A Branch and Bound Algorithm for Choquet Optimization in Multicriteria Problems

Abstract

This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial optimization with application to spanning tree problems and knapsack problems. After recalling basic notions concerning the use of Choquet integrals for preference aggregation, we present a condition (named preference for interior points) that characterizes preferences favouring well-balanced solutions, a natural attitude in multicriteria optimization. When using a Choquet integral as preference model, this condition amounts to choosing a submodular (resp. supermodular) capacity when criteria have to be minimized (resp. maximized). Under this assumption, we investigate the determination of Choquet-optimal solutions in the multicriteria spanning tree problem and the multicriteria 0-1 knapsack problem. For both problems, we introduce a linear bound for the Choquet integral, computable in polynomial time, and propose a branch and bound procedure using this bound. We provide numerical experiments that show the actual efficiency of the algorithms on various instances of different sizes.

Choquet optimal set in biobjective combinatorial optimization

“…We have only listed three previous works on this topic, none of them before 2009 [8][9][10]. Unlike the algorithm presented in this paper, these works only focus on the generation of one C-optimal solution given a capacity.…”

“…The WOWA optimal set is equal to the C-optimal set in biobjective optimization. In this example, the point (10,12) is the only point which is not WS optimal or OWA optimal but C-optimal.…”

“…In [10], Galand et al search for C-optimal solutions of multiobjective spanning trees problems and multiobjective knapsack problems. They present a condition (named preference for interior points) that characterizes preferences favoring well-balanced solutions: the capacity v has to be supermodular (in the case of maximization of the objectives), that is ∀A; B∈2 P ; vðA∪BÞ þ vðA∩BÞ≥vðAÞ þ vðBÞ.…”

“…If only two objectives are considered, we can wonder whether there is a difference between the Choquet integral and the WOWA operator. (10,12), with the capacity function vðf1gÞ ¼ 0 and vðf2gÞ ¼ 1 2 . For this capacity, the points (10,12), (13,11) and (9,13) have the same Choquet integral evaluations.…”

“…Some papers already deal with the optimization of the Choquet integral of multiobjective combinatorial optimization problems [8][9][10] when a specific capacity is given by the decision-maker. But to our knowledge, the development of a method to generate the whole set of Choquet optimal solutions has not yet been studied.…”