Interactive algorithms to solve biobjective and triobjective decision making problems

Abstract

We propose interactive algorithms to find the most preferred solution of biobjec-tive and triobjective integer programming problems. The algorithms can be used in any setting where the decision-maker has a general monotone utility function. They divide the image space of the problems into boxes and search them by solv-ing Pascoletti-Serafini scalarizations, asking questions to the decision-maker so as to eliminate boxes whenever possible. We also propose a cone based approach that can be incorporated into both algorithms if the decision-maker is assumed to have a non-decreasing quasiconcave utility function. We demonstrate the performances of the algorithms and their cone based extensions with computational experiments. The results of the experiments show that interactive algorithms are very useful in terms of solution time compared to a posteriori algorithms that find the whole Pareto set. The results of the experiments also show that the cone based approach leads to less interaction with the decision-maker.