decentralized multiechelon inventory control

PRODUCTION AND OPERATIONS MANAGEMENT Vol. 7, No. 4, Winter 1998

Printed in U.S.A.

DECENTRALIZED MULTIECHELON INVENTORY CONTROL *

JONAS ANDERSSON, SVEN AXSATER, AND JOHAN MARKLUND

Department of Industrial Engineering, Lund University, Lund, Sweden

This paper considers a model for decentralized control of an inventory system consisting of 1 central warehouse and a number of retailers. The cost structure includes holding costs at both echelons and shortage costs proportional to the time until delivery at the retailers. We analyze a procedure for coordinated but still decentralized control of the system. The procedure is based on a simple approximation, in which the stochastic lead times perceived by the retailers are replaced by their correct averages. The approximation enables us to decompose the considered multiechelon inventory problem into a number of single echelon problems, 1 for each installation. The information about how a certain decision at the warehouse affects the retailers is conveyed through the marginal cost increase with respect to a change of the expected lead time. This information about the retailer costs is used as a shortage cost at the warehouse. We show that a coordination procedure based on this information can be used for finding near-optimal reorder points for the system and provide bounds for the approximation errors. (INVENTORY /PRODUCTION; MULTIECHELON; STOCHASTIC; DECENTRALIZATION)

1. Introduction

One of the main difficulties in supply chain management is to achieve a highly decentralized yet efficient control of the different levels in the chain. What is required is an efficient method for coordinating the decisions at different installations using a limited amount of information. Our main purpose is therefore to develop a technique for coordination of highly autonomously governed installations. In this paper, we address this problem within the setting of a multiechelon inventory system consisting of 1 central warehouse and a number of retailers. More precisely, we investigate a decentralized control procedure, which enables us to decompose the multilevel inventory problem of finding near-optimal reorder points for all installations, into a number of coordinated single-level inventory problems.

The decomposition method is based on an approximate cost evaluation technique and a modified cost structure at the warehouse. The approximation we make is to replace the stochastic lead times perceived by the retailers by their correct averages. Subsequently, it is closely related to the well-known METRIC-~~~~~ (see Sherbrooke 1968). In a larger context, this means that we construct a model based on a limited amount of information. The main idea behind the modified cost structure is to charge the warehouse a stockout

* Received March 1996; revised August 1997 and March 1998; accepted April 1998.

370 1059-1478/98/0704/370$1.25

Copyright 0 1998, Production and Operations Management Society

DECENTRALIZED MULTIECHELON INVENTORY CONTROL 371

penalty cost for late deliveries to the retailers. This new cost structure, together with the lead time approximation, enable us to construct a decentralized control procedure, which can be used to find near-optimal reorder points for the installations in the system.

Cost evaluation and optimization of multiechelon inventory systems, under various control policies, have been studied extensively in the literature. See Axstiter ( 1993) and Federgruen (1993) for recent overviews. Existing models usually adopt a centralized approach in the sense that the optimal or near optimal values of the decision variables at each site are obtained by solving one large and complex problem. This approach is less appropriate to capture situations, in which the control decisions are highly decentralized. Other models, which, like ours, focus on coordination of highly autonomously governed installations, can be found in recent work by Lee and Whang ( 1994) and Axsater ( 1995). These papers both focus on decentralized control of an inventory system with stochastic demand. A similarity between these models and our model is that the coordination between different facilities is achieved by designing new cost structures; otherwise, the approaches are quite different. Lee and Whang consider a serial system, and their model is based on the well-known work by Clark and Scarf ( 1960). Axsater, on the other hand, focuses on a general framework for improving the coordination of decisions in a multilevel inventory system. There are also some more remotely related papers dealing with decentralized inventory control. Lee and Billington ( 1993) facilitate ,decentralized control of an inventory system by using service-level constraints for upstream installations. Muckstadt and Thomas ( 1980), followed up by Hausman and Erkip ( 1994), study a system with low- demand items. They compare decentralized control through single-echelon models with service level constraints to the performance of a multiechelon model. There are also a number of papers dealing with coordination mechanisms for deterministic systems. See, for example, Lee and Whang ( 1994) for a list of references.

The outlay of the paper is as follows. In Section 2, we define the considered system consisting of one warehouse and N

retailers. We also discuss some additional assumptions. Further, we define the exact and approximate model of the system.

In Section 3, we introduce the new cost structure, which enables us to decompose the approximate model into N + 1 coordinated single-level-models. We also present a simple iterative procedure that can be used to find a Nash equilibrium for the decomposed problem. For the special case of normally distributed demand, we show that this procedure can be used to find the optimal solution to the approximate model. Finally, we discuss two practical situations, in which our model can be especially useful.

Section 4 contains an analysis of the relationship between the costs of the optimal control policies in the approximate and the exact models. The main result is an upper bound for the performance ratio of the approximate model.

The solution procedure is numerically tested in Section 5 for the special case of identical retailers. Section 6 summarizes our conclusions. The more lengthy proofs are given in the Appendix at the end of the paper.

2. Problem Formulation

2.1. The Considered Inventory System

We consider an inventory system with one central warehouse and N retailers (Figure 1) . The customer demand takes place at the retailers, which replenish their stocks from the warehouse. The transportation times for such deliveries are constant, but additional delays may occur due to stockouts at the warehouse. The warehouse replenishes its stock from an outside supplier. The lead times for these deliveries are constant. Stockouts at both echelons are back-ordered and delivered on a first-come, first-served basis. All facilities apply continuous review installation stock (R, Q)-policies. This means that an

372 JONAS ANDERSSON, SVEN AXS;iTER, AND JOHAN MARKLUND

retailers

FIGURE 1. A Divergent Multiechelon Inventory System.

order of size Q is triggered when the inventory position (inventory on hand plus outstanding orders minus back orders) at the installation in question reaches the reorder point R. Such policies are common in practice. The cost evaluation includes inventory holding costs at both echelons and shortage costs proportional to the time until delivery at the retailers.

For the mathematical description, we need to introduce the following notation.

