Abstract
In some multi-criteria decision making problems, it is more convenient to express the decision maker preferences in bipolar scales. In such cases, the bipolar Choquet integral with respect to bi-capacities was introduced. In this paper, we address the problem of eliciting a bipolar Choquet integral with respect to a 2-additive bi-capacity. We assume that we are given a set of examples with (i) their scores distribution in regard to several criteria and (ii) their overall scores. We propose two types of optimization problems that allow identifying the parameters of a 2-additive bi-capacity such that the inferred bipolar Choquet integral is consistent with the given examples as much as possible. Furthermore, since the elicitation process we study has many relationships with problems in statistical machine learning, we also present the links between our models and concepts developed in the latter field.
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References
Choquet, G.: Theory of capacities. Annales de l’institut Fourier 5, 131–295 (1954)
Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92(2), 167–189 (1997)
Grabisch, M., Labreuche, C.: Fuzzy measures and integrals in MCDA. In: Multiple Criteria Decision Analysis: State of the Art Surveys. Int. Series in Op. Res. & Manag. Sci., vol. 78, pp. 563–604. Springer, New York (2005)
Meyer, P., Pirlot, M.: On the expressiveness of the additive value function and the choquet integral models. In: DA2PL 2012 From Multiple Criteria Decision Aid to Preference Learning (2012)
Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6(1), 1–44 (2008)
Grabisch, M., Labreuche, C.: Bi-capacities for decision making on bipolar scales. In: Proc. of the EUROFUSE 2002 Workshop on Information Systems, pp. 185–190 (2002)
Greco, S., Matarazzo, B., Slowinski, R.: Bipolar Sugeno and Choquet integrals. In: Proceedings of the EUROFUSE 2002 Workshop on Information Systems (2002)
Grabisch, M., Labreuche, C.: Bi-capacities-I: definition, Möbius transform and interaction. Fuzzy Sets and Systems 151(2), 211–236 (2005)
Grabisch, M., Labreuche, C.: Bi-capacities-II: the Choquet integral. Fuzzy Sets and Systems 151(2), 237–259 (2005)
Fujimoto, K.: New characterizations of k-additivity and k-monotonicity of bi-capacities. In: Joint 2nd Int. Conf. on Soft Computing and Intelligent Systems and 5th International Symposium on Advanced Intelligent Systems (2004)
Fujimoto, K., Murofushi, T.: Some characterizations of k-monotonicity through the bipolar möbius transform in bi-capacities. JACIII 9(5), 484–495 (2005)
Grabisch, M., Labreuche, C.: The choquet integral for 2-additive bi-capacities. In: EUSFLAT Conf., pp. 300–303 (2003)
Jacquet-Lagreze, E., Siskos, J.: Assessing a set of additive utility functions for multicriteria decision-making, the UTA method. European Journal of Operational Research 10(2), 151–164 (1982)
Grabisch, M., Kojadinovic, I., Meyer, P.: A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the kappalab R package. EJOR 186(2), 766–785 (2008)
Mayag, B., Rolland, A., Ah-Pine, J.: Elicitation of a 2-additive bi-capacity through cardinal information on trinary actions. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part IV. CCIS, vol. 300, pp. 238–247. Springer, Heidelberg (2012)
Waegeman, W., Baets, B.D., Boullart, L.: Kernel-based learning methods for preference aggregation. 4OR 7(2), 169–189 (2009)
Fujimoto, K., Murofushi, T., Sugeno, M.: k-additivity and c-decomposability of bi-capacities and its integral. Fuzzy Sets Syst. 158(15), 1698–1712 (2007)
Bouyssou, D., Marchant, T., Pirlot, M., Tsoukias, A., Vincke, P.: Evaluation and Decision models with multiple criteria: stepping stones for the analyst. Int. Series in Op. Res. & Manag. Sci., vol. 86. Springer (2006)
Marichal, J.L., Roubens, M.: Determination of weights of interacting criteria from a reference set. EJOR 124(3), 641–650 (2000)
Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)
Siskos, Y., Grigoroudis, E., Matsatsinis, N.: UTA methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol. 78, pp. 297–334. Springer, New York (2005)
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Ah-Pine, J., Mayag, B., Rolland, A. (2013). Identification of a 2-Additive Bi-Capacity by Using Mathematical Programming. In: Perny, P., Pirlot, M., Tsoukià s, A. (eds) Algorithmic Decision Theory. ADT 2013. Lecture Notes in Computer Science(), vol 8176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41575-3_2
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DOI: https://doi.org/10.1007/978-3-642-41575-3_2
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