inventory control in multi-echelon divergent systems with random lead times

OR Spektrum (1999) 21: 331–359

c© Springer-Verlag 1999

Inventory control in multi-echelon divergent systemswith random lead times

Lagerhaltungskontrolle in mehrstufigen, divergierendenSystemen mit stochastischen Wiederbeschaffungszeiten

Matthieu van der Heijden1, Erik Diks 2, Ton de Kok3

1 Department of Technology and Management, University of Twente, P.O. Box 217,NL-7500 AE Enschede, The Netherlands2 Department of Mathematics and Computer Science, Eindhoven University of Technology,Eindhoven, The Netherlands3 Department of Technology Management, Eindhoven University of Technology,Eindhoven, The Netherlands

Received: 29 April 1998 / Accepted: 25 November 1998

Abstract. This paper deals with integral inventory control in multi-echelon diver-gent systems with stochastic lead times. The policy considered is an echelon stock,periodic review, order-up-to (R, S) policy. A computational method is derived toobtain the order-up-to level and the allocation fractions required to achieve giventarget fill rates. Extensive numerical experimentation shows that the accuracy ofour approximate method is satisfactory. Further we show that variation in the leadtimes may have a significant effect on the stock levels required to achieve giventarget fill rates, hence this variation may not be ignored. Also, it appears that thecorrelation structure in the lead times may have a significant impact as well.

Zusammenfassung.Die Arbeit behandelt das Problem des Bestandsmanagementsin mehrstufigen, divergierenden Lagerhaltungssystemen mit stochastischen Wie-derbeschaffungszeiten. Die Steuerung erfolgt mittels einer systemweiten Bestell-grenzenregel bei periodischer Kontrolle. Es wird ein Verfahren zur Bestimmungvon Bestellgrenzen und Allokationsparametern zur Einhaltung vorgegebener (β-)Servicegrade konstruiert. Ausfuhrliche numerische Untersuchungen zeigen diezufriedenstellende Genauigkeit der heuristischen Vorgehensweise. Weiterhin wirdgezeigt, daß die Variabilitat der Wiederbeschaffungszeiten einen erheblichen Ein-fluß auf die erforderlichen Bestande haben kann und nicht vernachlassigt werdendarf. Außerdem stellt sich heraus, daß die Korrelationsstruktur in den Wiederbeschaf-fungszeiten ebenfalls von erheblicher Bedeutung sein kann.

Correspondence to: M. van der Heijden

332 M. van der Heijden et al.

Key words Inventory – Multi-echelon – Divergent – Allocation – Random leadtimes

Schlusselworter: Lagerhaltung – Mehrstufig – Divergierend – Allokation – Un-sichere Wiederbeschaffungszeiten

1 Introduction

In the last decade considerable progress has been made with respect to the analysisof multi-echelon models. In a number of recent review papers this progress hasbeen discussed. Nahmias and Smith (1993) give an overview of inventory modelsin one warehouse – N retailer systems. Axsater (1993) reviews the literature onmulti-echelon models with installation stock policies, i.e. each stockpoint is con-trolled based on local information about its inventory position and its demand. InFedergruen (1993) an overview is given of the papers focusing on the derivation ofdiscounted cost-optimal echelon stock control policies. Van Houtum et al. (1996)build on this review to discuss average cost-optimal multi-echelon models. In Dikset al. (1996) a review is given of multi-echelon models with service level constraints.They discuss both installation stock policies and echelon stock policies.

A problem that occurs when dealing with real-world supply chains is the ran-domness of lead times to stockpoints. Lead times to the most downstream stock-points are in fact transportation times and thereby usually reliable, but lead times tomore upstream stockpoints in a supply chain are random due to the fact that theselead times represent manufacturing throughput times. These throughput times arerandom variables due to e.g. capacity constraints, machine-breakdowns, rejects andbatching decisions. We have to be aware that the multi-echelon models representa supply chain for a set of products, but which share resources together with othersets of products. This sharing of resources is the major cause for randomness of leadtimes. So far no analysis is available of multi-echelon models with echelon stockpolicies and random lead times with the exception of the paper by Zipkin (1991),who deals with continuous review(S − 1, S) models. It follows from the analysisin De Kok (1990), Verrijdt and De Kok (1995) and De Kok et al. (1994) that ran-dom lead times to the most downstream stockpoints can easily be incorporated inthe analysis of multi-echelon models with constant lead times. However, the morepractical situation of random lead times to more upstream stockpoints cannot bestraightforwardly incorporated.

In this paper we consider divergent N-echelon models, where each stockpoint iscontrolled according to an echelon order-up-to-policy. We define the echelon inven-tory position as the amount of the stock in the stockpoint itself plus all downstreamstocks plus the stock in transit to the stockpoint minus all backorders at its down-stream stockpoints. Due to the fact that we deal with random lead times we have tobe more specific about the replenishment mechanism of the order-up-to-policies.The most upstream stockpoint follows a periodic review(R,S)-policy, i.e. at eachreview moment the echelon inventory position is raised to the order-up-to-levelS.However, a more downstream stockpoint raises its echelon inventory position tothe order-up-to-level upon arrival of a replenishment at the (unique) stockpoint that

Inventory control in multi-echelon divergent systems with random lead times 333

precedes this stockpoint. Hence the ordering decisions are taken at random pointsin time. The rate at which ordering decisions are taken is obviously equal to1/R.Through this way of timing the replenishment decisions we optimally synchronizethese decisions.

The lead times to the stockpoints are identically distributed random variables.Subsequent replenishment orders do not overtake. Such a model for the lead timesis realistic, because random delays in delivery are observed frequently in practice.For example, the supplier may have limited production capacity that is unknown tothe receiver. Delays as consequence of such a capacity restriction are observed asrandom by the receiver. Another example is limited transportation capacity. Limitedcapacities can be included explicitly in the model, but it only makes sense if thereceiver knows all information about capacity and (predicted) capacity utilization.Otherwise lead times should be modeled as random variables. If there are capacityrestrictions, it is also plausible that an order will not be delivered considerablyfaster than its predecessor, e.g. because both are influenced by the same capacityrestrictions. From a mathematical point of view, this implies that the subsequentlead times are correlated. In this paper we model this (auto)correlation assumingthat the lead times constitute an AR(1) process. Simulation experiments show thatthis modeling assumption yields good approximations for performance measureseven if the lead time process is not AR(1). The demand process is assumed to bestationary and time-homogeneous, i.e. the demand during an interval(s, s + t] isindependent of demand befores and aftert + s and not dependent on s itself.Since we consider divergent N-echelon systems we have to cope with situationswhere stock at a stockpoint is insufficient to satisfy the demand of its successors.In that case we use a linear rationing rule as defined in Van der Heijden (1997), theBalanced Stock rationing rule.

In this paper we concentrate on the determination of the control parameters,i.e. the order-up-to-levels and the parameters of the BS rationing rule, such thattarget fill rates at downstream stockpoints are achieved. Here the fill rate is definedas the fraction of demand satisfied from stock on hand. We assume that so-calledmaximum stock levels at intermediate stockpoints are given. The maximum stocklevel is the difference between the order-up-to-level at the stockpoint and the sumof the order-up-to-levels at its successors. By varying the maximum stock levels wecan vary the average physical stocks at all intermediate stockpoints. The motivationfor using a service measure concept instead of a penalty cost concept is the factthat due to the use of business information systems data are available in practiceabout target fill rates, whereas hardly any data are available about penalty costs.Therefore a service measure concept is more appropriate.

We develop a heuristic algorithm to compute the control parameters. Also,we give approximate expressions for the mean physical stock in all nodes in thenetwork. These expressions can be used to compare various scenarios for stockdistribution in the network for which the same fill rate is approximately attained.When holding costs are assigned to each node in the network, it is interestingto search for the minimum cost solution such that target fill rates are attained.In principle, one could use standard Newton methods for this purpose, (cf. Stoerand Bulirsch, 1993). However, practical optimization is not so straightforward and


actually an issue on itself. Optimization of a similar model for the special case ofdeterministic lead times is discussed in Van der Heijden (1997b). In this paper wefocus on stochastic lead times and we restrict ourselves to scenario analysis.

We have done extensive testing of the algorithm using discrete event simula-tion. These numerical experiments show that the algorithm performs quite well.Thereupon we use the algorithms to investigate whether the incorporation of au-tocorrelation of the lead times into the analysis is important for the accuracy ofthe algorithm. Furthermore we investigate the managerial issue of the impact ofrandomness of lead times on stock investments.

