for all lengths of series, leading to
good performance of Tree A.
Decision Tree B and the original GM protocol detect weak trended series
instead of strong trended ones, as expected. It is therefore necessary to compare the
forecasting performance of the protocols before drawing conclusions regarding
their applicability.
From the forecasting perspective, given the GRMSE as the performance
measure and taking into account only 1-step ahead forecasting horizon, the lowest
forecasting error is produced by the universal application of Holt’s linear method
regardless of the time series length (see Table 3.8). Furthermore, Table 3.8 shows
that the Decision Tree A performs better than the Tree B, while the performance of
the original GM protocol depends on the series length.
Similar conclusions hold for 3-steps ahead forecasting horizon (see Fig. 3.8).
Eighteen sets of the smoothing parameters and the damping parameter / were used
to generate series exhibiting damped trend. Decision trees A and B, original GM
and the SVC protocols were applied on the time series that follow the DTM model.
78 N. N. Atanackov and J. E. Boylan
Table 3.7 Percentage of LGM series correctly identified and misidentified by the protocols
(LGM model, a = 0.8 and b = 0.125)
Protocols Time series length
10 20 30 40 50 75 100 150
Percentage SSM identified
V(0)V(2), both trees 48.2 23.0 12.0 6.7 3.9 0.9 0.4 0.0
Original GM 22.5 6.1 1.5 0.5 0.1 0.0 0.0 0.0
SVC – – – – – – – –
Percentage DTM identified
Tree A 18.6 28.1 34.8 36.6 39.5 39.0 39.8 39.3
Tree B 50.0 76.9 88.0 93.3 96.1 99.1 99.6 100.0
V(0)V(2) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Original GM 75.7 93.9 98.5 99.5 99.9 100.0 100.0 100.0
SVC 38.7 37.2 40.6 39.8 41.4 39.5 40.0 39.3
Percentage LGM recognised
Tree A 33.3 48.9 53.2 56.7 56.6 60.1 59.9 60.7
Tree B 1.8 0.1 0.0 0.0 0.0 0.0 0.0 0.0
V(0)V(2) 51.8 77.0 88.0 93.3 96.1 99.1 99.6 100.0
Original GM 1.8 0.1 0.0 0.0 0.0 0.0 0.0 0.0
SVC 61.4 62.9 59.4 60.2 58.6 60.5 60.0 60.7
Table 3.8 GRMSE (1-step ahead) for the universal application of the Holt’s method, V(0)V(2)
protocol, Decision Trees, the original GM protocol and the SVC protocol
Protocols Time series length
10 20 30 40 50 75 100 150
LGM model, a = 0.4, b = 0.125
GRMSE 1-step ahead
Holt’s method 297.1 305.4 275.3 264.2 258.7 256.5 255.7 253.6
V(0)V(2) 297.7 315.6 296.1 281.6 271.4 264.0 262.0 259.4
Tree A 295.8 315.5 297.0 284.7 277.4 273.2 272.5 270.9
Tree B 296.3 315.7 298.4 286.9 279.9 275.7 274.9 273.2
Original GM 298.6 317.2 298.0 282.5 276.7 274.7 274.5 273.0
SVC 301.0 317.0 290.0 278.1 273.4 272.2 272.1 270.8
LGM model, a = 0.8, b = 0.125
GRMSE 1-step ahead
Holt’s method 59.0 67.4 64.7 60.6 59.5 58.0 57.9 57.7
V(0)V(2) 68.1 72.7 67.7 62.6 61.1 59.4 59.2 59.1
Tree A 68.0 73.5 68.3 63.0 61.6 59.9 59.8 59.6
Tree B 68.1 74.8 69.7 63.7 62.3 60.8 60.7 60.5
Original GM 69.4 77.8 70.7 63.8 62.3 60.8 60.7 60.5
SVC 68.3 74.2 68.6 63.0 61.6 59.9 59.8 59.6
The V(0)V(2) protocol is not applicable since it chooses between the non-trended
and trended time series regardless of the trend type. The results for four sets of the
parameters are presented in Table 3.9.
3 Decision Trees for Forecasting Trended Demand 79
120
100
GRMSE
80
60
40
20
0
10 20 30 40 50 75 100 150
Time series length
Holt's V0V2 Tree A Tree B GM SVC
Fig. 3.8 GRMSE (3-steps ahead) for the relevant protocols (LGM model, a = 0.8 and
b = 0.125)
Table 3.9 shows that Tree B and the original GM protocol selects the DTM
time series in a high percentage of cases for the time series containing more than
10 observations and the low values of the smoothing parameter b regardless of the
80 N. N. Atanackov and J. E. Boylan
Table 3.10 GRMSE (1-step ahead) for the universal application of the damped Holt’s method,
Decision Trees, the original GM protocol and the SVC protocol
Protocols Time series length
10 20 30 40 50 75 100 150
DTM model, / = 0.3, a = 0.35, b = 0.2
GRMSE 1-step ahead
Damped Holt’s 29.4 31.6 28.3 26.0 25.6 25.3 25.1 25.0
Tree A 29.4 32.5 29.2 26.8 26.4 26.1 26.0 25.9
Tree B 29.4 31.7 28.6 26.2 25.7 25.3 25.1 25.0
Original GM 30.2 32.5 28.6 26.2 25.7 25.3 25.1 25.0
SVC 29.4 32.1 29.0 26.7 26.4 26.1 26.0 25.9
DTM model, / = 0.9, a = 0.65, b= 0.1
GRMSE 1-step ahead
Damped Holt’s 489.8 550.2 506.4 466.8 455.2 448.0 446.5 445.6
Tree A 490.0 556.6 512.7 472.9 460.5 452.3 450.7 450.0
Tree B 492.6 551.1 506.9 466.8 455.4 448.0 446.6 445.6
Original GM 505.2 574.1 510.5 467.4 455.3 448.0 446.6 445.6
SVC 492.8 557.4 510.3 471.5 459.9 452.2 450.7 450.0
value of the damping parameter / and the smoothing parameter a. For the higher b
value of 0.4, the percentage recognition by the two protocols still remains high and
it is not less than 56% regardless of the series length. In those cases, the Tree A
recognises the DTM model for the series containing more than 10 observations.
From the forecasting perspective, using the GRMSE as the error measure and
taking into account 1-step ahead forecasting horizon, it follows that the lowest
forecasting error is produced by the universal application of damped Holt’s
method (see Table 3.10), as expected.
Even though the Decision Tree B and the original GM protocol have selected
the highest percentage of DTM time series, the difference in forecasting accuracy
is almost negligible.
The purpose of this section is to assess the forecasting performance of the pro-
tocols and decision trees for method selection as well as universal application of
forecasting methods using empirical data sets gathered from different sources. As
noted earlier, it is important to compare forecasting performance with universal
application of damped Holt’s method.
Since the focus of this chapter is the improvement of forecasting accuracy of the
non-seasonal exponential smoothing methods relevant for short-term forecasting,
it follows that the length of the forecasting horizon should be considered as an
important factor in the empirical analysis. This issue will be addressed accord-
ingly, using 1-, 2-, 3-, 4-, 5- and 6-step ahead horizons for the performance testing.
3 Decision Trees for Forecasting Trended Demand 81
During the experiment with the real data, single test periods will be used and two
error measures will be applied. There is no evidence that multiple test periods can
help in deciding which forecasting method produces the lowest error. The method
coefficients will be calibrated once, for every time series of interest. Errors are
calculated using the Mean Absolute Error (MAE) and the Mean Absolute Per-
centage Error (MAPE).
The entire M3-yearly data set has been used for testing the protocols and decision
trees since the data are neither slow (intermittent) nor seasonal. The M3-yearly
data set contains 645 time series of different lengths. The shortest time series
contain only 14 observations and the longest series contain 41 observations. As the
time series exhibit different lengths, the splitting rules defined in Sect. 3.6.2 were
adopted.
Table 3.11 presents the classification by the protocols and Table 3.12 presents
their forecasting performance using the MAPE as the error measure. Table 3.11
shows that the V(0)V(2) protocol, and consequently both decision trees, select 119
time series as non-trended, while the three protocols diagnose different types of
trend. Tree B favours the DTM model, while Tree A detects the DTM and LGM
models almost equally. The V(0)V(2) protocol, by construction, selects only the
LGM model for trended series.
Table 3.12 shows a forecasting accuracy comparison between the three uni-
versal methods and the method selection protocols.
Comparing the accuracy of the universal application of forecasting methods
among themselves, SES generates the most accurate forecasts for the horizons up
to 3-periods ahead, while for the longer horizons damped Holt’s is more accurate
than SES.
It is evident that Tree B outperforms the original GM procedure for all fore-
casting horizons. This confirms the results of the simulation experiment discussed
in the previous section of this chapter. Tree B is also more accurate than Tree A for
all horizons, with Tree A performing less well than the GM procedure for a
horizon of four periods or more. Overall, Tree B is the best performing method or
protocol. This result, shown above for Mean Absolute Percentage Error (MAPE) is
also confirmed for Mean Absolute Error (MAE).
The lack of a single universal method that dominates over all forecasting
horizons means that Tree B’s outperformance over SES and damped Holt’s is
greatest for different horizons: a gain of 0.63% over damped Holt’s for a one-step
ahead error, and a gain of 1.54% for a six-step ahead error.
variance procedure and Decision Tree B exhibited the same performance in terms
of number of series detected (only 2% of time series were detected as strong
trended). Furthermore, both the SVC protocol and Decision Tree A have exhibited
exactly the same performance and detected 38% of time series as damped trended
and 62% as strong trended.
Comparison of the universal application of single methods (see Table 3.14)
shows a clear advantage to the damped Holt’s method over SES for all forecast
horizons. It also shows a slight gain over Holt’s method (except for six-step ahead
forecasts, where the improvement is more marked).
None of the protocols perform badly, because none of them select SES, for
which there may be a strong penalty in forecast accuracy. There is little to choose
between the protocols (see Table 3.14), with the Tree B and the GM protocol
giving exactly the same result, because neither selects SES for any series. There is
a small gain in accuracy of Tree B over Tree A, widening as the forecast horizon
lengthens. Tree B is again the best protocol.
Unsurprisingly, these results show that if the damped Holt’s method is the best
method for almost all series, then there is little to be gained over universal
application of that method by employing a method selection protocol. However,
should any of the series have been allocated to SES by default, then the use of
Tree B, or one of the other protocols, would have been of some benefit.
Goodrich (2001) noted that in the M3-competition data set, there were no weekly
series, ‘‘despite their tremendous importance to business, e.g. for materials man-
agement and inventory control’’, (Goodrich 2001: 561). Therefore, a weekly data
set is included in this empirical study and the performance of the protocols will be
analysis and reported (Table 3.16).
Weekly data set contains 156 non-seasonal and non-slow-moving time series.
All series are the same length and they contain 125 observations each. Table 3.15
summarises the classification findings of the protocols.
In contrast to the previous dataset, this dataset is dominated by series for which
SES is the best method amongst the three smoothing methods. The SVC protocol
is of little relevance here, as it does not include SES in its pool of methods. The
performance of the protocols and universal application of forecasting methods is
presented in Table 3.16.
In this particular data set, the SES is the best performing method from 2- up to
6-periods-ahead forecasting horizon confirming that the trees have selected the
non-trended model correctly. Again, none of the protocols perform badly (except
the SVC), as they never or rarely select Holt’s method, for which the accuracy
penalty is greatest.
Unsurprisingly, these results show if Single Exponential Smoothing is the best
method for almost all the series, then there is little to be gained over universal
application of that method by employing a method selection protocol. However,
should any of the series have been allocated to Holt’s method by default, then the
use of Tree B, or one of the other protocols, would have been of some benefit.
3.8 Conclusions
Evidence is lacking on the comparison of decision trees for model selection with
the universal application of damped Holt’s method. The M3 competition shows
that this method is difficult to beat. This chapter has presented results comparing
decision trees with the damped Holt’s method, as an ,encompassing approach’.
This serves as a base for further studies, which should include other approaches
such as prediction validation and information criteria, discussed in Sect 3.3.
The simulation study showed that the Steady State Model may be detected more
accurately by the V(0)V(2) protocol (and, hence, by both decision trees) than by
the original GM protocol, although the performance of all protocols declines as the
length of series increases. Tree B’s forecasting performance is better than Tree A’s
in this case, because when it fails to recognise that data is non-trended, it is more
likely to allocate the series to the damped Holt’s method than Holt’s linear method.
The simulation study also showed that the Linear Growth Model (LGM) may be
detected more accurately by the V(0)V(2) protocol than the GM protocol. In this
case, Tree A performs better than Tree B in the sense of recognising more LGM
series. The best protocol, in terms of forecasting accuracy, is the V(0)V(2) method,
as it does not misclassify trended series as damped trend. There is little to choose
between Tree A and Tree B forecast accuracy, with Tree A having a slight edge.
For damped trend series, the simulation study showed that the original GM pro-
tocol classified the most series correctly, with Tree B being the next best classi-
fication method. However, the difference in forecasting performance is almost
negligible.
Empirical analysis of yearly data from the M3 competition shows that it is
possible for a protocol to give greater accuracy than universal application of the
damped Holt’s method. The weekly series confirm a small gain in accuracy for
Tree B. However, the telecommunications dataset shows almost identical accuracy
results between Tree B and damped Holt’s, as this method is diagnosed for almost
all series by Tree B. The decision tree produces no accuracy benefit in this case,
but neither does it yield any deterioration in performance, thus demonstrating its
robustness.
The M3 yearly data also show Tree B identifying series as damped trend much
more frequently than Tree A. For this dataset, Tree B gives greater forecast
accuracy than both Tree A and the original GM protocol. Although the benefits are
not large, they are consistent across forecasting horizons, with a general tendency
for Tree B to have greater accuracy over longer horizons. For the telecommuni-
cations data, Tree B and the GM protocol give the same results, which are better
than Tree A, particularly for longer horizons. The most natural explanation of this
difference is the tendency of Tree A to mis-specify damped trend as linear trend.
This tendency was identified in the simulation study, and it was shown that it
penalises forecast accuracy. These conclusions are restricted to non-seasonal
datasets. More research is needed on extending the approach to seasonal series,
including a detailed analysis of the Holt-Winters method.
86 N. N. Atanackov and J. E. Boylan
Overall, this study shows that Tree B is a promising approach for diagnosing
trend in non–seasonal time series. It can be implemented quite straightforwardly in
forecasting applications for service parts. As it is simple to understand and apply,
and produces robust accuracy results, this decision tree should be considered as a
potential approach to forecasting method selection along with other more
sophisticated approaches.
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3 Decision Trees for Forecasting Trended Demand 87
In the last 20 years companies have always paid great attention on managing
demand variability. Demand fluctuations are due to several reasons: quick changes
in the final customer’s preferences and taste are a common cause of demand
variability (e.g., in the fashion industry demand for a given color can change
dramatically from year to year). Marketing activities may also lead demand to
suddenly change e.g., when promotional activities are conducted due to the high
elasticity of demand to price. Competitors can also be a source of variability, since
their behavior can influence how the demand distributes on each single company
serving a specific market. The supply chain structure is also a significant cause of
demand unsteadiness: the bullwhip effect (Lee et al. 1997) is a common phe-
nomenon in different industrial contexts, leading to an increase in the variability of
the demand over supply chain stages.
A vast amount of the literature has addressed the issue of designing managerial
systems capable of coping with demand variability. This has been done by
focusing on different leverages: from demand forecasting, aiming at increasing the
capability of companies to understand variability, to production planning, trying to
design efficient planning systems, capable of reacting towards sudden changes in
the final demand, to inventory management, in order to manage the complex trade-
off between inventory cost and service level, and so on.
E. Bartezzaghi
Department of Management, Economics and Industrial Engineering,
Politecnico di Milano, Milano, Italy
e-mail: emilio.bartezzaghi@polimi.it
M. Kalchschmidt (&)
Department of Economics and Technology Management,
Università degli Studi di Bergamo, Bergamo, Italy
e-mail: matteo.kalchschmidt@unibg.it
This issue is common to all industrial contexts; however, a rather peculiar and
complex situation is faced in the case of spare parts. The problem of managing
spare parts demand is relevant for many reasons: first of all it influences the final
product business since it affects post sale service quality. Moreover, it is a relevant
business as the market is captive, thus very profitable and so firms have to pay
relevant attention towards this issue. However, it is a very difficult business to
cope with, since requirements are usually very dispersed over time and demand
uncertainty is frequently very high.
Spare parts, in fact, often show very sporadic demand patterns for long periods
of their life time. This is the case, for example, of service items that have to be
stored for years as long as repair service has to be guaranteed even for products
that have reached the end of their market life. Spare parts demand often tend to be
highly variable and sporadic showing frequently a very peculiar pattern called
lumpy demand.
Lumpy demand can be defined as (Bartezzaghi et al. 1999):
• variable, therefore characterized by relevant fluctuations (Wemmerlöv and
Whybark 1984; Wemmerlöv 1986; Ho 1995; Syntetos and Boylan 2005);
• sporadic, as historical series are characterized by many days with no demand at
all (Ward 1978; Williams 1982; Fildes and Beard 1992; Vereecke and Verst-
raeten 1994; Syntetos and Boylan 2005);
• nervous, thus leading to show differences between successive demand obser-
vation, so implying that cross time correlation is low (Wemmerlöv and Whybark
1984; Ho 1995; Bartezzaghi and Verganti 1995).
Managing inventories when demand is lumpy is thus a complex issue since
companies have to cope with both a sporadic pattern, that usually induces high
inventory investments, and highly variable order size, that make it difficult to
estimate inventory levels and may affect service levels. For this reason, companies
facing lumpy demand often experience both high inventory levels and unsatis-
factory service levels at the same time.
Lumpiness may emerge as the consequence of different structural character-
istics of the market. In particular, we may refer to the following main factors
(Bartezzaghi et al. 1999):
• low number of customers in the market. Fewer customers usually induce spo-
radic requests for the product unit and, therefore, demand lumpiness increases;
• high heterogeneity of customers. Heterogeneous requests occur when the
potential market consists of customers with considerably different sizes or
buying behaviors (i.e. customers that order for very different lot sizes or with
different frequencies); thus the higher the heterogeneity of customers, the higher
the demand lumpiness;
• low frequency of customers requests. The higher the frequency of requests from
a customer, the higher the number of different customers that ask for the unit in
a given time bracket. Thus lumpiness increases as the frequency of each cus-
tomer’s purchase decreases;
4 The Impact of Aggregation Level on Lumpy Demand Management 91
(1) One shall define the market he/she tries to forecast; e.g., one retailer might
want to forecast demand at the single store level, while a manufacturer might
be interested in the demand for the overall region or country; clearly the
former forecasting problem is harder to tackle than the latter;
(2) One shall define the product the demand refers to; e.g., for a given retailer it
might be fairly difficult to predict the demand for a given product at the style-
color-size-packaging level, whereas forecasting the total turnover for a given
product category might not be that hard (Wacker and Lummus 2002);
(3) Finally one needs to define the time frame of the forecasting problem, i.e., one
shall define the time bucket and the forecasting horizon; indeed forecasting
demand at the day level is much more complex than forecasting total yearly
demand.
The choice of the aggregation level is important since according to the specific
aggregation level chosen, demand variability may show specific peculiarities and
thus different techniques may apply, thus affecting forecasting and inventory
performances.
In the remainder of this work we will refer to the previous three dimensions as
the level of aggregation of the forecasting problem. The smaller the market, the
more detailed the definition of the product and the smaller the time bucket, the
more the forecasting problem is detailed.
This work focuses on the impact of the aggregation level of data on inventory
performance and we address in particular the specific case of lumpy demand. In
fact limited contributions can be found regarding how the aggregation level may
influence lumpy demand.
The remainder of this paper is thus structured as follows. In the next section the
level of aggregation of demand will be described and literature contributions on
this issue will be summarized. Then specific research objectives and methodology
will be described. In the last two sections, empirical results will be described, a
proper discussion of results will be provided and future research objectives will be
highlighted.
Data Forecast
aggregation aggregation
Demand data
No data No forecast
Forecast
aggregation/ aggregation/ Forecast
evaluation
disaggregation disaggregation
Data Forecast
disaggregation disaggregation
Forecast
Data aggregation Forecast aggregation
process evaluation process process
However, the level of aggregation at which the forecast has to be used (at thus
provided) is not necessarily the same of the level of aggregation at which the
forecast is evaluated. In particular, during the forecasting process, we face the
problem of aggregation in two different situation. First, when information (i.e.,
past demand data) is collected, one has to choose at which level of aggregation
these data should be used (we refer to this as the data aggregation process). Based
on what companies choose here, different forecasting methods may then be
selected, based on the characteristics of variability the demand shows at that level
of aggregation (e.g., if monthly data is used then seasonality may be an important
variability component to be considered; on the contrary if the same data is used at
the yearly level seasonality becomes irrelevant, at least for the forecast evaluation
process). Second, when the forecast has been evaluated it may need to be
aggregated or disaggregated in order to provide the final forecast at the required
aggregation level that the decision making process needs (e.g., if a market forecast
is needed for budget purposes, if we evaluate forecast at customer level we then
need to aggregate all these forecasts by simply summing up them). We refer to this
as the forecast aggregation process. Figure 4.1 summarizes these different situa-
tions. In this work we will focus on the data aggregation process.
As Fig. 4.1 exemplifies, given a certain aggregation level at which a forecast is
needed, one can decide to obtain this forecast though a more aggregate forecast
(thus a disaggregation process is needed to provide the final forecast), through a
more disaggregated process (thus an aggregation is needed) or without any
aggregation or disaggregation. Forecast evaluation can be performed at different
data aggregation levels. Data in fact can be aggregated before evaluating the
forecast or disaggregated (e.g., in the retail industry frequently companies estimate
the demand in different ways according to whether data collected on demand refer
to promotional periods or not).
Based on these options companies may then structure their forecasting process
differently. For example, one typical solution is when the forecast is evaluated at
the same level of aggregation at which the forecast is used. In this situation, based
on the aggregation level requested from the decision making process, the fore-
casting approach is selected according to data or information available, and no
94 E. Bartezzaghi and M. Kalchschmidt
1976; Barnea and Lakonishok 1980; Fliedner 1999) take a more contingent
approach and show that the choice between the aggregate and detailed approach
depends on the correlation among time series. Zotteri and Kalchschmidt (2007)
analytically demonstrate that in fact aggregation is preferable only under certain
circumstances (i.e., high demand variability, few periods of demand, etc.).
Limited contributions can, however, be found regarding aggregation in the case
of lumpy demand. Specifically, several of the mentioned contributions considered
frequently the case of stationary and continuous demand. Unfortunately this is not
always the case: spare parts usually show a lumpy pattern and it is not completely
clear whether literature findings still hold here.
Similarly, contributions on the aggregation level selection have mainly focused
on forecasting, i.e., the impact of aggregation on forecasting accuracy. Limited
contributions have consider simultaneously the impact on forecasting and inven-
tory management systems. In fact, literature on lumpy demand management has
argued and, sometimes, proved that designing an integrated forecasting and
inventory management system may be much more beneficial than focusing on just
one of the two (Kalchschmidt et al. 2003). In this situation the forecasting method
applied has to focus on estimating demand characteristics that the chosen inven-
tory system needs to define reorder politics.
This work aims at providing a better understanding of how aggregation may
influence inventory performance when demand is lumpy. In particular here
attention is devoted to the case of aggregation over time (temporal aggregation).
This choice is due to the fact that limited contributions can be found on this
specific issue (also in the stationary and stable demand case). As a matter of fact,
the vast amount of contributions on this topic usually refer to aggregation over
product and over market dimensions (see previously mentioned contributions),
while only limited contributions can be found regarding temporal aggregation for
non-lumpy demand (some contributions can be found in Johnston and Harrison
1986; Snyder et al. 1999; Dekker et al. 2004) and very few specific to the lumpy
demand case (Nikolopoulos et al. 2009). For all these reasons, in the reminder of
the paper only temporal aggregation will be considered.
The goal of this work is to study whether temporal aggregation of lumpy demand
may be beneficial in terms of impact on inventory performances. Specifically, the
objectives are:
(1) Analyze the impact of temporal aggregation level in a specific situation,
namely spare parts demand.
(2) Evaluate the impact of demand characteristics (e.g., lumpiness) on the choice
of the proper level of aggregation.
In order to achieve these goals a simulation analysis based on real demand data
has been considered. Demand data has been collected from the Spare Parts
96 E. Bartezzaghi and M. Kalchschmidt
values for these parameters. Even if differences arise when parameters change,
these do not affect significantly the results of our analyses. For this reason and
for briefness sake we omitted these analyses here.
(2) The safety stock is defined according to the actual demand variability and the
desired service level. Specifically we simulated different scenarios according
to different average service levels i.e., 94% (the average service level the
company was achieving) and 99.1% (the desired service level the company
was aiming to have). We considered these two levels of performance since
they represent what the company considered as reference. For briefness sake
we will show the results only for the former case.
(3) The reorder model considered is an order-up-to system with daily revisions of
inventory levels; backlog is allowed.
(4) Deliveries from suppliers are assumed constant and equal to 20 days for all
items. The company based its own reorder politics on this specific value. We
argue that according to the specific lead time suppliers provide, the impact of
the aggregation level on inventory performances may change. However, we
claim that the considerations we draw from our analyses are not affected
explicitly from this specific assumption. We discuss this issue in deeper detail
in the conclusions.
Each day of the simulation we update model parameters and evaluate inventory
performance in terms of inventory levels and service level (i.e., served quantity
compared to actual demand). If inventory is not enough to fulfill daily demand a
backlog is accounted and demand is served as soon as inventory is available.
Simulations were run according to five different aggregation levels of demand
data. Specifically we considered aggregation on a 1 day level (1d, data as it is), a
2 days level (2d, demand is aggregated between 2 subsequent days), 3 days level
(3d), 10 days level (10d) and 30 days level (30d). Other intermediate aggregation
levels were run but here they are omitted for sake of brevity. Since the perfor-
mances of the systems under investigation are based on two objectives (service
level and inventory level), in order to compare the different scenarios we run all
simulations so to guarantee a 94% service level on the average of the test period.
We then can directly compare inventory levels to identify the impact of the data
aggregation process.
Figure 4.2 shows the average inventory level of the considered items on the testing
period for different aggregation levels.
As it can be noted, the average inventory level required to guarantee a 94%
average service level reduces as we aggregate demand data. The extent to which
inventories benefit from aggregation is impressive (in particular when comparing
the more detailed levels with the more aggregate ones) and the benefit of further
aggregation tends to reduce on higher horizons. In order to verify that these
98 E. Bartezzaghi and M. Kalchschmidt
average results were consistent at SKUs level (and to avoid eventual bias due to
few peculiar cases, e.g., high volume skus) we ran nonparametric tests on the
equality of average inventories between the different simulation runs (we based
our analyses on Friedman’s test1). All tests were significant at 0.99 level, thus we
can claim than on a relevant portion of our SKUs, aggregating demand improves
inventory performances.
Even if on average the temporal aggregation seems to pay off, a more detailed
analysis showed that this is not true for all items. Table 4.2, in fact, highlights that
some items don’t benefit from aggregation but, on the contrary, face a worsening
of the inventory level. As we can see, among the considered SKUs, on average
almost 22% show worse performance when demand is aggregated, while almost
9% on average are not affected by the aggregation level.
The identified phenomenon seems to apply differently on the items considered,
thus we take a contingent approach to identify what are the key drivers that
influence the optimal aggregation level. In order to identify discriminant contin-
gent factors we ran multiple comparisons among three groups of items (namely
those for which aggregation improves performance, those were aggregation is
indifferent and those where aggregation lead to worse performance) for all the
considered aggregation levels. T tests on the equality of means were run among the
different groups on the following variables2:
• Average demand;
• Standard deviation of demand;
• Coefficient of variation of demand;
• Asymmetry of demand;
1
The Friedman test is the nonparametric equivalent of a one-sample repeated measures design
or a two-way analysis of variance with one observation per cell. Friedman tests the null
hypothesis that k related variables come from the same population. For each case, the k variables
are ranked from 1 to k. The test statistic is based on these ranks.
2
For space sake we omit all statistical analyses. All contingencies have been evaluated at daily
level since this was the most detailed level available.
4 The Impact of Aggregation Level on Lumpy Demand Management 99
Table 4.2 Distribution of SKUs for different aggregation levels, classified according to whether
they improve performance by aggregating demand, stay the same, or worsen
From 1 to 2 days From 2 to 3 days From 3 to 10 days From 10 to 30 days
Improvement 817 529 711 517
Indifference 59 80 76 105
Worsening 50 317 139 304
Total 926 926 926 926
CV 2
Lumpiness ¼
l LT
Where InvLev [i]k is the inventory level for item k at aggregation level i.
Based on these clusters we evaluated the average inventory reduction (AIR)
between the different aggregation levels within each cluster. Table 4.4 summarizes
this comparison.
These results show again that improvements in inventory performance are
widespread in the sample, and thus they confirm our previous evidence. Quite
interestingly, however the improvements obtained through a more aggregated
forecast are significant when the demand is not highly sporadic. In fact, when
sporadic behavior is limited, all reductions are significant (based on pair com-
parisons at SKU level). Significant benefits may occur also when the demand is
highly sporadic but only if the variability is limited; in fact, in most of the com-
parisons there is a significant reduction even if in one case a significant increase
can also be seen. When the demand is highly sporadic and the demand size is
highly variable, no significant improvements can be found; quite interestingly even
if on average some reductions can still be found here, they are not statistically
significant.
These results suggest that aggregating demand seems to be a reliable approach
when demand is lumpy. However when demand sporadic behavior and variability
are extremely high (i.e. HVS skus), this approach is not helpful and can in some
cases lead to worse performances. This result eventually suggests that in this last
situation, demand forecasting can be highly inefficient and one should design
inventory management solutions based on other approaches. In our case it should
also be noted that this situation affects only a limited part of the inventory problem
we are addressing. In fact these items account for no more than 10% of the
considered SKUs that are responsible for less than 2% of demand volumes and less
than 1% of the average inventory level.
Table 4.4 Average percentage inventory reduction for each cluster between the different
aggregation levels (AIR[i-j]: average inventory reduction with i days aggregation level compared
to j days aggregation level; * p \ 0.05, based on pair comparison of each SKU)
Demand size variability Interarrival
Sporadic Highly sporadic
High AIR[2–1]: -51.1%* AIR[2–1]: -20.0%
AIR[3–2]: -33.6%* AIR[3–2]: -10.9%
AIR[10–3]: -47.1%* AIR[10–3]: +4.1%
AIR[30–10]: -21.7%* AIR[30–10]: +9.5%*
Low AIR[2–1]: -39.5%* AIR[2–1]: -44.3%*
AIR[3–2]: -23.8%* AIR[3–2]: -12.5%
AIR[10–3]: -37.5%* AIR[10–3]: -19.8%*
AIR[30–10]: -44.3%* AIR[30–10]: +18.5%*
102 E. Bartezzaghi and M. Kalchschmidt
4.5 Conclusions
This work provides evidence that the temporal aggregation of data may be ben-
eficial in spare parts inventory management and forecasting. The presented results
show a clear effect of the aggregation of data over inventory performance, thus
they emphasize the importance of paying proper attention in defining the aggre-
gation level at which demand is managed. This consideration is coherent with
previous results on this topic in the case of non-lumpy demand (see literature
review for details) and provides evidence that also when demand is sporadic or
lumpy, this issue has to be taken in high consideration.
A second contribution relates to the contingent analysis of the impacts of
aggregation. The analyses show that even if the impact is usually significant, the
characteristics of the demand significantly influence the possibility of improving
inventory performance by leveraging on temporal aggregation. In particular,
results provide evidence that when the demand is sporadic, impressive inventory
reduction can be gained by leveraging on data aggregation. However when the
demand occurrence is extremely low (in our case less than four orders in one year),
leveraging on data aggregation may be effective if variability in demand size is not
extremely high. On the contrary, if both sporadic nature of demand and variability
of demand size are extremely high, impacts can be limited. This last situation,
however, is limited to few cases in our sample (almost 10% of the considered
SKUs). This result is coherent with other contributions in the field, claiming that
when demand lumpiness is too high, companies should not invest too much in
forecasting those patterns due to the extreme uncertainty of the situation.
This work also highlights some interesting issues that future studies should
devote attention to. First of all, it would be important to define criteria that can
provide companies with a clear a priori determination of the aggregation level they
should adopt. In fact this work, provides some guidelines for companies willing to
understand whether they should aggregate data or not. Such a contribution is
important for managers since it can provide them with some guidelines to better
manage their spare parts inventories.
We would also like to draw attention to some limitations of this work. First of
all, we considered a specific situation in terms of data (available from one single
company), thus one can doubt about the possibility to generalize these results. We
argue, however, that these results are at some extent of general validity since even
if the data come from a single company they represent a typical situation faced in
the spare parts business. Indeed future studies should consider other data sets from
other companies to verify these results. A second issue relates to the specific
forecasting technique that we adopted to manage demand. This work focuses on
one specific forecasting method (i.e. Syntetos and Boylan’s method). It would be
important to evaluate to which extent these results are method-specific and thus
how the selection of the aggregation level should take the forecasting method
adopted into account. We consider that some specificities of the applied method
may have an impact since different methods rely on the estimation of different
4 The Impact of Aggregation Level on Lumpy Demand Management 103
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Chapter 5
Bayesian Forecasting of Spare
Parts Using Simulation
5.1 Introduction
D. F. Muñoz (&)
Instituto Tecnológico Autónomo de México, Mexico, Mexico
e-mail: davidm@itam.mx
D. F. Muñoz
Stanford University, Stanford, CA, USA
e-mail: dkedmun@stanford.edu
not available). We propose two main steps for forecasting using simulation
(see Fig. 5.1). The first step consists in the assessment of parameter uncertainty
using available data (x) and a prior density (probability in the discrete case)
function p(h), so that parameter uncertainty is assessed through a resulting pos-
terior density function PðhjXÞ. In the second step we use the simulation model and
the posterior density to estimate the parameters related to the forecast of the
response variable (W). We now explain in detail this approach.
p (θ x )
forecasting using simulation Bayesian Estimation
Based on:
p (θ )
L (x θ )
Step II
Simulation Experiments
Based on:
p (θ x )
W = g (Y (s ) : 0 ≤ s ≤ T ; Θ )
r ( x ) = E [W X = x ]
[ q β / 2 (x ), q1 − β / 2 ( x )]
108 D. F. Muñoz and D. F. Muñoz
prior density function for the vector of parameters H, then the posterior (given the
data x) density function of H becomes
pðhÞLðxjhÞ
pðhj xÞ ¼ R ; ð1Þ
P pðhÞLðxjhÞdh
for 0 \ a \ 1. In order to asses the uncertainty on the point forecast r(x), a-quantiles
are useful, since for 0 \ b \ 1, a (1 - b)100% prediction interval in the form of
[qb/2(x), q1-b/2(x)] is usually constructed. This interval is called
a (1 - b)100%
prediction interval because p qb=2 ðxÞ W q1 b=2ðxÞjX ¼ 1 b, provided
FðwjxÞ is continuous at qb/2(x) and q1-b/2(x).
