Advertisement

New Directions in Ordinal Evaluation: Sugeno Integrals and Beyond

  • Miguel CouceiroEmail author
  • Didier Dubois
  • Hélène Fargier
  • Michel Grabisch
  • Henri Prade
  • Agnès Rico
Chapter
Part of the Multiple Criteria Decision Making book series (MCDM)

Abstract

This chapter provides a state-of-the-art account of the use of Sugeno integrals in decision evaluation, when it is difficult to use meaningful figures of merit when assessing the worth of a decision and when only a finite scale of, e.g., linguistic categories, can be used. Here, Sugeno integrals are thought of as idempotent lattice polynomial functions on a finite bounded chain, which makes it possible to assign importance weights to groups of criteria or states. Algebraic and behavioral characterizations of the Sugeno integral are presented and discussed, including the special cases of weighted minima and maxima. Extensions of this framework are also surveyed, namely: lexicographic refinements that increase the discrimination power of this approach; the use of local utility functions in order to cope with criteria having distinct rating scales; and the generalization of the criteria weighting scheme at work in Sugeno integrals. Another kind of extension considered is when ratings belong to a bipolar scale where good and bad figures are explicitly present, thus giving rise to the symmetric Sugeno integral or to the separate evaluation of pros and cons. Moreover, it is pointed out that Sugeno integrals encode decision rules and that this bridge leads to methods for extracting knowledge from qualitative data. The results of empirical studies of the latter are also presented and discussed, accordingly.

Keywords

Sugeno integral Lattice polynomial Bipolarity Qualitative decision theory Decision rule 

References

  1. Benferhat, S., Dubois, D., & Prade, H. (1999). Possibilistic and standard probabilistic semantics of conditional knowledge bases. Journal of Logic and Computation, 9, 873–895.CrossRefGoogle Scholar
  2. Benferhat, S., Dubois, D., Kaci, S., & Prade, H. (2006). Bipolar possibility theory in preference modeling: Representation, fusion and optimal solutions. Information Fusion, 7, 135–150.CrossRefGoogle Scholar
  3. Bennett, C. D., Holland, W. C., & Székely, G. J. (2014). Integer Valued Means. Aequationes Mathematicae, 88, 137–149.CrossRefGoogle Scholar
  4. Blaszczyński, J., Slowiński, R., & Szelag, M. (2011). Sequential covering rule induction algorithm for variable consistency rough set approaches. Information Sciences, 181(5), 987–1002.CrossRefGoogle Scholar
  5. Boczek, M., & Kaluszka, M. (2017). On conditions under which some generalized Sugeno integrals coincide: A solution to Dubois’ problem. Fuzzy Sets and Systems, 326, 81–88.CrossRefGoogle Scholar
  6. Bonnefon, J.-F., Dubois, D., & Fargier, H. (2008a). On the qualitative comparison of decisions having positive and negative features. J. Artificial Intelligence Research, 32, 385–417.Google Scholar
  7. Bonnefon, J.-F., Dubois, D., Fargier, H., & Leblois, S. (2008b). Qualitative heuristics for balancing the pros and cons. Theory and Decision, 65, 71–85.Google Scholar
  8. Borzová-Molnárová, J., Halčinová, L., & Hutník, O. (2015). The smallest semicopula-based universal integrals, part I. Fuzzy Sets and Systems, 271, 1–17.CrossRefGoogle Scholar
  9. Boutilier, C., Brafman, R. I., Domshlak, C., Hoos, H. H., & Poole, D. (2004). CP-nets: A tool for representing and reasoning with conditional ceteris paribus preference statements. J. Artificial Intelligence Research, 21, 135–191.CrossRefGoogle Scholar
