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A General Minimax Disparity OWA Operator Optimization Problem
International Journal of Fuzzy Logic and Intelligent Systems 2018;18(4):292-297
Published online December 31, 2018
© 2018 Korean Institute of Intelligent Systems.

Dug Hun Hong

Department of Mathematics, Myongji University, Yongin, Korea
Correspondence to: Correspondence to: Dug Hun Hong, (dhhong@mju.ac.kr)
Received June 26, 2018; Revised September 21, 2018; Accepted December 19, 2018.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

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.

Keywords : Fuzzy sets, OWA operator, Minimax disparity
1. Introduction

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 [111]. An OWA operator of dimension n is a mapping F: RnR that has an associated weighting vector W = (w1, ⋯, wn) of having the properties w1 + ⋯ + wn = 1, 0 ≤ wi ≤ 1, i = 1, ⋯, n, and such that

F(a1,,an)=i=1nwibi,

where bj is the jth largest element of the collection of the aggregated objects {a1, ⋯, an}. In [1], the author introduced a measure of “orness” associated with the weighting vector W of an OWA operator, defined as

orness(W)=i=1nn-in-1wi,

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:

Minimize         Maxi{1,,n-1}wi-wi+1subject to orness(W)=i=1nn-in-1wi=α,0α1,w1++wn=1,   0wi,   i=1,,n.

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:

* The general minimax disparity OWA operator problem

Minimize Maxi{1,,n-1}F(wi)-F(wi+1)subject to orness(W)=i=1nn-in-1wi=α,0α1,w1++wn=1,wi0,i=1,,n.

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 (1) with assuming continuous first differentiability of F instead of the second differentiability of F.

2. General Minimax Disparity OWA Operator Optimization 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:

F(0+)=-,   F(0+)<,

where limx→0+= F′(0+). In case of F′(0+) < ∞, we define F′(0+) = F′(0).

Theorem 2.1

Assume that F is strictly convex and F′(x) is continuous. The optimal solution for problem (1) with given orness level 0 < α < 1 is the weighting function

wi*=max{(F)-1(a*i+b*),0},

where a*, b* are determined by the constraints:

{iHn-in-1(F)-1(a*i+b*)=α,iH(F)-1(a*i+b*)=1,

and H = {i|(F′)−1(a*i + b*) > 0}.

Proof

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 W*=(1n,1n,,1n) for α = 1/2 and W* = (1, 0, ⋯, 0) for α = 1. It is easy to check that a* < 0 for the case of 1/2 ≤ α ≤ 1.. Indeed, we show that W*=(w1*,,ws*,0,,0), 2 ≤ sn is the unique optimum solution of the constrained optimization problem (1) that satisfies the orness constraint.

Let wi*=max{(F)-1(a*i+b*),0} such that

iwi*=n-(n-1)α   (i=1nn-in-1wi*=α),wi*=1.

Then we clearly have F(wi*)=a*i+b*, iH. If F′(0+) = −∞, then (F′)−1(x) > 0 and H = {1, 2, ⋯, n}. Hence

Maxi{1,,n-1}F(wi*)-F(wi+1*)=a*.

If F′ (0+) < ∞, then F(wi*)=max{a*i+b*,F(0)}. If wi*=(F)-1(a*i+b*)>0 and wi+1*=0, then (F′)−1(a*(i+ 1) + b*) ≤ 0, and hence a*(i + 1) + b*F′(0). Hence

Maxi{1,,n-1}F(wi*)-F(wi+1*)=a*.

Now, let wi, i = 1, ⋯, n satisfy

i=1niwi=n-(n-1)α,i=1nwi=1,   0wi,   i=1,,n.

Then we can prove that

Maxi{1,,n-1}F(wi)-F(wi+1)a*,

which shows that W* is an optimal solution of the constrained optimization problem (1). To prove (4), we assume the contrary, that is,

Maxi{1,,n-1}F(wi)-F(wi+1)<a*.

Then

w1<w1*.

If not, then w1w1* and hence F(w1)F(w1*)=a*+b*. By (5), we have

F(w2)>F(w1)-a*F(w1*)-a*=F(w1*)+a*=F(w2*).

Similarly, by induction, we have that F(wi)>F(wi*) for all i = 3, ⋯, s. Then we have that wi>wi* for all i = 1, ⋯, s and wiwi*=0 is trivial for i = s + 1, ⋯, n hence

i=1nwi>i=1nwi*=1,

which is contradictory to (3). It follows that w1<w1*. Here, we need a lemma.

Lemma 2.2

Let wi, i = 1, ⋯, n satisfy

i=1niwi=n-(n-1)α,i=1nwi=1,   0wi,   i=1,,n.

and suppose that

Maxi{1,,n-1}F(wi)-F(wi+1)a*.

Then w1w1*.

Proof

If not, then w1<w1*. Since i=1nwi=i=1nwi*=1, there exists i0s such that wi<wi* for i = 1, ⋯, i0 −1 and wi0wi0*. Then by (8), we have

F(wi0+1)F(wi0)-a*F(wi0*)-a*=F(wi0*)+a*=F(wi0+1*),

and hence wi0+1wi0+1*. Similarly, by induction, we have wiwi* for all i = i0 + 2, ⋯, s and wiwi* is trivial for i = s + 1, ⋯, n. Then we have that

i=1niwi-i=1niwi*=i=1ni(wi-wi*)=i<i0i(wi-wi*)+ii0i(wi-wi*)i<i0i(wi-wi*)+i0ii0(wi-wi*)=i<i0i(wi-wi*)-i0i<i0(wi-wi*)=i<i0(i-i0)(wi-wi*)>0,

where the second equality comes from the fact that i<i0(wi-wi*)=-ii0(wi-wi*). This means that i=1niwi>i=1niwi*=n-(n-1)α, which is contradictory to (7).

