Skip to main content
Log in

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

  • Invited Survey
  • Published:
4OR Aims and scope Submit manuscript

Abstract

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bana e Costa CA, Vansnick J-C (1994) A theoretical framework for measuring attractiveness by a categorical based evaluation TecHnique (MACBETH). In: Proceedings of XIth internatinal conference on multicriteria decision making, Coimbra, pp 15–24

  • Banae Costa CA, Vansnick J-C (1997) Applications of the MACBETH approach in the framework of an additive aggregation model. J Multicriteria Decis Anal 6: 107–114

    Article  Google Scholar 

  • Banae Costa CA, Vansnick J-C (1999) The MACBETH approach: basic ideas, software and an application. In: Meskens N, Roubens M(eds) Advances in decision analysis. Kluwer, Dordrecht, pp 131–157

    Google Scholar 

  • Berrah L, Mauris G, Montmain J (2008) Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation. Omega 36(3): 340–351

    Article  Google Scholar 

  • Bilbao JM, Fernandez JR, Jiménez Losada A, Lebrón E (2000) Bicooperative games. In: Bilbao JM(eds) Cooperative games on combinatorial structures. Kluwer, Dordrecht

    Google Scholar 

  • Bilbao JM, Fernández JR, Jiménez N, López JJ (2007) The value for bi-cooperative games. Ann Oper Res (to appear)

  • Bouyssou D, Marchant Th (2007a) An axiomatic approach to noncompensatory scoring methods in MCDM, I: The case of two categories. Eur J Oper Res 178: 217–245

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bouyssou D, Marchant Th, Pirlot M (2003) A conjoint measurement approach to the discrete sugeno integral. a note on a result of greco, matarazzo and słowiński. Ann LAMSADE 6: 39–60

    Google Scholar 

  • Choquet G (1953) Theory of capacities. Ann Inst Fourier 5: 131–295

    Google Scholar 

  • Combarro E, Miranda P (2003) A genetic algorithm for the identification of fuzzy measures from sample data. In: International fuzzy systems Association World Congress (IFSA 2003), Istanbul, pp 163–166

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

  • Dubois D, Ostasiewicz W, Prade H (2000a) Fuzzy sets: history and basic notions. In: Dubois D, Prade H(eds) Fundamentals of fuzzy sets, vol 1 of the handbooks of fuzzy sets. Kluwer, Dordrecht, pp 21–124

    Google Scholar 

  • Dubois D, Prade H, Sabbadin R (2000b) Qualitative decision theory with Sugeno integrals. In: Grabisch M, Murofushi T, Sugeno M(eds) Fuzzy measures and integrals—theory and applications. Physica Verlag, Heidelberg, pp 314–332

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Dujmović JJ (1974) Weighted conjunctive and disjunctive means and their application in system evaluation. Univ. Beograd. Publ. Elektrotechn. Fak., pp 147–158

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

    Article  Google Scholar 

  • Felsenthal D, Machove Mr (1997) Ternary voting games. Int J Game Theory 26: 335–351

    Google Scholar 

  • Fragnelli V, Moretti S (2007) A game theoretical approach to the classification problem in gene expression data analysis. Comput Math Appl (to appear)

  • Fujimoto K, Murofushi T (2000) Hierarchical decomposition of the Choquet integral. In: Grabisch M, Murofushi T, Sugeno M(eds) Fuzzy measures and integrals—theory and applications. Physica Verlag, Heidelberg, pp 94–104

    Google Scholar 

  • Fujimoto K, Kojadinovic I, Marichal J-L (2006) Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games Econ Behav 55: 72–99

    Article  Google Scholar 

  • Grabisch M (1995) A new algorithm for identifying fuzzy measures and its application to pattern recognition. In: Int. Joint Conf. of the 4th IEEE Int. Conf. on Fuzzy Systems and the 2nd Int. Fuzzy engineering symposium. Yokohama, pp 145–150

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Grabisch M (1997) Alternative representations of discrete fuzzy measures for decision making. Int J Uncertain Fuzziness Knowl Based Syst 5: 587–607