Quantiles are also useful to compute reorder points in inventory management.
In this particular application, when W is the demand during the lead time, the
a-quantile qa(x) can be interpreted as the reorder point for a 100a% (type-I) service
level (see e.g., Chopra and Meindl 2004).
When analytical expressions for the forecasting parameters of Eqs. 3 and 4 cannot
be obtained (or they are too complicated), simulation can be applied to estimate
these parameters. In Fig. 5.2 we illustrate a first algorithm, called posterior sam-
pling (PS), to estimate the required forecasting parameters using simulation. Under
PS we sample from the posterior density PðhjXÞ to obtain i.i.d. observations of the
response variable W, that allow us to estimate the point forecast r(x) by ^r ð xÞ; and
the a-quantile qa(x) by ^qa ð xÞ (as defined in the algorithm of Fig. 5.2). As is well
known, these estimators are consistent, which means that they approach the
where h 2 P is the fixed value of H. On the other hand, under our Bayesian
approach, the variance of the response variable is
r2W ¼ E W 2 jX ¼ x ðE½W jX ¼ xÞ2 ; ð8Þ
In order to implement the algorithm of Fig. 5.2, a valid method to generate samples
from the posterior density Pðhj XÞ is required, which can be available if the family
of distributions corresponding to Pðhj XÞ has been identified. However, in many
situations it is very difficult to obtain an analytic expression that allows us to
identify the family of distributions corresponding to the posterior density Pðhj XÞ,
and in this case we can apply a technique called Markov chain Monte Carlo
(MCMC), which does not require an algorithm to generate samples from Pðhj XÞ.
The algorithm of Fig. 5.3 is an implementation of MCMC that is called the
independence sampler, because the generation of a sample from an auxiliary
density q(h) (where q(h) [ 0 whenever p(h) [ 0) is required. As mentioned by
several authors (e.g., Asmussen and Glynn 2007), the algorithm of Fig. 5.3
performs better when q(h) is closer to the posterior density p(h|x).
Note that (although it is not required to compute the point estimators), we are
dividing the number of replications m into b batches of length mb. This is because
we are suggesting to use the method of batch means in order to produce asymptotic
confidence intervals for the point estimators. Although a CLT for the point
estimators can be established (under mild assumptions), we are not trying to
estimate the asymptotic variance of the point estimators, because finding consis-
tent estimators may be a difficult task and batch means may be also required
(see e.g., Song and Chih 2008). Instead we can choose a number of batches
b between 5 and 20 (as suggested by Schmeiser 1982), so that under suitable
In this section, we present two simple models that can be used to forecast the
demand for spare parts, and will help us illustrate how the techniques described in
the previous section can be applied. In both models we assume that failures occur
randomly, and the only difference is how the sample data is available, as we
explain below.
Under both models we have k machines, each has a critical part and failures
occur independently at the same rate H 2 P = (0, ?). There is uncertainty on the
failure rate H, so that p(h) is a prior density on H, and for i = 1,…, k,
Ni = {Ni(t):t C 0; H} denotes the failure process for component i, which are
assumed to be conditionally independent (see e.g., Chung 1974 for a definition)
relative to H, and
eHs ðHsÞ j
P½Ni ðt þ sÞ Ni ðtÞ ¼ jjH; Ni ðuÞ; 0 u t ¼
j!
j ¼ 0; 1; . . .; t; s 0; i ¼ 1; . . .; k
Under this model, the available sample information corresponds to the time
between failures of every part. Thus, the sample data x = (x1, x2,…, xn) comes
from a random sample X = (X1, X2,…, Xn) of the exponential density
hy
he ; y [ 0;
f ðyjhÞ ¼
0; otherwise,
P
n
n1
h xi
xi h e i¼1
i¼1
pðhj xÞ ¼ ; ð13Þ
ðn 1Þ!
P
which corresponds to the Gamma n; ni¼1 xi distribution, where, for b1, b2 [ 0,
Gamma(b1, b2) denotes a Gamma distribution with expectation b1b-1
2 .
114 D. F. Muñoz and D. F. Muñoz
X
k
W¼ Ni ðt0 Þ; ð14Þ
i¼1
given [X = x].
In order to obtain an analytical expression for the point forecast r(x), we can
apply Proposition 1 of Muñoz and Muñoz (2008), by taking into account that
r1(h) = E[W|H = h] = kt0h, so that
Z1 !1
X
n
r ð xÞ ¼ r1 ðhÞpðhj xÞdh ¼ kt0 n xi : ð15Þ
i¼1
0
For the case where t0 corresponds to the lead time for an order, the reorder
point for a 100a% service level is given by qa(x), as defined in Eq. 4. In this
case, we do not have a simple analytical expression for qa(x), for which using
the algorithm of Fig. 5.2 would be appropriate to estimate qa(x) using
simulation.
Now, let us suppose that we have Q spare parts at the beginning of a period of
length t0. Two performance measures of interest in this case are the type-I service
level (100a1%) and the type-II service level (100a2%), where
Let us suppose that times between failures are not recorded, however for each
period i = 1, 2,…, p, we record the number ki of machines in operation during
period i, and the number of failures
per machine during period i. In this case, the
sample data takes the form of x ¼ x11 ; . . .; x1k1 ; . . .; xp1 ; . . .; xpkp ; where xij is the
number of failures for the j-th machine Pp in operation during period i,
j = 1,…, ki; i = 1,…, p (in this case n ¼ i¼1 ki ).
In order to simplify our notation, we assume that each period has a length of 1
time unit, which means that the failure rate is expressed in the appropriate scale
(failures per time unit). Since the failure processes are assumed to be conditionally
5 Bayesian Forecasting of Spare Parts Using Simulation 115
eh hy
f ð y j hÞ ¼ ; y ¼ 0; 1; . . .;
y!
and it follows from (2) that the likelihood function is given by:
P
p P
p P
ki
h ki xij
e i¼1h i¼1 j¼1
LðxjhÞ ¼ : ð17Þ
‘ Qi
p k
xij !
i¼1 j¼1
An appropriate non-informative prior for the Poisson model (see e.g., Bernardo
and Smith 2000) is p(h) = h-1/2, so that it follows from (1) and (17) that
P
p P
ki P
p P
ki
xij þ1=2 xij 1=2
n i¼1 j¼1
h i¼1 j¼1
enh
pðhj xÞ ¼ ! ; ð18Þ
P
p P
ki
C xij þ 1=2
i¼1 j¼1
P P
p ki
which corresponds to the Gamma i¼1 j¼1 x ij þ 1=2; n distribution.
As in the previous model, we are interested in the number of failed components
during a time period of length t0, so that the response we want to forecast is the
same as in Eq. 14. By taking into account that under this model the posterior
density p pðhjxÞ now takes the form of Eq. 18, we can proceed as in Eq. 15 to
obtain an analytical expression for the point forecast r(x):
Z1 !
X
p X
ki
1
r ð xÞ ¼ r1 ðhÞpðhj xÞdh ¼ n kt0 xij þ 1=2 : ð19Þ
i¼1 j¼1
0
Also in this case, the use of the algorithm of Fig. 5.2 would be helpful to
estimate a reorder point qa(x), or a service level as defined in (15).
We would like to mention that in the two simple models presented we do not
require the use of the algorithm of Fig. 5.3, because we can recognize the family of
distributions corresponding to the posterior density (in both cases is the Gamma
family). However, if we do not use the objective (non-informative) prior in these
models, it may be the case that the family of distributions corresponding to the
posterior density p(h|x) may not be easily recognized, and the algorithm of Fig. 5.3
will be useful, as we will illustrate in the next section.
116 D. F. Muñoz and D. F. Muñoz
In this section we apply the model that uses censored data to forecast the demand
of clutches (from a particular model), that is experienced by a car dealer. We will
illustrate the use of the methods described in previous sections using demand data
obtained from a car and spare parts dealer in Mexico.
Car model A is the ongoing model of a very successful line, which continues to be
produced and offered to this day. Table 5.1 presents the data available for this
model, which include car and clutch demand for every month of 2008. To apply
the model with censored data, we assume that customers that acquire their cars at a
particular dealer also purchase spare parts from it. Thus, the number ki of cars in
operation at period i is equal to the amount of model A cars sold by the dealer up
to that period of time, so that ki = ki-1 ? Si, where Si is the amount of cars that
were sold in period i. As we can see from Table 5.1, model A began the year 2008
with 337 cars in operation (according to sales from previous years).
We are interested in forecasting the demand W, as given in Eq. 14, when
t0 = 0.5 and k = 500, since we are assuming that the lead time for an order of
clutches is approximately 15 days, and there are 500 cars in operation during the
forecasting period.
Using (19) we computed the point forecast for the demand of clutches during
the lead time, and obtained r(x) = 1.82701. Then, we applied the algorithm of
Table 5.1 Car sales and Month (i) Car Cars in Clutch Demand
clutch demand for Model Pki
Sales (Si) Operation (ki)
A in 2008 j¼1 xij
Fig. 5.2 in order to estimate q0.9(x), the reorder point for a 90% service level.
The results for the estimation of r(x) and q0.9(x) using m = 106 replications are
summarized in Table 5.2. In the case of r(x) the halfwidth indicates that ^r ð xÞ has
an error that is below 0.00228, although we already know that r ð xÞ and ^r ð xÞ are
equal within 4 decimal places.
In the case of the estimation of q0.9(x), we see from Table 5.2 that the halfwidth
is 0, which is not surprising if we take into account that in our application W is a
discrete random variable, and its c.d.f. is piecewise constant. Note also that
P½W qa ðxÞjX ¼ x is not necessarily equal to a (as is the case when W is a
continuous random variable), for which it would be of interest to estimate the true
service level corresponding to q0.9(x).
In Table 5.3 we report the estimated type-I service level (cumulative proba-
bility) for different values of Q, obtained from applying the PS algorithm with
m = 106 replications. From this table we see that the true service level for
q0.9(x) = 4 is approximately 95.74%.
As we can observe from Tables 5.2 and 5.3, the number of replications
m = 106 was large enough to obtain accurate estimators for all the estimated
parameters, which may not be true when the number of replications is small. To
illustrate how the number of replications affect coverage and halfwidths, we
repeated the estimation experiment M = 1000 times for different values of m, and
computed the empirical coverage, average halfwidth and standard deviation, mean
square error and bias for each set of replications. Since theoretical (true) values of
r(x) and q0.9(x) are required for these computations, we used the values of
r(x) = 1.82701 and q0.9(x) = 4 obtained before. The results of these experiments
are presented in Table 5.4.
Table 5.4 Performance of the PS algorithm from M=1000 experiments and different numbers of
replications m
Posterior Estimated Empirical Halfwidth Mean square Bias
sampling parameter coverage error
Average St.
dev.
m = 100 r(x) 0.878 0.2269 0.0187 0.0211 0.0064
q0.9(x) 0.679 0.5585 0.2467 0.3400 -0.3340
m = 400 r(x) 0.898 0.1139 0.0045 0.0047 0.0022
q0.9(x) 0.869 0.336 0.2348 0.1310 -0.1310
m = 1600 r(x) 0.904 0.0570 0.0011 0.0012 0.0003
q0.9(x) 0.987 0.1295 0.2191 0.0130 -0.0130
As can be observed from the results presented in Table 5.4, the empirical
coverage approaches the nominal 90% as the number of replications increases, and
for m = 1600 we already obtain an acceptable coverage. Also note that for
m = 1600 we obtained an over-coverage in the estimation of q0.9(x), which is
explained by the fact that the response variable W is discrete. It is worth men-
tioning that increasing the number of replications also increase the accuracy of the
point estimation, and consistently reduce the halfwidth, mean square error and
bias.
As explained in Sect. 5.3.2, when using the non-informative (Gamma) prior for
the model with censored data, we were able to identify that the posterior distri-
bution is also Gamma, for which we can apply the PS algorithm for the estimation
of r(x) and qa(x). In this section we illustrate how MCMC can be applied when
using a subjective prior for which we cannot identify the family of distributions
corresponding to the posterior density.
We considered a uniform prior on [a, b], where the values of a and b can be
supplied by the user. An analytical expression for the posterior density using this
prior is not easy to obtain, so that the use of the MCMC algorithm of Fig. 5.3 is
justified. For the implementation of this algorithm we considered a sampling
distribution q(h) that was obtained using the procedure suggested in Sect. 5.2.2.3,
as we explain below.
Since p(h) = 1, for a \ h \ b, it follows from (1) and (14) that:
!
X
p X
ki X
p a
p Y
ki
Ln ðhÞ ¼ log½pðhj xÞ ¼ xij logðhÞ h ki log xij ! ;
i¼1 j¼1 i¼1 i¼1 j¼1
5 Bayesian Forecasting of Spare Parts Using Simulation 119
0
for a \ h \ b, so that by solving Ln(mn) = 0 we obtain the mean
P
p P
ki
xij
i¼1 j¼1
mn ¼ : ð20Þ
Pp
ki
i¼1
00 2 Pp Pki
On the other hand, since Ln ðhÞ ¼ h i¼1 j¼1 xij ; we obtain the variance
h 00 i1 mn
2
Table 5.6 Simulated data Month (i) Machines in operation (ki) Failures Pki x
for 10 machines and a rate j¼1 ij
h=1 January (1) 10 8
February (2) 10 13
March (3) 10 13
April (4) 10 14
May (5) 10 9
June (6) 10 14
July (7) 10 12
August (8) 10 8
September (9) 10 11
October (10) 10 13
November (11) 10 13
December (12) 10 16
Total 120 144
In this chapter, we show how a simulation model can be used in conjunction with
Bayesian techniques to estimate point and variability parameters needed for
solving forecasting problems related to spare parts inventory. The advantage of
using simulation as a forecasting tool lies in the fact that even very complex
models that incorporate detailed information about the system under study may be
analyzed to produce reliable forecasts.
5 Bayesian Forecasting of Spare Parts Using Simulation 121
This chapter presents a Bayesian framework for forecasting using two different
methods. The first method, posterior sampling, requires a valid algorithm to
generate samples from the posterior density function p(h|x), whereas the second
one, Markov chain Monte Carlo, can be implemented when the family of distri-
butions corresponding to p(h|x) has not been identified. We illustrate the potential
applications of these methods by proposing two simple models, a model with
failure time data and a model with censored data, which can be applied under a
Bayesian framework to forecast the demand of spare parts.
We applied the model for censored data, using real data from a car and spare
parts dealer in Mexico, and illustrated the application of both the PS and MCMC
methodologies. In the case of the PS method, a non-informative prior was used to
establish the posterior distribution, whereas, for MCMC, a uniform (subjective)
prior was considered. The results of applying these methods allow us to arrive to
several interesting conclusions.
Firstly, the accuracy of the point estimators, as well as the halfwidth, bias and
mean square errors, are highly dependent of the number of replications m of the
simulation experiments. Thus, it is very important, from a practitioner’s point of
view, to establish the number of replications m according to the desired accuracy,
and the computation of the corresponding halfwidth is important to assess the
accuracy of the point estimators obtained using simulation.
122 D. F. Muñoz and D. F. Muñoz
Secondly, the results using real data show that having a large sample size let the
posterior distribution dominated by the data. Thus, the results obtained with an
objective prior are very similar to the ones obtained with a more informative prior,
since the priors have little influence on the posterior distribution. To illustrate this
phenomenon, another set of data was simulated using a smaller sample with a
higher failure rate. In this case, the influence of the prior is made evident and, thus,
the results are considerably different.
Finally, the relevance of the methodologies illustrated in this chapter depends
on the ability of the proposed model to imitate the real system, which is closed
related to an adequate selection of the parameters for which sample data is
available as well as the random components that incorporate the stochastic
uncertainty. In this direction, the models proposed in Sect. 5.3 can be modified to
resemble more accurately the system under study (e.g., assuming a different family
of distributions for the time between failures). Likewise, this framework could also
prove useful in forecasting problems related to supply chain management (see e.g.,
Kalchschmidt et al. 2006).
Acknowledgments This research has received support from the Asociación Mexicana de
Cultura A.C. Jaime Galindo and Jorge Luquin have also participated. They both shared their
knowledge on the auto-parts sector. As such, the authors want to express their most sincere
gratitude.
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forecasting processes. Int J Prod Econ 108:74–83
Chapter 6
A Review of Bootstrapping for Spare
Parts Forecasting
6.1 Introduction
Inventory forecasting for spare parts constitutes a very important operational issue
in many industries, such as automotive and aerospace. It has been reported that the
stock value for spare parts may account for up to 60% of the total stock value in
any industrial setting (Johnston et al. 2003). Since demand for spare parts arises
from corrective and preventive maintenance related activities, generally these
items are characterized by a highly variable demand size, with many periods when
there is no demand at all. Such patterns are often called sporadic or intermittent.
Intermittent demand patterns are built from constituent elements (demand sizes
and intervals) and they are very difficult to forecast. Croston’s method (Croston
1972) and its variants (in conjunction with an appropriate distribution) have been
reported to offer tangible benefits to stockists forecasting intermittent demand.
Nevertheless, there are certainly some restrictions regarding the degree of lump-
iness that may be dealt with effectively by any parametric distribution. In addition
to the average inter-demand interval, the coefficient of variation of demand sizes
has been shown in the literature to be very important from a forecasting per-
spective (Syntetos et al. 2005). However, as the data become more erratic, the true
demand size distribution may not comply with any standard theoretical distribu-
tion. This challenges the effectiveness of any parametric approach. When SKUs
exhibit a lumpy demand pattern one could argue that only non-parametric boot-
strapping approaches (that do not rely upon any underlying distributional
assumption) may provide opportunities for further improvements in this area.
M. Smith (&)
Winthrop University, Rock Hill, SC, USA
e-mail: smithm@winthrop.edu
M. Z. Babai
BEM Bordeaux Management School, Bordeaux, France
e-mail: Mohamed-zied.babai@bem.edu
During the 1977 Rietz Lecture at the Seattle joint statistical meetings, Professor
Bradley Efron introduced the bootstrap technique to estimate the sampling dis-
tribution of an observed sample. Later, the lecture was published in The Annals of
Statistics (Efron 1979). The paper presented the mathematical development for
bootstrapping as an extension of the jackknife method, which had first been
introduced by Quenouille for estimating bias, and then developed further by Tukey
for estimating variance, both of which were described by Miller (1974). Professor
Efron was asked in an interview if he had an application in mind when he
developed the bootstrap, but he indicated that he had been trying to determine why
the jackknife did not always give dependable variance estimates (Aczel 1995). His
study of the jackknife led him to the bootstrap, and while he modestly stated in the
abstract of the landmark work that the bootstrap worked ‘‘satisfactorily’’, he
demonstrated bootstrap’s superiority over the jackknife in estimating the sample
median, variance of a sample mean, and estimating error rates in a linear
6 A Review of Bootstrapping for Spare Parts Forecasting 127
discrimination problem. The key to the bootstrap was that it did not just use data
from a population (n), but it took a sample from the set with the mass of 1/n, and
replaced it in a set to form a new sample. The sample and replacement steps, or
making bootstrap samples, are repeated for a number of replications. These
bootstrap samples can then be analyzed for the distribution. Using direct calcu-
lations is too time consuming for most problems, but Efron presented a short
example with n = 13, using values of zero or one for the variables. If a Taylor
series expansion is used to evaluate the bootstrap samples, then the results are the
same as the jackknife. More information on the algorithmic steps related to the
application of Efron’s method is given in Appendix A.1. Hence with modern
computing technology (discussed later), the preferred method to generate the
bootstrap samples is to use a Monte Carlo simulation, and 50 replications were run
for each trial. In the example used, Efron was careful to note that the bootstrap
provided frequency statements, not likelihood statements.
While Efron published several articles, monographs, and books related to the
bootstrap, a limited number are reviewed here for their specific application to spare
parts inventory forecasting. These articles capture the major statistical models
developed from the bootstrap. In 1983, Efron and Gong presented the bootstrap,
jackknife, and cross-validation from a more ‘‘relaxed’’ perspective. That is, the
paper is more descriptive of the bootstrap technique, with limited mathematical
equations and proofs. The paper introduces a nonparametric example of LSAT
(law school admission test) scores and undergraduate GPA (grade point average).
In this problem 100 replications were run, and then 200 replications, but the
differences in the results were too small to be noted. If the reader is somewhat
rusty in mathematical and statistical theory, it may be better to start with this 1983
paper, and then move to the original 1979 paper.
Later Efron (1987) extended the original bootstrap concept with bias-corrected
confidence intervals, and further improved the confidence intervals to wider
problems. The law school student data introduced earlier was again used to
demonstrate the confidence intervals. Also, it was noted that while runs of 100
gave good results for the standard error estimates, trial and error indicated that
1,000 bootstrap replications are needed for determining confidence intervals. In
1990, Efron published bootstrap methods for bias estimates and confidence limits
that generate the bootstrap samples the same way, but require fewer computations
than the original models. After the bootstrap sample is generated, a probability
vector (P) is generated where the ith component is the proportion of bootstrap
sample equaling xi. The law school data were used again to illustrate the method,
using the sample correlation for P.
As an aside, Efron (2000) noted that the name bootstrap was taken from the
fables of Baron Muenchausen (also spelled Munchhausen and Munchausen).
These fables were published in the late 1700’s, and with centuries of time and
multiple translations, as well as fables published under the Baron’s name which he
did not write, an exact citation is difficult. However, in all the fables, the Baron has
extraordinary powers, such as our modern Superman, but also the phenomenal tool
making ability of Angus MacGyver from the 1980’s television series. Professor
128 M. Smith and M. Z. Babai
Roger Johnson (2009) has found a reference where the Baron falls a couple of
miles from the clouds (another tall tale), into a hole nine fathoms deep. The Baron
reports ‘‘Looking down, I observed that I had on a pair of boots with exceptionally
sturdy straps. Grasping them firmly, I pulled myself with all my might. Soon I had
hoist myself to the top and stepped out on terra firma without further ado.’’ The
Baron’s adventures were entertaining with no moral to the story, but the moral to
this story is to illustrate the power of the bootstrap statistical technique based on
the origins of its name.
The use of the computer has been integral to the bootstrap from the beginning,
with the bootstrap samples in the landmark paper being generated by Monte Carlo
simulation using a 370/168 computer. Efron reported the Stanford University
computer time for those runs cost $4.00. As his work progressed, computer
technology progressed, and by 1991 problems using up to 400 bootstrap samples
were being run on a personal computer (Efron and Tibshirani 1991). In 1994,
Thomas Willemain (1994) published a short paper describing how to use Lotus
1-2-3 to construct a basic spreadsheet for a bootstrap problem. The steps are well
documented, so the procedure could easily be adapted to Excel.
On October 26, 1999, Business Wire reported that Smart Software, Inc. was
releasing its new Intermittent Demand Forecasting for spare parts and other
intermittent demand (‘‘Smart Software brings…’’). Smart and Willemain (2000)
described the software, the bootstrapping methodology it uses and gave an
example problem in the June 2000 Performance Advantage. The pending patent
for the ‘‘system and method for forecasting intermittent demand’’ used by the
software was granted to Smart Software, Inc. March 20, 2001 (Smart Software
2001) (A later section provides more information about the bootstrap model used
in the patent). Today, Smart Software promises that SmartForecasts will ‘‘typically
reduce standing inventory by as much as 15–20% in the first year, increase parts
availability 10–20% and more, and reduce the need for associated costs of
emergency transshipments’’ (Smart Software ‘‘Intermittent Demand…’’).
In 2002b, Smart Software’s SmartForecasts was listed in a review of 52 fore-
casting packages. However, the review only listed support, price, and function-
alities, such as data entry, file export, graphic capabilities, and the statistical
methods used, but bootstrap was not listed as a choice for statistical technique and
would have only been captured in ‘‘Other’’. The review did not provide any
caparison of performance (Elikai et al. 2002). Sanders and Manrodt (2003) sur-
veyed practitioners regarding their use and satisfaction of marketing forecast
software, and their study could be used as a model for an updated study of
inventory forecasting software use, features, performance, and user satisfaction.
6 A Review of Bootstrapping for Spare Parts Forecasting 129
The intermittent demand pattern for spare parts means that traditional inventory
models, such as those with the economic order quantity (EOQ) or materials
requirements planning (MRP) as their foundations, are not applicable, since EOQ
models require a constant demand or a normal distribution, and MRP models
require fixed lot sizes. Most papers on spare parts inventory or intermittent demand
begin with a cursory introduction about the need for different models. These
models assume spare parts usually have periods of zero demand, then a part breaks
on some random interval, the part must be replaced, and the research issue is to be
able to predict the demand for the replacement parts. That is, the random time
between demands for spare parts and the number demanded do not tend to follow
patterns for parametric methods, which require a readily distinguishable statistical
distribution and meaningful measures of mean, standard deviations, etc. Kennedy
et al. (2002) began their literature review with a more extensive discussion of the
unique characteristics of maintenance inventories that would support the need for
non-parametric models. The special data related to these parts, includes supplier
reliability, stock-out objectives, inventory turn goals, age-based replacement of
working parts, spare parts obsolescence, and repairable items, as well as some
special cases such as emergency ordering.
Croston’s 1972 landmark work on intermittent demand uses parametric statis-
tics, but it serves as the benchmark for later non-parametric work using the
bootstrap. He extended the exponential smoothing method by adding variables for
the size of the non-zero demands when they occur and an estimated inter-arrival
time between non-zero demands that follow a Bernoulli process. It should be noted
that he assumed the demand variables and the time between demands were
independent. He used an example with 180 observations, with demand occurring
on average every six review periods, and the average demand 3.5 units with a
standard deviation of 1.0 unit. The Croston model prevented stockouts 95% of the
time, while the exponential smoothing model resulted in stockouts in 20% of the
periods.
Bookbinder and Lordahl (1989) developed a model using the bootstrap to
estimate the lead-time distribution (LTD) and determine re-order points. They then
compared their bootstrap model results to inventory models based on normal
distributions using simulated populations of varying shapes. The calculations were
done on a TRS (Model III) microcomputer, with only samples of n B 30 con-
sidered. Seven different probability density functions were used: uniform, trun-
cated normal, log normal, two-point, positively skewed bimodal, symmetric
bimodal, and positively skewed normal. The algorithmic steps for the imple-
mentation of Bookbinder and Lordahl’s method are given in Appendix A.2. The
researchers presented a comprehensive table of results of the acceptability of
service at .8, .9, and .95, which captured all seven tested distributions, different
130 M. Smith and M. Z. Babai
sample sizes and varying coefficients of variance. Their results show when the
bootstrap performed better, when the normal better, when both were acceptable,
and when neither was acceptable. Generally, the normal only performed better
than or as well as the bootstrap for the log normal and truncated normal distri-
butions. The bootstrap tended to dominate in the case of the two point distribution,
and negative skew binomial, which would better model the demand for spare parts.
Since the data was simulated, they could compare the known optimal costs to the
costs from the bootstrap model and the costs based on the normal theories. The
costs results were also presented for all the combinations of the service level
results. The bootstrap estimates gave better cost results for the distributions with
some type of bimodal data, as well as slightly better cost results for the uniform
data. The authors also performed a Newman–Keuls test for an analysis of variance
(ANOVA) for the bootstrap and normal procedures. The results showed that the
bootstrap was much less sensitive to the lead time distribution.
Snyder (2002) used data from an automobile company to illustrate four models
that used a parametric bootstrap for demand forecasting of slow and fast moving
inventory items. Like other researchers, he considered exponential smoothing and
Croston’s method. In his Monte Carlo simulation, he needed estimates of mean,
bias, and variance, so he used least squares estimates in their place making the
model a parametric bootstrap. He also reviewed the Croston method and identified
problems with how seed values are selected for the exponentially weighted mean
averages, and developed a modified version called MCROST. MCROST was then
modified for a log transformation of non-zero demands, giving the log-space
adaptation (LOG). Lastly, he experimented with a modification of Croston’s model
that used variances instead of mean absolute deviation and a second smoothing
variable was introduced to define how variable changes over time (AVAR). Details
related to the implementation of the MCROST, the LOG and the AVAR methods
are presented in Appendix A.3. The simulation used 10,000 replications with the
key performance measure being the order-up-to level (OUL) that represented the
ideal level of stock that achieved a 95% fill rate target while being as low as
possible. For the slow moving item, which would most closely represent the
typical demand for spare parts, the OUL’s were very close.
6.3.2 SmartForecasts
assessing the accuracy of the forecast. With respect to the use of the bootstrap, this
model expands the inventory applications of Bookbinder and Lordahl (1989) and
Wang and Rao (1992), since neither of these applications considered the special
case of intermittent demand. The paper presents general information about the nine
organizations used to develop the model, as well summary statistics for their
respective demand data.
The bootstrap was modified in three ways to better model the intermittent
inventory data with autocorrelation, frequent repeated values, and relatively short
series. First, positive and negative autocorrelation are added, since demand can run
in ‘‘streaks’’. These autocorrelations are modeled using a two state, first order
Markov process to get zero and non-zero demands. The next step is to generate
numerical values for the nonzero forecasts. If only the bootstrap replacement
values were used, then only past values could occur in the future. The model then
uses a patented ‘‘jittering’’ process to allow more variation around the nonzero
values. The example is given that a 7 may be replaced with a 7 or 6 or 10. The
authors report that the jittering was shown to improve the accuracy, especially for
small sample sizes. More information on the algorithmic steps for the imple-
mentation of Willemain et al.’s method is given in Appendix A.4.
The lead time distribution for the model was evaluated and compared to
exponential smoothing and Croston’s method. Results were examined on the upper
tail as a measure of the ability of the model to satisfy the availability requirement
and on the lower tail as a measure of holding costs. While the bootstrap method
was shown to be the most accurate forecasting method, all the methods were least
well calibrated in the tails of the distributions. The authors concluded their paper
with three significant suggestions for further work. First, they recognized that there
are problems with the jittering step, and it will not work for items that are sold in
different size cases. However, in the paper it is not clear exactly how one deter-
mines the standard normal random deviate for this step. The second issue is
nonstationarity, or the fact that the demand distributions for the items may be
changing due to seasonality, life cycle stage of the product, or some other com-
plication or trend. Finally, the authors acknowledge that they began the research
with the idea that the intermittent demand followed some kind of Poisson process.
While they were unable to find an acceptable Poisson based model, they still
consider this an avenue for future research.
Michael Lawrence, editor of the International Journal of Forecasting, pub-
lished a special commentary following the article described above. He noted the
unique aspect, perhaps the first ever, of publishing a patented technique in an
academic journal. One of the reviewers had expressed a concern about other
researchers being able to use and extend these contributions, and then publish their
comparisons with patent protected methods. In their response to these reviewer
comments, Willemain et al. (2005) stressed the importance of researchers being
able to patent their work and gain the potential economic benefits of the work.
Further, they note that discussions of patented algorithms do not result in legal
patent infringement. Finally, they offer that ‘‘it is easy to arrange a nominal
licensing fee for researchers doing non-competitive work’’ (Lawrence 2004).
6 A Review of Bootstrapping for Spare Parts Forecasting 133
More discussion ensued about the work described above. Gardner and Koehler
(2005) expressed reservations about the work because Willemain et al. had not
used some later published research for the simple exponential smoothing model
and Croston’s model. The comparison to the exponential smoothing method used
an estimated standard deviation based on work from 1959, but this estimate had
been improved by later researchers, including Snyder et al. (1999). Willemain
et al. (2004) assumed a normal distribution for the lead time demand in the
exponential smoothing model, but Gardner and Koehler suggested a bootstrap lead
time demand based on the work of Snyder et al. (2002). Gardner and Koehler also
noted published research that improved upon the original Croston model, but
Willemain et al. had only compared their work to the original Croston. In their
response, Willemain et al. acknowledge the use of the early versions of expo-
nential smoothing and Croston’s method, and they encourage researchers to
continue this stream of research by comparing the changes proposed by Gardner
and Koehler to the bootstrap method. However, they note that there should be
more research to determine how these proposed changes to exponential smoothing
and Croston would compare when their new accuracy metric was applied.
The patent granted to Smart Software, Inc. for the ‘‘System and Method for
Forecasting Intermittent Demand’’ provides more information than the articles
mentioned above. Some of the patent’s references are provided in this chapter, but
the patent references more book chapters and monographs, as well as more broadly
related topics such as forecasting accuracy and statistical smoothing methods,
which are outside the scope of this work.
The patent includes five high level flow diagrams (Figs. 6.1–6.5) of the com-
puter system, software, method of forecasting, selection of demand values, and the
method of forecasting using a subseries method. Figure 6.6 in the patent, the
experimental demand data, is the same as Table 6.2 in Willemain et al. (2004).
The patent includes subseries methods (normal and log normal) which approxi-
mate the distribution by using the sums of overlapping distributions. This com-
ponent of the patent is not featured in Willemain et al. (2004). However, the patent
reports that the subseries forecasts were more accurate than exponential smoothing
and Croston, and can be computed faster than their bootstrap. Figure 6.7 is the
same as Table 6.3 in Willemain et al. (2004), except Fig. 6.7 also includes
accuracy results for the subseries models. Figures 6.8, 6.9 and 6.10 show the
forecasting accuracy of exponential smoothing, Croston’s method, bootstrap, as
well as the subseries normal and subseries log normal, as measured by a table of
mean log likelihood ratios, chi-square values, and a graph of mean log likelihood
values, respectively.
The patent gives a small example to aid understanding the concepts, with
intermittent demand for 12 months presented in Table 6.1. Table 6.2 lists exam-
ples of five replications for possible demand and lead time distributions over
months 13–16 (Table 6.3 is the same as Table 6.1). Table 6.4 gives examples of
four different 12 month strings of zero and non-zero (1) data generated from
Markov transition probabilities based on the LTD in Table 6.1. Table 6.5 illus-
trates how bootstrap converts the non-zero values (the 1’s) in Table 4 to integers.
