Canonical form of ordered weighted averaging operators
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Canonical form of ordered weighted averaging operators LeSheng Jin1 · Radko Mesiar2,3 · Martin Kalina2
· Ronald R. Yager4
Accepted: 15 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Discrete Ordered Weighted Averaging (OWA) operators as one of the most representative proposals of Yager (1988) have been widely used and studied in both theoretical and application areas. However, there are no effective and systematic corresponding methods for continuous input functions. In this study, using the language of measure (capacity) space we propose a Canonical Form of OWA operators which yield some common properties like Monotonicity and Idempotency and thus serve as a generalization of Discrete OWA operators. We provide also a representation of the Canonical Form by means of asymmetric Choquet integrals. The Canonical Form of OWA operators can effectively handle some input functions defined on ordered sets. Keywords Aggregation function · OWA operator · Uncertain information · Measure of orness · Canonical form of OWA operator Mathematics Subject Classification 28E10 · 90B99
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Martin Kalina [email protected] LeSheng Jin [email protected] Radko Mesiar [email protected] Ronald R. Yager [email protected]
1
Business School, Nanjing Normal University, Nanjing, China
2
Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia
3
Faculty of Science, Department of Algebra and Geometry, Palacký University Olomouc, 17 Listopadu 12, 77146 Olomouc, Czech Republic
4
Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA
123
Annals of Operations Research
1 Introduction Discrete OWA operators (Yager 1988) are a powerful aggregation technique (Grabisch et al. 2009; Klement et al. 2000; Mesiar et al. 2018a, b; Paternain et al. 2018; Yager et al. 2011) used in a larger number of areas such as decision-making (Grabisch and Labreuche 2010; Liu and Da 2005), social choice (Llamazares 2007), etc. The OWA operators have been studied and extended from different aspects for the last three decades (Filev and Yager 1998; Jin et al. 2017; Liu 2005; Liu and Han 2008; Llamazares 2007; Mesiar et al. 2018a, b; Torra 1997; Yager and Filev 1994, 1999; Yager et al. 2011). A common discrete form of an OWA operator of dimension n ∈ {2, 3, . . . } is a piece-wise linear aggregation function OWAw : Rn → R, whose aggregation procedure can be split into four steps: (i) Select an input function f : {1, . . . , n} → R (alternatively, f = ( f 1 , . . . , f n ) is an input vector); (ii) Determine an underlying weighing function w : {1, . . . , n} → [0, 1] (alternatively, w = (w1 , . . . , wn ) is a weighing vector), called discrete OWA weighing function, which n satisfies i=1 w(i) = 1; (iii) Choose a suitable permutation τ : {1, . . . , n} → {1, . . . , n} such that f (τ (i)) ≥ f (τ ( j)) whenever i ≤ j; (iv) Aggregate f using w via τ to obtain a final aggregation result OWAw ( f ) =
n
w(i) · f (τ (i)).
(1)
i=1
It is commonly k
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