This paper is organized as follows. First we give a detailed model description(Sect. 2). In the Sects. 3–5 we derive the analysis of this model. We start with thesimplest case in Sect. 3, the two-echelon model with stockless central depot. Theanalysis is extended to two-echelon systems with central stocks in Sect. 4 and togeneral N-echelon systems in Sect. 5. A summary of the algorithm is given in Sect. 6.Next we validate our approximate method by comparison to simulation results, bothfor two-echelon and for three-echelon systems (Sect. 7). Some sensitivity analysisis shown in Sect. 8. Finally, we give our conclusions and suggest directions forfurther research in Sect. 9.

2 The mathematical model

To define the mathematical model, we first describe the network structure andthe material flow through the network (Sect. 2.1). Next, we describe the inventorycontrol policy (Sect. 2.2). Finally, we give the model assumptions and an overviewof notation (Sect. 2.3).

2.1 Network structure and material flow

We consider a single item, N-echelon divergent system as shown in Fig. 1. Thatis, each stockpoint receives products from exactly one supplier. These productsare either distributed further to one or more successive stockpoints or used to sat-isfy (stochastic) external demand. This demand takes place at theendstockpoints(numbers 4–12 in Fig. 1). The other nodes are calledintermediatestockpoints. Thematerial flows through the network from the single external supplier via one ormore stockpoints to the final customers. For a given stockpointi, we define theset ofupstreamstockpoints as all stockpoints between the external supplier andstockpointi. Then the set ofdownstreamstockpoints is defined as all stockpointsbetween stockpointi and the final customers. To describe the network structure,we use the following notation:

i = stockpoint index, wherei = 0 denotes the most upstream stockpoint,pre(i) = the single supplier of stockpointi, e.g.pre(8) = 2 in Fig. 1,succ(i) = the set of all stockpoints immediately supplied by stockpointi, e.g.

succ(2) = 7, 8, 9 in Fig. 1.


DemandSupply

DownstreamUpstream

12

11

10

9

7

6

8

5

4

3

2

1

0

Fig. 1. Divergent network structure

The lead times to move the material between two subsequent stages are randomvariables. Hence we have both demand- and supply uncertainty. The ultimate goalis to control the material flow such, that predetermined service levels to the finalcustomers are attained. We willnot optimize the stock levels at each node in thenetwork, but we will derive a method to calculate the stock required at the endstockpoints given the maximum stock levels at intermediate stockpoints.

The service measure that we use is thefill rate, i.e. the fraction of demand thatis satisfied immediately from stock on hand. We assume that demand that cannotbe satisfied immediately is backordered. We allow that different target fill rates areused for different end stockpoints.

2.2 Inventory control mechanism

The material flow in the network is controlled by an echelon stock(R,S) policy, aperiodic review, order-up-to policy. That is, the most upstream stockpoint releasesa replenishment order eachR time units, such that its echelon inventory position israised to the order-up-to levelS. A replenishment order arriving at an end stockpointcan be used to satisfy local customer demand. When a replenishment order arrivesat an intermediate stockpoint, one should decide what to do with it. Of course,the replenishment order can be sent on immediately to downstream stockpointsto satisfy local customer demand. Then one should decide how the order shouldbe allocated amongst the downstream stockpoints. Once this decision is taken, thedownstream stocks cannot be reallocated anymore (in practice this may be possible,but at high costs). When the allocation quantities per stockpoint are calculated,negative quantities are theoretically possible. This may occur if the demand duringthe last period at one stockpoint has been considerably higher then expected, whilethe demand at an other end stockpoint has been considerably lower then expected.Such a situation is calledimbalance. To reduce imbalance, one might keep backstocks at an intermediate stockpoint for allocation at a later point in time.


So the following decisions have to be taken after arrival of a replenishmentorder at an intermediate stockpoint:

– which part of the replenishment order should be kept back for allocation at alater point in time?

– how should the remaining part of the order be allocated amongst the downstreamstockpoints?

To explain the control mechanism, we will first describe the situation in which theentire replenishment order is allocated. Next we will add the option of keeping backsome stock for allocation later on.

Consider a replenishment order arriving at an intermediate stockpointi. Thisorder should be allocated to the succeeding stockpointsj ∈ succ(i). The allocationdecision is similar to the one described in Van der Heijden (1997a). This decisionis based on:

– the size of the replenishment orderQ,– the echelon inventory positions of the successorsj just before allocation, rep-

resented byzj , j ∈ succ(i),– a set of order-up-to levelsS∗

j , j ∈ succ(i),– a set of allocation fractionspj , j ∈ succ(i),

∑j∈succ(i) pj = 1

Now the allocation rule is as follows: Successorj gets an amount that raises itsechelon inventory position from the levelzj before allocation to the levelSj afterallocation, defined by

Sj = S∗j − pj

∑k∈succ(i)

(S∗k − zk) −Q

(1)

Hence stockpointj receives an amountSj −zj , which is a random variable becauseboth the inventory positions before and after allocation,zj andSj , are randomvariables. Note that summation of both sides of (1) overj yieldsΣSj = Q+Σzj ,which means that the replenishment order is allocated entirely indeed. We refer toVan der Heijden (1997a) for a further explanation of this rule.

If no stock is kept back at stockpointi, we have the relationS∗i =

∑j∈succ(i) S

∗j .

We can extend allocation rule (1) to situations with intermediate stocks by intro-ducing a parameter∆i, the maximum amount of stock to be hold in stockpointi.As mentioned in Sect. 2.1, we consider∆i as a given parameter. In fact, the relationbetween succeeding order-up-to levels changes to

S∗i = ∆i +

∑j∈succ(i)

S∗j . (2)

Now it may be possible that the size of the replenishment orderQ exceeds theamount needed to raise all inventory positions to their maximum levelsS∗

j . Negativerationing is not allowed, because then stock is not kept back at stockpointi, so rule(1) has to be changed to

Sj = S∗j − p∗

jmax

0,

∑k∈succ(i)

(S∗k − zk) −Q


= S∗j − p∗

jmax

0, S∗

i − (∆i +Q+∑

k∈succ(i)

zk)

. (3)

This rule will be used in the sequel. The problem is to find the control parameterspj , S

∗j for each stockpointj, given demand characteristics, lead time character-

istics, target fill rates at the end stockpoints and given maximum intermediate stocklevels∆i. For the most upstream stockpoint, say number 0, we have to find theorder-up-to levelS∗

0 with allocation fractionp0 = 1, of course.One additional remark about the inventory control in the network has to be

made. We assume that allocation decisions at intermediate stockpoints are takenimmediately after an order arrives. This seems to be reasonable at first sight, be-cause it is important to transfer products without delay to avoid shortages at theend stockpoints. It means also that the time between two subsequent shipmentsfrom intermediate stockpoints is a random variable as well (with mean R and pos-itive variance). This may not be desirable in all cases, for example with respect totransport planning.

Further this policy is not optimal in the case of stochastic lead times. As anexample, consider the arrival of two subsequent replenishment orders at an inter-mediate stockpoint. If the lead time of the first order is relatively short and thelead time of the second order is relatively long, the following may occur. Whenallocating the first order, the echelon inventory positionszj of the successors arerelatively high and a significant part of the replenishment order is kept back. If ittakes a long time before the next replenishment order arrives, shortages may haveoccurred downstream whilst sufficient stock would have been available upstream.On the other hand, if we would have chosen for a model in which allocation deci-sions are made at fixed points in time as well, it may occur that the stock is allocatedat the wrong moment, namely justbeforea replenishment order arrives instead ofjust after. This would give similar results. A solution to this problem might be amodel with fixed allocation epochs, but at a higher frequency in the downstreampart of the network. This is another model extension next to lead time variation,which should be subject for further research.

2.3 Assumptions and notation

The mathematical model uses the following assumptions

a) Customer demand only occurs at end stockpoints.b) The demand per period is random and stationary in time.c) The demand is both independent across end stockpoints and across periods in

time.d) All customer demand that can not be satisfied directly from stock on hand is

backlogged.e) Partial delivery of customer orders is allowed.f) Replenishment orders do not cross in time. This implies that successive lead

times can be correlated.


g) Lot sizing is not used, so any quantity can be ordered and delivered and anyallocation rule for material rationing to the local depots is allowed (as far as thequantity allocated is nonnegative).

h) There are no capacity constraints on production, storage or transport.

