optimal control of a divergent multi-echelon inventory system

EUROPEAN JOURNAL

OF ~~WI&O~AL

European Journal of Operational Research 111 (1998) 75-97

Theory and Methodology

Optimal control of a divergent multi-echelon inventory system

E.B. Diks *, A.G. de Kok Eindhooen Unicersity of Technology. Department of Mathematics and Computing Science, Hg 9.33. P.O. Box 513,

5600 MB Eindhooen. The Netherlands

Received 1 September 1996: accepted I July 1997

Abstract

Consider a divergent multi-echelon inventory system, such as a distribution system or a production system. At every facility in the system orders are placed (or production is initiated) periodically. The order arrives after a fixed lead time. At the end of each period linear costs are incurred penalty costs are incurred at the most downstream facilities for backorders. The objective is to minimize the expected holding and penalty costs per period. We prove that under the balance assumption it is cost optimal to control every facility by an order-up-to-policy. The optimal replenishment policy, i.e. the order-up-to-level and the allocation functions at each facility, can be determined by system decomposition. This decomposition reduces complex multi-dimensional control problems to simple one-dimensional problems. 0 1998 Elsevier Science B.V. All rights reserved.

Keywords: Inventory; Multi-echelon; Optimal control; Allocation; Rationing

1. Introduction

Research on multi-echelon inventory models has gained importance over the last decade mainly because integrated control of supply chains consisting of several processing and distribution stages has become fea- sible, through modern information technology. Multi-echelon inventory theory provides a means of mod- eling such supply chains, thereby enabling quantitative analysis and characterization of optimal control policies (cf. Clark and Scarf, 1960; Federgruen and Zipkin, 1984; Rosling, 1989; Langenhoff and Zijm, 1990).

The initial research on multi-echelon inventory models is generally attributed to Clark and Scarf (1960), who studied an N-echelon serial system operating under periodic review ordering policies. They introduced the concept of the echelon stock to prove that the optimal control policies for the N-echelon serial system with discounted penalty and holding costs, are characterized by N so-called echelon order-up-to-levels. The echelon stock of a stockpoint equals all stock at this stockpoint plus in transit to or on hand at any of its

*Corresponding author. Fax: +31 40 2465995: e-mail: [email protected].

0377-2217/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PIISO377-2217(97)00327-5

16 E.B. Diks. A. G. de Kok I European Journal of Operational Research III (1998) 75-97

downstream stockpoints minus the backorders at its downstream stockpoints. The echelon inventory position of a stockpoint equals its echelon stock plus all material in transfer to that stockpoint.

Eppen and Schrage (1981) analyzed a divergent two-echelon system in which a central depot supplies multiple end-stockpoints. Some restrictive assumptions are made. First, the depot cannot hold any stock. Second, the proportional holding and penalty costs at the end-stockpoints are identical, Third, the demand per period at each end-stockpoint is normally distributed. Fourth, the lead time of the end-stockpoints are identical. Finally, each time an allocation is made, the depot receives enough material from the supplier to be able to allocate the material to each end-stockpoint so that an equal fractile point is achieved on an appropriately chosen demand functions. This assumption is referred to as the allocation assumption. Nowa- days, however, this particular assumption is known as balanced inventories assumption (in the narrow sense). Simulation studies (cf. van Donselaar and Wijngaard, 1987; Lagodimos, 1992) indicate that this assumption is not a serious restriction. Under these assumptions, Eppen and Schrage (1981) derive an optimal order-up-to-policy at the depot, assuming no set-up costs. In case of fixed set-up costs, an approximately optimal policy is derived.

Federgruen and Zipkin (1984) extend the model considered by Eppen and Schrage (1981). Holding and penalty costs do not have to be identical across the end-stockpoints, and period demands at the end-stockpoints do not have to be normally distributed, but may belong to a larger class of demand distributions (e.g. Erlang and gamma distribution). The allocation of material at the depot is determined by solving a myopic allocation problem, which minimizes the expected costs in the period that the allocation actually takes ef- fect, ignoring costs in all subsequent periods. They also used an (approximate) dynamic program to prove that these myopic allocations are optimal given the allocation assumption. For an excellent overview concerning dynamic programming techniques to analyze multi-echelon systems we refer to the survey by Fed- ergruen (1993).

A more service-related approach to determine the control parameters is introduced by De Kok (1990) and Lagodimos (1992). In this approach, the objective to achieve some target service levels at different end-stockpoints replaces the objective of minimizing a cost function. In De Kok (1990) a planning procedure has been determined for a divergent two-echelon system with a stockless depot. This model is later extended in Seidel and De Kok (1990) and De Kok et al. (1994) by allowing the depot to hold stock.

Although much attention has been given to two-echelon systems, one seldom finds extensions to more general divergent N-echelon systems (e.g., Verrijdt and De Kok, 1995). In practice, however, large production and distribution networks are frequently encountered and therefore generalization of two-echelon policies is needed. In this paper we analyze a divergent N-echelon inventory system in which every stockpoint is allowed to hold stock. The objective is to minimize the average costs per period in the long run.

Our model can be regarded as an extension of those by Langenhoff and Zijm (1990) and van Houtum and Zijm (1991). Langenhoff and Zijm (1990) give exact decomposition results for a two-echelon assembly system, a two-echelon serial system and a divergent two-echelon system. The latter system is more thor- oughly analyzed in van Houtum and Zijm (1991). In this paper, we extend the analysis of the above stated papers by relaxing the following assumptions: (1) all end-stockpoint lead times are identical, (2) the penalty costs at all end-stockpoints are identical, (3) the model considered is a two-echelon model. A detailed review of this work, together with other related papers, is given by van Houtum et al. (1996).

The remainder of the paper is organized as follows. In Section 2, we describe the model under consid- eration. In Section 3, we decompose the system to obtain a tractable expression for the average holding and penalty costs per period as a function of the control policy used. Also some important properties of this cost function are derived. In Section 4, we use these properties to characterize the optimal replenishment policy. That is, we prove that the optimal order-up-to-level at each stockpoint satisfies a newsboy-type equation, and we make a classification of the optimal allocation function into four classes. In Section 5, we prove that, under certain conditions, the decomposition is exact (i.e. it results in control policies that are optimal). Since the decomposition only considers echelon order-up-to-policies, there exists an optimal policy in which

E.B. Diks, A.G. de Kok I European Journal of Operational Research III (1998) 75-97 II

every stockpoint is controlled by an echelon order-up-to-policy. Finally, in Section 6, we give a few con- cluding remarks.

2. Model description

We consider a single-item discrete-time multi-echelon inventory system where every stockpoint is allowed to hold stock. The system has an arborescent stockpoint structure, i.e., each stockpoint has a unique supplier. We refer to such systems as divergent multi-echelon systems, which can be described by a directed graph (see Fig. 1 for an example). The most upstream stockpoint in the material flow places orders to an external supplier, which can always meet the demand.