N = number of retailers, Q = common batch size for all retailers,

QO = warehouse batch size, in units of Q, hi = holding cost per unit and time unit at retailer i, h,, = holding cost per unit and time unit at the warehouse, pi = shortage cost per unit and time unit at retailer i, or, equivalently, according to

Little’s formula, per average number of back orders, & = constant lead time for an order to arrive at the warehouse, li = transportation time between the warehouse and retailer i,

Z+ = lead time for an order to arrive at retailer i, stochastic variable, Li = expected lead time for an order to arrive at retailer i,

Di (t) = customer demand at retailer i during the time period t, stochastic variable, pi = expected demand per time unit at retailer i,

D,(t) = retailer demand at the warehouse during the time period t, stochastic variable, Ri = reorder point for retailer i, R. = warehouse reorder point, in units of Q,

RF* = optimal reorder point for retailer i in the exact model, RE * = (Rf * , Rf * , . . . , RE* ) , RF* (R,) = optimal reorder point for a given Ro,

Rf * = optimal reorder point for retailer i in the approximate model, RA * = (Rf *, RA” . . . ) R$*), Rf * (R,) = optimal reorder point for a given Ro,

Rt * = o$imal warehouse reorder point in the exact model, in units of Q, Rt * = .optimal warehouse reorder point in the approximate model, in units of Q, Cf = expected cost per time unit at retailer i, Cf = expected cost per time unit at retailer i in the approximate model, CO = expected warehouse cost per time unit,

TCE = expected total system cost per time unit, TC, = expected total system cost per time unit in the approximate model,

r = TC,(R$*, R”*)/TC,(R f* RE*) = performance ratio. ,

We also make the following additional assumptions about the system in question.


1. We assume that all batch sizes are given and that all retailers use a common batch size Q.

2. The initial inventory position, the reorder point, and the batch size at the warehouse are assumed to be integer multiples of Q.

3. The cumulative demand faced by a retailer can be modeled by a nondecreasing stochastic process with stationary and mutually independent increments. It is well-known that such a process may always be represented as a limit of an appropriate sequence of compound Poisson processes. See, for example, Feller ( 1966). The cumulative demand has, at least approximately, a continuous distribution. See Browne and Zipkin ( 1991) for a detailed discussion of these assumptions.

4. The warehouse reorder point must satisfy, R. 2 - 1. Recall that R,, is expressed in units of Q. Since the inventory position at the warehouse is always at least R. + 1, it can then never become negative.

Assumptions 1 through 4 above, will have some important consequences, which need to be emphasized.

Assumption 1 means that the objective is to optimize the reorder points. The iterative coordination procedure that is suggested in Section 3 could, in principle, also include adjustments of the batch sizes. But since we have no convergence results for that case, we limit our attention to the reorder points. Note, however, that this is, in general, not a major restriction. Indications are that using deterministic lot sizing methods in a stochastic environment will only have a small effect on the expected cost, provided that the reorder points are adjusted accordingly. See, for example, Zheng (1992) and Axslter (1996). A common heuristic approach in connection with multiechelon inventory control is therefore to first use a deterministic model to’ determine the order quantities and then apply a stochastic model to determine the corresponding near- optimal reorder points. See, for example, Chen and Zheng (1997). Our procedure carries out the second step in a decentralized way. Further, physical restrictions, such as container or pallet sizes, often leave little room for varying the order quantities. The simplifying assumption that all retailers use a common batch size Q is of course restricting from a practical point of view. However, the main purpose with this paper is to introduce a basic mathematical model designed to handle a decentralized inventory. system. We therefore strive to avoid clouding the understanding of our model with more complicating features than necessary. By using a common batch size Q, we avoid dealing with rationing policies or partial deliveries at the warehouse. More precisely Assumptions 1 and 2 imply that the stock on hand at the warehouse is always a multiple of Q. Consequently, we do not need to consider situations in which the stock on hand at the warehouse partly covers a retailer order. Elaborate generalizations are left for future research.

Assumption 3 means that the continuous distribution of the inventory position at retailer i in steady state is uniform on [ Ri , Ri + Q] (see Zheng 1992). It is also possible to show that Assumptions 2 and 3 imply that the warehouse inventory position in steady state is uniform on the integers [R. + 1, R. + QO] (see, for example; Axsater 1998). Recall that Q0 as well as R. is expressed in units of Q.

Due to the first-come, first-served policy at the warehouse, retailer orders at the warehouse can never cross in time. From the retailer’s point of view, the orders are always effected sequentially. Together with Assumption 4, this means that the stochastic variable Li is limited to the interval [ li , Ei + Lo].

Assumption 4 also means that the lead time for a retailer order placed at time t is independent of the demand and the retailer orders occurring after time t. This means that we can treat the lead time Li as if it was an exogenously generated stochastic variable that is independent of the retailer demand. In the case when R,, < -1, the lead time for a retailer order will be dependent of retailer demand after the ordering epoch.

374 JONAS ANDERSSON, SVEN AXSiiTER, AND JOHAN MARKLUND

2.2. Exact and Approximate Problem

The total system cost can be divided into 2 parts, describing the cost at the warehouse and retailer levels, respectively.

TC, = CO(RO) + 5 C1”<Ri I &(&)I (1) i=l

When evaluating the cost in the approximate model, the stochastic lead time Li for re- plenishments to retailer i is replaced by its expected value L . The total approximate system cost can then be expressed as

TC, = CO (R,) + 5 C: (Ri &(R,)) (2) i=l

The approximation in itself is not relying on the specific Assumptions 1 through 4 made in the previous section. However, the assumptions are critical when evaluating the quality of the approximation. Also, note that the expected warehouse holding cost C, is exact in ( 1) as well as in (2). The warehouse holding cost is, of course, a function of the single decision variable RO.

We now turn to the problem of finding explicit expressions for the components in the cost expressions ( 1) and (2). In steady state, the warehouse inventory position y (in units of Q) is uniform over the integers [R, + 1, RO + QO] at some arbitrary time 7. The inventory level at time 7 + Lo is then yQ - DO(LO) since the outstanding orders at time r are on hand at time T + Lo. We obtain the following expression for Co(Ro):

Co(Ro) = ; ;i? ED,,L,,,[(YQ - &(Ld,))+l oy RO+l

where (x)+ = max (x, 0).

