Effect of correlated demands on safety stock centralization: Patterns of correlation versus degree of centralization

Journal of Business Logistics, 1999 by Das, Chandrasekhar, Tyagi, Rajesh

The effect of demand correlations on aggregate safety stock was first studied by Zinn, Levy, and Bowersox, who introduced the notion of portfolio effect (PE) for measuring savings in safety stock by centralization of customer demand.' They described the influence of correlation magnitudes and signs on aggregate safety stock and suggested investigating the effects of partial and complete centralization in order to maximize the PE. Partial centralization refers to grouping safety stocks at more than one location. They further suggested that determining these groupings is a combinatorial problem with no easy solution.

This article identifies situations in which PE is enhanced by partial centralization. This knowledge can help practitioners determine when to consider partial instead of complete centralization. We consider the sign and magnitude of the correlation coefficients to be important factors and attempt to generalize their effects on achievable savings in safety stock. A nonlinear optimization model, as shown in the Appendix, is used to carry out the investigation. In the next section, we illustrate the benefit of partial centralization using actual data from the apparel industry.

PARTIAL CONFIGURATION

Eppen initially used a newsboy type formulation to develop conditions under which centralization will lead to savings in safety stock.2 As noted, Zinn, Levy and Bowersox used PE to measure the saving due to consolidations of safety stock and showed that the saving is a function of the ratio of the standard deviations and the correlation coefficients of demands. Tallon extended this work to the case of uncertain leadtime and drew similar conclusions.3 Evers suggested an alternative form of the square root law applicable to correlated demands.4 General computational difficulties in solving such problems and a heuristic solution approach are outlined in Tyagi and Das.5

While the above research provides insight into savings in safety stock, there is little literature on the additional savings that may be derived by an optimal centralization involving more than one location. To illustrate, consider the U.S. apparel industry, for which quarterly shipments of select categories of garments during 1990 are shown in Figure 1. Due to the seasonal nature of demand for many of these categories, which include warm and cold weather garments, some of these products are expected to be positively and some others negatively correlated. This is confirmed by the correlation matrix (see Figure 1) derived from the quarterly sales data. For example, women's swimwear and women's shorts, both warm weather garments, display similar demand patterns and thus have a high positive correlation (0.75). Women's swimwear and women's sweater are negatively correlated (minus 0.93) because of their inverse seasonal demand patterns. Assuming a leadtime of one quarter and a 95 percent service level, the strategy of complete centralization requires a safety stock of $112.2 million. By consolidating the garment categories into two groups, as shown in Figure 2, the safety stock cost can be reduced to $111. I million. The group composition does not follow any conventional classification that is both, men's and women's apparel categories are included in each group. Also, the groups do not segregate into warm and cold weather garments. Although the actual saving in this case is only one percent, it establishes the fact that pooling customers into two or more groups rather than one can further reduce safety stock investment when demands are correlated.

SAFETY STOCK FOR CORRELATED DEMANDS

where L is the delivery lead time in weeks, K is the safety stock factor, and A, B, C, ..., are the customer groups to be formed. The complete mathematical model is included in the Appendix.

If customer demands were independent, then the covariance terms in the expression for VA, VB, and so on would be zero. In that case, the aggregate safety stock expression is minimized when all customers are assigned to a single group, leading to complete centralization. When demands are interdependent, however, the presence of the covariance terms affects the safety stock expression in two ways. First, the signs of the correlations affect the overall magnitude of safety stocks: Positive covariances increase the safety stock levels, and negative covariances decrease them. Second, the covariance term for a pair of demands, dj and dk, applies only when the two demands are assigned to the same group. It is, therefore, possible to eliminate large positive covariances by assigning such customers to different groups.

This observation leads us to investigate the benefit of partial centralization by forming multiple groups of customers. We hypothesize that the optimal degree of centralization will depend on the signs and magnitudes of the correlation coefficients. To test this hypothesis, we consider different patterns of correlations and determine the optimal degree of centralization for each pattern. For com parison purposes, the relative savings due to optimal centralization are shown for each case in the last column of Table 1.