Inventory is controlled by periodic review policies. That is, every R periods the most upstream stockpoint, 1 say, issues a replenishment order. For the sake of clarity, in the remainder of this paper we refer to the length of a review period as one period (R := 1). The replenishment order arrives after Lt periods, where Li is a fixed multiple integer of R. After this arrival the physical stock at stockpoint 1 is allocated immediately to its successors. There are two possibilities: 1. The physical stock is sufficient to raise the echelon inventory position of each successor to its order-up-

to-level. Then the required amounts are sent to the successors and excess stock is kept at stockpoint 1 to be allocated in the next occasion.

2. The physical stock is not sufficient to reach the order-up-to-levels. Then material rationing is required to allocate the available physical stock over its successors appropriately. For this purpose we introduce allocation functions in Section 3. To ration the stock as explained above, we require an assumption similar to the allocation assumption of

Eppen and Schrage (1981). This allocation assumption assumes that the available products are allocated such that each end-stockpoint faces an equal stockout probability after rationing. The phenomenon of not being able to achieve this is referred to as imbalance. Verrijdt and De Kok (1995) generalizes the definition by defining imbalance as the phenomenon where the rationing policy of an intermediate stockpoint allocates a negative quantity to at least one of its successors. As a result they introduce the balance assumption, which assumes that the rationing policy always allocates non-negative stock quantities. In the remainder of this paper, we assume this balance assumption. For a detailed discussion of alternative rationing policies together with the associated assumptions, the reader is referred to Diks et al. (1996).

Stage: 4 3 2 i

Fig. 1. Schematic representation of a divergent 4-echelon inventory system.

78 E. B. Diks, A. G. de Kok I European Journal of Operational Research I I I (I 998) 75-97

Without loss of generality, we assume that only the end-stockpoints face external customer demand. In case an intermediate stockpoint i faces external demand, we redirect this demand to a new successor j with lead time Lj := 0. By definition this successor j is an end-stockpoint. During one period the demand between end-stockpoints may be correlated, however, the demands in subsequent periods are i.i.d. With respect to the customer demand process, we assume that all demand which cannot be satisfied immediately is backordered.

The notation we used to describe the system lay-out is shown in Table 1 (the examples in brackets refer to the situation in Fig. 1).

Furthermore, we assign a low level code (LLC) to every stockpoint. By definition, for an end-stockpoint i we have LLC(I’): = 1, and for an intermediate stockpoint i we have LLC(I’): = l+maxjEK LLCQ).

At the end of each period both penalty and holding costs are incurred. The penalty costs equal pi for each backlogged product at end-stockpoint i. For a product at stockpoint i or in transfer to one of its successors the holding costs equals hi + CnEU, h,. Note that hi can be regarded as an additional holding cost due to value added in stockpoint i. No fixed ordering costs are assumed. The objective of the analysis is to determine a cost optimal replenishment policy. That is a policy, which minimizes the expected total costs in the long run.

3. Analysis cost function

In this section, we present an average cost analysis for the divergent N-echelon system. First, we derive an expression for the cost function, which yields the average system costs as a function of the control policy used (Section 3.1). Next, we assume that system decomposition, as in Langenhoff and Zijm (1990) and van Houtum and Zijm (1991), is a reasonable approach to determine the control parameters. To minimize the cost function we first derive important properties of: (1) the allocation function minimizing this cost function (Section 3.2) and (2) the cost function (Section 3.3). These properties are used in Section 4 to characterize the optimal control policy given decomposition. Also we apply these properties in Section 5 to prove that decomposition is exact (given the balance assumption), so it results in control policies that are cost optimal. Besides the notation of Table 1, we also introduce some additional notation in Table 2 with respect to the analysis of the system.

3.1. Cost function

At the end of each period, holding costs and penalty costs are incurred. Let xi denote the echelon stock of stockpoint i at the end of an arbitrary period. Then the total costs incurred equal

Table 1 Notation with respect to system lay-out

N ech(i)

pre(i) E E, M U K R

number of stages in inventory system (e.g., N = 4) stockpoints that constitute the echelon of stockpoint i (e.g., ech(5) = { 5,8,9}) preceding stockpoint of stockpoint i (e.g., pre(8) = 5) all end-stockpoints (e.g.. E = (2,6,8,9,10}) all end-stockpoints in ech(i) (e.g., ES = {6.8,9}) all intermediate stockpoints (e.g., M = { 1,3; 4,5,7}) all stockpoints on path from supplier to stockpoint i (e.g., U, = 0 and U6 = { 1,3}) all stockpoints which are supplied by stockpoint i (e.g., 6 = {2.3,4}) all stockpoints with low level code i (e.g., W, = {3,4})

E. B. Diks. A. G. de Kok I European Journal of Operational Research 111 (1998) 75-97 19

Table 2 Notation with respect to the analysis of the system

z,(x) the echelon inventory position of stockpoint j just after the allocation, if just before allocation the echelon stock of its supplier equals x

4: A, YY,

echelon order-up-to-level of stockpoint i maximum physical stock at stockpoint i, i.e., A, := y, - C,,I, JL all control parameters downstream of stockpoint i, i.e..

0. iEE.

,g(z,.y,.Y,j. iEA4

the expected total costs in ech(i) at the end of an arbitrary period given the echelon order-up-to-level J and F, (If Y, = 0 we suppress YE) cdf/pdf of demand at all end-stockpoints in E, during L periods (if L = I we suppress the index) incomplete convolution of L continuous distribution functions F;, i.e.,

P, h;

minimum penalty costs at a stockpoint in E,, i.e., min,,E,p, minimum value added at a successor of stockpoint i, i.e., min,,Kh,

V -I

7,

all successors of stockpoint i, which have the same minimum penalty costs as stockpoint i, i.e..

{jE KK =&I all successors of stockpoint i for which the minimum amount is added to a product, i.e., 0’ E Klh; = h,}

In Lemma 1 we use this expression to obtain an expression which associates costs to the echelon stock at each stockpoint, which constitutes the basis for the decomposition presented in Section 4.

Lemma 1. When at the end of an arbitrary period the echelon stock of a stockpoint i equals xi, the total costs incurred at the end of this period equal

+ C hixj.

iEM

Proof. In order to prove this we use that for all i E M

(1)

From Eq. (1) and the property that for the most upstream stockpoint i we have U; = 0, it follows that

(2)

From the definition of the inventory costs hi and penalty costs pi it easily follows that the expected total costs equals

80 E. B. Diks, A.G. de Kok I European Journal of Operational Research III (1998) 75-97

+ C hixi + x x hnxi - C C xi hi + C hn iEM IEM rsu, &M ( ) ntu, jEV

Rewriting the expression above completes the proof. 0

This lemma implies that costs hixi are incurred at each stockpoint i (independent of the sign of xi), where- as at an end-stockpoint extra costs are incurred when it is in a backlog position. These are both penalty costs pi and holding costs hi + CntU, h, per backlogged unit. Extra holding costs are incurred, since the echelon stock of an intermediate stockpoint j (for which i E Ej) does not coincide with the amount of products in echG), due to the backorders at the end-stockpoints i E Ej.

