Extension of the square-root law for safety stock to demands with unequal variances

Journal of Business Logistics, 1998 by Tyagi, Rajesh, Das, Chandrasekhar

Consider a distribution system in which safety stock is currently maintained at n decentralized locations. The demands at these locations are probabilistic with unequal variances. When inventories from these n locations are consolidated into m centralized locations, the savings in safety stock may be computed as (see Maister,1 or Zinn, Levy, and Bowersox2):

The purpose of this paper is to derive a method for allocating safety stock among any number of m so that the savings in safety stock are maximized. It will be shown that the maximum savings may be achieved without restricting the number or size of centralized locations. The motivation for this research stems from the following general conditions: The decentralized locations in practice may have unequal demand variances, and the centralized locations may have capacities of different magnitudes. The goal of our allocation rule is to meet these two general requirements without sacrificing the maximum savings in safety stock.

THE PORTFOLIO EFFECT

We make the following primary assumptions about the distribution system: All demands at the decentralized locations are uncorrelated, and lead-times and safety stock factors are known and equal for all centralized locations. Define:

CONCLUSION AND IMPLICATIONS

Our main intent is to extend the square-root law to situations in which customer demands represented by the decentralized locations have unequal variances. The main result in this regard is that optimal savings in aggregate safety stocks are achieved if each centralized location supplies the same fraction of each decentralized location's demand. In the literature, this result was known to be valid for customer demands with equal variances. We show that the same result applies in the case of unequal variances.

The generalized result has some important managerial implications. First, benefits from centralization can accrue even when customers are not homogeneous with respect to demand variability. Lack of uniformity in demand variance is, in fact, the rule rather than the exception in most practical circumstances. For instance, demands for repair parts and fashion goods are known to follow Poisson distribution, which has the property of equal mean and variance. Decentralized locations with unequal mean demands for such products are, therefore, expected to exhibit unequal demand variances as well. The allocation rule suggested here can now be applied to achieve savings from inventory centralization in such cases.

An important benefit of the allocation rule is that savings in aggregate safety stocks are not dependent on the number of centralized locations or their size. Theoretically, any number of facilities can be used to centralize inventory, and the fractional demand supplied by them may differ from one centralized location to another. A manager may exploit these two features to determine the number and size of centralized facilities to suit the application scenario. In some cases, we even foresee the possibility of increasing the number of centralized locations beyond the number of customer locations and still benefiting from inventory centralization. A case of this nature may be relevant in establishing centralized facilities for the supply of common parts to two large auto manufacturers. It is not necessary in this case to reduce the number of centralized locations to less than two (i.e., one) in order to benefit from centralization. The number can be more than two, and their capacities can be determined on the basis of location factors and distribution costs. For instance, an evaluation of the location factors may suggest four potential sites suitable for the centralized facilities. The capacities of these facilities can then be determined on the basis of their relative costs of distribution. To be specific, let us assume that the proportional allocation of the total capacity to the four facilities is ID%, 20%, 30%, and 40%, respectively. The suggested allocation rule then requires that centralized facility 1 should supply each of the two auto manufacturers exactly 10% of its demand. Similarly, centralized facility 2 should supply exactly 20% of each auto manufacturer's demand, and so on.

It is widely believed that centralization means consolidation of inventory in fewer locations. The allocation rule proposed here may seem to contradict that general notion. The contradiction is only apparent, because the proposed rule generates multiple solutions, including the option of fewer centralized locations. Solutions that allow many centralized locations would cease to be optimal if the inventory manager were to impose additional restrictions, such as sole-sourcing. The same will be true if there are strong economies of scale in inventory operations.

NOTES

1D. H. Maister, "Centralization of Inventories and the `Square Root Law,"' International Journal of Physical Distribution, 6: 3 (1976), pp. 124-34.

2W. Zinn, M. Levy, and D. J. Bowersox, "Measuring the Effect of Inventory Centralization/ Decentralization on Aggregate Safety Stock: The `Square Root Law' Revisited," Journal of Business Logistics, 10:1 (1989), pp. 1-14.