international journal of

production economics

ELSEVIER Int. J. Production Economics 35 (1994) 271-275

Multi-echelon inventory theory - A retrospective

Andrew J. Clark

79023 Sear.s Rd., Cottage Grow, OR 97424, USA

Abstract

One of the earliest studies of the multi-echelon inventory problem was by Clark-Scarf in 1969. An accounting is given of the events leading to this work and the main results are summarized. Subsequent extension of the multi-echelon inventory are outlined and areas of current and future applications are discussed.

1. Introduction

The general field of inventory theory is con- cerned with providing methods for managing and controlling inventories under different policy con- straints and environmental situations. In many such inventory environments, rational manage- ment decisions for inventories cannot be made without explicit consideration of interrelations among activities in an overall supply system and with transportation, manufacturing, or other pro- cesses. In particular recognition of such relations, multi-echelon inventory theory has evolved as an important specialty area.

A representative multi-echelon inventory system is illustrated in Fig. 1. This system, which might include inventories of a consumer item such as a TV set, consists of retail outlets at the lowest level supported by local distributors who order from regional distributors that receive shipments from a factory. In this example, it is assumed that each activity receives resupply from only one higher level activity and the system can therefore be described as

having an arborescence structure. This assumption does not always apply in the real world, but most work in multi-echelon inventory theory assumes the arborescence structure.

Two simplifying cases of the arborescence struc- ture have often been identified in the literature. In one case, two or more activities are arranged in series with only one activity at each level receiving resupply from the next higher activity. In the other case, there are two or more activities at the bottom level, all supported by a single activity at the next higher level. In both cases, as well as the general arborescence structure, it is usually assumed that exogenous demand occurs only at activities at the lowest level of the system, but this assumption is not necessary for the theory to be discussed later. The multi-echelon inventory problem can be defined as one of establishing rules for jointly con- trolling inventory levels at all activities in the system in order to satisfy some management policy. The most usual policy is to minimize the overall costs of carrying inventories in the system as a whole, per- haps subject to a customer service constraint.

0925.5273/94/$07.00 80 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)EOl 30-N

272 A.J. Clark: In!. J. Prochtion E~~onornks 35 (1994) 271-275

Fig. I. Structure of multi-echelon inventory systems

2. Retrospective

My interest in the multi-echelon inventory prob- lem began while I was employed at the RAND Corporation in Santa Monica, California. As my first full-time professional job, I became a member of the Logistics Department which was established at RAND in December, 1953, to study all aspects of USAF (United States Air Force) logistics support.

During my first couple of years with the Logistics Department, I helped in the design of broad-scale inventory control procedures based upon the use of large computers which were just then becoming available. As part of this project, we developed methods for setting stocks levels using the Wilson lot size formula and a variable safety level concept to allow for demand uncertainty and differences in unit prices. This work culminated in an implemen- tation program involving the management of repair parts for B-47, B-52 and KC-135 aircrafts, then the first line aircraft of the Strategic Air Command.

In setting item stock levels for this project, I be- came generally disturbed that each stocking facility (Air Force bases and a depot, arranged in the paral-

lel structure shown in Fig. 1) was being considered independently, ignoring the fact that they were inti- mately interconnected in the real world. For a given item of supply, we were suboptimizing for each activity without considering the stockage pos- ture at the other activities, particularly the high- er-level depot. My concern was reinforced by the fact that the supply performance of Air Force bases at that time was significantly degraded by lack of proper support from the depot. So I decided to attack this multi-echelon problem from an analytic point of view.

As the foundation of my analysis, I started with the classical AHM (Arrow-Harris-Marschak) model [l]. 1 was attracted to this approach pri- marily because it was able to accommodate time- dependent factors such as demand rates and also because it had been extended to consider resupply time delays and fixed reorder costs. At the start of my analysis, however, I imposed the most simplify- ing assumptions possible: a single user activity sup- ported by one higher level (depot) facility with unlimited resupply (from an implied factory), linear holding and shortage costs, linear ordering costs

A.J. Clark/Inr. J. Production Economics 35 (1994) 271-275 273

without resupply setup costs, zero resupply times, and stationary (but stochastic) demands only at the user activity with excess demand backlogged.