134 M. Smith and M. Z. Babai
Then Table 6.6 presents the demand values which were determined by jiggering
the bootstrap demand values from Table 6.5. The jittering process is presented in
general terms, but detailed enough to more clearly explain the specific data.
software package. The model for forecast adjustments and assessing forecast
performance incorporated concepts previously developed and tested by the
research team. The adjusted demand forecasts were better than the system forecast
for 61% of the SKUs studied, which suggests potential for expanding the model to
study judgment adjustments for spare parts forecasts. The comprehensive literature
review in Syntetos et al. (2009a) provides examples of related types of forecasting
models that have included judgment in the forecasts.
In 2008, Porras and Dekker published the results of a major study of re-order
points for spare parts inventory. The study used data covering a 5-year period from
a petrochemical company in the Netherlands with 60 plants, totaling 14,383 spare
parts. The current method is based in the SAP, which does not capture and
accommodate for the special case of the intermittent demand The parts were
classified into criticality classes, demand classes (based on demand pattern and
demand level), and price classes. The classes were combined, so that an item’s
three digit code indicated its criticality, demand, and price classes. First, classes
were optimized, and then items within classes were optimized. The simulation
model used an ex-ante approach, where once the distribution had been fitted, and
then a different distribution was used for testing. It also used an ex-post approach
where the same data set was used for fitting and testing. The performances of both
approaches were tested. The researchers tested four methods for determining LTD:
normal, Poisson, Willemain’s method, and a new empirical method. Their new
empirical method samples demands over blocks of time equal to the lead time
length (capturing implicitly underlying auto-correlation structures). Details related
to the implementation of this bootstrapping method are presented in Appendix A.5.
The optimization model determines the lot size (Q) and the smallest reorder point
that satisfies a 100% fill rate. Then the model determines cost savings using the
holding cost and ordering costs of the current model and the model being tested.
Porras and Dekker (2008) gleaned several interesting findings from their study.
When they used the ex-post approach, all of the tested models outperformed the
current approach. The normal LTD performed best overall, but Willemain’s model
and their empirical model were close. Their empirical model produced cost sav-
ings of 1.05 million euros and Willemain’s model saved .96 million euros over the
current model. The results for the ex-ante were similar. In their conclusion, they
reiterated the need for including high demands for preventative maintenance in
spare parts inventory models.
Varghese and Rossetti (2008) developed a model for use on the spare parts used
by the Naval Aviation Maintenance Program of the US Navy. The model is a
Markov Chain demand occurrence (MC) and auto regressive (ARTA) to any
demand amount (IDF), and it is a parametric bootstrap (PB); that is MC-ARTA-
IDF-PB). They compared their model to the work of Croston (1972) and Syntetos
(2001) on mean square error, mean absolute deviation, and mean absolute per-
centage error. With a bootstrap of 1,000 replications, they did not find any sig-
nificant differences between the three forecasting methods, leading them to
conclude that more work is needed on their estimation algorithm and to experi-
ment with using their model for differing inventory policies.
136 M. Smith and M. Z. Babai
Teunter and Duncan (2009) used UK Royal Air Force data for 5,000 items over
a 6-year period to compare forecasting methods for spare parts. The objective of
the system is to minimize stock levels, while meeting a service level constraint.
They compared six methods of forecasting demand. A zero forecast was the
benchmark, since traditional inventory models do not incorporate a forecast. The
traditional forecasting techniques of moving average and exponential smoothing
were used. They also considered Croston’s original model for intermittent
demands and the variation to Croston’s model that was developed by Syntetos and
Boylan (2005). Bookbinder and Lordahl’s (1989) bootstrapping method was also
included. To compare the first five methods, they used the relative geometric root
mean square error (RGRMSE), which calculates the performance of one method
compared to another as the ratio of the geometric mean of the squared errors. That
is, they calculated the average mean square error for each item, and then calculated
the geometric average of these numbers over the entire data set. When they used
the RGRMSE, the zero forecast gave the best results, but the authors concluded
that forecast performance cannot be measured strictly on error measures.
When Teunter and Duncan (2009) compared service levels, they used normal
and lognormal for lead time demand, except for the bootstrap method, which
generated its own lead time distribution. The six methods were compared for
service level and a combination of service level and average inventory level. The
zero forecast performed the worst. The bootstrap and the two Croston type
methods performed the best, with the results being very close. However, initially
all the methods gave results that were significantly below the ideal target. When
the researchers realized that an order in a period is triggered by a demand in that
period, they adjusted the lead time distribution to lognormal for the bootstrap and
Croston based methods. The improvement in service level accuracy was
significant.
Based on the review above, a few ideas for future work surface. First, an empirical
study of the commercial forecasting software packages that are available would be
timely. Such a study could follow Elikai et al. (2002), but also include capture
information about whether bootstrapping is used, and specifically whether the
forecasting techniques are applicable to intermittent demand. Other work, similar
to that of Sanders and Manrodt (2003) could examine the user satisfaction and
performance of commercially available software for forecasting of spare part
demand. A study like this was done by McNeill et al. (2007) with AMR Research,
but it does not appear to be in the public domain.
In their response to a reviewer comment regarding the ability of academics to
make comparisons of their work to the patented SmartForecasts (Lawrence 2004),
Willemain et al. offered to make their patented SmartForecasts available to aca-
demic researchers for a ‘‘nominal licensing fee’’ to encourage future scholarly
6 A Review of Bootstrapping for Spare Parts Forecasting 137
research in this area. So far, it does not appear that anyone has accepted the offer,
even though it could help researchers understand and extend the patented
‘‘jittering’’ step. In addition, Willemain et al. (2004) suggested work on examining
a Poisson based model for spare parts forecasting.
As noted above, the research to develop SmartForecasts used 28,000 inventory
items. Hence, the researchers decided against individual item analysis and pooled
data across items, which impacts error measures. However, the SmartForecasts
patent notes that the forecasting program is designed to run on any computing
hardware. So as hardware has improved, and continues to improve, then it is
assumed that the feasibility of performing comparisons and error analysis on per
item basis would increase.
Hsa et al. used plans for plant overhauls (scheduled preventative maintenance)
in their model, and other researchers have suggested models that can incorporate
other types of judgments (Bunn and Wright 1991; Syntetos et al. 2009b). As Hsa
et al. noted, while management information technology improves and the inte-
gration abilities increase, this should become easier. Also, Willemain et al. (2004)
had the considerations of seasonality, or other issues, as possible extensions of
their work.
Appendix
1. Obtain historical demand data in chosen time buckets (e.g., days, weeks,
months);
2. Express the lead-time L as an integer multiple of the time bucket;
3. From the demand history, determine least squares estimates of the parameters
a, l and r;
4. Generate binary values ðx1 Þ; ðx2 Þ . . . xL for the indicator variable xt from a
Bernoulli distribution with probability p;
5. Use Monte Carlo random number generation methods to obtain values of the
errors ðe1 Þ; ðe2 Þ . . . eL generated from a normal distribution with mean zero and
variance xtr2;
6. Generate realizations ðy1 Þ; ðy2 Þ . . . yL of future series values by using the
equations: ðyt Þ ¼ xt lt1 þ et ; lt ¼ lt1 þ aet and l0 ¼ l;
7. Sum the L values of yt;
8. Repeat steps 4–7 many times;
9. Sort and use the resulting distribution of LTD values.
The steps are the same as in the MCROST method except for the step 6, where the
smoothing equations are modified to not allow for negative values as follows:
logðyþ
t Þ if xt ¼ 1
yt ¼ xt lt1 þ et ; yþ
t ¼ xt expðyt Þ; yt ¼ ; et ¼ xt ðyt lt1 Þ;
arbitrary if xt ¼ 0
lt ¼ lt1 þ aet and l0 ¼ l
6 A Review of Bootstrapping for Spare Parts Forecasting 139
The steps are the same as in the MCROST method except for the steps 5 and 6,
where the equations are modified as follows:
et NID 0; r2t1 ; yt ¼ xt lt1 þ et ; yþ
t ¼ xt expðyt Þ; lt ¼ lt1 þ aet and l0 ¼ l
r2t ¼ r2t1 þ bxt e2t r2t1 ; r20 ¼ r2
6. Sum the forecast values over the horizon to get one predicted value of Lead
Time Demand (LTD);
7. Repeat steps 3–6 many times;
8. Sort and use the resulting distribution of LTD values.
140 M. Smith and M. Z. Babai
References
7.1 Introduction
This paper presents an optimization model for the planning of spare parts levels
used in the support of aircraft operations. In particular, the model addresses the
problem of planning rotable inventory: serialized items that are maintained and
restocked rather than discarded upon failure. This special problem can be char-
acterized as an operational situation where inventory levels do not change over the
planning period (typically a year) and the flow of inventory occurs in a closed
loop, from operation (installed on an aircraft), to the repair cycle, to spares
inventory and returning to operation. Thus inventory items classed in this way will
survive for the lifetime of the parent aircraft fleet. Changes in inventory are made
by buying and selling overhauled items in the case of a mature fleet. The trigger
for inventory activity in this operational situation is the failure of an item and its
removal from service: this then prompts a repair activity, which will later result in
replenishment. Thus demand is based purely on forecast failures, so there is no
need for ordering decisions in the short term. Inventory changes are reviewed and
planned in the medium term. Historically, inventory levels tend to creep up in
response to epidemics of failure. Given the pressure to maintain operational reli-
ability, it is usually considered acceptable to high levels of rotable stocks, even
though utilization of these stocks may be poor.
Treatment of the rotable problem in the literature is sparse: much of the prior
work addresses the forecasting of demand for consumable spares, even when the
operational context describes repairable assemblies. Much of the literature
M. MacDonnell (&)
School of Business, University College Dublin, Dublin 4, Ireland
e-mail: michael@ucd.ie
B. Clegg
Aston Business School, Aston University, Birmingham B4 7ET, England
e-mail: b.t.clegg@aston.ac.uk
addresses the problem at the item level, whereas the present work optimizes at a
system level. Given the stochastic nature of demand, the problem is re-stated to
give an objective of a service level (SL) target for an entire pool of inventory,
rather than applying a service level to each line item. In this manner it is possible
to skew inventory holdings by cost: the objective function changes from ‘‘exceed
target SL for each part’’ to ‘‘exceed target SL for the system of parts, at minimum
cost’’. One method in the literature, also observed in practice—Marginal Analy-
sis—addresses the problem at the system level, but is not an optimization.
A large-scale optimization model is developed and formulated for solution as a
binary integer linear program. This model is solved for a range of sensitivity
values and re-formulated for multiple operating scenarios, reflecting changing
operational criteria, such as increased fleet size, reduced repair times and varying
target SLs.
Current practice from the field and the literature is replicated and compared
with the new solution. The linear programming optimization is shown to produce
superior results to current practice: inventory investment can be reduced by 20% or
more without reducing SL. This is achieved through a coordinated increase in
availability of lower-cost parts with a reduction in holdings of higher-cost parts.
While the new solution is computationally intensive, its benefits exceed
implementation effort and the model may be applied to other operational settings
where expensive spares are held.
This paper looks at published model specifications and experience with specific
reference to the aircraft rotable inventory problem. Thus the focus is narrowed to
the rotable problem only, and does not consider the cases of consumable items,
which are better understood in mainstream practice. There are two main possible
approaches to classifying the rotable scheduling problem: planning parts at the
individual level, and planning for systems of parts where demand can be combined
and considered together. The latter is preferable since it more closely reflects the
reality of stochastic demand and is seen to give better results.
Demand for spares may arise in several ways (Ghobbar and Friend 2003a): due
to hard time constraints (for example, a landing gear assembly must be changed
after 500 flights), on condition (an item is inspected against a defined standard,
e.g., tyre tread depth) and condition monitoring (real time diagnosis of perfor-
mance, e.g., brake pad wear). Rotables can generally be considered as arising for
maintenance on condition, meaning that their performance is observed to be
deficient upon inspection, or often in operation. However, it is best to consider
rotables as arising for removal through condition monitoring, since their failure
will usually be observed during operation, so the removal does not typically result
from a planned inspection. Ghobbar’s analysis gives a detailed statistical analysis
of parts demand for maintenance items but there is no optimization involved.
7 A New Inventory Model for Aircraft Spares 145
indenture. Several stages of supply, or echelons, are also modeled. The model
consists of an aggregation of Poisson forecasts for individual part demand to meet
an overall SL target. However, there is no account of cost in the model, other than
aggregation: since an engine is effectively a hierarchical grouping of modules,
there is not much scope for cost optimization.
Adams (2004) compares item-level and system-level approaches to aircraft
rotable optimization, concluding that item-level forecasting is the least risky but
will over-provide spares. Meanwhile Marginal Analysis combines parts, takes
account of costs and can be modeled for multi-echelon scenarios. However, while
Marginal Analysis gives good results, they are not optimal. A genetic algorithm
approach is also tested—this is a complex approach, which may improve on
Marginal Analysis but is not necessarily optimal.
The Marginal Analysis method was first developed by Sherbrooke (1968) in a
military setting, in the Multi-Echelon Technique for Recoverable Item Control
(METRIC) model. It is interesting to note that the US Air Force investment in
recoverable items (rotables) is reported at $5bn in 1967. The METRIC model
addresses overall optimality of spares stocks and the balanced distribution of
spares in a network with two echelons, or levels of supply: bases and depots. Bases
are the locations from which aircraft operate (as in the supply chain schematic
Fig. 2.1) while depots are central inventory locations, usually with comprehensive
repair capabilities. The model represents failed requests for parts as back-orders,
so that a failed request survives until it is filled. The model is of type (s - 1, s),
i.e., a replacement is ordered when a part is taken from stock.
The approach adopted in this study differs from the METRIC model because:
(a) demand is presented here in mean terms, not recurring, so that behavior over a
planning period is represented by a SL;
(b) back-orders are not tolerated in commercial aviation—some action must be
taken to satisfy the demand, usually borrowing a part or expediting delivery in
the supply chain;
(c) Sherbrooke acknowledges that the marginal contribution of increasing part
numbers should be concave—marginal contribution should reduce as the
number of parts increases—but this is not the case and so the Marginal
Analysis approach is flawed.
The METRIC model is improved on with MOD-METRIC and VARI-METRIC
versions (Sherbrooke 1986), with better forecasting of expected back-order rates.
The ‘‘best’’ results, closest to optimal, are derived by simulation and the new
versions of the METRIC model are shown to be closer to those that are presumed
optimal.
General Electric Rotable Services claim massive savings in inventory through
the use of Marginal Analysis—see Fig. 7.4. Where current practice achieves 77%
SL with $50M in inventory, the same performance is claimed with $28M in
inventory, a 44% reduction, through the use of Logistechs k2s (knowledge to
spares) solution, in which GE holds a stake. Figure 7.4 also shows that, for the
existing $50M in inventory, the operator could increase SL from 77 to 90%
7 A New Inventory Model for Aircraft Spares 147
with parts with infrequent demand, namely marginal contribution that does not
diminish continually. Through its supply chain optimization project, which
included reducing inventory in the supply chain as well as in stores, Southwest cut
its 2003 budget for rotable purchase from $26 to $14M, identified $25M of excess
inventory and avoided repair costs of $2M, while increasing SL from 92 to 95%.
A model has been proposed using a small example for illustration (MacDonnell
and Clegg 2007), which involves using a linear programming model to pick an
optimum selection set of inventory levels for a connected set of parts. The
resulting solution should meet a target SL at the least overall cost. It is predicted
that this can be achieved on a larger scale by increasing stocks of cheap parts while
reducing stocks of expensive parts. The consequential equivalent SLs for the
individual inventory items will therefore deviate from the target SL but the global
SL is maintained. An issue that remains to be addressed is the different essentiality
levels of parts in the same inventory, as they have different SL requirements and
should contribute differently to a connected solution. A version of this model has
been tested with a large MRO (maintenance, repair and overhaul service provider)
SR Technics (Armac 2007), showing potential for a 25% reduction in capital
investment in inventory based on a 4-month trial reviewing new purchase requests.
As part of the study, the company showed that a 2-day reduction in component
repair cycle time would enable a reduction of $7.5M in their UK inventory.
Table 7.1 shows the values of a Poisson distribution for a range of mean values.
This is used as the basis for planning inventory levels at the line-item level in
current industry practice. Thus, if a given part number fails an average of 3 times a
year, then a stock of 6 spares is needed to ensure that at least 95% of requests for
spares are met. Note that demand is scaled down to allow for the return of stock:
for example, if a removed item is missing from stock (in the repair cycle) for one-
tenth of a year, then it is available for nine tenths of the year, so demand is scaled
down by a factor of ten. In such a case, the demand figures 2, 3, 4 and 5 in
Table 7.1 would equate to failure rates of 20, 30, 40 and 50.
The conventional approach described above is limited as follows:
for part number 1, and so on. The binary constraint and integer declaration
require that X1 and X2 will have exactly one quantity variable selected.
– The SL constraint in Fig. 7.2 requires that the combined number of satisfied
requests for inventory for the two parts will exceed 47.5, which is the total
number of expected demand events multiplied by the target SL.
– The objective function in Fig. 7.2 minimizes the total cost of the solution that
satisfies the constraints above.
Figure 7.3 shows that the solution calls for a quantity of 5 of part number 1 and
3 of part number 2. Note that, while part 1 is more expensive ($12,072 compared
with $1,429 from Fig. 7.2), it also has a higher fill rate, so a quantity of 5 of part 1
gives 31.68 fills, compared with 16.83 for 5 of part 2 (SL constraint in Fig. 7.2).
The total cost is $64,647 and the SL attained is derived from the number of
fills = 31.68 ? 16.32 = 48. If the target fill rate of 47.5 represents 95% SL, then
the SL attained can be calculated as: 0.95 * (48/47.5) = 96%, or 48 out of 50.
Several variants of the conventional and linear programming models are tested
to give a rigorous assessment; these are tested for a range of scenarios derived
from a common data set. In all, five models were tested on five scenarios, giving
25 formulations and sets of results, which are presented in the next section.
The data set used is a sample provided for Boeing 737 fleet support by a large
maintenance organization. The actual inventory holdings are also presented for
comparison. The sample contains 300 inventory line items, which is about one
tenth of a typical inventory package for an aircraft family.
Table 7.2 presents results for five models, each tested against a common group of
five data sets, numbered 1–5 for each model. The five data sets represent different
operational scenarios, such as larger fleet size and shorter repair time, to observe
Table 7.2 Results of multiple model tests for rotable inventory planning
Run SL Total cost Actual cost Total inventory Average item value
(%) ($M) ($M) count ($)
Actual 89 32.9 2325 14164
Poisson
P1 96 15.3 17.6 995 15362
P2 94 14.0 19.0 930 15006
P3 96 13.2 19.8 865 15220
P4 93 25.2 1649 15294
P5 94 21.0 1338 15713
Marginal
analysis
M1 96 11.1 21.8 1050 10597
M2 95 10.6 22.4 1006 10509
M3 96 9.5 23.4 916 10395
M4 99 25.2 2060 12227
M5 97 18.4 1623 11364
Cost-wise
skewed
C1 95 11.8 21.1 979 12063
C2 95 11.8 21.0 982 12178
C3 95 10.6 22.3 820 12984
C4 95 23.0 1634 14083
C5 95 18.4 1389 13260
LP
L1 94 10.3 22.6 1019 10106
L2 93 9.8 23.2 972 10035
L3 94 8.6 24.3 882 9807
L4 95 17.5 1785 9786
L5 95 14.6 1513 9665
LP3
L3-1 94 9.7 23.2 1006 9662
L3-2 91 8.7 24.2 932 9381
L3-3 94 8.3 24.6 854 9709
L3-4 94 16.4 1734 9469
L3-5 91 12.2 1381 8858
152 M. MacDonnell and B. Clegg
the effect of changing operational conditions and to add rigor to the evaluation.
The five scenarios are as follows:
(1) Base—operating conditions obtained from an actual fleet support case and
pertaining to the data set used
(2) Fewer—Airbus recommends lower SL values for non-essential parts than
Boeing does—{95, 89, 75%} against {95, 93, 90%}, so these values are used
as an alternative scenario.
(3) Faster—repair times are given as 20, 28 or 38 days in the test data set used. An
alternative scenario reduces this time by 5 days.
(4) Bigger—operators often question the increase in spare stock utilization that
would result from a greater fleet utilization, either by increasing flight activity
or pooling demand with another operator. This scenario assesses a doubling in
utilization.
(5) Best—this scenario combines Airbus recommended SLs, reduced repair times
and higher utilization.
The five models shown include conventional practice (Poisson), a published
system-level heuristic (Marginal Analysis), a simple heuristic sorting parts into
cost bands (Cost-wise skewed) and two linear programming (LP) models. LP
models differ in their treatment of parts with different target SLs. As explained
earlier, there are typically three SL values used to reflect three levels of
essentiality of parts. The LP model above comprises a single, large formulation
where demand for items with lower essentiality is scaled down relative to
demand for the parts with highest SL. This is an approximation but has the
advantage of being a single, large model so could provide a more efficient
solution. The LP3 model solves three separate formulations for different target
SLs so removing the approximation introduced in the LP model by scaling
demand for parts with lower SL.
With reference to the columns in Table 7.2, Run lists the solution sets gener-
ated, SL shows the SL attained, Total Cost is the extended cost of the solution,
Actual-cost is the improvement over the current inventory holding, Total inventory
count is the number of parts prescribed and Average item value is the total cost
divided by the inventory count. The values for Actual-cost left empty reflect
scenarios 4 and 5, which represent doubling the size of the supported fleet, so
comparison with the actual inventory holding is not meaningful.
Looking at the results for each model, the five rows represent different sce-
narios, so that P1 to P5 use the same methodology (industry standard Poisson
analysis) on the five different variations of operational conditions described.
Turning to the results, the first finding is that the Poisson model over-stocks:
for P1, where the target SLs are 95, 93 and 90%, the overall achieved is 96%.
This is because the process operates at the part level to exceed the target,
giving an excessive combined result. The Marginal Analysis model also over-
shoots substantially, the Cost-Wise Skewed model reaches the highest SL, 95%,
and the two LP models have lower aggregate results, meaning that they are
more precise.
7 A New Inventory Model for Aircraft Spares 153
Comparing overall results for the different models, it is striking that the
actual observed stock performance gives low SL but high cost—this is thought
to result from a practice over many years of over-ordering expensive items
following a low stock event. Interviewing managers in the organization, the
root cause appears to be poor management of repair times and the fact the
management priority is seen to be stock availability, not total inventory cost.
Also, there is a sense that low-cost items are less important and thus receive
less attention, although clearly their importance is the same as that of more
costly items.
The Poisson model is the most expensive for all scenarios, while LP3 is the
least expensive. Marginal Analysis gives good results for the first three scenarios
but over-stocks for the larger utilization levels, which is a recognized flaw in the
model, referred to as the need for a concave distribution, or diminishing marginal
return, which does not occur for higher demand rates.
The Cost-Wise Skewed model, a simple spreadsheet model that groups parts by
cost and lowers SL for the highest-cost items, gives better results than current
practice, so would be useful as an improvement. However, the LP models gives
significantly better results and should be recommended in all cases as superior.
The LP3 model, which solves separate formulations for different SL values, is
significantly superior to the LP model that combines SL using scaled-down
demand for non-essential items.
As well as better targeting of SLs, as can be seen from the overall SL values in
Table 7.2, it can be seen that, for each solution that gives a lower total cost than its
predecessor, the average item value is lower. Thus, while the total number of parts
tends to increase, they are less expensive parts. This is consistent with the aim of
reaching target inventory performance at the lowest cost. Therefore, having a
higher inventory count is not a negative: if average cost is lower, then the total cost
is also lower. Simply put, the aim is to ensure that, if a given number of demand
failures are expected to occur, then it is preferable that the stock-outs be for the
most expensive parts.
Having examined the aggregate results, which promise a potential cost reduc-
tion of up to 40% comparing different models, it is interesting to examine the
distribution of the results. Current practice performs a line-by-line set of calcu-
lations, so there is no relative analysis in relation to cost. The other models take
cost into account, so it is expected that their results prescribed high stock levels for
low-cost parts with stock levels (and consequent SLs) falling in proportion to cost
for the items with higher value.
Figure 7.4, which shows part numbers in order of increasing value, confirms
that low-value parts have high SL performance as they are highly stocked, while
high-value items have decreasing stock levels and SL performance. The curves
plotted are for 5 scenarios tested using model LP3. The noise in the curves is
attributed to rounding of small stock numbers.
It can be seen from Fig. 7.4 that there are two dominant gradients in the results,
with a steep slope for the rightmost portion of approximately one-fifth of the data
range. Fitting lines to the data gives two slopes for the first 80% and top 20% by
154 M. MacDonnell and B. Clegg
Fig. 7.4 SL performance for parts ranked by value, 5 cases for LP3 model, with two gradient
lines
value of the inventory list. The tangent of the left slope is approximated
as:ð0:9945 0:9482Þ=0:8 ¼ 0:058; giving anangle of 3:3 below horizontal:
The right-hand slope is approximated as:
This study shows that there is a clear benefit in planning rotable inventory stocks
by using a system-wide cost-oriented approach. This can achieve very significant
cost savings when compared with current practice.
The data set tested represents approximately one tenth of the inventory list held
in support of a large Boeing 737 fleet. The actual value of the inventory holding for
the part number selection tested is over $30M, so a total estimate of up to $300M
is given to this operational scenario. The potential to reduce this investment by up
to 40% gives strong motivation for any large operator to move beyond current
practice and implement an optimization model such as that presented here.
The model shown here has been implemented as an enterprise application and
tested on full sets of fleet data. This data and the results are not disclosed for
7 A New Inventory Model for Aircraft Spares 155
commercial reasons, but inventory value reductions of 20–40% have been pre-
dicted in several instances, without loss of performance (SL).
Without implementing the large-scale, complex and intensive LP solution, it is
possible to apply gradients derived from test results to give a scaled prioritization
of inventory holdings in order of cost. For example, it is feasible to prescribe that
the lowest-value parts seek 99.45% SL, with the highest-value in the band of 80%
of part numbers having an SL target of 94.82%, with a slope of 0.058 for part
numbers between these limits, and a similar graduation of parts in the top 20%
band with limits of 94.82 and 49.17% and a slope of 2.285. The results of both the
LP solution and the proposed gradient heuristic are easily checked by calculating
the total number of demand fills by each line item, and the resulting global SL
achieved.
In conclusion, current practice, which mainly employs the Poisson calculation,
is far from optimal and there are significant benefits to using a system-wide LP
solution for planning aircraft rotable inventory levels. Where it is not feasible to
formulate and run the LP model, the gradient heuristic will give a good approx-
imation of the LP results.
References
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http://www.cwhonors.org/Search/his_4a_detail.asp?id=4916. Accessed Jul 2008
Friend C, Swift A, Ghobbar AA (2001) A predictive cost model in lot-sizing methodology,
with specific reference to aircraft parts inventory: an appraisal. Prod Inventory Manag J
42(3/4):24–33
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evaluation of methods. J Aircr 41(3):665–673
Ghobbar A, Friend C (2003a) Evaluation of forecasting methods for intermittent parts demand in
the field of aviation: a predictive model. Comput Oper Res 30:2097–2114
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echelon availability models. Oper Res 34(2):311–319
Chapter 8
Forecasting and Inventory Management
for Spare Parts: An Installed Base
Approach
Stefan Minner
8.1 Introduction
As customers are more demanding with respect to after sales operations and
service level agreements put challenging availability targets on equipment uptime,
the provision and deployment of service parts becomes of focal interest for many
original equipment manufacturers. High demand variability and uncertainty driven
by different phases of a product’s and its critical components life-cycles make
spare parts demand forecasting and safety inventory management a major chal-
lenge. Boone et al. (2008) identify service parts demand forecasting as the
unanimously agreed challenge in service parts management in their Delphi study
with senior service parts managers. Parts management of a non-stationary demand
process over the phases of series production, end-of-production (EOP), end-of-
service (EOS), and end-of-life (EOL) is even further complicated by heteroge-
neous customers and different ages of equipment in use due to different points in
time of purchases and component replacements.
Mainstream stochastic inventory management approaches for spare parts make
use of distributional assumptions for demands. In reality, distributions and their
parameters are hardly known and need to be estimated. Practical suggestions
recommend the use of time series based forecasting approaches and derive safety
stock requirements from observed forecast errors. Boylan and Syntetos (2008) give
a recent excellent overview and classification in the area of service parts demand
forecasting. These ideas are supported by many Advanced Planning Systems and
commercial service parts supply chain solutions, e.g., SAP Service Parts Planning.
According to the literature, e.g., Dickersbach (2007), the majority of methods
offered by these systems are standard time series forecasting methods including
S. Minner (&)
University of Vienna, Brünner Straße 72, 1210 Vienna, Austria
e-mail: stefan.minner@univie.ac.at
Let Dt denote the original equipment sales in period t for the product under
consideration. As a modeling example, the demands might follow some theoretical
demand distribution where mean demand follows a suggested life-cycle pattern
function. In the literature, two popular functional representations are
1. Albach–Brockhoff formula (with parameters a, b, c)
replaced upon failure, but at the latest after a maximum age has been reached.
Under block replacement, a failed component is either repaired or replaced by a
new one, for an overview, see, e.g., Dohi et al. (2003). The knowledge about
customers’ maintenance and replacement policies therefore can yield a significant
improvement and this knowledge can be utilized in order to parameterize the
component demand models presented in the following.
We follow a discrete time approach and for ease of presentation assume that states
and dynamics are recorded at the end of the respective time periods t = 1, 2,…,
EOS. The installed base dynamics are driven by new product sales Dt entering the
base and end-of-use products leaving the base. The sequence of events assumed for
the following presentation of state dynamics is: demands occur, maintenance and
replacement of components is performed, end-of-use of products is recorded, and
finally, the installed base record is updated.
Let Bt and Bit denote the number of products in total and with age i at the end of
period t, respectively. For notational convenience, let B0;t1 ¼ Dt . Further, let Eit
denote the number of end-of-use products that leave the system with age i at the
end of period t prior to installed base recording. Then, the dynamics for the age-
specific number of products is
Bit ¼ Bi1; t1 Eit t ¼ 1; . . .; EOS; i ¼ 1; . . .; t: ð8:3Þ
Table 8.1 shows these dynamics over time and per age category.
Let pi denote the probability that a product of age i is taken out of use in an
arbitrary period. p1 denotes the probability that the sale of a period is no longer in
use in the next period. Assuming independence of end-of-use within and between
age categories, each component failure follows a Bernoulli process and the end-of-
use by age category follows a binomial distribution with parameters Bi-1,t-1 and pi.
Bi1; t1 e
PðEit ¼ eÞ ¼ pi ð1 pi ÞBi1; t1 e e ¼ 0; . . .; Bi1; t1 : ð8:4Þ
e
The total number of end-of-use products in period t is the sum of the age-
specific quantities for all ages up to t,
X
t
Et ¼ Eit : ð8:5Þ
i¼1
The distribution of total end-of-use Et given the installed base at the end of
period t - 1 is the sum of t (independent) binomially distributed random variables.
The convolution can numerically be determined recursively as follows.
X
e
PðEk ¼ eÞ ¼ PðEk1 ¼ jÞPðEkt ¼ e jÞ k ¼ 2; . . .; t and E1 ¼ E1t :
j¼0
ð8:6Þ
Starting with the distribution of products with age 1, E1t, the probability that
e products of age k and less are taken out of use is given by all possible combi-
nations of j products with maximum age k - 1 and e - j products of age k. This is
recursively repeated until the distribution of the number with age k = t has been
obtained.
products of age j in period t that are added to component age category 1 and leave
component age category i. Then, the component state dynamics are given by
8
>
> Dt E1t i ¼ 1; j ¼ 1
< Pj
Cijt ¼ Mkjt i ¼ 1; j ¼ 2; . . .; t : ð8:7Þ
>
> k¼1
:
Ci1; j1; t1 Mijt Eijt j ¼ 2; . . .; t; i ¼ 2; . . .; j
In the first row, the number of components in products of age 1 is equal to
demand less first-period end-of-use. In the second row, the number of components
of age 1 in products with an age j is equal to all component failures from products
of age j in period t. The last row shows component dynamics where the number of
components equals the number in the corresponding category at the end of the
previous period less the number of component failures and of components within
products that were terminated. Then, the total number of replaced components is
X
t X
j
Mt ¼ Mijt : ð8:8Þ
j¼1 i¼1
Assuming independence as done for the analysis on the product level, Mijt is
binomially distributed with parameters qi and Ci1; j1; t1: .
Ci1; j1; t1 m
PðMijt ¼ mÞ ¼ qi ð1 qi ÞCi1; j1; t1 m m ¼ 0; . . .; Ci1; j1; t1 :
m
ð8:9Þ
For the service parts demand in a period driven by failures of components
within that periods sales, we find
X
1
dm
PðM11t ¼ mÞ ¼ PðDt ¼ dÞqm
1 ð1 q1 Þ : ð8:10Þ
d¼m
Using (8.9) and (8.10) in (8.8) and applying the same convolution logic as
presented in the previous subsection, the probability distribution of total compo-
nent demands can be determined.
As a result of the above state analysis, we can derive the probability distributions
of future service parts requirements given the installed base information at the end
of period t. Note that this information can be used in a manifold way, to give a
single value demand forecast (e.g., mean, median or mode demand), give a
confidence interval, or directly use this distribution for inventory planning and
contracting purposes.
164 S. Minner
PðBiþ1; tþ1 ¼ 8
bjBit Þ
<P 1
PðDtþ1 ¼ dÞPðEiþ1; tþ1 ¼ Bit þ d bÞ i ¼ t; t EOP
¼ d¼0
:
PðEiþ1; tþ1 ¼ Bit bÞ i\t or t [ EOP
ð8:11Þ
The determination of the probability distribution of next period’s installed base
differs for periods before EOP and after EOP. Before EOP, the probability that
b units are in use, given that currently the base consists of Bit is given by all
combinations of new sales Dt+1 and end-of-use Et+1 that yield a new base of
b. After EOP with end-of-use only, it is determined by the probability that end-of-
use products equal the difference in installed base.
For the determination of the probability distribution of service parts demands in
the next period, we use (8.8). The convolution can be performed sequentially as
described above for total end-of-use in (8.5) using (8.10) and (8.9).
In order to illustrate the benefits of the above installed base dynamics analysis,
we use a simple comparison with exponential smoothing based methods in
the following. The proposed installed-base approach is benchmarked against
simple first-order exponential smoothing (though not being appropriate under a
life-cycle pattern driven random demand model from a theoretical perspective),
the naïve forecast that next period service parts demands will equal the observed
service parts demand from the current period, and an experience driven smoothing
forecast that updates service parts demand observed in a certain period t of the
life-cycle over the repetitions of the simulation.
The experimental design and the required parameters are set as follows.
