# SR-Fuzzy Sets and Their Weighted Aggregated Operators in Application to Decision-Making

Authors: Tareq M. Al-shami; Hariwan Z. Ibrahim; A. A. Azzam; Ahmed I. EL-Maghrabi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

An intuitionistic fuzzy set is one of the efficient generalizations of a fuzzy set for dealing with vagueness/uncertainties in information. Under this environment, in this manuscript, we familiarize a new type of extensions of fuzzy sets called square-root fuzzy sets (briefly, SR-Fuzzy sets) and contrast SR-Fuzzy sets with intuitionistic fuzzy sets and Pythagorean fuzzy sets. We discover the essential set of operations for the SR-Fuzzy sets along with their several properties. In addition, we define a score function for the ranking of SR-Fuzzy sets. To study multiattribute decision-making problems, we introduce four new weighted aggregated operators, namely, SR-Fuzzy weighted average (SR-FWA) operator, SR-Fuzzy weighted geometric (SR-FWG) operator, SR-Fuzzy weighted power average (SR-FWPA) operator, and SR-Fuzzy weighted power geometric (SR-FWPG) operator over SR-Fuzzy sets. We apply these operators to select the top-rank university and show how we can choose the best option by comparing the aggregate outputs through score values.


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## Body

## 1. Introduction

The idea of fuzzy sets was proposed by Zadeh [1] to handle the imprecise information. The notion of rough set theory was originally introduced by Pawlak [2], and it was applied to many different domains (see [3–5]). The concept of soft sets was first defined by Molodtsov [6] as a general mathematical tool for dealing with uncertain objects. The merging between fuzzy sets and some uncertainty approaches such as rough sets and soft sets have been discussed in [7–9].In several real-life situations, the degree of nonmembership is not obtained from the degree of membership. In these cases, the notion of intuitionistic fuzzy sets defined by Atanassov [10] worked very well. It is one of the interesting generalizations of fuzzy sets with best applicability. In various fields, the applications of intuitionistic fuzzy sets appear, including optimization problems, medical diagnosis, and decision-making [11–15]. However, there are numerous situations where the decision-maker may supply the degrees of membership and nonmembership of a specific attribute in such a way that their sum is greater than one. Therefore, Yager [16] put forward the concept of Pythagorean fuzzy sets which is a generalization of intuitionistic fuzzy sets, and it is a more powerful tool to solve uncertain problems. Ibrahim et al. [17] defined a new generalization of Pythagorean fuzzy sets called (3, 2)-Fuzzy sets. The main advantage of (3, 2)-Fuzzy sets is that they can characterize more vague cases than Pythagorean fuzzy sets, which can be exploited in many decision-making problems.The idea of intuitionistic fuzzy weighted averaging operators was proposed by Xu [18]. Some geometric weighted, geometric ordered weighted, and geometric hybrid operators under the environment of intuitionistic fuzzy sets were introduced by Xu and Yager [19]. In Refs. [20–25], many researchers worked in the area of intuitionistic fuzzy sets and established various aggregation operators which are applied to group decision-making. After the advent of the Pythagorean fuzzy sets, the operators of Pythagorean fuzzy aggregation have also become an important and interesting field for research. Yager and Abbasov [26, 27], in 2013, introduced the concepts of weighted geometric, weighted averaging, ordered weighted geometric, and ordered weighted averaging operators in the frame of Pythagorean fuzzy environment. The essential properties of Pythagorean fuzzy aggregation operators were investigated by the authors of [28]. Shahzadi et al. [29] established some aggregation operators under Pythagorean fuzzy data for assessing the distinct preferences of the choice among the decision-making process. By using Pythagorean fuzzy values, Rahman et al. presented many aggregation operators like weighted geometric [30], hybrid geometric [31], weighted averaging [32], and ordered weighted geometric operators [33] and also discussed their practical applications.The aims of writing this paper are (1) to present a novel extension of intuitionistic fuzzy set called SR-Fuzzy sets which is not obtained fromq-rung orthopair fuzzy sets, (2) to introduce novel types of weighted aggregation operators and discuss their main properties, and (3) to investigate a MCDM methods depending on these operators.In this paper, we define the concept of SR-Fuzzy sets and compare it with the other types of fuzzy sets in Section2. Then, we introduce the set of operations for the SR-Fuzzy sets and explore their main features in Section 3. Also, the concepts of weighted aggregated operators for SR-Fuzzy sets are investigated. Thereafter, we describe MADM problems under these operators in Section 4. Finally, we outline the main achievements of the paper and propose some upcoming works in Section 5.Before we present our main concepts and results, we recall the definitions of the intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS).Definition 1.
LetS be a universal set such that YΘ:S⟶0,1 and ΨΘ:S⟶0,1 are mapping. Then, the intuitionistic fuzzy set (IFS) [10] (resp., Pythagorean fuzzy set (PFS) [16]) is defined by the following:
(1)Θ=p,YΘp,ΨΘp:p∈S,including the condition 0≤YΘp+ΨΘp≤1 (resp., 0≤YΘp2+ΨΘp2≤1), where YΘp is the degree of membership and ΨΘp is the degree of nonmembership of every p∈S to Θ.


## 2. SR-Fuzzy Sets

In this section, we initiate the notion of SR-Fuzzy sets and study its features in detail. For computations, we use only six decimal places in the whole paper.Definition 2.
LetS be a universal set such that YΘ:S⟶0,1 and ΨΘ:S⟶0,1 are mapping. Then, the SR-Fuzzy set (briefly, SR-FS) Θ is defined as following:
(2)Θ=p,YΘp,ΨΘp:p∈S,where YΘp is the degree of membership and ΨΘp is the degree of nonmembership of p∈S to Θ, such that
(3)0≤YΘp2+ΨΘp≤1.
Then, there is a degree of indeterminacy ofp∈S to Θ defined by
(4)πΘp=1−YΘp2+ΨΘp.
It is obvious thatYΘp2+ΨΘp+πΘp=1. Otherwise, πΘp=0 whenever YΘp2+ΨΘp=1.In the interest of simplicity, we shall mention the symbolΘ=YΘ,ΨΘ for the SR-FS Θ=p,YΘp,ΨΘp:p∈S. The space of SR-Fuzzy membership grades is displayed in Figure 1.Example 1.
Assume thatYΘp=0.3 and ΨΘp=0.8 for S=p. Then, Θ=0.3,0.8 is not an intuitionistic fuzzy set because 0.3+0.8=1.1>1. In contrast, Θ=0.3,0.8 is an SR-FS because 0.32+0.8≈0.984427≤1.Figure 1
Grades space of SR-FSs.Note thatπΘp≈0.015573, and hence, YΘp2+ΨΘp+πΘp=1.Remark 3.
From Figure2, we get that
(1)
The space of Pythagorean membership grades is larger than the space of SR-Fuzzy membership grades(2)
The SR-Fuzzy and intuitionistic fuzzy sets intersect at the pointΘ=YΘ=−1+5/2,ΨΘ=3−5/2(3)
ForYΘ∈0,−1+5/2 and ΨΘ∈3−5/2,1, the space of SR-Fuzzy membership grades starts to be larger than the space of intuitionistic membership grades(4)
forYΘ∈−1+5/2,1 and ΨΘ∈0,3−5/2, the space of SR-Fuzzy membership grades starts to be smaller than the space of intuitionistic membership gradesDefinition 4.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs; then
(1)
Θ1=Θ2 if and only if YΘ1=YΘ2 and ΨΘ1=ΨΘ2(2)
Θ1≥Θ2 if and only if YΘ1≥YΘ2 and ΨΘ1≤ΨΘ2Example 2.
(1)
IfΘ1=0.2,0.9 and Θ2=0.2,0.9 for S=p, then Θ1=Θ2(2)
IfΘ1=0.2,0.9 and Θ2=0.1,0.91 for S=p, then Θ2≤Θ1Definition 5.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-Fuzzy sets (SR-FSs). Then
(1)
Θ1∩Θ2=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2(2)
Θ1∪Θ2=maxYΘ1,YΘ2,minΨΘ1,ΨΘ2(3)
Θ1c=ΨΘ14,YΘ14Figure 2
Comparison of grades space of IFSs, PFSs, and SR-FSs.Note thatΨΘ142+YΘ142=ΨΘ12+YΘ12≤1, so Θ1c is an SR-Fuzzy set. It is obvious that Θcc=ΨΘ4,YΘ4c=YΘ,ΨΘ.Example 3.
Assume thatΘ1=YΘ1=0.59,ΨΘ1=0.42 and Θ2=YΘ2=0.56,ΨΘ2=0.45 are both SR-FSs for S=p. Then
(1)
Θ1∩Θ2=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2=min0.59,0.56,max0.42,0.45=0.56,0.45(2)
Θ1∪Θ2=maxYΘ1,YΘ2,minΨΘ1,ΨΘ2=max0.59,0.56,min0.42,0.45=0.59,0.42(3)
Θ1c≈0.805030,0.121174Theorem 6.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs; then the following properties hold:
(1)
Θ1∩Θ2=Θ2∩Θ1(2)
Θ1∪Θ2=Θ2∪Θ1Proof.
From Definition5, we can obtain the following:
(1)
Θ1∩Θ2=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2=minYΘ2,YΘ1,maxΨΘ2,ΨΘ1=Θ2∩Θ1(2)
The proof is similar to (1)Theorem 7.
LetΘ1=YΘ1,ΨΘ1, Θ2=YΘ2,ΨΘ2 and Θ3=YΘ3,ΨΘ3 be three SR-FSs; then
(1)
Θ1∩Θ2∩Θ3=Θ1∩Θ2∩Θ3(2)
Θ1∪Θ2∪Θ3=Θ1∪Θ2∪Θ3Proof.
For the three SR-FSsΘ1,Θ2, and Θ3, according to Definition 5, we obtain the following:
(1)
Θ1∩Θ2∩Θ3=YΘ1,ΨΘ1∩minYΘ2,YΘ3,maxΨΘ2,ΨΘ3=minYΘ1,minYΘ2,YΘ3,maxΨΘ1,maxΨΘ2,ΨΘ3=minminYΘ1,YΘ2,YΘ3,maxmaxΨΘ1,ΨΘ2,ΨΘ3=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2∩YΘ3,ΨΘ3=Θ1∩Θ2∩Θ3(2)
The proof is similar to (1)Theorem 8.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs. Then
(1)
Θ1∩Θ2∪Θ2=Θ2(2)
Θ1∪Θ2∩Θ2=Θ2Proof.
From Definition5, we obtain the following:
(1)
Θ1∩Θ2∪Θ2=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2∪YΘ2,ΨΘ2=maxminYΘ1,YΘ2,YΘ2,minmaxΨΘ1,ΨΘ2,ΨΘ2=YΘ2,ΨΘ2=Θ2(2)
The proof is similar to (1)Theorem 9.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs; then
(1)
Θ1∩Θ2c=Θ1c∪Θ2c(2)
Θ1∪Θ2c=Θ1c∩Θ2cProof.
For the two SR-FSsΘ1 and Θ2, according to Definition 5, we obtain the following:
(1)
Θ1∩Θ2c=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2c=maxΨΘ14,ΨΘ24,minYΘ14,YΘ24=ΨΘ14,YΘ14∪ΨΘ24,YΘ24=Θ1c∪Θ2c(2)
The proof is similar to (1)