Assumption a) can be made without loss of generality, since we can add an endstockpoint with lead time zero for each intermediate stockpoint facing customerdemand. With respect to assumption c), remark that extension to demand that iscorrelated between end stockpoints is straightforward, see also Van der Heijden etal. (1997). This is however not true for demand that is correlated across periods intime. With respect to assumption f), we assume for a practical application that leadtimes can be measured, such that correlations between successive lead times canbe estimated. In Sect. 7 we will give numerical results on the effect of the lead timecorrelations.

Further we will use the following general balance assumption for the approxi-mation of the order-up-to levels:

General balance assumption:Allocation using rule (3) yields nonnegative al-location quantities only.

This assumption is usually made and simplifies the analysis considerably (see e.g.De Kok, 1990; De Kok et al., 1994; Van der Heijden, 1997a). Empirical evidenceshows that even rather strong violation of this assumption has only a limited effecton the quality of the approximations, unless the variation of demand during leadtime is very high (see Van Donselaar and Wijngaard, 1987).

For the mathematical analysis we use the notation as listed below.

Stockpoint status

zj = echelon inventory position of stockpointj, just before allocation of areplenishment order that has arrived at its supplieri = pre(j).

Sj = echelon inventory position of stockpointj, just after allocation of a re-plenishment order that has arrived at its supplieri = pre(j); for the mostupstream stockpoint 0 (see Fig. 1),S0 represents the echelon inventoryposition just after a replenishment order has been placed at the externalsupplier.

Lead times and demand

Lj = lead time between the stockpointspre(j) andj, a random variablewith meanE[Lj ], varianceVar[Lj ] and first-order autocorrelationρj .

Ljk = kth lead time between the stockpointspre(j) andj, this is used todistinguish subsequent replenishment orders in the analysis.

R = review period.Rj = the (stochastic) time period between the arrival of two replenishment

orders at stockpointj, the so-calledreplenishment cycle.Dij = demand per time unit at stockpointj, a random variable with meanµj

and standard deviationσj . If j is an intermediate stockpoint,Dj denotesall downstream demand, e.g.D2 is the period demand at the stockpoints7, 8 and 9.


Dj,Lj= demand at stockpointj during a lead timeLj .

Dj,Lj+Ri= demand at stockpointj during a lead timeLj plus a replenishment

cycleRi of its predecessor.X+ = maxX, 0.

Performance measures

βj = target fill rate for end stockpointj.ψj = mean physical stock at stockpointj.

Control parameters

pj = rationing fraction for stockpointj.S∗

j = maximum echelon inventory position of stockpointj.∆j = maximum physical stock allowed at stockpointj, a prespecified pa-

rameter.

Note that the system order-up-to level equalsS∗0 , because the external supplier is

assumed to have sufficient capacity and rationing is not necessary, see assumptionh).

3 Analysis of two-echelon systems with stockless central depot

In this section we will derive the mathematical expressions to calculate the systemcontrol parameters and the performance measures for the simplest model, namely atwo-echelon system with stockless central depot. We consider one upstream stock-point having index 0 and a number of end stockpointsj ∈ succ(0). In Sect. 4 wewill add the option that the central depot is allowed to hold stock. Extension of theresults to general N- echelon systems is discussed in Sect. 5.

The mathematical model is analyzed using the approach of Van der Heijden(1997a). However, we have to account for the stochastic, correlated lead times.In Sect. 3.1 we describe the calculation of the maximum inventory positionsS∗

j

assuming that the allocation fractionspj are known. It will appear that some aggre-gate demand characteristics are required for the calculations, which are derived inSect. 3.2. In Sect. 3.3 we derive an expression for the allocation fractionspj , suchthat an approximate expression for the mean system imbalance is minimized. It willappear that we can calculate these allocation fractions based on demand and leadtime characteristics only, soindependentlyof fill rate requirements and order-up-tolevels. This is attractive, both from an analytical and a computational point of view.Together, Sects. 3.1–3.3 give the control parameterspj , S

∗j of the system. Finally,

we will show how to calculate the mean physical stock in Sect. 3.4.


3.1 Maximum echelon inventory positionsS∗j

Our starting point is the simple singe location(R,Sj) system. It is well knownthat for this model the following equation should be solved forSj (see Diks et al.,1996):

E[(Dj,Lj+R − Sj)+] − E[(Dj,Lj− Sj)+]

Rµj= 1 − βj . (4)

The denominator equals the expected demand in a replenishment cycle. The nu-merator equals the difference between the expected shortage at the end and at thestart of a replenishment cycle. In many (but not all) practical situations, the secondterm in the numerator can be ignored (see e.g. Hadley and Whitin, 1963).

As shown in De Kok (1990) and Van der Heijden (1997a), the expression tocalculateS∗

j in a two-echelon system is a straightforward extension of (4), as-suming that the allocation parameterspj are known. Given the general imbalanceassumption as stated in Sect. 2.3, we find that we can obtainS∗

j by solving

E[Dj,Lj+R0+ pjD0,L0 − S∗

j +] − E[Dj,Lj+ pjD0,L0 − S∗

j +]

Rµj

= 1 − βj . (5)

This expression is valid for general demand- and lead time distributions. To solveequation (5) numerically, it is convenient to use some simple two-moment approx-imation for the stochastic components in both terms in the numerator,X1j :=Dj,Lj+R0

+ pjD0,L0 andX2j := Dj,Lj+ pjD0,L0 , see Appendix for details.

Then equation (5) can be solved forS∗j using bisection. In this way, the maximum

echelon inventory positionsS∗j can be calculated one-by-one for each end stock-

point j, if the allocation fractionspj are known. Next the order-up-to levelS∗0 is

obtained by summation of theS∗j , since the central depot does not hold stock (see

(2)).Now the problem is the derivation of the mean and variance ofX1j andX2j .

We find that

E[X1,j ] = E[Dj,Lj+R0] + pjE[D0,L0 ] , (6a)

Var[X1,j ] = Var[Dj,Lj+R0] + p2

jVar[D0,L0 ]

+2pjCov[Dj,Lj+R0, D0,L0 ] , (6b)

E[X2,j ] = E[Dj,Lj] + pjE[D0,L0 ] , (7a)

Var[X2,j ] = Var[Dj,Lj] + p2

jVar[D0,L0 ] + 2pjCov[Dj,Lj, D0,L0 ] . (7b)

We see that we need several demand characteristics to evaluate (6a)–(7b), amongwhich two covariances. In the next section we show how these characteristics canbe obtained.


3.2 Calculation of the demand characteristics

From the previous subsection we see that we need the following demand char-acteristics to calculate the control parametersS∗

j , given the allocation fractionspj :

(i) the mean and variance ofDj,Ljfor all end stockpointsj ∈ succ(0),

(ii) the mean and variance ofD0,L0 ,(iii) the mean and variance ofDj,Lj+R0

for all end stockpointsj ∈ succ(0),(iv) Cov[Dj,Lj+R0

, D0,L0 ] for all end stockpointsj ∈ succ(0),(v) Cov[Dj,Lj , D0,L0 ] for all end stockpointsj ∈ succ(0).

We can easily show thatCov[Dj,Lj , D0,L0 ] = 0. The demandDj,Lj relates tothe time interval[L01, L01 +Lj1], so the length of this time interval is independentof L01. Below we derive expressions for the other four demand characteristics.

First, we can easily derive the mean and variance ofDj,Ljfor each end stock-

point j by conditioning onLj , cf. Silver and Peterson (1985):

E[Dj,Lj ] = µjE[Lj ] , (8a)

Var[Dj,Lj] = σ2

jE[Lj ] + µ2jVar[Lj ] . (8b)

Second, the mean and variance ofD0,L0 can be calculated using the same equa-tions, once we have the mean and variance ofD0. The latter can easily be calculatedby summation, because the demand at different end stockpoints is independent, seeassumption c):

µ0 =∑

j∈succ(0)

µj and σ20 =

∑j∈succ(0)

σ2j . (9)

Note that correlation between demand at different end stockpoints could be includedeasily here.

