Relating the Approximability of the Fixed Cost and Space Constrained Assortment Problems

Customer choice models are a fundamental tool used by retailers to optimize inventory, pricing, and stocking decisions. One important optimization problem is the assortment problem in which the retailer wants to determine which products to offer to customers to maximize expected revenue. This paper relates two variants of this assortment problem: the space constrained assortment problem, in which the retailer has a limit on the total space of the offered assortment, and the fixed cost assortment problem, in which the retailer incurs a fixed cost for each offered product. In particular, we develop an approximation scheme for the space constrained problem for any random utility choice model that only relies on the ability to solve the corresponding fixed cost assortment problem. We then apply this technique to give the first constant factor approximation scheme for the space constrained assortment problem under a general choice model for vertically differentiated products. Last, we present computational results to show the efficacy of this approach