 # Elliptical Distributions-Based Weights-Determining Method for OWA Operators

The ordered weighted averaging (OWA) operators play a crucial role in aggregating multiple criteria evaluations into an overall assessment supporting the decision makers' choice. One key point steps is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then we propose a new approach for determining OWA weights by using the RIM quantifier. Motivated by the idea of normal distribution-based method to determine the OWA weights, we develop a method based on elliptical distributions for determining the OWA weights, and some of its desirable properties have been investigated.

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## 1 Introduction

Aggregation operators play important roles in the theory of fuzzy sets. The ordered weighted averaging operator (OWA) introduced by Yager (1988) is a fundamental aggregation operator which generalizes or and and aggregation operators. The OWA operators have been applied in diverse fields such as multicriteria and group decision making (Herrera (1995,1996), Liu et al. (2018), Yager and Alajlan (2018)), data mining (Torra, 2004), asset management and actuarial science (Casanovas et al. 2016, Merigó, 2018) and approximate reasoning (Dujmovic, 2006). The orness/andness measure of an OWA operator plays an important role in the studies of the OWA operators in both theoretical and applied areas; see Ahn (2006), Filev and Yager (1998), Jin (2015), Liu and Chen (2004), Liu and Han (2008), Jin and Qian (2015), Xu (2005), Yager et al. (2011), Reimann et al. (2017). Orness/andness measure reflects the extent of orlike/andlike of the aggregation result under an OWA operator.

The OWA operator provides a parameterized class of mean type operators which can be used to aggregate a collection of arguments. The parameterization is accomplished by the choice of the characterizing OWA weights that are multiplied by the argument values in a linear type aggregation. One important issue is the determination of the associated weights of the operator. During the last two decades, scholars proposed a large variety of weights determination methods for the OWA operators with numerous different constraints, such as maximizing deviation method (Wei and Feng (1998))and Gaussian distribution-based method (Xu (2005)), Sadiq and Tesfamariam (2007) extended this method by using the probability density functions, see also Lenormand (2018) for truncated distributions method. Liu (2007) proved the solution equivalence of the minimum variance problem and the minimax disparity problem. Wang and Xu (2008) introduced a method utilizing the binomial distribution for obtaining the OWA operator weighting vector. Wang et al. (2007) proposed least squares deviation and Chi-square models to produce the OWA weights with a given orness degree. Liu (2008) gave a more general form of the OWA operator determination methods with a convex objective function, which can include the maximum entropy and minimum variance problems as special cases. Merigó (2012a, 2012b) presented the probabilistic weighted average operator and the probabilistic OWA operator, respectively. The method of Lagrange multipliers to determine the optimal weighting vector was proposed by Sang and Liu (2014). Note also that there are several generalizations of the OWA operators, such as weighted OWA operator (Yager, 1993), induced OWA operator (Yager, 1996), generalized OWA operator (Beliakov, 2005), probabilistic OWA (POWA) operator (Merigó, 2012), and so on. Jin and Qian (2016) proposed a new tool called OWA generation function which is inspired from the generating function in probability theory. Recent extensions on OWA can be found in Yager (2017), Mesiar et al. (2018), Wang et al. (2018) and Yager and Alajlan (2018), among others.

In this paper, we first give a survey of the existing main methods and then develop two novel practical methods based on the quantifier functions and elliptical distributions for determining the OWA weights.

The rest of the paper is structured as follows: In Section 2 , we give a brief overview of the existing OWA literature. In Section 3, we propose a new method to determine weights by the quantifier function and compare the properties with usual weights determined by the same quantifier function. We describe an elliptical distribution-based method to determine the OWA weights in Section 4, which can be seen as the extension of Gaussian-based method in Xu (2005). The final section provides a summary and concludes the paper.

