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An axiomatic approach to the definition of the entropy of a discrete Choquet capacity ; Marichal, Jean-Luc ; in Information Sciences (2005), 172(1-2), 131-153 To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this ... [more ▼] To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this new concept on the basis of three axioms: the symmetry property, a boundary condition for which the entropy reduces to the Shannon entropy, and a generalized version of the well-known recursivity property. We also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy-like measure. Lastly, we provide an interpretation of it in the framework of aggregation by the discrete Choquet integral. [less ▲] Detailed reference viewed: 251 (6 UL)Sorting multiattribute alternatives: The TOMASO method Marichal, Jean-Luc ; ; in Computers and Operations Research (2005), 32(4), 861-877 We analyze a recently proposed ordinal sorting procedure (Tomaso) for the assignment of alternatives to graded classes and we present a freeware constructed from this procedure. We illustrate it by two ... [more ▼] We analyze a recently proposed ordinal sorting procedure (Tomaso) for the assignment of alternatives to graded classes and we present a freeware constructed from this procedure. We illustrate it by two examples, and do some testing in order to show its usefulness. [less ▲] Detailed reference viewed: 139 (5 UL)TOMASO, A solution in the presence of interacting points of views ; ; Marichal, Jean-Luc Article for general public (2004) This short article briefly presents the main features of the multiple criteria sorting tool TOMASO (Technique for Ordinal Multi-Attribute Sorting and Ordering) and its implementation. Its main ... [more ▼] This short article briefly presents the main features of the multiple criteria sorting tool TOMASO (Technique for Ordinal Multi-Attribute Sorting and Ordering) and its implementation. Its main particularities are the possibility to consider interacting points of view and the use of the Choquet integral as a discriminant function. The capacities are learnt through the use of protoypes, which are well known alternatives for the Decision Maker. [less ▲] Detailed reference viewed: 50 (4 UL)An axiomatic approach to the definition of the entropy of a discrete Choquet capacity ; Marichal, Jean-Luc ; in Proc. 9th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2002), Annecy, France, July 1-5, 2002 (2002, July) Detailed reference viewed: 224 (2 UL)The use of the discrete Sugeno integral in decision making: a survey ; Marichal, Jean-Luc ; et al in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (2001), 9(5), 539-561 An overview of the use of the discrete Sugeno integral as either an aggregation tool or a preference functional is presented in the qualitative framework of two decision paradigms: multi-criteria decision ... [more ▼] An overview of the use of the discrete Sugeno integral as either an aggregation tool or a preference functional is presented in the qualitative framework of two decision paradigms: multi-criteria decision-making and decision-making under uncertainty. The parallelism between the representation theorems in both settings is stressed, even if a basic requirement like the idempotency of the aggregation scheme should be explicitely stated in multi-criteria decision-making, while its counterpart is implicit in decision under uncertainty by equating the utility of a constant act with the utility of its consequence. Important particular cases of Sugeno integrals such as prioritized minimum and maximum operators, their ordered versions, and Boolean max-min functions are studied. [less ▲] Detailed reference viewed: 78 (4 UL)On a sorting procedure in the presence of qualitative interacting points of view Marichal, Jean-Luc ; in Klement, Erich Peter; Roubens, Marc (Eds.) Proc. 22nd Linz Seminar on Fuzzy Set Theory (LINZ 2001): Valued Relations and Capacities in Decision Theory (2001, February) We present a sorting procedure for the assignment of alternatives to graded classes. The available information is given by partial evaluations of the alternatives on ordinal scales representing ... [more ▼] We present a sorting procedure for the assignment of alternatives to graded classes. The available information is given by partial evaluations of the alternatives on ordinal scales representing interacting points of view and a subset of prototypic alternatives whose assignment is imposed beforehand. The partial evaluations of each alternative are embedded in a common interval scale by means of commensurateness mappings, which in turn are aggregated by the discrete Choquet integral. The behavioral properties of this Choquet integral are then measured through importance and interaction indices. [less ▲] Detailed