    Article  Google Scholar 

  • Grabisch M (2000a) Évaluation subjective. In: Bouyssou D, Dubois D, Pirlot M, Prade H (eds) Concepts et Méthodes pour l’Aide à la Décision, vol. 3: Analyse multicritère, IC2., Hermès, pp 175–232

  • Grabisch M (2000b) The interaction and Möbius representations of fuzzy measures on finite spaces, k-additive measures: a survey. In: Grabisch M, Murofushi T, Sugeno M(eds) Fuzzy measures and integrals—theory and applications. Physica Verlag, Heidelberg, pp 70–93

    Google Scholar 

  • Grabisch M (2002) Set function over finite sets: transformations and integrals. In: Pap E(eds) Handbook of measure theory. Elsevier, Amsterdam, pp 1381–1401

    Google Scholar 

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

    Article  Google Scholar 

  • Grabisch M (2003b) Modelling data by the Choquet integral. In: Torra V(eds) Information fusion in data mining. Physica Verlag, Heidelberg, pp 135–148

    Google Scholar 

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

    Article  Google Scholar 

  • Grabisch M (2004b) The Choquet integral as a linear interpolator. In: 10th international conference on information processing and management of uncertainty in knowledge-based systems (IPMU 2004). Perugia, pp 373–378

  • Grabisch M (2006) Some lexicographic approaches to the Sugeno integral. In: Proceedings of international conference on information processing and management of uncertainty (IPMU’06). Paris, pp 572–579

  • Grabisch M, Labreuche Ch (2001) How to improve acts: an alternative representation of the importance of criteria in mcdm. Int J Uncertain Fuzziness Knowl Based Syst 9: 145–157

    Google Scholar 

  • Grabisch M, Labreuche Ch (2005a) Bi-capacities. Part I: definition, Möbius transform and interaction. Fuzzy Sets Syst 151: 211–236

    Article  Google Scholar 

  • Grabisch M, Labreuche Ch (2005b) Fuzzy measures and integrals in MCDA. In: Figueira J, Greco S, Ehrgott M(eds) Multiple criteria decision analysis. Kluwer, Dordrecht, pp 563–608

    Chapter  Google Scholar 

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

    Article  Google Scholar 

  • Grabisch M, Nicolas JM (1994) Classification by fuzzy integral—performance and tests. Fuzzy Sets Syst Spec Issue Pattern Recogn 65: 255–271

    Article  Google Scholar 

  • Grabisch M, Raufaste E (2007) An empirical study of statistical properties of Choquet and Sugeno integrals. IEEE Trans Fuzzy Syst (to appear)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Grabisch M, Greco S, Pirlot M (2007) Bipolar and bivariate models in multicriteria decision analysis: descriptive and constructive approaches. Int J Intell Syst (to appear)

  • Grabisch M, Kojadinovic I, Meyer P (2008) 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

    Article  Google Scholar 

  • Greco S, Matarazzo B, Słowiński R (2002) Bipolar Sugeno and Choquet integrals. In: EUROFUSE workshop on informations systems. Varenna, pp 191–196

  • Greco S, Matarazzo B, Słowiński R (2004) Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. Eur J Oper Res 158(2): 271–292

    Article  Google Scholar 

  • Havrda J, Charvat F (1967) Quantification method of classification processes. concept of structural a-entropy. Kybernetika 3: 30–95

    Google Scholar 

  • Kojadinovic I (2007a) Minimum variance capacity identification. Eur J Oper Res 177: 498–514

    Article  Google Scholar 

  • Kojadinovic I (2007b) 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

    Article  Google Scholar 

  • Krantz DH, Luce RD, Suppes P, Tversky A (1971) Foundations of measurement, vol 1: additive and Polynomial Representations. Academic

    Google Scholar 

  • Kwon SH, Sugeno M (2000) 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. Physica Verlag, Heidelberg, pp 375–391

    Google Scholar 

  • Labreuche Ch (2004) Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation. In: International conference on information processing and management of uncertainty in knowledge-based systems (IPMU). Perugia, July 4–9, pp 609–616

  • Labreuche Ch (2005) 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). Barcelona, September 7–9, pp 482–487

  • Labreuche Ch (2006) Argumentation of the decision made by several aggregation operators based on weights. In: International conference on information processing and management of uncertainty in knowledge-based systems (IPMU). Paris, July 2–7, pp 683–691

