Maximum entropy ordered weighted averaging in the binomial decomposition framework

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, …,n are uniquely determined by the weighting structure of the OWA function. We observe that for low orness values Ω ∈[0, 0.5], the optimal weights are decreasing, whereas they are increasing for high orness values Ω ∈[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 Ω ∈[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 Ω ∈[0, 0.5] and then for high orness values Ω ∈[0.5, 1]. We observe that the optimal α_j, j = 1, …,n for low orness values Ω ∈[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 Ω ∈[0.5, 1] depend strongly on the dimension n, both in the complexity of their distribution and in the amplitude of their scale.