Managing single echelon inventories through demand aggregation and the feasibility of a correlation matrix

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Abstract

This paper examines the assertion that partial pooling of customers is sometimes favored over complete pooling on the sole basis of demand correlation. First, the management of inventory within a supply chain is discussed, with specific attention paid to risk-pooling. Then, the claim that partial pooling can dominate is theoretically discussed. The conditions under which previous research found that partial aggregation could, at times, be preferable are investigated next. From this, methods are proposed for checking correlation matrices to ensure their validity. It is concluded that partial pooling can never do better and that examples supporting partial aggregation are based on inconsistent correlation matrices.

Scope and purpose

Arising in part from uncertainty, inventory is a key component of the supply chain. This paper reviews alternative methods for managing demand uncertainty across various levels of the supply chain as well as within a single echelon of the system. Particular attention is paid to the fulfillment of customer demands from multiple stock-keeping facilities versus from a single location, with the superiority of the latter presented in terms of requisite safety stock levels. In doing so, methods for ensuring the feasibility of correlation matrices for simulation and numerical analyses are also highlighted.

Introduction

The mission of supply chain management is to coordinate the activities of all relevant entities for the purpose of providing goods and services to the end customer in an efficient and effective manner. One function that receives considerable attention in this alignment is the management of inventory. A typical goal is to increase supply chain velocity by minimizing total inventories within the supply chain, while maintaining an appropriate level of customer service. By doing so, products flow faster from raw materials sources to final consumers, leading to lower holding costs, less obsolescence, and reduced storage-space requirements, among other things.

Even in a fully functional supply chain, however, there is justification for holding inventory. No matter how good the demand forecasting system being used is, a firm usually cannot predict end customer orders with complete accuracy. Assuming that the customer is unwilling to wait for the product to be manufactured, inventories must be maintained to meet demand as it occurs. In other situations, it simply may not be economically feasible to produce on a just-in-time basis. Even in the case of Internet-based companies like e-tailers, the expected response time to requests may be such that items must be produced prior to actual customer requirements. As a consequence, methods are needed for efficiently maintaining inventories in general and, of particular interest here, inventories (commonly known as safety stocks) used to meet unexpected customer requirements.

In Tyagi and Das [1], a model and solution procedure for grouping customers in order to minimize total resource requirements such as safety stocks was proposed. Their work stemmed from the notion that demand correlations can be optimally combined in such a way that partial pooling of customers into multiple groups could, in some cases, result in fewer resource requirements than complete pooling of customers into one group. This paper, while not contesting their solution technique, objects to their fundamental contention that partial aggregation of customers can be better than complete aggregation based on demand correlations alone and argues that complete aggregation of demand is always as good as, if not better than, partial aggregation on the basis of minimized total resources. Further, this paper will establish that the results from Tyagi and Das [1] claiming to show partial aggregation to be better were based on mathematically inconsistent correlation matrices. To do so, this paper will demonstrate the existence of feasible and infeasible correlation structures and describe methods that allow researchers to check the validity of a correlation matrix for both numerical and simulation purposes. But before addressing these issues, the notion of pooling and its relationship to inventory management will be explained first.

Section snippets

Statistical economies of scale

Within supply chains, the principle of postponement relies heavily on statistical economies of scale, a term originally coined by Eppen and Schrage [2] and defined as “advantages that result from the pooling of uncertainty [3, p. 51].” Postponement recognizes that when changes in the form, identity, and/or location of a product are delayed, savings accrue because demand is easier to predict [4]. Not only does pushing inventory back up the supply chain lead to lower product valuations since

Partial pooling is never better

To understand why complete pooling can never do worse than partial pooling, consider the following as it relates to physical aggregation. Individual customers are initially classified into n distinct markets on some basis relevant to the firm (say, geographic area). As a starting point, each of these markets is assigned to a different stock-keeping location. Upon pooling, some or all of these markets are coupled together. Thus, pooling can range from one market per location (representing the

Results based on inconsistent correlation matrices

Having shown that partial pooling cannot do better than complete pooling, a logical question arises. How were Tyagi and Das [1] able to claim that partial aggregation was preferred in some cases? In particular, of the four different correlation matrices examined (R1,R2,R3, and R4), they found that partial aggregation dominated in three cases (R1,R2, and R3) while complete aggregation dominated in only one (R4). The source of these contrary findings stems not from their proposed solution

Conclusion

On the topic of pooling, it was shown that complete aggregation never performs worse than partial aggregation on the sole basis of demand correlation. There may, in fact, be cases where partial aggregation of inventory is preferred, but that entails the consideration of other factors not accounted for here, such as lead times or transportation costs. For example, inventory centralization at the wrong location could necessitate higher safety stocks to cover possible increases in average lead

Acknowledgements

Sections of this paper reviewing the risk-pooling literature were presented at the Workshop on Supply Chain Management Practice and Research: Status and Future Directions, in Rockville, MD, April 2001.

Kefeng Xu is an Assistant Professor of Operations Management at the University of Texas at San Antonio. He received his Ph.D. in operations management and logistics/transportation from the University of Maryland. His publications have appeared in journals such as Transportation Research, Journal of Business Logistics, Journal of Transportation Economics and Policy, and International Journal of Physical Distribution and Logistics Management. His research interests are in inventory management;

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  • Cited by (15)

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    Kefeng Xu is an Assistant Professor of Operations Management at the University of Texas at San Antonio. He received his Ph.D. in operations management and logistics/transportation from the University of Maryland. His publications have appeared in journals such as Transportation Research, Journal of Business Logistics, Journal of Transportation Economics and Policy, and International Journal of Physical Distribution and Logistics Management. His research interests are in inventory management; logistics, service and manufacturing strategy; supply chain modeling; transportation policy, service, cost and demand studies.

    Philip T. Evers is an Associate Professor of Logistics Management at the University of Maryland. He received his Ph.D. in logistics management from the University of Minnesota. His research interests include inventory management, transportation operations management, and intermodal transportation issues. He has published numerous articles in journals such as Transportation Research, Journal of Business Logistics, Transportation Journal, and Production and Inventory Management Journal.

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