Ordered Weighted Averaging (OWA) Aggregation Operators
Introduced by Ronald R. Yager, 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.
Formally an operator of dimension is a mapping that has an associated collection of weights lying in the unit interval and summing to one and with
- where bj is the jth largest of the ai
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.
The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below.
Bounded Monotonic if for Symmetric if is a permutation map Idempotent if all
Notable OWA Operators
- if and for
- if and for
Two features have been used to characterize the OWA operators. The first is the attudinal character(orness).
This is defined as
It is known that .
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 similarity of aggregation to OR operation(OR is defined as the Max).
The second feature is the dispersion. This defined as
An alternative definition is The dispersion characterizes how uniformly the arguments are being used
Type-1 OWA Aggregation Operators
The above Yager's OWA operators are used to aggregate the crisp values. Can we aggregate fuzzy sets in the OWA mechanism ? The type-1 OWA operators have been proposed for this purpose. So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.
The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows:
Given the n linguistic weights in the form of fuzzy sets defined on the domain of discourse , then for each , an α-level type-1 OWA operator with α-level sets to aggregate the α-cuts of fuzzy sets is given as
where , and is a permutation function such that , i.e., aσ(i) is the ith largest element in the set .
The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals : and where . Then membership function of resulting aggregation fuzzy set is:
For the left end-points, we we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
The paper has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.
. 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.
. Yager, R. R. and Kacprzyk, J., The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer: Norwell, MA, 1997.
. Liu, X., "The solution equivalence of minimax disparity and minimum variance problems for OWA operators," International Journal of Approximate Reasoning 45, 68-81, 2007.
. Emrouznejad (2009) SAS/OWA: ordered weighted averaging in SAS optimization, Soft Computing 
. Torra, V. and Narukawa, Y., Modeling Decisions: Information Fusion and Aggregation Operators, Springer: Berlin, 2007.
. Majlender, P., "OWA operators with maximal Rényi entropy," Fuzzy Sets and Systems 155, 340-360, 2005.
. Szekely, G. J. and Buczolich, Z., " When is a weighted average of ordered sample elements a maximum likelihood estimator of the location parameter?" Advances in Applied Mathematics 10, 1989, 439-456.
.S. -M. Zhou, F. Chiclana, R. I. John and J. M. Garibaldi, "Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers," Fuzzy Sets and Systems, Vol.159, No.24, pp.3281-3296, 2008 
.S. -M. Zhou, F. Chiclana, R. I. John and J. M. Garibaldi, "Alpha-level aggregation: a practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments," IEEE Transactions on Knowledge and Data Engineering, vol. 23, no.10, 2011, pp.1455 - 1468.
.S. -M. Zhou, R. I. John, F. Chiclana and J. M. Garibaldi, "On aggregating uncertain information by type-2 OWA operators for soft decision making," International Journal of Intelligent Systems, vol. 25, no.6, pp. 540-558, 2010.
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