Operations Research Letters 35 (2007) 567–572

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Developing a closed-form cost expression for an (R, s, nQ) policywhere the demand process is compound generalized Erlang

Christian Larsena, G.P. Kiesmüllerb,∗aDepartment of Business Studies, Aarhus School of Business, Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark

bFaculty of Technology Management, Technical University Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 12 June 2006; accepted 7 November 2006Available online 17 May 2007

Abstract

We derive a closed-form cost expression for an (R, s, nQ) inventory control policy where all replenishment orders have aconstant lead-time, unfilled demand is back-logged and inter-arrival times of order requests are generalized Erlang distributed.For given values of Q and R we show how to compute the optimal reorder level s.© 2006 Elsevier B.V. All rights reserved.

Keywords: Inventory control; Compound renewal process; Generalized Erlang distribution

1. Introduction

The analysis of an inventory system requires amodel for the demand process. Some contributionsconsider discrete-time models where the total de-mand in a time period can follow any probabilitydistribution. Others consider only the total demandin a lead-time, often assuming it to be normally dis-tributed. Note that both these approaches aggregatethe demand of the individual order requests. Finally,many contributions explicitly model the arrival pro-cess of the individual order requests. Most often the

∗ Corresponding author.E-mail addresses: chl@asb.dk (C. Larsen),

g.p.kiesmueller@tm.tue.nl (G.P. Kiesmüller).

0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2006.11.002

approach here is to assume the demand process is(compound) Poisson. We refer to [6,9] for a goodoverview of literature. Concerning the last approach,it is not always reasonable to assume that the timebetween order requests can be represented by an ex-ponential random variable. For instance, if the stockpoint is somewhere upstream in the supply chain witha few larger customers, all applying batching policies,it would probably then be more appropriate to modelinter-arrival times as a mixture of Erlang distributions.Moreover, it is known (see [7, Theorem 2.9.1]) that anypositive continuous random variable can be arbitrarilyclosely approximated with a mixture of Erlang distri-butions, all with the same scale parameter. Thereforeit is surprising that this very general demand model isnot studied very often.

568 C. Larsen, G.P. Kiesmüller / Operations Research Letters 35 (2007) 567–572

Inspired by the result of [7] we consider a demandprocess where the time between order requests is mod-eled by a random variable T that is a mixture of Er-lang distributions, all with the same scale parameter.We assume order requests are modeled by a positiveinteger-valued random variable X. In practice we can-not apply the result of [7] in its most general case; wemust approximate with a finite number of Erlang dis-tributions. Appendix B in [7] shows two methods toapproximate a positive random variable with a mixtureof two Erlang distributions, both with the same scaleparameter. These are denoted, by Tijms [7], as gen-eralized Erlang distributions. The first method showshow to approximate with a mixture of two Erlang dis-tributions with k and k + 1 phases, respectively. Thesecond method shows how to approximate with a mix-ture of an exponential distribution (which is an Erlangdistribution with 1 phase) and an Erlang distribution.It is noted by Tijms [7] that the first of these meth-ods probably has more practical relevance as it willgenerate a unimodal distribution, which resembles agamma distribution. Therefore we have decided in thepaper to feature the first of these two methods. Thus,we assume that the random variable T has the follow-ing distribution:

T ={

T1 prob p

T2 prob 1 − p(1)

where T1 is Erlang distributed with mean k/�, T2 isErlang distributed with mean (k + 1)/�, k is a posi-tive integer and � is a positive real number. The sec-ond method, which we do not explicitly feature in ourpaper, would be appropriate when the coefficient ofvariation of the random variable T is very high. How-ever, it would have been a fairly straightforward matterto modify the mathematical expressions in our paper,thereby also to have developed a mathematical modelof this case.

We analyze an (R, s, nQ) inventory control policywhere all replenishment orders have a constant lead-time L and unfilled demand is back-logged. It is a peri-odic review model, where the time between reviews isspecified by a given constant R. We find it more real-istic to consider a periodic review model rather than acontinuous one, because in reality there will always bean upper bound on how often it is possible to replenish.The control policy operates as follows. At each reviewinstance, if the inventory position is at or below s then

a replenishment order is made with size some integermultiple of Q, bringing the inventory position to be inthe interval from s +1 to s +Q. We assume Q is somepredetermined “practical” batch size, like a pallet or abox, but we have to find s. Since Q is given we do notinclude a fixed replenishment cost into our model, be-cause the expected number of replenishments per timeunit is fixed. We will, however, assume that there aregiven parameters of inventory and penalty costs. Wederive a closed-form cost expression of this (R, s, nQ)

policy from which one subsequently can find theoptimal reorder level s. We believe that many in theresearch community shy away from making inventoryanalyses assuming non-Poisson processes becausethey think it is either too difficult or impossible to de-rive closed-form cost expressions. Our conclusion isthat it is possible, and in fact not that difficult, thoughit requires some lengthy mathematical derivations.