In order to relate the inventory of a stockpoint at the beginning of a period with the costs incurred at the end of this period, we define the one-period cost function Ci(x).

C(x) := The end of period expected costs incurred at stockpoint i, when at the beginning of this period the echelon stock of stockpoint i is increased to x.

To evaluate Ci(X] we assume that the cdf of the total demand at all end-stockpoints in E, during L periods is given by F;( ) (if L = 1, we suppress the index). From Lemma 1 it follows that

1

Thi(x-u) dF;(u)+T h,+ C h +p. (u-x) a(u), iEE,

C,(X) I= O J n/ 1)

7 h;(x - u) dE:(u), i E M.

Before computing the expected costs of echelon i at the end of an arbitrary period, denoted by Di(y;, ‘Pi), we have to decide how to ration the available stock when a stockpoint has insufficient stock to satisfy all demand. Suppose that at the beginning of an arbitrary period (just before rationing), stockpoint i has an echelon stock of x products. All its successors j E I$ want to raise their echelon inventory position to yj. Hence, if x 2 CnEC: yn the echelon inventory position of stockpoint j just after rationing becomes yj, and the remainder x - CnEK y, is retained at stockpoint i. Otherwise, we have to deal with the main diffi- culties of divergent multi-echelon systems: how should we ration the available stock over these successors J?

For this purpose we define an allocation function “j(x). This means that the rationing policy allocates the products such that the echelon inventory position of stockpoint j after rationing equals “j(x). Since no products are retained at stockpoint i, we have

Throughout the analysis we assume that when a stockpoint i rations the available stock x over the stockpoints j E K, the echelon inventory position of every stockpoint j just after rationing is at least as large as it

E. B. Diks, A.G. de Kok I European Journal of Operational Research 1 I1 (1998) 75-97 81

was just before rationing. This assumption coincides with the balance assumption of Verrijdt and De Kok (1995). The next lemma enables us to determine the expected costs of echelon i, given the control policy C.Yi, yi).

Lemma 2. For replenishment policy (yi, Y’i) the average costs of echelon i equals

Proof. See Appendix A. ??

In the following analysis we assume that e(x) is a differentiable cdf for x > 0. Hence, from Lemma 2 it follows that Dibi, Yi) is also a differentiable function with respect to yi.

As already stated, the objective of this paper is to determine the control policy (~1, Yl) that minimizes Di (~1, Yi) (where index 1 denotes the most upstream stockpoint). To achieve this we will apply system decomposition, as in Langenhoff and Zijm (1990) and van Houtum and Zijm (1991). This means that, after the determination of the downstream control parameters, wejirst determine the allocation functions {zj}jcv, minimizing D, (_Yi, Yi) (given UjEc; (Jj, Yj)), and next determine the order-up-to-level yi which minimizes Di(yi, Yi) (given Yi). In Section 3.2, we derive properties of the allocation functions minimizing OiC_vi, Yi), which form the basis for an algorithm to determine these functions in Section 4.1. We also make a classification of the allocation functions based on their properties.

3.2. Properties of the allocation function

For convenience of presentation, we first introduce some definitions,

Definition 3. ?j is optimal when Dj(ij(x), Yj) is minimized for all x (given Yj).

Definition 4. Yi := Ujcc;(?j~jj, Yj) is locally optimal when: (i) For every j E I$ the allocation function ,?j is optimal (given Yj). (ii) For every j E K the order-up-to-level equals i; := argmin{yj 1 tIDjbj, Yj)/ay, = 0).

Definition 5. @, := Yi is optimal, if Yj is locally optimal for every j E ech(i).

In Definition 4, we defined the order-up-to-level i;. However, it is unclear whether there exists such a yj. Moreover, if indeed jj exists, it is unclear whether ~7~ corresponds to a minimum or a maximum of OjcV,, Yj). In Section 3.3, we prove that ~j exists and that it corresponds to a global minimum. Next, in Lemma 6 we present a very important property of the optimal allocation function ,?j, which is used throughout this paper.

Lemma 6. The optimal allocation functions {.Zj}jGK minimize Di(yi, Yi) if there exists a function A,(x) such that

aDjcV,> ‘yj) aYi

= A,(X) for every j E E and X E R. y,=i, (1)

(5)

82 E. B. Diks, A. G. de Kok I European Journal of‘ Operational Research 111 (1998) 75-97

Proof. We only give an outline of the proof here. When the echelon stock of stockpoint i, say X, is not sufficient to raise the echelon inventory positions of all successors j E 4 to their echelon order-up-to-level yj, rationing is needed. Lemma 2 implies that in order to do this optimally we have to solve

where 9 represents the set of all possible allocation functions. Using the Lagrange-multiplier technique, we can see that taking the derivative of the objective function CnE K D, (z,, (x): ul,) + Ai(x) ( CntC: z,(x) - x) with respect to “j(x) equals the Lagrange-multiplier n;(x). Hence, the allocation functions {Sj}iCK can only be optimal when Eq. (5) holds. 0

Lemma 6 shows that the derivative of Djbj, u/j) with respect to y, in yj = ij(x) is independent of stockpoint j. This is very important in order to characterize the optimal control policy. Based on this lemma we now are able to derive several properties, two of which are given below.

Lemma 7. Zf for every stockpoint j E 6 the cost function Djb,, Yj) is convex with respect to yj, then there exists a set of optimal allocation functions {ij}jcK such that

&Ax) dx

2 0 for all x, je F$.

Speczjically, if for every stockpoint j E I/; the cost function Dj(yj, Y,) is linearly decreasing for yj < xJ and strictly convex for yj > xi”, then for the set of optimal allocation functions it holds that

&jb) dx

>0 for x2x:, je 6,

where xi* equals 0 for each end-stockpoint i, and equals min{xlj,(x) > XT jbr j E K} for i E M.

Proof. See Appendix A. 0

Besides introducing an important property of an optimal allocation function, i.e. that it is a non-decreasing function, this lemma also is needed to prove that the cost function DicV,, ‘Pi) is convex with respect to yI (see Theorem 12).

Lemma 8. Zffor every stockpoint j E K the cost function Dj(yj, Yj) is convex with respect to yj, then for a local optimal Yi = UjcK(ij,jj, Y,) we have that jj(C,,Eeyn) = i;.


If the echelon stock of stockpoint i drops just below CnC K yn, say CnE K yn - t, then rationing is required. Hence, the echelon inventory position of a successor j E K (just after rationing) equals Sj(CnEKjn - E). However, if the echelon stock of stockpoint i equals or exceeds C,,,gn then the echelon stock position of successor j (just after rationing) equals jj. From Lemma 8 we know that lim,lo5j(C,EKj, - E) = i;, so the continuity of the amount of products allocated to successor j is guaranteed. This lemma is used throughout the paper, and plays an important role in the proof of Theorem 12 (concerning the convexity of the cost function).

