Capacitated two-stage facility location problem with fuzzy costs and demands
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ORIGINAL ARTICLE
Capacitated two-stage facility location problem with fuzzy costs and demands Shuming Wang • Junzo Watada
Received: 28 May 2011 / Accepted: 18 January 2012 / Published online: 15 February 2012 Ó Springer-Verlag 2012
Abstract In this study, we develop a two-stage capacitated facility location model with fuzzy costs and demands. The proposed model is a task of 0–1 integer two-stage fuzzy programming problem. In order to solve the problem, we first apply an approximation approach to estimate the objective function (with fuzzy random parameters) and prove the convergence of the approach. Then, we design a hybrid algorithm which integrates the approximation approach, neural network and particle swarm optimization, to solve the proposed facility location problem. Finally, a numerical example is provided to test the hybrid algorithm. Keywords Location Two-stage fuzzy programming Fuzzy variable Neural network Particle swarm optimization
1 Introduction Since the original study by Cooper [9], facility location problems (FLP) as a crucial and generic engineering optimization model have being attracting an increasing number of people. The key issue of FLP is to find the optimal sizes to open facilities among a given set of potential sites to meet the objective of profit maximization or cost minimization.
S. Wang (&) J. Watada Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu, Fukuoka 808-0135, Japan e-mail: [email protected] J. Watada e-mail: [email protected]
Regarding FLPs with deterministic parameters, capacitated FLP in which the capacities of facilities are limited, was originally discussed by Murtagh and Niwattisyawong [30] which is considered as one of studies of the most importance in the filed of FLP, and later on, more and more studies along this direction (capacitated FLP) have been reported in the literature [1, 4, 14]. Since it was proved by Megiddo and Supowit [29] that FLP is NP-hard, a series of analytical methods and programming techniques as well as heuristic algorithms have been developed to solve the FLPs. For instance, Love [27] discussed one-dimensional facility location–allocation problem using dynamic programming. Gong et al. [15] designed a hybrid evolutionary method for solving obstacle location–allocation problem. Lozano et al. [28] discussed the application of Kohonen maps to solve a class of location–allocation problems. Ernst and Krishnamoorthy [14] combined the simulated annealing and random descent method. In real applications, many parameters of FLP, such as demands of clients, the costs of operating the facilities, may be of uncertainties, e.g., randomness and fuzziness. These distinct uncertainties yields the stochastic FLP and fuzzy FLP, respectively. For stochastic scenarios, readers may refer to [5, 24–26]. Due to the development of fuzzy theory [12, 13, 21, 31, 40, 41], a number of pieces of research brought this tool into the FLP. Darzentas [11] discussed various facility location problems by
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