In the AHM model, future time is divided into N equal decision intervals which, for notational convenience, are numbered backwards from an assumed future termination date. A formula is developed for the total future costs incurred at the start of an arbitrary interval, n, and before making possible ordering decisions. As long as the total future costs are a convex function of the stock position, the optimal ordering policy is given by a single critical number, S, where an order is placed only if the stock position is less than S and for an amount needed to raise the stock level to S.

To facilitate my understanding of the AHM model, I plotted out the cost functions for the user activity before and after ordering, as shown in Fig. 2. With the simplifying assumptions, the total future costs at the beginning of an arbitrary time interval n is a convex function of the stock position before ordering, asymptoting on the right to a straight line with slope equal to the per unit holding cost (h) and on the left to a line with a slope equal to the per unit shortage cost (p). As illustrated in Fig. 2, the critical level, S, is found by fitting a line with slope equal to the per unit ordering cost (c) so as to be tangent to the before-ordering cost curve. The total future cost after ordering is then given by this line up to S and by the before-ordering cost function for stock positions greater than or equal to S.

Drawn in this fashion and with the multi-echelon problem in mind, it became apparent that the difference between the before-ordering and after- ordering cost functions represent the cost conse- quences to the user activity if insufficient amounts were available at the depot to satisfy the user activ- ity’s requirements. A notion then occurred to me that the extra costs incurred at the user activity, shown by the shaded area in Fig. 2, should some- how be charged to the depot activity as an “implied shortage cost”.

With this concept in hand, the remainder of the development fell into place. The AHM mode1 might be applied independently to the depot facility, with the implied shortage cost function from the user activity being added to the before-ordering cost function at the depot. However, several conse- quences of this approach then had to be resolved.

The most important issue concerned the stock position to be used in the computation for the depot. Upon reflection, the only thing that made sense was to consider the stock position not only at the depot itself but also on-hand amounts at the user activity. Thus, to generalize, I coined the term “echelon stock” for a given higher-level supply ac- tivity as consisting of all amounts on hand and intransit to the activity plus the on-hand and in- transit amounts at all support lower-level activities, or in other words, total system stock at a below the given higher-level activity/echelon.

Another issue concerned interpretations of the cost factors involved. In the multi-echelon

Fig. 2. The AHM model.

274 A.J. Clurk/Int. J. Production Eummks 35 (1994) 271-275

context, the per unit holding cost charged to a user activity should be the extra cost incurred by stocks held at the user activity instead of the next higher level. Similarly, the per unit ordering cost at a user activity should be the marginal cost of bringing a unit located at the next higher level down to that user activity. Finally, the natural per unit shortage cost (excluding the “implied shortage cost”) charged to each activity should consist of those costs directly incurred by the activity when a short- age occurs. For higher-level activities (the depot in the above example), natural shortage costs may be zero and only the “implied shortage costs” from the user activity are operative in the higher-level (depot) computations.

With these considerations resolved, I was then able to generalize the concept to the arbitrary serial multi-echelon system. The computation, using the AHM dynamic programming technique, commen- ces with the last period and a critical level S is computed for the lowest stockage facility. From this computation, an implied shortage cost function is derived and used to determine a critical level for the next higher activity. This procedure continues up the echelon structure until the top activity is reached where an unlimited resupply availability is assumed. The entire procedure is then repeated for the next-to-last time interval, and continued time period by time period to the current time. The end result is that an order-up-to level is determined for each activity and each future time period.

These results were presented in a RAND report in December of 1958 [2]. In addition to the basic methodology described above, I incorporated sev- eral factors important to the intended application to high-cost repair parts in the Air Force base- depot supply system. First, I was able to eliminate the assumption of zero resupply times due to the work of Karlin and Scarf [3]. Fixed reorder costs could also be included at the highest activity due to the work of Scarf 141. 1 also addressed the issue of reparable returns, where a portion (or all) of the user activity demands generate failed units which are sent to repair facility and subsequently inserted back into the supply flow. This feature was accom- modated by adjustments to the demand probability distributions used in the levels calculation for affec- ted activities.

