Skip to main content
SpringerLink
Log in

Fuzzy Measures and Integrals in MCDA

  • Chapter
Multiple Criteria Decision Analysis

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 233))

Abstract

This chapter aims at a unified presentation of various methods of MCDA based on fuzzy measures (capacity) and fuzzy integrals, essentially the Choquet and Sugeno integral. A first section sets the position of the problem of multicriteria decision making, and describes the various possible scales of measurement (cardinal unipolar and bipolar, and ordinal). Then a whole section is devoted to each case in detail: after introducing necessary concepts, the methodology is described, and the problem of the practical identification of fuzzy measures is given. The important concept of interaction between criteria, central in this chapter, is explained in detail. It is shown how it leads to k-additive fuzzy measures. The case of bipolar scales leads to the general model based on bi-capacities, encompassing usual models based on capacities. A general definition of interaction for bipolar scales is introduced. The case of ordinal scales leads to the use of Sugeno integral, and its symmetrized version when one considers symmetric ordinal scales. A practical methodology for the identification of fuzzy measures in this context is given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 349.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 449.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 449.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Angilella, S., Greco, S., Matarazzo, B.: Non-additive robust ordinal regression: a multiple criteria decision model based on the Choquet integral. Eur. J. Oper. Res. 201, 277–288 (2010)

    Article  Google Scholar 

  2. Bana e Costa, C.A., Vansnick, J.-C.: A theoretical framework for Measuring Attractiveness by a Categorical Based Evaluation TecHnique (MACBETH). In: Proceedings of XIth International Conference on MultiCriteria Decision Making, Coimbra, August 1994, pp. 15–24

    Google Scholar 

  3. Bana e Costa, C.A., Vansnick, J.-C.: Applications of the MACBETH approach in the framework of an additive aggregation model. J. Multicrit. Decis. Anal. 6, 107–114 (1997)

    Google Scholar 

  4. Bana e Costa, C.A., Vansnick, J.-C.: The MACBETH approach: basic ideas, software and an application. In: Meskens, N., Roubens, M. (eds.) Advances in Decision Analysis, pp. 131–157. Kluwer, Dordrecht (1999)

    Google Scholar 

  5. Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 1329–1379. Elsevier Science, Amsterdam (2002)

    Chapter  Google Scholar 

  6. Bilbao, J.M., Fernandez, J.R., Jiménez Losada, A., Lebrón, E.: Bicooperative games. In: Bilbao, J.M. (ed.) Cooperative Games on Combinatorial Structures. Kluwer, Boston (2000)

    Chapter  Google Scholar 

  7. Bilbao, J.M., Fernández, J.R., Jiménez, N., López, J.J.: The Shapley value for bi-cooperative games. Ann. Oper. Res. 158, 99–115 (2008)

    Article  Google Scholar 

  8. Bouyssou, D., Marchant, T.: An axiomatic approach to noncompensatory sorting methods in MCDM I: the case of two categories. Eur. J. Oper. Res. 178(1), 217–245 (2007)

    Article  Google Scholar 

  9. Bouyssou, D., Marchant, Th.: An axiomatic approach to noncompensatory scoring methods in MCDM, II: more than two categories. Eur. J. Oper. Res. 178, 246–276 (2007)

    Google Scholar 

  10. Carenini, G., Moore, J.D.: Generating and evaluating evaluative arguments. Artif. Intell. 170, 925–952 (2006)

    Article  Google Scholar 

  11. Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)

    Article  Google Scholar 

  12. Combarro, E.F., Miranda, P.: Identification of fuzzy measures from sample data with genetic algorithms. Comput. Oper. Res. 33, 3046–3066 (2006)

    Article  Google Scholar 

  13. Couceiro, M., Marichal, J.L.: Axiomatizations of Lovász extensions of pseudo-Boolean functions. Fuzzy Sets Syst. 181, 28–38 (2011)

    Article  Google Scholar 

  14. Couceiro, M., Marichal, J.L.: Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions. Aequationes Math. 82, 213–231 (2011)

    Article  Google Scholar 

  15. Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1994)

    Book  Google Scholar 

  16. Denneberg, D.: Non-additive measure and integral, basic concepts and their role for applications. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals—Theory and Applications, pp. 42–69. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  17. Denneberg, D., Grabisch, M.: Interaction transform of set functions over a finite set. Inf. Sci. 121, 149–170 (1999)

