On the sum-max bicriterion path problem

In network analysis, there are many applications in which several weights associated with traversing each arc are given, so it is natural to consider multiple-objective path problems. Usually, there is no path which is simultaneously optimal with respect to all objective functions, so it will be necessary to obtain efficient paths of interest, i.e. paths for which there exists no other path that yields an improvement in one of the objective functions without causing a degradation in the others. Methods for determining efficient paths form part of a very extensive literature on multiple-objective optimization, most of them being proposed for objectives defined by sum functions. Our aim is to study the particular case of a bicriterion problem whose objective functions are defined by a sum and a maximum. A very important aspect of this problem is to find a best compromise solution, which is shown to be equivalent to solve the quickest path problem, which has interesting applications in communication and transportation networks. In this paper, we study a special class of bicriterion path problems where the objective functions are defined by a sum and a maximum: The sum-max bicriterion path problem (SMBPP). After reviewing some special kinds of efficient paths, we propose some algorithms to generate these kinds of efficient paths, based on a progressive reduction of the original network. We analyse its relationship with the quickest path problem (QPP), showing that this is equivalent to the weighted problem associated to the SMBPP, which is also solved by a modification of an algorithm proposed for the QPP. A computational study is presented which shows the superiority of the algorithm proposed in this paper over other existing algorithms to generate the entire set E of efficient paths of the SMBPP.