Evaluating solution sets of a posteriori solution techniques for bi-criteria combinatorial optimization problems

The quality of an approximate solution for combinatorial optimization problems with a single objective can be evaluated relatively easily. However, this becomes more difficult when there are multiple objectives. One potential approach to solving multiple criteria combinatorial optimization problems when at least one of the single objective problems is NP-complete, is to use an a posteriori method that approximates the efficient frontier. A common difficulty in this type of approach, however, is evaluating the quality of approximate solutions, since sets of multiple solutions should be evaluated and compared. This necessitates the use of a comparison measure that is robust and accurate. Furthermore, a robust measure plays an important role in metaheuristic optimization for ‘tuning’ various parameters for evolutionary algorithms, simulated annealing, etc., which are frequently employed for multiple criteria combinatorial optimization problems. In this paper, the performance of a new measure, which we call Integrated Convex Preference (ICP), is compared to that of other measures appearing in the literature through numerical experiments – specifically, we use two a posteriori solution techniques based on genetic algorithms for a bi-criteria parallel machine scheduling problem and evaluate their performance (in terms of solution quality) using different measures. Experimental results show that the ICP measure evaluates the solution quality of approximations robustly (i.e., similar to visual comparison results) while other alternative measures can misjudge the solution quality. We note that the ICP measure can be applied to other non-scheduling multiple objective combinatorial optimization problems, as well.