    Article  Google Scholar 

  18. Denneberg, D., Grabisch, M.: Measure and integral with purely ordinal scales. J. Math. Psychol. 48, 15–27 (2004)

    Article  Google Scholar 

  19. Dubois, D., Fargier, H.: Bridging the gap between discrete Sugeno and Choquet integrals. Presented at RUD’2005 (workshop on Risk, Utility and Decision), Heidelberg, 22–24 June 2005

    Google Scholar 

  20. Dubois, D., Prade, H.: A class of fuzzy measures based on triangular norms. Int. J. Gen. Syst. 8, 43–61 (1982)

    Article  Google Scholar 

  21. Dubois, D., Prade, H., Sabbadin, R.: Qualitative decision theory with Sugeno integrals. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals—Theory and Applications, pp. 314–332. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  22. Dubois, D., Marichal, J.-L., Prade, H., Roubens, M., Sabbadin, R.: The use of the discrete Sugeno integral in decision making: a survey. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 9(5), 539–561 (2001)

    Google Scholar 

  23. Dubois, D., Prade, H., Sabbadin, R.: Decision-theoretic foundations of qualitative possibility theory. Eur. J. Oper. Res. 128, 459–478 (2001)

    Article  Google Scholar 

  24. Dyer, J.S., Sarin, R.K.: Measurable multiattribute value functions. Oper. Res. 27, 810–822 (1979)

    Article  Google Scholar 

  25. Fargier, H., Sabbadin, R.: Qualitative decision under uncertainty: back to expected utility. Artif. Intell. 164, 245–280 (2005)

    Article  Google Scholar 

  26. Felsenthal, D., Machover, M.: Ternary voting games. Int. J. Game Theory 26, 335–351 (1997)

    Article  Google Scholar 

  27. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multi-Criteria Decision Aid. Kluwer, Boston (1994)

    Book  Google Scholar 

  28. Fujimoto, K., Murofushi, T.: Some characterizations of k-monotonicity through the bipolar Möbius transform in bi-capacities. J. Adv. Comput. Intell. Intell. Inf. 9(5), 484–495 (2005)

    Google Scholar 

  29. Fujimoto, K., Murofushi, T., Sugeno, M.: k-additivity and \(\mathcal{C}\)-decomposability of bi-capacities and its integral. Fuzzy Sets Syst. 158, 1698–1712 (2007)

    Google Scholar 

  30. Grabisch, M.: A new algorithm for identifying fuzzy measures and its application to pattern recognition. In: International Joint Conference of the 4th IEEE International Conference on Fuzzy Systems and the 2nd International Fuzzy Engineering Symposium, Yokohama, March 1995, pp. 145–150

    Google Scholar 

  31. Grabisch, M.: The application of fuzzy integrals in multicriteria decision making. Eur. J. Oper. Res. 89, 445–456 (1996)

    Article  Google Scholar 

  32. Grabisch, M.: Alternative representations of discrete fuzzy measures for decision making. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 5, 587–607 (1997)

    Article  Google Scholar 

  33. Grabisch, M.: k-Order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)

    Google Scholar 

  34. Grabisch, M.: On the representation of k-decomposable measures. In: 7th IFSA World Congress, Prague, June 1997, pp. 478–483

    Google Scholar 

  35. Grabisch, M.: A graphical interpretation of the Choquet integral. IEEE Trans. Fuzzy Syst. 8, 627–631 (2000)

    Article  Google Scholar 

  36. Grabisch, M.: The Möbius function on symmetric ordered structures and its application to capacities and integrals. In: 9th International Conference on Information Proc. and Management of Uncertainty in Knowledge-Based systems (IPMU’2002), Annecy, July 2002, pp. 755–762

    Google Scholar 

  37. Grabisch, M.: Modelling data by the Choquet integral. In: Torra, V. (ed.) Information Fusion in Data Mining, pp. 135–148. Physica Verlag, Heidelberg (2003)

    Chapter  Google Scholar 

  38. Grabisch, M.: The symmetric Sugeno integral. Fuzzy Sets Syst. 139, 473–490 (2003)

    Article  Google Scholar 

  39. Grabisch, M.: The Möbius function on symmetric ordered structures and its application to capacities on finite sets. Discret. Math. 287(1–3), 17–34 (2004)

    Article  Google Scholar 

  40. Grabisch, M.: Representation of preferences over a finite scale by a mean operator. Math. Soc. Sci. 52, 131–151 (2006)

    Article  Google Scholar 

  41. Grabisch, M.: Some lexicographic approaches to the Sugeno integral. In: Proceedings of the International Conference on Information Processing and Management of Uncertainty (IPMU’06), Paris, July 2006, pp. 572–579

