An Interactive Polyhedral Approach for Multi-objective Combinatorial Optimization with Incomplete Preference Information

In this paper, we develop a general interactive polyhedral approach to solve multi-objective combinatorial optimization problems with incomplete preference information. Assuming that preferences can be represented by a parameterized scalarizing function, we iteratively ask preferences queries to the decision maker in order to reduce the imprecision over the preference parameters until being able to determine her preferred solution. To produce informative preference queries at each step, we generate promising solutions using the extreme points of the polyhedron representing the admissible preference parameters and then we ask the decision maker to compare two of these solutions (we propose different selection strategies). These extreme points are also used to provide a stopping criterion guaranteeing that the returned solution is optimal (or near-optimal) according to the decision maker's preferences. We provide numerical results for the multi-objective spanning tree and traveling salesman problems with preferences represented by a weighted sum to demonstrate the practical efficiency of our approach. We compare our results to a recent approach based on minimax regret, where preference queries are generated during the construction of an optimal solution. We show that better results are achieved by our method both in terms of running time and number of questions.