Third, the mean and variance ofDj,Lj+R0for all end stockpointsj should be

calculated. To do this, we should carefully specify the time interval of the demand.Without loss of generality, suppose that at time 0 the central depot issues replenish-ment order 1 that raises the echelon inventory position to the levelS∗

0 . This orderarrives after a random lead timeL01 and is immediately allocated. The amount al-located to stockpointj, together with the current physical stock and pipeline stock,should be sufficient to cover the demand until the arrival of the second replenish-ment order at stockpointj, which occurs at timeR + L02 + Lj2. Hence we areinterested in the distribution of the demand at end stockpointj in the time interval[L01, R + L02 + Lj2]. Now we can derive by conditioning on the interval lengthR+ Lj2 + L02 − L01 that

E[Dj,Lj+R0] = µj(R+ E[Lj ]) , (10a)

Var[Dj,Lj+R0] = σ2

j (R+ E[Lj ]) + µ2jVar[Lj ] + 2(1 − ρ0)Var[L0]. (10b)

Fourth, we derive an expression forCov[Dj,Lj+R0, D0,L0 ], again by carefully

specifying the time intervals of the demand. As stated above,Dj,Lj+R0is the de-

mand at stockpointj in the time interval[L01, R + L02 + Lj2]. The other term is


derived from the fact thatS∗j should also cover a fractionpj of the total demand

during the first replenishment lead timeL01. HenceD0,L0 is the total system de-mand in the time interval[0, L01]. From this we see thatDj,Lj+R0

andD0,L0 arenegatively correlated: If the first lead timeL01 is relatively long, the total demandin [0, L01] will be relatively high, but also[L01, R+ L02 + Lj2] is relatively shortand hence the demand at stockpointj during this interval will be relatively low. Ifwe ignore this negative correlation, the variance ofX1j = Dj,Lj+R0

+ pjD0,L0

will be overestimated, resulting in higher stock levels than actually necessary, seeequation (5). We will return on this subject at our numerical analysis in Sect. 8.

We obtain an expression forCov[Dj,Lj+R0, D0,L0 ] by conditioning onL01 and

using that in general

Cov[X1, X2] = E[Cov(X1, X2|Y )] + Cov(E[X1|Y ], E[X2|Y ]) . (11)

Expression (11) is derived analogously to the well-known equivalent for the vari-ance as derived in Mood et al. (1974). We have thatCov[Dj,Lj+R0

,D0,L0 |L01] = 0,since the demand in subsequent periods is independent (see assumption c). There-fore the first term of (11) is zero in this case. Assuming that the lead timesL0k followan AR(1) process with first order autocorrelationρ0, we have thatE[L02|L01] =ρ0L01 + (1 − ρ0)E[L0]. Using these facts, we find a negative correlation indeed:

Cov[Dj,Lj+R0, D0,L0 ] = −(1 − ρ0)µjµ0Var[L0] . (12)

3.3 Calculation of the allocation fractionspj

From the results in Sect. 3.1 we see that we can tune theS∗j to the target service

levelsβj for any arbitrary set of rationing parameterspj , provided that the generalbalance assumption is not violated. It can be expected that the approximationsare better if the general balance assumption is violated only slightly. This can beachieved choosing the allocation parameterspj such that the expected imbalance isminimized. We achieve this using the approach by Van der Heijden (1997a), takinginto account the stochastic, correlated lead times.

We define the amount of imbalanceΩj at stockpointj as the negative allocationquantity, soΩj = (zj − Sj)+, see also De Kok (1990). Assuming a balanced situ-ation at the previous allocation epoch, we find thatzj −Sj = pjD0[R,R+L02]−pjD0[0, L01]−Dj [L01, R+L02] where the numbers between brackets denote timeintervals. Analogously to Van der Heijden (1997a) and using the expressions forthe various demand characteristics from Sect. 3.2, we find that

E[zj − Sj ] = −Rµj , (13a)

Var[zj − Sj ] = 2p2jE[M0]σ2

0 + R− 2pjE[M0]σ2j

+2Var[L0](1 − ρ0)(µj − pjµ0)2, (13b)

whereM0 = minR,L0.Now two approaches are suggested in Van der Heijden et al. (1997). Firstly, we

can approximatezj −Sj by a normal distribution and derive an explicit expression


for E[Ωj ]. Using a numerical method, the allocation fractionspj can be obtainedsuch thatE[Ωj ] is minimized. Secondly, we can useVar[zj − Sj ] as a surrogateexpression for the imbalance and minimize (13b) subject toΣpj = 1. The latterapproach was suggested by Van Donselaar (1996) for the model with deterministiclead times. Numerical tests in Van der Heijden et al. (1997) revealed that the firstapproach is somewhat better, but the second approach is considerably simpler.Because our numerical experiments indicated that the difference between the twoapproaches decrease if stochastic lead times are introduced, we restrict ourselveshere to the second approach. Taking into account the variance and correlation ofL0, we can derive by minimizing (13b) subject toΣpj = 1 that

pj =12n

+E[M0]σ2

j + µ0(1 − ρ0)Var[L0]∗(2µj − µ0

n

)

2E[M0]σ20 + 2µ2

0(1 − ρ0)Var[L0], (14)

wheren denotes the number of end stockpoints. From (14) it can be shown that0 ≤ pj ≤ 1 for all j and thatΣpj = 1. Note thatE[M0] = E[minR,L0] shouldbe calculated to evaluate (14). This can be done by fitting e.g. a mixture of Erlangdistributions to the first two moments ofL0.

3.4 Physical stock

Just as forS∗j , we can find expressions for the mean physical stock per local depotψj

that are simple extensions of the single depot(R,S) model. We have approximatelythat

ψj ≈ E[stock at start of replenishment cycle] + E[stock at end of replenishment cycle]2

,

which can be written as

ψj ≈ S∗j − pjE[L0]µ0 − (E[Lj ] +

12R)µj (15)

+E[(Dj,Lj + pjD0,L0 − S∗

j )+] + E[(Dj,Lj+R0+ pjD0,L0 − S∗

j )+]

2.

The two expectations represent the mean shortage at the start and at the end of areplenishment cycle, which are already calculated in (5).

4 Two-echelon system with central stock

We can extend the results from Sect. 3 to the situation where the central depot isallowed to hold stock. The same subjects will be treated in the same order as inSect. 3.


Equation (5) can easily be modified such that the central depot holds central stockup to some prespecified level∆0 (see e.g. De Kok et al., 1994; Van der Heijden,1997a):


E[Dj,Lj+R0+ pj(D0,L0 − ∆0)+ − S∗

j +]−E[Dj,Lj + pj(D0,L0 − ∆0)+ − S∗j +]

Rµj

= 1 − βj . (16)

Again, equation (16) can be solved numerically using two-moment approximationsfor both terms in the numerator,X1j := Dj,Lj+R0

+pj(D0,L0 −∆0)+ andX2j :=Dj,Lj

+ pj(D0,L0 −∆0)+)+. For the mean and variance ofX1j andX2j we nowfind that

E[X1j ] = E[Dj,Lj+R0] + pjE[(D0,L0 −∆0)+] , (17a)

Var[X1j ] = Var[Dj,Lj+R0] + p2

jVar[(D0,L0 −∆0)+]

+2pjCov[Dj,Lj+R0, (D0,L0 −∆0)+] , (17b)

E[X2j ] = E[Dj,Lj ] + pjE[(D0,L0 −∆0)+] , (18a)

Var[X2j ] = Var[Dj,Lj] + p2

jVar[(D0,L0 −∆0)+]

+2pjCov[Dj,Lj, (D0,L0 −∆0)+] . (18b)

We see that we need several other demand characteristics to evaluate (17a)–(18b),among which two complex covariances. In the next section we show how thesecharacteristics can be obtained.


We need the following additional demand characteristics to calculate the controlparametersS∗


(i) the mean and variance of(D0,L0 −∆0)+,(ii) Cov[Dj,Lj+R0

, (D0,L0 −∆0)+] for all end stockpointsj ∈ succ(0),(iii) Cov[Dj,Lj

, (D0,L0 −∆0)+] for all end stockpointsj ∈ succ(0).

First, a common way to approximate the mean and variance(D0,L0 −∆0)+ ofis to approximateD0,L0 by a mixture of Erlang distributions (see Appendix).

Second, we need an expression forCov[Dj,Lj+R0, (D0,L0 −∆0)+]. By con-

sidering the time intervals and using (11), we can derive that

Cov[Dj,Lj+R0, (D0,L0 −∆0)+] = −(1 − ρ0)µjCov[L0, g(L0)] , (19)

whereg(L0) = E[(D0,L0 −∆0)+|L0]. This expression is difficult to evaluate ingeneral. Therefore we approximateCov[L0, g(L0)] by a first-order Taylor expan-sion aroundE[L0]: Cov[L0, g(L0)] ≈ g′(E[L0])∗Var[L0]. In this way we obtain

Cov[Dj,Lj+R0, (D0,L0 −∆0)+] ≈ −(1 − ρ0)µjVar[L0]∗g′(E[L0]) . (20)


Although some approximation forg′(.) is possible, this function gets increasinglycomplicated if the number of echelons in the network increase, see Sect. 5. Thereforewe calculateg′(.) using a numerical derivative:

g′(E[L0]) ≈ g(E[L0] + ε) − g(E[L0])ε

, (21)

whereε is some small number (we tookε = 0.001∗E[L0]).Third, we need an expression forCov[Dj,Lj

, (D0,L0 −∆0)+]. Using the samereasoning as in Sect. 3.2, we can show that this covariance is zero.