(3)

The exact expected retailer cost for retailer i , Ck ( Ri 1 Li (R,) , as well as the approximate expected retailer cost, C: (Ri 1 Li (R,), are functions of both R. and Ri . The warehouse reorder point R. affects the retailer cost because it affects the retailer lead time distribution. However, note that the cost at an arbitrary retailer is not affected by the reorder points at other retailers since changes in other reorder points will not affect the demand process at the warehouse.

In the approximate retailer cost evaluation, we have a constant lead time equal to L (R,) . The cost can then in analogy with (3) be obtained as follows (also see Zheng 1992):

Cf(Ri ILi(Ro)) = i lRT+Q Eo,c~,,[hi(Y-Di(~i))++pi(Di(Z;i)-Y)+ldY (4)

An important observation is that in the case of nonidentical retailers, even though they use the same order quantity Q, the expected lead times Zi (R,) are, in general, different for different retailers. The reason for this is that the ordering frequency of a retailer depends on the customer demand process as well as on Q. In order not to obscure our theoretical reasoning even further, we will not deal with this problem here. Our main purpose with this paper is, as mentioned before, to introduce a basic theoretical model suitable for a system with decentralized control. Extensions that will enhance the flexibility of the model are left for future research. In Section 5, we will therefore restrict our numerical study to the special case of identical retailers. This means that all the retailers have identical expected lead times that can easily be obtained from Little’s formula. In case of nonidentical retailers, a reasonable approximation, especially when there are many retailers, would be to ignore the difference in lead times and set them all equal to the


average expected lead time. An exact determination of different values of Ei (R,) is relatively complex.

Observe that Cf (Ri 1 Ei (R,) Q/pi is the expected cost per batch with a constant lead time. In the exact case when the lead time Li (R,) is stochastic, the batches are ordered at the same time epochs and meet the same customer demands as in the deterministic case. Now, since the lead time for an ordered batch is independent of the lead time demand and can be regarded as exogenously generated, we obtain the exact cost per batch simply by taking the expected value of the cost over all possible lead times, as defined by the lead time distribution, as follows:

ck tRi I Li (RI))

EDj(Li)[hi(Y -Di(Li))+ +Pi(Di(Li)-Y)+ILildY I

(5)

3. Coordination Procedure

In.the previous section we supplied exact and approximate cost evaluation formulas for the inventory system under consideration. In this section, we will introduce a new cost structure, which enables us to decompose the approximate problem into N + 1 coordinated single-level inventory problems. We also present an iterative procedure, which, under certain conditions, will find a stationary solution, or equivalently a Nash equilibrium, for the decomposed problem.

We distinguish between the following two cases: the general case and the case with normally distributed demand. In the general case, we consider an inventory system in the setting described in Section 2. For this case, we are unable to give any useful convergence properties for the suggested procedure. For the special case when the lead time demand is normally distributed, the results are stronger. We can prove that by using the coordination procedure, we will find the optimal solution to the approximate problem.

In the last part of this section, we will consider two interesting interpretations of our model with practical implications. First of all, the coordination procedure can be seen as a negotiation process, which ends in a stationary solution, in which all parties are satisfied, that is, a Nash equilibrium. Secondly, we can view the procedure as a means to obtain a shortage cost for the warehouse, which, when it is used, will lead to a fair and efficient cost allocation between the warehouse and the retailers.

3.1. General Case

Our basic idea is to split the total system into the following 2 levels: the warehouse level and the retailer level. This corresponds to a decomposition of the approximate problem into N + 1 subproblems, one for each installation in the original system. Consequently, there will be N retailer problems, which will all be of a standard single-level type, that is, in our notation, MRm Cf (Ri 1 zi (R,)) .

For the warehouse problem, we need a new cost structure, which takes into account the impact that the warehouse service level has on the retailer costs. We do this by introducing a shortage penalty cost (pi) per unit and time unit for delayed retailer orders. If we define pi in this way, it will be directly comparable to the shortage cost pi at retailer i. The decomposed warehouse problem can then be denoted Min C,,(R,) = Min ( Co(&) + c7Pi/4(L(RCl) - li>l. 47 RO

The penalty costs pi, i = 1, . . . , N, are determined through the retailer problems and are set equal to the expected marginal retailer cost per time unit with respect to a change in the lead time Ei (R,). The optimal cost for a given lead time & will be denoted Cf (Zi ). Note that the decision variable Rj is excluded from this expression since we assume that Ri is implicitly chosen to its optimal value, RT (Li), for each value of L7:.

376 JONAS ANDERSSON, SVEN AXSATER, AND JOHAN MARKLUND

We now obtain the penalty costs as pi = (dC4 (Zi (R,))IdLi (R,)). l/pi. The factor l/ pi is included because pi is defined as the marginal cost per unit and time unit.

Now we turn to the coordination procedure. In order to find candidate solutions to the decomposed approximate problem, we can use a simple procedure in which we deal with the warehouse problem and the N retailer problems in a sequential iterative manner.

First, set each of the N replenishment lead times L to an initial value between Zi and L,, + Zi. Given these lead times 6, we can solve the N retailer problems independently of each other. When we know the optimal solutions to the retailer problems we can evaluate pi for each retailer. Using these shortage penalty costs, we can solve the warehouse problem and find an optimal value of RO . This R,, determines new expected lead times L (R,,) , which enable us to reoptimize the N retailer problems and to find new values of pi, i = 1, . . . , N. If the procedure converges, we obtain a Nash equilibrium for the decomposed problem; that is, neither the warehouse nor the retailers want to change their policies from the current solution. In the general case, we can, unfortunately, not guarantee the existence of a Nash equilibrium for the decomposed problem. If there is no Nash equilibrium, the procedure can obviously not converge. In case of normally distributed demand (Section 3.2), we are able to show that Cf (Li ) is concave, which is a sufficient condition for the existence of a Nash equilibrium for the coordination procedure. It is then easy to obtain the optimal solution to the approximate problem. Note that neither the decomposition method nor the iterative procedure are at this stage relying on Assumption 4 in Section 2.

