A demand-shifting feasibility algorithm for Benders decomposition

Abstract

Benders decomposition is a popular method for solving problems by resource-directive decomposition. Often, the resource allocations from the master problem lead to infeasible subproblems, as resources are insufficient to meet demand. This generally requires the use of feasibility cuts to reach a feasible solution, which can be computationally expensive. For problems in which subproblems have limited capacity, we propose an efficient algorithm which shifts excess demand to other sources of capacity. The advantages of the algorithm are that it is quick, requires only one solution of each subproblem in each Benders iteration, and does not add any feasibility cuts into the master problem. A computational study is performed on a fleet sizing problem to illustrate the algorithm’s efficiency when compared to the traditional feasibility cut method.

Introduction

Benders decomposition (Benders, 1962), also known as Benders’ partitioning or outer linearization, is a popular method for solving problems by resource-directive decomposition. It has been applied, with success, to both linear and mixed integer programming problems in a variety of applications. These include solving two-stage recourse problems (Van Slyke and Wets, 1969), the multi-commodity distribution system design problem (Geoffrion and Graves, 1974), the quadratic assignment problem (Bazaraa and Sherali, 1980), hierarchical production planning problems (Aardal and Larsson, 1990) and the parallel replacement problem (Chen, 1998), among others.

In general, Benders decomposition works in an iterative fashion. First, the configurative variables (usually the complicating variables) are determined in a master problem and then the remaining, lower-level decision variables are found in subproblems. At each iteration, optimality cuts are generated from the subproblem dual variables and added into the master problem. It is well known that solving the Benders master problem can be computationally intensive. Magnanti and Simpson (1978) and Magnanti and Wong (1981) proposed finding “pareto-optimal” cuts to accelerate Benders decomposition, with multiple optimal solutions from the subproblems. Minoux (1984) suggested solving the master problem only to obtain a feasible solution and a lower bound, due to the disadvantage of degeneracy and round-off errors in the simplex method. Aardal and Larsson (1990) priced out the Benders primal cuts such that the remaining Lagrangean relaxation problem could be solved easily by dynamic programming. However, Holmberg (1994) showed that relaxing the Benders primal cuts does not yield a lower bound that is as good as the bound from Lagrangean relaxation of the original problem.

In this paper, we are concerned with the feasibility of the subproblems once the master problem has been solved and allocations to the subproblems have been made. This is not always a problem. In Geoffrion and Graves (1974), it is assumed that the transportation subproblem always has capacity that is greater than the total demand, so as to preclude the possibility of an infeasible solution. The Benders’ subproblems of the hierarchical production planning model are always feasible (Aardal and Larsson, 1990), as unlimited overtime production is allowed. Similarly, as the assumption of constant demand implies that the total number of machines is constant over time in (Chen, 1998), the subproblem for each asset group does not encounter infeasibility. The power plant expansion problem has complete recourse, provided that at least one zero-leadtime technology is available (Birge and Louveaux, 1997). In this paper, we are concerned with problems in which the subproblems have fixed, limited capacity. Specifically, we illustrate this problem with a rental fleet sizing model which decomposes the fleet by truck type and age. As it is assumed that only new trucks can be purchased, the subproblems with older trucks have an upper bound on capacity. While it is possible to penalize the acquisition of additional capacity with arbitrarily high prices, it has been our experience that the convergence of the Benders procedure may be slow with this assumption, thus further motivating the development of our algorithm.

Classical Benders decomposition deals with the infeasibility of subproblems by means of generating feasibility cuts (Van Slyke and Wets, 1969). Obviously, solution of the master problem is more computationally expensive with the addition of both optimality and feasibility cuts. We are therefore motivated to propose a feasibility algorithm which shifts excess demand to other sources of capacities (from the infeasible subproblem to a different subproblem) such that a feasible solution can be found quickly, and no additional computational burden is placed on solving the master problem in the context of our rental fleet sizing problem. The algorithm, by construction, also has the benefit of only requiring one solution of each subproblem in each iteration of the Benders procedure. Thus, the master problem need not be reformulated and solved repeatedly until all the subproblems are feasible.

We review the traditional feasibility cut approach of Benders decomposition and present the details of the demand-shifting feasibility algorithm in Section 2. In Section 3, a case study on a fleet sizing problem illustrates how the demand-shifting algorithm works effectively on dynamic problems of multiple dimensions (space and time). The computational effectiveness of the algorithm is analyzed, along with a comparison to a traditional, feasibility-cut generating scheme. Conclusions and directions for future research are given in the final section.