On the Computation of Choquet Optimal Solutions in Multicriteria Decision Contexts

We study in this paper the computation of Choquet optimal solutions in decision contexts involving multiple criteria or multiple agents. Choquet optimal solutions are solutions that optimize a Choquet integral, one of the most powerful tools in multicriteria decision making. We develop a new property that characterizes the Choquet optimal solutions. From this property, a general method to generate these solutions in the case of several criteria is proposed. We apply the method to different Pareto non-dominated sets coming from different knapsack instances with a number of criteria included between two and seven. We show that the method is effective for a number of criteria lower than five or for high size Pareto non-dominated sets. We also observe that the percentage of Choquet optimal solutions increase with the number of criteria.

[1]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[2]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[3]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[4]  Alain Chateauneuf,et al.  Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion , 1989, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[5]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .

[6]  Thibaut Lust,et al.  Choquet optimal set in biobjective combinatorial optimization , 2013, Comput. Oper. Res..

[7]  R. Mesiar,et al.  Aggregation Functions (Encyclopedia of Mathematics and its Applications) , 2009 .

[8]  Patrick Meyer,et al.  On the expressiveness of the additive value function and the Choquet integral models , 2012 .

[9]  Andrea Pacifici,et al.  On multi-agent knapsack problems , 2009 .

[10]  Patrice Perny,et al.  Choquet-based optimisation in multiobjective shortest path and spanning tree problems , 2010, Eur. J. Oper. Res..

[11]  M. Grabisch.  The application of fuzzy integrals in multicriteria decision making , 1996 .

[12]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[13]  Toby Walsh,et al.  Principles and Practice of Constraint Programming — CP 2001: 7th International Conference, CP 2001 Paphos, Cyprus, November 26 – December 1, 2001 Proceedings , 2001, Lecture Notes in Computer Science.

[14]  Michel Grabisch,et al.  A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid , 2010, Ann. Oper. Res..

[15]  Xavier Gandibleux,et al.  Preferred solutions computed with a label setting algorithm based on Choquet integral for multi-objective shortest paths , 2011, 2011 IEEE Symposium on Computational Intelligence in Multicriteria Decision-Making (MDCM).

[16]  Jacques Teghem,et al.  The multiobjective multidimensional knapsack problem: a survey and a new approach , 2010, Int. Trans. Oper. Res..

[17]  G. Choquet.  Theory of capacities , 1954 .

[18]  Patrice Perny,et al.  A Branch and Bound Algorithm for Choquet Optimization in Multicriteria Problems , 2008, MCDM.

[19]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decision-making , 1988 .

[20]  Vesa Ojalehto,et al.  Interactive Software for Multiobjective Optimization: IND-NIMBUS , 2007 .

[21]  Vicenç Torra,et al.  The weighted OWA operator , 1997, Int. J. Intell. Syst..

[22]  Michel Grabisch,et al.  A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package , 2008, Eur. J. Oper. Res..

[23]  David Manlove,et al.  A Constraint Programming Approach to the Stable Marriage Problem , 2001, CP.

[24]  Judy Goldsmith,et al.  The AI conference paper assignment problem , 2007, AAAI 2007.

[25]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..