Capacitated assortment optimization and pricing problems under mixed multinomial logit model

We study capacitated assortment optimization problem under mixed multinomial logit model where a retailer wants to choose the set of products to offer to various customer segments with the goal of maximizing revenue while satisfying different capacity constraints. Each customer segment is identiffed with a unique purchase behaviour modelled by multinomial logit demand. We consider three general cases of capacity constraints: single resource constraint, multiple resource constraints and multiple cardinality constraints. This problem is NP-hard and there exist two approaches to find exact solutions: formulating the problem as a mixed integer linear program (MILP) or a mixed integer conic quadratic program (CONIC). For each constraint structure, we develop new efficient procedures to derive McCormick valid inequalities. We provide extensive numerical studies the results of which demonstrate that when the CONIC model is accompanied with the McCormick inequalities, the problem can be solved effectively even for large sized instances using a commercial optimization software. We also study joint pricing and assortment optimization problem with a single cardinality constraint and establish a new procedure to construct McCormick inequalities. We then present the related numerical studies which indicate that the CONIC formulation accomplishes the best outcome in the presence of the McCormick inequalities.