  10. Bouyssou, D., Marchant, T., & Pirlot, M. (2009). A conjoint measurement approach to the discrete Sugeno integral. In: S. Brams, W. V. Gehrlein, & F. S. Roberts (Eds.) The Mathematics of Preference, Choice and Order; Essays in Honor of Peter C. Fishburn (pp. 85–109). Berlin, London: Springer.Google Scholar
  11. Brabant, Q., Couceiro, M., Dubois, D., Prade, H., Rico, A. (2018). Extracting decision rules from qualitative data via Sugeno Utility functionals. In Medina, J. et al. (Eds.)Proceedings International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU 2018) (pp. 253–265). Springer, CCIS 853.Google Scholar
  12. Cacioppo, J. T., & Berntson, G. G. (1994). Relationship between attitudes and evaluative space: A critical review, with emphasis on the separability of positive and negative substrates. Psychological Bulletin, 115, 401–423.CrossRefGoogle Scholar
  13. de Campos, L. M., & Bolaños, M. J. (1992). Characterization and comparison of Sugeno and Choquet integrals. Fuzzy Sets and Systems, 52, 61–67.CrossRefGoogle Scholar
  14. de Campos, L. M., Lamata, M. T., & Moral, S. (1991). A unified approach to define fuzzy integrals. Fuzzy Sets and Systems, 39(1), 75–90.CrossRefGoogle Scholar
  15. Chateauneuf, A. (1996). Decomposable capacities, distorted probabilities and concave capacities. Mathematical Social Sciences, 31(1), 19–37.CrossRefGoogle Scholar
  16. Chateauneuf, A., Grabisch, M., & Rico, A. (2008). Modeling attitudes toward uncertainty through the use of the Sugeno integral. J. Math. Econ., 44(11), 1084–1099.CrossRefGoogle Scholar
  17. Cohen, M., & Jaffray, J. Y. (1980). Rational behavior under complete ignorance. Econometrica, 48(5), 1281–1299.CrossRefGoogle Scholar
  18. Couceiro, M., Dubois, D., Prade, H., & Rico, A. (2017a). Enhancing the expressive power of Sugeno integrals for qualitative data analysis. In Kacprzyk, J. et al. (Eds.) Advances in Fuzzy Logic and Technology 2017 (Proc. EUSFLAT 2017). Advances in Intelligent Systems and Computing (Vol. 641, pp. 534–547). SpringerGoogle Scholar
  19. Couceiro, M., Dubois, D., Prade, H., & Waldhauser, T. (2016). Decision-making with sugeno integrals—Bridging the gap between multicriteria evaluation and decision under uncertainty. Order, 33(3), 517–535.CrossRefGoogle Scholar
  20. Couceiro, M., Foldes, S., & Lehtonen, E. (2006). Composition of post classes and normal forms of Boolean functions. Discrete Mathematics, 306, 3223–3243.CrossRefGoogle Scholar
  21. Couceiro, M., & Grabisch, M. (2013). On the poset of computation rules for nonassociative calculus. Order, 30(1), 269–288.CrossRefGoogle Scholar
  22. Couceiro, M., & Grabisch, M. (2017). On integer-valued means and the symmetric maximum. Aequationes Mathematicae, 91(2), 353–371.CrossRefGoogle Scholar
  23. Couceiro, M., Lehtonen, E., Marichal, J.-L., & Waldhauser, T. (2011). An algorithm for producing median normal form representations for Boolean functions. In The Proceedings of the Reed-Muller Workshop 2011 (pp. 49–54).Google Scholar
  24. Couceiro, M., & Marichal, J.-L. (2010a). Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices. Fuzzy Sets and Systems, 161(5), 694–707.Google Scholar
  25. Couceiro, M., & Marichal, J.-L. (2010b). Representations and characterizations of polynomial functions on chains. Journal of Multiple-Valued Logic and Soft Computing, 16(1–2), 65–86.Google Scholar