The derivation of near-optimal allocation fractions gets more complicated in thepresence of central stock. Therefore we will use the allocation fractions (14) asobtained from the situation with stockless central depot. Van der Heijden et al.(1997) showed that this approach yields satisfactory results.

4.4 Physical stock

For the mean physical stock in the end stockpoints, only a small modification ofexpression (15) is required. We find that

ψj ≈ S∗j − pjE[(D0,L0 −∆0)+] − (E[Lj ] +

12R)µj (22)

+E[(Dj,Lj +pj(D0,L0 −∆0)+−S∗

j )+]+E[(Dj,Lj+R0+pj(D0,L0 −∆0)+−S∗

j )+]

2

Again, all the terms required are already available.However, now we have to calculate the mean physical stock in the central depot

as well. Note that the central stock remains constant between the arrival of twosuccessive replenishment orders, so at first sight we only need the mean physicalstock at the central depot just after allocation,E[(∆0−D0,L0)

+]. However, the timeperiod between two order arrivalsR0 is a random variable now, which is correlatedwith the stock level just after allocation. If the first replenishment order lead timeis relatively short, then:

– the time between two successive order arrivals is relatively long, and– the total demand during the first lead time is relatively low, so that the central

stock just after allocation is relatively high.

Hence we should account for the length of the time between order arrivalsR0 =R+L02 −L01 as well. The appropriate expression for the mean physical stock atthe central depot is

ψ0 ≈ E[(R+ L02 − L01)∗(∆0 −D0,L0)+]

R. (23)

Conditioning onL01, we find

ψ0 ≈ E[(∆0 −D0,L0)+] − 1 − ρ0

RCov[L0, (∆0 −D0,L0)

+] . (24)

The latter covariance can be approximated analogously to (20)–(21).


5 General N-echelon systems

In this section we extend our results to general N- echelon models. For sake ofconvenience, the expressions are given for three-echelon systems. Modification tomore than three echelons is straightforward.


We consider an end-stockpointj with pre(j) = i andpre(i) = 0. Van der Heijdenet al. (1997) show that then the following modification of (16) is required:

E[(X1j − S∗j )+] − E[(X2j − S∗

j )+]Rµj

= 1 − βj , (25)

where X1j = Dj,Lj+Ri+ pj(pi(D0,L0 −∆0)+ +Di,Li

−∆i)+

and X2j = Dj,Lj+ pj(pi(D0,L0 −∆0)+ +Di,Li

−∆i)+ ,

which can be solved numerically using two-moment approximations forX1j andX2j . Eliminating zero covariances, we find for the mean and variance ofX1j andX2j :

E[X1j ] = E[Dj,Lj+Ri] + pjE[(pi(D0,L0 −∆0)+ +Di,Li

−∆i)+], (26a)

Var[X1j ] = Var[Dj,Lj+Ri] + p2

jVar[(pi(D0,L0 −∆0)+ +Di,Li−∆i)+]

+2pjCov[Dj,Lj+Ri, (pi(D0,L0 −∆0)+ +Di,Li

−∆i)+] ,

(26b)

E[X2j ] = E[Dj,Lj] + pjE[(pi(D0,L0 −∆0)+ +Di,Li

−∆i)+], (27a)

Var[X2j ] = Var[Dj,Lj] + p2

jVar[(pi(D0,L0 −∆0)+ +Di,Li−∆i)+]. (27b)


We need the following additional demand characteristics to calculate the controlparametersS∗


(i) the mean and variance ofDj,Lj+Rifor all end stockpointsj ∈ succ(i),

(ii) the mean and variance of(pi(D0,L0 −∆0)+ +Di,Li −∆i)+,(iii) Cov[Dj,Lj+Ri

, (pi(D0,L0 − ∆0)+ + Di,Li− ∆i)+] for all end stockpoints

j ∈ succ(i).

First, the mean and variance ofDj,Lj+Rican be obtained analogously to the

two-echelon system. Now we need the demand characteristics in a slightly differenttime interval, namely[L01 +Li1, R+L02 +Li2 +Lj2] instead of[L01, R+L02 +


Lj2] as for the two-echelon system. Again, by conditioning on the interval lengthR+ Lj2 + Li2 − Li1 + L02 − L01, we find

E[Dj,Lj+R0] = µj(R+ E[Lj ]), (28a)

Var[Dj,Lj+R0] = σ2

j (R+ E[Lj ]) (28b)

+µ2jVar[Lj ] + 2(1 − ρ0)Var[L0] + 2(1 − ρi)Var[Li] .

Extension to general N-echelon systems is straightforward.Second, the approximation of the mean and variance of(pi(D0,L0 −∆0)+ +

Di,Li− ∆i)+ is a straightforward extension of the two-echelon system. First we

approximateD0,L0 by an Erlang mixture. Next we calculate the first two momentsof (D0,L0 − ∆0)+ as described in the Appendix. Using these two moments, weapproximatepi(D0,L0 − ∆0)+ + Di,Li by an Erlang mixture. Finally we obtainthe desired mean and variance of(pi(D0,L0 − ∆0)+ + Di,Li − ∆i)+ using theAppendix once more. It is clear that this approach can be extended to generalN-echelon systems.

Third, we need an expression forCov[Dj,Lj+Ri, (pi(D0,L0 −∆0)+ +Di,Li

−∆i)+]. By considering the time intervals and using (11), we can derive that

Cov[Dj,Lj+Ri, (pi(D0,L0 −∆0)+ +Di,Li

−∆i)+]

= −(1 − ρ0)µjCov[L0, g0(L0)] − (1 − ρi)µjCov[Li, gi(Li)] , (29)

whereg0(L0) = E[(pi(D0,L0 −∆0)+ +Di,Li −∆i)+|L0]

andgi(L0) = E[(pi(D0,L0 −∆0)+ +Di,Li −∆i)+|Li].

Similarly to (19), this expression can be approximated using first-order Taylorexpansions. Also these calculations can straightforwardly be extended to generalN-echelon systems.


Similar to Van der Heijden et al. (1997), the allocation fractions are calculatedaccording to (14), ignoring the effect of lead times in earlier stages. For the inter-mediate stockpoints, we also use (14) making the appropriate substitutions forµj ,σ2

j , µ0 andσ20 .

5.4 Physical stock

Similar to the preceding sections, we can make the following extension for themean physical stock in the end stockpoints:

ψj ≈ S∗j − pjE[(pi(D0,L0 −∆0)+ +Di,Li

−∆i)+] − (E[Lj ] +12R)µj

+E[(X1j − S∗

j )+] + E[(X1j − S∗j )+]

2(30)


whereE[(X1j − S∗j )+] andE[(X2j − S∗

j )+] denote the expected shortage at theend and at the start of a replenishment cycle respectively, see equation (25).

For the mean physical stock in the intermediate stockpoints we find analogouslyto Sect. 4.4:

Ψi ≈ E[(∆i −Di,Li− pi(D0,L0 −∆0)+)+]

−1 − ρ0

RCov[L0, (∆i −Di,Li − pi(D0,L0 −∆0)+)+]

−1 − ρi

RCov[Li, (∆i −Di,Li − pi(D0,L0 −∆0)+)+] . (31)

The latter covariances can be approximated analogously to (29).Finally note that expression (24) for the stock in the most upstream stockpoint

Ψ0 remains valid.

6 Summary of the algorithm

The purpose of our algorithm is to calculate the allocation parameterspj and max-imum echelon inventory positionsS∗

j for all stockpoints in the network, such that(different) target fill ratesβj for the end stockpoints are attained using minimalinventory imbalance. Then ordering decisions can take place using the order-up-tolevelS∗

0 and allocation decisions can be made using bothpj andS∗j according to

(3).We need the following information as a starting point for our algorithm:

– the lengthR of the review period,– the meanµj and standard deviationσj of the demand at each end stockpointj,– the target fill ratesβj for all end stockpointsj,– the meanE[Lj ], varianceVar[Lj ] and first-order autocorrelationρj of all lead

times,– the maximum inventory levels∆i for each intermediate stockpointi.