## 2 Review of the OWA operators

In this section we review the notion and some facts about the OWA operator and its generalization. The ordered weighted averaging (OWA) operator was introduced by Yager (1988) which is a generation of arithmetic mean, maximum and minimum operators. It provides a parameterized family of aggregation operators which can be defined as follows:

Definition 2.1. (Yager (1988))  An OWA operator of dimensions is a mapping OWA: such that

 OWA(a1,⋯,an)=n∑i=1wia(i),

where is a permutation of such that , i.e., is the th largest element of the collection of the aggregated objects and the ’s are weights satisfying and .

Each weight is associated with the ordered position rather than the argument . Yager (1988) defined two important measures associated with an OWA operator. The first measure, called the dispersion (or entropy) of an OWA vector is defined as:

 Disp(W)=−n∑i=1wilnwi.

The second measure called “orness” is defined as:

 orness(W)=n∑j=1n−jn−1wj. (2.1)

The measure of “andness” associated with an OWA operator is the complement of its “orness”, which is defined as It is noted that different OWA operators are distinguished by their weighting function. Yager (1993) discussed various different examples of weighting vectors. For example, and , which correspond to the , , and . Obviously, , and . It has been established in Yager (1988) that the OWA operator has the following properties:
Commutativity: for any permutation ;
Monotonicity: if for each ;
Boundedness: ;
Idempotency: OWA.

A very useful approach for obtaining the OWA weights is the functional method introduced by Yager (1996). A fuzzy subset of the real line is called a Regular Increasing Monotone (RIM) quantifier if and if . We shall call a quantifier regular unimodal (RUM) if ; l for some ; there exist two values and such that if then ; if then . These functions were also denoted as basic unit-interval monotonic (BUM) functions in Yager (2004). Examples of this kind of quantifier are , , , (Yager, 1996). The quantifier is represented by the fuzzy subset , the quantifier , , is defined as , where if the event is true, or otherwise.

Given a RIM quantifier, Yager (1988) generated the OWA weights by

 wi=Q(in)−Q(i−1n),i=1,2,⋯,n.

The quantifier guided aggregation with the OWA operator is

 OWAQ(a1,⋯,an)=n∑i=1wia(i),

where is the th largest element of the collection of the aggregated objects . We refer the reader to Yager (1988) for the definition and more examples.

Yager (1996) extended the orness measure of the OWA operator (2.1), and defined the orness of a RIM quantifier:

 orness(Q)=limn→∞1n−1n−1∑j=1Q(jn)=∫10Q(r)dr.

We have , , and , where .

Note that, sometimes, as in fuzzy set theory, it is better to use in the definition a mapping OWA: . In this special case, the OWA also has the following properties and .

Definition 2.2. (Yager, 1992)  Assume that is an OWA weighting vector of dimension . We shall say that is its dual weighting vector if . A weighting vector is said to be symmetric if .

The dual OWA operator has the property , and in the special case when (see also Yager and Alajlan (2016)):

 OWAW(a1,⋯,an)=1−OWA^W(1−a1,⋯,1−an).

To establish the corresponding relationship between the OWA operator and the RIM quantifier, a generating function representation of the RIM quantifier was proposed by Liu (2005). A function on is called the generating function of RIM quantifier , if it satisfies , and Obviously, . Here we will list some properties of the quantifier generating function and the OWA operator:

###### Proposition 2.1.

(Liu and Han (2004))  For the RIM quantifiers and which are determined by and respectively, if , then .

###### Proposition 2.2.

(Liu and Han (2008))  For the RIM quantifiers and , and (for all ) if and only if for every rational point .

A number of different methods have been suggested for obtaining the weights associated with the OWA operator (Xu, 2005). Moreover, Xu (2005) introduced a procedure for generating the OWA weights based on the normal distribution (or Gaussian distribution). Consider a normal distribution , where

 μn=n+12,σ2n=1nn∑i=1(i−μn)2=n2−112.

The associated OWA weights is defined as:

 wi=e−(i−μn)2/2σ2n∑nj=1e−(j−μn)2/2σ2n,i=1,2,⋯,n.

It is clear that and and is symmetric, that is, . The prominent characteristic of the developed method is that it can relieve the influence of unfair arguments on the decision results by assigning low weights to those “false” or “biased” ones.