E. 8. Diks. A. G. de Kok I European Journal of Operational Research 111 (1998) 7S-97 83

3.3. Properties of the cost function

In this section, we focus on obtaining properties of the cost function Di bi, ‘Pi). To determine the optimal order-up-to-level A, we need a tractable expression for alibi> Y’i)/@i. In Theorem 9, we derive such an expression. With the result presented in Theorem 10, we simplify this expression considerably (see Theorem 11). Finally, in Theorem 12, we derive properties (such as convexity) of the cost function to characterize the behavior of an optimal allocation function (Section 4.2).

Theorem 9. For every stockpoint i E M with a locally optimal Y, it holds that:

x

aDiC_Yi7 yi) = hi + aYi J’

tli(,Vi - U) dlfL’)(u) for Ai < 0,

0

where Ai is dejined in Eq. (5).

Proof. Substitution of the definition of Ci(x) in Eq. (4) gives

Since Yyi is locally optimal, {,+}jEK satisfy Eq. (5). Applying Lemma 6 yields

aDiC&, yi) a.vi

= hi + M &(yi - u) 1 y J d$‘+). A, PICK X==Y,_U Since equality (3) holds for every allocation function, using this property yields

cc

“t; lyl) = h, +

J &(_Y; - 24) d&JL”(U). (6) 1

A,

When Ai < 0 the proof of the theorem is trivial from Eq. (6), since @‘(u) = 0 when u < 0. For the case Ai 3 0, writing U* for u - Ai, after substitution of U* in Eq. (6) we obtain

84 E. B. Diks, A. G. de Kok I European Journal of’ Operational Research I1 I (I 998) 75-97

aDi(.Vi, Yi) cx J

8Yl = hi +

I /l;(yi - U* - A;) dI$(L’)(~* + Ai) = hi + J ju;(vi - U* -AI) d($Lo)AC(u*).

0 o+

Since Yi is locally optimal, from Lemma 8, we know that I&+ - A,) = 0. 0

The expression for alibi, Yi)/$; in Theorem 9 can be rewritten in a more insightful manner. Gong et al. (1994) proved that the optimal order-up-to-level ji of a stockpoint i in a serial system satisfies a newsboy- type equation. We will show that this result also applies to divergent systems, so we first derived an expression for the non-stockout probability of an end-stockpoint.

Theorem 10. Let c$(yi) denote the non-stockout probability of an end-stockpoint k in a divergent echelon system, in which the most upstream stockpoint i has an order-up-to-level y,. Iffor every allocation function in stockpoint i it holds that zj(C,,,zK y,,) = yj, then

i E E,

iEM, A; < 0,

i E M, Ai 2 0,

(7)

for every j E 6 and k E Ej.

Proof. See Appendix A.

From Theorems 9 and 10 we can now derive an explicit expression for aDi@i, @;)/a~,, when ‘Pi is op-

timal.

Theorem 11. For every DiCJ,, S‘i) with optimal ii it holds that

aDi(yi, ii) ayi =-[zh.+Pk) + (h*+~h,+a)&(%) forevery kEE;.

Proof. This theorem can be proved by induction on i. For i E E it immediately follows from Eq. (A.8). Next, consider a stockpoint i E M. Assume the theorem holds for every successor j E 6 (induction assumption). Let yi < CnEC:jn. From Theorem 9 it follows

“‘;; “) = hi + j A& - u) dF;,‘L’)(u). 0

Substitution of the definition of nib - U) above yields

aDi(.Yi, *i) m aDj(,Yj, @j)

aYi = hi +

J’ aYj dFJL’) (a) for every j E 6 I

n Y,=&Cv-u)

E. B. Diks. A. G. de Kok / European Journal of Operational Research Ill (1998) 75-97 85

Using the induction assumption we have

The use of the result in Theorem 10 directly completes the proof for y, < CnEI/; j,,. The proof is completely analogous for yi 3 CnEI: jn. 0

Finally, in Theorem 12, we derive some additional properties of the cost function Dibi, ‘Pi), which pro- vide insight into the behavior of the optimal allocation function (Section 4.1) and are required to prove the optimality of the decomposition (Section 5).

Theorem 12. For each end-stockpoint i we define XT := 0, and for each intermediate stockpoint i we define XT := min {x Isj(x) > x/* for every j E v}. Then, the cost function Di(yi, i,) with optimal ‘Pi is.

(i) linear decreasing with slope -(CnEU, h, + pi’) for y, < XT, (ii) convex with respect to yi, specifically, if for every end-stockpoint k E Ei the demandfunction Fk(x) is strictly increasing for x 2 0, then Di(yi, Yi) 2s a strict convex function Of yi 2 XI, (iii) aDi(V;, !Pi)/a yI ten d s to h, when yi goes to infinity.


Note that from Theorem 12, it follows that, if hi is positive, a minimum of the function Di(yi, ‘?i) exists for yi > x;. Specifically, if at every stockpoint k E Ei the demand function Fk(x) is strictly increasing in x 3 0, the uniqueness of this minimum is guaranteed. However, if hi = 0 the minimum is attained in infinity.

4. Optimal control policies

From the results in Section 3, we are now able to determine optimal control policies for the multi-echelon systems we study, specifically, in Sections 4.1 and 4.2 we present how to determine the optimal allocation function and the optimal order-up-to-level, respectively.

4.1. Optimal allocation functions

Consider the situation where we already determined the optimal ‘?j for j E I$. We wish to determine the optimal allocation functions {=?j}jtK. From Theorem 11, we know that

8DjcV,, ‘P.) ay, ’ =-( zhn+pk) + (nk+zhn+pk)dbj); kEEj.

Substitution of the above in Lemma 6 yields (after some elementary algebra)

ni(X) + CnE~, h, +pk ‘(‘i(x)) = hk + xnEUk h, +pk ’

_ic K, k E Ej.

The /Ii(x) is determined such that CnEI/; in(x) = X. This can be done by a one-dimensional bisection procedure where -(hi + CnEuz h, +p,*) 6 Ai( x < h; (see Eq. (A.lO)). The whole procedure works as depicted by ) the algorithm in Fig. 2.

86 E. B. Diks, A. G. de Kok I European Journal of Operational Research III (1998) 75-97

(i) Initialize z, and t > 0. (ii) Choose -(h; + C h, +p:) L Xi(Z) 5 h,f.

nw, (iii) Determine +(z) from

(iv) if C nEV, .&(z) < z - 6 then increase Xi(z) and return to step (ii). if CnEV 2, x > x - E then decrease Xi x and return to step ii .

Fig. 2. Algoriththm to determine the optimal allocation functions {S,}.

