Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice
This volume presents the state of the art of new developments, and some interesting and relevant applications of the OWA (ordered weighted averaging) operators. The OWA operators were introduced in the early 1980s by Ronald R. Yager as a conceptually and
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Abstract. We give a survey on the relations between nonadditive integrals (Choquet integral, Sugeno integral) and the OWA operator and its variants. We give also some behavioral indices for the OWA operator, as orness, veto and favor indices, etc. Finally, we propose the use of p-symmetric capacities for a natural generalization of the OWA operator.
1 Introduction This paper offers a survey on the relations between the OWA operators (its classical definition and its variants) and the so-called fuzzy integrals (or more exactly nonadditive integrals). Although the fact that the original OWA operator was a particular case of Choquet integral was discovered, with some surprise, only several years after its birth in 19881, their close relation appears rather obviously if one considers that both operators are linear up to a rearrangement of the arguments in increasing order. Later variants of the original definition, since still based on some rearrangement of the arguments, remain closely related to nonadditive integrals. One may then consider that, since all OWA operators are more or less nonadditive integrals, these operators are no longer useful and there is no need to consider them any more. On the contrary, they provided useful and meaningful families of operators among the vast and unexplored realm of aggregation operators based on nonnaditive integrals (we refer the reader to some chapters of the recent monograph [10] for a detailed account of this Michel Grabisch Centre d’Economie de la Sorbonne, Universit´e Paris I 106-112, Bd de l’Hˆ opital, 75013 Paris, France e-mail: [email protected] 1
Up to our knowledge, this was noticed by Murofushi and Sugeno in a 1993 paper [16].
R.R. Yager et al. (Eds.): Recent Developments in the OWA Operators, STUDFUZZ 265, pp. 3–15. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
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question). In this respect, OWA operators provide aggregation operators with a clear interpretation. Moreover, by some crossfertilization process, many indices defined for nonadditive integrals can be applied to OWA operators to bring new insights into their behavioral properties. Our survey will try to emphasize these issues. Due to size limitation, results are given without proofs. The reader is referred to the bibliography for more details.
2 Capacities and Nonadditive Integrals Let us denote by N := {1, . . . , n} the index set of arguments to be aggregated (scores, utilities, etc.). For simplicity we consider here that scores to be aggregated lie in [0, 1]. Hence, all integrals will be defined for functions f : N → [0, 1], thus assimilated to vectors in [0, 1]n . In the whole paper, we use ∧, ∨ for min and max. Definition 1. A capacity [2] or fuzzy measure [20] on N is a mapping μ : 2N → [0, 1] satisfying (i) μ(∅) = 0, μ(N ) = 1 (normalization) (ii) A ⊆ B implies μ(A) ≤ μ(B) (monotonicity). Definition 2. A capacity μ on N is symmetric if for all A, B ∈ 2N such that |A| = |B|, we have μ(A) = μ(B). Definition 3. Let μ be a capacity on N . The dual (or conjugate)
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