3.2. Normally Distributed Demand

As indicated above, a special case, in which we can show that the optimal solution to the approximate problem can be found by solving the decomposed approximate problem, is when the cumulative lead time demand Di ( t) is normally distributed with mean pit and standard deviation gi t I”. This is a convenient observation since a demand processes with the independence properties described in Section 2 will render cumulative lead time demand, which is approximately normally distributed when the lead time is long enough. Through the Berry-Esseen theorem, we can further conclude that the lead time demand distribution converges in distribution towards the normal distribution, at least at a rate proportional to the inverse of z,!‘* (see, for example, Petrov 1995). Note also that even if the approximation errors are not completely negligible, it is anyway, in general, most practical to use the normal distribution as an approximation. In that case, the coordination procedure still works in the same way.

We now turn to the evaluation of retailer costs in the case of normally distributed demand. It is well known (see, for example, Silver and Peterson 1985) that we, in the case of normally distributed demand, can express the retailer cost (4) as

Cf(Ri IIS;( = hi

+ Chi +Pi) d$[~(“-?) -H(“~ -o;$;2+Q)] (6)

where H(u) = (v* + l)(l - @J(U)) - vcp(u), a,(.) is the standard normal distribution function; and cp( .) is the normal density function.

If we optimize Cf (Ri 1 zi (R,)) with respect to Ri , the first-order optimality condition is

aCf(RiIG(&)) dRi

= hi - (hi +pi)

where


G(u) = s

m (x - u)p(x)dx = cp(u) - lJ(1 - @p(u)) = - $3’(v) (8) v

By using (7), it is possible to show that

Proposition 1 gives the following important result regarding the cost Cf (Li ) .

PROPOSITION 1. Given that pi 2 hi, Cb (Zi ) is a concave function.

A proof of Proposition 1 can be found in Appendix A. The condition pi Z- hi is not very restricting and should be satisfied in most real world situations.

We are now ready to analyze the behavior of the decomposed approximate problem and to show that its optimal solution can be found in the set of stationary solutions. Furthermore, we will show that the optimal solution to the decomposed problem equals the optimal solution to the approximate problem.

Let (Ri, R”) be a stationary solution or, equivalently, a Nash equilibrium for the decomposed problem. For each retailer, Rf is the solution to the corresponding retailer problem, given the constant lead time Zi (RG) . At the same time, Ri is the solution to the warehouse problem with the shortage costs pi = (dC7 ( zi (RG))/dLi (Rh)) .l /pi per unit and time unit for late deliveries to retailer i.

The optimization of the warehouse cost under the new cost structure can be viewed as an optimization of the total approximate problem, in which the retailer costs Cf (Zi (R,) are replaced by approximations, which are linear in the average retailer lead time. The optimal RO in this problem will, of course, only depend on the constant derivatives of these linear functions; that is, the constant parts of these retailer costs are irrelevant for the optimization. Consider the retailer cost, as follows: - - - cf (h (&)I = Cf (L (R8)) + dCa(Li(R’)) (Li(Ro) - L,(R&)) 2 Cf(Zi(Ro)) dzi (RG) (10)

The inequality in ( 10) follows because C:’ (Zi ) is concave. Since the linear function I?: IZi (R,)] has the prescribed slope and Cf (Ii (Ri)) is a constant, choosing RO to minimize holding costs and expected shortage costs at the warehouse, is equivalent to minimizing TcA = C,(R,) + IZy C;” (Zi (R,)) . Consequently, the warehouse is minimizing a cost function that gives an upper bound for the correct approximate cost, TC,. Evidently, in a stationary point, where RO = R; , the cost TcA is equal to the approximate cost TC, . This follows directly from ( 10). With the discussion above in mind, we can show the following result.

PROPOSITION 2. The optimal solution to the original approximate problem (Rt * , R* * ) must be a Nash equilibrium for the decomposed problem.

ProoJ Rt* is by definition the optimal solution to the approximate problem, min TCA(Ro). We can then state the following relation: min TCA( Ro) 2 min TCA(R,) = TCA(Rt*) = TCA(Rt *). The last equality follows from ( 10) when Rf, = R;f *. The conclusion must be that if we use the reference point Rt *, that is R; = Rt*, and minimize TCA (R,) , the optimal solution will be Rt * , that is, when reached we remain in the point Rt * . Conseque n 1 t y , the optimal approximate solution must also be a stationary solution or equivalently a Nash equilibrium to the decomposed problem. Q.E.D.

Now consider the coordination procedure outlined in the previous section. By using the result in Proposition 1, we can device a method for finding all the stationary solutions to the decomposed problem.

Start by setting the initial lead times zp = Zi . Now solve the N retailer problems given these initial lead times. Since li is the lowest possible lead time and Cf (Zi ) is concave,


the corresponding shortage costs at the warehouse are as large as possible. The warehouse now solves the warehouse problem with these shortage costs, that is, with pi = (dCf (zy )Idzy ). 1 Ipi per unit and time unit for late deliveries to retailer i. The resulting RA is evidently an upper bound for the value of RO in all Nash equilibriums. This implies that zi’ 2 Ep and that the shortage penalty costs will decrease. Naturally, this implies that R% 5 Rh. Note, however, that the shortage costs are still upper bounds for a Nash equilibrium since RA is an upper bound. This means that Ri is also an upper bound. Continuing in the same way, it is evident that zf is increasing with k and that Rg is decreasing. Since Lf 5 Zi + L+,, it is clear that the procedure will converge to a Nash equilibrium after a finite number of steps (recall that RO is discrete). Let us denote the resulting RO by R;. We know that R[ is an upper bound for the warehouse reorder point in all Nash equilibriums and, hence, for the optimal solution to the approximate problem Rt * . Obviously, the corresponding retailer reorder points will constitute lower bounds for the values of Ri in all Nash equilibriums.

If we instead initiate the procedure with the maximum possible lead time EQ = Zi + L,,, a corresponding iterative procedure will lead to a lower bound Rb to Rt *. Our findings can be summarized in the following proposition.

PROPOSITION 3. The suggested coordination procedure gives upper and lower bounds for the optimal reorder points in the approximate problem.

