Ordered Weighted Averaging (OWA) Aggregation Operators

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The Ordered Weighted Averaging operators, commonly called OWA operators, provide a parameterized class of mean type aggregation operators. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions.

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[edit] Definition

Formally an \ OWA operator of dimension \ n is a mapping F: R_n \rightarrow R that has an associated collection of weights \ W = [w_1, ..., w_n] lying in the unit interval and summing to one and with

F(a_1, ..., a_n) = \sum_{j=1}^n w_j b_j where bj is the jth largest of the aj

By choosing different W we can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj.

[edit] Properties

The OWA operator is a mean operator. It is

Bounded: Min(a_1, ..., a_n) \le F(a_1, ..., a_n) \le Max(a_1, ..., a_n)
Monotonic: F(a_1, ..., a_n) \ge F(g_1, ..., g_n) if a_i \ge g_i
Symmetric: F(a_1, ..., a_n) = F(a_\boldsymbol{\pi(1)}, ..., a_\boldsymbol{\pi(n)}) if \boldsymbol{\pi} is a permutation map
Idempotent: \ F(a_1, ..., a_n) = a if all \ a_i = a

[edit] Notable OWA Operators

\ F(a_1, ..., a_n) = Max(a_1, ..., a_n) if \ w_1 = 1 and \ w_j = 0 for j \ne 1
\ F(a_1, ..., a_n) = Min(a_1, ..., a_n) if \ w_n = 1 and \ w_j = 0 for j \ne n

[edit] Characterizing Features

Two features have been used to characterize the OWA operators. The first is the attitudinal character.

This is defined as

A-C(W) = A-C(W) = \frac{1}{n-1} \sum_{j=1}^n (n - j).

It is known that A-C(W) \in [0, 1] .

In addition A-C(Max) = 1, A-C(Ave) = A-C(Med) = 0.5 and A–C(Min) = 0. Thus the A-C goes from 1 to 0 as we go from Max to Min aggregation. The attitudinal character characterizes the type of aggregation.

The second feature is the dispersion. This defined as

H(W) = -\sum_{j=1}^n w_j \ln (w_j).

An alternative definition is E(W) = \sum_{j=1}^n w_j^2 . The dispersion characterizes how uniformly the arguments are being used


[edit] References

  • Yager, R. R., "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man and Cybernetics 18, 183-190, 1988.
  • Liu, X., "The solution equivalence of minimax disparity and minimum variance problems for OWA operators," International Journal of Approximate Reasoning 45, 68-81, 2007.
  • Majlender, P., "OWA operators with maximal Renya entropy," Fuzzy Sets and Systems 155, 340-360, 2005.
  • Torra, V. and Narukawa, Y., Modeling Decisions: Information Fusion and Aggregation Operators, Springer: Berlin, 2007.