    Google Scholar 

  42. Grabisch, M.: How to score alternatives when criteria are scored on an ordinal scale. J. Multi-Criteria Decis. Anal. 15, 31–44 (2008)

    Article  Google Scholar 

  43. Grabisch, M., Labreuche, Ch.: Bi-capacities. In: Joint International Conference on Soft Computing and Intelligent Systems and 3d International Symposium on Advanced Intelligent Systems, Tsukuba, October 2002

    Google Scholar 

  44. Grabisch, M., Labreuche, Ch.: Bi-capacities for decision making on bipolar scales. In: EUROFUSE Workshop on Informations Systems, Varenna, September 2002, pp. 185–190

    Google Scholar 

  45. Grabisch, M., Labreuche, Ch.: Bi-capacities. Technical Report 2003/002, LIP6, UPMC. http://www.lip6.fr/reports/lip6.2003.002.html (2003)

  46. Grabisch, M., Labreuche, Ch.: Bi-capacities—I: definitions, Möbius transform and interaction. Fuzzy Sets Syst. 151, 211–236 (2005)

    Article  Google Scholar 

  47. Grabisch, M., Labreuche, Ch.: Bi-capacities—II: the Choquet integral. Fuzzy Sets Syst. 151, 237–259 (2005)

    Article  Google Scholar 

  48. Grabisch, M., Roubens, M.: An axiomatic approach to the concept of interaction among players in cooperative games. Int. J. Game Theory 28, 547–565 (1999)

    Article  Google Scholar 

  49. Grabisch, M., Nguyen, H.T., Walker, E.A.: Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference. Kluwer, Dordrecht (1995)

    Book  Google Scholar 

  50. Grabisch, M., Marichal, J.-L., Roubens, M.: Equivalent representations of set functions. Math. Oper. Res. 25(2), 157–178 (2000)

    Article  Google Scholar 

  51. Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals. Theory and Applications (edited volume). Studies in Fuzziness. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  52. Grabisch, M., Dia, S., Labreuche, Ch.: A multicriteria decision making framework in ordinal context based on Sugeno integral. In: Joint 9th IFSA World Congress and 20th NAFIPS International Conference, Vancouver, July 2001

    Google Scholar 

  53. Grabisch, M., Duchêne, J., Lino, F., Perny, P.: Subjective evaluation of discomfort in sitting position. Fuzzy Optim. Decis. Mak. 1(3), 287–312 (2002)

    Article  Google Scholar 

  54. Grabisch, M., Labreuche, Ch., Vansnick, J.-C.: On the extension of pseudo-Boolean functions for the aggregation of interacting bipolar criteria. Eur. J. Oper. Res. 148, 28–47 (2003)

    Article  Google Scholar 

  55. Grabisch, M., Kojadinovic, I., Meyer, P.: A review of capacity identification methods for Choquet integral based multi-attribute utility theory—applications of the Kappalab R package. Eur. J. Oper. Res. 186, 766–785 (2008)

    Article  Google Scholar 

  56. Greco, S., Matarazzo, B., Słowinski, R.: Conjoint measurement and rough set approach for multicriteria sorting problems in presence of ordinal criteria. In: Colorni, A., Paruccini, M. (eds.) A-MCD-A Aide Multicritère à la Décision—Multiple Criteria Decision Aiding, pp. 117–144. JRC (2001)

    Google Scholar 

  57. Greco, S., Matarazzo, B., Słowinski, R.: Bipolar Sugeno and Choquet integrals. In: EUROFUSE Workshop on Informations Systems, Varenna, September 2002, pp. 191–196

    Google Scholar 

  58. Greco, S., Matarazzo, B., Słowinski, R.: The axiomatic basis of Sugeno integral and associative aggregation operator. In: 4th International Workshop on Preferences and Decisions, pp. 33–38, Trento, September 2003

    Google Scholar 

  59. Greco, S., Matarazzo, B., Słowinski, R.: Ordinal regression revisited: multiple criteria ranking with a set of additive value functions. Eur. J. Oper. Res. 191, 416–436 (2008)

    Article  Google Scholar 

  60. Greco, S., Słowinski, R., Figueira, J., Mousseau, V.: Robust ordinal regression. In: Ehrgott, M., Figueira, J., Greco, S. (eds.) New Trends in Multiple Criteria Decision Analysis. Springer (2016)

    Google Scholar 

  61. Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer, New York (1968)