If Rb = Rg, we have found the optimal solution of the approximate problem. If this is not the case, we need to check the finite number of values of RO in the interval RA 5 RO 5 Rf;.

3.3. Suggested Applications

So far, the coordination procedure has been analyzed from a pure mathematical point of view. In this section, we will change perspective and present some ideas of how our results can be applied in practice.

Consider a situation in which a warehouse and N retailers want to coordinate their actions in order to improve the total system efficiency without compromising the decentralized decision-making process. This can be achieved by introducing a shortage penalty cost, pi, in much the same way as in Sections 3.1. and 3.2. The difference is that the warehouse will now actually reimburse the retailers for late deliveries in accordance with the penalty cost. This means that the cost perceived by the warehouse can be expressed as l$(Ro) = CO(RO) + Ey Pi~i(Li(Ro) - li) and the actual cost at retailer i will be ef(Zli(Ro)) = Cf(Li(R,)) - Pipi(Zi(Ro) - Zi).

We can now translate the iterative procedure in the previous section into a negotiation process in which retailer i conveys its shortage cost, pi, for the current value of Ei to the warehouse. Assume that the system at hand is coherent with the one described in Section 3.2. We then know that the negotiation process will lead to a Nash equilibrium, in which all parties are satisfied in the sense that a renegotiation will not change the situation. This corresponds to a stationary solution for the decomposed problem.

When the negotiation process starts, a natural assumption for a retailer to make is that the replenishment lead time is equal to the transportation time. We can then conclude that the negotiations will lead to a Nash equilibrium, [ RE, RA *( RI)], in which the warehouse service level will be higher than in any other equilibrium situation reached through the negotiation process. A nice property, which is a direct consequence of Proposition 1, is that for each value of pi, both the warehouse cost &(R,) and the retailer costs cq (& (R,) are reduced (or at least kept constant) when choosing the optimal reorder points R,, and Rf , respectively. Consequently, if it is advantageous for the warehouse to change RO, this is then also an advantage for the retailers. This means that each party has an incentive to participate in the process. It should be noted though that there is no guarantee that &’ (& (R,) will decrease in successive negotiations when pi is reevaluated.


However, since we know that the total cost of the system will decrease with each negotiation, the warehouse can always reimburse the different retailers in such a way that all parties reduce their costs in each iteration.

Another setting in which our results can be of use is when we have a supply chain manager, whose main purpose is to enable an efficient coordination of the different stages in a supply chain, for example, between a wholesaler, which owns a central warehouse, and N independent retailers.

If this system satisfies the conditions described in Section 3.2, the supply chain manager can use our iterative procedure to find the shortage penalty costs pi associated with the optimal solution to the approximate problem. The involved parties can thereafter be in- structed to use these penalty costs in their dealings with each other. The result will be a system that operates at the lowest possible total cost according to the approximate model.

4. Error Bounds

In the previous section, we showed that the suggested coordination procedure, in the case of normally distributed retailer demand, can be used to find the optimal solution to the approximate problem. We now need to investigate the performance of the policy rendered by optimization of the approximate cost. We will do this by deriving a performance bound, r = TC,(Rt*, RA*)ITCE(RE*, R E * ) , valid in the case of nondecreasing retailer demand with independent increments. Unfortunately, the normal distribution does not satisfy this assumption. Still, if the mean lead time demand per time unit, pi, is sufficiently large compared to the standard deviation per time unit, Oi, the probability of negative demand is negligible. Also, if we know that the normal approximation in itself will cause a maximum relative deviation Q in any cost, an upper bound for the performance ratio r, with normally distributed demand can be derived, r,, = r( 1 + a)/( 1 - a).

We shall start by analyzing how the retailer cost (4) depends on a deterministic lead time i. Since an arbitrary constant lead time i is not necessarily associated with a certain Ro, we shall use the notation Cp ( Rj Ii) for the retailer cost when using the reorderpoint Ri together with the constant lead time i.

PROPOSITION 4. C: (Ri Ii) is convex in e for a given value of Ri .

Proposition 4 is proven in Appendix B. By using the fact that the retailer cost in the approximate problem is convex in the lead time variable, we can conclude that the minimum retailer cost in the approximate model will always be a lower bound for the true minimal retailer cost.

PROPOSITIONS. Cf[Rf*(R,)I&(R,)] 5 Ck[Rf*(R,)IL,(R,)]

Pro05 Since Ro 2 - 1, Ck [ Ri 1 Li (R,)] = ELi(R,,) { Cf [ Ri I Li (R,)] } . From Proposition 4, we know that Cf (Ri Ii) is convex in i. By applying Jensen’s inequality, we obtain Cf[R,IZ,(R,)] 5 Cf[RJLi(R,)]. Thi s is true for all values of Ri, in partid~, for Ri = RB*(Ro).Finally,wecanconcludethatC~[Rq*(Ro)IZ;i(R,)]~C~[RF*(R,)IZi(R,)] 5 Cf[Rf*(R,)ILi(R,)] Q.E.D. It is easy to prove a similar statement regarding the cost for the entire system.

PROPOSITION 6. TCA(R$“, RA*) 5 TC,(Rt*, RE*)

Pro05 It is an immediate consequence of Proposition 5 that TCA[RE*, R”*(Rt*)] I TC,(Rf*, RE*). By definition, we also know that TCA(R$*, RA*) 5 TCA[Rf*, RA*(R$*)]. Q.E.D. Equipped with this result, we can construct an upper bound for the performance ratio of our approximation, r = TCE(Rt*, RA*)ITCE(Rt*, RE*).

