Branch and peg algorithms for the simple plant location problem

Abstract

The simple plant location problem is a well-studied problem in combinatorial optimization. It is one of deciding where to locate a set of plants so that a set of clients can be supplied by them at the minimum cost. This problem often appears as a subproblem in other combinatorial problems. Several branch and bound techniques have been developed to solve these problems. In this paper we present two techniques that enhance the performance of branch and bound algorithms. The new algorithms thus obtained are called branch and peg algorithms, where pegging refers to fixing values of variables at each subproblem in the branch and bound tree, and is distinct from variable fixing during the branching process. We present exhaustive computational experiments which show that the new algorithms generate less than 60% of the number of subproblems generated by branch and bound algorithms, and in certain cases require less than 10% of the execution times required by branch and bound algorithms. Scope and purpose Locational decision problems of choosing the location of facilities to satisfy the demand of a set of clients at minimum costs constitute a very important class of practical problems. The simple plant location problem is a subclass of such problems, in which the facilities are supposed to have infinite capacities. These problems are used to model, among others, bank account location problems and in cluster analysis (refer, for example, to Cornuejols et al., in Discrete Location Theory, Wiley-Interscience, New York, 1990, p. 119) and also, in recent times, to problems related to electronic business (refer to Fourer and Goux, Interfaces 31 (2001) 130). Not only are these problems important in themselves, but they also appear as subproblems in a much wider class of combinatorial problems, for example in crew scheduling, vehicle despatching, assortment, etc. The simple plant location problem is NP-Hard and there is much research on good algorithms for these problems. Effective solution techniques based on branch and bound algorithms have been suggested in the literature. In this paper we suggest techniques that substantially reduce the execution times of such algorithms. We think that these techniques can be used to enhance the performance of almost all the available solution algorithms for this problem. We hope that this paper will motivate research to apply the techniques suggested here to other problems.