An algorithm for the biobjective integer minimum cost flow problem

Abstract

In this paper, we study the single commodity flow problems, optimizing two objectives simultaneously, where the flow values must be integer values. We propose a method that finds all the efficient integer points in the objective space. Our algorithm performs two phases. In the first phase, all integer points on the efficient boundary are found and in the second phase, the efficient integer points that do not lie on the efficient boundary are calculated. In addition, we carry out a computational experiment showing that the number of efficient integer solutions that do not lie on the efficient boundary is greater than the number of integer solutions on the efficient boundary. Scope and purpose In many combinatorial optimization problems, the selection of the optimum solution takes into account more than one criterion. For example, in transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival times at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc. Often, these criteria are in conflict and for this reason, a multiobjective network flow formulation of the problem is necessary. The solution to this problem is searched for among the set of efficient points. Although multiobjective network flow problems can be solved using the techniques available for the multiobjective linear programming problem, network-based methods are computationally better. The multicriteria minimum cost flow problem has already merited the attention of several authors and the case which has been considered in literature is that which has two objectives, where the continuous flow values are permissible. However, the integer case of the biobjective minimum cost flow problem has scarcely been studied. Whereas, in many real network flow problems, integer values on flow values are required. In this paper, we propose an approach to solve the biobjective integer minimum cost flow problem. An algorithm to obtain all efficient integer solutions of this problem is introduced. This method is characterized by the use of the classic resolution tools of network flow problems, such as the network simplex method. It does not utilize the biobjective integer linear programming methodology. Furthermore, the method does not calculate dominated solutions, so it is not necessary to incorporate tools to eliminate dominated solutions.