## 3 Determine weights by the quantifier function

A fuzzy subset of the real line is called a Regular Increasing Monotone (RIM) quantifier if and if . RIM quantifiers can be considered as a starting point for constructing families of the OWA operators. Thus, constructions of the RIM quantifiers are very important in producing various OWA operators. One way to generate RIM quantifier is to use the method of mixing along with finitely RIM quantifiers or infinitely many RIM quantifiers. Specifically, if () is a one-parameter family of the RIM quantifier, is an increasing function on such that , then the function is a RIM quantifier. In particular, if is discrete distribution, then (.

The concept of dual OWA operators can be found in Yager (1992). We now use a RIM quantifier to give the definition of dual OWA operators.

Definition 3.1. Let be a RIM quantifier. An operator of dimension is a mapping : such that

 ˜OWAQ(a1,⋯,an)=n∑i=1~wia(i),

where is the th largest element of the collection of the aggregated objects and the ’s are weights satisfying

 ~wi=Q(1−i−1n)−Q(1−in),i=1,2,⋯,n.

The orness of a RIM quantifier is defined as:

 ˜orness(Q)=limn→∞n∑j=1n−jn−1~wj.

Note that and . In addition,

 ˜orness(Q)=limn→∞1n−1n−1∑i=1(1−Q(1−in))=1−∫10Q(r)dr. (3.1)

We have , , and , where . It is clear that and thus the operator is just the dual OWA operator. Consequently, the operator has the following properties as the operator: Commutativity, Monotonicity, Boundedness and Idempotency. In the special case, when the arguments , and also has the following properties and . It is obvious that if the small aggregated objects have big weights for the operator, then the big aggregated objects have big weights for the operator, if the big aggregated objects have big weights for the operator, then the small aggregated objects have big weights for the operator.

Example  If , where , , and

 Q2(u)={0,ifu∈[0,1),1,ifu=1,
 Q3(u)={0,ifu=0,1,ifu∈(0,1],
 Q4(u)={0,ifu

and

 Q5(u)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0,ifu<1n,nn−2x−1n−2,if1n≤u

Then for , we have

 ~wi=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩α1n+α3,ifi=1,α1n+α4+α5n−2,if1

and

 ˜OWAQ(a1,⋯,an)=α11nn∑i=1ai+α2a(n)+α3a(1)+α4a(k)+α51n−2n−1∑i=2a(i).

In particular, when we recover the results (22) and (23) in Xu (2005); when and we obtain, respectively, the orlike and andlike aggregations in Yager and Filev (1994); when , we get the so-called olympic aggregators in Yager (1998) (where ).

Motivated by the notion of WOWA (Weighted OWA) in Torra (1997) (see also Llamazares (2016)), we define the operator which is a direct extension of . Let and be two weighting vectors and let be a quantifier generating the weighting vector . The operator associated with , and is the function given by

 ˜WOWAP,Q(a1,⋯,an)=n∑i=1~qia(i),

where the weight is defined as:

 ~qi=Q(1−i−1∑j=1pj)−Q(1−i∑j=1pj).

The operator can also be written as:

 ˜WOWAP,Q(a1,⋯,an)=a(1)+n∑i=2Q(1−i−1∑j=1pj)(a(i)−a(i−1)). (3.2)

In particular, when , we get

 ˜OWAQ(a1,⋯,an)=a(1)+n∑i=2Q(n+1−in)(a(i)−a(i−1)).

The following property was considered for and in Liu and Han (2008).

###### Theorem 3.1.

For RIM the quantifiers and , for every rational point if and only if , and for any aggregated arguments ,

 ˜WOWAP,Q1(a1,⋯,an)≥˜WOWAP,Q2(a1,⋯,an).

Proof. If for every rational point , then by using (3.1) one has

 ˜orness(Q1)−˜orness(Q2)=∫10(Q2(x)−Q1(x))dx≥0,

which is .