## 3. Aggregation of SR-Fuzzy Sets and Its Properties

In this section, we introduce some new operations on SR-Fuzzy sets. Besides, we study the SR-Fuzzy aggregation operators and some attracted properties are indicated in detail.


### 3.1. Some Operations On SR-Fuzzy Sets

Definition 10.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs and ρ be a positive real value (ρ>0). Then their operations are defined as follows:
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22(3)
ρΘ=1−1−YΘ2ρ,ΨΘρ(4)
Θρ=YΘρ,1−1−ΨΘρ2Example 4.
Suppose thatΘ1=YΘ1=0.43,ΨΘ1=0.64 and Θ2=YΘ2=0.26,ΨΘ2=0.81 are both SR-FSs for S=p. Then
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2=0.432+0.262−0.4320.262,0.640.81≈0.489899,0.5184(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22=0.430.26,0.64+0.81−0.640.812=0.1118,0.9604(3)
ρΘ1=1−1−YΘ12ρ,ΨΘ1ρ=1−1−0.4323,0.643≈0.677095,0.262144, for ρ=3(4)
Θ1ρ=YΘ1ρ,1−1−ΨΘ1ρ2=0.433,1−1−0.6432=0.079507,0.984064, for ρ=3Theorem 11.
IfΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 are two SR-FSs, then Θ1⊕Θ2 and Θ1⊗Θ2 are also SR-FSs.Proof.
For SR-FSsΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2, the following relations are evident:
(5)0≤YΘ12≤1,0≤ΨΘ1≤1,0≤YΘ12+ΨΘ1≤1,0≤YΘ22≤1,0≤ΨΘ2≤1,0≤YΘ22+ΨΘ2≤1.
Then, we have(6)YΘ12≥YΘ12YΘ22,YΘ22≥YΘ12YΘ22,1≥YΘ12YΘ22≥0,ΨΘ1≥ΨΘ1ΨΘ2,ΨΘ2≥ΨΘ1ΨΘ2,1≥ΨΘ1ΨΘ2≥0,which indicates that
(7)YΘ12+YΘ22−YΘ12YΘ22≥0,whichmeansthatYΘ12+YΘ22−YΘ12YΘ22≥0,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ2≥0,whichmeansthatΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≥0.
SinceYΘ22≤1 and 0≤1−YΘ12, then YΘ221−YΘ12≤1−YΘ12, and we get YΘ12+YΘ22−YΘ12YΘ22≤1, and hence, YΘ12+YΘ22−YΘ12YΘ22≤1.
Similarly, we get(8)ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1.
It is obvious that(9)0≤ΨΘ1≤1−YΘ12and0≤ΨΘ2≤1−YΘ22.
Then we get(10)YΘ12+YΘ22−YΘ12YΘ222+ΨΘ1ΨΘ2≤YΘ12+YΘ22−YΘ12YΘ22+1−YΘ121−YΘ22=1.
Therefore(11)0≤YΘ12+YΘ22−YΘ12YΘ22≤1,0≤ΨΘ1ΨΘ2≤1,and
(12)0≤YΘ12+YΘ22−YΘ12YΘ222+ΨΘ1ΨΘ2≤1.
Similarly, we have(13)0≤YΘ1YΘ2≤1,0≤ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1,and
(14)0≤YΘ1YΘ22+ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1.
These indicate that both ofΘ1⊕Θ2 and Θ1⊗Θ2 are SR-FSs.Theorem 12.
LetΘ=YΘ,ΨΘ be an SR-FS and ρ be a positive real value. Then, ρΘ and Θρ are also SR-FSs.Proof.
Since0≤YΘ2≤1, 0≤ΨΘ≤1, and 0≤YΘ2+ΨΘ≤1, then
(15)0≤ΨΘ≤1−YΘ2⇒0≤1−YΘ2ρ⇒1−1−YΘ2ρ≤1⇒0≤1−1−YΘ2ρ≤1=1.
It is obvious that0≤ΨΘρ≤1; then we get
(16)0≤1−1−YΘ2ρ2+ΨΘρ≤1−1−YΘ2ρ+1−YΘ2ρ=1.
Similarly, we also get(17)0≤YΘρ2+1−1−ΨΘρ2≤1.
Therefore,ρΘ and Θρ are SR-FSs.Theorem 13.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs. Then the following properties hold:
(1)
Θ1⊕Θ2=Θ2⊕Θ1(2)
Θ1⊗Θ2=Θ2⊗Θ1Proof.
From Definition10, we obtain the following:
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2YΘ22+YΘ12−YΘ22YΘ12,ΨΘ2ΨΘ1=Θ2⊕Θ1(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22=YΘ2YΘ1,ΨΘ2+ΨΘ1−ΨΘ2ΨΘ12=Θ2⊗Θ1Theorem 14.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs. Then
(1)
ρΘ1⊕Θ2=ρΘ1⊕ρΘ2, for ρ>0(2)
ρ1+ρ2Θ=ρ1Θ⊕ρ2Θ, for ρ1,ρ2>0(3)
Θ1⊗Θ2ρ=Θ1ρ⊗Θ2ρ, for ρ>0(4)
Θρ1⊗Θρ2=Θρ1+ρ2, for ρ1,ρ2>0Proof.
For the three SR-FSsΘ,Θ1, and Θ2, and ρ,ρ1,ρ2>0, according to Definition 10, we obtain the following:
(1)
ρΘ1⊕Θ2=ρYΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2=1−1−YΘ12−YΘ22+YΘ12YΘ22ρ,ΨΘ1ΨΘ2ρ=1−1−YΘ12ρ1−YΘ22ρ,ΨΘ1ρΨΘ2ρ.
AndρΘ1⊕ρΘ2=1−1−YΘ12ρ,ΨΘ1ρ⊕1−1−YΘ22ρ,ΨΘ2ρ=1−1−YΘ12ρ+1−1−YΘ22ρ−1−1−YΘ12ρ1−1−YΘ22ρ,ΨΘ1ρΨΘ2ρ=1−1−YΘ12ρ1−YΘ22ρ,ΨΘ1ρΨΘ2ρ=ρΘ1⊕Θ2(2)
ρ1+ρ2Θ=ρ1+ρ2YΘ,ΨΘ=1−1−YΘ2ρ1+ρ2,ΨΘρ1+ρ2=1−1−YΘ2ρ11−YΘ2ρ2,ΨΘρ1+ρ2=1−1−YΘ2ρ1+1−1−YΘ2ρ2−1−1−YΘ2ρ11−1−YΘ2ρ2,ΨΘρ1ΨΘρ2=1−1−YΘ2ρ1,ΨΘρ1⊕1−1−YΘ2ρ2,ΨΘρ2=ρ1Θ⊕ρ2Θ(3)
Θ1⊗Θ2ρ=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22ρ=YΘ1YΘ2ρ,1−1−ΨΘ1−ΨΘ2+ΨΘ1ΨΘ2ρ2=YΘ1ρYΘ2ρ,1−1−ΨΘ1ρ1−ΨΘ2ρ2=YΘ1ρ,1−1−ΨΘ1ρ2⊗YΘ2ρ,1−1−ΨΘ2ρ2=Θ1ρ⊗Θ2ρ(4)