Note that the way to adapt Li( x as suggested in step (io) of Fig. 2 is valid, due to the convexity of the ) cost function Djcyj, Yj) for j E E. Unfortunately, we have to apply the procedure above for every x. Hence, it is not suited for practical purposes. However, it does give insight into the behavior of these optimal allocation functions.

Based on the results in Section 3, we can now derive some interesting properties of {i/}jEv. These properties are formalized in Theorem 13. After presenting this theorem, we elaborate on the behavior of an optimal allocation function, also giving examples of the behavior of such a function.

Theorem 13. For every set of optimal allocation functions it holds that (i) .2j(x) E [$,I$] for x <xi’ = xkE4: x; + xkEKi4: xf: and j E 6 \ Vi, Cj E X \ Y denotes j E X and

j # Y>. (ii) 2j(X) < Xj for every x and j E E \ vi, where.

xj := argmin aDjcVj, ‘u,) = h* X l3Yj i

for j E I: \ Vi.

Proof. See Appendix A. ??

This theorem states that, if the echelon stock of stockpoint i just before rationing, x say, is less than or equal to XT, the optimal allocation policy ensures that only the echelons j with the lowest penalty cost (p:) get a small (or even negative) echelon inventory position after rationing. If x is very large, the optimal allocation policy ensures that the major part of x is allocated to the successors j with the lowest added value (0

It is interesting to note that when the cdf of the customer demand is strictly increasing for every end- stockpoint, it can be shown that Z$ = 5;. So in that case 5,(x) = $ for j @ yi and x < XT. In the more general case where there is no strict monotonicity of the customer demand cdf, it can be shown that there exists a

E. B. Diks, A.G. de Kok I European Journal of Operational Research 111 (1998) 75-97 81

set of optimal allocation functions such that ,?i(x) = $ for j $ E and x < XT. In the remainder of this paper we consider this specific set.

From Theorem 13, it is possible to distinguish between four classes of allocation functions. Fig. 3 depicts an example of an allocation function for each class. In Fig. 3(a) and (b) the amount of products allocated to successor j tends to a limit when x goes to infinity, although the allocation functions in Fig. 3(c) and (d) do not have an upperbound.

Concentrating on Fig. 3(a) and (b), suppose stockpoint i has to allocate many products, x say. Since high penalty costs are incurred for every backordered product, we would like to allocate as much as possible to successor j. However, holding costs are also incurred which assures that not too much stock is kept at the various stockpoints. It is clear that there is a trade-off between penalty and holding costs. If the amount x which needs to be allocated is large, then the allocation decisions are mainly based on the holding costs, since the penalty costs are very small. Specifically, if x goes to infinity the reduction of the penalty costs of echelon j by allocating an extra product to j equals 0, while the additional holding costs are hj. There- fore, an extra product will be allocated to those successors with minimal added value.

Also note that in Fig. 3(b) and (d) the amount of products allocated to successor j is fixed when x is sufficiently small, although, this is not true in Fig. 3(a) and (c). If x is very small (or even negative), there is an incentive to allocate the major part to the successors j which (indirectly) supply end-stockpoints with high penalty costs (j $ 5). The actual amount allocated to a successor j 6 4: is such that the marginal costs in ech@ equals the marginal costs of a successor j E &..

4.2. Optimal order-up-to-levels

From Theorem 11 it follows that the optimal order-up-to-level jj has to satisfy

hzj(x)

F St. J --__-_____--

XT X

‘j(x)

:

Kj --__-----______.

Xf X

Fig. 3. The behaviour of an allocation function for the four different classes.

88 E. B. Diks. A. G. de Kok I European Journal of‘ Operational Research I I1 (I 998) 75-97

Lx;&) = C,EU, hn +Pk hk + C,,cq hn +Pk

for every k E Ei.

Note that condition (8) resembles the classical newsboy result, which prescribes the optimal critical ratio for a single-location inventory system to be p/(p + h). Similar newsboy-type results are derived in Rogers and Tsubakitani (1991), for the problem of minimizing the penalty costs in a divergent two-echelon system, sub- jected to a budget constraint.

Diks and De Kok (in press) proved that when a specific set of linear allocation functions are used, instead of the optimal functions, then the newsboy-type formula in Eq. (8) also yields the optimal order-up- to-level. Hence, Eq. (8) covers a larger class of allocation functions. For computational results we refer to the above cited paper.

5. Optimal decomposition approach

We now apply the previous results to develop an optimization scheme for (,C,, !?I), where index 1 represents the most upstream stockpoint of the N-echelon system. We show that the problem of finding this optimal (j$ , !k,) can be decomposed into solving one-dimensional problems subsequently.

5. I. Decomposition approach

(i) n := 1. (ii) -Set Pi := 0 for every i E E.

-Determine the optimal order-up-to-level ji for every i E E. (iii) n := n + 1. (iv) Determine for every stockpoint i E Wn:

-The optimal allocation functions {~j}j~~ for every X. -ii := U,,,(i,,i;, $j) for i E W,. -The optimal order-up-to-level j$

(iv) If n < N then return to step (iv). Now we prove that the above approach yields the optimal policy.

Definition 14. Let SO := yi U Ujts (zj,,vj, u/j) be an arbitrary replenishment policy (where index 1 represents the most upstream stockpoint). Then s, is defined as the replenishment policy in which for every stockpoint i E Wn+l: 1. The order-up-to-levels and allocation functions upstream of i are identically defined as in the policy SO. 2. The order-up-to-level and the allocation functions of i are identically defined as in the policy SO and the

optimal policy i,, respectively. 3. The order-up-to-level and the allocation functions downstream of i are identically defined as in the op-

timal policy ii.

Lemma 15. Let g,,(s,,) denote the expected total costs of the stockpoints in UT=, 4 when the multi-echelon system is controlled by policy s,. For n E { 1,. . , N} holds g,(s,) < gn(sn_l).


Now we prove in Theorem 16 that the decomposition is exact (given the balance assumption).

E. B. Diks, A.G. de Kok I European Journal of Operational Research 111 (1998) 75-97 89

Theorem 16. Zf the balance assumption holds, then the decomposition approach yields the optimal replenishment policy, that is the policy which minimizes the long run average costs.

Proof. This proof is very similar to the proof in van Houtum (1990). Consider an arbitrary replenishment policy so. For both s,, and s,-1 the behavior of the stock at all stockpoints in u,“I=,+, 4 are identical. Hence, forn=l,...,Nitholds

gN(&l) -&A%) = gN(%I) -g&-1).

Using Eq. (9) and Lemma 15 yields

gAJ(%v) <giV(.N1) G “’ <gfv(so).

