Ordered weighted averaging aggregation operator

In applied mathematics – specifically in fuzzy logic – the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager. 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.
Definition.
Formally an OWA operator of dimension formula_1 is a mapping formula_2 that has an associated collection of weights formula_3 lying in the unit interval and summing to one and with 
where formula_5 is the "j"th largest of the formula_6.
By choosing different "W" one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the "b""j".
Properties.
The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below.
Characterizing features.
Two features have been used to characterize the OWA operators. The first is the attitudinal character(orness).
This is defined as
It is known that formula_19.
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 formula_21 The dispersion characterizes how uniformly the arguments are being used
ÀĚ
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 formula_22 in the form of fuzzy sets defined on the domain of discourse formula_23, then for each formula_24, an formula_25-level type-1 OWA operator with formula_25-level sets formula_27 to aggregate the formula_25-cuts of fuzzy sets formula_29 is given as
where formula_31, and formula_32 is a permutation function such that formula_33, i.e., formula_34 is the formula_35th largest
element in the set formula_36.
The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals formula_37:
formula_38 and formula_39
where formula_40. Then membership function of resulting aggregation fuzzy set is:
For the left end-points, we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
This paper has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.