Θρ1⊗Θρ2=YΘρ1,1−1−ΨΘρ12⊗YΘρ2,1−1−ΨΘρ22=YΘρ1+ρ2,1−1−ΨΘρ1+1−1−ΨΘρ2−1−1−ΨΘρ11−1−ΨΘρ2=YΘρ1+ρ2,1−1−ΨΘρ1+ρ22=Θρ1+ρ2Theorem 15.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs and ρ>0. Then
(1)
ρΘ1∪Θ2=ρΘ1∪ρΘ2(2)
Θ1∪Θ2ρ=Θ1ρ∪Θ2ρProof.
For the two SR-FSsΘ1 and Θ2, and ρ>0, according to Definitions 5 and 10, we obtain the following:
(1)
ρΘ1∪Θ2=ρmaxYΘ1,YΘ2,minΨΘ1,ΨΘ2=1−1−maxYΘ12,YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ.
AndρΘ1∪ρΘ2=1−1−YΘ12ρ,ΨΘ1ρ∪1−1−YΘ22ρ,ΨΘ2ρ=max1−1−YΘ12ρ,1−1−YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ=1−1−maxYΘ12,YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ=ρΘ1∪Θ2(2)
The proof is similar to (1)Theorem 16.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs, and ρ>0. Then
(1)
Θ1⊕Θ2c=Θ1c⊗Θ2c(2)
Θ1⊗Θ2c=Θ1c⊕Θ2c(3)
Θcρ=ρΘc(4)
ρΘc=ΘρcProof.
For the three SR-FSsΘ,Θ1 and Θ2, and ρ>0, according to Definitions 5(3) and 10, we obtain the following:
(1)
Θ1⊕Θ2c=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2c=ΨΘ1ΨΘ24,YΘ12+YΘ22−YΘ12YΘ224=ΨΘ14ΨΘ24,YΘ12+YΘ22−YΘ12YΘ222=ΨΘ14,YΘ14⊗ΨΘ24,YΘ24=Θ1c⊗Θ2c(2)
Θ1⊗Θ2c=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22c=ΨΘ1+ΨΘ2−ΨΘ1ΨΘ224,YΘ1YΘ24=ΨΘ1+ΨΘ2−ΨΘ1ΨΘ2,YΘ14YΘ24=ΨΘ14,YΘ14⊕ΨΘ24,YΘ24=Θ1c⊕Θ2c(3)
Θcρ=ΨΘ4,YΘ4ρ=ΨΘ4ρ,1−1−YΘ2ρ2=1−1−YΘ2ρ,ΨΘρc=ρΘc(4)
ρΘc=ρΨΘ4,YΘ4=1−1−ΨΘρ,YΘ4ρ=YΘρ,1−1−ΨΘρ2c=ΘρcTheorem 17.
LetΘ1=YΘ1,ΨΘ1, Θ2=YΘ2,ΨΘ2, and Θ3=YΘ3,ΨΘ3 be three SR-FSs. Then
(1)
Θ1∩Θ2⊕Θ3=Θ1⊕Θ3∩Θ2⊕Θ3(2)
Θ1∪Θ2⊕Θ3=Θ1⊕Θ3∪Θ2⊕Θ3(3)
Θ1∩Θ2⊗Θ3=Θ1⊗Θ3∩Θ2⊗Θ3(4)
Θ1∪Θ2⊗Θ3=Θ1⊗Θ3∪Θ2⊗Θ3Proof.
By Definitions5 and 10, we obtain the following:
(1)
Θ1∩Θ2⊕Θ3=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2⊕YΘ3,ΨΘ3=minYΘ12,YΘ22+YΘ32−YΘ32minYΘ12,YΘ22,maxΨΘ1,ΨΘ2ΨΘ3=1−YΘ32minYΘ12,YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3.
AndΘ1⊕Θ3∩Θ2⊕Θ3=YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ3∩YΘ22+YΘ32−YΘ22YΘ32,ΨΘ2ΨΘ3=minYΘ12+YΘ32−YΘ12YΘ32,YΘ22+YΘ32−YΘ22YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3=min1−YΘ32YΘ12+YΘ32,1−YΘ32YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3=1−YΘ32minYΘ12,YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3.
Thus,Θ1∩Θ2⊕Θ3=Θ1⊕Θ3∩Θ2⊕Θ3(2)
The proof is similar to (1)(3)
Θ1∩Θ2⊗Θ3=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2⊗Θ3=minYΘ1,YΘ2YΘ3,maxΨΘ1,ΨΘ2+ΨΘ3−ΨΘ3maxΨΘ1,ΨΘ22=minYΘ1YΘ3,YΘ2YΘ3,1−ΨΘ3maxΨΘ1,ΨΘ2+ΨΘ32.
AndΘ1⊗Θ3∩Θ2⊗Θ3=YΘ1YΘ3,ΨΘ1+ΨΘ3−ΨΘ1ΨΘ32∩YΘ2YΘ3,ΨΘ2+ΨΘ3−ΨΘ2ΨΘ32=YΘ1YΘ3,1−ΨΘ3ΨΘ1+ΨΘ32∩YΘ2YΘ3,1−ΨΘ3ΨΘ2+ΨΘ32=minYΘ1YΘ3,YΘ2YΘ3,max1−ΨΘ3ΨΘ1+ΨΘ32,1−ΨΘ3ΨΘ2+ΨΘ32=minYΘ1YΘ3,YΘ2YΘ3,1−ΨΘ3maxΨΘ1,ΨΘ2+ΨΘ32.
Thus,Θ1∩Θ2⊗Θ3=Θ1⊗Θ3∩Θ2⊗Θ3(4)
The proof is similar to (3)Theorem 18.
LetΘ1=YΘ1,ΨΘ1, Θ2=YΘ2,ΨΘ2, and Θ3=YΘ3,ΨΘ3 be three SR-FSs. Then
(1)
Θ1⊕Θ2⊕Θ3=Θ1⊕Θ3⊕Θ2(2)
Θ1⊗Θ2⊗Θ3=Θ1⊗Θ3⊗Θ2Proof.
(1)
Θ1⊕Θ2⊕Θ3=YΘ1,ΨΘ1⊕YΘ2,ΨΘ2⊕YΘ3,ΨΘ3=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2⊕YΘ3,ΨΘ3=YΘ12+YΘ22−YΘ12YΘ22+YΘ32−YΘ32YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ22+YΘ32−YΘ12YΘ22−YΘ12YΘ32−YΘ22YΘ32+YΘ12YΘ22YΘ32,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ32−YΘ12YΘ32+YΘ22−YΘ22YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ3⊕YΘ2,ΨΘ2=Θ1⊕Θ3⊕Θ2(2)
The proof is similar to (1)In order to rank SR-FSs, we define the score function and accuracy function of the SR-FS:Definition 19.
(1)
The score function of an SR-FSΘ=YΘ,ΨΘ can be represented as scoreΘ=YΘ2−ΨΘ(2)
The accuracy function of an SR-FSΘ=YΘ,ΨΘ can be represented as accuracyΘ=YΘ2+ΨΘExample 5.
For an SR-FSΘ=0.3,0.8, we find that scoreΘ≈−0.804427 and accuracyΘ≈0.984427. In particular, if Θ=0,1, then scoreΘ=−1, and if Θ=1,0, then scoreΘ=1.Theorem 20.
The suggested score function of any SR-FSΘ=YΘ,ΨΘ, denoted by scoreΘ lies in −1,1.Proof.
Since for any SR-FSΘ, we have YΘ2+ΨΘ≤1, hence YΘ2−ΨΘ≤YΘ2≤1 and YΘ2−ΨΘ≥−ΨΘ≥−1. Therefore, −1≤YΘ2−ΨΘ≤1. Hence, scoreΘ∈−1,1.Remark 21.
The suggested accuracy function of any SR-FSΘ=YΘ,ΨΘ, denoted by accuracyΘ, lies in 0,1.