In Section 2 we analyze the compound renewalprocess. In Section 3 we develop our cost expressionfor the (R, s, nQ) policy and present our algorithmto compute the optimal s. In Section 4 we illus-trate our derivations by a numerical example. Finally,Section 5 contains some concluding remarks.

2. Analysis of the renewal process

One can find some, but not many, expositions oncompound renewal processes and inventory theory, cf.[1,5]. Our exposition follows the analysis in [4]. (Notall the lengthy mathematical derivations are stated inthe paper, often only the end results. However, theycan be found in the accompanying working paper[3].) Define:

T (m) the random variable which is the sum of m inter-arrival times. As per definition T (1) ≡ T andT (0) ≡ 0.

�m(t) the distribution function of T (m). It is the prob-ability that at least m customers have arrived ina time interval of length t that began immedi-ately after a demand instance.

pm(t) the probability that exactly m customers havearrived in a time interval of length t that beganimmediately after a demand instance.

T̃ the random variable which is the residual timefrom a randomly chosen time point until thenext demand instance.

C. Larsen, G.P. Kiesmüller / Operations Research Letters 35 (2007) 567–572 569

�̃(t) the density function of T̃ .�̃m(t) the distribution function of T̃ + T (m − 1). It is

the probability that at least m customers havearrived in a time interval of length t where thestart time is randomly chosen.

p̃m(t) the probability that exactly m customers havearrived in a time interval of length t where thestart time is randomly chosen.

In order to evaluate �m(t) we need to look at allthe possible ways in which m inter-arrival times canbe generated. With probability b(m, i) = (m!/i!(m −i)!)pi(1 − p)m−i when generating m inter-arrivaltimes, we generate i inter-arrival times from distribu-tion T1 and m − i inter-arrival times from distributionT2. In all, the number of phase completions must inthis case be at least ik+ (m− i)(k+1)= (k+1)m− i,which has probability e−�t

∑∞j=(k+1)m−i (�t)j /j !, as

all phase completions have durations that are expo-nentially distributed with mean 1/�.

By collecting all these probabilities we get

�m(t) = e−�tm∑

i=0

b(m, i)

∞∑j=(k+1)m−i

(�t)j

j ! ,

m = 1, 2, . . . , (2)

where as per definition �0(t) = 1.It follows from [4] that

�̃(t) = �

k + 1 − pe−�t

⎡⎣k−1∑

j=0

(�t)j

j ! + (1 − p)(�t)k

k!

⎤⎦ .

(3)

From [4] we also have

�̃m(t) =∫ t

0�m−1(t − x)�̃(x) dx (4a)

which can be rewritten as (see [3] for details)

�̃m(t) = e−�t

(k+1)m−1∑r=k(m−1)+1

�̃m(r)(�t)r

r!

+ e−�t∞∑

r=(k+1)m

(�t)r

r! , (4b)

where the coefficients �̃m(r) are given as follows (see[3] for details): When k(m − 1) + 1�r �km,

�̃m(r) = 1

k + 1 − p

m−1∑i=max{(k+1)(m−1)−r+1,0}

b(m − 1, i)

× (r − (k + 1)(m − 1) + i). (5a)

When km + 1�r �(k + 1)m − 1,

�̃m(r) = 1

k + 1 − p

n−1−r+km∑i=max{(k+1)(m−1)−r+1,0}

b(m − 1, i)

× (r − (k + 1)(m − 1) + i)

+m−1∑

i=(k+1)m−r

b(m − 1, i). (5b)

Here we use the convention that b(0, 0) = 1.Because p̃m(t)=�̃m(t)−�̃m+1(t) (and thus p̃0(t)=

1 − �̃1(t)), it can be written (see [3] for details) incondensed form as

p̃m(t) =

⎧⎪⎪⎨⎪⎪⎩

e−�tk∑

r=0�̃0 (r)

(�t)r

r! , m = 0

e−�t(k+1)(m+1)−1∑r=k(m−1)+1

�̃m(r)(�t)r

r! , m = 1, 2 . . . ,

(6)

where the coefficients �̃m(r) are given as follows:

�̃0(r) ={

1, r = 0,

1 − �̃0(r), 1�r �k,(7a)

�̃1(r) ={

�̃1(r), 1�r �k,

1 − �̃2(r), k + 1�r �2k + 1,(7b)

and for m�2,

�̃m(r) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�̃m(r), k(m − 1)

+1� r �km,

�̃m(r) − �̃m+1(r), km + 1� r

� (k + 1)m − 1,

1 − �̃m+1(r), (k + 1)m� r � (k + 1)

(m + 1) − 1.