We use two life-cycle settings with a short and a long production cycle. The
parameters that represent dynamic mean demand development are LC-1 (life-cycle
8 Forecasting and Inventory Management for Spare Parts 165
Figure 8.2 shows an example of a single simulation run and illustrates original
product sales, service parts demand, and total installed base volume for each
period until EOS.
Table 8.2 shows the installed base specified by age of the built-in components
at the end of period 10, taken from one single simulation example.
In the following we benchmark the installed-base driven inventory planning
approach against three simple smoothing based methods.
Fig. 8.2 Simulation example: sales, service parts demand, and installed base
targets, short term, operational lead times are negligible. For the installed base
method, we use the derived service parts demand distribution and set the required
inventory level such that a non-stockout probability of 90% is guaranteed.
Table 8.3 shows the average inventory increase of the respective smoothing
method over the installed base approach. On average over all considered problem
instances, the information on installed base and its incorporation into tactical target
inventory planning offers an inventory reduction potential of 16% (over the
learning approach) to 50% (over simple first-order exponential smoothing). The
other rows in Table 8.3 show the respective increases of inventory levels detailed
by design parameters, that is for all instances having the same characteristics or
parameter as indicated in the first column.
As an inherent shortcoming from the exponential smoothing based methods, the
ex-ante learning capability lags behind the life-cycle development until end-of-
production and in the after-production phase, fails to properly adapt for end-of-use
and component failures that drive service parts demands. As a consequence, these
approaches do not meet the required service target early and overachieve the target
later. In order to prevent this happening, knowledge about the life-cycle has to be
incorporated. A detailed comparison for each varied parameter characteristic
shows that especially life-cycles with irregular pattern and early end-of-production
(LC-1) and late EOS have the largest impact on inventory improvement using
installed base information. The coefficient of demand variation and failure char-
acteristics do not show a clear indication of differences in improvements.
For all forecasting methods, Fig. 8.3 shows the average inventories over all 500
replications of the instance with life-cycle type 2, EOS = 15, cv = 0.4, pt = 0.1,
and qt = 0.2 detailed by the period within the life-cycle.
During the early production cycle, all smoothing methods adapt too slowly to
the service parts demands whereas in the after-sales phase with increasing
replacements and end-of-use, they do not adapt fast enough to the decreasing
number of components in use.
Table 8.3 Average relative inventory differences: time series versus installed base
Exp. smoothing (%) Naive (%) Learning (%)
LC-1 64 43 22
LC-2 36 17 9
EOS = EOP ? 5 39 23 13
EOS = EOP ? 10 66 37 16
cv = 0.2 49 29 13
cv = 0.4 44 26 24
cv = 0.8 56 34 9
p = 0.1, q = 0.2 27 17 12
p = 0.1, q = 0.1 39 24 15
p = 1/(10 - t), q = 1/(5 - t) 76 45 19
p = 1/(10 - t), q = 1/(10 - t) 59 35 16
All instances 50 30 16
168 S. Minner
We have presented a framework for using information about product and com-
ponent installed-base, component age distributions according to previous main-
tenance and replacement that is provided by recent information technology and
equipment monitoring tools. We illustrated the value of this knowledge for
medium term inventory forecasts, e.g., in negotiations and volume contracting
with component suppliers after end-of-production. Compared to (admittedly not
appropriate but widely used for dynamic service parts demand forecasting) simple
time series methods for predicting service parts demands, the framework offers a
substantial inventory reduction potential, especially for longer life-cycles. The
approach offers a framework to unlock the value of information in service parts
management by transforming the knowledge about sales, maintenance and
replacement characteristics and the current age-structure of components in the
product base into a statistical model for mid-term inventory planning.
The presented framework can be adapted and extended into different directions.
Most obvious, a comparison with more sophisticated time series methods is
required to assess the full benefits of the proposed framework. In addition, other
error measures than MSE can be utilized when comparing the different forecasting
methods. The exemplary analysis can further be refined by introducing different
customer classes with individual product use and component replacement distri-
butions. As an example, consider customers replacing certain components under a
maintenance contract and age replacement. Several assumptions about the avail-
ability and observability of information are quite strong, especially about the end-
of-use characteristics and customer loyalty to OEM parts when components need
to be replaced. Depending on the industry and product under consideration, the
available data and its quality might differ. Some or all of the parameters that were
assumed to be given need to be estimated too, e.g., within a Bayesian framework.
Another fruitful application is to extend the methodology to multi-step forecasts,
e.g., when the EOL procurement decision needs to be taken (e.g., Teunter and
Fortuin 1999). However, then the determination of the total service parts demand
distribution over several future periods is somewhat more complicated as service
parts requirements of consecutive periods are correlated (through the dependence
of base dynamics and component dynamics). As a venue for computational
8 Forecasting and Inventory Management for Spare Parts 169
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Chapter 9
A Decision Making Framework
for Managing Maintenance Spare Parts
in Case of Lumpy Demand: Action
Research in the Avionic Sector
9.1 Introduction
Table 9.1 Example of processes involved in outsourcing contracts (adapted from Kumar et al.
2004)
Candidate processes Main activities
Operative Includes all the activities at the operational level, such as the equipment
maintenance repair, intervention preparation, intervention reporting, and so forth
Maintenance Includes all the activities at the management level, such as the work order
management requests management, work order assignment, operative data storage,
resources management, and so forth
Contractual Includes all the activities needed to transfer administrative information,
management contracts and documents between customers and provider
Data management Includes all the activities related to the acquisition and transfer of data from
the equipment in a customer’s site to the provider’s service workshop,
for their further analysis in maintenance engineering, condition
monitoring, and so forth
Spare parts Includes the acquisition and storage of spare parts and analysis on data
management regarding spare parts logistics
Maintenance Includes all the activities aimed at improving the performances of the
engineering maintenance contract by leveraging on preventive (cyclic and condition
based) maintenance and proactive maintenance
KPI monitoring Includes the monitoring of maintenance activities, at different level of
analysis (e.g., KPI at operative, tactical, strategic level)
Training activities Includes the activities offered to the customer’s maintenance personnel in
order to enable the transfer of information and knowhow required by
the maintenance interventions
system to manage differences between the SL and the effective outcomes, measured
by a KPIs’ dashboard covering the main aspects included in the outsourcing scope.
A critical issue that may arise when discussing about maintenance outsourcing
regards the identification of the processes that can be effectively outsourced. In
order to provide a quick reference, Table 9.1—adapted from Kumar et al.
(2004)—presents, with no claim of exhaustiveness, a range of candidate processes
for outsourcing.
In this chapter we consider the case of an Original Equipment Manufacturer
(OEM) contractually committed—in its role of service provider—to execute four
of the above mentioned processes:
Operative Maintenance: the OEM performs repair activities on pieces of equip-
ment sent from the customer to a centralized service workshop.
Maintenance Management: the OEM manages work order requests for the repair
activities coming from different customers’ sites.
Contractual Management: all interventions are reported and any document
required in the contract is duly completed to be available for customer’s audits.
Spare Parts Management: whenever a defect is detected at the service workshop,
the component must be replaced; the OEM is responsible for the availability of
the proper components at the repairing premises.
Focusing in particular on the Spare Parts Management process, a structured
procedure supporting the management of a service contract from the OEM
standpoint is presented. Such a procedure tackles three main decisions:
9 Decision Making Framework for Managing Maintenance Spare Parts 173
Spare parts management is undoubtedly not a novel topic in the academic and
industrial world. Several approaches, coming mainly from logistics, operations
management and operations research communities, have been proposed in the past.
Given the plethora of contributions, there have been many systematic overviews
and comprehensive surveys, among which it is worthwhile to cite the works of
Guide and Srivastava (1997) and Kennedy et al. (2002).
A visible fall-out of such an intensive strive has been the development and
commercial proposal of numerous vertical IT solutions, often hyped to the
potential industrial customers as the real panacea for their day-by-day issues in
managing their spare parts inventory. Not neglecting the substantial added value
provided by some of these solutions, what seems really missing in the industrial
practice is the capability to follow a sound and consistent logical procedure in
tackling the spare part management problem. This motivated the same authors to
propose in Cavalieri et al. (2008) a stepwise decision making path in order to
orienteer an industrial manager on how to pragmatically handle the management
of maintenance spare parts in an industrial company. The framework is organized
into five sequential steps: (i) Part coding, (ii) Part classification, (iii) Part demand
forecasting, (iv) Stock management policy selection, (v) Policy test and validation.
A summary of the objectives of each step is provided in Table 9.2.
Since in the performed action research the part coding resulted as a non-critical
step, the remainder of this chapter will specifically address the other four phases.
The identification of criticality of spare items is the primary output of part clas-
sification. This is normally obtained by considering different aspects of the parts
and the considered environment. It is worth noticing that there are several defi-
nitions of criticality; hence it is useful to delineate criticality in a meaningful way
for the purposes of the chapter.
According to Dekker et al. (1998), criticality is the level of importance of a
piece of equipment for sustaining production in a safe and efficient way. A clas-
sification of level of equipment criticality can be used to put evidence on the
critical spare parts which deserve more attention and are to be kept in stock to
sustain production. Along with Huiskonen (2001), the criticality of a spare item is
related to the consequences caused by the failure of a part on the process in case a
replacement is not readily available. Hence, it could be named as process criti-
cality. Therefore, a spare part can be considered critical for the process when it
causes either long lasting or frequent stoppages (see also Schultz 2004), while no
alternative production facility is available in order to guarantee the production
continuity (see also Gajpal et al. 1994).
9 Decision Making Framework for Managing Maintenance Spare Parts 175
Table 9.2 The five decision making steps of the framework (adapted from Cavalieri et al. 2008)
Steps Objectives and contents
Part coding A specific spare item coding system has to provide a prompt
understanding of the technical features of the item, the equipment
tree it refers to, the involved supplier (especially for specific and
on-design parts) and, for stocked items, their physical location in
the warehouse. A part coding system is mandatory in order to
make decisions properly, since it realizes a rationalized set of data
on which decisions can rely upon
Part classification A proper classification of spare items is needed because of the high
variety of materials used for maintenance and repair purposes;
their technical and economical features can be highly different. A
proper classification system should give fundamental information
for establishing the criticality and, as a consequence, the strategies
for developing the logistics for different classes of maintenance
spare parts
Part demand forecasting Special forecasting techniques are required for some types of spare
items. Neglecting consumables, a common feature of many spare
items is their relative low level of consumption, due to breakdown
or preventive maintenance. Sometimes, time intervals between
requests span over several years. Moreover, the consumption rate
of a spare part is highly dependent on the number of pieces of
equipment where the part is installed, as well as its intrinsic level
of reliability
Stock management policy A stock management policy customized upon each class of spare
selection items is required; it might range from no-stock and on-demand
policies to the traditional EOQ-RL (Economic Order Quantity -
Reorder Level) approach
Policy test and validation Test and validation of the results achieved applying the above
mentioned steps are to be accomplished and refinement may be
applied when necessary. This can be obtained by what-if analysis
carried out in different scenarios of consumption and supply/repair
of the spare items
Beside aspects concerning the consequences on the process, other aspects are
related to the possibilities to control the logistics in a given service supply chain
setting. To this concern, the same Huiskonen (2001) suggests a so called control
criticality, including aspects more related to criticality in managing the spare
parts logistics as: (i) the availability of spare part suppliers, (ii) the supply lead-
times, (iii) the presence of standard versus user-defined features of a spare item,
(iv) the purchasing and inventory holding costs, (v) the demand patterns (either
easing or not the predictability of the maintenance events). Other authors mention
aspects related to the control criticality as the case of Haffar (1995), Dhillon
(2002), Mukhopadhyay et al. (2003), Syntetos (2001) and Ghobbar and Friend
(2002).
Part classification methods represent a support in considering all the aspects
needed to characterize both the process and the control criticality. A part classi-
fication may adopt both quantitative methods, implying the adoption of drivers
176 M. Macchi et al.
Spare parts demand time series can show diversified patterns, depending upon the
type of part considered and the specific industry. In most cases, spare parts demand
is characterized by a sporadic behavior, which implies a large proportion of zero
values (i.e., periods in which there is no demand at all) and a great variability of
demand size, when it occurs. The consumption rate is not stationary; hence the
demand statistical properties are not independent by the time.
Describing the patterns of sporadic demand for forecasting purposes, the terms
erratic, intermittent and lumpy are often used as synonyms. A better specification
of these terms can be made by introducing two explicit measures of the demand
patterns (Syntetos 2001):
1. The average time between two consecutive orders of the same part, evaluated
through the Average Demand Interval (ADI) coefficient.
2. The variation of the demand size, evaluated through the square of the Coeffi-
cient of Variation (CV).
Depending on the values of these two indicators demand patterns can be
classified into four categories:
1. Smooth demand, which occurs randomly with few or none periods with no
demand and with modest variation in the demand size.
2. Intermittent demand, which appears randomly with many time periods having
no demand, but without a substantial variation in demand size.
3. Erratic demand, which is highly variable in the demand size and presents few or
none periods with no demand.
4. Lumpy demand, with many periods having no demand and high variability in
the demand size.
Table 9.4 provides an outline of the most common methods for forecasting
spare parts. In particular, two primary classes of techniques are:
– Reliability based forecasting (RBF), to be used when the installed base, that is
the number of current installations and their own technical operating conditions,
is known.
– Time series forecasting (TSF), suitable when the only available data are related
to the time series of the spare items consumption or repair records, while no
information about the reliability of the installed base is retrievable.
178 M. Macchi et al.
Based on the result of the forecasting step, this step aims at selecting an inventory
model (implementing a stock management policy or, briefly, a stocking policy),
and subsequently defining the stock size in each warehouse for those items that is
advisable to detain. Well known inventory models are:
– the continuous review, with fixed reorder point (r) and fixed order quantity (Q),
referred to as (Q, r);
– the continuous review, with fixed reorder point (s) and order-up-to level (S),
referred to as (s, S);
– the periodic review, with fixed ordering interval (T) and order-up-to level (R),
referred to as (T, R);
– the continuous review and order-up-to level (S) in a one-for-one replenishment
mode, referred to as (S - 1, S).
Specific models have been proposed in many case studies, mainly based on a set
of rules and algorithms tailored for each single case. As an example of this kind of
approach, special applications in many sectors like computers (Ashayeri et al. 1996),
airline (Tedone 1989), bus fleets (Singh et al. 1980), power generation (Bailey and
Helms 2007), and military (Rustenburg et al. 2001) are reported. Being industry-
specific, the portability of these models to other industrial settings is generally poor.
In general, a clear application grid of the different stocking policies with an
unambiguous understanding of the assumptions, starting hypothesis, area of
application (in terms of classes of items) and expected performance (in terms of
inventory or hidden costs) is still lacking today. According to Guide and
Srivastava (1997) and Cavalieri et al. (2008), the most important and critical
drivers that should be considered in selecting a proper inventory model for stock
sizing decisions are: (i) the demand pattern (deterministic or stochastic), (ii) the
degree of repairability of the spare items, (iii) the level of centralization/decen-
tralization (with inventory located either in a central site, in multiple decentralized
sites or with a mixed configuration between centralized and decentralized sites).
By combining the possible values of these drivers specific models can be
properly selected, as reported in Table 9.5.
Last step of the decision making process involves simulation in order to ensure that
the selected stocking policy is the most appropriate. Simulation is suitable to verify
whether the stock sizing decision taken at the previous step is robust and consistent
under different stochastic scenarios of consumption and supply/repair of the spare
items. In fact, it can be used to generate and register, on a simulated time scale, the
stochastic behaviors of repairable and non repairable systems. Based on a random
generation mechanism—the well known Monte Carlo sampling, see (Dubi 1999)
180 M. Macchi et al.
Table 9.5 Short review of models for stock management policy selection
Types of models Main references
Inventory models for stochastic demand with Archibald and Silver (1978): Inventory policies
non-repairable items for a continuous review system with discrete
compound Poisson demand. The paper
presents a recursive formula to calculate the
cost for any pair (s, S) and relations among s,
S, S - s and the cost that leads to an efficient
determination of the optimal s and S
Dekker et al. (1998) and Jardine and Tsang
(2006): Poisson model for calculating the
stock size S of a (S - 1, S) inventory model,
such that the demand may be directly
fulfilled from stock on hand at a given target
fill rate (i.e. probability of not running out of
stock when a failure occurs) during the
replenishment lead time T (which is a
constant supply time of the new item)
The Poisson model can be adopted as a good
approximation for the stock sizing of spare
parts when the demand rate in a period T is
‘‘not very high’’. In practice, this is valid for
the slow moving parts
Inventory models for stochastic demand with Jardine and Tsang (2006): The Poisson model
repairable items can be also adopted for the case of
repairable items, considering the
replenishment lead time T as a constant time
to repair (of the repairable items). This
means to assume an infinite repair capacity
Several issues related to the repair activities
should be included in the model, like (i) the
replacement by new items, when the
repairable items are worn beyond recovery,
so that they cannot be repaired anymore and
have to be condemned (Muckstadt and Isaac
(1981), Schaefer (1989)); (ii) the finite
repair capacity of the repair shop (Balana
et al. (1989), Ebeling (1991))
Inventory models for deterministic demand Cobbaert and Van Oudheusden (1996):
with non repairable items Modified EOQ models for fast moving parts
undergoing the risk of unexpected
obsolescence. These models are applicable
mainly to fast moving consumable items
with regular demand volumes; they can be
thought as modifications of existent models
for inventory management in manufacturing
(continued)
9 Decision Making Framework for Managing Maintenance Spare Parts 181
This study has been carried out using Action Research (AR) as the empirical
methodology for applying, testing and evaluating the consistency and support to
industrial decision makers of the proposed spare parts management framework.
182 M. Macchi et al.
Table 9.6 Short review of applications of the simulation method for policy test and validation
Potential applications References (only representative examples)
Verification of the service levels guaranteed by Dekker et al. (1998): Simulation is adopted to
a stocking policy verify the service level in a system, where
the stocks are already established based on a
critical-level policy sized by means of
analytic approximation
Verification of the availability of a system De Smidt-Destombes et al. (2006): The
based on different settings of maintenance availability of a k-out of-n system with
logistics support deteriorating components and hot standby
redundancy can be influenced by the
combined decision on different variables
concerning the maintenance logistics
support: the conditions to initiate a
maintenance task, the spare parts inventory
levels, the repair capacity and repair job
priority settings
Joint verification of the age replacement and Zohrul Kabir and Farrash (1996) and Sarker and
spare parts provisioning policy Haque (2000): a joint optimal age
replacement and spare parts provisioning
policy is searched for, based on the analysis
of the effects resulting from factors like age
based item replacement, shortage and
inventory holding costs, order supply lead
time
but situational; (ii) the action researcher is immersed in the problem setting and not
a mere observer of phenomena; (iii) the research situation has demanded
responsiveness, as the research occurs in a changing environment in real-time.
On the other hand, as Conboy and Kirwan (2009) assert, there are some evident
limitations a researcher should be well aware of when applying such a method-
ology. In particular, AR is much harder to report and implies an heavy involve-
ment of the action researcher in the research situation, with the opportunity for
good learning, but at the potential cost of sound objectivity.
The action research deals with a company producing and servicing radars, whose
customers are military air forces operating a relatively wide number of jet fighters.
The OEM is contractually committed to execute the repairing of the radars
installed on customers’ jet fighters. Since the radar is a critical part of the jet
fighter (a jet cannot be operated if the radar is not available) it represents an
important cause of Aircraft On Ground (AOG) time when radar replacements are
unavailable.
In order to reduce the AOG risk, the customer itself can stock radar modules to
be substituted on field: these are the so called Line Replaceable Units (LRUs) and
are directly replaced at military hangars without the direct intervention of the
OEM. A customer requires support from the OEM on other modules which are
sub-components of a LRU: these are the so called Shop Replaceable Units (SRUs).
Figure 9.1 outlines a simplified bill of material of a generic radar.
SRU and LRU (the latter in case that the damaged or malfunctioning SRU
cannot be identified at the customer’s premise) must be sent to the OEM in order to
be repaired. Each air force tends to accumulate a batch of failed SRUs/LRUs, in
order to send them all together for being repaired, reducing the number of
shipments needed, and to control the costs of the outbound logistics. However, due
to the scarce integration of the OEM with its customers, the decision on the batch
size is up to the customer. Along with the relatively low number of customers, the
customers batching policy is recognized as a source of lumpy demand (Bart-
ezzaghi et al. 1999).
At OEM’s premises the broken SRUs/LRUs are first sent to the Failure Anal-
ysis unit where they are tested for defects. After detecting a malfunction, the SRU
is sent to the Repair unit if it is assumed that it can be repaired. The LRU without
SRU (i.e., incomplete LRU) is sent to the Control unit, where it should be reas-
sembled together with the repaired SRU.
Another possibility is that the LRU or SRU is not repairable. Then, a complete
new product has to be sent to the customer. Moreover, it is possible that the Failure
Analysis unit did not detect any malfunction. Then the ‘‘not broken or failed’’
SRU/LRU is sent to the Control unit for the Final Inspection.
Figure 9.2 reports in a IDEF format the main logistics flows involved in the
radar repair process.
The overall logistics performance is evaluated by measuring the Time to
Restore (TTR). Along with the active time required for isolating and replacing the
fault components (that is diagnostics and repair time in Fig. 9.3), there are specific
time components which are due to the logistics support to the maintenance
activities.
As an example, if the spare part is not detained, there could be a supply time
needed in order to contact and negotiate with the supplier the delivery of the
components to be replaced during the repair activity. Also the administrative and
logistics delay counts for the outbound logistics: this mainly refers to the move-
ment of the batch of SRUs from and to the customer’s site.
9 Decision Making Framework for Managing Maintenance Spare Parts 185
Fig. 9.3 Typical time components of the Time to Restore (TTR) of a repairable item
Clearly a long TTR negatively impacts on the performance of the OEM and the
service level provided to the customer, since it increases the risk of AOG
(Fig. 9.3).
The OEM is responsible for a portion of the whole TTR, that in turn is com-
posed by three elements:
– the active repair time (ART);
– the supply time (SUT) for any spare item required for replacement;
– any administrative and logistics delay (ALD) registered within the scope of its
service workshop, that is from the entrance until the exit of a batch of SRUs,
before moving to the customer’s site.
The SLA established in the contract is applied to this part of the TTR (hereafter
referred to as TTROEM).
A fast delivery from the OEM (that is, a low TTROEM) entails a reduction of AOG
risk. Hence, in order to induce quicker repair activities, the OEM is rewarded with
a bonus when it keeps the TTROEM lower than an established threshold (in the
remainder this is referred as TTRLowerBound
OEM ). Conversely, the OEM has to pay a
malus, as a penalty cost, whenever it exceeds a threshold (in the remainder referred
to as TTRUpperBound
OEM ). If the TTROEM falls between these thresholds the customers
pay the regular price defined in the contract. TTRUpperBound
OEM and TTRLowerBound
OEM are
the service levels agreed by the OEM in the contract.
The following expressions provide the mathematical formulation of the bonus/
malus. Firstly, the number of repaired items generating a bonus (NB) or a malus
186 M. Macchi et al.
(NM) can be evaluated starting from the repair orders delivered back by the OEM
to its customers (i being the index of each order, see Eqs. 1 and 2).
Order Sizei if TTROEM TTRLowerBound
OEM
NBðiÞ ¼ ð1Þ
0 if TTROEM [ TTRLowerBound
OEM
Order Sizei if TTROEM [ TTRUpperBound
OEM
NM ðiÞ ¼ ð2Þ
0 if TTROEM TTRUpperBound
OEM
NB(i) and NM(i) are indexed in accordance to the number of the repair orders
delivered back to the customer, starting from i = 1 (first order delivered back to
the customer) until i = I(T) (last order delivered back to the customer). I(T) is
dependent on the horizon length T considered for decision making. Therefore,
based on such a set of orders, it is possible to define the cash flow (CFT) of bonus/
malus (Eq. 4), which, as well, depends on the horizon length T.
I ðT Þ
X
CFT ¼ ½Bonus NBðiÞ Malus NM ðiÞ ð4Þ
i¼1
In this case the bonus/malus are constants, not dependent on the type of items
under repair, the overall number of repaired items (no discount or ‘‘economy of
scale’’-like effects) and the customer.
The objective of the OEM is to maximize the cash flow, either by increasing the
bonus gained or reducing the malus to be paid. The leverages to this end are
ARTOEM, SUTOEM and ALDOEM (Fig. 9.3). Keeping constant the capacity of the
repairing shop floor, the inbound logistics and the administrative management, the
only time component that might change the performances is the SUTOEM. This can
be reduced through a sound management of spare parts inventory.
The following objective function (OF) is then introduced (Eq. 5).
OFT ¼ CFT IHCT ð5Þ
Different spare items s can be kept in stock (up to N types), each spare item
having its own level of inventory (S(t,s)); hence, the Inventory Holding Cost
(IHCT) is evaluated by integrating, over the horizon T, the value progressively
reached by the level of Inventory at each time t for each spare item s kept in stock
(S(t, s)) (see the following Eqs. 6 and 7).
N Z
T
X
IHCT ¼ IHCðsÞ Sðt; sÞdt ð6Þ
s¼1
0
where IHs inventory level for spare item s, representing the decision variable of the
problem; D(t,s) total number of spare items s demanded by the repair orders issued
over the horizon t; R(t,s) total number of spare items s restored over the horizon t;
IHC(s) inventory holding cost per unit of stocked item and unit of time for spare
items.
In order to guarantee the profitability of the contract, the objective for the OEM
is to achieve a good balance between (i) the improvement of the total bonus and
reduction of the total malus and (ii) the reduction of the inventory holding costs
(Eq. 5).
D(t, s) and R(t, s) are the two stochastic variables which makes this balance
probabilistic in nature. As better discussed later, indeed, D(t,s) has been considered
as the most influencing variable, so a more robust study of its behavior is needed.
After having described the industrial context of the case study and the main issues
and motivations for the company to change its current practice of managing its
spare parts logistic system, the following subsections will provide a detailed
explanation on how the decision making framework has been applied by following
the logical steps illustrated in Table 9.2. During this phase of the action research,
the industrial counterpart personnel has been involved by the action researcher,
according to the specific competences and role required.
indeed, the TTROEM is higher than the upper bound set in the contract clause.
Conversely, a bonus can be expected in the ‘‘to-be’’ scenario, with inventory
holding of SRU A: TTROEM is reduced to the diagnostics time (which is part of the
ART component) required for the first failure analysis plus some administrative
and logistics delays (the ALD time component); these are far below the lower
bound of the contract clause.
SRU C is ranked lower in criticality; hence it is classified as Essential and a
malus is never incurred, on average, also without inventory holding in the ‘‘as-is’’
situation. Last but not least, SRU M and SRU N differs only for the confidence that
they have to reach the bonus: both guarantee a bonus, even without inventory
holding; however, SRU M is closer to the lower bound than SRU N, hence the
decision maker is less confident that the bonus can be achieved; in this concern,
SRU M is considered Desirable, while SRU N is not considered at all in further
steps of the analysis.
Control criticality—Decisions about SRUs stocks impact on inventory holding
costs differently if compared to keeping stocks of other components at lower levels
of the BOM, like modules of electromechanical components of SRUs or, directly,
the electromechanical components. At this step, however, it is not easy to provide
a detailed assessment for balancing costs and grants, since the alternative stocking
policies, possible at different levels of the product BOM, lead to a combinatorial
problem. After having carried out the criticality analysis, the stocking policy is
selected working only on a subset of spare items—i.e. the critical items thus
reducing the size of the search space.
In a first step, an ABC classification is performed, by assuming the IHC(s) (for
unit of stocked item and time of item s) as the main driver for the cumulated Pareto
analysis: this provides a criticality ranking of most valuable SRUs (i.e. items
having the highest financial impact when kept on stock). This is clearly not
complete, even if it enables to identify, specifically amongst the SRUs, the class A
items, critical for their high values with respect to other SRUs (the B and C class
items), for which is less expensive keeping inventories.
In addition, when the demand is lumpy, the intermittent occurrences and high
variation of the order sizes finally lead to criticality in logistics, since the demand
predictability is challenging. According to the classification method presented in
Sect. 9.2.1 and the cut off values provided in Syntetos (2001), herein just con-
sidered as a reference benchmark, the demand is lumpy if CV2 [ 0,49 and
ADI [ 1,32. In this concern, all the SRUs shown in Fig. 9.4 represent critical
items, because they are expression of a lumpy demand.
Multi-dimensional criticality analysis—Three types of methods have been
applied leading to three independent spare parts criticality rankings. The three
rankings are now combined together for selecting the SRUs that deserve attention
in the further steps of the decision making process.
– The VED classification is helpful in order to define a priority list to be con-
sidered in the following steps by the OEM, according to the process criticality.
Keeping into account the maximization of the profitability of the contract,
190 M. Macchi et al.
cv2
8,00 SRU e
SRU b
SRU a SRU f
6,00 SRU g
SRU h
4,00 SRU d SRU h
2,00
0,00
0,00 2,00 4,00 6,00 8,00 10,00 12,00
ADI
specifically the grants in the bonus-malus formula, the stock management policy
is selected starting first from all vital items, which guarantee better grants
incoming from achievement of bonus and avoidance of malus.
– Referring to the control criticality, the ABC classification, based on the IHC(s)s
(for each unit of stocked item and time of spare item s), is also helpful: it would
be preferable to stock first the SRUs pertaining to class B and C, which cause
less financial loads.
– Finally, three different classes of control criticality can be also defined based on
the demand pattern and demand predictability, in particular considering only the
average demand interval (ADI): preference for selecting a stocking policy is
assigned to the most frequent issues of repair orders, equivalent to lower ADIs
(so a reduced intermittency of order arrivals). Three classes are defined to this
concern: a class with more frequent issues (less intermittent arrivals), which, in
the action research, is set to when ADI is lower than 6 months; a class with less
frequent demand (more intermittent arrivals), with ADI higher than 6 months
but lower than 12 months; a class with very low frequency, i.e. ADI higher than
12 months. The last class is not kept into account in the remainder, having
considered more than 12 months a long time span to wait for a new order to
consume the spare item.
The flow chart in Fig. 9.5 summarizes the combined adoption of the three
methods. On one hand, the priorities are assigned to SRUs whose repair orders
occur more frequently (i.e., ADI lower than 6 months, which guarantees a less
intermittent consumption of inventories), and less valuables with respect to their
unit inventory holding cost (B and C classes in the ABC classification). On the
other hand, only if budget is still available, the remaining vital and desirable items
are considered for stock management policy, also amongst the most valuable items
(class A of IHC(s)s) and less frequent (ADI more than 6 months but lower than
12 months).
Only the SRUs selected from criticality analysis, according to the flow chart in
Fig. 9.5, have been considered from now on: indeed, the results, achieved until the
second criticality ranking lead to select eight critical SRUs.
9 Decision Making Framework for Managing Maintenance Spare Parts 191
Fig. 9.5 Flow chart for combining the methods adopted for spare parts criticality analysis
The partial or null visibility on how the Radars, and so their SRUs, are actually
utilized on field by the military customers on their jet fighters make forecasting
activities extremely hard: number of operating hours, operating conditions and
mission profiles are unknown, due to the military reluctance to reveal data.
Considering the uncertainties on this operational information, a RBF method (see
Table 9.4) based on life data analysis of different items on field is not applicable.
The scarce information about the real conditions of use and the duty cycle of
each item operating on field obstacle, in practice, also the RBF method based on
data banks. The last option is the time series forecasting (TSF), since data related
to the time series of the spare items consumption and repair records are available
in the Maintenance Information System and are duly completed during the con-
tractual management of the service; indeed, the commitment to proper contractual
management can be considered a relevant factor for guaranteeing the good quality
of the data records. The main issue to solve regards demand predictability. A clear
definition of which forecasting method is more suitable for the demand type of
each quadrant of the classification matrix of Fig. 9.4 is not an easy task in general
and this is particularly true for the lumpy demand quadrant, for which the demand
predictability is the most challenging. In the scope of this action research, it has
been preferred to test the stocking policies for the critical SRUs in different
simulated demand scenarios, instead of making use of a forecasting method to
obtain accurate spare parts forecasts. In this concern, policy test and validation
(step 5 of the decision making framework) has become relevant. The reason for
this preference is strictly subsequent to the high character of lumpiness observed
192 M. Macchi et al.
for the demand, especially due to the high variation of the order sizes: CV2 is in
the range between 5.21 and 13.62 (Fig. 9.4) revealing a quite relevant variation in
the demand size, combined with its high intermittency. This certainly imposes
high requirements on the forecasting method, potentially asking for a complex one,
with the hidden risk of only a partial capability to precisely fit, in the forecast, the
repair order distribution in time and quantity.
Accordingly, it has been decided to rely on the empirical probability density
functions of two primary stochastic variables, in the remainder symbolized as DI
(Demand Interval) and ROS (repair order size). These functions can be easily
extracted from the time series, expressed in the form of histograms of frequencies:
indeed, the length of the period kept under observation (i.e., 6 years) guarantees a
robust sample for building the histograms.
Looking ahead at the next steps, averages—like ADI—are the only statistics
being adopted for the rough stock sizing (step 4), before passing to policy test and
validation (step 5), wherein the histogram of frequencies themselves is directly
being exploited. This last issue means that other summary statistics, characterizing
the probability density functions (as CV2), have been considered during policy test
and validation. The empirical probability density functions of each critical SRU—
hence, the histograms and their related summary statistics—are the basis to enable
a Monte Carlo simulation of different demand scenarios. Further details about that
are provided in the step of policy test and validation (see Sect. 9.2.4).
Figure 9.6 summarizes the links between spare parts forecasting and the other
steps of the decision making framework, by showing the respective expressions of
the empirical observations used by each further step.
Among the different models available in literature, the continuous review and
order-up-to level (S) in a one-for-one replenishment mode, referred to as (S - 1, S),
Histograms of
frequencies
Fig. 9.6 From spare parts forecasting to next steps in the decision making framework
9 Decision Making Framework for Managing Maintenance Spare Parts 193
is more advisable for this case. It should be kept into account that the repair order
size varies, depending on each customer: so the one-for-one replenishment is
applied directly to the repair order that is received, meaning that one batch of SRUs
is repaired and is used, without being splitted, to replenish the stock level on hand,
previously reduced of the same quantity delivered to the same customer.
The selection of the (S - 1, S) model is in line with many references in lit-
erature of similar examples where the demand rate in a period T is assumed ‘‘not
very high’’ (see for example Jardine and Tsang 2006). Also in our case, as the ADI
shown in Fig. 9.4 demonstrates, the situation implies a demand rate in a period
T which is ‘‘not very high’’, leading to a slow moving character of the spare items.
