This paper provides an improvement of the general minimax disparity ordered weighted averaging (OWA) operator problem under more generalized assumption of the OWA operator. The work builds upon previous work regarding a general model of OWA aggregation of a given “orness” level.
Yager [1] introduced a new aggregation technique based on the ordered weighted averaging (OWA) as a method for aggregating multiple units within max and min operators. Previous studies have suggested a number of approaches for obtaining the associated weights in different areas such as decision making, neural networks, expert systems, date mining, approximate reasoning, fuzzy system and control [1–11]. An OWA operator of dimension n is a mapping F: R^{n} → R that has an associated weighting vector W = (w_{1}, ⋯, w_{n}) of having the properties w_{1} + ⋯ + w_{n} = 1, 0 ≤ w_{i} ≤ 1, i = 1, ⋯, n, and such that
where b_{j} is the jth largest element of the collection of the aggregated objects {a_{1}, ⋯, a_{n}}. In [1], the author introduced a measure of “orness” associated with the weighting vector W of an OWA operator, defined as
and characterizes the degree to which the aggregation is like an or operation. Here, “orness” is interpreted as the mode of decision-making circumstances by conferring a meaning to the weights used in the process aggregation. An important issue in the theory of OWA operators is the determination of the associated weights, such as the methodology developed by O’Hagan [9]. In this methodology, the vector of the OWA weights can be calculated for a previously defined level of “orness”.
Wang and Parkan [11] first proposed a minimax disparity OWA operator optimization problem:
The minimax disparity approach obtains OWA operator weights based on the minimization of the maximum difference between any two adjacent weights. The new approach was determined to be simpler than existing approaches due the requirement of only the formulation and solution of a linear programming model compared to other methods that require complex non-linear equations.
Recently, Liu [8, Theorem 6] considered the general minimax disparity OWA operator optimization problem:
*
where F is a strictly convex function on [0,∞), and it is at least two order differentiable.
In this model, a consistent OWA operator is obtained allowing the aggregation value of the operator to monotonically changed with the given “orness” level. Further, a general regular increasing monotone quantifier determination was proposed and solved using the optimal control technique. The work of Liu demonstrated a key optimization model to incorporate background knowledge or required characteristics of aggregation problems.
In this paper, we give an improved version of the result for the problem (
It is noted that F is strictly convex if and only if F′ is strictly increasing. If F′(x) is strictly increasing, then there are two possible cases:
where lim_{x}_{→0+}= F′(0^{+}). In case of F′(0^{+}) < ∞, we define F′(0^{+}) = F′(0).
Assume that F is strictly convex and F′(x) is continuous. The optimal solution for problem (
where a^{*}, b^{*} are determined by the constraints:
and H = {i|(F′)^{−1}(a^{*}i + b^{*}) > 0}.
By the dual property [Theorem 5, 7] of OWA operator to generate OWA operator weights, it suffices only to prove the case of 1/2 ≤ α ≤ 1. We clearly have
Let
Then we clearly have
If F′ (0^{+}) < ∞, then
Now, let w_{i}, i = 1, ⋯, n satisfy
Then we can prove that
which shows that W^{*} is an optimal solution of the constrained optimization problem (
Then
If not, then
Similarly, by induction, we have that
which is contradictory to (
Let w_{i}, i = 1, ⋯, n satisfy
and suppose that
Then
If not, then
and hence
where the second equality comes from the fact that
We also have
and hence
and suppose that
By Lemma 2.2, we have
and hence
We consider a OWA operator F which is not differentiable to at least the second order but F′ is absolutely continuous and find a optimal solution of two OWA problem.
Let a OWA operator F be
Then
Hence F(x) is strictly convex and F′(x) is continuous, but F(x) is not second order differentiable. Then we cannot apply the result of Liu [8, Theorem 6] but our result, Theorem 2.1. By the dual property [8, Theorem 5] of OWA to generate OWA operate weights, it suffices only to prove the case
where m ∈ H, m − 1∉ H and
Since
If
where m ∈ H and m + 1 ∉ H. From
we can get
Let α_{1} and α_{2} be solutions of
Since a^{*} + b^{*} is increasing and
This implies thatm < 4.
If m = 3 then,
and if m = 2, then
So
for
for
If
we can get
Hence
and so
for 1 ≤ i ≤ m and
and
For example, if n = 5, then we have for
for
for
and for
An improvement of the general minimax disparity OWA operator problem developed by Liu [8] is presented. A more generalized assumption of the OWA operator is utilized to build upon previous work regarding a general aggregation model for a given “orness” level.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03027869).
The OWA operator weihts for
E-mail: dhhong@mju.ac.kr