Inventory Models for Substitutable Products: Optimal Policies and Heuristics

In this paper, we examine the nature of optimal inventory policies in a system where a retailer manages substitutable products. We first consider a system with two products 1 and 2 whose total demand is D and individual demand proportions are p and (1-p). A fixed proportion of the unsatisfied customers for 1(2) will purchase item 2 (1), if it is available in inventory. For the single period case, we show that the optimal inventory levels of the two items can be computed easily and follow what we refer to as "partially decoupled" policies, i.e. base stock policies that are not state dependent, in certain critical regions of interest both when D is known and random. Furthermore, we show that such a partially decoupled base-stock policy is optimal even in a multi-period version of the problem for known D for a wide range of parameter values. Using a numerical study, we show that heuristics based on the de-coupled inventory policies perform well in conditions more general than the ones assumed to obtain the analytical results. The analytical and numerical results suggest that the approach presented here is most valuable in retail settings for product categories where there is a moderate level of substitution between items in the category, demand variation at the category level is not too high and service levels are high.