reference viewed: 48 (3 UL)The use of the discrete Sugeno integral in decision making: a survey ; Marichal, Jean-Luc ; et al in Colorni, A.; Paruccini, M.; Roy, B. (Eds.) AMCDA - Aide Multi Critère à la Décision (Multiple Criteria Decision Aiding) (2001) An overview of the use of the discrete Sugeno integral as either an aggregation tool or a utility function is presented in the qualitative framework of two decision paradigms: multi-criteria decision ... [more ▼] An overview of the use of the discrete Sugeno integral as either an aggregation tool or a utility function is presented in the qualitative framework of two decision paradigms: multi-criteria decision-making and decision-making under uncertainty. The parallelism between the representation theorems in both settings is stressed, even if a basic requirement like idempotency should be explicitely stated in multi-criteria decision-making, while its counterpart is implicit in decision under uncertainty by equating the utility of a constant act with the utility of its consequence. Important particular cases of Sugeno integrals such as prioritized minimum and maximum operators, their ordered versions, and Boolean max-min functions are studied. [less ▲] Detailed reference viewed: 56 (5 UL)On a sorting procedure in the presence of qualitative interacting points of view Marichal, Jean-Luc ; in Chojcan, J.; Leski, J. (Eds.) Fuzzy sets and their applications (2001) We present a sorting procedure for the assignment of alternatives to graded classes. The available information is given by partial evaluations of the alternatives on ordinal scales representing ... [more ▼] We present a sorting procedure for the assignment of alternatives to graded classes. The available information is given by partial evaluations of the alternatives on ordinal scales representing interacting points of view and a subset of prototypic alternatives whose assignment is imposed beforehand. The partial evaluations of each alternative are embedded in a common interval scale by means of commensurateness mappings, which in turn are aggregated by the discrete Choquet integral. The behavioral properties of this Choquet integral are then measured through importance and interaction indices. [less ▲] Detailed reference viewed: 42 (4 UL)Entropy of discrete fuzzy measures Marichal, Jean-Luc ; in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (2000), 8(6), 625-640 The concept of entropy of a discrete fuzzy measure has been recently introduced in two different ways. A first definition was proposed by Marichal in the aggregation framework, and a second one by Yager ... [more ▼] The concept of entropy of a discrete fuzzy measure has been recently introduced in two different ways. A first definition was proposed by Marichal in the aggregation framework, and a second one by Yager in the framework of uncertain variables. We present a comparative study between these two proposals and point out their properties. We also propose a definition for the entropy of an ordinal fuzzy measure, that is, a fuzzy measure taking its values in an ordinal scale in the sense of measurement theory. [less ▲] Detailed reference viewed: 111 (3 UL)Determination of weights of interacting criteria from a reference set Marichal, Jean-Luc ; in European Journal of Operational Research (2000), 124(3), 641-650 In this paper, we present a model allowing to determine the weights related to interacting criteria. This is done on the basis of the knowledge of a partial ranking over a reference set of alternatives ... [more ▼] In this paper, we present a model allowing to determine the weights related to interacting criteria. This is done on the basis of the knowledge of a partial ranking over a reference set of alternatives (prototypes), a partial ranking over the set of criteria, and a partial ranking over the set of interactions between pairs of criteria. [less ▲] Detailed reference viewed: 128 (7 UL)On the entropy of non-additive weights Marichal, Jean-Luc ; Scientific Conference (2000, July) We consider a Choquet capacity, that is a set function which describes the importance of every subset of criteria in a MCDA problem. The following question is approached: what is the generalized ... [more ▼] We consider a Choquet capacity, that is a set function which describes the importance of every subset of criteria in a MCDA problem. The following question is approached: what is the generalized counterpart of the Shannon entropy (defined for a probabilistic measure) for such a capacity? The extension that is proposed depends on the scale type. In the cardinal case, the entropy is defined in terms of the first derivatives of the non-additive measures. In the ordinal case, it refers to the cardinality of the scale values that appear in the set of all capacities. Both generalized