Inventory control under substitutable demand

Many retailers are trying to expand their product lines to compete for market share. This is not only true for an individual retailer it is also true across the retailers. As the retailers offer more similar products, they find that these products are substitutable in the eyes of their customers. When selling substitutable products demand for a particular product depends on the inventory positions of the other products. Thus, the effective demand for a particular product is correlated to the demand for the other products. These dependencies make it challenging for the retailers to generate accurate demand forecast for each product and to determine the right order quantities so as to maximize its own profit. Survey results reported at the Harvard/Wharton Merchandising Effectiveness project have found that these subjective forecasts tend to have an average forecasting error of 50% or more. As a result, some retailers buy too little of some products resulting in lost sales and profit margin, and some retailers buy too much of some products resulting in excess supply that must be marked down after a while, frequently to the point where the product is sold at a loss. In this research, we study the multi-period multi-product inventory control problem of a single or multiple retailers in which a product can be used to substitute for a portion of the unmet demand of another out-of-stock product. We consider both the competitive case where each retailer sells a dedicated product that is substitutable with his competitor's products, and the cooperative case where a single retailer sells multiple substitutable products. Competitive substitutable product inventory management problem is analyzed using the concepts of stochastic game theory. Multiple retailers offering substitutable products try to maximize their own profit over infinite time horizon. Demand for each product is random, and a fixed portion of demand could be met by another item in case of stock out. Inventory position is observed periodically, and then the ordering decision of each retailer is given simultaneously. This process, repeated over time, can be modeled as a stochastic game. In the case of competing retailers, under the discounted payoff criterion, this problem is formulated as a nonzero-sum stochastic game. In these games, we need to concentrate on the inventory control policies that correspond to a Nash equilibrium. In such an equilibrium, unilateral deviations of either of the retailers from its Nash strategy do not improve its expected profit. Thus, each retailer's policy is optimal against those of the others. We compare the optimal order quantities obtained when substitution is possible versus when there is no product substitution. We investigate the structure of the optimal policy under substitution. In the case of two retailers and linear ordering cost, it can be shown that there exists a unique Nash equilibrium characterized by stationary base stock strategies for the infinite horizon problem. We also consider the case of cooperating retailers, which dominates the non-cooperative solution alternatives, in the sense of giving lower expected payoff than the sum of each retailer's payoff in the non-cooperative case.