PROPOSITION 7. The performance ratio r is bounded from above by


li +Lo-&(Rt*) Cf(Rf* I&) +

&(R;:*) - Zi

Lo Lo Cf(Rf* /Zi +L) 1

i-5 TCA(R$*,RA*)

(11)

ProoJ: Recall that

TCdR; *, RA*) = Co(R$*) + i E’Li~R~){ Cf[Rf* 1 Li(Rt*)]} (12) i=l

We know from Proposition 4 that Cf (Rj If,) is convex in ,?. for a given Ri . Furthermore, Zi zs Li zs Zi + Lo. Therefore, if we replace the distribution of Li by a two- point distribution with the same mean and defined in the extreme values Zi and Zi + Lo, that is, with probabilities (Zi + Lo - L,)IL, and (& - Zi)lLo, respectively, we obtain an upper bound. Consequently,

- TG(Rt*, RA*) 5 Co (R;“) + i

i=l

li + Lo iLicR8’) Cf(Rf* Iii)

+ Ei (R$*) - Zi

r, Cf(Rf* l/i + &) 1 (13)

By combining ( 13) with Proposition 6, we obtain ( 11). Q.E.D. The bound ( 11) is always relatively easy to determine since it is obtained solely from

the approximate problem. The bound is affected both by the error caused when the real distribution of Li is replaced by a two-point distribution and by the difference between the approximate and the exact cost (Proposition 6). If the bound is close to one, we know that our approximation is good. On the other hand, if we do not get a tight upper bound, the approximation may still be very good. The tightness is dependent on how well the true lead time distribution is approximated by the two-point distribution discussed above. If the bound is not sufficiently close to one, the quality of the approximation needs to be evaluated in some other way, for example, by simulation.

If we summarize how the different propositions relate to each other, we can conclude that Proposition 4 is needed for the proof of Proposition 7. Propositions 5 and 6 are more or less direct consequences of Proposition 4. However, they could also be proved by using the results in Song ( 1994) if the demand processes that we consider are interpreted as limits of sequences of compound Poisson processes.

5. Numerical Results

In this section, we will evaluate the coordination procedure and the error bound ( 11)) through a number of test problems based on a system with identical retailers facing normally distributed demand. In the case of identical retailers, it is easy to determine the average warehouse delay by using Little’s formula. It is therefore the simplest situation to deal with numerically. Handling the case with nonidentical retailers numerically would, as explained earlier, require a more elaborate way of determining the retailer lead times, &. One possibility would be to consider a heuristic in which all retailer lead times were approximated by the average expected lead time, as given by Little’s formula. However, the problem is that we thereby introduce yet another level of approximation into our model, which makes it difficult to assess the performance of the coordination procedure. Also, the bound in its present form would be useless as an indicator of the quality of the solution to the approximate problem. The conclusion is that, for our purposes, the identical retailer case is a reasonable situation to study.


We shall first describe the numerical computations in more detail. The coordination procedure iterates between the warehouse and the retailers. Let us first consider the warehouse problem. The holding costs at the warehouse are obtained according to (3). The distribution of Do (Lo), that is, the retailer demand at the warehouse during the warehouse lead time, is determined by convolving the distributions of the warehouse demand ema- nating from each retailer. The distribution of the warehouse demand from a single retailer can be determined in the following way. Let qj equal the probability that the number of orders triggered at a retailer during & is less or equal toj, or equivalently, that the demand is less or equal to j Q . We then have

(j+ l>Q-x-Pi& Ui L~12

dx

((j+ l)Q-//,i&)@

where x is the uniformly distributed customer demand at the retailer since the preceding retailer order was initiated. Next, we consider the average retailer lead time for a given Ro. Let B,(R,) equal the expected back orders at the warehouse. In complete analogy with (3 ) , we have

1 Ro+Qo

Bo(Ro) = e,,=-& ED,,,,[(Do(J% - yQ)+l (15)

According to Little’s formula, we can then determine the expected retailer lead time as

(16)

The preceding iteration step at the retailers resulted in an artificial warehouse shortage cost pi = [ dC4 (Ei )ldZi I.1 lpi per unit and time unit. TO obtain the new Ro, we optimize the sum of holding and shortage costs,

$on (Co(Ro) + N/3ipi(zi(Ro) - li>) = mion ICo(Ro> + PiBo(Ro)) (17)

After determination of the new Ro, we turn to the retailers. The optimal Ri for a given Li is obtained from (7). We can then get the new shortage penalty cost, pi , from (9). After that, we go back to the warehouse and determine a new Ro.

Table 1 shows the numerical results when applying the procedure to 32 test problems with 10 identical retailers. The expected demand per time unit at all retailers, pi, is set to 50 units per time unit. The holding costs at the warehouse and at the retailers are all equal t0 1 (ho = hi = 1; i = 1, *. . 3 N), and the shortage costs at every retailer is equal to 10 (pi = 10; i = 1, . . . ) N). The problems in Table 1 cover all combinations of Q. = 1 or 2, Q = 300 or 600, & = 1 or 2, Zi = 1 or 2, and gi = 10 or 20. Table 2 shows some additional statistics concerning the solutions of the test problems.

For all problems in Table 1, the coordination procedure converges in just a few steps, and, in all cases, upper and lower bounds were identical; that is, Rf; = R$ = Rb. We


have also found problems, though, for which this is not the case. For these problems, the gap Rg - Rb has been very small ( 1 or 2 units).

The upper bound for the relative error is quite small for most of the problems in Table 1. However, it is not difficult to find problem instances when the error bound is more or less useless. Recall that the bound is constructed by approximating the real retailer lead time distribution by a two-point distribution. This is a good approximation if there are large variations in the retailer lead times. In general, this is the case if there are many retailers with large batch quantities. For example, if we change the data in Table 1 so that N = 20 and /.Li = 25, the average demand at the warehouse will remain the same, but the variations will be larger. The result is that the error bounds are reduced by about 75%. If we, on the other hand, go in the other direction and consider N = 5 and pi = 100, the error bounds are approximately 4 times higher. It is obvious and reasonable that the bound is increasing with the warehouse lead time, &. We can also conclude that the bound, in general, is not very sensitive to demand variations at the retailers.