### 3.2. Aggregation of SR-Fuzzy Sets

Definition 22.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs and w=w1,w2,⋯,wmT be the weight vector of Θi with wi>0 and ∑i=1mwi=1. Then an SR-Fuzzy
(1)
Weighted average (SR-FWA) operator is a functionSR-FWA:Θm⟶Θ, where(18)SR−FWAΘ1,Θ2,⋯,Θm=∑i=1mwiYΘi,∑i=1mwiΨΘi(2)
Weighted geometric (SR-FWG) operator is a functionSR-FWG:Θm⟶Θ, where(19)SR−FWGΘ1,Θ2,⋯,Θm=∏i=1mYΘiwi,∏i=1mΨΘiwi(3)
Weighted power average (SR-FWPA) operator is a functionSR-FWPA:Θm⟶Θ, where(20)SR−FWPAΘ1,Θ2,⋯,Θm=∑i=1mwiYΘi21/2,∑i=1mwiΨΘi2(4)
Weighted power geometric (SR-FWPG) operator is a functionSR-FWPG:Θm⟶Θ, where(21)SR−FWPGΘ1,Θ2,⋯,Θm=1−∏i=1m1−YΘi2wi12,1−∏i=1m1−ΨΘiwi2Example 6.
Suppose thatΘ1=0.53,0.49,Θ2=0.52,0.51,Θ3=0.26,0.76,Θ4=0.51,0.53,Θ5=0.50,0.54, and Θ6=0.22,0.86 are six SR-Fuzzy sets, and let w=0.12,0.32,0.22,0.13,0.10,0.11T be a weight vector of Θi (i=1,2,⋯,6). Then
(1)
SR−FWAΘ1,Θ2,⋯,Θ6=0.53×0.12+0.52×0.32+0.26×0.22+0.51×0.13+0.50×0.10+0.22×0.11,0.49×0.12+0.51×0.32+0.76×0.22+0.53×0.13+0.54×0.10+0.86×0.11=0.4277,0.6067(2)
SR−FWGΘ1,Θ2,⋯,Θ6=0.530.12×0.520.32×0.260.22×0.510.13×0.500.10×0.220.11,0.490.12×0.510.32×0.760.22×0.530.13×0.540.10×0.860.11≈0.404460,0.593219(3)
SR−FWPAΘ1,Θ2,⋯,Θ6=0.532×0.12+0.522×0.32+0.262×0.22+0.512×0.13+0.502×0.10+0.222×0.111/2,0.12×0.49+0.32×0.51+0.22×0.76+0.13×0.53+0.10×0.54+0.11×0.862≈0.446369,0.599778(4)
SR−FWPGΘ1,Θ2,⋯,Θ6=1−1−0.5320.12×1−0.5220.32×1−0.2620.22×1−0.5120.13×1−0.5020.10×1−0.2220.111/2,1−1−0.490.12×1−0.510.32×1−0.760.22×1−0.530.13×1−0.540.10×1−0.860.112≈0.452616,0.632933Remark 23.
Note that the ordered values induced from the different operators introduced in Definition22 need not be an SR-FS. To validate this matter, take the ordered values 0.452616,0.632933 which are given in (4) of the above example. By calculating, we find that 0.4526162+0.632933=1.0004>1 which means that SR-FWPGΘ1,Θ2,⋯,Θ6 is not an SR-FS.Theorem 24.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be a weight vector of Θi with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(2)
SR−FWGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(3)
SR−FWPAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWPAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(4)
SR−FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗ΘProof.
We will display the proof of (1) and (4). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(22)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘiYΘ,ΨΘi+ΨΘ−ΨΘiΨΘ2≥2ΨΘiΨΘ−ΨΘiΨΘ2=ΨΘiΨΘ,that is
(23)∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2≥∑i=1mwiYΘiYΘ,
and(24)∑i=1mwiΨΘi+ΨΘ−ΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By Definition10 (1) and (2), we have
(25)SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−ΨΘiΨΘ2.
Therefore, from (25), the proof is proved
(4)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(26)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘi2YΘ2⇒1−YΘi2+YΘ2−YΘi2YΘ2≤1−YΘi2YΘ2⇒1−YΘi2+YΘ2−YΘi2YΘ2wi≤1−YΘi2YΘ2wi⇒∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi≤∏i=1m1−YΘi2YΘ2wi⇒1−∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi≥1−∏i=1m1−YΘi2YΘ2wi.
Similarly(27)⇒1−∏i=1m1−ΨΘi+ΨΘ−ΨΘiΨΘwi≥1−∏i=1m1−ΨΘiΨΘwi.
Now, by (1) and (2) of Definition10, we have
(28)SR−FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=1−∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi1/2,1−∏i=1m1−ΨΘiΨΘwi2,SR−FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ=1−∏i=1m1−YΘi2YΘ2wi12,1−∏i=1m1−ΨΘi+ΨΘ−ΨΘiΨΘwi2.
Hence,SR-FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ.Theorem 25.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1; then
(1)
SR-FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWAΘ1,Θ2,⋯,Θm⊗Θ(2)
SR-FWGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWGΘ1,Θ2,⋯,Θm⊗Θ(3)
SR-FWPAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPAΘ1,Θ2,⋯,Θm⊗Θ(4)
SR-FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPGΘ1,Θ2,⋯,Θm⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(29)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘiYΘ,that is
(30)∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2≥∑i=1mwiYΘiYΘ.
Similarly,(31)∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By (1) and (2) of Definition10, we have
(32)SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1,Θ2,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2.
Hence, the desired result is proved.Theorem 26.
LetΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m be two values of SR-FSs, and w=w1,w2,⋯,wmT be a weight vector of them with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(2)
SR−FWGΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWGΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(3)
SR−FWPAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWPAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(4)
SR−FWPGΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWPGΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗KmProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m, we get(33)YΘi2+YKi2−YΘi2YKi2≥2YΘi2YKi2−YΘi2YKi2=YΘiYKi,that is
(34)∑i=1mwiYΘi2+YKi2−YΘi2YKi2≥∑i=1mwiYΘiYKi.
Similarly(35)∑i=1mwiΨΘi+ΨKi−ΨΘiΨKi2≥∑i=1mwiΨΘiΨKi.
By (1) and (2) of Definition10, we have
(36)SR−FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km=∑i=1mwiYΘi2+YKi2−YΘi2YKi2,∑i=1mwiΨΘiΨKi,SR−FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km=∑i=1mwiYΘiYKi,∑i=1mwiΨΘi+ΨKi−ΨΘiΨKi2.
Thus,SR-FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR-FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km.Theorem 27.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1 and ρ≥1; then
(1)
SR−FWAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ(2)
SR−FWGρΘ1,ρΘ2,⋯,ρΘm≥SR−FWGΘ1ρ,Θ2ρ,⋯,Θmρ(3)
SR−FWPAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWPAΘ1ρ,Θ2ρ,⋯,Θmρ(4)
SR−FWPGρΘ1,ρΘ2,⋯,ρΘm≥SR−FWPGΘ1ρ,Θ2ρ,⋯,ΘmρProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m, we have(37)SR−FWAρΘ1,ρΘ2,⋯,ρΘm=∑i=1mwi1−1−YΘi2ρ,∑i=1mwiΨΘiρ,SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ=∑i=1mwiYΘiρ,∑i=1mwi1−1−ΨΘiρ2.
LetfYΘi=1−1−YΘi2ρ−YΘi2ρ, and we have to show fYΘi≥0. Using the Newton generalized binomial theorem, we are able to get
(38)1−YΘi2ρ+YΘi2ρ≤1−YΘi2+YΘi2ρ=1.
Thus,fYΘi≥0, that is
(39)1−1−YΘi2ρ−YΘi2ρ≥0⇒1−1−YΘi2ρ≥YΘi2ρ⇒1−1−YΘi2ρ≥YΘiρ⇒∑i=1mwi1−1−YΘi2ρ≥∑i=1mwiYΘiρ.
Similarly,(40)∑i=1mwi1−1−ΨΘiρ2≥∑i=1mwiΨΘiρ.
ThereforeSR−FWAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ.Theorem 28.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be a weight vector of Θi with ∑i=1mwi=1 and ρ≥1. Then
(1)
SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(2)
SR−FWGρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWGΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(3)
SR−FWPAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWPAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(4)
SR−FWPGρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWPGΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we have(41)SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ=∑i=1mwi1−1−YΘi2ρ1−YΘ2,∑i=1mwiΨΘiρΨΘ,SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ=∑i=1mwiYΘiρYΘ,∑i=1mwi1−1−ΨΘiρ1−ΨΘ2.
LetfYΘi=1−1−YΘi2ρ1−YΘ2−YΘi2ρYΘ2, and we have to show that fYΘi≥0. At first we indicate gYΘi=1−YΘi2ρ+YΘi2ρ and take the derivative of gYΘi; then
(42)g′YΘi=−2ρYΘi1−YΘi2ρ−1+2ρYΘiYΘi2ρ−1=2ρYΘiYΘi2ρ−1−1−YΘi2ρ−1.
Therefore, ifYΘi>1/2, then gYΘi is monotonic increasing, and if YΘi<1/2, then gYΘi is monotonic decreasing, so gYΘi≤gYΘimax=maxg0,g1=1. Since 1−YΘi2ρ1−YΘ2+YΘi2ρYΘ2≤1, hence
(43)fYΘi=1−1−YΘi2ρ1−YΘ2−YΘi2ρYΘ2≥0⇒∑i=1mwi1−1−YΘi2ρ1−YΘ2≥∑i=1mwiYΘiρYΘ.
Similarly(44)∑i=1mwi1−1−ΨΘiρ1−ΨΘ2≥∑i=1mwiΨΘiρΨΘ.
Hence,SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ.
To prove the following three results, we suppose that the values obtained from the introduced operators are an SR-FS (see Remark23).Theorem 29.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1; then
(1)
SR-FWAΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWAΘ1,Θ2,⋯,Θm⊗Θ(2)
SR-FWGΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWGΘ1,Θ2,⋯,Θm⊗Θ(3)
SR-FWPAΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWPAΘ1,Θ2,⋯,Θm⊗Θ(4)
SR-FWPGΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWPGΘ1,Θ2,⋯,Θm⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(45)∑i=1mwiYΘi2+YΘ2−∑i=1mwiYΘi2YΘ2≥2∑i=1mwiYΘi2YΘ2−∑i=1mwiYΘi2YΘ2=∑i=1mwiYΘiYΘ.
Similarly(46)∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By (1) and (2) of Definition10, we obtain
(47)SR−FWAΘ1,Θ2,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−∑i=1mwiYΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1,Θ2,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2.
Hence, the desired result is proved.Theorem 30.
LetΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m be two values of SR-FSs and w=w1,w2,⋯,wmT be a weight vector of them with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1,Θ2,⋯,Θm⊕SR−FWAK1,K2,⋯,Km≥SR−FWAΘ1,Θ2,⋯,Θm⊗SR−FWAK1,K2,⋯,Km(2)
SR−FWGΘ1,Θ2,⋯,Θm⊕SR−FWGK1,K2,⋯,Km≥SR−FWGΘ1,Θ2,⋯,Θm⊗SR−FWGK1,K2,⋯,Km(3)
SR−FWPAΘ1,Θ2,⋯,Θm⊕SR−FWPAK1,K2,⋯,Km≥SR−FWPAΘ1,Θ2,⋯,Θm⊗SR−FWPAK1,K2,⋯,Km(4)
SR−FWPGΘ1,Θ2,⋯,Θm⊕SR−FWPGK1,K2,⋯,Km≥SR−FWPGΘ1,Θ2,⋯,Θm⊗SR−FWPGK1,K2,⋯,KmTheorem 31.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1 and ρ≥1; then
(1)
ρSR−FWAΘ1,Θ2,⋯,Θm≥SR−FWAΘ1,Θ2,⋯,Θmρ(2)
ρSR−FWGΘ1,Θ2,⋯,Θm≥SR−FWGΘ1,Θ2,⋯,Θmρ(3)
ρSR−FWPAΘ1,Θ2,⋯,Θm≥SR−FWPAΘ1,Θ2,⋯,Θmρ(4)
ρSR−FWPGΘ1,Θ2,⋯,Θm≥SR−FWPGΘ1,Θ2,⋯,Θmρ