(7c)

In finishing this section, we also define the randomvariable:

D̃(t) the total demand in a time interval of length twith a randomly chosen start time.

570 C. Larsen, G.P. Kiesmüller / Operations Research Letters 35 (2007) 567–572

It has probability distribution

P(D̃(t) = x) =

⎧⎪⎨⎪⎩

p̃0(t), x = 0,x∑

m=1p̃m(t)

P (X(m) = x), x = 1, 2, . . . ,

(8)

where X(m) is the random variable denoting the to-tal demand of m customers, where as per definitionX(1) ≡ X.

3. The (R, s, nQ) inventory control policy

Note that we let the length of the review period beR and not be scaled to one as in many expositions.We find it most natural to derive the cost expressionas generally as possible. In a later contribution wealso, as in [8], intend to let this (R, s, nQ) policy be asubpart of a multi-item inventory control policy wherethe length of the review period is a policy variable.We assume the cost parameters:h inventory cost rate� penalty cost rate� fixed cost per item back-logged.

Let x be the inventory position at a review instancejust after a replenishment decision has been made (thusx belongs to the interval from s + 1 to s +Q). Denoteby G(x) the total costs incurred during the reviewperiod when starting with an inventory position of x.Then following the derivation of the cost function ofModel 3 in [4] we get (see [3] for details):

If x�0

G(x) = v

[−xR + �E[X]

k + 1 − pR

(L + R

2

)]

+ ��E[X]

k + 1 − pR. (9a)

If x > 0

G(x) = v

[−xR + �E[X]

k + 1 − pR

(L + R

2

)]

+ ��E[X]

k + 1 − pR

+ (v + h)

x−1∑y=0

(x − y)

∫ R

0P(D̃(L+) = y) d

× �x−1∑y=0

(x − y)[P(D̃(L+R) = y)

− P(D̃(L) = y)]. (9b)

These are the same cost expression as that in [4, Equa-tion 14]. Due to our more numerical approach we havespecifically chosen to write everything as finite sums.

In order to make the cost expression computable westill need to evaluate the integral

∫ R

0P(D̃(L+) = y) d. (10)

For a Poisson process with rate �, let the random vari-able NA denote the number of arrivals in a time inter-val of length A. Thus NA is Poisson distributed withmean �A.

By repeated partial integration it holds that (see [3]for details)

∫ R

0e−�(L+) (�(L + ))r

r! d

= P(NL �r) − P(NL+R �r)

�. (11)

Consequently, if y > 0,

∫ R

0P(D̃(L+) = y) d

= 1

�

y∑m=1

P(X(m) = y)

(k+1)(m+1)−1∑r=k(m−1)+1

�̃m(r)

× [P(NL �r) − P(NL+R �r)], (12a)

and if y = 0,

∫ R

0P(D̃(L+) = y) d

= 1

�

k∑r=0

�̃0(r)[P(NL �r) − P(NL+R �r)]. (12b)

When combining (12a)–(12b) with (9a)–(9b) andusing (5a)–(5b), (6), (7a)–(7c) and (8), we get a closed-form expression of the cost function. The inventoryposition, immediately after a replenishment decision,is uniformly distributed on s + 1, . . . , s + Q. For amathematical proof see Exercise 6.5 in [9]. Therefore,the relevant cost expression to be optimized (with re-spect to s) is

∑s+Qx=s+1G(x)

Q. (13)

C. Larsen, G.P. Kiesmüller / Operations Research Letters 35 (2007) 567–572 571

This is accomplished by the following algorithm,stated here in pseudo-programming language. It issimilar to the algorithm in [2].