    Book  Google Scholar 

  62. Ishii, K., Sugeno, M.: A model of human evaluation process using fuzzy measure. Int. J. Man-Mach. Stud. 22, 19–38 (1985)

    Article  Google Scholar 

  63. Keeney, R.L., Raiffa, H.: Decision with Multiple Objectives. Wiley, New York (1976)

    Google Scholar 

  64. Klein, D.A.: Decision Analytic Intelligent Systems: Automated Explanation and Knowledge Acquisition. Lawrence Erlbaum Associates, Mahwah (1994)

    Google Scholar 

  65. Kojadinovic, I.: Minimum variance capacity identification. Q. J. Oper. Res. (4OR) 12, 23–36 (2006)

    Google Scholar 

  66. Kojadinovic, I.: A weight-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral. Eur. J. Oper. Res. 179, 498–517 (2007)

    Article  Google Scholar 

  67. Kojadinovic, I., Labreuche, Ch.: Partially bipolar Choquet integrals. IEEE Trans. Fuzzy Syst. 17, 839–850 (2009)

    Article  Google Scholar 

  68. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement. Additive and Polynomial Representations, vol. 1. Academic, New York (1971)

    Google Scholar 

  69. Kwon, S.H., Sugeno, M.: A hierarchical subjective evaluation model using non-monotonic fuzzy measures and the Choquet integral. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals—Theory and Applications, pp. 375–391. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  70. Labreuche, Ch.: Argumentation of the results of a multi-criteria evaluation model in individual and group decision aiding. In: International Conference of the Euro Society for Fuzzy Logic and Technology (EUSFLAT), pp. 482–487, Barcelona, 7–9 September 2005

    Google Scholar 

  71. Labreuche, Ch.: Identification of a fuzzy measure with an L1 entropy. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Malaga, 22–27 June 2008

    Google Scholar 

  72. Labreuche, Ch.: On the robustness for the Choquet integral. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pp. 484–493, Dortmund, 28 June to 02 July 2010

    Google Scholar 

  73. Labreuche, Ch.: Construction of a Choquet integral and the value functions without any commensurateness assumption in multi-criteria decision making. In: International Conference of the Euro Society for Fuzzy Logic and Technology (EUSFLAT), Aix Les Bains, 18–22 July 2011

    Google Scholar 

  74. Labreuche, Ch.: A general framework for explaining the results of a multi-attribute preference model. Artif. Intell. 175, 1410–1448 (2011)

    Article  Google Scholar 

  75. Labreuche, Ch.: An axiomatization of the Choquet integral and its utility functions without any commensurateness assumption. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Catania, 9–13 July 2012

    Google Scholar 

  76. Labreuche, Ch., Grabisch, M.: Bi-cooperative games and their importance and interaction indices. In: 14th Mini-EURO Conference on Human Centered Processes (HCP’2003), pp. 287–291, Luxembourg, May 2003

    Google Scholar 

  77. Labreuche, Ch., Grabisch, M.: The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets Syst. 137, 11–26 (2003)

    Article  Google Scholar 

  78. Labreuche, Ch., Grabisch, M.: The Shapley value and the interaction indices for bi-cooperative games. Technical Report 2003/001, LIP6, UPMC, http://www.lip6.fr/reports/lip6.2003.001.html (2003)

  79. Labreuche, Ch., Grabisch, M.: Definition of importance indices for bicapacities in mcda. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Paris, 2–7 July 2006, pp. 292–299

    Google Scholar 

  80. Labreuche, Ch., Grabisch, M.: Generalized Choquet-like aggregation functions for handling bipolar scales. Eur. J. Oper. Res. 172, 931–955 (2006)

    Article  Google Scholar 

  81. Labreuche, Ch., Grabisch, M.: The representation of conditional relative importance between criteria. Ann. Oper. Res. 154, 93–122 (2007)

    Article  Google Scholar 

  82. Labreuche, Ch., Grabisch, M.: A value for bi-cooperative games. Int. J. Game Theory 37, 409–438 (2008)

    Article  Google Scholar 

  83. Labreuche, Ch., Le Huédé, F.: Inconsistencies in the determination of a capacity. In: European Conference on Artificial Intelligence (ECAI) - Workshop on Advances in Preference Handling, Riva del Garda, 28 August to 1 September 2006, pp. 73–80