For 2 of the test problems in Table 1, the error bound is above 20%. In order to assess the quality of our approximation in these 2 cases, we conducted a simulation study. For practical

TABLE 1

Test Problem Solutions

QO

Optimal Approx. Problem Data Policy Approx.

cost

Q Lo 6 OL RY Rf” Tc,

1 300 1 1 10 0 47.6 1439 1 300 1 1 20 0 49.5 1515 1 300 1 2 10 0 97.9 1457 1 300 1 2 20 0 101.9 1574 1 300 2 1 10 0 93.5 1413 1 300 2 1 20 0 97.3 1526 1 300 2 2 10 0 143.9 1430 1 300 2 2 20 0 149.9 1580 1 600 1 1 10 0 ‘10.6 2991 1 600 1 1 20 0 10.7 3026 1 600 1 2 10 0 60.5 3000 1 600 1 2 20 0 61.1 3062 1 600 2 1 10 0 45.1 2842 1 600 2 1 20 0 45.4 2897 1 600 2 2 10 0 95.1 2852 1 600 2 2 20 0 96.0 2932 2 300 1 1 10 -1 60.2 1420 2 300 1 1 20 -1 62.7 1506 2 300 1 2 10 -1 110.5 1437 2 300 1 2 20 -,l 115.1 1564 2 300 2 1 10 -1 108.3 1415 2 300 2 1 20 0 83.2 1526 2 300 2 2 10 -1 158.7 1432 2 300 2 2 20 0 135.8 1582 2 600 1 1 10 -1 28.0 2868 2 600 1 1 20 -1 28.2 2913 2 600 1 2 10 -1 78.0 2877 2 600 1 2 20 -1 78.7 2949 2 600 2 1 10 -1 70.3 2799 2 600 2 1 20 -1 70.9 2866 2 600 2 2 10 -1 120.3 2808 2 600 2 2 20 -1 121.6 2901

Bound Rel. Error UB(r - 1)

0.08 0.07 0.07 0.06 0.19 0.17 0.19 0.15 0.02 0.02 0.02 0.02 0.08 0.08 0.08 0.07 0.05 0.05

0.04 0.10 0.23 0.09 0.21 0.02 0.02 0.02 0.02 0.06

0.06 0.05

iV = 10; pi = 50; ho = hi = 1; pi = 10; UB = Upper Bound.


TABLE 2

Test Problem Statistics

Parameters

Pi (Li - l&

Minimum Value

0.018 0.303

Average Value

0.059 0.608

Maximum Value

0.125 0.853

reasons, in these simulations, we replaced the normally distributed demand by a compound Poisson process with the same mean and variance. What we found was that, in both problems, the relative excess cost (1. - 1) was between 3 and 4%. This illustrates that the quality of our approximation, as expected, is much better than the error bound indicates.

From Table 2, we can see that the value of pi is significantly lower than the shortage cost pi for all our test problems. The conclusion must be that the most cost effective strategy is to have a low service level at the warehouse and, instead, allocate more stock to the retailers.

TABLE 3

Warehouse Penal@ Costs and Error Bounds for DifSerent Shortage Costs

QO

Problem Data pi = 5 pi= 10 pi = 20

Q Lo li u, Pr UB(r - 1) Pi UB(r - 1) PI UB(I. - 1)

1 300 1 1 10 0.02 0.05 0.04 0.08 0.06 0.12 1 300 1 1 20 0.08 0.05 0.12 0.07 0.18 0.10 1 300 1 2 10 0.02 0.05 0.04 0.07 0.05 0.11 1 300 1 2 20 0.08 0.04 0.11 0.06 0.15 0.08 1 300 2 1 10 0.02 0.15 0.04 0.19 0.06 0.25 1 300 2 1 20 0.08 0.14 0.11 0.17 0.17 0.36 1 300 2 2 10 0.02 0.15 0.03 0.19 0.05 0.24 1 300 2 2 20 0.07 0.13 0.10 0.15 0.14 0.31 1 600 1 1 10 0.01 0.01 0.02 0.02 0.03 0.03 1 600 1 1 20 0.04 0.01 0.07 0.02 0.12 0.03 1 600 1 2 10 0.01 0.01 0.02 0.02 0.03 0.03 1 600 1 2 20 0.04 0.01 0.07 0.02 0.11 0.03 1 600 2 1 10 0.01 0.05 0.02 0.08 0.03 0.13 1 600 2 1 20 0.04 0.05 0.07 0.08 0.11 0.12 1 600 2 2 10 0.01 0.05 0.02 0.08 0.03 0.13 1 600 2 2 20 0.04 0.05 0.07 0.07 0.11 0.11 2 300 1 1 10 0.02 0.04 0.04 0.05 0.06 0.07 2 300 1 1 20 0.08 0.03 0.12 0.05 0.17 0.06 2 300 1 2 10 0.02 0.04 0.03 0.05 0.05 0.07 2 300 1 2 20 0.07 0.03 0.11 0.04 0.15 0.05 2 300 2 1 10 0.02 0.08 0.03 0.10 0.05 0.11 2 300 2 1 20 0.08 0.17 0.12 0.23 0.16 0.30 2 300 2 2 10 0.02 0.08 0.03 0.09 0.05 0.11 2 300 2 2 20 0.07 0.16 0.11 0.21 0.14 0.26 2 600 1 1 10 0.01 0.01 0.02 0.02 0.03 0.03 2 600 1 1 20 0.04 0.01 0.07 0.02 0.12 0.03 2 600 1 2 10 0.01 0.01 0.02 0.02 0.03 0.03 2 600 1 2 20 0.04 0.01 0.07 0.02 0.11 0.03 2 600 2 1 10 0.01 0.04 0.02 0.06 0.03 0.08 2 600 2 1 20 0.04 0.04 0.07 0.05 0.11 0.07 2 600 2 2 10 0.01 0.04 0.02 0.06 0.03 0.08 2 600 2 2 20 0.04 0.04 0.07 0.05 0.10 0.07

N = 10; pi = 50; ho = hi = 1; UB = Upper Bound.


To see how the warehouse shortage penalty cost and the error bound depends on the service requirements for the system, we also solved all test problems for both a lower and a higher retailer shortage cost, pi = 5 and pi = 20. Table 3 shows our results. The warehouse penalty cost pi is, as expected, always increasing with pi, but not linearly. In Table 3, the ratio pi/pi is varying between 0.001 and 0.02. The error bound declines for higher shortage costs. We can also conclude that pi is quite sensitive to an increase of ai, the standard deviation of customer demand. Although not illustrated by Table 3, we also noted that when increasing the retailer shortage costs, the relative increase of retailer stock was higher than the relative increase of warehouse stock.

