Variety and Substitution

We study two major issues in retail inventory management: (1) The optimal set of variants (the assortment problem), that is, how many and which variants should be carried by retailers; and (2) the optimal inventory policy, given a set of variants. It is well known that it is difficult to reach analytically tractable solutions solving these problems because of the randomness in the total market demand, in consumer choices, and the resulting substitution behaviors. To the best of our knowledge, this is the first study that provides structural results to a large family of substitution problems. We start by considering a single product category and a single profit-maximizing retailer. Demands for each of the variants are uncertain and are generated from a consumer choice model. For each customer, a realization of the consumer choice model is a ranking of the variants, from the most desirable variant to the least desirable one. Also, every consumer has a critical utility (no-purchasing-utility) such that if the utility resulting from the purchase of a certain variant is below the critical utility level, then the consumer will not purchase this variant. As a result, a consumer may have a set of variants that he or she is unwilling to purchase. When a certain variant that consumers desire to purchase is not available, they purchase the highest-ranked available variant with a utility that is higher than the no-purchase utility. We assume that each customer purchases at most one unit of the product, and we are able to determine an inventory policy which results in the optimal expected cost with a very high probability. We are also able to determine the optimal number of variants with a very high probability. Variety and Substitution Feng and Bassok 2