The Poisson model can be adopted as a good approximation for the stock sizing of
spare parts in such a situation (as already outlined in Table 9.5).
Applying the Poisson distribution, a stock size S to be kept can be calculated,
based on a target level of fill rate R (i.e. the probability of not running out of stock
when a failure occurs). Equation 8 expresses the general formula of the Poisson
model.
X
S1
ðd T Þi
Prfs1 þ þ sn [ T g ¼ edT R ð8Þ
i¼0
i!
where s1, … , sn represent the times to each failure, requiring the spare part (until
the n-th request), and are assumed to be independent positive random variables;
d is the demand rate per unit period and it is, in general, estimated according either
to a reliability based or a time series based forecasting; T expresses the time
interval taken as a reference for the target fill rate.
Applying the formula to the data of the case study:
– due to the simplification done for spare parts forecasting, at this step d is esti-
mated only on average, simply as 1 repair order for each ADI (1/ADI), so using
only part of the summary statistics available for each SRU;
– T is defined by considering the Poisson model calculated in the case of
repairable items (according to the classification illustrated in Table 9.5), hence
it represents the MTTROEM for the repair orders;
– the stock level S is fixed at a number of items close to the Average Repair Order
Size (AROS) keeping into account the fact that, at each ADI, the number of
items requested from stock is AROS.
Table 9.8 shows the rough stock sizing resulting from applying the Poisson
model: S has been rounded to the closer integer number (lower or higher); R is the
resulting target rate for this choice.
It is worth pointing out that R represents an analytic approximation that should
be further verified; besides, R means the instantaneous or point reliability, pro-
viding that spares are available on demand, at any given moment in time; while the
interval reliability (i.e., spares are available at all moments in a given interval) is
not estimated and would be ‘‘more demanding’’ (Jardine and Tsang 2006): the
following step of policy test and validation may help to provide more accurate
194 M. Macchi et al.
Table 9.8 Stock sizing Part code AROS (# items) S (# items, rounded) R (%)
based on the Poisson model
SRU a 1.33 1 61.8
SRU b 1.22 1 88.1
SRU c 1.63 2 99.6
SRU d 3.13 3 99.7
SRU e 1.88 2 99.9
SRU f 3 3 99.9
SRU g 2 2 99.9
SRU h 1.27 1 98.5
estimates of the performances expected by the service workshop, both for R and
other additional measures.
Furthermore, S is clearly not the final solution: it is instead an initial and rough
solution that will be improved again in the next step; hence, rounding to the lower
or upper integer is an acceptable choice at the current phase.
Policy test and validation enables to verify the stocking policies selected for the
SRUs. The initial stock sizing provided in the previous step 4 is at this phase verified
and progressively improved thanks to a search procedure enacted together with the
decision maker and the aid of the simulation used as a test bench tool (Fig. 9.7).
9 Decision Making Framework for Managing Maintenance Spare Parts 195
Table 9.10 Rules associated Part code ADIs (calendar months, ascendant order) Action
to the spare parts criticality
ranking based on the ADIs SRU b 3.38 Increase
SRU d 4.00 Increase
SRU c 4.80 Increase
SRU h 5.00 Keep
SRU a 5.25 Keep
SRU e 6.00 Keep
SRU f 7.34 Decrease
SRU g 10.25 Decrease
The search procedure is derived directly from the multi dimensional criticality
analysis played out during the step of part classification.
In order to improve the solution, three actions are defined: increase or decrease
the stock size (by one or more items) or keep it constant. Each action is associated
with the spare parts criticality rankings of the critical items (Tables 9.9, 9.10 and
9.11) resulting from the multi dimensional criticality analysis of part classification.
The associations express the behavior of the logistics expert who generally
tends to:
196 M. Macchi et al.
1. decrease stocks of items when they have a high IHC(s) or, vice versa, increase
stocks of items having a low IHC(s)s (Table 9.9 shows the rules associated to
the ABC classification, cumulated Pareto analysis, of IHC(s));
2. increase stocks of items frequently requested, while decrease those not fre-
quently requested (Table 9.10 reports items sorted from low ADIs—orders
more frequently requested—up to high ADIs—orders less frequently reques-
ted—rules are likewise associated);
3. increase or at least keep the stocks when the items are vital or essential, to gain
grants from the bonus-malus contract (the ranking in Table 9.11 is descendent,
being the top ranking occupied by the item with the highest MTTROEM, so the
highest risk, on average, to pay a malus and not be rewarded with a bonus).
The rules are then combined in order to form an integrated rule set (Table 9.12)
as a guideline for deciding on how to improve an initial solution (i.e. the stock
size). Using this rule set, a trade off may emerge in some cases. In order to manage
in a simple way these uncertain cases, alternative criteria have been selected: (i) a
random criterion is the simplest one; (ii) a criterion where either the process
criticality or the control criticality are dominant in the multidimensional analysis.
Table 9.12 demonstrates an example of improvement of an initial solution by
means of the search procedure, passing from an initial stock SI to a final stock SF:
(i) in clear situations prevail the most voted action (as an example, when at least 2
out of 3 increase are present, increase is selected; see these situations highlighted
in italic in the same table); (ii) uncertain situations are the remainder (i.e., three out
of eight SRUs highlighted in bold); in this specific case, the decrease or keep
action are prevailing and the control criticality is considered dominant. It is
however worth noticing also that, when trying a possible improvement, it has been
kept in mind the criticality of the eight items, as a binding rule: in this context, it
has been decided that a ‘‘minimum stock’’ should anyhow be guaranteed (i.e. S
equal to 1 item as minimum). This applies to one case in the example, SRU b,
where, due to this binding rule, the keep decision prevailed.
Table 9.12 Finding out a solution during the search procedure driven by the integrated rule set
I ðT Þ
X
TNMT ¼ NM ðiÞ ð10Þ
i¼1
I ðT Þ
X
TNPT ¼ NPðiÞ ð11Þ
i¼1
I ðT Þ
X
TNNPT ¼ NNPðiÞ ð12Þ
i¼1
These performances are clearly defined in order to be aligned with the profit
function (see the previous Eqs. 4–6). Indeed:
– the Total Number of Bonus and Total Number of Malus (Eqs. 9 and 10) directly
influence the grants coming from the cash flow of bonus/malus (Eq. 4): when
assuming, for example, the same value for the bonus and malus of an item,
profitability is guaranteed if the total number of bonus (i.e., items achieving the
bonus) is higher than the total number of malus (i.e. items achieving the malus);
a similar reasoning can be applied when knowing that the ratio r between the
bonus and malus parameters of an item is different than 1; in this case, profit-
ability is guaranteed if the total number of bonus is higher than 1/r times the
total number of malus;
– on the other hand, inventory holding costs are also part of the profit function
(Eqs. 5 and 6); hence, the Total Number of Profitable items (Eq. 11) is used in
order to measure a subset of Total Number of Bonus; this subset counts only
those items for which the achieved Bonus is higher than the Inventory Holding
Cost spent for the time that the same items are kept in stock before being used;
vice versa, in the case of Total Number of Non Profitable items (Eq. 12), the
achieved Bonus is lower than the Inventory Holding Cost; hence, these
performances are also aligned to the profit function, now having a specific
concern to the ratio between benefits (the bonus achieved by each item) and
costs (the IHC(s)s spent for each item on stock for some given units of time).
The results of the progressive search, as a whole, are shown in Table 9.14: the
marginal improvement progressively observed for the selected performances
decreases, passing from the ‘‘minimum stock’’ situation (S1) to the last trial (S4)
for the ‘‘to-be’’ scenario. The last trial in the table provides a solution which,
amongst all the trials, is characterized by the highest service level: it shows the
highest number of bonus TNB (high service due to quick response) and a limited
number of malus TNM (low service due to slow response). Besides, it is worth
noticing how this last trial guarantees that, among the orders reaching the bonus,
the total number of profitable items TNP is clearly higher than TNNP. These are
possible ‘‘to-be’’ solutions and have to be compared, of course, with the ‘‘as-is’’
case, without inventory holding, so to assess how much improvement each solu-
tion could guarantee, with respect to the existent operation.
Most of the chapter has been devoted to a detailed description of the way the
multi-step decision making procedure has been planned to be properly applied in
the specific industrial context. The following main outcomes emerged from the
feedbacks gotten from the management involved in the action research.
– Complex, analytical models sometimes assume stringent hypothesis and a
reduced scope of application. Due to the peculiarities of the presented process in
the avionic sector, an analytical model would have been too complex and dif-
ficult for industrialists to tackle with reasonable effort. On the other hand, a
combination of simple models, organized within a well structured procedure,
can overcome the practical issues imposed by larger models providing a robust
decision support.
– The judgment of the decision-making operators can be effectively supported by
quantitative measures that reduce the ambiguity of qualitative decisions and
improve the effectiveness of the choice. As an example in the action research,
the assignment of criticality of spare items is one relevant step in order to
initially define the strategies for developing the maintenance spare part logistics,
and there is little room for subjectiveness thanks to the quantitative measures
adopted.
– A simulation model is a powerful and user-friendly tool suitable for the analysis
and validation of the decisions taken about stock sizing. Besides, its combined
use with the judgment of the operators can effectively improve the outcome
deriving from first rough solutions.
– From the managerial point of view, the usage of a well defined decision making
framework can improve the overall performance by aligning the decisions to the
specific business scenario, giving sound evidence to the results of the strategic
200 M. Macchi et al.
managerial choices in a company. The proper definition of the stock sizes can
definitely lead to a better exploitation of the financial capital, while preserving
(or even improving) the required logistic service level.
Considering the main theoretical knowledge gained from this empirical
research, some further observations can be drawn.
First, the presented framework can be classified as a rule based one. Hence it is
quite easy to implement in the real practice of a company without the need of
complex and costly software or hardware supports: the way the framework was
implemented in the Action Research can be considered a live demonstration of its
implementation in corporate practice. Moreover, the overall framework has been
designed in order to let the human operator ‘‘keep the control’’ on the final
decision, so to avoid the negative feeling that a completely automated decision-
making system can sometimes generate.
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Chapter 10
Configuring Single-Echelon Systems
Using Demand Categorization
Spare parts planning is a complex task involving a large number of SKUs (stock-
keeping units) with many zero demand periods, which makes forecasting and
inventory control difficult. Intermittent SKUs may comprise about 60% of total
inventory in many industrial settings (Johnston and Boylan 1996; Johnston et al.
2003); for example in the aircraft industry, the demand for a specific jet engine, as
spare part, may show many zero demand periods, leading to a so called inter-
mittent demand pattern. Regarding the size of the intermittent SKU group, an
efficient selection of the best inventory methods implicates huge cost reductions
and service level improvements. However, the large amount of spare part SKUs
held in companies also implies that the inventory system cannot be configured
manually on an individual basis. Therefore, recent studies propose a sub-grouping
of intermittent demand patterns by a categorization scheme. Categorization
schemes provide the inventory manager with a better overview of the large number
of SKUs to be dealt with, similar to the ABC-analysis by Dickie (1951). Forming
sub-groups with similar inventory management requirements comprises the
opportunity to develop inventory rules for each sub-group and subsequently allow
an automated configuration of the sub-groups’ single-echelon inventory systems.
The application of an effective categorization scheme represents the foundation
of an efficiently managed spare part inventory system. Recent studies pursue the
development of a universally applicable categorization scheme. These studies give
Low High
Non-intermittent Intermittent
High
Erratic Erratic Lumpy
Coefficient of
variation of
demand size
Non-erratic Smooth Slow
Low
Smooth This category comprises items with relatively few zero demand periods
and low demand size variability. Due to the comparably low variability
of these demand patterns, the forecast and stock control system should
lead to good results. However, many of these spare parts SKUs still
have a non-normal demand pattern, so normality assumptions might not
be valid.
Erratic Erratic SKUs have relatively few zero demand periods, but the demand
size variability is high. Erratic patterns are difficult to forecast, which
implies a relatively high forecast error, and therefore, this pattern often
tends to have excessive stock.
Slow Items with many zero demand periods and low demand size variability
are named slow SKUs. The denomination slow refers to the slow
turnover of these SKUs, as there are only a few periods with demand
greater zero, and when demand occurs, it usually equals one in the
context of spare parts management, leading to low demand size
variability. Boylan et al. (2008) propose mean demand size as third
factor to define slow-moving units. However, in the context of spare
parts inventories in general, low demand sizes can be expected if
demand size variability is low.
Lumpy SKUs categorized as lumpy have high demand size variability and a
high level of intermittence. These SKUs represent the biggest challenge
for spare parts inventory management as they often tend to have
excessive stocks and low CSLs at the same time. The slow category and
the lumpy category are likely to comprise most of the spare parts SKUs
and therefore should be in the focus of the inventory manager.
These definitions should avoid ambiguity when comparing the results of dif-
ferent academic studies and when implementing a categorization scheme in a spare
part inventory system.
Williams (1984) is the first to examine intermittent demand patterns. In his work
he presents a classification scheme based on an idea called variance partition,
meaning that the lead time variance is divided into its constituent parts, namely
variance of the order sizes, transaction variability and variance of the lead-times.
Assuming random demand arrivals (meaning that the number of arrivals per period
206 D. Bucher and J. Meissner
n are Poisson distributed with mean k) and constant lead times, Williams (1984)
derives the following formula from the variance partition equation:
2 1 CVx2
CVDDTL ¼ þ ð1Þ
kL kL
This formula shows how the squared coefficient of variation of demand during
2
lead time, denoted as CVDDTL , can be written as the sum of two statistical
meaningful terms. The first term represents the reciprocal value of the product of
the mean number of arrivals per period, denoted by k, and the mean lead time,
denoted by L. The second term represents the ratio of the squared coefficient of
variation of the distribution of the demand sizes, denoted by CX2 , and the product of
the mean number of arrivals k and the mean lead time L. The first term represents
the mean number of lead times between demands whereas the second term relates
to the lumpiness of demand. These two terms are used as measures to categorize
11,000 SKUs of a public utility with individual demands of the SKUs being small.
The cut-off values are chosen based on the considered empirical data set. Williams
(1984) presents the following classification scheme (Fig. 10.2).
Category A is called smooth. Thus continuous-demand stock-control techniques
are recommended by Williams (1984). The empirical demand distributions of the
A-category SKUs are further examined. A chi-test is conducted to show that the
Gamma distribution approximates well the A-category demand distributions. It is
suggested that category C and D1 are managed in a similar way; however, the
negative binomial distribution is mentioned as an alternative to the Gamma dis-
tribution. Category B consists of slow-moving items, having infrequent demand
arrivals and low coefficients of demand size variation. A chi-test conducted to test
the B-category demand distributions against a Poisson distribution shows that the
Poisson distribution adequately describes the empirical demand distributions.
A C
Intermittence:
0.7
D1
λ–
1
L
B 2.8
D2
10 Configuring Single-Echelon Systems Using Demand Categorization 207
The probability for D2 items to have more than one order per lead time is very low.
Therefore, these items are managed with a method developed by Williams (1982),
which is based on a Gamma distribution and the assumption that there is no more
than one order per lead time.
Williams (1984) introduces two factors to group intermittent demand patterns
in sub-groups, with both being dependent on the average lead-time. Using average
lead-time to differ intermittent demand patterns is an important contribution
incorporated in further studies. However, cut-off values are determined based on
the underlying empirical data, and therefore, this approach does not account for
universal validity.
Johnston and Boylan (1996) compare the performance of the Croston forecasting
method (Croston 1972) with EWMA (Exponentially Weighted Moving Average),
thereby introducing the measure ‘‘average inter-demand interval’’. Their work
shows that the Croston method outperforms the EWMA method robustly over a
wide range of parameter settings, when the average inter-demand interval is
greater than 1.25 forecast revision periods. This result represents the first inventory
rule pursuing universal validity and additionally redefines intermittence by
showing that the method developed explicitly for intermittent demand (Croston
1972) outperforms the method for non-intermittent demand patterns (EWMA)
when the average inter-demand interval is greater than 1.25 forecasting periods.
Eaves (2002) analyzes data of the Royal Air Force using the categorization method
of Williams (1984). He concludes that the demand variability is not properly
described by the categorization scheme. In particular, Eaves (2002) criticises the
differentiation of regular demand and the rest solely based on the variability of
demand. Subsequently, Eaves presents a new categorization of intermittent
demand using three measures to group SKUs: transaction variability, demand size
variability and lead time variability. In contrast to the categorization scheme by
Williams (1984), Eaves considers lead time variability to allow for a finer cate-
gorization. The three measures are chosen based on the lead time variance partition
formula derived by Williams (1984) for the case of variable lead times, stated as
2 Cn2 Cz2
CVDDTL ¼ þ þ CL2 ð2Þ
L nL
208 D. Bucher and J. Meissner
where Cz is the coefficient of variation for the demand size, Cn is the coefficient of
variation for the transactions per unit time, CL is the coefficient of variation for the
replenishment lead time, n is the mean number of transactions per unit time and L
is the mean replenishment lead time.
Figure 10.3 shows the categorization scheme according to Eaves (2002). The
SKUs with low variability of demand intervals are grouped into groups A (smooth)
and C (irregular), depending on variability of demand size. Variability of lead time
is solely used to group SKUs into D1 (erratic) and D2 (highly erratic). SKUs of
category B are called slow-moving. The cut-off values are derived from the data
set. This approach neglects universal validity like Williams (1984).
A C
Transaction
variability
0.74
D1
Lead time
variability
B 0.53
D2
10 Configuring Single-Echelon Systems Using Demand Categorization 209
With l and r2 being mean and variance, respectively, of the demand sizes when
demand occurs. The average inter-demand interval is represented by p as the
number of forecast review periods including the demand occurring period. a is
the common smoothing constant value used, with b = 1 - a for EWMA. For
Crostons’ method and the SBA-method (Syntetos and Boylan Approximation) a is
used for the smoothing of the intervals and the demand size. Equations 3 and
approximations (4) and (5) assume a Bernoulli process of demand occurrence,
and therefore, inter-demand intervals are geometrically distributed.
Comparing the MSE of the Croston method with the MSE of the Syntetos
and Boylan Approximation, the following inequality is derived, assuming a fixed
lead time of L C 1.
for p [ 1, 0 B a B 1.
From inequality (6) theoretical rules can be derived based on two criterions,
namely the squared coefficient of variation CV2 and the average inter-demand
interval p. Depending on the setting of the control parameters a, l, p and r2,
cut-off values can be developed. Inequality (6) holds for any p [ 1.32, implying
that superior performance is expected by the SBA method for any average inter-
demand interval greater than 1.32 forecast review periods. This result shows that
the SBA method delivers lower MSEs than the Croston method when there are
many zero demand periods. If p B 1.32, it depends on the value of the squared
coefficient of variation CV2. If CV2 [ 0.48, the MSE of the SBA method is
210 D. Bucher and J. Meissner
expected to be smaller, and thus, the SBA method is chosen. If CV2 B 0.48, there
is a cut-off value for p with 1 \ p B 1.32, with the Croston method chosen, when
p is below the cut-off value. With CV2 increasing, the p cut-off value increases up
to 1.32 for CV2 = 0.001. Thus, the SBA method is also preferable for SKUs with
fewer zero demand periods but relatively high changes of the demand size. Syn-
tetos et al. (2005) show that these results are valid for a smoothing constant value
of a = 0.15 and approximately true for other realistic a values.
Comparisons of the Croston method and the SBA method with EWMA showed
that the MSE of EWMA is always theoretically expected to be higher than the
MSE of Croston and SBA. The EWMA method is discarded from the categori-
zation scheme.
Based on these results a categorization scheme with four factor ranges is
modelled. For ranges with p [ 1.32 and/or CV2 [ 0.49 the Syntetos and Boylan
method is shown to perform theoretically better. For the factor range of p B 1.32
and CV2 B 0.49 neither method is shown to perform better in all cases.
A numerical result conducted by Syntetos et al. (2005) indicates that the Croston
method is expected to work better in that range, thus the Croston method is
assigned to this factor range of indecision.
Figure 10.4 shows the developed categorization scheme with the following
groups: A-erratic (but not very intermittent), B-smooth, C-lumpy, D-intermittent
(but not very erratic).
Developed for intermittent demand SKUs are expected to perform theoretically
better than conventional forecast methods. For fast-moving demand items, the
Croston method seems to be more appropriate. The Syntetos and Boylan
Approximation performs best for more intermittent and/or more irregular demand
items.
The categorization scheme of Syntetos et al. (2005) is the first one to appear in
the literature pursuing general validity. By theoretically comparing the mean
Erratic Lumpy
SBA SBA
Demand variability
A C
CV =0.49
2
Smooth Slow
Croston SBA
B D
10 Configuring Single-Echelon Systems Using Demand Categorization 211
squared errors of the EWMA method, the Croston method and the newly presented
Syntetos and Boylan Approximation, cut-off values are derived. The values are
expected to have general validity for a wide range of realistic control parameters.
This work currently forms the base of a new line of research pursuing a complete
categorization scheme for items with intermittent demand patterns. Extensions of
this promising approach are presented in the following.
Boylan et al. (2008) propose the first extension of the categorization of Syntetos
et al. (2005) by implementing stock control policies using different theoretical
statistical distributions to achieve a predetermined service level (see Fig. 10.5).
The stock control policies are assigned to each sub-group based on professional
experience. Thus, the categorization represents rather a best-practice approach
than an analytical rule. A theoretical comparison of the statistical distributions and
a subsequent derivation of areas of superior performance analogues to the work
conducted by Syntetos et al. (2005) is not conducted. The categorization system
developed in this case-study based paper is, however, thoroughly tested with a data
set provided by an inventory management software manufacturer.
Boylan et al. (2008) choose the continuous re-order point, order quantity (s, Q)
control policy. The authors, however, explicitly state that no significant difference
is expected when using other stock control policies, such as the periodic order-up-
to level (T, S) policy or the periodic order point order-up-to-level (T, s, S) policy.
This argument is supported by the work of Sani and Kingsman (1997), who use
empirical data to show that there are only minor differences in the use of different
control policies in the context of intermittent demand. Thus, the order quantity Q is
Lumpy
Demand size variability
SBA
(s,Q)-policy
NBD
CV =0
Non-intermittent
2
Slow
SBA
(s,Q)-policy
Poisson
212 D. Bucher and J. Meissner
determined by the cumulative forecast over the lead-time using the Syntetos and
Boylan Approximation. Safety stock s is determined by choosing an appropriate
statistical distribution. As mentioned earlier, the distributions are assigned to each
factor range based on professional experience. The Poisson distribution is decided
to be appropriate for items with slow demand patterns. According to Boylan et al.
(2008), this is an obvious choice for slow moving items and is already implied in
the inventory control software tool of the software manufacturer under concern
in the case study. For the category of lumpy demand patterns, the negative
binomial distribution was decided to be the most appropriate choice as it satisfies
both theoretical and empirical criteria (Syntetos and Boylan 2006).
Compared to the categorization scheme presented by Syntetos et al. (2005), two
alterations regarding the used criteria are undertaken. Due to the data set examined
in their case study, Boylan et al. (2008) decided to reset the cut-off value of CV2
from 0.49 to 0, as almost 50% of the first data set had zero variance of demand
size. Thus, this group is named slow instead of intermittent as in Syntetos et al.
(2005), and the remainder is called lumpy. The criterion inter-arrival interval
p used in Syntetos et al. (2005) was replaced by a criterion used by the software
manufacturer. The criterion is the number of zero demand periods over the last 13
periods, with the cut-off value being three periods. This criterion is used to sep-
arate normal demand from intermittent demand. A thorough discussion of the
nomenclature of different non-normal demand patterns is provided in Sect. 10.1.
Based on this categorization framework, Boylan et al. (2008) examine the
inventory management performance of the developed single-echelon inventory
systems for three data sets comprising about 16,000 SKUs coming from the
automotive, aerospace and chemical industry. In a first step, different combinations
of the forecast methods under concern, namely the Croston method, the Syntetos
and Boylan Approximation for the intermittent demand categories and Simple-
Exponential-Smoothing (SES) and Simple-Moving-Average (SMA) for the non-
intermittent categories, are compared for different cut-off values of zero demand
periods in the last 13 months. The results show that in the underlying data set the
forecast error, measured with the geometric root mean squared error (GRMSE) and
the average mean absolute error (MAE), shows little sensitivity towards the
selection of the cut-off values r for ranges from r = 2 to r = 6. In the range with
r = 7 to r = 13, the forecast accuracy is highly sensitive to the cut-off value with
a fast raising forecast error.
In the next step, implications of the chosen forecast methods on the stock
control performance are examined. For the group of slow moving items, a com-
parison of SMA with the SBA shows that SBA tends to slightly undershoot the
target customer service level (CSL), whereas the positive biased SMA results in a
significant overshoot of the CSL. This leads to considerable higher stock values for
SMA compared to the SBA method. It was decided that the savings in inventory
costs occurring when using the SBA method overcompensate for the slight
undershoot of the CSL. For the category of lumpy items no forecast method gets
close to the target CSL. Nevertheless, the usage of the negative binomial
10 Configuring Single-Echelon Systems Using Demand Categorization 213
In this case study, Syntetos et al. (2008a, b) show more empirical proof for the
need of an effective categorization scheme for the management of spare parts. The
European spare parts logistics operations of a Japanese electronics manufacturer
are centralized and within the same project, new and more effective classification
rules for the spare parts management are implemented. Before the project, the
decentralized spare part inventories in 15 countries in Europe showed poor service
levels of an average of 78.6% as well as high numbers of obsolete stocks. A highly
simplified inventory categorization scheme was in use based solely on order fre-
quencies and arbitrarily chosen cut-off values. The objective of the project was to
improve the service level of up to 95% as well as reducing inventory by 50%.
More elaborate categorization schemes, as discussed in the previous sections, were
thought to be necessary to be implemented. However, due to the short duration of
the project, only a simplified categorization scheme could be implemented. This
categorization scheme also takes into account the demand value of an SKU, which
is the value per item times the annual demand rate of the SKU, similar to the
classical ABC categorization scheme by Dickie (1951).
Although a more elaborate categorization scheme could not yet be implemented
due to limited time, the slightly improved categorization scheme increased the
service level up to 92.7%, the inventory investment was reduced by roughly 40%.
This case study shows that there is a very high potential in improving the inventory
performance by using more elaborate inventory categorization schemes. The
implementation of a very simple but meaningful categorization scheme for the
spare parts management of the Japanese manufacturer significantly reduced
inventory investments and assured a more accurate accomplishment of the target
customer service level. The results of this case study may encourage other com-
panies to pay more attention on the categorization scheme applied on spare parts
management, as it constitutes a simple and powerful mean to control stocks and
increase inventory performance considerably. In the next section, a rough guide
will be given on how to implement an effective categorization scheme for spare
parts management.
214 D. Bucher and J. Meissner
2 2
CV > 0.49? yes yes CV > 0.49?
no no
Erratic Lumpy
SBA SBA
(t,S)-policy (t,S)-policy
Normal NBD
CV =0.49
2
Smooth Slow
Croston SBA
(t,S)-policy (t,S)-policy
Normal Poisson
216 D. Bucher and J. Meissner
This guideline aims to show the basic steps that need to be undertaken when
implementing a categorization scheme for spare part inventories. However, it is
important to consider that this framework does not constitute an approach of
universal validity in a technical sense, as only for the choice of a (parametric)
forecasting technique results that come close to universal validity are available
(Syntetos et al. 2005). Nevertheless, the application of best-practices for the choice
of the statistical distribution and the inventory policy should lead to adequately
configured inventory systems in many industrial settings.
This approach also simplifies the inventory system configuration. In ERP sys-
tems such as SAP ERP, decisions have to be made on a large variety of system
parameters. This guideline focuses solely on the core issues in forecasting and
inventory control of intermittent demand.
Another potential obstacle, when implementing this approach is that it might
create a black box of how every SKU is managed. Therefore, it is important for the
inventory management to understand the applied forecast and inventory methods
and to question their appropriate application in the company’s inventory system.
Further, it is worth mentioning that manual managerial judgements are often
important in the context of spare parts management, as a wide range of information
may be very important. A recent study by Syntetos et al. (2008a, b) shows that the
combination of parametric forecast methods with managerial judgement often
leads to a considerable improvement of the inventory performance.
The basic categorization scheme presented in the previous section can be expan-
ded and combined with various other methods to enhance the inventory system.
Possible additional components are briefly discussed in the following.
The classical ABC-analysis introduced by Dickie (1951) is the most popular
inventory categorization scheme and still widely applied. Applying this catego-
rization scheme in combination with a categorization scheme for intermittent
demand patterns will allow the inventory manager to focus on high value spare
part SKUs and manage them manually when necessary and leave less valuable
C-items to the demand categorization scheme. This combination of categorization
schemes accounts for the fact that few SKUs are of high importance and therefore
need special attention of the inventory manager, while a large number of SKUs
account for only a small percentile of the usage and therefore can be left to the
demand categorization scheme.
The LMS-categorization which differentiates items by their volume require-
ments when stocked (large, medium, small) can also be combined with the demand
categorization scheme. This might be of particular interest for spare parts with
considerable differences in size and will account for the high differences in
stocking costs. Items of the L-category might be managed manually, or the safety
stock will be adjusted downwards due to the high storage costs.
10 Configuring Single-Echelon Systems Using Demand Categorization 217
As discussed in the literature review there are only few studies published in the
area of categorization of demand. New studies were conducted showing the high
potential of demand categorization schemes to efficiently manage SKUs with non-
normal demand patterns. First steps have been made towards a universal appli-
cable inventory system based on demand category schemes. Nevertheless, these
studies form only the beginning of an emerging research area which needs more
academic attention. This section aims to highlight research areas which are of
particular importance to enhance the theoretic understanding of the complex inter-
relations in the inventory system, as well as to accelerate the dispersal of effective
inventory systems based on demand categorization in practice.
Syntetos et al. (2005) compare different parametric forecast methods and
subsequently define areas of superiority. This approach represents the first cate-
gorization scheme giving theoretically derived suggestions about which forecast
method to use. As a second step, the derivation of areas of superiority for different
theoretic distributions is desirable. This can be undertaken following a similar
procedure as used in Syntetos et al. (2005).
The final goal of a categorization scheme is to achieve an overall optimization
of the inventory system, including forecast and inventory control. The determi-
nation of an optimum of this complex inventory system constitutes a challenge for
researchers, and it may not be feasible. Nevertheless, a valid formulation of the
overall optimization problem would be of great value to gain insights on how the
different modules of the inventory systems interact and what the right configura-
tion of such a system would be. For a recent case study in the context of the
German automobile industry see Bucher and Meissner (2010).
Most of the work presented in the literature review focuses solely on variability
of the demand patterns. However, the determination of an appropriate safety stock
needs to account for lead time variability as well, especially when lead times are
218 D. Bucher and J. Meissner
10.6 Conclusion
Spare parts planning is a very complex task involving a wide variety of methods in
forecasting and inventory control. Demand categorization schemes allow an effi-
cient, automated selection of these methods for each SKU leading to an adequate
single-echelon inventory system configuration. Although decisions on how to
combine forecast and inventory control methods have a great impact on the
inventory performance, there is a limited number of studies concerning this issue.
A new line of research appeared with the goal to develop a universal applicable
demand categorization scheme.
Recent empirical studies show that although the research is still at the begin-
ning, the application of a demand categorization scheme may lead to a consid-
erable improvement of inventory performance. This work provides practitioners
with a guideline on how to implement a simple demand categorization scheme
which may leverage the inventory performance. The categorization scheme can be
run automatically and frees time of the spare parts inventory manager to focus on
the most important SKUs. To enhance the inventory performance, the presented
categorization scheme can be combined and extended with other categorization
methods as discussed in Sect. 10.4.2.
Approaches which are applicable in practice recently appeared in the literature,
including best-practices and analytically derived rules. There is still a great need
for further research in the area of demand categorization for intermittent demand
patterns. Particular research gaps are outlined in Sect. 10.5. Improving the
understanding of the interrelation between forecasting and inventory control and
how to truly optimize the single-echelon inventory system are still subject to
further research. This work as a collection of most recent studies shall represent a
base from which further research can be conducted.
10 Configuring Single-Echelon Systems Using Demand Categorization 219
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35:939–948
Chapter 11
Optimal and Heuristic Solutions
for the Spare Parts Inventory
Control Problem
Ibrahim S. Kurtulus
Abstract One inventory problem that has not changed much with the advent of
supply chains is the control of spare parts which still involves managing a very
large number of parts that face erratic demand which occurs far in between. For
most items it has not been feasible to establish a control system based on the
individual item’s demand history. Hence either the parts have beenbundled into
one group as Wagner did and only one demand distribution has been used for all or
they have been divided into different groups and group distributions have been
used as in Razi. We picked two popular rules by Wagner and extensively tested
their performance on data obtained from a Fortune 500 company. Comparisons
were made with the optimal solution and optimal cost obtained from our procedure
based on the Archibald and Silver’s optimizing algorithm. For some problems with
very small mean demand and small average number of demand occurrences,
finding the optimal solution required an inordinate amount of CPU time, thus
justifying the need to use heuristics.
11.1 Introduction
The problem studied is an (s, S) type of inventory control system for a single item,
which is subject to sporadic demand that is also highly variable in size. There is a
fixed (replenishment) lead time of L. Item replenishment costs consist of a setup
cost K and a unit cost c. Units carried in excess of demand incur a holding cost of h
I. S. Kurtulus (&)
School of Business, Virginia Commonwealth University, 301 W. Main St., Richmond
VA, USA
e-mail: ikurtulu@vcu.edu
per unit and demand not satisfied is backordered and incurs a penalty cost of d per
unit. The objective function is to minimize the expected value of the sum of
carrying, backordering and (fixed) ordering costs. Under these assumptions, it has
been shown (Iglehart 1963) that a policy of type (s, S) will minimize the undis-
counted expected cost per period over an infinite horizon, where ordering deci-
sions are restricted to demand occurrences (Archibald 1976 and Silver 1978;
Beckmann 1961; Ehrhard 1979; Vienot and Wagner 1965). A replenishment order
of size (S - y) is placed when inventory on hand plus on order minus backorders y
is less than or equal to s.
