The Comparison of Multi-objective Preference Inference Based on Lexicographic and Weighted Average Models

In this paper, we consider the effect of different order relations on the solutions of Multi-Objective Constraint Optimization Problems (MOCOP) with tradeoffs, where the tradeoffs are given in the form of elicited or observed preferences over alternatives. In MOCOP, alternatives are evaluated on a number of objectives (utility scales) and thus correspond to utility vectors, the set of optimal solutions corresponds to the set of undominated alternatives with respect to some order relation on the utility vectors. Thus, the choice of an order relation on the utility vectors is crucial, a strong order relation results in a smaller set of solutions which can be helpful for the decision maker. Our focus lies on the comparison between Pareto, weighted average and lexicographic orderings. We show that every inference that can be made from a set of given preferences considering weighted average orders can be made for lexicographic orders as well. Further results on the relation between the sets of optimal solutions corresponding to lexicographic and weighted average orders are established under the distinction between strict and non-strict preferences. For solving MOCOP, we apply variants of Preference Inference for the different order relations as dominance checks. Our experimental results show that lexicographic orders give much stronger inferences than Pareto and weighted average orders. However, the lexicographic order based algorithm also results in a longer running time than the other two.

I. Introduction

In decision making based on a number of objectives, the number of Pareto optimal solutions can sometimes be very large; it can be helpful for the user to have some way of focusing on a smaller set. However, it may not be satisfactory to collapse the different objectives to a single scale of utility, since we may not know which objectives are regarded as more important by the decision maker. An alternative approach is to attempt to make use of a (possibly small) number of elicited preferences, and generalise from these to infer further preferences, which then can be used to reduce the number of undominated solutions. The number of preference inferences depends on the assumptions one makes on the user model. If one makes weak assumptions, then the inferences will be more reliable, but we may obtain little reduction of the undominated set of solutions; if, on the other hand, one makes strong assumptions, then one may be able to reduce the undominated set much more, although the inferences are less reliable.