Our algorithm consists of the following steps:

1. Calculate the following demand characteristics, working from the end stock-points in the upstream direction:(a) the mean and standard deviation ofDi for all intermediate stockpointsi

using (9),(b) the mean and variance ofDj,Lj

for all intermediate and end stockpointsjusing (8a–b),

(c) the mean and variance ofDj,Lj+Rifor all end stockpoints using (28a-b),

(d) the covariances required for the evaluation of the fill rate expressions (25),like Cov[Dj,Lj+R0

, (pi(D0,L0 −∆0)++Di,Li−∆i)+] for a three- echelon

system, wherei = prec(j) and0 = prec(i). These can be obtained from(29) using a Taylor expansion and numerical differentiation. In the caseof stockless depots, the simple expression (12) with extension to multipleechelons can be used.


(e) similarly, the covariances required to evaluate intermediate stock levels,like Cov[L0, (∆i −Di,Li

− pi(D0,L0 −∆0)+)+], Cov[Li, (∆i −Di,Li−

pi(D0,L0 −∆0)+)+] andCov[L0, (∆0 −D0,L0)+)] for three-echelon sys-

tems, see (24) and (31).2. Determine the allocation fractionspj for all stockpointsi from equation (14),

where stockpoint 0 is replaced by stockpointi = prec(j).3. Determine the maximum echelon inventory positionsS∗

j for each end stockpointj:(a) Approximate the distributions ofX1j andX2j as given by (25) using (suc-

cessive) approximation by Erlang mixtures as described in Sect. 5.1. Again,i = prec(j) and0 = prec(i). The mean and variance ofX1j andX2j aregiven by (26a)–(27b). The two-moment fitting is described in the Appendix.

(b) Use bisection to find from (25) the maximum echelon inventory positionsS∗

j that match the target fill ratesβj .4. Determine maximum echelon inventory positions for the intermediate stock-

points using (2), working from end stockpoints in the upstream direction, untilwe have found the order-up-to levelS∗

0 .5. Determine the mean physical stock per end stockpoint from equation (30) using

the expected shortage at the start and at the end of a replenishment cycle as foundin step 3.

6. Finally determine the mean physical stock per intermediate stockpoint fromequations (24) and (31).

7 Numerical validation

The algorithm that we developed is approximate, therefore we establish the accuracyof our approximations using extensive numerical experimentation, both for two-echelon systems (experimental design in Sect. 7.1, results in Sect. 7.2) and for threeechelon systems (experimental design in Sect. 7.3, results in Sect. 7.4). We use thedifference between target fill rate and actual (simulated) fill rate as a measure ofaccuracy. Next to the mean absolute deviation from the target fill rate, we alsoconsider the maximum deviation as a measure of robustness. Further we examinethe accuracy of our expressions for the mean physical stock in the network. Thisis relevant, because the same fill rates can be attained using various combinationsof intermediate stock levels. To make a trade-off between various options, accuratestock estimates are needed. This is both true when analyzing scenario’s and whenapplying some optimization algorithm (see e.g. Van der Heijden, 1997b) for thespecial case of deterministic lead times).

7.1 Experimental design for two-echelon models

In the first experiment we test two-echelon systems, in which one central depotsupplies products to two so-calledservice groups. For sake of convenience we referto these groups as service group A and B respectively. A service group consistsof a number of local stockpoints with the same service, demand and lead time


characteristics. Both service groups consist of three end stockpoints, so we havesix end stockpoints in total. To normalize time and quantities, we made the followingchoices for all test runs:

– the review period equalsR = 1– the mean demand per time unit for each local stockpoint in service group A

equalsE[DA] = 10

A special point of attention is the stochastic process by which successive leadtimes are generated. Assumption f) in Sect. 2.3 stated that orders may not cross.This is reasonable in practice, but it complicates simulation. If we simply generateindependent lead times, we cannot prevent that orders cross in general. Therefore,we choose the following mechanism. A replenishment order first enters a singleserver queue. If the order leaves the queue, it passes a pipeline with deterministicsojourn time. In this way, orders can not cross, while also a large range of leadtime distributions can be modeled (see Diks and Van der Heijden, 1997). Whatremains is the choice how to divide the lead time in the stochastic queuing sojourntimeLS and the deterministic pipeline sojourn timeLD. We found that the divisionLS = cLE[L] is a good choice, wherecL denotes the coefficient of variation ofL(=ratio of standard deviation and mean). This division is only valid for0 ≤ cL ≤ 1,sinceLD = E[L] − LS < 0 otherwise. However, lead times withcL > 1 areexceptional.

Now we proceed as follows. First, we split the lead time in a deterministic anda stochastic part according toLS = cLE[L]. Next, we determine the parametersof the queuing model such, that the sojourn time of a customer in the single serverqueuing system plusLD has meanE[L] and varianceVar[L]. This is done usingthe method as described in Diks and Van der Heijden (1997). We simulate this leadtime process to estimate the first-order autocorrelationρ. This value is used in ourapproximate method to determine the control parameterspj andS∗

j . Finally wesimulate the two-echelon system with the same stochastic lead time processes andthe control parameterspj andS∗

j to estimate the approximation accuracy. Note thatthis method is only used for validation purposes. Once we know that our method isaccurate, we only have to measure the mean, variance and first-order autocorrelationof all lead times from actual data.

Let us return to the choice of the parameter values. Unfortunately, the numberof parameters that can be varied in our experiment is still quite large. To keep thesize of the experiment within reasonable limits, we take the following parametersfixed:

– The mean lead time to the central depot equalsE[L0] = 3. Reason for this is thatmost upstream (production) lead times are usually larger than a review period.Besides, the experiments for deterministic lead times in Van der Heijden et al.(1997) revealed that the accuracy of the approximation was better forL0 = 1than forL0 = 3, so the latter should be tested.

– The downstream lead times are usually small, because these lead times representusually order picking, handling and transport times. Therefore we takeE[Lj ] =1 in all test runs.


Table 1.Parameter values in the experiment with two-echelon systems

Parameter Description Values in test runs

E[DB ] the mean demand per period at an end

stockpoint in service group B 10, 30

c[DA] coefficient of variation of demand per period

at an end stockpoint in service group A 0.4, 0.8

c[DB ] coefficient of variation of demand per period

at an end stockpoint in service group B 0.4, 0.8

βA target fill rate at an end stockpoint

in service group A (%) 90, 99

βB target fill rate at an end stockpoint

in service group B (%) 90, 99

c[L0] coefficient of variation ofL0 0, 0.25, 0.5

c[Lj ] coefficient of variation ofLj 0, 0.5

a0 constant, describing the level of stock

at the central warehouse∆0 = a∗0E[D0,L0 ] 0, 1.2

Eight other parameters are varied in our experiment. We choose two differentvalues for each parameter (see Table 1), except for the variation in the upstreamlead timeL0. Since the variation ofL0 is largest in practice, we should carefullyexamine the effect of this variance. To choose the maximum amount of centralstock∆0, we proceed as follows. Equation (24) shows that the amount of centralstock heavily depends on(∆0 − D0,L0)

+. Therefore it is convenient to express∆0 in the mean system demand during the lead timeL0, say∆0 = a∗

0E[D0,L0 ]for some constanta0. We have significant central stock ifa0 > 1, so we choosea0 = 1.2. Also, we consider the situation with stockless central depot,a0 = 0.Using the value of the constanta0, we determine the appropriate value of∆0 foreach case.

We tested all possible parameter combinations, yielding3∗27 = 384 cases. Theperformance of the algorithm is tested by an extensive simulation of 75,000 timeperiods for each case.