Thanks to (3.2), we get

 ˜WOWAP,Q1(a1,⋯,an) − ˜WOWAP,Q2(a1,⋯,an)=n∑i=2Q1(1−i−1∑j=1pj)(a(i)−a(i−1)) − n∑i=2Q2(1−i−1∑j=1pj)(a(i)−a(i−1)) = n∑i=2{Q1(1−i−1∑j=1pj)−Q2(1−i−1∑j=1pj)}(a(i)−a(i−1))) ≥ 0.

On the other hand, letting , then from the nonnegativity of

 ˜WOWAP,Q1(a1,⋯,an) − ˜WOWAP,Q2(a1,⋯,an)=Q1(1−i−1∑j=1pj)−Q1(1−i∑j=1pj) − Q2(1−i−1∑j=1pj)+Q2(1−i∑j=1pj),

we get the recursive formula for :

from which we get

 Q1(1−i∑j=1pj)≤Q2(1−i∑j=1pj),i=1,2,⋯,n.

It follows that for every rational point . This completes the proof of Theorem 3.1.

###### Corollary 3.1.

If , then , and for any aggregated arguments , we have

 ˜OWAQ1(a1,⋯,an)≥˜OWAQ2(a1,⋯,an).

Note that the identity quantifier is the smallest concave RIM quantifier and also the largest convex RIM quantifier, and when , , , we can easily get the following from Corollary 3.1:

###### Corollary 3.2.

For any RIM quantifier , we have

(i) if is convex function, then and .

(ii) if is concave function, then and .

###### Remark 3.1.

For two RIM quantifiers and , the following two sufficient conditions for were proposed on generating functions in Liu (2005):

(i) , for any such that ;

(ii) , for any such that ;

(iii) If and is differentiable, .

A special class of the OWA operator with monotonic weights was investigated by Liu and Chen (2004). The following gives the relationship between monotonic weights and the concavity/convexity of RIM quantifier. Let us start with the standard definitions of convexity and concavity.

Definition 3.2. Let be an interval in real line . Then the function is said to be convex if for all and all , the inequality

 f(αx+(1−α)y)≤αf(x)+(1−α)f(y)

holds. If this inequality is strict for all and , then is said to be strictly convex. A closely related concept is that of concavity: is said to be (strictly) concave if, and only if, is (strictly) convex.

###### Theorem 3.2.

Let and be two weighting vectors and let be a quantifier generating the weighting vector . We define

 qi=Q(i∑j=1pj)−Q(i−1∑j=1pj),i=1,2,⋯,n

and

 ~qi=Q(1−i−1∑j=1pj)−Q(1−i∑j=1pj),i=1,2,⋯,n.

(i)  If is convex, then is monotonic increasing; if is concave, then is monotonic decreasing.
(ii)  If is convex, then is monotonic decreasing; if is concave, then is monotonic increasing.

Proof.  If is convex, then

 qi+1−qi=Q(i+1∑j=1pj)+Q(i−1∑j=1pj)−2Q(i∑j=1pj)≥0,

and

 ~qi+1−~qi=qn−i−qn−i+1≤0.

If is concave, then

 qi+1−qi=Q(i+1∑j=1pj)+Q(i−1∑j=1pj)−2Q(i∑j=1pj)≤0,

and

 ~qi+1−~qi=qn−i−qn−i+1≥0.

This ends the proof of Theorem 3.2.

Letting , we get the following corollary.

###### Corollary 3.3.

Let be a weighting vector generated by the quantifier . We define

 wi=Q(in)−Q(i−1n),i=1,2,⋯,n

and

 ~wi=Q(1−i−1n)−Q(1−in),i=1,2,⋯,n.

(i)  If is convex, then is monotonic increasing; if is concave, then is monotonic decreasing.
(ii)  If is convex, then is monotonic decreasing; if is concave, then is monotonic increasing.

###### Remark 3.2.

For any RIM quantifier , if is convex, then the operator has the property: small aggregated objects have big weights and big aggregated objects have small weights; if is concave, then the operator has the property: small aggregated objects have small weights and big aggregated objects have big weights. Contrarily, if is convex, then the operator has the property: small aggregated objects have small weights and big aggregated objects have big weights; if is concave, then the operator has the property: small aggregated objects have big weights and big aggregated objects have small weights.