  26. Couceiro, M., & Marichal, J.-L. (2012). Polynomial functions over bounded distributive lattices. Journal of Multiple-Valued Logic and Soft Computing, 18, 247–256.Google Scholar
  27. Couceiro, M., Mercuriali, P., Péchoux, R., & Saffidine, A. (2017b). Median based calculus for lattice polynomials and monotone Boolean functions. In 47th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2017) (pp. 37–42 ). IEEE Computer SocietyGoogle Scholar
  28. Couceiro, M., Mercuriali, P., Péchoux, R., & Saffidine, A. On the complexity of minimizing median normal forms of monotone Boolean functions and lattice polynomials (23 pp). https://hal.inria.fr/hal-01905491
  29. Couceiro, M., Waldhauser, T. (2011). Axiomatizations and factorizations of Sugeno utility functionals. International Journal Uncertainity Fuzziness Knowledge Based Systems, 19(4), 635–658Google Scholar
  30. Couceiro, M., & Waldhauser, T. (2014). Pseudo-polynomial functions over finite distributive lattices. Fuzzy Sets and Systems, 239, 21–34.CrossRefGoogle Scholar
  31. Deschamps, R., & Gevers, L. (1978). Leximin and utilitarian rules: a joint characterization. J. of Economic Theory, 17, 143–163.CrossRefGoogle Scholar
  32. Doyle, J., & Thomason, R. (1999). Background to qualitative decision theory. The AI Magazine, 20(2), 55–68.Google Scholar
  33. Dubois, D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets. Computers and Artificial Intelligence, 5(5), 403–416.Google Scholar
  34. Dubois, D., Durrieu, C., Prade, H., Rico, A., & Ferro, Y. (2015). Extracting Decision Rules from Qualitative Data Using Sugeno Integral: A Case-Study. ECSQARU, 2015, 14–24.Google Scholar
  35. Dubois, D., & Fargier, H. (2009a). Making Discrete Sugeno Integrals More Discriminant. International Journal of Approximate Reasoning, 50, 880–898.Google Scholar
  36. Dubois, D., Fargier, H. (2009b). Capacity refinements and their application to qualitative decision evaluation. In C. Sossai & G. Chemello (Eds.) Symbolic and quantitative approaches to reasoning with uncertainty (ECSQARU 2009) (pp. 311–322). Springer, LNAI 5590.Google Scholar
  37. Dubois, D., Fargier, H. (2010). Qualitative bipolar decision rules: Toward more expressive settings. dans: preferences and decisions—Models and applications. In S. Greco, R. M. Peirera, M. Squillante, R. R. Yager, & J. Kacprzyk (Eds.) Studies in fuzziness and soft computing (Vol. 257, pp. 139–158). Springer.Google Scholar
  38. Dubois, D., Fargier, H., & Prade, H. (1996). Refinements of the maximin approach to decision-making in a fuzzy environment. Fuzzy Sets Systems, 81(1), 103–122.CrossRefGoogle Scholar
  39. Dubois, D., Fargier, H., Prade, H., Sabbadin, R. (2009). A survey of qualitative decision rules under uncertainty. In D. Bouyssou, D. Dubois, M. Pirlot, & Prade H. (Eds.) Decision-making process—Concepts and methods (Chap. 11, pp. 435–473). Wiley.Google Scholar
  40. Dubois, D., & Fortemps, P. (1999). Computing improved optimal solutions to max-min flexible constraint satisfaction problems. European Journal of Operational Research, 118, 95–126.CrossRefGoogle Scholar
  41. Dubois, D., Le Berre, D., Prade, H., & Sabbadin, R. (1999). Using possibilistic logic for modeling qualitative decision: ATMS-based algorithms. Fundamenta Informaticae, 37, 1–30.Google Scholar
  42. Dubois, D., Marichal, J.-L., Prade, H., Roubens, M., & Sabbadin, R. (2001). The use of the discrete Sugeno integral in decision making: a survey. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9, 539–561.Google Scholar
