Operational aspects of inventory consolidation decision making

Journal of Business Logistics, 1998 by Evers, Philip T, Beier, Frederick J

Recent logistics research has focused attention on the effects of consolidating multilocation inventory facilities on safety stock. This discussion has centered around the square-root law and the portfolio effect. The body of literature has now reached the point at which alternative models produce substantially different results. The purpose of this paper is twofold: (1) to analyze two contrasting theoretical approaches, both incorporating lead times, for estimating the portfolio effect and (2) to examine the effect of lead times, in part using actual firm data, on the portfolio effect. First, pertinent multilocation literature relating to the portfolio effect is reviewed. Second, two portfolio effect models are discussed and compared. Third, industrial data are used to explore consolidation issues. Finally, sensitivity analysis of the preferred model is performed, and managerial implications are identified.

THE MULTILOCATION LITERATURE

Multilocation research focuses on the effect of inventory centralization across locations at the same channel position. The square-root law, which links the number of stocking locations with the level of inventory, represents a traditional way of analyzing inventory consolidations. An initial version of the square-root law was developed by Maister.1 In the process of formulating the portfolio effect model, Zinn, Levy, and Bowersox also identified certain assumptions of the square-root law.2 Evers and Beier, showing that the Maister model when applied to safety stocks was not applicable in certain circumstances, constructed a slightly altered version of the law and clarified additional assumptions underlying the analysis.3 Evers extended the law even farther by suggesting alternative models based on cycle stock considerations.4 While specifying all the law's assumptions, Evers noted the low likelihood that these would all hold at the same time in practice. Because of the potentially severe limitations of the square-root law, a more general, robust model is needed to examine inventory consolidation. Zinn, Levy, and Bowersox showed that the portfolio effect model, which measures the percentage reduction in total safety stock due to inventory consolidation, represents a generalization of the square-root law of safety stocks.5 This model incorporates the standard deviation of demand and the correlation of demand between locations into the analysis of inventory consolidations. This interest in correlated demands stemmed from the work of Eppen, who first recognized that the higher the correlation, the lower is the savings resulting from consolidation.6 Eppen's work, dealing with the well-known newsboy problem, has been extended by others to include such issues as concave holding- and penalty-cost functions7 and transshipments.*

Considerable work has been published on the initial portfolio effect model developed by Zinn, Levy, and Bowersox. Ronen considered an alternative method for determining safety stock levels.9 Mahmoud provided the portfolio quantity effect model, which measures inventory consolidation in terms of quantity (as opposed to percentage) reductions in aggregate safety stock." Mahmoud also proposed the portfolio cost effect model, which accounts for the influence that consolidation may have on other parts of the logistics system, including transportation and procurement. Tallon factored lead time uncertainty into the portfolio effect model. tI Finally, since earlier models had focused on consolidation to a single location, Evers and Beier incorporated a "proportion" variable into the analysis to allow for consolidation to multiple locations.'2 They also included lead time uncertainty into the portfolio effect model, but in a manner substantially different from that of Tal]on. The models of interest for this paper are that of Tallon, on the one hand, and that of Evers and Beier, on the other. COMPARISON OF ALTERNATIVE PORTFOLIO EFFECT MODELS Conceptually, the fundamental difference between the two models is the treatment of lead times at the consolidated location. Tallon implicitly assumes the lead time at the centralized location to be a function of those at the decentralized locations. As a result, his model does not allow for specific, exogenous lead time means and standard deviations at the centralized location. The Evers and Beier model does allow for such lead time means and standard deviations.13As will be discussed later, the ability to define externally the lead time parameters is more realistic, making the Evers and Beier model a more useful and preferred alternative.

A second difference is that the models require significantly different correlation terms. Based on the method of substitution Tallon employed to derive his equation, his model requires the correlation of demand during lead time between decentralized locations.14 In contrast, Evers and Beier used the correlation of demand between decentralized locations. Compared to the correlation of demand, the correlation of demand during lead time is difficult to estimate. Within any given period, two decentralized facilities (implying that they operate independently of each other) are likely to order and receive inbound shipments at different times, thus making it difficult to match shipments in order to calculate the correlation of demand during lead time. Moreover, as opposed to demand (which can easily be measured on an hourly, daily, or some other period basis), demand during lead time is difficult to define in terms of an appropriate dimension that can be accurately measured and interpreted. While the preference of one measure of correlation over the other is debatable and left to future research, it would seem that correlations among individual data points (such as demand) would be preferable to correlations among compound data points (such as demand during lead time). Analytically, in order to compare the two models while staying within the narrower scope of the Tallon model, the remainder of this paper only considers consolidations that are made to a single location. Also in parallel with Tallon, throughout the paper all safety factors (k) used in the kor approach to setting safety stocks are assumed to be equal at all locations, both centralized and decentralized. Relative to the Evers and Beier model, the Tallon model underestimates the reduction in safety stock from a consolidation at Chicago and overestimates the reduction from a consolidation at St. Louis. In fact, the Evers and Beier model suggests that safety stocks will increase as a result of a consolidation at St. Louis. Although Zinn, Levy, and Bowersox claim that at worst a consolidation will have no effect on inventory levels, their model specifically excludes lead times from consideration.21 Both models compared in this section make it clear that consolidation savings are a function of lead times. The difference between the two is that the Evers and Beier model appears more precise than the Tallon model. In other words, the Evers and Beier model recognizes that a consolidation in St. Louis would subject all the heretofore Chicago shipments to inferior lead times, resulting in the need for more safety stock. This is an extremely important observation, as it indicates that lead time service associated with a consolidation site is an influential variable in the determination of safety stock savings. As demonstrated by this example, consolidations may produce either increases or decreases in inventory depending upon the particular lead time parameters.