## 3.1. Some Operations On SR-Fuzzy Sets

Definition 10.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs and ρ be a positive real value (ρ>0). Then their operations are defined as follows:
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22(3)
ρΘ=1−1−YΘ2ρ,ΨΘρ(4)
Θρ=YΘρ,1−1−ΨΘρ2Example 4.
Suppose thatΘ1=YΘ1=0.43,ΨΘ1=0.64 and Θ2=YΘ2=0.26,ΨΘ2=0.81 are both SR-FSs for S=p. Then
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2=0.432+0.262−0.4320.262,0.640.81≈0.489899,0.5184(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22=0.430.26,0.64+0.81−0.640.812=0.1118,0.9604(3)
ρΘ1=1−1−YΘ12ρ,ΨΘ1ρ=1−1−0.4323,0.643≈0.677095,0.262144, for ρ=3(4)
Θ1ρ=YΘ1ρ,1−1−ΨΘ1ρ2=0.433,1−1−0.6432=0.079507,0.984064, for ρ=3Theorem 11.
IfΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 are two SR-FSs, then Θ1⊕Θ2 and Θ1⊗Θ2 are also SR-FSs.Proof.
For SR-FSsΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2, the following relations are evident:
(5)0≤YΘ12≤1,0≤ΨΘ1≤1,0≤YΘ12+ΨΘ1≤1,0≤YΘ22≤1,0≤ΨΘ2≤1,0≤YΘ22+ΨΘ2≤1.
Then, we have(6)YΘ12≥YΘ12YΘ22,YΘ22≥YΘ12YΘ22,1≥YΘ12YΘ22≥0,ΨΘ1≥ΨΘ1ΨΘ2,ΨΘ2≥ΨΘ1ΨΘ2,1≥ΨΘ1ΨΘ2≥0,which indicates that
(7)YΘ12+YΘ22−YΘ12YΘ22≥0,whichmeansthatYΘ12+YΘ22−YΘ12YΘ22≥0,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ2≥0,whichmeansthatΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≥0.
SinceYΘ22≤1 and 0≤1−YΘ12, then YΘ221−YΘ12≤1−YΘ12, and we get YΘ12+YΘ22−YΘ12YΘ22≤1, and hence, YΘ12+YΘ22−YΘ12YΘ22≤1.
Similarly, we get(8)ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1.
It is obvious that(9)0≤ΨΘ1≤1−YΘ12and0≤ΨΘ2≤1−YΘ22.
Then we get(10)YΘ12+YΘ22−YΘ12YΘ222+ΨΘ1ΨΘ2≤YΘ12+YΘ22−YΘ12YΘ22+1−YΘ121−YΘ22=1.
Therefore(11)0≤YΘ12+YΘ22−YΘ12YΘ22≤1,0≤ΨΘ1ΨΘ2≤1,and
(12)0≤YΘ12+YΘ22−YΘ12YΘ222+ΨΘ1ΨΘ2≤1.
Similarly, we have(13)0≤YΘ1YΘ2≤1,0≤ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1,and
(14)0≤YΘ1YΘ22+ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22≤1.
These indicate that both ofΘ1⊕Θ2 and Θ1⊗Θ2 are SR-FSs.Theorem 12.
LetΘ=YΘ,ΨΘ be an SR-FS and ρ be a positive real value. Then, ρΘ and Θρ are also SR-FSs.Proof.
Since0≤YΘ2≤1, 0≤ΨΘ≤1, and 0≤YΘ2+ΨΘ≤1, then
(15)0≤ΨΘ≤1−YΘ2⇒0≤1−YΘ2ρ⇒1−1−YΘ2ρ≤1⇒0≤1−1−YΘ2ρ≤1=1.
It is obvious that0≤ΨΘρ≤1; then we get
(16)0≤1−1−YΘ2ρ2+ΨΘρ≤1−1−YΘ2ρ+1−YΘ2ρ=1.
Similarly, we also get(17)0≤YΘρ2+1−1−ΨΘρ2≤1.
Therefore,ρΘ and Θρ are SR-FSs.Theorem 13.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs. Then the following properties hold:
(1)
Θ1⊕Θ2=Θ2⊕Θ1(2)
Θ1⊗Θ2=Θ2⊗Θ1Proof.
From Definition10, we obtain the following:
(1)
Θ1⊕Θ2=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2YΘ22+YΘ12−YΘ22YΘ12,ΨΘ2ΨΘ1=Θ2⊕Θ1(2)
Θ1⊗Θ2=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22=YΘ2YΘ1,ΨΘ2+ΨΘ1−ΨΘ2ΨΘ12=Θ2⊗Θ1Theorem 14.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs. Then
(1)
ρΘ1⊕Θ2=ρΘ1⊕ρΘ2, for ρ>0(2)
ρ1+ρ2Θ=ρ1Θ⊕ρ2Θ, for ρ1,ρ2>0(3)
Θ1⊗Θ2ρ=Θ1ρ⊗Θ2ρ, for ρ>0(4)
Θρ1⊗Θρ2=Θρ1+ρ2, for ρ1,ρ2>0Proof.
For the three SR-FSsΘ,Θ1, and Θ2, and ρ,ρ1,ρ2>0, according to Definition 10, we obtain the following:
(1)
ρΘ1⊕Θ2=ρYΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2=1−1−YΘ12−YΘ22+YΘ12YΘ22ρ,ΨΘ1ΨΘ2ρ=1−1−YΘ12ρ1−YΘ22ρ,ΨΘ1ρΨΘ2ρ.
AndρΘ1⊕ρΘ2=1−1−YΘ12ρ,ΨΘ1ρ⊕1−1−YΘ22ρ,ΨΘ2ρ=1−1−YΘ12ρ+1−1−YΘ22ρ−1−1−YΘ12ρ1−1−YΘ22ρ,ΨΘ1ρΨΘ2ρ=1−1−YΘ12ρ1−YΘ22ρ,ΨΘ1ρΨΘ2ρ=ρΘ1⊕Θ2(2)
ρ1+ρ2Θ=ρ1+ρ2YΘ,ΨΘ=1−1−YΘ2ρ1+ρ2,ΨΘρ1+ρ2=1−1−YΘ2ρ11−YΘ2ρ2,ΨΘρ1+ρ2=1−1−YΘ2ρ1+1−1−YΘ2ρ2−1−1−YΘ2ρ11−1−YΘ2ρ2,ΨΘρ1ΨΘρ2=1−1−YΘ2ρ1,ΨΘρ1⊕1−1−YΘ2ρ2,ΨΘρ2=ρ1Θ⊕ρ2Θ(3)
Θ1⊗Θ2ρ=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22ρ=YΘ1YΘ2ρ,1−1−ΨΘ1−ΨΘ2+ΨΘ1ΨΘ2ρ2=YΘ1ρYΘ2ρ,1−1−ΨΘ1ρ1−ΨΘ2ρ2=YΘ1ρ,1−1−ΨΘ1ρ2⊗YΘ2ρ,1−1−ΨΘ2ρ2=Θ1ρ⊗Θ2ρ(4)
Θρ1⊗Θρ2=YΘρ1,1−1−ΨΘρ12⊗YΘρ2,1−1−ΨΘρ22=YΘρ1+ρ2,1−1−ΨΘρ1+1−1−ΨΘρ2−1−1−ΨΘρ11−1−ΨΘρ2=YΘρ1+ρ2,1−1−ΨΘρ1+ρ22=Θρ1+ρ2Theorem 15.