7.2 Results for two-echelon models

The accuracy of our algorithm is shown in the Tables 2–4 below. The accuracy ofthe expressions for the mean physical stock is expressed as the mean percentagedeviation from simulation results and as the maximum percentage deviation over all384 cases. The fill rate accuracy is expressed by three measures: The mean deviationfrom target in percent points, the mean absolute deviation and as the maximumabsolute deviation. Note that for each case the deviation is measured as the averagedifference between simulated fill rate and target fill rate over all stockpoints in aservice group (the simulated fill rate for separate stockpoints in the same service


Table 2.Fill rate accuracy per target fill rate

Target Mean Mean Maximum

fill rate (β) deviation absolute deviation absolute deviation

90% 0.45 0.52 1.8099% −0.29 0.41 1.06ALL 0.08 0.45 1.80

Table 3.Fill rate accuracy per central stock level

Central stock Mean Mean absolute Maximum absolute

level deviation deviation deviation

a0 = 0 −0.20 0.39 1.06

a0 = 1.2 0.35 0.54 1.80

ALL 0.08 0.45 1.80

group may slightly vary because of simulation inaccuracy). The maximum absolutedeviation of the fill rate refers to the maximum absolute deviation found over allservice groups in all 384 cases. The mean deviation shows to which extend theactual fill rate is a systematically underestimated (positive value) or underestimated(negative value).However, the mean and maximumabsolutedeviation are betterindicators for the approximation accuracy itself, as positive and negative valuesmay compensate

Because a deviation from the target service level has usually more seriousconsequences in the case of a high target service level, we separately give therationing policy performance for each fill rate level in Table 2. Also we show theperformance for stockless and stock holding central depot separately (Table 3).Reason for this is that we may expect a better performance in the case of a stocklesscentral depot, since we do not need a Taylor expansion then, cf. equations (12) and(20). To examine other possible systematic errors, Table 4 shows the approximationaccuracy for each service group, depending on demand characteristics. Finally weshow the accuracy of the approximation for the system stock in Table 5, expressed asthe mean and maximum relative deviation from simulation results. Here the systemstock is defined as the sum of the mean physical stock in the central depot and theend stockpoints plus the pipeline stock between central depot and end stockpoints.The pipeline stock from the external supplier to the central depot isnot included,because this is usual for external account.

The overall results show that the accuracy is sufficient for practical applications,although the accuracy for the stock levels is clearly better ifa0 = 0. Of course,zero central stock is easy to ‘approximate’. According to Table 3, there seem to beno severe systematic errors in the approximation for the fill rate, although it seemsthat low fill rates are slightly underestimated and high fill rates are slightly over-estimated. Table 4 shows that there are no severe systematic errors depending onservice group, although there seems to be a relation to the amount of demand varia-


Table 4.Fill rate accuracy per service group

E[DA], c[DA], Mean Mean Mean relative Mean relative

E[DB ], c[DB ] deviation (A) deviation (B) deviation (A) deviation (B)

10, 0.4, 10, 0.4−0.09 −0.07 0.51 0.50

10, 0.4, 10, 0.8−0.08 0.14 0.49 0.40

10, 0.4, 30, 0.4−0.06 −0.06 0.51 0.50

10, 0.4, 30, 0.8 0.12 0.16 0.53 0.40

10, 0.8, 10, 0.4 0.19 −0.04 0.42 0.48

10, 0.8, 10, 0.8 0.20 0.18 0.42 0.40

10, 0.8, 30, 0.4 0.19 −0.09 0.43 0.49

10, 0.8, 30, 0.8 0.41 0.12 0.53 0.42

ALL 0.11 0.04 0.48 0.45

Table 5.Stock accuracy per central stock level

Central stock level Mean relative deviation Maximum relative deviation

a0 = 0 0.4% 1.6%

a0 = 1.2 2.2% 5.0%

ALL 1.3% 5.0%

tion. That is, the actual fill rate tends to be slightly underestimated (overestimated)if the demand coefficient of variation is high (low). For more detailed numericalanalysis we refer to Sect. 8.

7.3 Experimental design for three-echelon models

To design an experiment for three-echelon models, we proceed from the design thatis developed in Van der Heijden et al. (1997) for the situation with deterministiclead times. We consider a system consisting of one central depot, supplying 4intermediate stockpoints. Each intermediate stockpoint supplies 6 end stockpoints,so the system consists of 24 end stockpoints totally. This is the largest systemconsidered in Van der Heijden et al. (1997), showing the highest approximationerrors in the case of deterministic lead times. Because the accuracy is usually betterfor smaller systems, we chose this large three-echelon system with1+4+24 = 29stockpoints.

To keep the number of test runs within reasonable limits, we take the followingparameters fixed:

– the review period equalsR = 1– the lead time to the central depot 0 has meanE[L0] = 3 and coefficient of

variationc[L0] = 0.5– the lead time to each intermediate stockpointi from its supplierpre(i) = 0 has

meanE[Li] = 1


Table 6.Parameter values in the experiment with three-echelon systems

Parameter Description Values in test runs

E[D] the mean demand per period

at an end stockpoint 10, 30

c[D] coefficient of variation of demand

per period at an end stockpoint 0.4, 0.8

β target fill rate at an end stockpoint (%) 90, 99

c[Li] coefficient of variation of the lead time

Li to each intermediate stockpointi 0, 0.5

c[Lj ] coefficient of variation of the lead time

Lj to each end stockpointj 0, 0.5

a0 constant, describing the level of stock

at the central warehouse∆0 = a∗0E[D0,L0 ] 0, 1.2

ai constant, describing the level of stock

at each intermediate stockpoint

∆i = a∗i E[Di,Li + pi(D0,L0 − ∆0)+] 0, 1.2

Table 7.Fill rate accuracy per target fill rate

Target fill rate Mean Mean relative Maximum relative

(β) deviation deviation deviation

90% 0.87 1.14 2.48

99% −0.50 0.75 1.60

ALL 0.18 0.95 2.48

– the lead time to each end stockpointj from its supplierpre(j) has meanE[Lj ]= 1

For the other parameters, we selected the values as shown in Table 6.For the demand and service characteristics, we used the experimental design

as described in Van der Heijden et al. (1997), where it is shown that 87 parametersets are sufficient to cover this part of the design. We combined these 87 sets withall possible combinations ofc[Li], c[Lj ], a0 andai, resulting in24∗87 = 1392 testruns. The accuracy of our approximation is tested by a simulation of 25 000 timeperiods for each case.

7.4 Results for three-echelon models

Similar to Sect. 7.2, we present results on the accuracy of our algorithm for thethree-echelon systems in the Tables 7–9 below. We give the fill rate accuracy bothper target fill rate (Table 7) and per intermediate stock level (Table 8). The stockaccuracy per intermediate stock level is shown in Table 9.


Table 8.Fill rate accuracy per intermediate stock level

Central and intermediate Mean Mean absolute Max. absolute

stock level deviation deviation deviation

(a0, ai) = (0, 0) −0.28 0.78 1.69

(a0, ai) = (0, 1.2) 0.43 0.96 2.48

(a0, ai) = (1.2, 0) 0.21 1.01 2.01

(a0, ai) = (1.2, 1.2) 0.41 1.04 2.40

ALL 0.18 0.95 2.48

Table 9.Stock accuracy per intermediate stock level

Central and intermediate Mean absolute Maximum absolute

stock level deviation deviation

(a0, ai) = (0, 0) 2.0 2.9

(a0, ai) = (0, 1.2) 4.6 6.3

(a0, ai) = (1.2, 0) 3.8 5.5

(a0, ai) = (1.2, 1.2) 4.3 5.6

ALL 3.7 6.3

We see that the approximation errors increase compared to the two-echelonsystem. Also, systematic errors seem to be more severe. Low service levels areunderestimated and high service levels are overestimated. Although the results areworse then for the two-echelon model, we think that our method is still accurateenough for practical applications. This is especially true for systems with stocklessintermediate stockpoints, such as the hierarchical planning procedure as describedby De Kok (1990).

7.5 Computational effort

It appears that the approximate method is fast. CPU time using a Pentium 100 MHzPC equals 0.13 seconds per case on average for the two-echelon systems and 0.30seconds on average for the three- echelon systems. Little computational effort isimportant, because often control rules have to be established for hundreds or thou-sands of products.

8 Sensitivity analysis

Now that a tool is available, it can be used for some sensitivity analysis to get someinsight in the effects of lead time variation and autocorrelation. We focus on thefollowing two issues:

1. Which effect has lead time variation on the amount of stock required to obtainprespecified target service levels?


Table 10.Average number of time units stock, depending on lead time variation

c[Li] = 0 c[Li] = 0.5 ALL

c[L0] = 0 3.4 3.7 3.5

c[L0] = 0.25 4.1 4.4 4.3

c[L0] = 0.5 5.4 5.6 5.5

ALL 4.3 4.6 4.4

Table 11.Effect of lead time variation and demand variation on the average stock, expressedin time units to cover mean total customer demand

c[D] → 0.4, 0.4 0.4, 0.8 0.8, 0.8 ALL

c[L0] ↓0 2.8 3.5 4.3 3.5

0.25 3.6 4.3 4.9 4.3

0.5 4.9 5.5 6.1 5.5

ALL 3.8 4.4 5.1 4.4

2. Is it really important to include correlations in the approximate method?

We will answer these questions by more detailed analysis of the results for two-echelon models only to keep the results clear.

To start with the first issue, we take a look at Table 10 in which the averageamount of system stock is shown for various combinations of lead time variation. Tomake the various cases comparable, we expressed the average stock as the numberof time units during which mean total customer demand can be covered (e.g., ifdemand is specified per week, the numbers refer to the stocks in weeks of totalcustomer demand).

