Multi-location assortment optimization under capacity constraints

We study the assortment optimization problem in an online setting where a retailer determines the set of products to carry in each of its distribution centers under a capacity constraint so as to maximize its expected profit (revenue minus the shipping costs). It is assumed that each distribution center is primarily responsible for a geographical location whose customers' choice is governed by a separate multinomial logit model. A distribution center can satisfy a demand of a region that it is not primarily responsible for, but this incurs an additional shipping cost for the retail company. We consider two variants of this problem. In the first variant, customers have access to the entire assortment in all locations but in the second variant, the online retail company can select which product to show to each region. Under each variant, we first assume that there is a constant shipping cost for all products between any two location. In the second case, we allow the shipping costs to differ based on the origin and destination. We develop conic quadratic mixed integer programming formulations and suggest a family of valid inequalities to strengthen these formulations. Numerical experiments show that our conic approach, combined with valid inequalities over-perform the mixed integer linear programming formulation and enables us to solve large instances optimally. Finally, we study the effect of various factors such as no-purchase preference, capacity constraint and shipping cost on company's profitability and assortment selection.