  43. Dubois, D., & Prade, H. (1980). Fuzzy Sets and Systems. Theory and Applications: Academic Press.Google Scholar
  44. Dubois, D., & Prade, H. (1984). A theorem on implication functions defined from triangular norms. Stochastica, 8, 267–279.Google Scholar
  45. Dubois, D., & Prade, H. (1985). Evidence measures based on fuzzy information. Automatica, 21, 547–562.Google Scholar
  46. Dubois, D., & Prade, H. (1986). Weighted minimum and maximum operations. Information Sciences, 39, 205–210.CrossRefGoogle Scholar
  47. Dubois, D., & Prade, H. (1988). Possibility Theory: An Approach to Computerized Processing of Uncertainty. New York: Plenum Press.CrossRefGoogle Scholar
  48. Dubois, D., & Prade, H. (2004). Possibilistic logic: a retrospective and prospective view. Fuzzy Sets and Systems, 144(1), 3–23.CrossRefGoogle Scholar
  49. Dubois, D., & Prade, H. (2008). An introduction to bipolar representations of information and preference. Int. J. Intelligent Systems., 23(8), 866–877.CrossRefGoogle Scholar
  50. Dubois, D., Prade, H. (2015). Possibility theory and its applications: where do we stand? In J. Kacprzyk & W. Pedrycz (Eds.) Springer handbook of computational intelligence (pp. 31–60). Springer.Google Scholar
  51. Dubois, D., Prade, H., & Rico, A. (2014). The logical encoding of Sugeno integrals. Fuzzy Sets and Systems, 241, 61–75.CrossRefGoogle Scholar
  52. Dubois, D., Prade, H., & Rico, A. (2015). Representing qualitative capacities as families of possibility measures. International Journal Approximate Reasoning, 58, 3–24.CrossRefGoogle Scholar
  53. Dubois, D., Prade, H., & Rico, A. (2016). Residuated variants of Sugeno integrals. Information Sciences, 329, 765–781.CrossRefGoogle Scholar
  54. Dubois, D., Prade, H., Rico, A., & Teheux, B. (2017). Generalized qualitative Sugeno integrals. Information Sciences, 415, 429–445.Google Scholar
  55. Dubois, D., Prade, H., & Sabbadin, R. (1998). Qualitative decision theory with Sugeno integrals. In Proceedings of 14th Conference on Uncertainty in AI (pp. 121–128)Google Scholar
  56. Dubois, D., Prade, H., & Sabbadin, R. (2000). Qualitative decision theory with Sugeno integrals. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.) Fuzzy measures and integrals. Theory and applications, studies in fuzziness and soft computing (pp. 314–322). Physica-VerlagGoogle Scholar
  57. Dubois, D., Prade, H., & Sabbadin, R. (2001). Decision theoretic foundations of qualitative possibility theory. European Journal of Operational Research, 128, 459–478.CrossRefGoogle Scholar
  58. Dubois, D., & Rico, A. (2018). New axiomatisations of discrete quantitative and qualitative possibilistic integrals. Fuzzy Sets and Systems, 343, 3–19.CrossRefGoogle Scholar
  59. Dvořák, A., & Holčapek, M. (2012). Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices. Information Sciences, 185, 205–229.CrossRefGoogle Scholar
  60. Fargier, H., Lang, J., Schiex, T. (1993). Selecting preferred solutions in fuzzy constraint satisfaction problems. In Proceedings 1st European Congress on Fuzzy and Intelligent Technologies (EUFIT ’93) (pp. 1128–1134). Aachen, Germany.Google Scholar
  61. Fargier, H., & Sabbadin, R. (2005). Qualitative decision under uncertainty: Back to expected utility. Artificial Intelligence, 164, 245–280.CrossRefGoogle Scholar
  62. Franklin, B., Letter to Priestley, J. B. 1772. (1887). In J. Bigelow (Ed.)The Complete Works. New York: Putnam.Google Scholar
