Facility Location Problems With Equilibrium Constraints: Models And Algorithms

This paper develops a universal generalized bilevel programming model forthe facility location problem with the supply chain network equilibriumconstraints, which describe the competition behaviors of three kinds of decision-makers, manufacturers, retailers and consumers in a supply chain network for aproduct, respectively. The upper level problem is an mixed-integer nonlinearproblem that aims to optimize a utility function by identifying facility locations toproduce a kind of product. However, the lower problem, that is a variationalinequality problem characterizing the supply chain network equilibrium condition,determines the supply prices and shipments from these new facilities to retailers,which are necessary inputs of the upper level problem. Two alterative approachesfor solving the supply chain network equilibrium model developed by variationalinequalities are present. One is the modified projection method. The other is toapply transfer the variabtional inequality formulations into a nonlinearcomplementary problem. A merit function based on Fisher function can be appliedto convert the nonlinear complementary problem to an unconstrained minimizationproblem. Then quasi-newton method can be applied to solve this problem. Tosolve the generalized bilevel programming model involving binary locationdecision variables, a genetic algorithm, that incorporates a modified projectionmethod for solving the variational inequality problem of the lower level problem,is then suggested. Finally, this study employs a hypothetical numerical example todemonstrate the availability of the proposed model and algorithms.