6. Conclusions

Our main purpose with this paper has been to present a model that can be useful for decentralized control of a multiechelon inventory system with one central warehouse and an arbitrary number of retailers. More precisely, we have suggested and analyzed a simple approximate procedure that enables the facilities to use standard single-level techniques to determine their reorder points. The approximate solution can, under certain conditions, be obtained by applying a simple iterative procedure. We have also derived a bound for the relative excess cost of using the approximation. Since the bound is completely based on the approximate solution, it is easy to compute.

The bound can be used in a first test for two-echelon inventory problems. If our simple bound is satisfactory, we know that our decentralized approximate technique is performing well, and there is no need for more. sophisticated approaches.

From a practical point of view, explicit use of the new cost structure and the associated penalty costs offers a simple and, in many cases, efficient way, to coordinate the decisions made at different installations in the system.

Appendix A. Proof of Proposition 1

In order to prove Proposition 1, we need the results in Lemma Al and A2.

LEMMA Al. G(u) is decreasing and convex. The proof is almost trivial and is therefore omitted.

LEMMA A2. pi 2 hi implies R,* = p& - Q/2. ProofI Let u = (R,* - ~i~,)l(~i~~‘2) and IJ = Ql(a,zt’*). Note that v > 0. The optimality condition (7),

together with pz = hi, implies that

G(u) - G(u + u) 1 5-

u 2

Since C’(.) = a(,) - 1, we can rewrite (Al) as

(A21

The left-hand side of (A2) is increasing with u. We obtain equality for u = -u/2 and can therefore conclude that RT 2 piLi - Q/2. Q.E.D.

PROPOSITION 1. Given that pi 2 hi, C:’ (L, ) is a concave function. Pro05 Let

a = (R, - i&)/Q

b = Ql(o~i;:“)

a = (h, + pi)c7;/(2Q)

y = hi/(hi + P,) (A3)


where (Y and y are simply positive constants. Note also that a 2 -i , due to Lemma A2, and that b > 0. From (9), we have

dC: 6) - = a{ @[b(a + l)] - +(ba))

dL,

and for each value of Li , (7) implies that

~(a, b) = i [G(ba) - G(b(a + I))] = y

Consider a small increase in &. Evidently, this corresponds to a decrease in b [see (A3)]. If we keep bu at a constant level, a decrease in b will lead to an increase in x( a, b) However, x( a, b) must be held constant [ follows from (7)]; to achieve this, ba must increase: when b decreases. This follows from Lemma Al and (A5).

We know that a(x) is increasing and concave for x > 0 (convex for n < 0). Using the symmetry of Q(x) and the condition a 2 -a, we can conclude that d C! (z, )/d& must decrease when b decreases and bu increases, or, equivalently, dC$ (L,)/d& is decreasing with t,. Q.E.D.

Appendix B. Proof of Proposition 4

Let us first state 2 simple lemmas. The proofs are omitted.

LEMMA A3. Consider x < y and z > 0. Then

(x - z)’ + y+ 2 x+ + (y - z)+

LEMMA A4. Consider a function f( y). If, for any y and for any x > 0,

f(Y +x) -f(y) Zf(Y) -f(y -xl (A7)

(A61

then fis convex.

PROPOSITION 4. C$ (& 1 i) is convex in e for a given value of R, . Proof

C;4(Rili) =$[‘” Eo,,i, [hi(y - Q(i))+ + p,(dJ - y)+l&

[(hi +~iW~,,i,[(y - &@))+I +pi(/d - y)ldy (A8)

It is sufficient to show that E,,,i,[ (y - Di @))‘I is convex in i [see (A8)]. We know that Di (2) is independent of D, (A) if e and A are nonoverlapping time intervals. This follows from Assumption 3 in Section 2. From the same argument, we also know that Df (A) and D?(A), as depicted in Figure Al must be independent, identically distributed and nonnegative.

L=et g(t) = ED,(,,(y - Di (t)) + . We want to show that g(t) is convex. The following inequality is a consequence of Lemma A3.

g(i + 2A) + g(i) = &,(L+zA) ([Y-D,(~+~A)I+)+E,,,~,~[Y-D~(~)I+}

= &,c~~Eo~p,-G~a,~ [Y - D,(i) -D!(A) -@(A)l+ + [Y - &@)I+ 1

2 J%,&D+v%~A,~ [Y - D,(i) - Df (A)]+ + [Y --&@I - D:(A,l+ 1

= ~ED~&+A,[ [y - Di(i + A)]+ ) = 2g(i + A)

According to Lemma A4, (A9) implies that g(r) is convex. Q.E.D.

(A9)

DiC’) D:(A) D:(A) 4 t

il A A

FIGURE A 1. Time Intervals and Corresponding Demands.


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Jonas Andersson is a Ph.D. student at the Department of Industrial Engineering at Lund University in Sweden. He received his BSc. from Vaxjo University and his Lit. Eng. from Lund University. Andersson’s research is oriented towards developing algorithms for operational decision making in production and inventory control, specifically, control methods for multiechelon inventory systems with stochastic demand.

Sven Axsiiter has been Professor of Production Management at Lund University since 1993. Before that, he held professorships at LinkSping Institute of Technology and Lulea University of Technology. He got his degree from the Royal Institute of Technology in Stockholm. Past and current research interests include hierarchical production planning, lot sizing, and, most recently, multiechelon inventory systems. Sven Axsater is President of the International Society for Inventory Research and a member of the Royal Swedish Academy of Engineering Sciences. He is serving as Associate Editor of Operations Research and Area Editor of Production and Operations Manage- ment. He is also member of the editorial board of several other scientific journals. He has published widely in various journals.

Johan Marklund is a Ph.D. student at the Department of Industrial Engineering at Lund University in Sweden. He received his M.Sc. degree in Industrial Engineering and Management from Linktjping Institute of Technology 1994. He also holds a B.B.A. from Lund University and a Lit. Eng. from Lund University. Marklund’s research is oriented towards developing models for operational decision making in the fields of production and inventory management, specifically, control methods for multiechelon inventory systems with stochastic demand.

decentralized multiechelon inventory control

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