LetΘ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be two SR-FSs and ρ>0. Then
(1)
ρΘ1∪Θ2=ρΘ1∪ρΘ2(2)
Θ1∪Θ2ρ=Θ1ρ∪Θ2ρProof.
For the two SR-FSsΘ1 and Θ2, and ρ>0, according to Definitions 5 and 10, we obtain the following:
(1)
ρΘ1∪Θ2=ρmaxYΘ1,YΘ2,minΨΘ1,ΨΘ2=1−1−maxYΘ12,YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ.
AndρΘ1∪ρΘ2=1−1−YΘ12ρ,ΨΘ1ρ∪1−1−YΘ22ρ,ΨΘ2ρ=max1−1−YΘ12ρ,1−1−YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ=1−1−maxYΘ12,YΘ22ρ,minΨΘ1ρ,ΨΘ2ρ=ρΘ1∪Θ2(2)
The proof is similar to (1)Theorem 16.
LetΘ=YΘ,ΨΘ, Θ1=YΘ1,ΨΘ1 and Θ2=YΘ2,ΨΘ2 be three SR-FSs, and ρ>0. Then
(1)
Θ1⊕Θ2c=Θ1c⊗Θ2c(2)
Θ1⊗Θ2c=Θ1c⊕Θ2c(3)
Θcρ=ρΘc(4)
ρΘc=ΘρcProof.
For the three SR-FSsΘ,Θ1 and Θ2, and ρ>0, according to Definitions 5(3) and 10, we obtain the following:
(1)
Θ1⊕Θ2c=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2c=ΨΘ1ΨΘ24,YΘ12+YΘ22−YΘ12YΘ224=ΨΘ14ΨΘ24,YΘ12+YΘ22−YΘ12YΘ222=ΨΘ14,YΘ14⊗ΨΘ24,YΘ24=Θ1c⊗Θ2c(2)
Θ1⊗Θ2c=YΘ1YΘ2,ΨΘ1+ΨΘ2−ΨΘ1ΨΘ22c=ΨΘ1+ΨΘ2−ΨΘ1ΨΘ224,YΘ1YΘ24=ΨΘ1+ΨΘ2−ΨΘ1ΨΘ2,YΘ14YΘ24=ΨΘ14,YΘ14⊕ΨΘ24,YΘ24=Θ1c⊕Θ2c(3)
Θcρ=ΨΘ4,YΘ4ρ=ΨΘ4ρ,1−1−YΘ2ρ2=1−1−YΘ2ρ,ΨΘρc=ρΘc(4)
ρΘc=ρΨΘ4,YΘ4=1−1−ΨΘρ,YΘ4ρ=YΘρ,1−1−ΨΘρ2c=ΘρcTheorem 17.
LetΘ1=YΘ1,ΨΘ1, Θ2=YΘ2,ΨΘ2, and Θ3=YΘ3,ΨΘ3 be three SR-FSs. Then
(1)
Θ1∩Θ2⊕Θ3=Θ1⊕Θ3∩Θ2⊕Θ3(2)
Θ1∪Θ2⊕Θ3=Θ1⊕Θ3∪Θ2⊕Θ3(3)
Θ1∩Θ2⊗Θ3=Θ1⊗Θ3∩Θ2⊗Θ3(4)
Θ1∪Θ2⊗Θ3=Θ1⊗Θ3∪Θ2⊗Θ3Proof.
By Definitions5 and 10, we obtain the following:
(1)
Θ1∩Θ2⊕Θ3=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2⊕YΘ3,ΨΘ3=minYΘ12,YΘ22+YΘ32−YΘ32minYΘ12,YΘ22,maxΨΘ1,ΨΘ2ΨΘ3=1−YΘ32minYΘ12,YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3.
AndΘ1⊕Θ3∩Θ2⊕Θ3=YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ3∩YΘ22+YΘ32−YΘ22YΘ32,ΨΘ2ΨΘ3=minYΘ12+YΘ32−YΘ12YΘ32,YΘ22+YΘ32−YΘ22YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3=min1−YΘ32YΘ12+YΘ32,1−YΘ32YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3=1−YΘ32minYΘ12,YΘ22+YΘ32,maxΨΘ1ΨΘ3,ΨΘ2ΨΘ3.
Thus,Θ1∩Θ2⊕Θ3=Θ1⊕Θ3∩Θ2⊕Θ3(2)
The proof is similar to (1)(3)
Θ1∩Θ2⊗Θ3=minYΘ1,YΘ2,maxΨΘ1,ΨΘ2⊗Θ3=minYΘ1,YΘ2YΘ3,maxΨΘ1,ΨΘ2+ΨΘ3−ΨΘ3maxΨΘ1,ΨΘ22=minYΘ1YΘ3,YΘ2YΘ3,1−ΨΘ3maxΨΘ1,ΨΘ2+ΨΘ32.
AndΘ1⊗Θ3∩Θ2⊗Θ3=YΘ1YΘ3,ΨΘ1+ΨΘ3−ΨΘ1ΨΘ32∩YΘ2YΘ3,ΨΘ2+ΨΘ3−ΨΘ2ΨΘ32=YΘ1YΘ3,1−ΨΘ3ΨΘ1+ΨΘ32∩YΘ2YΘ3,1−ΨΘ3ΨΘ2+ΨΘ32=minYΘ1YΘ3,YΘ2YΘ3,max1−ΨΘ3ΨΘ1+ΨΘ32,1−ΨΘ3ΨΘ2+ΨΘ32=minYΘ1YΘ3,YΘ2YΘ3,1−ΨΘ3maxΨΘ1,ΨΘ2+ΨΘ32.
Thus,Θ1∩Θ2⊗Θ3=Θ1⊗Θ3∩Θ2⊗Θ3(4)
The proof is similar to (3)Theorem 18.
LetΘ1=YΘ1,ΨΘ1, Θ2=YΘ2,ΨΘ2, and Θ3=YΘ3,ΨΘ3 be three SR-FSs. Then
(1)
Θ1⊕Θ2⊕Θ3=Θ1⊕Θ3⊕Θ2(2)
Θ1⊗Θ2⊗Θ3=Θ1⊗Θ3⊗Θ2Proof.
(1)
Θ1⊕Θ2⊕Θ3=YΘ1,ΨΘ1⊕YΘ2,ΨΘ2⊕YΘ3,ΨΘ3=YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2⊕YΘ3,ΨΘ3=YΘ12+YΘ22−YΘ12YΘ22+YΘ32−YΘ32YΘ12+YΘ22−YΘ12YΘ22,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ22+YΘ32−YΘ12YΘ22−YΘ12YΘ32−YΘ22YΘ32+YΘ12YΘ22YΘ32,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ32−YΘ12YΘ32+YΘ22−YΘ22YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ2ΨΘ3=YΘ12+YΘ32−YΘ12YΘ32,ΨΘ1ΨΘ3⊕YΘ2,ΨΘ2=Θ1⊕Θ3⊕Θ2(2)
The proof is similar to (1)In order to rank SR-FSs, we define the score function and accuracy function of the SR-FS:Definition 19.
(1)
The score function of an SR-FSΘ=YΘ,ΨΘ can be represented as scoreΘ=YΘ2−ΨΘ(2)
The accuracy function of an SR-FSΘ=YΘ,ΨΘ can be represented as accuracyΘ=YΘ2+ΨΘExample 5.
For an SR-FSΘ=0.3,0.8, we find that scoreΘ≈−0.804427 and accuracyΘ≈0.984427. In particular, if Θ=0,1, then scoreΘ=−1, and if Θ=1,0, then scoreΘ=1.Theorem 20.
The suggested score function of any SR-FSΘ=YΘ,ΨΘ, denoted by scoreΘ lies in −1,1.Proof.
Since for any SR-FSΘ, we have YΘ2+ΨΘ≤1, hence YΘ2−ΨΘ≤YΘ2≤1 and YΘ2−ΨΘ≥−ΨΘ≥−1. Therefore, −1≤YΘ2−ΨΘ≤1. Hence, scoreΘ∈−1,1.Remark 21.
The suggested accuracy function of any SR-FSΘ=YΘ,ΨΘ, denoted by accuracyΘ, lies in 0,1.