(9)

Because replenishment policy sN does not have costs larger than an arbitrary policy, specifically the optimal policy, sN should be optimal. 0

6. Conclusions

The objective of the analysis presented in this paper was to determine a cost optimal replenishment policy for a divergent multi-echelon inventory system under periodic review order-up-to-policies. It was proved that a decomposition approach, similar to the one proposed in Langenhoff and Zijm (1990) can be used to deal with divergent N-echelon systems, given the balance assumption. Hence the complex multi-dimensional problem of determining the cost optimal policy reduces to the problem of separately determining (1) the optimal order-up-to-level at every stockpoint and (2) the optimal allocation functions to its successors.

From our analysis, we can easily determine the optimal order-up-to-level by solving a one-dimensional problem closely resembling the classical newsboy problem. For the allocation functions we derived several properties which lead to an algorithm to determine the optimal allocation functions, and a classification scheme of optimal allocation functions.

Despite being exact, it is cumbersome and time-consuming to determine the set of optimal allocation functions following the full procedure proposed in this paper. Therefore, there is a need for a more approximate approach to determine a replenishment policy that is almost cost optimal, but easy and fast to determine in practical applications. Diks and De Kok (1998) developed such an approach using linear allocation functions.

Appendix A. Omitted proofs

A. I. Proof of Lemma 2

Consider a replenishment policy bi, Yi). Suppose that at the beginning of an arbitrary review period t the echelon inventory position of stockpoint i is raised to yi and the total demand in the periods t up to and including period t + Li - 1 equals u. Hence at the end of period t + Li - 1 (just after arrival order) the echelon stock of stockpoint i equals y, - u. Therefore at time t + Lj the expected costs incurred for echelon i equals Ci(yi - u). If at the beginning of period t + Li we have that yi - u > CnEY; y,, every stockpoint j E % raises its echelon inventory position to yj. Therefore, at time t + Li + Lj, the expected costs incurred for echelon j equals Djbj, Yj). However, if at the beginning of period t + Li we have yi - u < CnEK y,,, every stockpoint j E I$ raises its echelon inventory position to zj(Ji - ~4). Therefore, at time t + L, + Lj, the

expected costs incurred for stockpoint j equals D,j(Zjbi - u), Yj).

90 E.B. Diks, A. G. de Kok I European Journal of Operational Research 111 (1998) 75-97

By conditioning on u and using the above mentioned relations the lemma follows.

A.2. Proof of Lemma 7

Let DjcV,, Yj) be convex with respect to Yj for every j E 6. Assume for every set of optimal allocation functions {ij}jg y there exists an m E K such that &,(x)/dx < 0 for some X. Then there exists an E > 0 such that i,(x + E) C’&(X). Let us distinguish between two cases:

(1) There exists a stockpoint k E K such that

From Lemma 6 it follows that

aO,C.Yj> ‘J’j) aDjbj> Yj) aYj

> a.&

for every j E 6. J’,=?,(X+t) _“, =i, (X)

Since Di(Jj, Y,) is convex with respect to Yj we know that jj(x + E) > ij(x) for every j E F$. This contradicts our earlier made assumption.

(2) For every successor j E K holds

(A.11

Suppose there exists a stockpoint k E 6 such that

Then, applying Lemma 6 using the convexity of DjcV,, Y,), yields s/(x + E) < ii(x) for every j E K. So, nEV 2,(x + c) < CrIEK Sn(x). This contradicts Eq. (3), which states CnEK 2,(x + E) = x + E > x = ,,ei i,,(x). Hence, Eq. (A. 1) reduces to

aDjCYj> Yj) = aDicV,, yj) aYj aYj

for je 6. (A.l*) y,=?,(x+r) J; =2, (x)

Next, we construct a set of optimal allocation functions {2j}jtK:, for which i;(x + C) > i,(x) for every j E E. This contradicts our earlier made assumption that for every set of optimal allocation functions {?j}jCv there exists an m E r/; and c > 0 such that im(x + t) < im(x). Let us define the allocation functions Zj identical to .?j for every j E K, except in x f t:

Zj(X + E) := {

2j(x),

sj(X> + 4jG

j E 4, j E Bi

with

A; := G E Eliij(x + C) < Sj(x)}, B; := {j E K Iij(x + E) > i/(x)},

ij(X + (5) - ?j(X) qj := -y&, (&((x + 6) - in(X)) > O7 j E Bi.

E.B. Diks, A. G. de Kok / European Journal of Operational Research 111 (1998) 75-97 91

From ,&(x + E) < &m(x) and Eq. (3), we conclude that, both Ai and Bi are non-empty sets. We prove that Zj is an optimal allocation function in x + E. First, it can be shown that CnEK Z,,(x + E) = x + E. Next, we prove that Eq. (5) is satisfied in x + E.

(1) We have

(2) We have

Since Dj(yj, Yj) is convex with respect to yj, and ii(X) < Zj(x + t) < i/(x + E) we know from Eq. (A.l*)

aDJ.vj, ul,) = aD+& ‘Ii;) ayi y,=i,(x+c) ayj y,=i, c-4

SO, {Zj}jEq is a set of optimal allocation functions. However, i;.(x + t) > Z,(x) for all j E E contradicts our earlier made assumption.

Assume that for every stockpoint j E K the cost function Dj(X, Yj) is linear decreasing for x < x/’ and strictly convex for x > xj”. Let XT := min{x Iii(x) $ xJ* for j E x}. Now we prove that ,?j(x) is strictly increasing for x 2 XT. Assume for a stockpoint m E F holds d&,,(x)/dx < 0 for a x 2 XT. Then there exists an E > 0 for which &,(x + e) 6 5,,,(x). Since D,,,(y,,,, Y,,,) is a convex function, we obtain

Using Lemma 6 and Eq. (A.2), we obtain

aDJyj, ‘J’j) < aDd.Yji yjui) 39

\ ayj

for every j E Vi. y,=i,(x+t) I, =i, (x)

Since x 2 XT and ij(x) is non-decreasing in X, it follows ?j(x) > XT. The cost function Dj(_V,, Yj) is strictly convex in Yj = ij(X) 2 Xf . Hence, ?j(x + 6) < ij(x). SO, CnEK z,(x + E) < CnEK in(x). This contradicts Eq. (3), which states CnEC: i,,(x + 6) = x + e > x = CnEK in(x). Hence, d&(x)/dx > 0 for x > xf. 0

A.3. Proof of Lemma 8

This proof is by contradiction. Assume Dj@j, Yj) is convex with respect to yj for every j E E. Suppose there exists a stockpoint k E F$ for which &(CnEK pn,) < A. From the definition of yk and the convexity of D~(J+, Yk), we conclude

mkbk, yk)

aYk

< aDk(.Yk, Yk)

aYk = 0.

Yk =.?J

Hence, from Lemma 6, it follows that &(J&y jn) < 0. Since xjEK .?j(CnEK pn) = cjCK i;, there has to be a stockpoint, m say, for which &(CnGKFn) > ;i,. Using similar arguments as above it can be shown that I_i(C,EKiR) 2 0. This leads to a contradiction. Hence, ik(c,# jQ 2 j$. Analogous we can prove ?kk(CnEE j,,) <jk. Thus ~k(~,&n) = .+k.