Algorithm.

s := argmin{G(x) : x�0}s := s − 1q := 1While q < Q do

If G(s) < G(s + q + 1) thens := s − 1

End Ifq := q + 1

End whileEnd AlgorithmEach time the while-loop is entered we have found

the points s+1, . . . , s+q, where the function G(x) at-tains its least values. The justification for this observa-tion follows from Rosling [4] who under very generalconditions show that G(x) is quasi-convex. Becauseq < Q we must enlarge this range. If G(s) < G(s +q + 1) then we should enlarge this range in a down-ward direction, and accordingly decrement s by 1.Otherwise s remains the same. In any case q is incre-mented by 1. In this manner we continue until q =Q.

4. A numerical example

We have developed a computer code, programmedin Visual Basic for Excel, which computes (9a)–(9b)and finds the optimal value of s. In order to demon-strate this we present a numerical example. We havecost parameters h = 2, v = 4, � = 30. The lead-timeL = 0.5, and the length of review period R = 1. Thecharacteristics of the demand process is k=4, p=0.2,� = 8 while the random variable X can attain values1, 2, 3 and 4 with equal probability. Table 1 lists ourresults for values of Q ranging from 1 to 6. We havealso simulated the inventory system using the com-puted control policy, because in [4] it is acknowledgedthat Model 3 is an approximation. Here we believe,however, that [4] is very cautious, which also seemsto be tacitly assumed in [4], since otherwise the fol-lowing rigorous analysis of the cost function in [4]seems odd. We believe that for all practical purposesa review instance must be considered as a randomlychosen time point with respect to the renewal process.

Table 1Optimal solutions for the data set when Q ranges from 1 to 6

Q Optimal s Minimum costs (13) Simulated costs

1 10 15.8385 15.8469 (0.0308)

2 9 15.9479 15.9561 (0.0408)

3 9 16.1961 16.2038 (0.0349)

4 8 16.6359 16.6326 (0.0378)

5 8 16.9368 16.9301 (0.0317)

6 8 17.4249 17.4288 (0.0356)

The simulation was conducted in Arena, Version 9.0. Eachsimulation lasted 120,000 time units and the initial inventoryposition and net inventory were both set equal to s + Q. Thesimulation experiment was replicated 10 times. In parenthesis isstated the 95% confidence interval.

Therefore the cost expression of Model 3 in [4] shouldbe almost exact.

As expected when comparing the simulation resultsto the derived cost expressions, we see a nice fit inTable 1.

5. Concluding remarks

It is, of course, also possible to conduct a similaranalysis for the case of continuous review (s, nQ)

inventory system, taking a point of departure inModel 2 in [4]. In our paper we have assumed thatthe order size distribution X is positive and integervalued. We could also have developed our analysisassuming P(X = 0) > 0 as well as assuming X to bea continuous random variable. It is also clear that itis possible to further generalize the model by lettingT to be mixtures of more than two Erlang distribu-tions. The derived mathematical expressions will theninvolve multinomial expressions instead of binomialexpressions, but they remain computable.

Acknowledgment

This research was made while the first author spenthis sabbatical leave at Technical University Eind-hoven, The Netherlands, partly supported by a grantfrom Otto MZnsteds Fond.

References

[1] M. Beckmann, An inventory model for arbitrary interval andquantity distributions of demand, Management Sci. 8 (1961)35–57.

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[2] A. Federgruen, Y. Zheng, An efficient algorithm to compute anoptimal (r, Q) policy in continuous review inventory systems,Oper. Res. 40 (1992) 808–813.

[3] C. Larsen, K.P. Kiesmüller, Developing a closed-form costexpression for an (R, s, nQ) policy where the demandprocess is compound generalized Erlang, Working Paper L-09,Logistics/SCM Research Group, Department of Accounting,Finance and Logistics, Aarhus School of Business, 2006.〈http://www.hha.dk/bs/wp/log/L_2006_09.pdf〉.

[4] K. Rosling, Inventory cost rate functions with nonlinearshortage costs, Oper. Res. 50 (2002) 1007–1017.

[5] I. Sahin, Regenerative Inventory Systems: OperatingCharacteristics and Optimization, Springer, Berlin, 1990.

[6] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management andProduction Planning and Scheduling, Wiley, New York, 1998.

[7] H. Tijms, Stochastic Models: an Algorithmic Approach, Wiley,New York, 1994.

[8] S. Viswanathan, Periodic review (s, S) policies for jointreplenishment inventory problems, Management Sci. 43 (1997)1447–1454.

[9] P.H. Zipkin, Foundations of Inventory Management, McGraw-Hill, New York, 2000.