In testing the performance of heuristics and optimal procedures, various
assumptions have been made with respect to demand. Common ones included
Poisson (Schultz 1987; Gelders and Looy 1978; Hadley and Whitin 1963) and
normal distributions (Croston 1972; Bartakke 1981; Porteus 1985; Vereecke and
Verstraeten 1994; Sani and Kingsman 1997). Unfortunately, test of fit of certain
parametric distribution to actual demand data has been difficult to show and
rarely reported in the literature. So far the most convincing arguments have
been made in favor of the compound Poisson distribution, which is what we
have used in this research. The assumption of random demand occurrence
(usually assumed to be Poisson as in queuing theory) and a nonparametric
distribution of demand size given as a frequency distribution are combined to
form the compound Poisson distribution (Adelson 1966; Feller 1968; Vienot
and Wagner 1965; Feeney and Sherbrooke 1966; Archibald and Silver 1978;
Razi 1999).
Another distribution recommended by researchers (i.e., Dunsmuir and Snyder
1989; Segerstedt 1994; Yeh 1997) has been a positively skewed gamma distri-
bution with a large spike at zero. It has been recommended both for demand size
and occurrence.
Two variations of the problem have also been investigated. One shows how the
algorithm is simplified if orders are placed when the reorder point s is reached
exactly (Zheng and Federgruen 1991). Since it is a model more appropriate for
continuous review systems, it is not used in this paper. And yet another addresses
the issue of nonstationary (variable) mean demand and develops a myopic heu-
ristic (Bollapragada and Morton 1999).
Over the years, various heuristics have also been recommended to solve the
problem. We pick two that were developed by Wagner with very realistic and
convenient assumptions for managers. Both heuristics can also be considered
variations of the same method. When the exact probability distribution (for
demand) is used Wagner has called his heuristic the algorithmic method and when
the probability distribution is approximated, the normal approximation. In past
research, heuristics’ performance has been tested by using simulation (i.e., Porteus
1985; Sani and Kingsman 1997). This paper deviates from this approach and
compares heuristics’ performance with the optimal cost obtained when the com-
pound Poisson distribution is used.
11 Optimal and Heuristic Solutions 223
could define it). (3) Different parameters used for k, l, K and h affected the CPU
time needed to find the optimal solution, which clearly indicated that they would
also affect the number of runs needed to reach a steady state, or some version of it.
The actual data used in the study was obtained from a Fortune 500 Company (Razi
1999). The manager in charge of spare parts inventory kept monthly demand
history for a 2 year period for each of the 22,500 parts. Those items with no
demand history (i.e., 30% of the items.) in the 2 year period were excluded from
the study. Of the remaining items, any item with total demand of 50 units or less
(in the 2 year period) was classified as slow moving. Then these items were
divided into 12 groups based on replenishment lead time (2, 3, 4, 6 weeks) and
total demand (i.e., 1–10, 11–20, 21–50) for the 2 year period. Finally, a frequency
distribution for demand size was generated for each group. All of these probability
distributions for demand were extremely skewed to the right, with peaks occurring
at one. Plots of the five distributions we picked are provided in Appendix 1.
The manager estimated ordering cost as $50 per order, carrying cost as 15% of the
item cost, backordering cost (cost of delay) as $20 if the item’s cost was less than
$100 and 30% of the cost of the item if its cost was more. Implicit in these com-
putations was the fact that the cost of work stoppage was not a problem and hence was
not considered as part of the cost of backordering. In the worst case scenario the part
could be obtained without work stoppage by Federal Express. Lambdas, the average
number of demand occurrences per year, were computed as follows: First we found
the average number of occurrences per month based on the 24 month demand history
for each item. We picked the minimum and the maximum in each group. Averaged
them and multiplied the average by 12 to find the yearly figure. Item cost used for
each group was also a simple average of the minimum and maximum in the group.
The data is summarized in Tables 11.1 and 11.2 in Appendix 1.
Peterson (1987) and Peterson et al. (2000) reported that Poisson distribution
adequately reflects the demand distribution for spare parts inventory. He used
ordering cost of ($3 or $20) and carrying cost of ($0.1, $0.2, $0.7) which he reports as
being similar to those used by the US Air Force inventory policies. His two back-
ordering cost levels (49 carrying cost, 99 carrying cost) correspond to 80 and 90%
service levels and represent the percentage of time an optimally controlled system
(in his simulations) incurs no backorders. Also, these service levels are around the
86% service level the Air Force targeted at the time. His lead times are
(0, 2, 4 time units) and mean demands are (0.1 and 1.0) and Poisson distributed.
When compared to Razi (1999), his ratios involving ordering-to-carrying cost and
ordering-to-backordering cost were higher and his mean demands were much lower.
11 Optimal and Heuristic Solutions 225
The results involving the data given in Table 11.1 and the two replenishment lead
times, 2 weeks (L = 0.04) and 4 weeks (L = 0.083) with WNOR are summarized
in Tables 11.3 and 11.4 (Appendix 1). In the two tables, the average increase in
cost (from optimal) are 97.9 and 71.04%, respectively. When the ratio of ordering
cost to holding cost is increased (or h is reduced) by 12-fold, the results improve to
76.9 and 66.4% as shown in Tables 11.5 and 11.6. The best results are obtained
when the ratio of ordering to holding cost is high and the replenishment lead time
is long (i.e., Table 11.6).
As expected, WEX does better than WNOR under both replenishment lead
times, giving only an average 42.8% increase from the optimal in Table 11.7 and
27.2% in Table 11.8 when the data in Table 11.1 is used. When the ratio of
ordering cost to holding cost is increased (or h is reduced) by 12-fold, the results
improve to 15.0 and 20.1% in Tables 11.9 and 11.10, respectively.
We are puzzled with the improvement in performance of both rules when the
re-supply lead time is doubled and the ratio of ordering to holding cost is not high.
In Tables 11.4 and 11.8, in case of groups 1, 2, 6, and 14, the percent increase from
optimal cost is less when L = 0.083, and approximately equal with group 3.
Under WEX, the best results are obtained when the ratio of ordering to holding
cost is high but when the replenishment lead time is short (i.e., Table 11.9). The
short lead time part is contrary to what we had observed in other cases. We believe
it is due to interaction affects not yet defined. Under WEX, in all cases Group 3
provides the worst results. However, the same cannot be said for WNOR. Hence
Group 3 cannot be treated as an outlier?
Given that all actual group distributions were skewed to the right with peak at
one, can we use a triangular distribution that has a peak at 1 and uses the range of
values for demand comparable to actual data being simulated. If successful, using
a triangular distribution with easily definable parameters, will free the practitioner
from the task of actually developing the exact distributions required by WEX?
Future research will answer some of these questions.
Appendix 1
Table 11.1 Characteristics of the original Razi data used in the study
Groups Ord$ Mean Variance Hold$ BckOrd$ Lambda
1 50 1.734 1.050 38.52 77.0 2.76
2 50 2.745 9.293 82.64 165.3 9.30
3 50 3.854 14.68 6.34 20.0 18.30
6 50 2.740 8.167 54.00 108.0 7.74
14 50 3.110 13.654 261.15 522.5 13.02
226 I. S. Kurtulus
Table 11.2 Ratios of ordering and backordering to holding cost used in the study
Groups Regular Ord$/ Regular BckOrd$/ Reduced Reduced Ord$/ Reduced BckOrd$/
Hold$ Hold$ Hold$ Hold$ Hold$
1 1.30 2 3.21 15.6 24
2 0.60 2 6.89 7.3 24
3 7.90 3.2 0.53 94.3 37.7
6 9.3 2 4.50 11.1 24
14 0.19 2 21.76 2.3 24
Appendix 2
Wagner’s Heuristics
Wagner (1975) has developed his heuristics as probabilistic extensions of the basic
EOQ model. If the exact distribution is known, Wagner calls his method
the algorithmic solution since it has to go through a number of iterations to
determine s and Q(S = s ? Q). We will call it the exact method (WEX). The
solution will be optimal with respect to the objective function assumed in the
model. Let pL(xL) be the probability mass functionP of demand (nonparametric)
during lead time and lL its mean and PL ðsÞ ¼ sxL ¼0 pL ðxL Þ: Then the exact
method goes through the following steps to find the solution:
Step 1: Let initial trial value of
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q¼ 2KlL =h ð1Þ
Step 2: Using the trial value of Q, compute:
hQ
R¼1 ð2Þ
hlL =2 þ lL d
Find a trial value for s such that it is the smallest positive integer for
which:
PL ðsÞ R ð3Þ
From the definition of (3) we can think of the value of PL(s) as the
minimum service level acceptable to the company.
Step 3: Stop if the new trial value of s is the same as before. Otherwise, calculate
a new trial value for Q by using (4) below and go back to Step 2.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X ffi
Q ¼ ð2KlL Þ=h þ ðlL þ ð2lL d=hÞÞ ðxL sÞpL ðxL Þ ð4Þ
xL s
11 Optimal and Heuristic Solutions 229
and find the fs value of the unit loss function (or standardized normal loss
integral) IN(f) such that:
IN ðfs Þ ¼ RN ð7Þ
Step 3: If l \ 0.8888 (K/h), then let s be determined by:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s ¼ ðL þ 1Þl þ fs ðL þ 1Þr ð8Þ
and
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii
S ¼ ðL þ 1Þl þ min fs ðL þ 1Þr þ EOQ; fv ðL þ 1Þr ð13Þ
For further discussion and justification for both WEX and WNOR, please refer to
Wagner (1975).
230 I. S. Kurtulus
References
12.1 Introduction
Many studies describe challenges facing large manufacturers who must efficiently
control an inventory of tens of thousands of finished products, maintenance and
replacement or spare parts (Ward 1978; Gelders and Van Looy 1978; Dunsmuir
and Snyder 1989; Hua et al. 2007). Wagner and Lindemann (2008) have urgently
called for future research on strategic spare parts management. When stocking
spare parts, a few parts often represent the bulk of the investment and the majority
of the demand. However, it is important to be able to forecast the demand rate for
the slow-moving items as well as the heavily used parts. If a product has not had a
demand over a specified duration of time, its demand would be projected to be zero
based on many of the popular forecasting models, such as simple exponential
smoothing or moving averages. Yet, this product may still be required and be
worth carrying, particularly if the inventory cost is well managed.
This study examines the demand for these types of products and develops a
methodology to address related issues. In particular, we will propose a method-
ology to determine a one-sided prediction interval for predicting demand rates for
intermittent or slow-moving spare parts that is adapted from statistical procedures
developed for software reliability. The one-sided prediction interval is an upper-
sided interval and the upper endpoint could be compared to a threshold value so
that a decision can be made on whether to continue carrying a group of products.
In essence, a stopping rule can be employed to determine whether to stop stocking
certain products. A stopping rule procedure is a rule used in a decision-making
M. Lindsey (&)
Stephen F. Austin State University, Nacogdoches, TX, USA
e-mail: lindseymd@sfasu.edu
R. Pavur
University of North Texas, Denton, TX, USA
e-mail: pavur@unt.edu
Browne and Pitts (2004) state that a stopping rule can be used in the decision-
making process to make a judgment based on the information gathered about the
sufficiency of that information and the need to acquire additional information. That
is, a stopping rule is some test or heuristic invoked by the decision-maker to
determine the sufficiency of the information obtained. They remark that stopping
rules have been investigated extensively in decision-making research. Brown and
Zacks (2006) studied a stopping rule for the problem of quick detection of a
change-point in the intensity of a homogeneous ordinary Poisson process. In
general, finding optimal stopping rules is quite difficult.
236 M. Lindsey and R. Pavur
Ranges of these parameters over which the two-sided prediction intervals were
reliable were noted.
The current study differs from Lindsey and Pavur (2009) in that the reliability
of one-sided prediction intervals is investigated. Although it may seem that the
parameters under which two-sided predictions for products with no observed
demand are the same as that for one-sided prediction intervals, this is not the case
as is illustrated in a comparison of the two in the simulation study section of this
paper. The two-sided prediction interval allows for more error in one tail of the
prediction interval to be compensated by less error in the other tail. In addition,
this paper investigates an extension in which the demand rate of products which
display no more than one unit of demand are investigated. Although the prediction
intervals are adapted from Ross (1985a, b, 2002)’s estimators, he did not assess the
distribution or reliability of these estimators.
If a set of Poisson random variables are observed over a specified time frame
and no observations occur, then an estimate of zero may be inappropriate
simply because the time frame may not have been long enough. This type of
problem has appeared in the software reliability literature. Ross (1985a, b,
2002) derived an estimator for the future demand of bugs in a software
package that has no occurrences. However, he did not examine the distribution
of this estimator or the reliability of prediction intervals constructed from this
estimator. He also did not extend this estimator to the case of estimating the
future demand rate of Poisson random variables with zero or one observed
occurrences. Many models based on the Poisson distribution have been con-
sidered for estimating failure rates in software (Abdel-Ghaly et al. 1986;
Kaufman 1996).
A description of this estimator will now be presented using the context of errors
in a software package during testing. Then a description will follow in which the
context will be translated into demand for a group of products with zero or one
observations. Suppose that there are n bugs contained in a software package.
The number of errors caused by bug i is assumed to follow a Poisson distribution
with a mean of ki, i ¼ 1; 2; . . .; n. Ross (2002) defines Wi(t) = 1 if bug i has not
caused a detected error by time t [ 0 and 0 otherwise, i ¼ 1; 2; . . .; n. These
indicator variables allow the future error rate of the bug with no observed error to
P
be KðtÞ ¼ ni¼1 ki Wi ðtÞ. This expression has unknown rate parameters, ki, that are
difficult to estimate without using a time frame that is long enough to provide an
accurate estimate. In the context of software applications, a high error rate by this
expression would be unacceptable to the customer. Ross (1985a, b) used the
following notation.
238 M. Lindsey and R. Pavur
n Number of errors
ki Error rate for bug i, i ¼ 1; 2; . . . ; n
t Length of time period over which errors are observed
Wi(t) Equal to 1 if bug i has not caused an error and 0 otherwise
K(t) Theoretical error rate of bugs
Mj(t) Number of bugs that have caused j errors by time t
of K(t). For it to be a good estimator of K(t), its difference with K(t) should be
small. The second moment of KðtÞ M1tðtÞ is the same as the expected value of
M1 ðtÞþ2M2 ðtÞ
, which is a function of M1(t) and M2(t). Therefore, the mean squared
t2
difference between K(t) and M1tðtÞ can be estimated by M1 ðtÞþ2M 2 ðtÞ
.
t2
M1 ðtÞ
The following results in Eq. 12.1 summarize that t is an unbiased estimator
of K(t) and that the variance of the difference, or equivalently the expected squared
difference, between the estimator and the unknown population rate K(t) decreases
over time.
Xn
E½M1 ðtÞ ¼ ki teki t
i¼1
M1 ðtÞ
E ¼ E½KðtÞ
t
1X n
E½M2 ðtÞ ¼ ðki tÞ2 eki t ð12:1Þ
2 i¼1
( ) X n 2 ki t
M1 ðtÞ 2 ki e þ ki eki t
E KðtÞ ¼
t i¼1
t
E½M1 ðtÞ þ 2M2 ðtÞ
¼
t2
The underlying assumptions for occurrences of bugs are the same assumptions
often made for occurrences of demand for products. That is, the demand for
products can be assumed, as is frequently stated in the literature, to follow a
Poisson process. In addition, an assumption can be made that one product’s
12 Reliable Stopping Rules for Stocking Spare Parts 239
Inventory managers may need a forecast for not only spare parts with no demand
but also spare parts that had a few demand occurrences. That is, a one-sided
prediction interval for slow-moving spares with less than some minimum number
of demands over a specified time period may be desired. For this reason, a one-
sided prediction interval is considered for the case in which there is zero or one
demand. Although this extension could be carried further, a simulation study of
many different one-sided prediction intervals would become too involved.
Two-sided prediction intervals were considered in Lindsey and Pavur (2009).
Since the upper limit of the prediction interval may be of more importance in
decision making on liquidating or no longer carrying a product, one-sided pre-
diction intervals (OSPIs) are proposed in this section for the case of estimating
future demand for products with no observed demand or with no more than one
observed demand. The proposed OSPIs with 100(1 - a)% confidence will be the
240 M. Lindsey and R. Pavur
estimate of future demand plus the appropriate 100(1 - a)% normal distribution
percentile multiplied by an estimate of the standard error of the estimate. However,
for certain parameters this OSPI may not be reliable. For example, the estimator
M1 ðtÞ
t for the demand rate of products with no demand may be too skewed to be
assumed to be approximately normally distributed. Under certain parameter values
such as a small product group size, the normal approximation may make the upper
limit of an OSPI too small. That is, the nominal (stated) confidence level of the
OSPI will no longer hold if the normal distribution is not a good approximation of
a distribution that is skewed or heavy tailed. Thus, the reliability of such one-sided
intervals will be assessed over a variety of demand rates and numbers of products.
The upper endpoint of a proposed OSPI for the case of estimating the future
demand of products with no observed demand is as follows.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M1 ðtÞ M1 ðtÞ þ 2M2 ðtÞ
þ Za ð12:2Þ
t t2
A proposed estimator for the future sales rate of products having no more than
one sale over a specified time frame is M1 ðtÞþ2M
t
2 ðtÞ
since it is an unbiased estimator
of the underlying demand rate. The unbiased estimator for the expected squared
difference of this estimator and the future demand rate is M1 ðtÞþ2M22ðtÞþ6M3 ðtÞ. The
t
upper end point of the proposed OSPI for the future demand rate of products
having no more than one sale by time period t are as follows.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M1 ðtÞ þ 2M2 ðtÞ M1 ðtÞ þ 2M2 ðtÞ þ 6M3 ðtÞ
þ Za : ð12:3Þ
t t2
Equation 12.2 will be referred to as the Zero Sales prediction intervals. That is,
the Zero Sales prediction intervals determine future demand rate for products
exhibiting no sales over a specified time frame. Equation 12.3 will be referred to
as the Zero and One Sales prediction interval. These prediction one-sided intervals
determine the future demand rate for spare parts having no more than one sale unit
of demand over a specified time frame. A value of zero can be used for the lower
end-point for these OSPIs.
A Monte Carlo simulation with 5,000 replications was conducted to assess the
reliability of OSPIs for the demand rate of slow-moving products. One group of
slow-moving products that have not exhibited any demand over the specified time
frame will be referred to as the Zero Sales group. Another group of slow-moving
products that have exhibited no more than one demand will be referred to as the
12 Reliable Stopping Rules for Stocking Spare Parts 241
Zero and One Sales group. The reliability of the proposed OSPIs will be assessed
by simulating their Type I error rate. The next section will provide the empirical
Type I error rates for the Zero Sales OPSIs across various Product Group Sizes and
MTBDs. This section is followed by the results for the Zero and One Sales OSPIs.
Next, a comparison of the reliability of the two different OSPIs is presented.
Finally, an illustration is provided as to how the Zero and One Sales OSPIs
compare to the TSPIs with respect to their reliability across a variety of MTBDs.
Monte Carlo parameters similar to those selected in Lindsey and Pavur (2009)
in studying the reliability of two-sided prediction intervals for the Zero Sales group
are used in this study. A specified time frame of 100 units of time was selected for
the entire simulation study as 100 units of time could be converted into hourly,
daily or weekly data, but not longer periods, such as monthly or yearly. One
hundred time units would be roughly equal to 3 months or about a quarter if the
unit was a day or about 2 years if the unit was a week. The time frame of 100 units
was arbitrary, but it is reasonable to assume it would be a common time frame to
study for data either collected on a daily or weekly basis. This amount of time
would be an appropriate amount of time in which a manager would need to make
critical decisions if products were not moving. The total number of products in a
group that are observed for their demand range from 10 to 1,000. MTBD range
from 10 to 1,000 and include a case in which there is a mixture of MTBDs.
The confidence levels for the OSPIs are selected to be at the 90%, 95%, and
99% levels. That is, the nominal Type I error rates (alphas) are 10%, 5%, and 1%.
A Type I error occurs if a known demand rate is not within the one-sided pre-
diction interval constructed from the simulated demand of the products. An
empirical Type I error is considered near its nominal alpha value if this error is
within plus or minus two standard deviations of the nominal value (Zwick 1986;
Harwell 1991). For example, at the 99% confidence level with a nominal Type I
error rate of 1%, the empirical Type I error for 5,000 simulations must be between
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:01 2 ð0:01Þð10:01Þ
5;000 or from 0.007 to 0.013 for the prediction interval to be
considered reliable. For the 95 and 90% confidence levels, these intervals are
0.044 to 0.056 and 0.092 to 0.108, respectively.
The next section will provide the empirical Type I error rates for the Zero Sales
OPSIs across various Product Group Sizes and MTBDs. This section is followed
by the results for the Zero and One Sales OSPIs. Next, a comparison of the
reliability of the two different OSPIs is presented. Finally, an illustration is pro-
vided as to how the Zero and One Sales OSPIs compare to the TSPIs with respect
to their reliability across a variety of MTBDs.
Empirical Type I errors of OSPIs for a group of products with zero sales are
assessed across a variety of conditions for the number of products and the demand
242 M. Lindsey and R. Pavur
Empirical Type I errors of OSPIs for a group of products with zero and one sales
are assessed in a similar fashion to the OSPIs for a group of products with zero
sales. The parameters used in the simulation in the previous section were used to
determine the reliability of the OSPIs for zero and one sales. Figure 12.2 illustrates
that for MTBDs of 100 and the mixture of MTBDs, that the Zero and One Sales
OSPIs maintain their nominal Type I error rate as the product group size increases.
A product group size of at least 200 should be used with these MTBDs. For low
demand rates (high MTBDs), the demand may be too slow to provide sufficient
information, to provide accurate statistics for a reliable OSPI for the demand rate
of products which have exhibited one demand.
12 Reliable Stopping Rules for Stocking Spare Parts 243
Fig. 12.1 Empirical Type I error versus product group size for OSPI of Zero Sales group
Since the Zero and One Sales OSPIs are estimating a demand rate for potentially
more products than the Zero Sales prediction interval, the Zero and One Sales
OSPI is expected to perform better with products having higher demand, since
more demand will occur more often even for the slowest moving products. To
assess this characteristic, a comparison is made between these two prediction
intervals.
Figure 12.3 reveals the empirical Type I error rates for the Zero Sales OPSIs
and the Zero and One Sales OPSI across MTBDs ranging from 10 to 1,000.
A group product size of 200 was selected since this group size appears to be where
the OSPIs start to show robustness with respect to maintaining their nominal Type
I error rates. In general, the empirical Type I error rates for the Zero and One Sales
OSPIs are lower than those of the Zero Sales OSPIs for the first half of the graph
244 M. Lindsey and R. Pavur
Fig. 12.2 Empirical Type I One-Sided Zero and One Sales Prediction
error for Zero and One Sales Intervals Illustrating Effect of Product Group
OSPIs across product group Size
size 0.14
0.12
0.1
0.08
0.06
0.04
95% Prediction Interval
99% Prediction Interval
0.02
0
50 100 200 300 400 500 600 700 800 900 1000
Product Group Size
and then reverse. These results indeed reveal that for high demand rates, such as
MTBD = 15 and MTBD = 20, that the Zero and One Sales prediction intervals
have empirical Type I error rates closer to their nominal Type I error rates. This
occurs because the variability of the demand rate estimates for the Zero and One
Sales OPSIs tend to be lower at relatively higher demand rates. However, as the
MTBD increases, the Zero and One Sales prediction interval does appear to have
higher empirical Type I error rates than the Zero Sales prediction interval.
Although these empirical Type I errors tend to improve with an increase in the
number of products, the general pattern showing the relationship between these
two OSPIs appears to remain.
Two-sided prediction intervals for the demand rate of products exhibiting zero
demand over a fixed time frame were the focus of the study by Lindsey and Pavur
(2009). To illustrate that a separate simulation study investigating OSPIs is needed
12 Reliable Stopping Rules for Stocking Spare Parts 245
Fig. 12.3 Empirical Type I error for Zero and One Sales and Zero Sales OSPIs
and that OSPIs should be used more cautiously then the two-sided prediction
intervals, a comparison of the empirical Type I errors of the two types of pre-
diction intervals for Zero and One Sales is illustrated in Fig. 12.4. The parameters
selected are the same as that used in Fig. 12.3 to compare the performance of the
Zero and One Sales OSPIs and the Zero Sales OSPIs.
Generally, the empirical Type I error rates for the two-sided prediction intervals
are lower than those for the Zero and One Sales OSPIs. Clearly, at the 95% and
99% confidence levels, an MTBD of 10 yields a greatly inflated Type I error. The
90% two-sided prediction interval is generally robust across the MTBDs in
maintaining its nominal Type I error rate. For the 95% and 99% two-sided and
one-sided prediction intervals, very large MTBDs can easily affect the reliability
of these intervals.
246 M. Lindsey and R. Pavur
0.13
0.12
0.11
0.1
Empirical Type I Error Rate
0.09
90% Prediction Interval
0.08
0.07
0.06
0.05
0.04
95% Prediction Interval 99% Prediction Interval
0.03
0.02
0.01
0
10 15 20 30 50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000
MTBD
Fig. 12.4 Empirical Type I error for Zero and One Sales OSPIs and two-sided prediction
intervals
Estimation of the future demand rate of a group of products without sales or with
no more than one sale over a specified time period is difficult due to lack of data.
There are limited demand rate estimation procedures for this type of slow-moving
inventory. The proposed prediction intervals for the future demand rate of these
slow-moving products are unique in that one-sided prediction intervals addressing
12 Reliable Stopping Rules for Stocking Spare Parts 247
this problem have not been presented. The Monte Carlo simulation results pre-
sented in this paper provide insight into conditions under which these prediction
intervals are reliable. Knowledge of reliable upper endpoints of a one-sided pre-
diction interval allows managers to compare this value to some threshold value for
decision-making purposes.
The simulation study suggests that the OSPIs are not as robust with respect to
maintaining their nominal Type I error as the two-sided prediction intervals.
Generally, the OSPIs are reliable for higher demand rates (shorter MTBD) with
relatively larger product group sizes. The OSPIs should be used with caution for
product group sizes below 300 or with very low demand rates. Reliable OSPIs can
be obtained for use in a stopping rule. The conditions under which OSPIs should
be considered reliable include product group sizes that are large and an observed
time period that should approximate or be close to the MTBD of the product
group. The Zero and One Sales OSPI provides an interval for a demand rate for
potentially more products than the Zero Sales OSPI. This proposed prediction
interval performs better with products having higher demand.
One-sided prediction intervals are applicable to stopping rules to help deter-
mine when product demand rates are below threshold limits set by managers for
carrying the merchandise. Knowledge of the upper endpoint of an OSPI allows
managers to compare this value to some threshold value for decision-making
purposes. As long as estimated future demand rates are above an acceptable
minimum determined by management, products will likely be kept in stock. Once
the minimum demand rate (threshold value) is reached, products may be consid-
ered for liquidation. There are few options for managers to use in estimating the
future demand rate of products with no demand or with little demand. This study
provides an additional tool that can be used in the decision-making process of
deciding whether to continue carrying spare parts with little demand.
Only a finite number of experimental conditions were investigated for the pro-
posed methodology. Additional simulations should be completed to extend this
research for values outside the ranges tested and even between the parameter
values selected. For example, the proposed prediction intervals were studied over a
specified range of product group sizes. General trends were identified, but running
additional simulations with group sizes in between the points selected would
support the general trend or possibly identify potential anomalies resulting from
some particular group size.
Modified prediction intervals for future demand should be investigated to
determine approaches to making them reliable to a wider range of demand rates
and product group sizes. There may be a correction factor that may be developed
248 M. Lindsey and R. Pavur
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12 Reliable Stopping Rules for Stocking Spare Parts 249
Y. Sui (&)
MicroStrategy, Inc, McLean, VA, USA
e-mail: sui11yi3@yahoo.com
E. Kutanoglu J. W. Barnes
The University of Texas at Austin, Austin, TX, USA
e-mail: erhank@me.utexas.edu
J. W. Barnes
e-mail: wbarnes@mail.utexas.edu
computational effort. For the smaller problems, the tabu search solution is
identical or very close to the optimal solution provided by classical optimization-
based methods. For the larger problems, RTS obtains solutions superior to those
obtained by classical approaches.
13.1 Introduction
Some classical network design and facility location problems such as the set
covering location problem (Toregas et al. 1971), the p-median problem (Hakimi
1964), and the uncapacitated facility location (UFL) problem (Kuehn and
Hamburger 1963) have been widely studied. Magnanti and Wong (1984)
summarized early work on the location problem and Drezner (1995) gave a survey
of applications and methods for solving the facility location problem. Melkote and
Daskin (2001a) provided a comprehensive overview of network location models.
Some research papers in this area also study problems with service constraints or
reliability issues. Simchi-Levi (1991) studied the traveling salesman location
problem with a capacitated server at the service location. Melkote and Daskin
(2001b) examined a combined facility location/network design problem in which
the facilities have limited capacities on the amount of demand they can serve.
There is a significant amount of inventory management literature. In this study,
we focus only on literature closely related to the research documented here which
includes papers for specialized SPL inventory models. Cohen et al. (1988) pre-
sented a model of an (s, S) inventory system with two priority classes of cus-
tomers, Chen and Krass (2001) investigated inventory models with minimal
service level constraints, and Agrawal and Seshadri (2000) derived the bounds to
the fill-rate constrained (Q, r) inventory problem. Early work in the area of multi-
facility inventory models includes Sherbrooke (1968, 1986) and Muckstadt (1973).
SPL systems have been successfully implemented in different industries
(Cohen et al. 1990, 1999, 2000) and in the military domain (Rustenburg et al
2001). Cohen et al. (1997) summarized a benchmark study of after-sales service
logistics systems.
In recent years, the integrated problem of facility location and inventory
stocking has attracted some attention. Barahona and Jensen (1998) studied the
integrated model considering a low level of inventory for computer spare parts.
Nozick and Turnquist (1998) investigated a distribution system model with
inventory costs as part of the facility fixed cost while maximizing the service level.
A variation of this model minimizes the cost subject to the service constraints
(Nozick 2001). Daskin et al. (2002) introduced a facility location model which
incorporates working inventory and safety stock inventory costs at the facilities.
They also proposed a Lagrangian relaxation solution algorithm for this model.
Shen et al. (2003) considered a joint location inventory problem involving a single
supplier and multiple retailers and developed a column generation method to solve
this model. Jeet (2006) solved the integrated facility location and inventory
stocking problem while considering part commonality by using a fill rate outer
approximation scheme. Candas and Kutanoglu (2007) showed that determining the
254 Y. Sui et al.
This section introduces the MIP model for the integrated SPL problem. Stochastic
demands are included in terms of fill rate to represent the part availability. The
limitations of the mathematical programming model are listed at the end of this
section as well.
The service level requirements imposed on the SPL system motivate the simul-
taneous modeling of the network design and inventory stocking decisions. Our
integrated model development is based on Jeet et al. (2009). Similar to their study,
we make the following assumptions:
13 Reactive Tabu Search for Large Scale SPL 255
1. We assume that the SPL network has only one echelon and all the stocking
facilities serve the customers directly. We also assume these facilities can be
replenished by a central warehouse with unlimited-capacity (i.e., infinite sup-
ply) and no time delay.
2. We use a one-for-one replenishment policy for all the stocking facilities. Since
the demand is assumed to be very low and lead time is relatively short, we do
not need to consider the batch ordering where replenishment quantity is more
than one. This is a very common assumption in low demand inventory systems
such as SPL.
3. We assume that all the customer service level requirements are aggregated to
obtain the system target service level.
4. The demands from different customers are governed by statistically indepen-
dent Poisson distributions. The Poisson distribution is a good approximation of
the low-demand distribution (Muckstadt 2005).
5. Customer demand can be only assigned to one facility with at least one unit of stock.
In case of a facility ‘‘stock-out,’’ the customer demand is passed to the central
warehouse, which satisfies the customer demand with a direct shipment. For the
‘‘stocked-out’’ facility, the demand is considered lost, hence we use the lost-sales fill
rate. (Long run average fill rates are computed using the lost sales formula derived
from the steady state behavior of the M/G/s/s queuing model (Zipkin 2000).
6. There is only one time-window over which the target aggregate service level is
to be satisfied. When the facility can provide the part to the customer in need
within the time window (depending on the distance between the facility and the
customer), the satisfied unit of demand is counts towards its target service level.
• Parameters
fi Annual cost of operating facility i (assumed constant once the facility is open).
cij Cost of shipping one unit of the part from facility i to customer j.
dj Mean annual demand for customer j.
hi Annual inventory holding cost at facility i.
t Stock replenishment (resupply) lead time (assumed identical for all facilities).
a Time based service level (defined as the percentage of demand that is to be
satisfied within the time window).
256 Y. Sui et al.
• Variables
The model objective is to minimize the total annual cost of open facilities,
transportation, and inventory stocking. The constraints of our model draw mainly
from the UFL model and integrate the inventory part of the problem (fill rates and
service levels):
X X X X
minimize fY þ
i2I i i i2I
c dX þ
j2J ij j ij
hS
i2I i i
ð13:1Þ
X
subject to i2I
Xij ¼ 1; 8j 2 J ð13:2Þ
ksi =s 4
bi ¼ 1 Psi i ni ; 8i 2 I ð13:6Þ
n¼0 ki =n!
demand for each facility which is used to calculate the facility’s fill rate in con-
straints (6). Constraints (6) compute facility fill rates using the lost sales formula
(Zipkin 2000). Constraints (7) state that every facility that has been allocated some
demand must have at least one unit of stock. Constraints (8) state that the variables
Xij and Yi are 0 or 1, and constraints (9) confine the range and enforce the inte-
grality of the stock levels for all the facilities. For more details of the mathematical
model, see Jeet et al. (2009).
As we can see from the mathematical model above, variables ki are dependent on
integer demand assignment variables Xij in constraints (5) and fill rate variables bi
are calculated from mean lead time demand variables ki in constraints (6). How-
ever, on the left hand side of service level constraint (4), the variables bi, which are
already originally computed from Xij, are multiplied by Xij. These interactions
make the problem non-linear and strongly coupled on variables Xij and make it
impossible to solve practically sized problems with classical MIP methods. Jeet
(2006) tested the above model for different smaller problem sizes, and created
benchmark results for these problem sets. These problems were modeled using
CPLEX 9.0 on a microcomputer with dual Xeon 1.8 GHz processors with 1 GB
RAM running the Suse Linux operating system. The largest data set had 24
potential facilities and 158 customers. Some of the problems were not solved to
optimality within 30 min of computational effort. A custom method specifically
developed for the integrated model using an outer approximation and a strong
lower bound had to solve a still time consuming lower bounding problem multiple
times. In real applications, SPL networks may have hundreds of potential facilities
and thousands of customers requiring far more scalable approaches. The inability
of classical approaches required the development of an efficient heuristic meth-
odology that would provide high quality solutions within acceptable amounts of
computational effort.