In the RAND paper, I also presented a method for applying the procedure to the parallel-echelon structure as illustrated in Fig. 1. In this method, the implied shortage cost functions are determined for all user activities (for a given time interval) and then combined into a single implied shortage cost func- tion to be assessed against the depot. The single function is derived by ranking, in ascending order, the per unit implied shortage costs across all user activities and then assigning, in order, the cost from the ranked list to successive depot echelon stocks less than the sum of the order-up-to levels for all bases. The caveat here is that the depot echelon stocks are assumed to be freely available to all activities in the echelon and that a shortage at one base can be covered by on-hand stock at another. However, I noted that, in practice, sufficient items become available each period from depot-level re- pair or production so that base protection can be equalized through the normal echelon flow without inter-base shipments.

After publication of the RAND report, there was considerable interest in the approach but the next stage of development occurred elsewhere. In 1959, I left the RAND Corporation and joined a logistics research and development team at Planning Re- search Corporation (PRC) to continue work on the multi-echelon problem.

Arrangements were made for Prof. Herbert Scarf from Stanford University, one of the leaders in mathematical inventory theory, to spend a week at PRC as a consultant. His assignment was to pro- vide assistance in the overall work of the logistics group and, in particular, to help prove optimality (or non-optimality) of the multi-echelon model. During his visit, Herb initially became busy with other work and did not turn his attention to the multi-echelon problem until the last day. Then, working at a blackboard, he started by writing a total system cost function for the simple one base, one depot problem. Incorporating the implied shortage cost function and echelon definitions that I described in the RAND report, he was able to group the total system costs into the sum of two functions, one for the base and one for the depot. He noticed that the cost functions for both the base and depot were convex and each dependent upon only one decision variable, thereby decomposing

A.J. Clarkilnt. J. Production Economics 35 (19941 271-275 215

a two-variable dynamic programming problem into two one-variable problems which provided the sought-after proof of optimality. Herb sub- sequently wrote a first draft of a paper covering the basic results, to which I added a section covering the reparable return case as discussed in the RAND report. The paper was published by Management Science in 1960 [S].

After this breakthrough, work on the multi-eche- lon problem intensified. Herb and I wrote a paper [6] in which bounds were found for optimal solu- tions in the case of fixed reorder costs at lower echelons. Fukuda, a member of the PRC team, extended the theory to consider optimal disposal (of excess stock) policies in addition to ordering policies [7]. He also extended previous single-activ- ity results for Bayesian and maximum likelihood policies to the multi-echelon case [S]. More recent developments are given in books by Schwarz [9] and Axsater et al. [lo].

Since the original development, there has been no significant implementation of the multi-echelon policies due primarily, in my opinion, to difficulties in explaining the theory to managers and also fears that the amount of computation involved would be prohibitive. However, the time may now be at hand for practical implementation of the theory de- veloped over the last three decades.

3. Conclusions

In this paper, the conceptual approach taken by Clark-Scarf in their early work on the multi-eche- lon inventory problem has been summarized. In the intervening years, this approach has led to increased interest in this problem, resulting in substantial extensions and additions to the optim- ization theory. However, practical implementation of the theory has been quite limited due, in part, to the extensive computational requirements. With

the current availability of very high-speed com- puters, the time may now be at hand for practical use of the theory, not only for establishing integ- rated inventories in a multi-echelon distribution system but also to schedule factory production in consonance with dynamic and stochastic demands for a product.

References

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inventory policy, Econometrica, XIX: 250-272.

[2] Clark, A., 1958. A dynamic, single-item, multi-echelon in- ventory model, RM-2297, The RAND Corporation, Santa

Monica, CA.

[3] Karlin, S. and Scarf, H., 1958. Inventory models of the Arrow-HarrissMarschak type with time lag. in: K.J. Ar-

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[IO] Axsater, S., Scheeweiss Ch. and Silver E. (Eds.), 1986.

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