Value-at-risk-based two-stage fuzzy facility location problems

Reducing risks in location decisions when coping with imprecise information is critical in supply chain management so as to increase competitiveness and profitability. In this paper, a two-stage fuzzy facility location problem with Value-at-Risk (VaR), called VaR-FFLP, is proposed, which results in a two-stage fuzzy zero-one integer programming problem. Some properties of the VaR-FFLP, including the value of perfect information (VPI), the value of fuzzy solution (VFS), and the bounds of the fuzzy solution, are discussed. Since the fuzzy parameters of the location problem are represented in the form of continuous fuzzy variables, the determination of VaR is inherently an infinite-dimensional optimization problem that cannot be solved analytically. Therefore, a method based on the discretization of the fuzzy variables is proposed to approximate the VaR. The Approximation Approach converts the original problem into a finite-dimensional optimization problem. A pertinent convergence theorem for the Approximation Approach is proved. Subsequently, by combining the Simplex Algorithm, the Approximation Approach, and a mechanism of genotype-phenotype- mutation-based binary particle swarm optimization (GPM-BPSO), a hybrid GPM-BPSO algorithm is being exploited to solve the VaR-FFLP. A numerical example illustrates the effectiveness of the hybrid GPM-BPSO algorithm and shows its enhanced performance in comparison with the results obtained by other approaches using genetic algorithm (GA), tabu search (TS), and Boolean BPSO (B-BPSO).