The Capacitated Multi-facility Weber Problem (CMWP) is concerned with locating I capacitated facilities so as to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a non-convex optimization problem and is difficult to solve. This work focuses on a multi-commodity extension and considers 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 distance between customers and facilities, depend on the commodity type. A mathematical programming formulation of the problem is presented and two alternate location-allocation heuristics and a discrete approximation method are proposed and subsequently used to statistically estimate confidence intervals on the optimal objective function values. Computational experiments on standard and randomly generated test instances are also presented.