Efficient Lower and Upper Bounds for the Multi-commodity Capacitated Multi-facility Weber Problem with Rectilinear Distances

Abstract

The capacitated multi-facilityWeber problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a nonconvex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped to the customers subject to capacity and demand constraints. Customer locations, demands and capacities for each commodity are known a priori. The transportation costs, which are proportional to the rectilinear distance between customers and facilities, depend on the commodity type. We first present three different equivalent mathematical programming formulations of the problem. Then we propose Lagrangean relaxation schemes for these formulations to obtain lower bounds on the problem. Upper bounds are produced by using an alternate location-transportation heuristic. Computational experiments on randomly generated test instances are also reported.