    Google Scholar 

  84. Labreuche, Ch., Miranda, P., Lehuédé, F.: Computation of the robust preference relation combining a Choquet integral and utility functions. In: European Conference on Artificial Intelligence (ECAI) - Workshop on Advances in Preference Handling, Lisbon, 28 August to 1 September 2010, pp. 49–54

    Google Scholar 

  85. Marichal, J.-L.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Trans. Fuzzy Syst. 8(6), 800–807 (2000)

    Article  Google Scholar 

  86. Marichal, J.-L.: On Sugeno integral as an aggregation function. Fuzzy Sets Syst. 114, 347–365 (2000)

    Article  Google Scholar 

  87. Marichal, J.-L.: An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework. IEEE Trans. Fuzzy Syst. 9(1), 164–172 (2001)

    Article  Google Scholar 

  88. Marichal, J.-L., Roubens, M.: Dependence between criteria and multiple criteria decision aid. In: 2nd International Workshop on Preferences and Decisions, Trento, 1998, pp. 69–75

    Google Scholar 

  89. Mauris, G., Montmain, J., Akharraz, A.: Knowledge management as a support for collective decision-making and argumentation processes. In: 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based systems (IPMU’2002), Annecy, July 2002, pp. 91–98

    Google Scholar 

  90. Mayag, B.: Elaboration d’une démarche constructive prenant en compte les interactions entre critères en aide multicritère à la décision. Ph.D. thesis, University of Paris I (2010)

    Google Scholar 

  91. Mayag, B., Grabisch, M., Labreuche, Ch.: Un algorithme de détermination de la capacité pour l’intégrale de Choquet 2-additive. In: LFA’2008, Actes des Rencontres Francophones sur la Logique Floue et ses Applications, Lens, 16–17 October 2008

    Google Scholar 

  92. Mayag, B., Grabisch, M., Labreuche, Ch.: An interactive algorithm to deal with inconsistencies in the representation of cardinal information. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Dortmund, 28 June to 02 July 2010, pp. 148–157

    Google Scholar 

  93. Mayag, B., Grabisch, M., Labreuche, Ch.: A representation of preferences by the Choquet integral with respect to a 2-additive capacity. Theor. Decis. 71, 297–324 (2011)

    Article  Google Scholar 

  94. Mcsherry, D.: Explanation in recommender systems. AIR 24, 179–197 (2005)

    Google Scholar 

  95. Miranda, P., Grabisch, M.: p-Symmetric bi-capacities. In: International Summer School on Aggregation Operators and their Applications, Alcala, July 2003, pp. 123–129

    Google Scholar 

  96. Miranda, P., Grabisch, M., Gil, P.: p-Symmetric fuzzy measures. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 10(Suppl.), 105–123 (2002)

    Google Scholar 

  97. Modave, F., Grabisch, M.: Preference representation by a Choquet integral: the commensurability hypothesis. In: 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), Paris, July 1998, pp. 164–171

    Google Scholar 

  98. Montmain, J., Mauris, G., Akharraz, A.: Elucidation and decisional risk in a multi criteria decision based on a Choquet integral aggregation: a cybernetic framework. Int. J. Multi-Criteria Decis. Anal. 13, 239–258 (2005)

    Article  Google Scholar 

  99. Mori, T., Murofushi, T.: An analysis of evaluation model using fuzzy measure and the Choquet integral. In: 5th Fuzzy System Symposium, Kobe, 1989, pp. 207–212 (in Japanese)

    Google Scholar 

  100. Moulin, H.: Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge (1988)

    Book  Google Scholar 

  101. Murofushi, T.: Lexicographic use of Sugeno integrals and monotonicity conditions. IEEE Trans. Fuzzy Syst. 9(6), 783–794 (2001)

    Article  Google Scholar 

  102. Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (III), interaction index. In: 9th Fuzzy System Symposium, Sapporo, May 1993, pp. 693–696

    Google Scholar 

  103. Murofushi, T., Sugeno, M.: Fuzzy measures and fuzzy integrals. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals – Theory and Applications, pp. 3–41. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  104. Onisawa, T., Sugeno, M., Nishiwaki, Y., Kawai, H., Harima, Y.: Fuzzy measure analysis of public attitude towards the use of nuclear energy. Fuzzy Sets Syst. 20, 259–289 (1986)

    Article  Google Scholar 

  105. Pignon, J.P., Labreuche, Ch.: A methodological approach for operational and technical experimentation based evaluation of systems of systems architectures. In: International Conference on Software & Systems Engineering and their Applications (ICSSEA), Paris, 4–6 December 2007

