Using a Genetic Algorithm to Solve the Benders’ Master Problem for Capacitated Plant Location

Abstract

The capacitated plant location problem (CPL) consists of locating a set of potential plants with capacities, and assigning a set of customers to these plants. The objective is to minimize the total fixed and shipping costs while at the same time demand of all the customers can be satisfied without violating the capacity restrictions of the plants. The CPL is a well-known combinatorial optimization problem and a number of decision problems can be obtained as special cases of CPL. There are substantial numbers of heuristic solution algorithms proposed in the literature (See Rolland et al., 1996; Holmberg & Ling, 1997; Delmaire et al., 1999; Kratica et al., 2001; He et al., 2003; Uno et al., 2005). As well, exact solution methods have been studied by many authors. These include branch-and-bound procedures, typically with linear programming relaxation (Van Roy & Erlenkotter, 1982; Geoffrion & Graves, 1974) or Lagrangiran relaxation (Cortinhal & Captivo, 2003). Van Roy (1986) used the Cross decomposition which is a hybrid of primal and dual decomposition algorithm, and Geoffrion & Graves (1974) considered Benders’ decomposition to solve CPL problem. Unlike many other mixed-integer linear programming applications, however, Benders decomposition algorithm was not successful in this problem domain because of the difficulty of solving the master system. In mixed-integer linear programming problems, where Benders’ algorithm is most often applied, the master problem selects values for the integer variables (the more difficult decisions) and the subproblem is a linear programming problem which selects values for the continuous variables (the easier decisions). If the constraints are explicit only in the subproblem, then the master problem is free of explicit constraints, making it more amenable to solution by genetic algorithm (GA). The fitness function of the GA is, in this case, evaluated quickly and simply by evaluating a set of linear functions. In this chapter, therefore, we discuss about a hybrid algorithm (Lai et al., 2010) and its implementation to overcome the difficulty of Benders’ decomposition. The hybrid algorithm is based on the solution framework of Benders’ decomposition algorithm, together with the use of GA to effectively reduce the computational difficulty. The rest of