On the exact calculation of the mean stock level in the base stock periodic review policy
Eugenia
Babiloni,
Manuel Cardós,
Ester Guijarro
Universitat
Politècnica
de València
(SPAIN)
Received
December 2010
Accepted
May 2011
Babiloni,
E., Cardós,
M., & Guijarro,
E. (2011).
On the exact
calculation of the mean level in the base stock periodic review policy.
Journal of
Industrial Engineering and
Management, 4(2), 194205. doi:10.3926/jiem.2011.v4n2.p194205

Abstract:
Purpose: One of the most usual indicators to measure the performance of any inventory policy is the mean stock level. In the generalized base stock, periodic review policy, the expected mean stock during the replenishment cycle is usually estimated by practitioners and researchers with the traditional HadleyWhitin approximation. However it is not accurate enough and exact methods suggested on the related literature focus on specific demand distributions. This paper proposes a generalized method to compute the exact value of the expected mean stock to be used when demand is modelled by any uncorrelated, discrete and stationary demand pattern.
Design/methodology/approach: The suggested method is based on computing the probability of every stock level at every point of the replenishment cycle for which it is required to know the probability of any stock level at the beginning of the cycle and the probability transition matrix between two consecutive periods of time. Furthermore, the traditional HadleyWhitin approximation is compared with the proposed exact method over different discrete demand distributions
Findings: This paper points out
the lack of accuracy that
the HadleyWhitin
approximation shows over a wide
range of service levels and discrete demand distributions.
Research
limitations/implications: The
suggested method
requires the availability of appropriate tools as
well as a sound mathematical background. For this reason,
approximations to it
are the logical further research of this work.
Practical
implications: The use of the
HadleyWhitin
approximation instead of an exact method can lead to
underestimate systematically the expected mean stock level. This fact
may increase
total costs of the inventory system.
Originality/value: The original derivation of an exact method to compute the expected mean stock level for the base stock, periodic review policy when demand is modelled by any discrete function and backlog is not allowed.
Keywords: mean stock level, periodic review, discrete demand distribution

1 Introduction and related literature
The control procedure of
the traditional periodicreview, base stock
policy consists on examining the status of an item every R
fixed time
periods and launching a replenishment order which raise the inventory
position
(onhand stock plus onorder stock minus backorders) to the order up to
level S.
This policy is commonly denoted by (R,
S)
policy. In this paper,
we focus on the exact estimation of the expected mean onhand stock
level in
the (R,
S)
policy when demand is modelled by any discrete
distribution function. For a more easy reading of the paper the onhand
stock
or physical stock will be refer just as stock further on.
In practice,
an inventory policy is designed to satisfy a predefined target service
level,
such as the cycle service level or the fill rate, whereas the mean
stock level
is minimized. Another approach consists on the minimization of total
costs of
the systems, but even then, an accurate enough estimation of the mean
stock
level is necessary. Traditionally, the estimation of the mean
stock level in the (R,
S)
policy has been fulfilled by using the easytocompute HadleyWhitin
approximation, which derivation can be found in Hadley and Whitin (1963). This approximation is
computing by
means of the following the expression (1).