## 3.2. Aggregation of SR-Fuzzy Sets

Definition 22.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs and w=w1,w2,⋯,wmT be the weight vector of Θi with wi>0 and ∑i=1mwi=1. Then an SR-Fuzzy
(1)
Weighted average (SR-FWA) operator is a functionSR-FWA:Θm⟶Θ, where(18)SR−FWAΘ1,Θ2,⋯,Θm=∑i=1mwiYΘi,∑i=1mwiΨΘi(2)
Weighted geometric (SR-FWG) operator is a functionSR-FWG:Θm⟶Θ, where(19)SR−FWGΘ1,Θ2,⋯,Θm=∏i=1mYΘiwi,∏i=1mΨΘiwi(3)
Weighted power average (SR-FWPA) operator is a functionSR-FWPA:Θm⟶Θ, where(20)SR−FWPAΘ1,Θ2,⋯,Θm=∑i=1mwiYΘi21/2,∑i=1mwiΨΘi2(4)
Weighted power geometric (SR-FWPG) operator is a functionSR-FWPG:Θm⟶Θ, where(21)SR−FWPGΘ1,Θ2,⋯,Θm=1−∏i=1m1−YΘi2wi12,1−∏i=1m1−ΨΘiwi2Example 6.
Suppose thatΘ1=0.53,0.49,Θ2=0.52,0.51,Θ3=0.26,0.76,Θ4=0.51,0.53,Θ5=0.50,0.54, and Θ6=0.22,0.86 are six SR-Fuzzy sets, and let w=0.12,0.32,0.22,0.13,0.10,0.11T be a weight vector of Θi (i=1,2,⋯,6). Then
(1)
SR−FWAΘ1,Θ2,⋯,Θ6=0.53×0.12+0.52×0.32+0.26×0.22+0.51×0.13+0.50×0.10+0.22×0.11,0.49×0.12+0.51×0.32+0.76×0.22+0.53×0.13+0.54×0.10+0.86×0.11=0.4277,0.6067(2)
SR−FWGΘ1,Θ2,⋯,Θ6=0.530.12×0.520.32×0.260.22×0.510.13×0.500.10×0.220.11,0.490.12×0.510.32×0.760.22×0.530.13×0.540.10×0.860.11≈0.404460,0.593219(3)
SR−FWPAΘ1,Θ2,⋯,Θ6=0.532×0.12+0.522×0.32+0.262×0.22+0.512×0.13+0.502×0.10+0.222×0.111/2,0.12×0.49+0.32×0.51+0.22×0.76+0.13×0.53+0.10×0.54+0.11×0.862≈0.446369,0.599778(4)
SR−FWPGΘ1,Θ2,⋯,Θ6=1−1−0.5320.12×1−0.5220.32×1−0.2620.22×1−0.5120.13×1−0.5020.10×1−0.2220.111/2,1−1−0.490.12×1−0.510.32×1−0.760.22×1−0.530.13×1−0.540.10×1−0.860.112≈0.452616,0.632933Remark 23.
Note that the ordered values induced from the different operators introduced in Definition22 need not be an SR-FS. To validate this matter, take the ordered values 0.452616,0.632933 which are given in (4) of the above example. By calculating, we find that 0.4526162+0.632933=1.0004>1 which means that SR-FWPGΘ1,Θ2,⋯,Θ6 is not an SR-FS.Theorem 24.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be a weight vector of Θi with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(2)
SR−FWGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(3)
SR−FWPAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWPAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ(4)
SR−FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR−FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗ΘProof.
We will display the proof of (1) and (4). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(22)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘiYΘ,ΨΘi+ΨΘ−ΨΘiΨΘ2≥2ΨΘiΨΘ−ΨΘiΨΘ2=ΨΘiΨΘ,that is
(23)∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2≥∑i=1mwiYΘiYΘ,
and(24)∑i=1mwiΨΘi+ΨΘ−ΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By Definition10 (1) and (2), we have
(25)SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−ΨΘiΨΘ2.
Therefore, from (25), the proof is proved
(4)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(26)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘi2YΘ2⇒1−YΘi2+YΘ2−YΘi2YΘ2≤1−YΘi2YΘ2⇒1−YΘi2+YΘ2−YΘi2YΘ2wi≤1−YΘi2YΘ2wi⇒∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi≤∏i=1m1−YΘi2YΘ2wi⇒1−∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi≥1−∏i=1m1−YΘi2YΘ2wi.
Similarly(27)⇒1−∏i=1m1−ΨΘi+ΨΘ−ΨΘiΨΘwi≥1−∏i=1m1−ΨΘiΨΘwi.
Now, by (1) and (2) of Definition10, we have
(28)SR−FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=1−∏i=1m1−YΘi2+YΘ2−YΘi2YΘ2wi1/2,1−∏i=1m1−ΨΘiΨΘwi2,SR−FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ=1−∏i=1m1−YΘi2YΘ2wi12,1−∏i=1m1−ΨΘi+ΨΘ−ΨΘiΨΘwi2.
Hence,SR-FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPGΘ1⊗Θ,Θ2⊗Θ,⋯,Θm⊗Θ.Theorem 25.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1; then
(1)
SR-FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWAΘ1,Θ2,⋯,Θm⊗Θ(2)
SR-FWGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWGΘ1,Θ2,⋯,Θm⊗Θ(3)
SR-FWPAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPAΘ1,Θ2,⋯,Θm⊗Θ(4)
SR-FWPGΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ≥SR-FWPGΘ1,Θ2,⋯,Θm⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(29)YΘi2+YΘ2−YΘi2YΘ2≥2YΘi2YΘ2−YΘi2YΘ2=YΘiYΘ,that is
(30)∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2≥∑i=1mwiYΘiYΘ.
Similarly,(31)∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By (1) and (2) of Definition10, we have
(32)SR−FWAΘ1⊕Θ,Θ2⊕Θ,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−YΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1,Θ2,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2.
Hence, the desired result is proved.Theorem 26.
LetΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m be two values of SR-FSs, and w=w1,w2,⋯,wmT be a weight vector of them with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(2)
SR−FWGΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWGΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(3)
SR−FWPAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWPAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km(4)
SR−FWPGΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR−FWPGΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗KmProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m, we get(33)YΘi2+YKi2−YΘi2YKi2≥2YΘi2YKi2−YΘi2YKi2=YΘiYKi,that is
(34)∑i=1mwiYΘi2+YKi2−YΘi2YKi2≥∑i=1mwiYΘiYKi.
Similarly(35)∑i=1mwiΨΘi+ΨKi−ΨΘiΨKi2≥∑i=1mwiΨΘiΨKi.
By (1) and (2) of Definition10, we have
(36)SR−FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km=∑i=1mwiYΘi2+YKi2−YΘi2YKi2,∑i=1mwiΨΘiΨKi,SR−FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km=∑i=1mwiYΘiYKi,∑i=1mwiΨΘi+ΨKi−ΨΘiΨKi2.
Thus,SR-FWAΘ1⊕K1,Θ2⊕K2,⋯,Θm⊕Km≥SR-FWAΘ1⊗K1,Θ2⊗K2,⋯,Θm⊗Km.Theorem 27.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1 and ρ≥1; then
(1)
SR−FWAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ(2)
SR−FWGρΘ1,ρΘ2,⋯,ρΘm≥SR−FWGΘ1ρ,Θ2ρ,⋯,Θmρ(3)
SR−FWPAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWPAΘ1ρ,Θ2ρ,⋯,Θmρ(4)
SR−FWPGρΘ1,ρΘ2,⋯,ρΘm≥SR−FWPGΘ1ρ,Θ2ρ,⋯,ΘmρProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m, we have(37)SR−FWAρΘ1,ρΘ2,⋯,ρΘm=∑i=1mwi1−1−YΘi2ρ,∑i=1mwiΨΘiρ,SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ=∑i=1mwiYΘiρ,∑i=1mwi1−1−ΨΘiρ2.
LetfYΘi=1−1−YΘi2ρ−YΘi2ρ, and we have to show fYΘi≥0. Using the Newton generalized binomial theorem, we are able to get
(38)1−YΘi2ρ+YΘi2ρ≤1−YΘi2+YΘi2ρ=1.
Thus,fYΘi≥0, that is
(39)1−1−YΘi2ρ−YΘi2ρ≥0⇒1−1−YΘi2ρ≥YΘi2ρ⇒1−1−YΘi2ρ≥YΘiρ⇒∑i=1mwi1−1−YΘi2ρ≥∑i=1mwiYΘiρ.
Similarly,(40)∑i=1mwi1−1−ΨΘiρ2≥∑i=1mwiΨΘiρ.
ThereforeSR−FWAρΘ1,ρΘ2,⋯,ρΘm≥SR−FWAΘ1ρ,Θ2ρ,⋯,Θmρ.Theorem 28.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be a weight vector of Θi with ∑i=1mwi=1 and ρ≥1. Then
(1)
SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(2)
SR−FWGρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWGΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(3)
SR−FWPAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWPAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ(4)
SR−FWPGρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWPGΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we have(41)SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ=∑i=1mwi1−1−YΘi2ρ1−YΘ2,∑i=1mwiΨΘiρΨΘ,SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ=∑i=1mwiYΘiρYΘ,∑i=1mwi1−1−ΨΘiρ1−ΨΘ2.
LetfYΘi=1−1−YΘi2ρ1−YΘ2−YΘi2ρYΘ2, and we have to show that fYΘi≥0. At first we indicate gYΘi=1−YΘi2ρ+YΘi2ρ and take the derivative of gYΘi; then
(42)g′YΘi=−2ρYΘi1−YΘi2ρ−1+2ρYΘiYΘi2ρ−1=2ρYΘiYΘi2ρ−1−1−YΘi2ρ−1.
Therefore, ifYΘi>1/2, then gYΘi is monotonic increasing, and if YΘi<1/2, then gYΘi is monotonic decreasing, so gYΘi≤gYΘimax=maxg0,g1=1. Since 1−YΘi2ρ1−YΘ2+YΘi2ρYΘ2≤1, hence
(43)fYΘi=1−1−YΘi2ρ1−YΘ2−YΘi2ρYΘ2≥0⇒∑i=1mwi1−1−YΘi2ρ1−YΘ2≥∑i=1mwiYΘiρYΘ.
Similarly(44)∑i=1mwi1−1−ΨΘiρ1−ΨΘ2≥∑i=1mwiΨΘiρΨΘ.
Hence,SR−FWAρΘ1⊕Θ,ρΘ2⊕Θ,⋯,ρΘm⊕Θ≥SR−FWAΘ1ρ⊗Θ,Θ2ρ⊗Θ,⋯,Θmρ⊗Θ.
To prove the following three results, we suppose that the values obtained from the introduced operators are an SR-FS (see Remark23).Theorem 29.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs, Θ=YΘ,ΨΘ be SR-FS, and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1; then
(1)
SR-FWAΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWAΘ1,Θ2,⋯,Θm⊗Θ(2)
SR-FWGΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWGΘ1,Θ2,⋯,Θm⊗Θ(3)
SR-FWPAΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWPAΘ1,Θ2,⋯,Θm⊗Θ(4)
SR-FWPGΘ1,Θ2,⋯,Θm⊕Θ≥SR-FWPGΘ1,Θ2,⋯,Θm⊗ΘProof.
We will display the proof of (1). The other affirmations are proved in a similar fashion.(1)
For anyΘi=YΘi,ΨΘii=1,2,⋯,m and Θ=YΘ,ΨΘ, we get(45)∑i=1mwiYΘi2+YΘ2−∑i=1mwiYΘi2YΘ2≥2∑i=1mwiYΘi2YΘ2−∑i=1mwiYΘi2YΘ2=∑i=1mwiYΘiYΘ.
Similarly(46)∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2≥∑i=1mwiΨΘiΨΘ.
By (1) and (2) of Definition10, we obtain
(47)SR−FWAΘ1,Θ2,⋯,Θm⊕Θ=∑i=1mwiYΘi2+YΘ2−∑i=1mwiYΘi2YΘ2,∑i=1mwiΨΘiΨΘ,SR−FWAΘ1,Θ2,⋯,Θm⊗Θ=∑i=1mwiYΘiYΘ,∑i=1mwiΨΘi+ΨΘ−∑i=1mwiΨΘiΨΘ2.
Hence, the desired result is proved.Theorem 30.
LetΘi=YΘi,ΨΘi and Ki=YKi,ΨKii=1,2,⋯,m be two values of SR-FSs and w=w1,w2,⋯,wmT be a weight vector of them with ∑i=1mwi=1. Then
(1)
SR−FWAΘ1,Θ2,⋯,Θm⊕SR−FWAK1,K2,⋯,Km≥SR−FWAΘ1,Θ2,⋯,Θm⊗SR−FWAK1,K2,⋯,Km(2)
SR−FWGΘ1,Θ2,⋯,Θm⊕SR−FWGK1,K2,⋯,Km≥SR−FWGΘ1,Θ2,⋯,Θm⊗SR−FWGK1,K2,⋯,Km(3)
SR−FWPAΘ1,Θ2,⋯,Θm⊕SR−FWPAK1,K2,⋯,Km≥SR−FWPAΘ1,Θ2,⋯,Θm⊗SR−FWPAK1,K2,⋯,Km(4)
SR−FWPGΘ1,Θ2,⋯,Θm⊕SR−FWPGK1,K2,⋯,Km≥SR−FWPGΘ1,Θ2,⋯,Θm⊗SR−FWPGK1,K2,⋯,KmTheorem 31.
LetΘi=YΘi,ΨΘii=1,2,⋯,m be a value of SR-FSs and w=w1,w2,⋯,wmT be the weight vector of Θi with ∑i=1mwi=1 and ρ≥1; then
(1)
ρSR−FWAΘ1,Θ2,⋯,Θm≥SR−FWAΘ1,Θ2,⋯,Θmρ(2)
ρSR−FWGΘ1,Θ2,⋯,Θm≥SR−FWGΘ1,Θ2,⋯,Θmρ(3)
ρSR−FWPAΘ1,Θ2,⋯,Θm≥SR−FWPAΘ1,Θ2,⋯,Θmρ(4)
ρSR−FWPGΘ1,Θ2,⋯,Θm≥SR−FWPGΘ1,Θ2,⋯,Θmρ


