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.

We study the assortment optimization problem in an online setting where a retailer uses multiple distribution centers (DC) to fulfill orders from multiple regions. Customer choice in each region follows a multinomial logit model. Each DC can carry up to a pre‐specified number of products. Outbound shipping cost to a region depends on the DC that ships the order. The problem is to determine which products to carry in each DC and which products to offer for sale in each region to maximize the expected profit. We first show that the problem is NP‐complete. We develop a conic quadratic mixed integer programming formulation and suggest a family of valid inequalities. We also show that a special case with identical choice models can be solved as a linear program. This LP solution approach can be used to develop heuristics for the general case. Numerical experiments show that our conic approach outperforms the mixed integer linear programming formulation and enables us to solve moderately sized instances optimally. The experiments also show that not allowing cross‐shipments or not considering them in assortment decisions may lead to substantial losses and LP‐based heuristics can be effective in practice.