A hybrid modified PSO approach to VaR-based facility location problems with variable capacity in fuzzy random uncertainty Shuming Wang, Junzo Watada * Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan article info Article history: Received 12 August 2009 Received in revised form 23 January 2010 Accepted 8 February 2010 Available online xxxx Keywords: Facility location Fuzzy random variable Value-at-Risk Particle swarm optimization Mixed 0–1 integer fuzzy random programming Variable capacity abstract This paper studies a facility location model with fuzzy random parameters and its swarm intelligence approach. A Value-at-Risk (VaR) based fuzzy random facility location model (VaR-FRFLM) is built in which both the costs and demands are assumed to be fuzzy random variables, and the capacity of each facility is unfixed but a decision variable assuming con- tinuous values. Under this setting, the VaR-FRFLM is inherently a two-stage mixed 0–1 integer fuzzy random programming problem, to which analytical nonlinear programming methods are not applicable. A hybrid modified particle swarm optimization (MPSO) approach is proposed to solve the VaR-FRFLM. In this hybrid mechanism, an approximation algorithm is utilized to com- pute the fuzzy random VaR objective, a continuous Nbest–Gbest-based PSO and a geno- type–phenotype-based binary PSO vehicles are designed to deal with the continuous capacity decisions and the binary location decisions, respectively, and two mutation oper- ators are incorporated into the PSO to further decrease the possibility of becoming trapped in the local optima. A numerical experiment illustrates the application of the proposed hybrid MPSO algorithm and lays out its robustness to the system parameter settings. The comparison shows that the hybrid MPSO exhibits better performance than that when hybrid regular continuous-binary PSO and genetic algorithm (GA) are used to solve the VaR-FRFLM. Ó 2010 Published by Elsevier Inc. 1. Introduction Facility location selection is a category of optimization problems that aim to maximize the return or minimize the costs via determining the locations of facilities to open from a set of potential sites. Various kinds of facility location problems with uncertainty have been investigated in the literature. The first category is the stochastic facility location problems which deal with the cases when the uncertain parameters, like customers’ demands and operating costs of plants, are characterized by random variables. For instance, Logendran and Terrell [22] developed a stochastic uncapacitated transportation plant loca- tion-allocation model with the objective of maximizing the expected profits, and they proposed some heuristics to solve the problem. Louveaux and Peeters [23] discussed a dual-based procedure for a stochastic facility location problem with re- course. Laporte et al. [14] formulated a class of capacitated facility location problems with random demands by using sto- chastic integer linear programming, and proposed a branch and cut solution approach. Schutz et al. [29] considered a stochastic facility location problem with general long-run costs and convex short-run costs, and solved the problem through 0020-0255/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.ins.2010.02.014 * Corresponding author. E-mail addresses: smwangips@gmail.com (S. Wang), junzow@osb.att.ne.jp (J. Watada). Information Sciences xxx (2010) xxx–xxx Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins ARTICLE IN PRESS Please cite this article in press as: S. Wang, J. Watada, A hybrid modified PSO approach to VaR-based facility location problems with var- iable capacity in fuzzy random uncertainty, Inform. Sci. (2010), doi:10.1016/j.ins.2010.02.014