92 E.B. Diks. A. G. de Kok I European Journal of Operational Research II I (1998) 75-97

A.4. Proof of Theorem 10

This proof is by induction on i. When i E E it is trivial. Suppose equality (7) holds for a divergent system with most upstream stockpoint j (induction assumption). Then the non-stockout probability of an end- stockpoint k equals O$(yj), where yj equals the order-up-to-level of this stockpoint j. Next, consider a divergent system with most upstream stockpoint i, and j E E. In order to determine the non-stockout probability of stockpoint k in this system we introduce some additional notation. Let CI; , denote the non-stockout probability of this stockpoint k as a result of the rationing decision at the beginning of period t, and Dr_L,,r equals the total demand at all end-stockpoints during [t - L;, t).

When at the beginning of an arbitrary period t the echelon stock of stockpoint i is less than the sum of all the order-up-to-levels of its successors, rationing is necessary. This means that every successor j E E gets its appropriate share zj(n - Dt-~,,t) instead of order-up-to-level yj. From the induction assumption, we obtain that for j E I$ and k E Ej

However, if at time t the echelon stock at stockpoint i is sufficient, all its successors raise their echelon inventory positions to their order-up-to-levels. From the induction assumption, we obtain that for j E I$ and k E Ej

Yi - Dt-~,,t > CYn * @&,(.I+) = d,bj). (A.4) Kc:

When the demand process is stationary we may suppress the index t in a6,,(yi). From Eqs. (A.3) and (A.4) it can be verified that for j E K and k E E,,:

‘XI

A, < 0 =+ &(,vJ = J’

$k(ZjcV, - U)) d&p)(U), (A.5) 0

A,

Al > 0 * $(yi) = J’

Qyj) dF;‘L’)(u) + 7 d(zj@i - u)) dF;‘L”(U). (A.6) 0 A,

Rewriting Eq. (A.6), using the assumption that zj(Cn,=Kyn) = y,, yields that for j E 6 and k E Ej

(A.7) 0

From Eqs. (A.5) and (A.7) it follows that equality Eq. (A.7) also holds for i. 0


This proof is by induction on i. First, consider a stockpoint i E E, and define xr := 0. Substituting the definition of C,(x) in Eq. (4), and differentiating with respect to x, we obtain

WV

E.B. Diks. A. G. de Kok I European Journal of Operational Research Ill (1998) 75-97 93

Recall that $‘+‘) (x) = 0 for x < 0. This proves (i). Result (ii) immediately follows from

d2DicV,) _= (hr+~h.ipr)i;(i~+~lhi) 20.

Finally, we note that when x goes to infinity, F;(L’+‘) (x) increases to 1, which proves (iii). Next, consider a stockpoint i E M, and define XT := min {X 1 ii(x) 2 x/” for every j E K}. Assume the the-

orem holds for every successor j E K (induction assumption). First we prove (i). Let x < XT. Then, there exists a stockpoint m E c for which i,,,(x) <XL. Using the induction assumption and Lemma 6, we obtain that iuI(x) = -(C,tr_,, h, +&). From the definition of ii(x) we have

(A.9)

Substitution of n,(x) in the formulas of Theorem 9 yield

Next, we prove pi = p:. Suppose there exists a j E I$ for which p: > pi’. Using the fact that Dj(Yj, ii/j) is convex with respect to yj and the induction assumption, we have

But, since pk > p; and U, = qi, we also know

This contradicts Eq. (A.9). Thus, a necessary condition for m is

pz <?T for every j E I$.

Hence, pi = pf , which provesA (i). Next, we prove that Die/1, Yi) is convex with respect toy;. From the induction assumption we have that

Difjvi, Yi) is convex with respect to yj for every j E I$. Then, from Lemma 8, we know that .2j(CnEK ~2,,) = i;. Using this result, after substitution of x = Cntc: Gn in Eq. (6), yields iI(C,,K A) = 0. This property and Eq. (6) yields

d*Di b+ > *i) ‘CA a? =

&j(X - U) $DjcV,, @j) d#‘)(u) for every j E V. I I

v,=l,(x-u)

From Lemma 7 (using the induction assumption), it immediately follows that a*Di(Yi, !Pi)/aYf 2 0. Specif- ically, let x 2 x:. It can be shown that CnEI/; jjn 2 XT. Thus, there exists u 3 0 such that x-&&,2&x-Xxf. Then, from Lemma 7 and the induction assumption, it follows that Di(Y,, pi) is strictly convex for yi > XT.

Finally, we prove that when y, goes to infinity aDiij[, 'Pi)/aYi tends to hi. From the induction assumption and Lemma 6, it follows

94 E.B. Diks, A. G. de Kok I European Journal of Operational Research 111 (1998) 75-97

- (hi+zhn+pT) <h(x)<hr.

From Eqs. (6) and (A.lO) it follows

(A.lO)

When Yi tends to infinity both the lower and upper bound converge to h,. 0


(i) Suppose a stockpoint i has to ration its echelon stock x < XT over its successors. Consider a successor, m say, for which p; > p:. So, m E K \ vi. We prove (i) by showing that 5,,,(x) > & and i,,,(x) 6 xi, respectively. For both cases the proofs are given by contradiction.

I (1) Assume &,(x) < x_~. From the convexity of Dm(y,,,, i,) in y,, and the definition of & we obtain

%,(Y,,,, %z) 8Yt?? I+n=%&)

< RDm~~~B..‘~~~=i=-(~h.+d).

From Lemma 6, we obtain

Ai(x) <-(hi+zb+P;). (A.ll)

Next, we consider a stockpoint, j say, for which pi* = p,*. So, j E vi. From the convexity of DjcV,, @j) and Theorem 12, it follows

aDjdv,l ‘%.) ayj J 2 - (zh.+P;) =-(z,hn+P;).