  63. Gérard, R., Kaci, S., Prade, H. (2007). Ranking alternatives on the basis of generic constraints and examples—A possibilistic approach. In M. M. Veloso (Ed.) Proceedings 20th International Joint Conference on Artificial Intelligence (IJCAI 2007) (pp. 393–398). Hyderabad.Google Scholar
  64. Goodstein, R. L. (1965/1967). The solution of equations in a lattice. In Proceedings of Royal Society Edinburgh Sect. A, 67, 231–242.Google Scholar
  65. Giang, P. H., & Shenoy, P. P. (2000). A qualitative utility theory for Spohn’s theory of epistemic beliefs. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence (pp. 220–227).Google Scholar
  66. Giang, P. H., & Shenoy, P. P. (2005). Two axiomatic approaches to decision-making using possibility theory. European Journal of Operational Research, 162, 450–467.CrossRefGoogle Scholar
  67. Gigerenzer, G., & Todd, P. M. (1999). The ABC group: Simple heuristics that make us smart. Oxford University Press.Google Scholar
  68. Grabisch, M. (1996). The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research, 89(3), 445–456.CrossRefGoogle Scholar
  69. Grabisch, M. (2003). The symmetric Sugeno integral. Fuzzy Sets and Systems, 139, 473–490.CrossRefGoogle Scholar
  70. Grabisch, M. (2004). The Moebius transform on symmetric ordered structures and its application to capacities on finite sets. Discrete Mathematics, 287, 17–34.CrossRefGoogle Scholar
  71. Grabisch, M. (2016). Set functions, games and capacities in decision-making. Springer.Google Scholar
  72. Grabisch, M., & Labreuche, C. (2010). A decade of application of the Choquet and Sugeno intégrals in multi-criteria décision aid. Annals of Operations Research, 175, 247–286.CrossRefGoogle Scholar
  73. Grabisch, M. Marichal, J.-L., Mesiar, R., & Pap, E. (2009). Aggregation functions. Cambridge University Press.Google Scholar
  74. Grabisch, M., Murofushi, T. Sugeno, & M., (1992). Fuzzy measure of fuzzy events defined by fuzzy integrals. Fuzzy sets and systems, 50(3), 293–313.Google Scholar
  75. Greco, S., Matarazzo, B., & Slowinski, R. (2004). Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. European Journal of Operational Research, 158, 271–292.CrossRefGoogle Scholar
  76. Gutiérrez, P. A., Pérez-Ortiz, M., Sánchez-Monedero, J., Fernández-Navarro, F., & Hervás-Martínez, C. (2016). Ordinal regression methods: Survey and experimental study. IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146.Google Scholar
  77. Kandel, A., & Byatt, W. J. (1978). Fuzzy sets, fuzzy algebra and fuzzy statistics. Proceedings of the IEEE, 68, 1619–1639.CrossRefGoogle Scholar
  78. Kaufmann, A. (1978). Le calcul des admissibilités. Une idée nouvelle à partir de la théorie des sous-ensembles flous. In Proceedings Colloque International sur la Théorie et les Applications des Sous-Ensembles Flous (Vol. I, 14 p.). Marseilles.Google Scholar
  79. Lewis, D. (1973). Counterfactuals and comparative possibility. Journal of Philosophical Logic, 2(4), 418–446.CrossRefGoogle Scholar
  80. Marichal, J.-L. (2000). On Sugeno integral as an aggregation function. Fuzzy Sets and Systems, 114, 347–365.CrossRefGoogle Scholar
  81. Marichal, J.-L. (2009). Weighted Lattice Polynomials. Discrete Mathematics, 309(4), 814–820.CrossRefGoogle Scholar
  82. Mesiar, R. (1997). $k$-order pan-discrete fuzzy measures. In Proceedings 7th IFSA World Congress (Vol. 1, pp. 488–490). Prague.Google Scholar
  83. Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(2), 81–97.Google Scholar
  84. Mitchell, T. (1982). Generalization as search. Artificial Intelligence, 18, 203–226.Google Scholar