TS, a metaheuristic search algorithm (Glover 1989, 1990), controls a hill climbing
heuristic using a tabu memory structure to manage the search and allows non-
improving solutions during the search to permit escape from local optima. Battiti
and Tecchiolli (1994) first introduced Reactive Tabu Search (RTS) to dynamically
change the tabu tenure by monitoring the repetitions of visited incumbent solutions
and allowed to search to escape from chaotic attractor basins. The solutions visited
during the search and the corresponding iteration numbers are preserved in
memory. When a candidate solution is considered, the repetitions of that solution
258 Y. Sui et al.
and the number of iterations between the two adjacent visits are checked. The RTS
increases the tabu tenure when solutions are often repeated to diversify the search
and decreases the tabu tenure to intensify the search when there are few repeated
solutions.
The total solution cost is the sum of the fixed facility opening costs, the inventory
costs, and the transportation costs. The first two of these are easily calculated using
the solution representation. For transportation costs, customer demands must be
assigned to facilities. The assignment of customers to facilities also directly affects
the service level achieved. To this end, assignment decisions (represented by Xij’s
in the MIP model) are made carefully by paying attention to both transportation
costs and service level contribution of an assignment. We define ui ¼
P
bi j2J dij dj bi Xij to be the service level contribution of facility i. Summing the ui
over all i yields the left hand side of the service level constraint (4). ui is the
expected demand satisfied within the time window (over a year period) at facility
i. If a customer within a facility i’s time window is assigned to that facility, ui will
13 Reactive Tabu Search for Large Scale SPL 259
the service level increment and pick the customer with the least ratio. Continue
moving customers to open higher fill rate facilities until the solution either
reaches feasibility or a maximum number of customers are moved.
The tabu memory structure consists of a matrix of size jIjXSmax with the rows
corresponding to the facility indices ( fac_index) and the columns to the facility
stock levels ( fac_stock). A neighbor solution is tabu if the stock level for that
facility has been changed from the neighbor solution’s proposed stock level in the
last tabu tenure iterations. For a swap move, both facilities stock levels must
satisfy this tabu criterion for the neighbor solution to be tabu.
The Initialization function initializes the data structures for the tabu memory
structure and the initial solution is generated by a heuristic method. Viewing the
construction of the initial solution as a weighted set covering problem, we desire to
open the facility, i, that can cover the largest amount of unassigned demand. To
determine the appropriate inventory level for that facility, we compute the ratio of
264 Y. Sui et al.
the opening cost, fi, and the cost of providing one unit of inventory, hi. If fi,/
hi C Smax, opening facility i is more expensive than stocking more units in the
currently open facilities which indicates the stock level of the newly opened
facility i should be ‘‘high,’’ hence is set to its maximum level. Otherwise, we set
the inventory level of facility i to one unit. This facility opening procedure is
repeated until all the customers are assigned. Function Reactive_Tabu_Search,
presented in Fig. 13.5, summarizes the RTS–SPL algorithm.
Two of our test problem sets are from Jeet (2006) and the other two larger sets are
randomly generated for this study using the same approach in Jeet (2006).
13 Reactive Tabu Search for Large Scale SPL 265
RTS–SPL was executed on the same computer used to generate Jeet (2006) results:
Dual Xeon 1.8 GHz processors with 1 GB RAM. During all problem runs, all
RTS-SPL parameters are held constant at the following values: INCREASE = 1.2,
DECREASE = 0.8, REP = 2, CHAOS = 5, CYCLE_MAX = 10, W1 = 0.1,
W2 = 0.9.
The small problems have 15 facilities and 50 customers (15 9 50) with customer
and facility locations generated randomly on a 150 9 150 grid. The transportation
costs, cij, or equal to one tenth of the Euclidean distance between facility i and
customer j, rounded to a positive integer. The annual mean demand values, dj, are
uniformly distributed over the range from 1 to 3. Replenishment lead times for all
facilities are 7 (days). Time window indicators, dij, are set to one if the Euclidean
distance between facility i and customer j is less than 40, and zero, otherwise. Smax
266 Y. Sui et al.
is set to 5, which is more than enough to obtain fill rates very close to 100% at all
facilities regardless of the demand assigned to any facility.
Five levels of holding cost, same for all facilities, hi (h = 1, 10, 20, 50, and
100) and three service levels (40, 60, or 80% of the maximum possible service) are
investigated. Fixed costs, fi, of opening all facilities are set to either 0 or 1,000.
Generating one problem for every possible combination of holding costs, service
levels, and facility fixed costs yields 30 problems. Three such problem sets were
generated.
Tables 13.1, 13.2, and 13.3 present the comparative results including the RTS–
SPL solution values, MIP solution values (Jeet 2006), the percentage difference,
D = 100(RTS - MIP)/MIP%, the number of open facilities, and the total
inventory across all facilities. A bracketed value after the facility or inventory
numbers indicates the difference between the RTS–SPL and MIP solutions. For
example, 15(-1) means the RTS-SPL solution has 15 units of stock, 1 less than the
MIP solution’s total inventory. Similarly, 17(+1) means that the RTS-SPL’s total
stock level is 17, 1 more than the MIP solution’s.
For Tables 13.1, 13.2, and 13.3, when the fixed cost is 0 (top halves of the
tables), RTS–SPL obtained the same or better solutions than the MIP for all
problems (Note that MIP solutions may not be optimal due to time limit used for
both methods, 900 s in this case). When the fixed cost is 1000 (bottom halves of
the tables), RTS–SPL found slightly inferior solution values in 8 of the 45 prob-
lems when compared to the MIP solutions with a maximal difference of 1.06% and
superior solutions in 10 out of the 45 instances with a maximal difference of 2.81%
improvement.
The computational time for RTS–SPL for any problem was \1 s while the
average time for MIP to run for one instance was 34 s with maximum time of
806 s. The RTS–SPL obtains near optimal solutions in markedly less computa-
tional effort.
The medium sized problem sets were based on real data provided by a service
parts logistics group at a large computer hardware manufacturer. We use six
representative networks (customer and facility locations), representing different
regions in the United States. The sizes of the networks are different depending on
the region. We use the actual transportation costs and dij values provided in the
real data. The problems were solved for service time-windows of 2 hours (TW1)
and 12 hours (TW2). Longer time-windows yield more dij = 1 indicating that the
demands are more easily satisfied towards time-based service since more candidate
facilities can provide service to those customers. The maximum stock level for
each facility is 5.
Three levels of holding cost hi (50, 100, or 200) and three service levels (40, 60,
or 80%) were investigated. Facility opening costs were all set to 0, focusing on the
13 Reactive Tabu Search for Large Scale SPL 267
Table 13.1 RTS and MIP solutions for small data set A
Instance Solution
n h a (%) f RTS MIP D (%) Open fac Total inventory
1a 1 40 0 232 232 0.00 12 12
2a 1 60 0 232 232 0.00 12 12
3a 1 80 0 232 232 0.00 12 12
4a 10 40 0 314 314 0.00 7 7
5a 10 60 0 314 314 0.00 7 7
6a 10 80 0 321 321 0.00 9 9
7a 20 40 0 369 369 0.00 5 5
8a 20 60 0 369 369 0.00 5 5
9a 20 80 0 411 411 0.00 9 9
10a 50 40 0 491 491 0.00 4 4
11a 50 60 0 519 519 0.00 5 5
12a 50 80 0 652 652 0.00 7 8
13a 100 40 0 665 665 0.00 3 3
14a 100 60 0 769 769 0.00 5 5
15a 100 80 0 1052 1052 0.00 7 8
16a 1 40 1000 2444 2444 0.00 2 5
17a 1 60 1000 3366 3366 0.00 3 7
18a 1 80 1000 4300 4300 0.00 4 9
19a 10 40 1000 2489 2489 0.00 2 5
20a 10 60 1000 3429 3435 -0.17 3 7
21a 10 80 1000 4381 4381 0.00 4 9
22a 20 40 1000 2539 2539 0.00 2 5
23a 20 60 1000 3499 3505 -0.17 3 7
24a 20 80 1000 4471 4471 0.00 4 9
25a 50 40 1000 2689 2689 0.00 2 5
26a 50 60 1000 3709 3715 -0.16 3 7
27a 50 80 1000 4741 4741 0.00 4 9
28a 100 40 1000 2939 3024 -2.81 2 5
29a 100 60 1000 4059 4065 -0.15 3 7
30a 100 80 1000 5191 5191 0.00 4 9
Table 13.2 RTS and MIP solutions for small data set B
Instance Solution
n h a (%) f RTS MIP D (%) Open fac Total inventory
1b 1 40 0 279 279 0.00 15 15
2b 1 60 0 279 279 0.00 15 15
3b 1 80 0 280 280 0.00 15 16
4b 10 40 0 368 368 0.00 9 9
5b 10 60 0 368 368 0.00 9 9
6b 10 80 0 388 388 0.00 9 11
7b 20 40 0 443 443 0.00 7 7
8b 20 60 0 443 443 0.00 7 7
9b 20 80 0 490 490 0.00 8 10
10b 50 40 0 583 583 0.00 4 4
11b 50 60 0 628 629 -0.16 6 6
12b 50 80 0 790 790 0.00 8 10
13b 100 40 0 783 783 0.00 4 4
14b 100 60 0 928 929 -0.11 6 6
15b 100 80 0 1290 1290 0.00 8 10
16b 1 40 1000 2535 2535 0.00 2 5
17b 1 60 1000 3449 3449 0.00 3 9
18b 1 80 1000 6341 6342 -0.02 6 15(-1)
19b 10 40 1000 2580 2580 0.00 2 5
20b 10 60 1000 3530 3530 0.00 3 9
21b 10 80 1000 6464 6464 0.00 6 13
22b 20 40 1000 2630 2630 0.00 2 5
23b 20 60 1000 3620 3620 0.00 3 9
24b 20 80 1000 6594 6594 0.00 6 13(+1)
25b 50 40 1000 2780 2780 0.00 2 5
26b 50 60 1000 3890 3890 0.00 3 9
27b 50 80 1000 6984 6954 0.43 6 13(+1)
28b 100 40 1000 3030 3030 0.00 2 5
29b 100 60 1000 4340 4340 0.00 3 9
30b 100 80 1000 7634 7554 1.06 6 13(+1)
medium size data sets with 12 h time windows are virtually identical, hence are
not presented here.
The run time limits of RTS-SPL are 5 s for the first three data sets and 10 s for
the last three data sets. The actual average total run time over 18 instances for each
of the six networks was 1.814, 2.844, 3.957, 8.361, 6.453, and 7.809 seconds and
the time when the best solution was first encountered was shorter. The average run
times of MIP for the six sets are 0.28, 1.67, 69.78, 4.11, 19.83, and 3.78 s. There is
no apparent advantage for RTS–SPL for some data sets, but the MIP run times are
highly variable while the RTS is much more consistent. The longest MIP run times
are 1, 18, 386, 36, 183, and 17 s for each set. RTS–SPL found very good or
optimal solutions in acceptable times for all the instances.
13 Reactive Tabu Search for Large Scale SPL 269
Table 13.3 RTS and MIP solutions for small data set C
Instance Solution
n h a (%) f RTS MIP D (%) Open fac Total inventory
1c 1 40 0 290 290 0.00 11 11
2c 1 60 0 290 290 0.00 11 11
3c 1 80 0 297 298 -0.34 11 18
4c 10 40 0 366 366 0.00 6 6
5c 10 60 0 366 366 0.00 6 6
6c 10 80 0 438 438 0.00 7 14
7c 20 40 0 416 416 0.00 5 5
8c 20 60 0 426 426 0.00 6 6
9c 20 80 0 578 579 -0.17 7 14
10c 50 40 0 533 533 0.00 3 3
11c 50 60 0 606 606 0.00 6 6
12c 50 80 0 998 999 -0.10 7 14
13c 100 40 0 683 683 0.00 3 3
14c 100 60 0 906 906 0.00 6 6
15c 100 80 0 1698 1699 -0.06 7 14
16c 1 40 1000 2462 2462 0.00 2 6
17c 1 60 1000 3390 3385 0.15 3 7(-4)
18c 1 80 1000 6336 6326 0.16 6 17(+1)
19c 10 40 1000 2516 2530 -0.55 2 6(+1)
20c 10 60 1000 3453 3453 0.00 3 7
21c 10 80 1000 6485 6470 0.23 6 16
22c 20 40 1000 2576 2580 -0.16 2 6(+1)
23c 20 60 1000 3523 3523 0.00 3 7
24c 20 80 1000 6645 6630 0.23 6 16
25c 50 40 1000 2729 2730 -0.04 2 5
26c 50 60 1000 3733 3733 0.00 3 7
27c 50 80 1000 7125 7110 0.21 6 16
28c 100 40 1000 2979 2980 -0.03 2 5
29c 100 60 1000 4083 4083 0.00 3 7
30c 100 80 1000 7925 7910 0.19 6 16
Table 13.4 RTS and MIP solutions for medium data set A
Instance Solution
n h a (%) TW RTS MIP D (%) Open Total
fac inventory
1a 50 0.4 TW1 8404 8404 0.00 9 9
2a 100 0.4 TW1 8854 8854 0.00 9 9
3a 200 0.4 TW1 9738 9738 0.00 8 8
4a 50 0.6 TW1 8404 8404 0.00 9 9
5a 100 0.6 TW1 8854 8854 0.00 9 9
6a 200 0.6 TW1 9738 9738 0.00 8 8
7a 50 0.8 TW1 8454 8454 0.00 9 10
8a 100 0.8 TW1 8954 8954 0.00 9 10
9a 200 0.8 TW1 9954 9954 0.00 9 10
270 Y. Sui et al.
Table 13.5 RTS and MIP solutions for medium data set B
Instance Solution
n h a (%) TW RTS MIP D (%) Open Total
fac inventory
1b 50 0.4 TW1 8437 8437 0.00 12 12
2b 100 0.4 TW1 9012 9012 0.00 11 11
3b 200 0.4 TW1 10010 10010 0.00 9 9
4b 50 0.6 TW1 8437 8437 0.00 12 12
5b 100 0.6 TW1 9012 9012 0.00 11 11
6b 200 0.6 TW1 10010 10010 0.00 9 9
7b 50 0.8 TW1 8487 8487 0.00 12 13
8b 100 0.8 TW1 9112 9212 -1.09 11 12(-1)
9b 200 0.8 TW1 10312 10437 -1.20 11(-1) 12(-1)
Table 13.6 RTS and MIP solutions for medium data set C
Instance Solution
n h a (%) TW RTS MIP D (%) Open Total
fac inventory
1c 50 0.4 TW1 7397 7396 0.01 7 7
2c 100 0.4 TW1 7651 7651 0.00 5 5
3c 200 0.4 TW1 8151 8151 0.00 5 5
4c 50 0.6 TW1 7397 7446 -0.66 7(-1) 7(-1)
5c 100 0.6 TW1 7747 7751 -0.05 7(+2) 7(+1)
6c 200 0.6 TW1 8351 8351 0.00 5 6
7c 50 0.8 TW1 7547 7546 0.01 7 10
8c 100 0.8 TW1 7999 7999 0.00 6 9
9c 200 0.8 TW1 8899 8899 0.00 6 9
Table 13.7 RTS and MIP solutions for medium data set D
Instance Solution
n h a (%) TW RTS MIP D (%) Open Total
fac inventory
1d 50 0.4 TW1 13468 13468 0.00 17 17
2d 100 0.4 TW1 14246 14246 0.00 15 15
3d 200 0.4 TW1 15656 15656 0.00 12 12
4d 50 0.6 TW1 13468 13468 0.00 17 17
5d 100 0.6 TW1 14246 14246 0.00 15 15
6d 200 0.6 TW1 15656 15656 0.00 12 12
7d 50 0.8 TW1 13518 13518 0.00 17 18
8d 100 0.8 TW1 14346 14377 -0.22 15(-1) 16(-1)
9d 200 0.8 TW1 15946 16077 -0.81 15(-1) 16(-1)
13 Reactive Tabu Search for Large Scale SPL 271
Table 13.8 RTS and MIP solutions for medium data set E
Instance Solution
n h a (%) TW RTS MIP D (%) Open Total
fac inventory
1e 50 0.4 TW1 10055 10055 0.00 14 14
2e 100 0.4 TW1 10708 10708 0.00 13 13
3e 200 0.4 TW1 11863 11863 0.00 11 11
4e 50 0.6 TW1 10055 10055 0.00 14 14
5e 100 0.6 TW1 10708 10708 0.00 13 13
6e 200 0.6 TW1 11863 11863 0.00 11 11
7e 50 0.8 TW1 10205 10205 0.00 14 17
8e 100 0.8 TW1 11008 11008 0.00 13 16
9e 200 0.8 TW1 12511 12511 0.00 12 15
Table 13.9 RTS and MIP solutions for medium data set F
Instance Solution
n h a TW RTS MIP D (%) Open Total
(%) fac inventory
1f 50 0.4 TW1 17040 17040 0.00 19 19
2f 100 0.4 TW1 17891 17891 0.00 16 16
3f 200 0.4 TW1 19480 19480 0.00 14 14
4f 50 0.6 TW1 17040 17040 0.00 19 19
5f 100 0.6 TW1 17891 17891 0.00 16 16
6f 200 0.6 TW1 19480 19480 0.00 14 14
7f 50 0.8 TW1 17040 17090 -0.29 19 19(-1)
8f 100 0.8 TW1 17990 18022 -0.18 19(+2) 19(+1)
9f 200 0.8 TW1 19691 19822 -0.66 16(-1) 17(-1)
The restart and escape features do not play very important role in the medium
size data sets. The instances are ‘‘easy’’ for the RTS-SPL since there are certain
patterns embedded in the real data. Some limitations are incorporated in the data
itself to prevent some customers from being assigned to certain facilities and these
limitations make the search easier. The iteration when the best solution was first
found was small implying that the search finds the best solution very quickly and
leaves very little room of improvement for the restart and escape mechanisms.
13.5.4 Results for the Large and Extra Large Problem Sets
The two data sets discussed in this section were generated the same way as the
small data sets. The three large data sets have 50 facilities and 200 customers
randomly generated on a 150 9 150 grid and the three extra large data sets have
272 Y. Sui et al.
100 facilities and 500 customers randomly generated on a 200 9 200 grid. Dif-
ferent random number seeds were used for the problem generations. The service
radius is set to 40 as in the small data set. The maximum stock level for the large
problem set is 7 and for the extra large problem set 10.
Three values for holding cost (1, 20, 100) and three target service levels (40, 60,
80%) were investigated. Two values for the facility fixed cost (0, 1000) were also
considered. Combining all possible interactions of these factors produces 18
instances for each problem set. Thus, there are 54 instances associated with either
the large problem sets or extra large problem sets.
The results of the three large data sets presented in Tables 13.10, 13.11, and
13.12 show that the average time for obtaining the best solution is 3.175 s with a
300-s limit on run time. For the extra large data sets (Tables 13.13, 13.14, and
13.15) the average run time is 117 s with a 600-s limit.
The problem sizes considered here made it impossible to obtain an optimal
or near optimal solution for any of these problems using classical MIP
approaches. Hence, we cannot compare RTS–SPL with a classical MIP
approach. Nevertheless, we compare the RTS–SPL to its variations without
escape and without restart procedures. Tables 13.10, 13.11, 13.12, 13.13, 13.14,
and 13.15 show the results of (1) the standard RTS, (2) the RTS without escape
mechanism (NoEsc), and (3) the RTS without restart mechanism (NoRes). The
D after NoEsc or NoRes means the percentage differences between their
solutions and RTS solutions. All the versions of TS have the same parameter
settings and the same run time limit. For the large data sets, there are 29 and 6
Table 13.13 RTS and TS solutions for extra large data set A
Instance RTS solution Compare
n h a (%) f RTS Best iter Best time(ms) NoEsc D (%) NoRes D (%)
1a 1 40 0 1572 75 3850 1572 0.00 1572 0.00
2a 1 60 0 1572 83 4650 1572 0.00 1572 0.00
3a 1 80 0 1574 91 7750 1574 0.00 1574 0.00
4a 20 40 0 2455 211 18800 2457 0.08 2455 0.00
5a 20 60 0 2455 209 18590 2457 0.08 2455 0.00
6a 20 80 0 2771 1767 205600 2858 3.14 2775 0.14
7a 100 40 0 3919 2603 177460 3933 0.36 3925 0.15
8a 100 60 0 4700 100 8960 4762 1.32 4700 0.00
9a 100 80 0 6095 507 45400 6151 0.92 6095 0.00
10a 1 40 1000 8332 86 9400 8643 3.73 8332 0.00
11a 1 60 1000 8963 39 3720 9075 1.25 8963 0.00
12a 1 80 1000 10491 130 14330 10532 0.39 10491 0.00
13a 20 40 1000 8714 31 3150 9023 3.55 8714 0.00
14a 20 60 1000 9533 39 4080 9626 0.98 9533 0.00
15a 20 80 1000 11358 458 48070 11406 0.42 11358 0.00
16a 100 40 1000 10314 31 3600 10623 3.00 10314 0.00
17a 100 60 1000 11933 39 4080 11946 0.11 11933 0.00
18a 100 80 1000 14958 458 50480 15086 0.86 14958 0.00
Table 13.14 RTS and TS solutions for extra large data set B
Instance RTS solution Compare
n h a (%) f RTS Best iter Best time(ms) NoEsc D (%) NoRes D (%)
1b 1 40 0 1618 62 3360 1618 0.00 1618 0.00
2b 1 60 0 1618 66 3510 1618 0.00 1618 0.00
3b 1 80 0 1621 110 10850 1622 0.06 1621 0.00
4b 20 40 0 2440 146 12500 2442 0.08 2440 0.00
5b 20 60 0 2440 156 15120 2442 0.08 2440 0.00
6b 20 80 0 2758 4717 516530 2799 1.49 2756 -0.07
7b 100 40 0 3853 4259 355130 3866 0.34 3854 0.03
8b 100 60 0 4568 7223 598410 4613 0.99 4613 0.99
9b 100 80 0 5877 6016 575340 6030 2.60 5925 0.82
10b 1 40 1000 7911 46 5460 8045 1.69 7911 0.00
11b 1 60 1000 8838 5836 557400 8997 1.80 8839 0.01
12b 1 80 1000 10292 43 3960 10408 1.13 10292 0.00
13b 20 40 1000 8386 46 5470 8501 1.37 8386 0.00
14b 20 60 1000 9409 99 10880 9586 1.88 9409 0.00
15b 20 80 1000 11052 43 3950 11320 2.42 11052 0.00
16b 100 40 1000 10037 174 19610 10214 1.76 10037 0.00
17b 100 60 1000 11809 91 10830 12066 2.18 11809 0.00
18b 100 80 1000 14252 43 4200 15160 6.37 14252 0.00
13 Reactive Tabu Search for Large Scale SPL 275
Table 13.15 RTS and TS solutions for extra large data set C
Instance RTS solution Compare
n h a (%) f RTS Best iter Best time(ms) NoEsc D (%) NoRes D (%)
1c 1 40 0 1644 67 3460 1644 0.00 1644 0.00
2c 1 60 0 1644 72 4320 1644 0.00 1644 0.00
3c 1 80 0 1647 133 16250 1648 0.06 1647 0.00
4c 20 40 0 2467 146 11830 2478 0.45 2467 0.00
5c 20 60 0 2461 475 49650 2481 0.81 2461 0.00
6c 20 80 0 2768 3928 462650 2853 3.07 2769 0.04
7c 100 40 0 3772 1486 103140 3798 0.69 3772 0.00
8c 100 60 0 4595 5933 560910 4621 0.57 4608 0.28
9c 100 80 0 5921 5009 477140 6084 2.75 6025 1.76
10c 1 40 1000 8049 1042 109980 8122 0.91 8054 0.06
11c 1 60 1000 8929 4205 439850 8971 0.47 8959 0.34
12c 1 80 1000 11086 105 10230 11330 2.20 11086 0.00
13c 20 40 1000 8567 28 3930 8616 0.57 8567 0.00
14c 20 60 1000 9518 2976 311770 9579 0.64 9548 0.32
15c 20 80 1000 11799 80 8190 12147 2.95 11799 0.00
16c 100 40 1000 10157 62 7560 10189 0.32 10157 0.00
17c 100 60 1000 11998 3788 405300 12139 1.18 12028 0.25
18c 100 80 1000 14759 80 8030 15587 5.61 14759 0.00
out of 54 instances with better results when compared to the RTS–SPL without
escape and without restart separately, respectively. Only 1 instance saw
improvement for the RTS–SPL without escape or restart (red highlighting). For
the extra large data sets, only 7 instances for the RTS–SPL without escape
obtained the same result as the full version of the RTS. The other 47 instances
obtained inferior results when the escape feature was disabled and 2 of them
were inferior by more than 5%. Disabling the restart mechanism affects the
solution quality for total 13 instances and only 1 instance improves. Hence, it
appears that the larger the data set size, the more important the restart and
escape mechanisms become.
RTS–SPL provides very good solutions for various size problems as tested com-
putationally. However, during the search procedure, RTS–SPL visits a large
number of solutions. To ‘‘validate’’ RTS–SPL, we randomly generated the same
number of solutions that were visited by RTS–SPL and compared their best
solution value found with the RTS–SPL final solution. We performed this exercise
using the small problems, conjecturing that the random solutions would have the
highest chance to ‘‘hit’’ a near optimal solution for this data set and hence would
be more competitive compared to RTS–SPL solutions. For the small data sets,
276 Y. Sui et al.
there are around 2,000 iterations processed for each instance and the number of
solutions visited at each iteration is about 30. Therefore, we randomly generate
60,000 solutions for each instance and compare the best of these with RTS
solutions. As presented in Sui (2008), the RTS solutions were always better than
the best of random ones and the improvement was greater than 100% for some
instances. The average improvement was about 28% for the zero fixed cost and
56% for the fixed cost of 1000.
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Chapter 14
Common Mistakes and Guidelines
for Change in Service Parts Management
Andrew J. Huber
A. J. Huber (&)
Xerox Services, Xerox Corporation, 100 Clinton Avenue South,
Rochester, 14644 NY, USA
e-mail: Andrew.Huber@xerox.com
In recent decades the level of investment in service supply chains has increased due to
the recognition of their importance. Service parts management is ripe with oppor-
tunity for academicians, practitioners and managers due to several emerging trends:
1. Product commoditization and price pressure
2. Cost reduction opportunities through technology upgrades
3. Increased demand for differentiated service offerings
4. The need for improvement in underperforming service supply chains
This chapter will focus primarily on the need for improvement. There is a
wide disparity between organizations in the level of sophistication brought to the
practice of service parts management. On the sophisticated side of the spectrum,
the United States military has long employed the use of sophisticated techniques
and systems to manage over $100 billion worth of service parts that are nec-
essary to maintain the readiness of its weapon systems. In the commercial world
there are large organizations that have very sophisticated processes and systems
for the management of parts, however there are many more who do not. Before
we list some of the more common mistakes and methods available for correcting
them, it is helpful to understand some misperceptions that cause the common
mistakes and often lead to poor performing service networks.
When a company sells a product, it must convince its customers to purchase their
product rather than its competitors. The company’s effectiveness is largely a
14 Common Mistakes and Guidelines 281
function of its ability to effectively execute the ‘‘4 Ps’’ of marketing: product,
price, place and promotion. As product markets mature, competition leads to
product commoditization, where the products are relatively indistinguishable from
one another. This leads to competition based on price that benefits consumers but
reduces profits to the supplier.
For long life cycle maintenance intensive products, the profitability associated
with the service of a product is typically much larger than that achieved on the
sale. For one thing, the sale of the product occurs only once, but the sale of parts,
service, supplies, and sometimes the operation of the product represent a stream of
payments where profits are made year after year. The manager of a medical
equipment manufacturer once demonstrated to me that his company realizes eight
times as much profit from post sale services and supplies than from the sale of the
product itself. Similar parallels exist in many industries. Where maintenance is
involved and the parts required are uniquely engineered to the specific product,
service parts revenues are often a significant source of profitability.
For the uninitiated, service parts management is viewed as a simple series of tasks:
people place orders on suppliers, shipments are put into inventory, and orders are
filled to customers. All that is needed is to monitor supplies on hand and order
more when needed. As a result of this perception of simplicity, rudimentary
systems are used and people with insufficient skills are employed to perform the
repetitive tasks of ordering. This is where the trouble starts!
Suppliers de-prioritize orders for small quantities and shipments arrive late. End
consumers incur backorders while their machines are down. Planners are blamed
for poor performance and stock up on items that they’ve been ‘‘burned’’ on before.
Inventory grows yet shortages persist. Millions of dollars off write-offs occur on
obsolete inventories. Customers incur costs of not having their equipment avail-
able and have to resort to alternatives like redundant machines, demanding their
own inventory often on consignment, sending jobs out while waiting for equip-
ment to be restored. Jobs don’t get done on time, planes don’t take off on time,
medical tests are delayed, and penalties are paid. ‘‘Planning’’ becomes just a word
on an organization chart as most of the people spend most of their time expediting
rather than planning.
1. Metrics
What performance goals will be applied and how will they be measured?
2. Forecasting
How will demand forecasts be generated and from what data? Different
methods are generally appropriate at the pre-launch, post-launch, mid-life and end
of life cycles.
3. Replenishment methods
What replenishment rules will be applied and how will they be calculated?
Examples include use-one-replenish-one; economic order quantities, or periodic
replenishment.
4. Replenishment order prioritization
Which parts will be repaired first? How will manufacturing prioritize alloca-
tions between production requirements versus service needs? How will critical
shortage situations be managed?
5. Stock allocation
How will arriving shipments be distributed throughout the stocking network?
Will transshipments between locations be possible and under what circumstances
will they occur? When shortages exist, will inventory be reserved to meet critical
customer needs?
284 A. J. Huber
Traditionally, advanced analytical methods have not been typically viable for
transactional systems at the operational level. Today however, several classes of
resources have become more prevalent making the potential for operational
decision making in real time an expanding area of opportunity. These resources
include:
• inexpensive storage
• data from transactional systems
• communications bandwidth
• inexpensive and rapid processing power has created an environment where the
application of techniques that can respond in real time can improve perfor-
mance. Real time analytical methods are fertile ground for research and appli-
cation. In time, real time analytical modules could supplement or replace rules
and thresholds developed by tactical solutions. Imagine a world where a diag-
nostic system indicates an item has failed, and instantaneously a work order to
replace the unit is generated, the return ship instructions are provided, the part
repair facility schedules the repair of the part, and components are immediately
replenished from the manufacturer. The technology for this type of world exist
today, but to a large degree has not been exploited.
Here are some examples of operational decisions for service parts.
1. Whether to retain, repair or scrap a failed item.
2. To which repair facility a returned item should be sent.
3. When items should be transshipped between facilities.
4. Immediate updates to forecasts triggered from transactional activity. Examples
include:
a. Orders for parts
b. Installation of equipment
c. Relocation of equipment
d. Removals of equipment
e. Changes in the usage or mission of equipment
On the day of this writing, a Google search on the words ‘‘improve demand
forecasting’’ yielded the following results on the first search results page:
• 10 articles on forecasting and planning software packages
• 2 articles containing the essential proposition was that judgmental forecasting is
‘‘bad’’ and statistical forecasting is ‘‘good’’
• 1 web site of a panel of experts listing their findings on the challenge of fore-
casting global demand for health care services
14 Common Mistakes and Guidelines 285
purchased prior to launch then the statistical forecasting would take over and
perform adequately until an all time buy had to be purchased. With the digital
products however, the life cycles were much shorter. Typically by the time the
product was on retail shelves manufacturing was already planning the last pro-
duction runs and the parts planner was being asked to place all time buy orders.
Here the statistical methods that simply projected past history for the analog
products could no longer perform for the shorter life-cycle digital products. A
statistical model was in fact constructed for this situation, and it incorporated
channel sales volumes, sell-through time lags, and time to failure distributions
derived from previous generation but similar products. This example illustrates
that forecasting requires the use of both statistical methods as well as judgment
derived from knowledge of the market matched to knowledge of forecasting
methods.
Now that we understand some common misperceptions about service parts man-
agement, let us turn to some of the mistakes that lead to poor performance.
Mistake #1: Failure to Recognize the Strategic Importance of Service to the
Profitability of the Company
I have introduced the misperception by many that equipment service is not
highly profitable. This misperception results in organizations to forgo profit
opportunities in either of two ways:
1. Failure to exploit service revenue potential by the development and marketing
of differentiated services offerings
2. Failure to apply technologies and an information infrastructure to achieve
efficiency and effectiveness.
customer a machine may be used for work that is not critical and a service offering
that provides next business day response and repair within 48 h may meet their
needs. For another customer, the same type of machine may be used in the
operation of their business and if it cannot be repaired the same day that customer
may lose business. In these environments it is common to offer three levels of
service with significantly uplifted prices for higher levels of service.
When companies tailor service offerings to unique customer needs, they create
win-win business for both them and their customer: they realize the greater rev-
enue and margins from customers with the greatest need for equipment avail-
ability, and they retain the business from customers who would otherwise turn to
lower cost competitors. Differentiated offerings and flexibility in terms along with
consistency in delivery to those terms enable long term business relationships
based on customer loyalty that enable consistent and predictable levels of
profitability.
Many companies traditionally thought of as product manufacturers have been
transforming themselves to become total solutions providers comprised largely of
three components:
1. The products
2. The maintenance of those products
3. The operation of the product and delivery of the services produced
Thus a manufacturer of printers may sell the printers, or the management of a
fleet of printers including maintenance and supplies replenishment, or it may be a
provider of printing services where the product, supplies and maintenance are only
the means to produce prints.
For repair and maintenance contracts, service offerings may be differentiated
along three dimensions:
1. The days and hours of coverage when service is available
2. The services that are included and excluded from the offering
3. The Service Level Agreement (SLA) performance guarantees
Differentiation in service offerings represent an opportunity for profits yet a
challenge for service parts managers. Consider a simple example of three tiered
offerings:
• mission critical with 2 h guaranteed onsite response
• same day service
• next day service
It is clear that a very high level of parts availability must be maintained for
customers who have purchased the ‘‘mission critical’’ service offering. Service
level targets would be less stringent for the same-day customers whereas overnight
shipment for the next day customers sufficient if problems can be diagnosed
remotely. In this example, answers to important strategic, tactical and operational
questions become more complicated compared to a ‘‘one size fits all’’ service
environment where every customer is entitled to the same level of service. Where
288 A. J. Huber
will what parts be stocked? For whom will they be stocked? How will orders be
filled? How can we ensure that parts for mission critical customers are always
readily available without having to provide the same high levels of service to
same-day and next-day customers? Service marketing consultant Al Hahn
emphasizes that to effectively market service, ‘‘It is important to deliver the level
of service you have sold the customer, and no more.’’ If you don’t deliver to the
SLAs the customer paid for, you will pay penalties and risk losing that customer.
