The binomial decomposition of OWA functions, the 2-additive and 3-additive cases in<i>n</i>dimensions

Bortot, Silvia; Pereira, Ricardo Alberto Marques; Nguyen, Thuy Hong

doi:10.1002/int.21947

Cited by 3 publications

(1 citation statement)

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“…In the binomial decomposition framework, due to Calvo and De Baets and later discussed by Bortot et al in the context of generalized Gini welfare functions, any ordered weighted averaging (OWA) function with weights

w_{i}

with

i = 1, \dots, n

can be represented as a linear combination of binomial OWA functions

C_{j}

with

j = 1, \dots, n

whose coefficients

α_{j}

with

j = 1, \dots, n

are subject to boundary and monotonicity conditions. The coefficients

α_{j}

with

j = 1, \dots, n

are uniquely determined by the weighting structure of the OWA function, and vice versa.…”

Section: Introductionmentioning

confidence: 99%

Maximum entropy ordered weighted averaging in the binomial decomposition framework

2018

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We consider the maximum entropy constrained optimization problem associated with ordered weighted averaging (OWA) in the binomial decomposition framework. We begin by reviewing the analytic solution of the maximum entropy method proposed by Filev and Yager in 1995, and later by Fullér and Majlender in 2001. Next, we briefly review the binomial decomposition framework, which allows for an alternative parametric description of the OWA functions. The values of the binomial coefficients α j , j = 1 ,-0.15em … , n are uniquely determined by the weighting structure of the OWA function. We observe that for low orness values normalΩ ∈ [ 0 , 0.5 ], the optimal weights are decreasing, whereas they are increasing for high orness values normalΩ ∈ [ 0.5 , 1 ]. Moreover, we notice that the optimal values of the first and last weights have a wide range in [ 0 , 1 ], whereas the values of the other weights have more restricted ranges. As for the optimal α j , j = 1 , … , n coefficients, we find that their behavior with respect to orness values normalΩ ∈ [ 0 , 1 ] is very different for low/high orness. We illustrate graphically the optimal α j , j = 1 , … , n coefficients in two parts, first for low orness values normalΩ ∈ [ 0 , 0.5 ] and then for high orness values normalΩ ∈ [ 0.5 , 1 ]. We observe that the optimal α j , j = 1 , … , n for low orness values normalΩ ∈ [ 0 , 0.5 ] are all nonnegative and take values in the unit interval, independently of the dimension n. On the contrary, the optimal values of the α j , j = 1 , … , n coefficients for high orness values normalΩ ∈ [ 0.5 , 1 ] depend strongly on the dimension n, both in the complexity of their distribution and in the amplitude of their scale.

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