(1) 
where L
is the replenishment lead time and µ_{R+L}
and µ_{R}
represents the mean demand during R+L
and L
consecutive
periods respectively.
Apart from
the HadleyWhitin
approximation, there is a handful
of works that are dedicated to derive exact methods to estimate the
mean stock
level in the (R,
S)
policy. Nevertheless, they apply only under
specific demand distributions, such as the Hadley and Whitin
(1963) method that
apply only when
demand is Poisson distributed; the Nahmias and Smith
(1994) method
that calculates the mean stock level based on the two parameters of a
negative
binomial demand distribution in a multilevel inventory system; or the van der Heijden and de
Kok (1992) method
that applies for compound Poisson demand process with gamma distributed
customers. This method also appears in van der Heijden and
de Kok (1998) where
the performance of the traditional HadleyWhitin
approximation is
assessed for this particular demand context showing that it can be very
inaccurate,
especially for low customer service levels.
To the best of our
knowledge, no
authors have
actually suggested a generalized and exact method to compute the mean
stock
level able to be applied when demand is modelled by using any discrete
distribution function. Note that
exact methods
referenced above apply only under specific demand conditions.
Furthermore we
assume that unsatisfied demands are lost which is the most common
situation in
the retail sector as Johansen
(2005) points out.
Therefore,
this paper has a twofold
objective: (1) to derive an exact method to compute the mean stock
level when
demand process is
stationary, independent and identically distributed (i.i.d.)
and modelled with any discrete distribution function; and (2) to point
out the
lack of accuracy the HadleyWhitin
approximation shows under several discrete demand
distributions and over a wide range of service levels.
This paper is organized
as follows. Section 2 dedicates to describing
the basic assumptions and the derivation of the exact method to compute
the
expected mean stock level. Section 3 shows the comparison between the
exact and
the HadleyWhitin
approximation of the mean stock
level followed by the discussion of the results. Finally conclusions
and
further research are briefly pointed out in Section 4.
2 Exact calculation of the mean stock level in the periodic review policy
As known,
the (R,
S)
policy place
replenishment orders every R
fixed time periods which
is received L
time periods after being launched. Figure 1 shows an
example of the evolution of the onhand stock in a periodic review
system.
Figure 1. “Example of the stock evolution in a periodic review system”.
The mathematical
derivation of the proposed method is based on the following
assumptions: (i)
the time is discrete and is organized in a numerable and
infinite succession of equispaced
instants; (ii) L
is constant; (iii) backordering is not
allowed and
therefore L<R;
(iv) the replenishment order is added to
the inventory at the end of the period in which the order is received;
(v)
demand during a period is fulfilled with the inventory at the beginning
of that
period; and (vi) the demand process is assumed to be stationary with a
known,
discrete and i.i.d.
distribution function. Notations
used in this paper are listed as follows:
S 
= 
order up to level, 
R 
= 
review period and
replenishment cycle corresponding to the time between two consecutive
deliveries, 
L 
= 
lead time for the
replenishment order, 
z_{t} 
= 
onhand stock at time t
from the first reception, 
D_{t} 
= 
accumulated demand
during t
consecutive periods of time, 
D_{[t, t1]} 
= 
demand that occurs
between t
and n
consecutive periods of time, 
µ_{t} 
= 
mean demand during t
periods of time, 
f_{t}(·) 
= 
probability density
function of demand at time t, 
F_{t}(·) 
= 
cumulative distribution
function of demand during t
periods of time, 
Y_{t} 
= 
expected stock level at
time t, 
Y 
= 
expected mean
stock level during
the replenishment cycle. 
In general, the expected
stock level at any time of the cycle depends on
the probability of every feasible stock value. Then obviously:

(2) 
where


and


Therefore,
the expected mean stock level during the replenishment is

(3) 
However,
the problem lies
in computing the probability vector of every stock level in each period
of the
cycle, .
Note
that the probability of depends
on the stock level at the precious
period, and on the demand that
appears
between two consecutive periods
.
Then

(4) 
where

(5) 
Expressing (4)
as matrixes

(6) 
Where 


(7) 
is defined as
the
probability
transition matrix between stock levels of two consecutive periods and
according
to (5):

(8) 
Therefore it
is possible to compute the probability of every stock level at any
point of the
cycle if we know the probability of every feasible stock level at the
beginning
of the cycle

(9) 
And
therefore, the expected mean stock level during the replenishment cycle
is
calculated by using the following expression:

(10) 
that can
be applied to any discrete, stationary and i.i.d.
demand pattern and leads to the exact value of the mean stock level.
The
estimation of the probability of every stock level at the beginning of
the
cycle can
be obtained
following the approach suggested by Cardós,
Miralles, and Ros (2006).
3 Experimentation
3.1 Experiment desing
This section is dedicated to compare the exact
method and the
traditional HadleyWhitin
approximation in order to
analyse deviations which arise from using the latest. For that purpose,
we
design an experiment that follows two steps: (1) Given a known discrete
demand
distribution and a target cycle service level, CSL,
we compute the base stock that guarantees reaching the target;
and (2) with the base stock from step 1, we compute the value of the
mean stock
level using the expression (10) and the HadleyWhitin
approximation (expression (1)).
In general,
the selection of the most appropriate discrete distribution function of
demand
is a difficult task. Syntetos and Boylan
(2006) recommend using the
Negative
Binomial distribution, Teunter, Syntetos, and
Babai (2010) suggest using a
compound Bernoulli
distribution, Janssen, Heuts, and de
Kok (1998) and Strijbosch, Heuts, and
van der Schoot (2000) proposes using a
Compound Bernoulli
Distribution, whereas Silver, Pyke, and
Peterson (1998) recommend the Poisson
distribution
whenever the item is considered strategic but slow moving. In keeping
with this,
we have selected the following discrete distributions: (1) Bernoulli(p);
(2) Binomial(n,
p);
and (3) Poisson(l). Taking into account
that
Bernoulli distribution is equivalent to Binomial distribution with n=1,
the
experiment considers the Binomial and the Poisson distribution with the
appropriate set of parameters. The parameters of each distribution are
selected
to meet the smooth and the intermittent demand pattern suggested by Syntetos, Boylan, and
Croston (2005). Regarding the
parameters of the
inventory system, an extensive range of values are selected to consider
as many
different and realistic contexts as possible. Table 1 summarizes the
parameters
and values used in the experiment which was programmed in Java.
The feasible
combination of values of Table 1 leads to 19.866 different cases. The
estimating errors arising from estimating the mean stock level using the
HadleyWhitin
approximation instead or the exact method are analyzed by using the
relative
error expressed in terms of per 100, by means of the following
expression:

(11) 
Note that
computing relative errors as in expression (11) enables not only to
know the
magnitude of the error but also the type of error by analyzing its
sign. A
negative relative error means that the approximation overestimates the
expected
mean stock. Otherwise, if the relative error is positive, the
approximation is
underestimating the real mean stock level.
Factor 

Values 
Demand
distribution 


Binomial (n, p) 
n= 
1,2,3,4,5,6,7,8,9,10,15 
p= 
0.1,0.25,0.5,0.75,0.8,0.9,0.99 

Poisson (l) 
l= 
0.01,0.1,0.5,1,2,3,5,7,10 
Cycle
service level,
CSL 

0.50,0.55,0.60,0.65,0.70,0.75,0.80,0.85,0.90,0.95,0.99 
Inventory policy 


Review Period, R 

2,3,4,5,10,15,20,30 
Lead time, L 

1,3,5,10 
Table 1. “Experiment factors and values”.
3.2 Results and discussion
Table 2
shows the maximum, minimum, average and standard deviation, sorted by
the
target CSL,
of the relative errors
the HadleyWhitin
approximation fall into. The
relative errors for every of the 19,866 cases will be available upon
request. A first look into it reveals that
the
relative errors are always
positive and therefore the HadleyWhitin
approximation tends to underestimate the mean stock level. In practice
the
underestimation of the expected mean stock has important consequences
for the
system since the stock level is above the expected and therefore, based
on the
HadleyWhitin
estimation, managers would plan wrongly
its material and logistic requirement.
As expected, the HadleyWhitin approximation improves its performance when the CSL increases. This result was pointed out by van der Heijden and de Kok (1998) for gamma distributed demands. However, we realize that average relative errors are not near to zero when the CSL is near 1. This fact is easier observable through Figure 2, where the relative errors from the 19,866 cases are represented versus the target CSL. We observe how the size of the relative errors decreases when the CSL increases. However, when CSL>0.8, the relative error per target CSL shows an almost similar pattern. Therefore we can affirm that the HadleyWhitin approximation is biased and not an enough accurate approximation of the mean stock level, even if the target CSL is near to 1.
CSL 
Number of cases 
Maximum relative
error 
Minimum relative
error 
Average relative
error 
Standard deviation 
0.50 
1,806 
100.00% 
0.51% 
21.43% 
16.08% 
0.55 
1,806 
80.99% 
0.51% 
18.77% 
13.42% 
0.60 
1,806 
65.61% 
0.51% 
16.20% 
11.03% 
0.65 
1,806 
64.03% 
0.51% 
14.64% 
9.90% 
0.70 
1,806 
56.33% 
0.51% 
12.75% 
8.33% 
0.75 
1,806 
36.24% 
0.51% 
11.38% 
7.37% 
0.80 
1,806 
35.37% 
0.51% 
10.15% 
6.75% 
0.85 
1,806 
32.67% 
0.38% 
9.05% 
6.30% 
0.90 
1,806 
32.67% 
0.26% 
8.10% 
5.96% 
0.95 
1,806 
32.67% 
0.26% 
7.20% 
5.73% 
0.99 
1,806 
32.67% 
0.18% 
6.36% 
5.57% 
Table 2. “Maximum, minimum, average and standard deviation of relative errors”.
Figure 2. “Relative errors versus the target CSL for the 19.866 resulted cases”.
4 Conclusion and further research
This paper derives
an exact method to calculate the expected mean stock level suitable for
any
stationary, discrete, and i.i.d.
demand pattern for
the periodic review, base stock policy (R,
S).
The suggested exact method is
based on computing the probability transition matrix of stock levels
between
two consecutive periods of time in the replenishment cycle. This method
is
appropriate for any discrete demand distribution and therefore it do
not depend
on parameters of the distribution.
Traditionally,
the computation of the expected stock level has been done using the
HadleyWhitin
approximation. However, this paper shows that: (1)
it systematically underestimates the mean stock level and therefore it
is
biased, and (2) its accuracy decreases when the cycle service level
decreases.
Hence, in practice, using the HadleyWhitin
approximation could lead to fall into significant errors not only for
the
inventory system itself but also could influence decisions related with
the
purchase and production policy and as a result, to increasing the total
costs
of the system.
Further
extensions of this work should be focused on applying the easy to
compute
estimation of the probability of every stock level at the beginning of
the
cycle suggested by Cardós and
Babiloni (2011) in order to evaluate
its
performance when using it for the mean stock level estimation.
This paper
is part of a wider research project devoted to identify the most simple
and
effective stock policy
to properly manage any particular
demand pattern based on the characteristics of demand
itself.
Acknowledgements
The authors would like to thank the two anonymous referees their constructive comments on earlier versions of this paper.
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Journal of Industrial Engineering and Management, 2011 (www.jiem.org)
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