However, if you over-deliver on the 2nd tier contracts there will be no incentive to
buy the higher priced (and more profitable) mission-critical contract.
One well-known equipment provider in the information technology industry
solves this challenge by maintaining two separate sets of inventories with dif-
ferent service targets: one for its business customers, and one for its consumers.
Most readers of this book however will quickly recognize that maintaining two
separate inventories of the same parts is inherently inefficient. Savings could be
realized if both customer sets can be supplied from the same pool of parts
inventory.
Another executive I spoke with from a large service provider also in the
information technology industry told me that his company manages to 82 sep-
arate sets of SLAs depending on what was negotiated with each customer. His
company’s problem has become even more complex. How many parts must be
stocked to meet the needs of all customers? At what point do you put an order
from a lower priority customer on backorder to reserve inventory for the higher
priority customers? When must a replenishment order be shipped by air versus
ground? To which locations should parts in short supply be replenished first if
each location supports a different combination of high, medium and lower pri-
ority customers?
As companies traditionally thought of as product manufacturers expand their
scope towards providers of total solutions, new challenges (and hence opportunities)
have arisen for service parts managers and researchers. Aviation and defense
industries are an example. Aircraft operators like Air France, United Airlines and
Delta Airlines have large maintenance organizations that they leverage for other
airlines as well as their own. Aircraft manufacturers Boeing and Lockheed offer
Performance-Based Logistics (PBL) contracts to commercial and government cus-
tomers. SLA targets associated with these contracts are typically based on targets for
uptime or operational availability of aircraft systems. The challenge for the service
parts manager is to provide a supply of parts that provide high probabilities of
achieving the SLA targets, but at lowest possible cost. The ‘‘right’’ level of cost in
these complex environments include all costs incurred in support of the PBL
contract, such as the number and capacity of stock locations and part repair
facilities, the operating costs of those facilities, the information technology
infrastructure, and the cost of transportation. As new methods are developed and
new systems that apply them are deployed, costs can continue to decrease with no
degradation or even higher levels of performance. As a result, the ‘‘right’’ levels of
cost are continuously moving targets.
14 Common Mistakes and Guidelines 289
Companies who are best in class at marketing their service can are not necessarily
the most efficient in delivering those services. Conversely, companies who are best
in class in delivery of service do not necessarily fully exploit the market value of
their capabilities.
One large global high technology company is often cited as best in class for its
service portfolio of offerings. This company appropriately places extremely high
priority on ensuring that each customer consistently receives the contracted levels
of service. Everyone in the supply chain organization knows the importance of
having a full compliment of parts readily available locally for its most important
customers. While it does a credible job in achieving high levels of service, it does
so at tremendous levels of inefficiency. While it makes very good profits from
service, it clearly could do much better.
Oftentimes companies fail to achieve efficiencies that are possible because they
are focused on relatively small areas of opportunity while overlooking larger
opportunities. The high technology company mentioned above has been in a multi-
year battle for cost efficiency due to intense price competition with its competitors.
The price competition has dramatically impaired the profitability of both com-
panies. One of the opportunities it has pursued is the off-shoring of business
process services. The service parts supply chain planning groups were one of the
business processes identified and the company will undoubtedly reduce its labor
costs by training offshore providers to perform planning functions that had pre-
viously been done primarily in the United States and Europe. While the company
is clearly benefiting from the labor cost reduction, it did so before pursuing even
higher cost reductions that may have been achieved by reengineering the manner
in which it manages its service parts supply chain. First, there existed opportunities
to reduce global parts inventories by reducing its number of stocking locations and
pooling of inventories, the savings from which could have surpassed the entire
cost of its planning organization. Secondly, the planning processes were highly
manual. New systems had been put in place to consolidate multiple sets of legacy
applications it had accumulated through acquisitions. The new systems largely
supported the old manual processes, whereas much of the manual decision making
could have been automated reducing the need for the planning labor. In my
assessment, increasing the level of automation had the potential to reduce the total
labor required by more than half. Had the company consolidated its stock locations
and improved its inventory management practices before it moved the work off-
shore, much greater savings could have been realized.
Why didn’t the company reengineer its inventory management practices? It
certainly had the opportunity when it was consolidating its legacy applications and
developing new systems. To me the answer was simple: the people on the project
team did not know how to transform the supply chain. This leads me to the next
mistake.
290 A. J. Huber
teams do not perpetuate suboptimal past practices, thought leaders with deep
knowledge of available research and current technologies should be key contrib-
utors to strategies for change. The power of knowledge and the value of outside
perspective should not be underestimated. It is critical that people with an
expanded perspective be participants in the decision process. Sometimes the views
of experts can be overruled by project leaders unable to grasp the potential of the
ideas of the expert or they reject them because they don’t fit within their mental
model of the current process. This can be a warning sign: either the expert is really
no expert at all, or the expert’s ideas have been pushed aside and the value of those
ideas will not be realized.
While the presence of experts is important, operational people are best able to
evaluate the practicality of new approaches. Operational people must have an open
mind and be able to see possibilities beyond current practices. To ensure ideas are
practical, operational people must identify what additional considerations are
required, based on their knowledge of customers, suppliers and products.
The need for systems thinkers and process and technology experts in driving
transformational change to achieve dramatic levels of improvement cannot be
overemphasized. If change agents do not drive efforts, new systems will typically
be enablers of past business processes and practices. Because most operational
people have the current process as their point of reference, they are likely to ask
for functionality that duplicates what they already have or support their current
practices with some minor improvements that are unlikely to achieve dramatic
levels of improvement.
Supply chain and information technology transformation projects must be
coordinated if their potential is to be realized. Substituting one set of systems for
another under the assumption of ‘‘like for like’’ functionality substitution generally
represents a missed opportunity for improvement. (Upgrading from version x to
Version x.1 of a commercial off the shelf application may be an exception.) Major
information technology implementations are projects where it is critical to involve
experts, otherwise the end result will be to continue current processes and (at best)
continue current levels of performance.
One symptom of failure to apply systems thinking to the structure and operation
of the supply chain is the inflexibility imposed by not planning for change. Major
information technology projects can take several years from initial planning to full
deployment. Cross-functional teams of people with business subject matter
knowledge are teamed with information technologists. The business people make
it a point to demand the capabilities they use today, plus issues that have chal-
lenged them in the past, plus challenges they are currently facing (a new product
launch for example). By the time the project is fully deployed, the technology
supports business needs that were in place several years ago, but do not meet the
unanticipated challenges of today. There may be new technology in place, yet the
business continues to be paralyzed by inflexibility. That is why it is important to
think strategically from a systems perspective. What types of products might the
organization need to support, what are the range of associated services that may be
offered, how will service levels be defined, and what are might be the implications
292 A. J. Huber
A common mantra of the service parts manager is to have the right part in the right
place at the right time at the right cost. Another common business mantra is ‘‘what
gets measured gets done’’. Metrics like fill rates and inventory turnover ratios are
convenient and easy to produce. But are they accurate reflections of the ‘‘right’’
parts, places, times and costs?
At its most fundamental level, the role of the service parts manager is to provide
parts availability when needed. The end consumer, the user of the equipment, or
the consumer of the service that the equipment produces, desires that the service be
available when needed. Thus the traveler wants no flight delays, the pilot wants to
fly the plane when needed, the mechanic wants to perform maintenance when
scheduled, and he needs the tools, knowledge, time and service parts to perform
the maintenance. Similarly, the military wants the weapon system to be available
when a mission must be performed, the printer wants to complete print jobs on
time, and the homeowner wants the furnace or air conditioner to be operating when
its needed most. In each of these scenarios, the customer desires the product to be
operational when needed. Thus it is the uptime of the product during the times the
user is likely to need it that is the only relative metric to the end consumer. The
customer only cares about service parts management if the parts are not available,
they cost too much, or they do not perform as intended. In other words, they don’t
care about fill rates or inventory turns.
This is not to say that fill rates and inventory turns are not useful. Fill rates are a
convenient way to gauge changes in what it represents: the percentage of demands
that can be filled immediately from inventory. If the objective is to improve uptime
by placing more critical parts closer to the customer, then it may be useful to
measure fill rates of critical components at field locations.
Military supply chains have long targeted service parts management around
operational availability metrics (see Sherbrooke 2004). PBL contracts also com-
monly use operational availability to define service levels required. In the com-
mercial world outside the airlines, uptime-based service level agreement targets
are routinely specified in contracts, yet many service supply chains continue to use
metrics like fill rates and turns as their primary performance targets. Having a high
fill rate is of no value to the end consumer if they are unable to use their equipment
when needed. A better approach is to have a mix of service-level metrics that are
more relevant to what the customer wants. Examples include backorders, back-
order duration, outstanding backorders with down machines, aircrafts on ground,
equipment uptime, ‘‘hard’’ machine down occurrences (where the equipment is not
14 Common Mistakes and Guidelines 293
Much has been written about the ‘‘bullwhip effect’’ in supply chains (Lee et al.
1997). When a systems approach is not applied to the structure of the entire supply
chain and each parts facility is operated as if it were an independent entity, it is
common to track demand and generate forecasts based on orders from downstream
locations as if they were independent demands from external customers, when in
reality they are not. In fact, the bullwhip effect will result in more variability as
you move up the echelon structure, making the orders less predictable and harder
to forecast, resulting in excess inventory, variable workloads, and shortages.
At a large global diesel powered vehicle manufacturer I worked with, inde-
pendent repair shops purchase parts from local dealers. Local dealers purchase
parts for from the manufacturer for their own service facilities as well as for sales
to the local repair shops. The manufacturer supplies the dealers from a global
network of large distribution facilities which in turn are resupplied from a central
warehouse located within the grounds of their expansive manufacturing plant. The
distribution network for truck and bus parts is multi-enterprise: dealers had no
visibility to consumption of the repair shops, and the manufacturer had no visi-
bility to the consumption of the dealers, only their replenishment orders. As a
result, replenishment orders from the dealers to the manufacturer warehouses
upstream were erratic, making them difficult to forecast and resulted in the need
for expensive safety stock.
In order to improve levels of service to the dealers, the company transformed
the way they replenished dealers using a more collaborative systems approach.
Each dealer provided from their own inventory system daily demands for each part
it used or sold. The dealers and the manufacturers agreed on the levels of service
that would be provided to the end customers. Consistent levels of service and
‘‘minimum truck on lift’’ time for long-haul truckers is a competitive advantage for
both the manufacturer and its dealer network. The logistics arm of the
294 A. J. Huber
manufacturer developed and provided to its dealers a system to set target stock
levels of every part based on its high standards for service. Critical repair parts are
given priority over deferrable maintenance items with different service objectives.
Advanced inventory optimization methods are used to enable the dealers to pro-
vide high levels of service at minimal inventory investment.
The primary objective of the collaborative system with its dealers was to ensure
consistently high levels of parts availability throughout the dealer network so that
its customers could deliver its goods and passengers on time with minimal cost of
downtime. While it certainly achieved its goal of consistently high service, it was
able to do so at significantly reduced inventory, both for the dealer and for the
manufacturer. Because it now forecasts based primarily on consumption demand
data rather than replenishment orders, forecasts are more accurate, safety stock is
reduced, and the work levels in the warehouses are less variable. Dealer inventory
was reduced due to the use of advanced inventory optimization. Warehouse work
levels reflected scheduled replenishments to the dealers rather than reacting to
orders from disparate dealer systems and practices. As a result, overtime cost at the
warehouse was also reduced.
At a company producing high technology equipment, the global parts supply
chain was largely a set of independently operated operations. Local service pro-
viders ordered from facilities in each country. In Europe country stocks were
replenished from a central facility. The European facility was replenished from the
US central warehouse using largely manual ordering processes. The US facility
had highly erratic order patterns that were difficult to forecast accurately. When
demand is not forecast at the point of consumption and the inventory replenish-
ments are not coordinated throughout the entire network, the result is high levels of
inventory and imbalances with overages in some locations while others incur
backorders of the same items.
Statistical forecast methods that apply historical demand data to predict future
demands are perfectly acceptable in some but not all situations. When demand is
relatively steady or in some way predictable and changes come gradually over
time and the lead time to replenish inventory is short relative to the time in
which demand levels change, simple methods like moving averages or expo-
nential smoothing may be all that is required. If no reliable causal data is
available and the value of increases in forecast accuracy is high, it may be
worthwhile to apply more advanced and sophisticated methods that can detect
seasonal and cyclical patterns.
Oftentimes however, demand can be highly dynamic and causal data is readily
available. Forecasts and inventories can be highly responsive to changes in causal
data in dynamic environments yet many companies continue to apply simple time
series-based statistical methods. Consider a field stock location that supports a
14 Common Mistakes and Guidelines 295
Significant value from a systems approach can often be obtained when applying
the following tactical and operational practices:
• Forecast demand at the point of consumption.
• Apply relevant causal factors in the forecasting process.
• Optimize inventory targets using multi-item multi-echelon inventory
optimization
• Automate routine replenishment transactions.
A manufacturer of printing systems wanted to improve its local availability of
parts for one of its products to reduce customer downtime. A team of engineers
and a parts planning specialist gathered data on historical demand patterns,
equipment placements, parts reliability, and engineering changes. The team came
up with 3 sets of field stocking recommendations; one each for large, medium
and small cities. The process took about a month and was reasonably successful
in improving the local supplies for that one product. Later however an automated
solution was put in place that had a more dramatic effect. A system was
developed that tracked part usage and the supported contracted equipment at the
technician level. Demand rates were calculated and inventory targets set auto-
matically for the technicians van stock, with some allowable discretion based on
their local knowledge. The system computed demand rates for the local field
stock locations after consideration of the inventory carried by each technician.
Field stock locations were reduced to fewer, larger facilities in strategic locations
with delivery services available when needed. As a result, field inventory levels
were reduced by 2/3 while local fill rates improved from roughly 85 to 95%.
Replenishment of both technician van stock and field stock locations was fully
automated. All that was needed was to put away the parts that were shipped and
return items on the return list (parts no longer required due to changes in the
install base).
Levels of performance improvement possible at companies that make these
fundamental mistakes (often without knowing it) can be so dramatic as to be
unbelievable. An engine manufacturer sells parts to a network of independent
distributors who in tern sell parts to dealers and repair shops. Like the large
vehicle manufacturer already discussed, this engine producer set out to establish
a collaborative system with its customers. In this companies case, its customers
were the independent distributors. High levels of service were deemed to be the
most critical requirement of the product. The manufacturer adopted software
that utilized advanced inventory optimization methods to set inventory levels
whereas their previous system applied a simplistic set of single-item safety
stock calculations with targets set by traditional ABC classifications. When the
new system was being configured, the company used its historically high fill
rate values as its performance objective. When the resulting recommended value
of the inventory was only one third the current inventory, the managers could
not believe it was possible. In fact, they assumed it was an error in the
software.
14 Common Mistakes and Guidelines 297
Systems approaches are needed to make the best of a bad situation when it comes
to short supply situations. A clear hierarchy of needs should be defined in advance
as part of a strategic design exercise. Many organizations fall into the trap of
simply reacting and expediting when shortages occur.
An example of a more effective approach is to first supply the most important
customer when their machine is down and when they need it the most. If all
machine-down situations can be fulfilled, then position remaining inventory in
locations where demands from machine-down customers are most likely to occur.
As more parts become available, distribute them throughout the network using
optimization that maximizes the expected number of critical orders filled until the
next replenishment arrives. When replenishments arrive late and inventory
decreases, reverse the process.
When automated controls are not put in place, human behavior can cause part
shortages to go from bad to worse. Spread a rumor among the field technician
workforce that a part is in short supply and hoarding behavior will ensue. Fearing
that they need to protect inventory for ‘‘my own customers’’, each inventory
manager refuses to transship parts to other locations in dire need. Inventory
increases, service gets worse.
When advanced parts management techniques are applied, when systems are
put in place to automate processes and when audits are put in place to ensure
accurate data, there is little need for manual overriding of individual transactions.
In fact environments where individuals examine and override optimized inventory
targets, order quantities and order timing more typically make performance worse
rather than better. When automated methods are effective, it reduces the need for
labor and frees people to work the root causes of supply problems while the system
makes best use of the inventory available at the time.
298 A. J. Huber
If service levels are not meeting requirements, clear priority must be placed on
finding the root causes of delays in providing parts and causes of shortages and
methods to avoid them. Often times, opportunities to improve service through
process and technology improvement will simultaneously bring about opportuni-
ties to reduce cost. If processes are not changed, service levels can only be
improved at increased cost, largely by increasing inventory to mask the ineffi-
ciencies that are occurring. This may be necessary in the short term, but it is not an
effective strategy for the long term.
If cost is the primary objective, it is helpful to have team members define the
overall cost levels and the components of cost, including any hidden components. It is
critical that the focus is on all relevant costs, not just those included in the current set
of performance metrics and not just those in the accounting entries. Obvious costs
include operational costs of planning, repair facilities, transportation, warehousing,
order processing. Inventory levels represent opportunity cost of capital. Not so
obvious are the costs associated with shortages and delays. For example, the average
mission capable availability of aircraft in the US Air Force fleet was 72% in 2001.
Fully 14% of the fleet on average was not mission capable due to supply problems
300 A. J. Huber
(GAO 2001). Given the size of the Air Force fleet and the cost of each aircraft, this
represents and opportunity cost of billions of dollars.
Although it is important to define the problem and understand the causes of
delays and shortages as well as the operational, inventory and opportunity costs;
do not limit the idea gathering to only those that directly address the stated
problems at hand. While this may seem counterintuitive, I emphasize this because
there may be opportunities for new processes and technologies that may not be
apparent. For example, the techniques described throughout this book represent
ideas worthy of evaluation that will be new to most readers. Other new ideas may
be gleaned from conferences, recent publications, consultants and staff.
Ideas can come from anywhere. When I managed a parts supply chain, the
warehouse manager expressed that my planning organization must be doing a bad
job of planning because there were so many expensive parts in the warehouse that
were not moving at all. That led us to adopt a practice of what we called the ‘‘dusty
parts’’ test. When he found high cost parts with a ‘‘thick layer of dust’’ on them, he
would bring them to my attention and I would have the planning group find out
when were they purchased and why were they purchased at that time. On several
occasions we uncovered process improvement opportunities that we could go after
to prevent wasted expense in the future.
When assemblies are removed and replaced in the field, technicians return them
in the box the new assembly was shipped in using a preprinted return label from
the shipping provider. The damaged assemblies are then sent to the receiving area
of the central parts warehouse, also on the grounds of the manufacturing plant.
The service parts planning organization determines the disposition of each
returned item. During the early and mid life cycle stages, almost all assemblies are
sent to the appropriate repair facility for refurbishing. During the later stages of the
product life cycle, the planner may choose to scrap some units or store them in
their damaged condition until more are needed.
In the field, technicians are expected to return each item within a week. To save
trips, technicians typically keep the return parts in their vans and stop by UPS to
drop off them off when they happen to be near a UPS facility. Currently, about
65% of the parts arrive at the central warehouse within 2 weeks, 85% within a
month, some take much longer, and each year about 4% are never accounted for.
The receiving workload at the central warehouse is highly variable. The wait
time for a returned part to be received can vary from same day to as long as two
weeks, with three days being typical. When items are received, the warehouse
system instructs the receiving clerk whether the item is to be scrapped (in which
case it is sent to the recycler), put away in its unrepaired state, or sent to the repair
facility for refurbishment.
The repair facility typically runs with 30 days of work in process. About 15
percent of the time, items wait longer because not all the required component parts
are available. Some units can wait as long as 5 months before they are repaired.
Repaired parts are returned to the central warehouse where they are returned to stock.
The manager of the service parts operation saw the reverse logistics process as a
big opportunity for improvement. She was frustrated with the large amount of
unrepaired inventory in both the receiving area, the storage area for unrepaired
items, and within the repair facility. Although the used parts did not show on the
books, she knew that her planners were buying newly manufactured parts when
good used ones were tied up in technician vans, the warehouse, and the repair
facility. A team was commissioned to study the problem and make recommenda-
tions. Two university professors of Operations Management were brought in to
work with the team and make recommendations. They interviewed managers and
specialists from the planning department, warehouse and repair center. They were
provided with data from the planning and transaction processing systems. After a
few months of part time effort, the professors made a preliminary recommendation.
The professors proposed that a real-time decision support model be developed
by them over the summer that could be incorporated into the company’s existing
warehouse, repair center and field parts management systems. Data from the
planning application would feed the decision making model that would disposition
‘‘optimal’’ courses of action for each failed repairable part as the failures occurred.
The model would be developed by the professors and some of their students who
would deliver a working prototype that the company’s IT organization could
transform to a production application. The new decision model would be inte-
grated with the transaction processing systems.
302 A. J. Huber
Here’s how the application would work. At the time a part failed, a transaction
would be sent to the real time decision model along with data on that part and each
of its substitutes from the planning system. The model would determine in real
time where the part should be returned to and what method of transportation to use.
The decision alternatives would be to send it to a reclamation facility, to the repair
vendor, or to the central warehouse for long term storage. Once the model rec-
ommended scrap, store or repair; it would determine the level of urgency and
recommend air or ground transportation. Technicians would be expected to deliver
emergency air shipments before taking their next service call, ground shipments
would be processed every Friday afternoon.
For parts in all time buy status, a complimentary forecast model would estimate
the remaining lifetime supply required incorporating current the current install
base and its trend rate of decline as well as actual repair yields. These lifetime
supply requirements forecasts would be used to feed the real-time model in
deciding whether to scrap or return parts in lifetime buy status.
Since the needed parts would go directly to the repair facility, they would no
longer be tied up at the central warehouse. For the company’s repair facility, the
professors made two recommendations. First, there were bottlenecks that occurred
frequently for several classes of expensive critical parts. For those, additional
capacity would be added by purchasing additional test equipment and cross
training technicians from other stations. Second, the professors would develop a
2nd set of algorithms that would optimize the order that each waiting part would
be repaired in. Since it would receive real-time notification of every failed part as
it happened, the repair center could prioritize the work with an optimization
objective of minimizing priority-weighted customer delay times.
The professors quantified the projected reductions in cycle times, work in
process, and inventory of both repaired and unrepaired parts, yielding a substantial
savings. They also pointed out how quickly critical parts in short supply could flow
through the entire reverse logistics supply chain. This would significantly reduce
critical customer backorders and reduced customer downtime.
After the professors presented to the service parts management team, the idea
could be evaluated using the four questions simple questions.
Here the answer is obviously yes, since the professors were brought in specifically
to address the reverse supply chain of unrepaired inventory. However, one can
imagine other scenarios where an idea like this would be proposed:
• The idea may be published in a book like the one you are reading now.
• The concept may be published in a leading journal.
14 Common Mistakes and Guidelines 303
• The supply chain manager may present the concept and accomplishments post-
implementation at an industry conference.
• The ideas may be incorporated into a commercial off the shelf software
company.
• The method may be offered as a service by a transportation company with repair
facilities or by a contract repair provider.
Let’s assume that the professors completed the project and published their work
in a leading journal. An inventory specialist at a major airline engine repair facility
might have read the article with interest and compared it to the methods his
company used to perform engine overhauls for their own and other airlines. While
the method may not have been 100% applicable to his own engine repair opera-
tion, there certainly would be parallels. Because of the high cost of engines, the
airlines may already move engines directly from aircraft to the repair facility and
the process would begin almost immediately. If however the engine repairs con-
sisted both of planned and unplanned activity, the concept of a model like the one
that the professors proposed to optimize the order of repairs would be applicable
using the same or a similar approach.
For our medical equipment manufacturer we have already described how they
currently manage the problem. Other approaches in practice are similar to the
situation we have described and others are radically different.
An information technology hardware provider has a strategy of outsourcing all
of its manufacturing to contract manufacturers. They also require that these
manufacturers supply service parts, thereby bypassing the need to have a signifi-
cant investment in a service parts management with the exception of an investment
in field parts inventory. In this type of environment, each replaced assembly might
be returned to the manufacturer for return credit, then the challenge of managing
the part repair process becomes the contract manufacturer’s challenge.
Regardless of the contractual arrangement however, the fundamental process of
return and repair for resale still exists, and one can envision the adoption of the
technique by the contract manufacturer. One can imagine the manufacturer
maintaining one or several repair facilities and a parts warehouse where it
refurbishes returned parts from all its customers and returns them to stock for
resale.
In the aircraft engine example, the maintenance facility may receive the engines
from its customers, perform an initial inspection to determine the parts required,
order any missing parts, and begin the refurbishment operation only once all
component parts are received and assigned to the repair order, and a workstation is
available to begin the work.
304 A. J. Huber
There may be alternatives to the approach recommended by the professors that the
medical equipment manufacturer may consider. Rather than make the higher cost
investment in the development and integration of the two real-time decision
support models into their existing systems, it may instead consider a less costly
simpler approach that would achieve a portion of the results. For example, it could
establish some heuristic thresholds to automate the decision to scrap, retain, return
or air ship the return from the field. An example might include the following:
• If the supply exceeds the forecasted remaining life cycle requirements adjusted
for yield rates and some level of safety stock, scrap the item.
• If [X months supply on hand exist, retain the item for future repair.
• If on hand inventory \ safety stock, air ship the return to the repair facility.
• If an outstanding backorder exists, expedite the repair upon receipt.
14 Common Mistakes and Guidelines 305
Of course, both ideas might make sense, the rules-based heuristics listed above
for a ‘‘quick fix’’, followed by a more elegant solution to realize more cost
reduction and service level improvement.
Another possibility might be consideration of more strategic alternatives. One
alternative might be to discontinue the medical device manufacturer’s internal
repair operation and outsource it to a contract repair provider that has a higher
scale operation, such as the example of the IT hardware provider’s relationship
with the contract manufacturer who maintains supplementary repair capabilities. A
contract repair provider may have the scale to leverage information technology
investments to achieve a higher return on investment and share a portion of the
savings with their customers while increasing its own profitability.
An alternative method that the aircraft engine manufacturer might pursue could
arise through a root cause analysis of its repair cycle times and causes of delays.
For example, perhaps much of its cycle times are due to times awaiting parts to
complete engine overhauls. A closer look may reveal that component part supplies
are managed using an independent-demand forecasting and ordering model opti-
mized to a fill rate objective. Perhaps time awaiting parts could be reduced through
the incorporation of a dependent-demand forecast approach coupled with an
inventory optimization model whose objective function is to minimize repair cycle
times.
The four-question approach while simple, may be useful for reducing the number
of investment alternatives under consideration, and may lead to investigation of
other alternatives that would not have been considered previously. Examples
include the case of the outsourcing alternative for the medical device provider or
the dependent-demand approach for the aircraft engine repair facility.
It can be helpful to keep a list of improvement ideas that survive the evaluation
process described in Step 2. First, consider service level improvements. I have
proposed that if service levels fall below requirements, fix them first.
One way to prioritize competing approaches to improve service level is to
identify what service metric will be used, and then rank ideas by cost divided by
service improvement. For example, a retail parts supplier may use fill rate as the
performance metric if customers take their purchases elsewhere if orders cannot be
filled immediately. For each competing idea, you may evaluate the project cost per
expected increases in orders filled.
For the medical equipment provider, backorder time may be a superior service
metric to fill rate, since its customers may have no alternatives to placing a
backorder when stock-outs occur. Here for every day the customer must wait for
backorders to be filled, their medical device may be unavailable for use. Therefore,
competing projects might be ranked by their investment cost per expected
reductions in backorder days.
306 A. J. Huber
If cost reduction is the objective, then a similar method could be used in mea-
suring investment required per dollar of annual spend reduction. Computation of net
present value for each project using time-phased expenditures and time-phased
return on investment will enable comparison of projects with the greatest value. If
investment funds are limited, internal rate of returns should also be considered.
To project inventory savings, it will often be helpful to apply Little’s Law
where L = k * w where L = inventory, k = demand rate and w = cycle times
(see Muckstadt 2005; Fredendall and Hill 2001). Little’s Law is helpful in
translating cycle time improvements to inventory reductions. (It is also useful in
translating the number of backorders to backorder delay times.)
Because service parts supply chains often perform very inefficiently due to the
types of mistakes that have been outlined here, many improvement projects will
often result in service level improvement and cost reduction. Here it may be useful
to add the value of service improvement and cost reduction per dollar of invest-
ment. It is not straightforward to value service improvement due to the behavioral
questions of what levels of service improvement will increase customer loyalty,
improve the company’s reputation to attract more business, and similar arguments.
For simplicity’s sake, it can be helpful to arbitrarily assign a value to the service
metric, such as the value of a lost order or the value of a backorder day. Don’t
forget that losing an order may also involve loss of repeat business. While there
may be no ‘‘right’’ value to place on these service metrics (don’t ask the
accountants to answer it for you), you can construct your spreadsheet with a
parameter for service value, then increase and decrease it until it seems to make
the appropriate trade-off between cost reduction and service level improvement. In
other words, ask yourself how much you would be willing to invest in a project for
service level improvement to derive a reasonable value. In this manner, it can be
convenient to rank projects that improve service, reduce cost, or both.
14.4 Conclusion
Service parts supply chains can yield dramatic improvements for reasonable levels
of investments because it is an area where current practice does not incorporate the
processes and technologies that are currently available to run them more effec-
tively and efficiently.
The initial and perhaps most significant challenge for the service parts manager,
practitioner or consultant is to get the attention of senior management and show
convince them of the magnitude of the opportunities that exist. Part of that selling
process should include consideration for whether the senior managers are afflicted
with the misperceptions I’ve identified here. It should also look at the motives of
the manager and translation of those motives into the implications and opportu-
nities for service parts management. The two most common are to maintain the
uptime of the customer’s equipment, and to provide the consistently profitable
annuity streams that aftermarket support businesses represent.
14 Common Mistakes and Guidelines 307
The consultant must craft a convincing argument that educates senior man-
agement on the link between their objectives and the improvement opportunities
you are requesting permission to pursue. The education must be quick and con-
vincing. ‘‘Deep dive’’ discussions into arcane inventory theory can quickly lose the
interest of many. Focus your proposal and presentation on the conclusions you
want the executive to draw. Here are three desirable reactions that may be
worthwhile aiming for:
• That’s my problem!
• These ideas will go a long way to helping solve my problem.
• This proposal seems low-risk.
Finally, nothing breeds confidence more than success. Smaller projects tackled
first that deliver results quickly will build confidence both by the project sponsors
and by the teams implementing them.
Acknowledgments To Robert G. Brown, whose teaching, consulting and text Advanced Service
Parts Inventory Management (Brown 1982) when I was a young analyst working with Eastman
Kodak’s service parts organization inspired me to apply analytical methods and would one day
lead to my assignment there as Service Parts Supply Chain Director. To my friends Peter L.
Jackson and John A. Muckstadt of Cornell University whose knowledge and understanding of the
elements involved in optimizing supply chains under the uncertain conditions of service envi-
ronments convinced me that ‘‘It’s not rocket science, it’s much harder than that!’’ Many of the
issues I’ve highlighted in this chapter would have been overlooked by me were it not for the
mentoring of Professor Muckstadt and working prototypes developed by Professor Jackson.
References
Brown R (1982) Advanced service parts inventory control. Materials Management Systems,
Norwich
Fredendall L, Hill E (2001) Basics of supply chain management. St. Lucie Press, Boca Raton
Kotter J (1996) Leading change. Harvard Business School Press, Boston
Lee H, Padmanabhan V, Whang S (1997) The bullwhip effect in supply chains. Sloan
Management Review 38:93–102
Muckstadt J (2005) Analysis and algorithms for service parts supply chains. Springer, New York
Sherbrooke C (2004) Optimal inventory modeling of systems. Springer, New York
United States General Accounting Office (2001) Air Force inventory: parts shortages are
impacting operations and maintenance. Report: 01-587
Index
309
310 Index
D (cont.) G
Interval, 3–10, 12, 15, 16, 20, 22, 29, 38, Gardner-McKenzie protocol, 63
41, 43, 131, 177, 190, 192, 199, Geometric distribution, 3, 5, 10, 15
207–210, 215 General Electric, 146
Lumpy, 35, 89, 90–92, 95, 102, 103, Genetic algorithm, 146
105, 106, 125, 171, 176–178, Goodness-of-fit, 37, 40
184, 191, 235 GRMSE, 71, 212
Slow, 212
Smooth, 14, 177
Dendogram, 100 H
Discriminant analysis, 59 Halfwidth, 110, 112, 117–121
Distributions Hannan-Quinn criterion, 58
Assumptions, 31, 32, 39, 43, 47, 157 Heuristics, 222, 228
Demand, 164, 166, 167, 169, 206, 223, 224 Holt’s linear method, 56
Exponential, 7, 35
Gamma, 34, 35, 39, 42, 46, 47, 51, 113,
115, 206, 207, 222 I
Geometric, 3, 5, 7, 10, 14, 15, 35, 44, 45, Independence sampler, 112
209 Inductive rule, 43
Lognormal, 136 Installed base, 157, 161
Negative binomial, 32, 35, 41, 47, 206, Integer programming, 254
212, 215 Integrated forecasting method, 134
Normal, 35, 36, 39, 41–43, 46, 47, 60, 61, Intermittent demand, 1, 31
68, 110, 112, 129, 130, 138, 169, Inter-demand intervals, 5, 15, 125
215, 240 Inventory
Poisson, 35, 36, 39, 41, 44, 45, 47, 50, 115, Re-order point, 34
120, 145, 146, 148, 193, 206, 212, Optimization, 295
215, 222, 224, 235, 237, 255 Order-up-to-Level, 34, 211
Dynamic service parts management, 159
J
E Jackknife, 126–127
Efron, 126, 127 Jittering, 132
End-of-life, 157
End-of-production, 157, 167
End-of-service, 157 K
End-of-use, 165 Kolmogorov-Smirnov test, 37
EWMA, 207, 210
Expediting, 281
Exponential smoothing L
Estimates, 13 Linear growth model, 60, 76
Method, 2 Linear programming, 149–150
Non-seasonal, 54 Line replaceable unit, 183
Single, 55 Little’s law, 306
LMS categorization, 216
Log-space adaptation, 138
F Logistic growth function, 159
Failure
Process, 113
Simulated data, 119 M
Time, 113 Maintenance, 160
Forecast aggregation, 93 MAPELTD, 134
Forecasting origin, 70 Marginal analysis, 144–147, 152
Friedman’s test, 98 Markov chain, 111
Index 311