Lower and Upper Bounds for the Rectilinear Distance Single Source Capacitated Multi-facility Weber Problem M. Emre Demircio ˘ glu 1 , M. Hakan Akyüz 1 , Temel Öncan 1 , ˙ I. Kuban Altınel 2 1 Department of Industrial Engineering, Galatasaray University Ortaköy, ˙ Istanbul, 34357, TÜRK ˙ IYE {edemircioglu,ytoncan,mhakyuz}@gsu.edu.tr 2 Department of Industrial Engineering, Bo ˘ gaziçi University Bebek, ˙ Istanbul, 34342, TÜRK ˙ IYE altinel@boun.edu.tr Mots-clés : Location Allocation Problem, Lagrangean Relaxation, Heuristics. 1 Introduction Given the locations of J customers and their demands, the Capacitated Multi-facility Weber Pro- blem (CMWP) is concerned with locating I facilities with limited capacities (known a priori) in the plane and satisfying the demands of the customers at minimum total cost. The objective function of the CMWP is neither convex nor concave [4], which makes it hard to solve exactly. It becomes the so-called Multi-facility Weber Problem (MWP) when the capacity restrictions on facilities are igno- red. The MWP further reduces to the well known (single facility) Weber Problem (WP) when the objective is to determine optimal location of a single facility. The CMWP is a multi-source problem which is shown to be NP-Hard by Sherali and Nordai ([8]). A vast amount of literature is devoted to the CMWP starting from the seminal work by Cooper ([4]) for the Euclidean distance CMWP (ECMWP). Single source CMWP (SCMWP) is a special case of the CMWP where each customer satisfies all its demand from a single facility. This case may arise in real-world problems such as the Single source Transportation Problem (STP) and the Covering Assignment Problem (CAP). In the STP, which has first been addressed by DeMaio and Roveda ([5]), the customers must be served by exactly one facility. Foulds and Wilson ([6]) studied the CAP, in the context of allocation problems in the New Zealand dairy industry, where the milk demand of dairy companies is supplied by different farms. The CAP deals with the allocation of farms to factories such that each farm supplies exactly one factory. Consider a set of J customers whose known locations are denoted by a j = ( a j1 a j2 ) T with a demand of q j for j = 1,..., J . Given a set of I facilities whose unknown locations are denoted by x i = ( x i1 x i2 ) T with a capacity of s i for i = 1,..., I . Let binary decision variable w ij equals to 1 if and only if the demand of customer j is completely satisfied from facility i, with a shipment cost of c ij per unit distance where the distance is measured by d (x i , a j )= |x i1 − a j1 | + |x i2 − a j2 | between facility i and customer j. A mathematical formulation of the SCMWP can be stated as follows : min Z = I ∑ i=1 J ∑ j=1 w ij q j c ij d (x i , a j ) (1) subject to J ∑ j=1 q j w ij ≤ s i i = 1,..., I (2)