Abstract
Fuzzy measures can flexibly describe the relative importance of decision criterion as well as their interactions in multicriteria decision making. Based on the diamond pairwise comparison, a new identification method of 2-order additive fuzzy measure is proposed. The relative weight and the interaction degree can be obtained simultaneously for every pair of criteria in the diamond pairwise comparison. The Choquet integral-based equivalent alternative curve can help the decision maker estimate the interaction degrees between criteria. The overall importance of each criterion is obtained by the maximum eigenvector method of AHP. According to the maximum fuzzy measure entropy principal, a nonlinear programming is constructed to identify the interaction indices among criteria. Finally, an illustrative example shows the feasibility and validity of the proposed identification method.
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Choquet G. (1954) Theory of capacities. Annales de l’ Institut Fourier 5: 131–295
Combarro E. F., Miranda P. (2006) Identification of fuzzy measures from sample data with genetic algorithms. Computers & Operations Research 33(10): 3046–3066
Grabisch, M. (1995a). A new algorithm for identifying fuzzy measures and its application to pattern recognition. In Proceedings of international joint conference of the fourth IEEE international conference on fuzzy systems and the second international fuzzy engineering symposium (pp. 145–50). Yokohama, Japan.
Grabisch M. (1995b) Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69(3): 279–298
Grabisch M. (1996) The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research 89(3): 445–456
Grabisch M. (1997) K-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92(2): 167–189
Grabisch M. (2000a) Application of fuzzy integrals in multicriteria decision making. In: Grabisch M, Murofushi T., Sugeno M. (eds) Fuzzy measures and integrals: Theory and applications. Physica-Verlag, Heidelberg, New York, pp 348–374
Grabisch M. (2000b) A graphical interpretation of the Choquet integral. IEEE Transactions on Fuzzy Systems 8(5): 627–631
Grabisch M., Labreuche C. (2005) Fuzzy measures and integrals in MCDA. In: Figueira J., Greco S., Ehrgott M. (eds) Multiple criteria decision analysis. Springer Science+Bussiness Media, New York, pp 563–608
Grabisch M., Kojadinovic I., Meyer P. (2008) A review of methods for capacity identification in Choquet integral based multi-attribute utility theory Applications of the Kappalab R package. European Journal of Operational Research 186(2): 766–785
Grabisch, M., Murofushi, T., Sugeno, M. (eds) (2000) Fuzzy measure and integrals: Theory and applications. Physica-Verlag, Heidelberg, New York
Ishii K., Sugeno M. (1996) A model of human evaluation process using fuzzy measure. International Journal of Man-Machine Studies 67(1): 242–257
Kojadinovic I. (2007a) Minimum variance capacity identification. European Journal of Operational Research 177(1): 498–514
Kojadinovic I. (2007b) Quadratic distances for capacity and bi-capacity approximation and identification. 4OR: A Quarterly Journal of Operations Research 5(2): 117–142
Kojadinovic I., Marichal J. L., Roubens M. (2005) An axiomatic approach to the definition of the entropy of a discrete Choquet capacity. Information Sciences 172(1–2): 131–153
Labreuche C., Grabisch M. (2003) The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets and Systems 137(1): 11–16
Marichal J. L., Roubens M. (2000) Determination of weights of interacting criteria from a reference set. European Journal of Operational Research 124(3): 641–650
Marichal J. L. (2000) An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Transactions on Fuzzy Systems 8(6): 800–807
Marichal J. L. (2002) Entropy of discrete Choquet capacities. European Journal of Operational Research 137(3): 612–624
Marichal J. L. (2004) Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. European Journal of Operational Research 155(3): 771–791
Meyer P., Roubens M. (2005) Choice, ranking and sorting in fuzzy multiple criteria decision aid. In: Figueira J., Greco S., Ehrgott M. (eds) Multiple criteria decision analysis: State of the art surveys. Springer, New York, pp 471–506
Miranda P., Grabisch M. (1999) Optimization issues for fuzzy measures. International Journal of Uncertainty, Fuzziness, and Knowledge Based Systems 7(6): 545–560
Miranda P., Grabisch M., Gil P. (2002) p-Symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness, and Knowledge Based Systems 10(supplement): 105–123
Pap E. (1995) Null-additive set functions. Kluwer Academic Publisher, Boston
Sugeno, M. (1974). Theory of fuzzy integral and its applications. Ph.D. Dissertation, Tokyo Institute of Technology.
Takahagi E. (2008) A fuzzy measure identification method by diamond pairwise comparisons and phi(s) transformation. Fuzzy Optimization and Decision Making 7(3): 219–232
Wang Z., Klir G. J. (1992) Fuzzy measure theory. Plenum Publishing Corporation, New York
Wang Z., Klir G. J. (2009) Generalized measure theory. Springer Science+Bussiness Media, New York
Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1(1): 3–28
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Wu, JZ., Zhang, Q. 2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle. Fuzzy Optim Decis Making 9, 435–453 (2010). https://doi.org/10.1007/s10700-010-9086-x
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DOI: https://doi.org/10.1007/s10700-010-9086-x