Exact and Asymptotically Exact Solutions for a Class of Assortment Problems

Mass customization requires us to select a few types of resources to produce heterogeneous classes of products. In the assortment problem addressed here, a resource unit of type j yields, at a cost c_ij, a batch of a_ij product units of type i. The problem, a generalization of the p-median, calls for (i) choosing a restricted subset of resource types and (ii) assigning resource units to products so as to fulfill a given demand vector at a minimum cost. For this problem, we develop a branch-and-price scheme that can either be used to find optimal solutions, or tuned by choosing columns in a suitable class so as to get approximate solutions. The solutions obtained in the second case approach the optimum by a ratio that asymptotically reduces to zero as the demand of the least-required product increases. A comparative analysis of the features of the algorithm is discussed for a wide set of large problem instances.