Back to the proof of Theorem 2.1

We also have w1w1* by Lemma 2.2, which is contradictory to (6). It follows that

Maxi{1,,n-1}F(wi)-F(wi+1)a*,

and hence W*=(w1*,,ws*,0,,0), 2 ≤ sn is a optimum solution of the constrained optimization problem (1). Now, we show that W* is the unique optimum solution of the constrained optimization problem (1). Let wi, i = 1, ⋯, n satisfy

i=1niwi=n-(n-1)α,i=1nwi=1,0wi,i=1,,n.

and suppose that

Maxi{1,,n-1}F(wi)-F(wi+1)=a*.

By Lemma 2.2, we have w1w1*. Then by (9), we have

F(w2)F(w1)-a*F(w1*)-a*=F(w1*)+a*=F(w2*)

and hence w2w2*. Similarly, by induction, we have that wiwi* for i = 3, ⋯, s, and wiwi* is trivial for i = s+1, ⋯, n. Now, since wi-wi*0, i = 1, 2, ⋯, n and 0=i=1n(wi-wi*), we have that wi=wi* for all i = 1, ⋯, n, which completes the proof.

3. Numerical Example

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

F(x)={x22,if 0x<12,x2-12x+18,if 12x1.

Then

F(x)={x,if 0x<12,2x-12,if 12x1.

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 12α1. Then the OWA operator vector has the form

W*=(w1*,w2*,,wm*,0,0,,0),

where mH, m − 1∉ H and

wi*={(F)-1(a*i+b*),1im,0,m<i<n.

Since (F)-1(x)>12 for x(12,1], there exist only two cases.

Case 1)

F(w1*)=a*+b*>12

If a*+b*>12 then

wi*=(F)-1(a*i+b*)={12a*+12b*+14,i=1,a*i+b*,2im,

where mH and m + 1 ∉ H. From

{i=1nn-in-1wi*=α,i=1nwi=1,

we can get

a*=-3{3m(m-1)-4(1-α)(n-1)(2m-1)}2m(m-1)(m2-m+1),b*=3m(m-1)(m+1)m(m-1)(m2-m+1)-6(1-α)(n-1)(m2+m-1)m(m-1)(m2-m+1).

Let α1 and α2 be solutions of a*+b*=12 and wm*=a*m+b*=0, respectively. Then

α1=m2-7m-8+12n12(n-1),α2=-m2+4m-1-4mn4m(n-1).

Since a* + b* is increasing and wm* is decreasing as α is increasing, a*+b*>12 and wm*>0 implies α1 < α < α2. Therefore

α2-α1=-m2+4m-1-4mn4m(n-1)-m2-7m-8+12n12(n-1)=-(m-4)(m2-m+1)4(n-1)(m-1)>0.

This implies thatm < 4.

If m = 3 then,

α2=3n-53(n-1),a*=-9-10(1-α)(n-1)14,b*=12-11(1-α)(n-1)7,

and if m = 2, then

α2=8n-118(n-1),a*=-3-6(1-α)(n-1)2,b*=3-5(1-α)(n-1).

So

wi*={11-6(1-α)(n-1)14,i=1,-9-10(1-α)(n-1)14i+12-11(1-α)(n-1)7,i=2,3,0,3<in,

for α(3n-53(n-1),8n-118(n-1)], and

wi*={1-(1-α)(n-1),i=1,(1-α)(n-1),i=2,0,2<in,

for α(8n-118(n-1),1].

Case 2)

F(w1*)=a*+b*12

If a*+b*12 then m ≥ 4 and (F′)−1 (a*i + b*) = a*i + b* for all 1 ≤ im. From

{i=1nn-in-1wi*=α,i=1nwi=1,

we can get

a*=-6{m-1-2(1-α)(n-1)}m(m-1)(m+1),b*=2{2(m-1)-3(1-α)(n-1)}m(m-1).

Hence wm+1*0<wm* implies

1+3(1-α)(n-1)<m2+3(1-α)(n-1),

and so

wi*=-6{m-1-2(1-α)(n-1)}m(m-1)(m+1)i+2{2(m-1)-3(1-α)(n-1)}m(m-1)

for 1 ≤ im and wi*=0 for m < in, where

m=min{n,[2+3(1-α)(n-1)]},

and 3n-m-23(n-1)<α3n-m-13(n-1).

For example, if n = 5, then we have for α(2932,1], m = 2

wi*={4α-3,i=1,4-4α,i=2,0,2<i5,

for α(56,2932], m = 3

wi*={24α-1314,i=1,-40α-3114i+44α-327,i=2,3,0,3<in,

for α(34,56], m = 4

wi*={-8α-510i+2α-1,1i4,0,i=5,

and for α[12,34], m = 5

wi*=-2α-15i+6α-25,   1i5.
4. Conclusion

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.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03027869).


Figures
Fig. 1.

The OWA operator weihts for F(x) with n = 5.


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Biography

Dug Hun Hong received the B.S. and M.S. degrees in Mathematics from Kyungpook National university, Daegu Korea in 1981 and 1983, respectively. He received the M.S. and Ph.D. degrees from the University of Minnesota in 1988 and 1990, respectively. His research interests include probability theory and fuzzy theory with applications.

E-mail: dhhong@mju.ac.kr