Appl. Comput. Math., V.11, N.1, 2012, pp. LAGRANGIAN HEURISTIC FOR THE TWO-STAGE CAPACITATED FACILITY LOCATION PROBLEM I.S.LITVINCHEV 1 , M.MATA 2 , L.OZUNA 3 Abstract. In the two-stage capacitated facility location problem a single product is produced at some plants in order to satisfy customer demands. The product is transported from these plants to some depots and then to the customers. The capacities of the plants and depots are limited. The aim is to select cost minimizing locations from a set of potential plants and depots. This cost includes fixed cost associated with opening plants and depots, and variable cost associated with both transportation stages. In this work a Lagrangian relaxation is analyzed and a Lagrangian heuristic producing feasible solutions is presented. The results of a computational study are reported. Keywords: Two Stage Capacitated Facility Location, Lagrangian Relaxation, Lagrangian Heuris- tic, Subgradient Technique. AMS Subject Classification: 90C59. 1. Introduction The two-stage capacitated facility location problem can be defined as follows: a single product is produced at plants and then transported to depots, both having limited capacities. From the depots the product is transported to customers to satisfy their demands. The use of the plants/depots incurs a fixed cost, while transportation from the plants to the customers through the depots results in a variable cost. We need to identify what plants and depots to use, as well as the product flows from the plants to the depots and then to the customers such that the demands are met at a minimal cost. The literature on facility location problem and its applications is quite extensive. We refer here only for a few recent review papers [8, 14, 24, 26]. In what follows we focus more on approximate techniques to solve facility location problems. For exact approaches see, e.g., [1, 15] and the references therein. Approximate approaches can be roughly divided into two large groups: metaheuristics and Lagrangian based techniques [30]. Metaheuristic approaches to the problem like tabu search, GRASP, are discussed in [9, 28]. A cross entropy-based metaheuristic algorithm for the capaci- tated facility location problem was studied in [5]. An algorithm for large instances is presented in [2], they used a heuristic procedure that produces a feasible integer solution and used a Lagrangian relaxation to obtain a lower bound on the optimal value. A Lagrangian based heuristic for solving the capacitated plant location problem with side constraints was presented in [27]. Approaches and relaxations proposed in the literature for the capacitated facility location problem are compared in [7]. Fenchel cutting planes methodology *Work is supported by RFBR Grant No. 09-01-00592 1 Dorodnicyn Computing Centre of RAS, Vavilov str. 40, Moscow, 119333, Russia, e-mail:litvin@ccas.ru 2 Faculty of Mechanical and Electrical Engineering, UANL Pedro de Alba S/N, San Nicolas de los Garza NL 66450, Mexico 3 Faculty of Mechanical and Electrical Engineering, UANL Pedro de Alba S/N, San Nicolas de los Garza NL 66450, Mexico Manuscript received 29 October 2009. 1