## 4. Application of SR-FSs to Select the Top-Rank University

In this section, we apply the SR-FWA, SR-FWG, SR-FWPA, and SR-FWPG operators to select the top-rank university among different universities.One of the following techniques to handle the multicriteria decision-making (in short, MCDM) problems is based on the different types of fuzzy weighted operators. Herein, we, first, show the steps used in the proposed methodology for MCDM:Step 1.
Represent a MCDM problem under study using the SR-Fuzzy decision matrix.Step 2.
Transmit SR-Fuzzy decision matrix into the normalized SR-Fuzzy decision matrix.Step 3.
Compute for each alternative all kinds of SR-Fuzzy weighted operators.Step 4.
Compute the scores and accuracy functions for each alternative (as we showed in Remark23, the ordered values induced from the different operators need not be an SR-FS; however, we apply the formulas of scores and accuracy functions given in Definition 19 for those ordered values).Step 5.
Compare the given alternatives based on the score function.Step 6.
If the score functions are equal for some alternatives, then compare between them in terms of accuracy function.Step 7.
Determine the optimal ranking order of the alternatives and recognize the optimal alternative(s).In the next example, we illustrate how the above steps are applied to select the top-rank university among different universities.Example 7.
LetU=U1,U2,U3 be a set of alternatives (universities) and P=P1,P2,P3,P4,P5,P6,P7 be a set of seven attributes for the selection of universities, where
(48)P1=representsAcademicStaffs,P2=representsScientificResearch,P3=representsNationalandInternationalScientificActivities,P4=representsStudentSatisfaction,P5=representsQualityAssurance,P6=representsCulturalandCommunityActivitiesandP7=representsLibrary.
Suppose that the weight vector of the attributes given by the decision-maker isw=0.1,0.37,0.14,0.03,0.27,0.06,0.03T. Obviously, ∑i=17wi=1. The SR-F values Ypi,Ψpi of the alternatives according to different attributes are given in Table 1, where Ypi is the positive membership degree for which alternative obeys the given attribute and Ψpi is the membership degree for which alternative does not obey the given attribute such that 0≤Ypi2+Ψpi≤1 and Ypi,Ψpi∈0,1.Table 1
SR-Fuzzy values.
UniversitiesP1P2P3P4P5P6P7U1(0.7, 0.2)(0.4, 0.5)(0.4, 0.7)(0.5, 0.5)(0.4, 0.5)(0.7, 0.1)(0.8, 0.1)U2(0.6, 0.3)(0.3, 0.6)(0.3, 0.8)(0.6, 0.4)(0.4, 0.7)(0.6, 0.1)(0.7, 0.2)U3(0.8, 0.1)(0.5, 0.4)(0.5, 0.5)(0.6, 0.4)(0.7, 0.2)(0.8, 0.1)(1, 0)Applying the proposed aggregation operators given in Definition22, score and accuracy functions, we find, as demonstrated in Table 2, that the optimal ranking order of the three universities is U3≻U1≻U2, and thus, the top alternative is U3.Table 2
Evaluation of scores and accuracy with SR-Fuzzy aggregation operators.
U1U2U3rank orderSR−FWA(0.463, 0.462)(0.396, 0.577)(0.62, 0.3)Score-0.465337-0.602789-0.163323U3≻U1≻U2Accuracy0.8940750.9164210.932123U3≻U2≻U1SR−FWG(0.449660, 0.413739)(0.379393, 0.521591)(0.606024, 0)Score-0.441031-0.5782740.367265U3≻U1≻U2Accuracy0.8454200.8661520.367265U2≻U1≻U3SR−FWPA(0.479479, 0.440997)(0.414970, 0.553320)(0.634823, 0.273865)Score-0.434176-0.571655-0.120321U3≻U1≻U2Accuracy0.8939760.9160550.926321U3≻U2≻U1SR−FWPG(0.496476, 0.475544)(0.425919, 0.605327)(1, 0)Score-0.443109-0.5966211U3≻U1≻U2Accuracy0.9360850.9594351U3≻U2≻U1Note that the score functions for the data given in Table2 are unequal, so they are enough to determine the optimal alternative.The method adopted in this application is illustrated in Figure3.Figure 3
Flow chart to select the high-rank university.


## 5. Conclusions

This paper contributes to the fuzzy set theory in which interest in it grew since the moment Zadeh launched it. To handle some real-life issues which are difficult to solve using fuzzy set theory, some researchers extended this theory to other fuzzy models; the most important are IFSs and PFSs.In this paper, we have proposed a new shape of fuzzy sets called an SR-Fuzzy set and revealed its relationship with other types of the generalizations of fuzzy sets. Then, some operators on SR-Fuzzy sets have been defined, and their relationships have been presented. Furthermore, we have introduced four new weighted aggregated operators over SR-Fuzzy sets and discussed their properties in detail. Moreover, we have shown this procedure with one practical fully developed example.In future works, further applications of SR-Fuzzy sets may be investigated, and also, SR-Fuzzy soft sets may be explored. Also, we will try to generate the topology from the collection of SR-Fuzzy sets and introduce the ideas of connectedness and compactness in SR-Fuzzy topology.



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