  85. Osgood, C. E., Suci, G. J., & Tannenbaum, P. H. (1957). The measurement of meaning. Chicago: University of Illinois Press.Google Scholar
  86. Öztürk, M., & Tsoukiás, A. (2008). Bipolar preference modeling and aggregation in decision support. International Journal of Intelligent Systems, 23(9), 970–984.CrossRefGoogle Scholar
  87. Prade, H., Rico, A., Serrurier, M. (2009a). Elicitation of Sugeno integrals: A version space learning perspective. In J. Rauch, Z. W. Ras, P. Berka, & T. Elomaa (Eds.) Proceedings 18th International Symposium on Foundations of Intelligent Systems (ISMIS ’09) (pp. 392–401). Prague, Czech Rep., Sept. 14–17, Springer LNCS 5722.Google Scholar
  88. Prade, H., Rico, A., Serrurier, M., Raufaste, E. (2009b). Elicitating Sugeno integrals: Methodology and a case study. In C. Sossai & G. Chemello (eds.) Proceedings 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU ’09). Verona, Italy, July 1–3, LNCS 5590.Google Scholar
  89. Ralescu, D., & Sugeno, M. (1996). Fuzzy integral representation. Fuzzy Sets and Systems, 84, 127–133.CrossRefGoogle Scholar
  90. Rico, A. (2002). Modélisation des Préférences pour l’Aide à la Décision par l’Intégrale de Sugeno. Ph. D. Thesis, Université Paris I Sorbonne.Google Scholar
  91. Rico, A., Grabisch, M., Labreuche, C., & Chateauneuf, A. (2005). Preference modeling on totally ordered sets by the Sugeno integral. Discrete Applied Mathematics, 147, 113–124.CrossRefGoogle Scholar
  92. Roy, B. (1996). Multicriteria methodology for decision aiding. Nonconvex optimization and its applications (Vol. 12). Kluwer Academic Publishers: DordrechtGoogle Scholar
  93. Schmeidler, D. (1972). Cores of exact games. Journal of Mathematical Analysis and Applications, 40(1), 214–225.CrossRefGoogle Scholar
  94. Shilkret, N. (1971). Maxitive measure and integration. Indagationes Mathematicae, 33, 109–116.CrossRefGoogle Scholar
  95. Slovic, P., Finucane, M., Peters, E., & MacGregor, D. G. (2002). Rational actors or rational fools? implications of the affect heuristic for behavioral economics. The Journal of Socio-Economics, 31, 329–342.CrossRefGoogle Scholar
  96. Snow, P. (1999). Diverse confidence levels in a probabilistic semantics for conditional logics. Artificial Intelligence, 113(1–2), 269–279.CrossRefGoogle Scholar
  97. Sugeno, M. (1974). Theory of fuzzy integrals and its applications. Ph.D. Thesis, Tokyo Institute of Technology.Google Scholar
  98. Sugeno, M. (1977). Fuzzy measures and fuzzy integrals: A survey. In: M. M. Gupta, et al. (Eds.)Fuzzy automata and decision processes (pp. 89–102). North-Holland.Google Scholar
  99. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.CrossRefGoogle Scholar
  100. Weng, P. (2006). An axiomatic approach in qualitative decision theory with binary possibilistic utility. In Proceedings of the 17th European Conference on Artificial Intelligence (ECAI 2006) (pp. 467–471). Riva del Garda, Italy, IOS Press.Google Scholar
  101. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Miguel Couceiro
    • 1
    Email author
  • Didier Dubois
    • 2
  • Hélène Fargier
    • 2
  • Michel Grabisch
    • 3
  • Henri Prade
    • 2
  • Agnès Rico
    • 4
  1. 1.Université de Lorraine, CNRS, Inria, LORIANancyFrance
  2. 2.IRIT, CNRS, Université Paul SabatierToulouseFrance
  3. 3.Paris School of Economics, Université Paris I Panthéon-SorbonneParisFrance
  4. 4.ERIC, Université Claude Bernard Lyon 1VilleurbanneFrance

Personalised recommendations