Product line optimization with multiples sites.

We consider the problem faced by a retail chain that must select what mutual-substitute items to display each one of its stores to maximize revenues. The number of items cannot exceed the limit space capacity of each store. Customers purchase the one product that maximizes their utility, which depends on the product price, travel cost to the store, and reservation price, known to the retailer. The retailer can set different price markdowns at different stores and products. The retailer considers the decisions of customers, and solves mixed-integer bilevel optimization problem, which can be formulated as a single-level optimization problem by using optimality conditions for the lower level. We propose Branch and Cut and Cut and Branch methods and include a family of valid inequalities to solve the problem. We compare the results with those of a Benders decomposition method. Our computational results show that the proposed Cut and Branch method obtains the best performance and improves the current state of the art.