Technical Note—Leveraging the Degree of Dynamic Substitution in Assortment and Inventory Planning

We study the joint assortment and inventory planning problem with stockout-based substitution. In this problem, we pick the number of units to stock for the products at the beginning of the selling horizon. Each arriving customer makes a choice among the set of products with remaining on-hand inventories. Our goal is to pick the stocking quantities to maximize the total expected revenue from the sales net of the stocking cost. We develop a rounding scheme that uses the solution to a fluid approximation to generate stocking quantities with performance guarantees that improve earlier results. Letting T be the number of time periods in the selling horizon and n be the number of products, when customers choose under a general choice model, we show that we can round the solution to the fluid approximation to obtain stocking quantities with an optimality gap of O ( n T ) , improving earlier optimality gaps by a logarithmic factor. More importantly, when customers choose under the multinomial logit model, by leveraging the degree of substitution, we show that our rounded fluid solution is within an optimality gap of O ( log T T log T ) . The optimality gap that we give under the multinomial logit model is the first one that does not depend on the number of products. Such an optimality gap has important practical implications. Earlier results cannot guarantee that the stocking quantities generated by the fluid approximation perform well when both the demand volume and the number of products are large, which is a regime becoming more relevant for online retail applications with large product variety. In contrast, we can guarantee that stocking quantities generated by our rounding scheme perform well when both the demand volume and the number of products are large.