Estimation of Inventory Re-Order Levels Using the Bootstrap Statistical Procedure

Abstract

It can be difficult to set the reorder point in an inventory system because often one does not have much knowledge of the lead-time demand (LTD)distribution. A frequent practice is to assume a “standard” distribution, such as the Normal. The reorder point is then taken as the p-th fractile of that standard distribution, where (1 −p) is the specified probability of stockout during a replenishment cycle.

When the desired service level p is high and the “true” LTD distribution is skewed, previous research has shown that the reorder point and inventory costs are strongly affected by the shape of the assumed LTD distribution. Ideally, no assumptions about this distribution should be necessary.

One such “distribution-free” approach is the bootstrap procedure. Beginning with a single sample W = {x₁x₂,…,x_n} of lead-time demand, the bootstrap repeatedly samples with replacement from W, A family of bootstrap samples of size n is thereby created, each sample furnishing an estimate X_p∗ of the p-th fractile of the LTD distribution. The probability mass function of these values yields the bootstrap estimate of the reorder point [[Xcirc]_p = E(X_p)].

We employ the bootstrap procedure to determine the reorder point s in an (s,Q) system for desired service levels p = 0.8, 0.9 and 0.95. The bootstrap and normal approaches are compared using LTD data simulated from a number of populations with varying tail shapes.

Analyses of variance are performed to compare the estimated reorder points across various levels of distribution form, coefficient of variation, sample size and service level. The relative performance of the bootstrap and normal approaches is also discussed in terms of a cost model.

We found it preferable to use the bootstrap procedure in any situation where a “non-standard’ LTD distribution, e.g. a bimodal one, is suspected. If the distribution actually is a standard shape, such as normal or lognormal, the bootstrap estimate is often as good as the normal. Situations are also identified where the bootstrap procedure should not be used, and suggestions are made for further research.