  • Labreuche Ch (2007) Modeling the context in multi-criteria decision analysis. In: International conference of the Euro society for fuzzy logic and technology (EUSFLAT), Ostrava, September 11–14

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

    Article  Google Scholar 

  • Labreuche Ch, Grabisch M (2004) Partially unipolar bi-capacities in MCDM. In: SCIS-ISIS. Yokohama, Japan, September 21–24

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

    Article  Google Scholar 

  • Labreuche Ch, Grabisch M (2006b) Definition of importance indices for bicapacities in MCDA. In: International conference on information processing and management of uncertainty in knowledge-based systems (IPMU). Paris, July 2–7, pp 292–299

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

    Article  Google Scholar 

  • Labreuche Ch, Grabisch M (2007b) A value for bi-cooperative games (submitted)

  • Labreuche Ch, Le Huédé F (2005) Myriad: a tool suite for MCDA. In: Int. Conf. of the Euro Society for Fuzzy Logic and Technology (EUSFLAT). Barcelona, September 7–9, pp 204–209

  • Labreuche Ch, Le Huédé F (2006) Inconsistencies in the determination of a capacity. In: European conference on artificial intelligence (ECAI)—workshop on advances in preference handling. Riva del Garda, August 28–September 1, pp 73–80

  • Liginlal D, Ow TT (2005) On policy capturing with fuzzy measures. Eur J Oper Res 167: 461–474

    Article  Google Scholar 

  • Lovász L (1983) Submodular functions and convexity. In: Bachem A, Grötschel M, Korte B(eds) Mathematical programming. The state of the art. Springer, Heidelberg, pp 235–257

    Google Scholar 

  • Marichal J-L (1998) Aggregation operators for multicriteria decision aid. PhD Thesis, University of Liège

  • Marichal J-L (2001) 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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Marichal J-L (2002) Aggregation of interacting criteria by means of the discrete Choquet integral. In: Calvo T, Mayor G, Mesiar R(eds) Aggregation operators: new trends and applications, vol 97 of Studies in Fuzziness and Soft Computing. Physica Verlag, Heidelberg, pp 224–244

    Google Scholar 

  • Marichal J-L (2004) Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. Eur J Oper Res 155(3): 771–791

    Article  Google Scholar 

  • Marichal J-L (2007) Weighted lattice polynomials. Discrete Math (submitted)

  • Marichal J-L (2007) k-intolerant capacities and Choquet integrals. Eur J Oper Res 177(3): 1453–1468

    Article  Google Scholar 

  • Marichal J-L, Mathonet P (2007) Approximations of lovász extensions and their induced interaction index. Discrete Appl Math. doi:10.1016/j.dam.2007.07.011

    Google Scholar 

  • Marichal J-L, Roubens M (2000) Determination of weights of interacting criteria from a reference set. Eur J Oper Res 124: 641–650

    Article  Google Scholar 

  • Marichal J-L, Meyer P, Roubens M (2005) Sorting multi-attribute alternatives: the TOMASO method. Comput Oper Res 32: 861–877

    Article  Google Scholar 

  • 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. Kluwer, Dordrecht, pp 471–506

    Chapter  Google Scholar 

  • Miranda P, Grabisch M (1999) Optimization issues for fuzzy measures. Int J Uncertain Fuzziness Knowl Based Syst 7(6): 545–560

    Article  Google Scholar 

  • Miranda P, Grabisch M, Gi Pl P (2002) p-symmetric fuzzy measures. Int J Uncertain Fuzziness Knowl Based Syst 10(Suppl): 105–123

    Article  Google Scholar 

  • Montmain J (2007) Interprétation de la contribution d’un critère dans le contexte d’une amélioration multicritère basée sur une intégrale de Choquet. In: Logique Floue et Applications (LFA). Nîmes, France

  • Moulin H (1988) Axioms of cooperative decision making. Wily

  • Murofushi T (1992) A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure (in Japanese). In: 2nd Fuzzy Workshop. Nagaoka, October 1992, pp 39–48