    Google Scholar 

  106. Rico, A., Grabisch, M., Labreuche, Ch., Chateauneuf, A.: Preference modelling on totally ordered sets by the Sugeno integral. Discret. Appl. Math. 147, 113–124 (2005)

    Article  Google Scholar 

  107. Roberts, F.S.: Measurement Theory. Addison-Wesley, Reading (1979)

    Google Scholar 

  108. Rota, G.C.: On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 2, 340–368 (1964)

    Article  Google Scholar 

  109. Roubens, M.: Ordinal multiattribute sorting and ordering in the presence of interacting points of view. In: Bouyssou, D., Jacquet-Lagrèze, E., Perny, P., Slowinsky, R., Vanderpooten, D., Vincke, Ph. (eds.) Aiding Decisions with Multiple Criteria: Essays in Honor of Bernard Roy, pp. 229–246. Kluwer, Dordrecht (2001)

    Google Scholar 

  110. Roy, B.: Multicriteria Methodology for Decision Aiding. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

  111. Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)

    Article  Google Scholar 

  112. Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games. Number 28 in Annals of Mathematics Studies, vol. II, pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

  113. Shapley, L.S.: Simple games: an outline of the descriptive theory. Behav. Sci. 7, 59–66 (1962)

    Article  Google Scholar 

  114. Simon, H.: Rational choice and the structure of the environment. Psychol. Rev. 63(2), 129–138 (1956)

    Article  Google Scholar 

  115. Šipoš, J.: Integral with respect to a pre-measure. Math. Slovaca 29, 141–155 (1979)

    Google Scholar 

  116. Slowiński, R., Greco, S., Matarazzo, B.: Axiomatization of utility, outranking and decision-rule preference models for multi-criteria classification. Control Cybern. 31(4), 1005–1035 (2002)

    Google Scholar 

  117. Soria-Frisch, A.: Unsupervised construction of fuzzy measures through self-organizing feature maps and its application in color image segmentation. Int. J. Approx. Reason. 41, 23–42 (2006)

    Article  Google Scholar 

  118. Sugeno, M.: Theory of Fuzzy Integrals and Its Applications. Ph.D. thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  119. Torra, V.: Learning aggregation operators for preference modeling. In: Fürnkranz, J., Hüllermeier, E. (eds.) Preference Learning, pp. 317–333. Springer, New York (2010)

    Chapter  Google Scholar 

  120. Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992)

    Article  Google Scholar 

  121. Verkeyn, A., Botteldooren, D., De Baets, B.: Genetic learning of fuzzy integrals accumulating human-reported environmental stress. Appl. Soft Comput. 11, 305–314 (2011)

    Article  Google Scholar 

  122. Wakker, P.: Additive Representations of Preferences. Kluwer, Dordrecht (1989)

    Book  Google Scholar 

  123. Wang, Z., Leung, K.S., Wang, J.: A genetic algorithm for determining nonadditive set functions in information fusion. Fuzzy Sets Syst. 102, 462–469 (1999)

    Article  Google Scholar 

  124. Wang, X.-Z., He, Y.-L., Dong, L.-C., Zhao, H.-Y.: Particle swarm optimization for determining fuzzy measures from data. Inf. Sci. 181, 4230–4252 (2011)

    Article  Google Scholar 

  125. Weber, S.: ⊥ -Decomposable measures and integrals for Archimedean t-conorms ⊥ . J. Math. Anal. Appl. 101, 114–138 (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Paris School of Economics, Université Paris I - Panthéon-Sorbonne, 106-112, Bd de l’Hôpital, 75013, Paris, France

    Michel Grabisch

  2. Thales Research & Technology, 1 avenue Augustin Fresnel, 91767, Palaiseau Cedex, France

    Christophe Labreuche

Authors
  1. Michel Grabisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christophe Labreuche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Grabisch .

Editor information

Editors and Affiliations

  1. Department of Economics and Business; Portsmouth Business School, Operations & Systems Management, University of Catania; University of Portsmouth, Catania; Portsmouth, Italy

    Salvatore Greco

  2. Department of Management Science, Lancaster University, Lancaster, United Kingdom

    Matthias Ehrgott

  3. Universidade de Lisboa, CEG-IST, Instituto Superior Técnico, Lisboa, Portugal

    José Rui Figueira

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media New York

About this chapter

Cite this chapter

Grabisch, M., Labreuche, C. (2016). Fuzzy Measures and Integrals in MCDA. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, vol 233. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3094-4_14

Download citation

Publish with us

Policies and ethics

Search

Navigation