We see that especially the upstream lead time variation has a significant impacton the amount of stock required to reach the target fill rates. The effect of the leadtime variation between central depot and intermediate stockpoints is less, whichcan partly be explained by the fact thatE[L0] = 3 andE[Li] = 1, soVar[L0] islarger thanVar[Li]. Anyway, we see that we can not ignore the lead time variation,because in practice significant lead time variation is present in the upstream part ofthe network.

Now the question raises what has more impact, demand variation or lead timevariation? Usually the attention is focused on demand variation, but is this justified?This question can be answered by looking at Table 11. Here we consider the mostimportant source of lead time variationc[L0] and the various combinations ofdemand variation in the two service groups.

Table 11 shows that in our experiment the effect of lead time variation evendominates the effect of demand variation on the stock levels required to achievethe target fill rates! Although this can be parameter dependent, it is clear that bothsupply and demand variation should be taken into account when determining safetystock levels in multi-echelon divergent systems. This conclusion is in line with theresults obtained by Gross and Soriano (1969) for a single stockpoint.


Table 12.The effect of neglecting correlations

Performance measure c[L0] Original Ignore correlations

mean absolute deviation 0.25 0.45 1.39

from target fill rate 0.5 0.67 1.54

maximum absolute deviation 0.25 1.41 4.03

from target fill rate 0.5 1.80 3.45

The second issue is important to judge whether our method can be simplifiedconsiderably by excluding the correlations, such as given by the equations (12),(20) and (29). We recalculated the control parameterspj , S

∗j for all 384 cases,

ignoring all these correlations. Also, we simulated the 384 cases with the modifiedcontrol parameters. In Table 12 below we compare the accuracy of the simplifiedmethod to the original results. As can be seen from the formulas, this difference isonly relevant ifc[L0] > 0.

We see that the performance of our method is considerably better if we takeinto account the correlation effects. The price paid is that of considerable additionalcomplexity.

9 Conclusions

In this paper we developed an algorithm to analyze multi-echelon divergent net-works with integral(R,S) inventory control under both stochastic demand and leadtimes. Validation of our method by extensive comparison to simulation results, bothfor two-level and for three-level systems, shows that our algorithm is sufficientlyaccurate for practical applications. This is in particular true for situations wherethe intermediate depots do not carry stocks and are only used as allocation points.An example is the hierarchical planning procedure as described by De Kok (1990).Using our method, it is easy to include stochastic production lead times in thisprocedure, yielding accurate results.

Our method can be simplified by ignoring the (complicating) correlations in-volved, but at considerable loss of accuracy, as is shown by numerical experi-mentation. Further we showed the importance of including lead time variation inthe model. Although frequently the attention is focused on demand variation, weshowed that the effect of lead time variation may be larger than the effect of demandvariation on the stock levels required to obtain prespecified target fill rates.

One of the differences between our model and practical situations is the fact thatthe replenishment frequency is usually not the same throughout the logistic chain inpractice. For example, production orders are released monthly, while transport fromwarehouses to local stockpoints may be carried out weekly or multiple times a week.Therefore, subsequent research will be focused on the inclusion of differentiatedreplenishment frequencies in our model.


Appendix. Approximation of E[(X − S)+] and Var[(X − S)+]

In this appendix we give a method to calculate the first two moments of(X−S)+,whereX is a random variable with meanE[X] and varianceVar[X] andS is anonnegative constant. We approximate the density ofX by a mixture of Erlangdensities with the same scale parameter, see e.g. Tijms (1994):

Hr,q,λ(x) = qFr − 1, λ(x) + (1 − q)Fr−1,λ(x), (A.1)

where the Erlang distributionFr,λ(x) is defined by

Fr,λ(x) = 1 −r−1∑i=0

(λx)ie−λx

i!. (A.2)

Defining the squared coefficient of variation of a random variableX asc2[X] =Var[X]/E2[X], we can determine the parameters of the Erlang mixture using atwo-moment fit as follows (see e.g. Tijms (1994)):

q =r c2[X] − √

r(1 + c2[X]) − r2c2[X]1 + c2[X]

, (A.3)

λ =r − q

E[X], (A.4)

wherer is chosen such, thatr−1 ≤ c2[X] < (r − 1)−1. If some random variableY has a pure Erlang distributionFr,λ(x), we obtain from (A1)–(A2) that

E[(Y − S)+] def= E(1)r,λ(S) =

r

λ[1 − Fr+1,λ(S)] − S[1 − Fr,λ(S)] (A.5)

and

E[(Y − S)+2] def= E(2)r,λ(S) =

r(r + 1)λ2 [1 − Fr+2,λ(S)]

+2Srλ

[1 − Fr+1,λ(S)] − S2[1 − Fr,λ(S)] . (A.6)

For a mixed-Erlang distributed random variableX we simple have

E[(Y − S)+] = qE(1)r−1,λ(S) + (1 − q)E(1)

r,λ(S)] (A.7)

and

E[(Y − S)+] = qE(2)r−1,λ(S) + (1 − q)E(2)

r,λ(S)] (A.8)

Of course, we have the variance of(X − S)+ now that we have the first twomoments.

References

Axsater S (1993) Continuous review policies for multi-level inventory systems with stochas-tic demand. In: Graves SC, Rinnooy Kan AHG, Zipkin PH (eds) Logistics of productionand inventory. Handbooks in Operations research and management science, Vol 4. Else-vier, Amsterdam


Diks EB, de Kok AG, Lagodimos AG (1996) Multi-echelon systems: a service measureperspective. European Journal of Operational Research 95: 241–263

Diks EB, van der Heijden MC (1997) Modeling stochastic lead times in multi-echelonsystems, Probability in the Engineering and Informational Sciences 11: 469–485

Donselaar K van, Wijngaard J (1987) Commonality and safety stocks. Engineering Costsand Production Economics 12: 197–204

Donselaar K van (1996) Personal communicationFedergruen A (1993) Centralized planning models for multi-echelon inventory systems un-

der uncertainty. In: Graves SC, Rinnooy Kan AHG, Zipkin PH (eds) Logistics of produc-tion and inventory. Handbooks in Operations research and management science, Vol 4.Elsevier, Amsterdam

Gross D, Soriano A (1969) The effect of reducing leadtime on inventory levels – simulationanalysis. Management Science 16: 61–76

Hadley G, Whitin TM (1963) Analysis of inventory systems. Prentice-Hall, EnglewoodCliffs, NJ

Heijden MC van der (1997a) Supply rationing in multi-echelon distribution systems. Euro-pean Journal of Operational Research 101: 532–549

Heijden MC van der (1997b) Near-optimal inventory control policies for divergent networksunder fill rate constraints. Working paper PR-15, Institute for Business Engineering andTechnology Application, Enschede (submitted)

Heijden MC van der, Diks EB, de Kok AG (1997) Allocation policies in general multi-echelon distribution systems with(R, S) order-up-to policies. International Journal ofProduction Economics 49(2): 157–174

Houtum GJ van, Inderfurth K, Zijm WHM (1996) Materials coordination in stochastic multi-echelon systems. European Journal of Operational Research 95(1): 1–23

Kok AG de (1990) Hierarchical production planning for consumer goods. European Journalof Operational Research 45: 55–69

Kok AG de, Lagodimos AG, Seidel HP (1994) Stock allocation in a two-echelon distributionnetwork under service constraints. Working paper, Eindhoven University of TechnologyTUE/BDK/LBS/94-03

Mood AM, Graybill FA, Boes DC (1974) Introduction to the theory of statistics, 3rd edn.McGraw-Hill, Tokyo

Nahmias S, Smith AA (1993) Mathematical models of retailer inventory systems. In: SarinRK (ed) Perspectives in operations management. Kluwer, Dordrecht

Silver EA, Peterson R (1985) Decision systems for inventory management and productionplanning. Wiley, New York

Stoer J, Bulirsch R (1994) Introduction to numerical analysis, 2nd edn. Springer, BerlinHeidelberg New York

Tijms HC (1994) Stochastic models: an algorithmic approach. Wiley, ChichesterVerrijdt JHCM, de Kok AG (1995) Distribution planning for a divergent N-echelon network

without intermediate stocks under service restrictions. International Journal of ProductionEconomics 38: 225–243

Zipkin P (1991) Evaluation of base-stock policies in multiechelon inventory systems withcompound-Poisson demands. Naval Research Logistics 38: 397–412

inventory control in multi-echelon divergent systems with random lead times

Documents