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

    Article  Google Scholar 

  • Murofushi T, Soneda S (1993) Techniques for reading fuzzy measures (III): interaction index (in Japanese). In: 9th fuzzy system symposium. Sapporo, pp 693–696

  • Murofushi T, Sugeno M (2000) Fuzzy measures and fuzzy integrals. In: Grabisch M, Murofushi T, Sugeno M(eds) Fuzzy measures and integrals—theory and applications. Physica Verlag, Heidelberg, pp 3–41

    Google Scholar 

  • Näther W, Wälder K (2007) Applying fuzzy measures for considering interaction effects in root dispersal models. Fuzzy Sets Syst 158: 572–582

    Article  Google Scholar 

  • Pignon JP, Labreuche Ch (2007) 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, December 4–6

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

    Article  Google Scholar 

  • Roberts FS (1979) Measurement theory. Addison-Wesley

    Google Scholar 

  • Roubens M (2001) Ordinal multiattribute sorting and ordering in the presence of interacting points of view. In: Bouyssou D, Jacquet-Lagrèze E, Perny P, Slowinsky D, Vanderpooten R, Vincke Ph(eds) Aiding decisions with multiple criteria: essays in honor of Bernard Roy. Kluwer, Dordrecht, pp 229–246

    Google Scholar 

  • Saito K, Watanabe Y, Hashimoto H, Uchiyama A (2007) An application of fuzzy integral model for the clinical diagnosis. J Biomed Fuzzy Syst Assoc 1(1): 17–24

    Google Scholar 

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

    Article  Google Scholar 

  • Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57(3): 571–587

    Article  Google Scholar 

  • Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW(eds) Contributions to the theory of games, vol. II, number 28 in annals of mathematics studies. Princeton University Press, New Jersey, pp 307–317

    Google Scholar 

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

    Article  Google Scholar 

  • Singer I (1984) Extensions of functions of 0-1 variables and applications to combinatorial optimization. Numer Funct Anal Optim 7(1): 23–62

    Article  Google Scholar 

  • Slovic P, Finucane M, Peters E, MacGregor DG (2002) The affect heuristic. In: Gilovitch T, Griffin D, Kahneman D(eds) Heuristics and biases: the psychology of intuitive judgment. Cambridge University Press, Cambridge, pp 397–420

    Google Scholar 

  • Słowiński R, Greco S, Matarazzo B (2002) Axiomatization of utility, outranking and decision-rule preference models for multiple-criteria classification problems under partial inconsistency with the dominance principle. Control Cybern 31(4): 1005–1035

    Google Scholar 

  • Sugeno M (1973) Construction of fuzzy measures and evaluation of similarity patterns by fuzzy integral. Trans. S.I.C.E. (Keisoku Jidōseigyō Gakkai) 9:361–368, 1973 (in japanese)

    Google Scholar 

  • Sugeno M (1974) Theory of fuzzy integrals and its applications. PhD Thesis, Tokyo Institute of Technology

  • Sugeno M (1977) Fuzzy measures and fuzzy integrals—a survey. In: Gupta MM, Saridis GN, Gaines BR (eds) Fuzzy automata and decision processes. North Holland, Amsterdam, pp 89–102

  • Sugeno M, Kwon SH (1995) A clusterwise regression-type model for subjective evaluation. J Japan Soc Fuzzy Theory Syst 7(2): 291–310

    Google Scholar 

  • Tanaka A, Murofushi T (1989) A learning model using fuzzy measures and the Choquet integral (in Japanese). In: 5th Fuzzy System Symposium. Kobe, June 1989, pp 213–218

  • Torra V, Narukawa Y (2007) The h-index and the number of citations: two fuzzy integrals. IEEE Trans on Fuzzy Syst (to appear)

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Wendling L, Tabbone S (2003) Recognition of arrows in line drawings based on the aggregation of geometric criteria using the choquet integral. In: Seventh international conference on document analysis and recognition—ICDAR2003 . Edinburgh, August, pp 299–303

  • Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18: 183–190

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michel Grabisch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grabisch, M., Labreuche, C. A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6, 1–44 (2008). https://doi.org/10.1007/s10288-007-0064-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10288-007-0064-2

Keywords

MSC Classification (2000)

Navigation