Again using Lemma 6, we obtain

Ai(x) 2 - (4-%h.+pT)

Eqs. (A. 11) and (A. 12) lead to a contradiction. Hence 5,,,(x) 3 &. (2) Assume 5,,,(x) > x_;. Similar to Eq. (A.1 1) it can be shown that

After substitution of Eq. (5) in Eq. (A.13) we have that for every stockpoint j E K \ vi

(A.12)

(A.13)

E.B. Diks. A.G. de Kok I European Journal of Operational Research Ill (1998) 75-97 95

Since oi(.Yj, *j) is convex with respect to Yj, we know that ij(x) > ~7. Using this and equality Eq. (3, we obtain

So, there exists a stockpoint j E vi for which ij(x) < xi. From Theorem 12 (i) we know that for this stockpoint j

Again using Lemma 6, we obtain

(A.14)

Eqs. (A. 13) and (A. 14) lead to a contradiction. Hence .5*(x) 6 x_i. So srn (x) E [g,,, &I. (ii) Suppose some stockpoint i has to ration its echelon stock x (with x <XT) over its successors. Consider

such a successor, m say, for which h, > h:. So, m E K \ Vi. Assume that im(x) > Y,. Since D,,, (y,, Y,) is convex with respect to y,,,, we have

a&cVm, @,) 8Y*

~ a&(,~,,,, 9,) 8Ym -

= h;.

y!n=ffn(x) .vl?? =x,

So, from Lemma 6, it follows that n,(x) > hT. From Theorem 12, it is clear that for a stockpoint j E Vi holds 8Djbj, li/i)/& < hj = h:. Then, Lemma 6 leads to a contradiction, since we obtain ii(x) < hi’. This proves z,(x) < X,. 0

A.7. Proof of Lemma 1.5

Suppose SO is an arbitrary replenishment policy. Let us consider a stockpoint i E F&+1. For our convenience, we introduce some additional notation. Let gi(s,) denote the expected total costs in Ujcs ech(j), due to the replenishment orders placed by policy s, at the beginning of period t. The allocation function from stockpoint i to successor j when applying policy s, and s,_t is denoted by ij and zj, respectively. The order- up-to-level of j when applying policy s, and s,_r is denoted by i; and yj. We distinguish two cases for CjEf(Yj < CjEKi; and Cjt:t(Yi > CjEKi;’

??CYj < x.9;. iEK /EC:

If at the beginning of period t the echelon stock of stockpoint i, Zi say, is sufficient to raise the echelon inventory positions of all successors j E K to their order-up-to-level, the expected costs of echelon j equals Djci;., ‘?j) and Djtjj, !kj) for replenishment policy s, and s,_t, respectively. Hence, from the fact that jj minimizes Di(yj, ii/i) we obtain

1: 3 Cj; + gf(Sn) = C Djci;, ‘kj) < C Djbj, !kj) = g;(s,_l). iEK iEK iEK

However, if at the beginning of period t the echelon stock of stockpoint i is not sufficient, all this echelon stock is allocated to the successors j E E

96 E. B. Diks, A. G. de Kok / European Journal of Operational Research II I (1998) 75-97

Finally, we analyze the situation where cjCr: yj < Zi < xjEK i;. Then policy s, rations all the echelon stock I:’ over the stockpoints j E K;, while policy s,_r raises the echelon inventory positions of stockpoints j E Z$ to their order-up-to-levels yj, and the remainder (Ii - CjEC:yj) is retained at stockpoint i.

The allocation function Zj, defined by

Zj(X) :=yj - qj with qj := _&~(~~y~ yk)

has the property that yj < 5j(Zj) < jj if yj <i;, and i; < sj(Zi) < y, if yj > jj. From this property and the fact that DjcV,, ‘Pj) is convex with respect to yj (with the minimum in yj = j;) it follows

(A.15)

Since ,2j is at least as good as Zj, and Eq. (A. 15) it follows that

Analogously, we can prove that:

Note that when CjGc:yj = cj,,jj, the convexity of Dj(yj, @j) with respect to yj (with the minimum in yj = jZ> immediately results in gi(s,) < gi(s,_t ). So, for an arbitrary period t and an arbitrary stockpoint i E Wn+l it holds that g;(s,) <gf(s,_t). Cl

References

Clark, A.J., Scarf, H., 1960. Optimal policies for a multi-echelon inventory problem. Management Science 6, 475490. Diks, E.B., de Kok, A.G., Lagodimos, A.G., 1996. Multi-echelon systems: A service measure perspective. European Journal of

Operational Research 95, 241-263. Diks, E.B., de Kok, A.G., 1998. Computational results for the control of a divergent N-echelon inventory system. International

Journal of Production Economics (in press). Eppen, G., Schrage, L., 1981. Centralized ordering policies in a multi-warehouse system with lead times and random demand. In: L.B.

Schwarz (Ed.), Multi-level Production/Inventory Control Systems: Theory and Practice. North-Holland, Amsterdam, pp. 51-67. Federgruen, A., Zipkin, P., 1984. Approximations of dynamic multilocation production and inventory problems. Management Science

30. 69-84.

E. B. Diks, A. G. de Kok I European Journal of Operational Research I1 1 (I 998) 75-97 91

Federgruen, A., 1993. Centralized planning models for multi-echelon inventory systems under uncertainty. In: Graves, SC., Rinnooy Kan, A.H.G., Zipkin, P.H. (Eds.), Logistics of production and inventory. Handbooks in Operations Research and Management Science, vol. 4, ch. 3. Elsevier, Amsterdam, pp. 1333173.

Gong, L., de Kok, A.G., Ding, J., 1994. Optimal leadtimes planning in a serial production system. Management Science 40, 629-632. de Kok, A.G., 1990. Hierarchical production planning for consumer goods. European Journal of Operational Research 45, 55-69. de Kok, A.G., Lagodimos, A.G., Seidel, H.P., 1994. Stock allocation in a two-echelon distribution network under service-constraints.

Department of Industrial Engineering and Management Science, Eindhoven University of Technology, EUT 94-03. Lagodimos, A.G., 1992. Multi-echelon service models for inventory systems under different rationing policies. International Journal of

Production Research 30, 939-958. Langenhoff, L.J.G., Zijm, W.H.M., 1990. An analytical theory of multi-echelon production/distribution systems. Statistica

Neerlandica 44, 149-174. Rogers, D.F., Tsubakitani, S., 1991. Newsboy-style results for multi-echelon inventory problems: Backorders optimization with

intermediate delays. Journal of Operational Research Society 42, 57-68. Rosling, K., 1989. Optimal inventory policies for assembly systems under random demands. Operations Research 37, 565-579. Seidel, H.P., de Kok, A.G., 1990. Analysis of stock allocation in a 2-echelon distribution system. Technical Report 098, CQM

Eindhoven. van Donselaar, K., Wijngaard, J., 1987. Commonality and safety stocks. Engineering Costs and Production Economics 12, 197-204. van Houtum, G.J., 1990. Theoretische en numerieke analyse van multi-echelon voorraadsystemen met lijn- of assemblagestruktuur.

Master’s Thesis, Department of Mathematics and Computing Science, Eindhoven University of Technology. van Houtum, J., Zijm, W.H.M., 1991. Theoretische en Numerieke Analyse van Multi-Echelon Voorraadsystemen met

distributiestruktuur. Technical report, Eindhoven University of Technology (in Dutch). van Houtum, J., Inderfurth, K., Zijm, W.H.M., 1996. Materials coordination in stochastic multi-echelon systems. European Journal of

Operational Research 95, l-23. Verrijdt, J.H.C.M., de Kok, A.G., 1995. Distribution planning for a divergent N-echelon network without intermediate stocks under

service restrictions. International Journal of Production Economics 38, 225-243.

optimal control of a divergent multi-echelon inventory system

Documents