## 4 Elliptical distributions-based weights-determining method

Xu (2005) introduced a procedure for generating the OWA weights based on the normal distribution (or Gaussian distribution). Yager (2007) referred to these as Gaussian weights and was described in the following. This is a specific case of the centered OWA operators. Consider a normal distribution , where

 μn=n+12,σ2n=1nn∑i=1(i−μn)2=n2−112.

The associated OWA weights are defined as:

 wi=e−(i−μn)2/2σ2n∑nj=1e−(j−μn)2/2σ2n,i=1,2,⋯,n.

It is clear that and and is symmetric, that is .

The Xu’s method on the normal type OWA weighting vector inspires us to consider a more general class of OWA aggregation operators of this type. We shall refer to these as elliptical OWA operators.

Definition 4.1.  Let

be the continuous random variable, we say

belonging to the class of elliptical distributions if its density can be expressed as:

 fX(x)=Cσg[(x−μσ)2],−∞

for some so-called density generator (which is a function of non-negative variables) satisfying the condition:

 0<∫∞0x−12g(x)dx<∞,

and a normalizing constant given by

 C=[∫∞0x−12g(x)dx]−1.

For the normal distribution , it is straightforward to show that its density generator has the form In general, elliptical distributions can be bounded or unbounded, unimodal or multimodal, the class of elliptical distributions consists of the class of symmetric distributions.

For details, see Landsman and Valdez (2003).

Following Xu (2005), we define

 μn=1+n2,σ2n=1nn∑i=1(i−μn)2,

and

 wi=g[(i−μnσn)2]∑nj=1g[(j−μnσn)2],i=1,2,⋯,n. (4.1)

It can be shown that these weights satisfy the conditions and .

Similar to Theorem 2 in Xu (2005), we have

###### Theorem 4.1.

(1) The weights are symmetrical, that is,

 wi=wn+1−i,i=1,2⋯,n. (4.2)

(2) If is non-increasing, then for all , and for all where is the usual round operation.
(3) If

is odd, then the weight

reaches its maximum when ; if is even, then the weight reaches its maximum when or .
(4) Orness(w)=0.5.

Proof. (1) It follows that (4.1) that

 wn+1−i = g[(n+1−i−μnσn)2]n∑j=1,j≠ig[(j−μnσn)2]+g[(n+1−i−μnσn)2] = = g[(i−μnσn)2]n∑j=1g[(j−μnσn)2]=wi,i=1,2,⋯,n.

(2) Because

 (i−μnσn)2>(i+1−μnσn)2fori=1,2,⋯,[1+n2],

and

 (i−μnσn)2<(i+1−μnσn)2fori=[1+n2]+1,⋯,n,

and note that is non-increasing, then

 g((i−μnσn)2)≤g((i+1−μnσn)2),i=1,2,⋯,[1+n2],

and

The result follows since the function is increasing function, where is a constant.

(3) Obvious.

(4) This result can be derived directly from Theorem in Yager (2007) and (4.2). This ends the proof of Theorem 4.1.

As pointed out by Su et al. (2016), the above method is simple and straightforward, but sometimes there exists some problems in actual applications. For example, the same value may have different weights because of their different positions; For details see Su et al. (2016). In order to avoid the issue above, Wang and Xu (2008) introduced a weighting method for the weighted arithmetic aggregation (WAA) operator, which is based on the normal probability density function and the given arguments simultaneously: Given a collection of n preference values , and let be the weighting vector of the WAA operator, then Wang and Xu (2008) gave the following formula:

 wi=e−(ai−μ)2/2σ2∑nj=1e−(aj−μ)2/2σ2,i=1,2,⋯,n, (4.3)

where is the mean of the collection of , and

is the standard deviation of

, i.e.,

 μ=∑nj=1ajn,σ= ⎷1nn∑j=1(aj−μ)2.

Motivated by the method above, for any density generator , we define

 wi=g[(ai−μσ)2]∑nj=1g[(aj−μσ)2],