entropies are symmetric functions of the capacities and their extreme values (max entropy and min entropy) are characterized. An application to the determination of weights is given when interacting criteria are considered. [less ▲] Detailed reference viewed: 39 (0 UL)Entropy of discrete fuzzy measures Marichal, Jean-Luc ; in Proc. 3rd Int. Workshop on Preferences and Decisions (TRENTO 2000) (2000, June) The concept of entropy of a discrete fuzzy measure has been recently introduced in two different ways. A first definition was proposed by Marichal [10] in the aggregation framework, and a second one by ... [more ▼] The concept of entropy of a discrete fuzzy measure has been recently introduced in two different ways. A first definition was proposed by Marichal [10] in the aggregation framework, and a second one by Yager [25] in the framework of uncertain variables. We present a comparative study between these two proposals and point out their properties. We also propose a definition for the entropy of an ordinal fuzzy measure, that is, a fuzzy measure taking its values in an ordinal scale in the sense of measurement theory. [less ▲] Detailed reference viewed: 101 (2 UL)Equivalent representations of set functions ; Marichal, Jean-Luc ; in Mathematics of Operations Research (2000), 25(2), 157-178 This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius transformation, and the interactions between elements of any subset of a given set as ... [more ▼] This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation. [less ▲] Detailed reference viewed: 119 (15 UL)Entropy of discrete fuzzy measures Marichal, Jean-Luc ; Scientific Conference (2000, April) Detailed reference viewed: 71 (0 UL)Aide à la décision en présence de données ordinales Marichal, Jean-Luc ; Presentation (1999, October 22) Detailed reference viewed: 39 (1 UL)Consensus with ordinal data Marichal, Jean-Luc ; in Proc. 7th Eur. Congr. on Intelligent Techniques and Soft Computing (EUFIT'99) (1999, September) We present a model allowing to aggregate decision criteria when the available information is of qualitative nature. The use of the Sugeno integral in this model is justified by an axiomatic approach. An ... [more ▼] We present a model allowing to aggregate decision criteria when the available information is of qualitative nature. The use of the Sugeno integral in this model is justified by an axiomatic approach. An illustrative example is also provided. [less ▲] Detailed reference viewed: 45 (1 UL)Ordinal aggregation with qualitative scales Marichal, Jean-Luc ; in Proc. 50th Meeting of the Eur. Working Group "Multiple Criteria Decision Aiding" (MCDA 50), Cerisy-la-Salle, France, Sep. 28 - Oct. 2, 1999 (1999, September) Detailed reference viewed: 43 (3 UL)Entropy of a Choquet capacity Marichal, Jean-Luc ; in Mayor, Gaspar; Suñer, Jaume (Eds.) Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain, September 22-25, 1999. (1999, September) Detailed reference viewed: 42 (3 UL)About the Choquet integral as a tool to aggregate utilities in the presence of interacting points of view Marichal, Jean-Luc ; Scientific Conference (1999, August) One basic assumption to consider an additive utility function is the preferential independence. When interacting criteria are considered, this condition might be violated and a substitute to the classical ... [more ▼] One basic assumption to consider an additive utility function is the preferential independence. When interacting criteria are considered, this condition might be violated and a substitute to the classical weighted mean has to be adopted. The Choquet integral seems to be an adequate aggregation operator that extends the weighted mean and the ordered weighted average (OWA). The axiomatics that supports the Choquet integral is presented as well as its behavioral analysis with regards to veto and favor effects, degree of disjunction and measure of dispersion. One illustrative example of its application in the field of MCDM is provided. [less ▲] Detailed reference viewed: 38 (3 UL)Equivalent representations of a set function with applications to game theory and multicriteria decision making ; Marichal, Jean-Luc ; in de Swart, Harrie (Ed.) Proc. of the Int. Conf. on Logic, Game theory and Social choice (LGS'99), Oisterwijk, the Netherlands, May 13-16, 1999 (1999, May) This paper introduces four alternative representations of a set function: the M¨obius transformation, the co-M¨obius transformation, and the interactions between elements of any subset of a given set as ... [more ▼] This paper introduces four alternative representations of a set function: the M¨obius transformation, the co-M¨obius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation. [less ▲] Detailed reference viewed: 54 (3 UL) |
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