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

Al-shami, Tareq M.; Ibrahim, Hariwan Z.; Azzam, A. A.; EL-Maghrabi, Ahmed I.

doi:https://doi.org/10.1155/2022/3653225

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Fuzzy Sets and Their Applications in Mathematics

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Volume 2022 | Article ID 3653225 | https://doi.org/10.1155/2022/3653225

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

Tareq M. Al-shami,¹Hariwan Z. Ibrahim,²A. A. Azzam,^3,4and Ahmed I. EL-Maghrabi⁵

Academic Editor: Muhammad Gulzar

Received29 Dec 2021

Accepted13 Feb 2022

Published11 Mar 2022

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.

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 from -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 Section 2. 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. Let be a universal set such that and are mapping. Then, the intuitionistic fuzzy set (IFS) [10] (resp., Pythagorean fuzzy set (PFS) [16]) is defined by the following: including the condition (resp., ), where is the degree of membership and is the degree of nonmembership of every to .

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. Let be a universal set such that and are mapping. Then, the SR-Fuzzy set (briefly, SR-FS) is defined as following: where is the degree of membership and is the degree of nonmembership of to , such that Then, there is a degree of indeterminacy of to defined by It is obvious that . Otherwise, whenever .

In the interest of simplicity, we shall mention the symbol for the SR-FS . The space of SR-Fuzzy membership grades is displayed in Figure 1.

Example 1. Assume that and for . Then, is not an intuitionistic fuzzy set because . In contrast, is an SR-FS because .

Note that , and hence, .

Remark 3. From Figure 2, 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 point (3)For and , the space of SR-Fuzzy membership grades starts to be larger than the space of intuitionistic membership grades(4)for and , the space of SR-Fuzzy membership grades starts to be smaller than the space of intuitionistic membership grades

Definition 4. Let and be two SR-FSs; then (1) if and only if and (2) if and only if and

Example 2. (1)If and for , then (2)If and for , then

Definition 5. Let and be two SR-Fuzzy sets (SR-FSs). Then (1)(2)(3)

Note that , so is an SR-Fuzzy set. It is obvious that .

Example 3. Assume that and are both SR-FSs for . Then (1)(2)(3)

Theorem 6. Let and be two SR-FSs; then the following properties hold: (1)(2)

Proof. From Definition 5, we can obtain the following: (1)(2)The proof is similar to (1)

Theorem 7. Let , and be three SR-FSs; then (1)(2)

Proof. For the three SR-FSs , and , according to Definition 5, we obtain the following: (1)(2)The proof is similar to (1)

Theorem 8. Let and be two SR-FSs. Then (1)(2)

Proof. From Definition 5, we obtain the following: (1)(2)The proof is similar to (1)

Theorem 9. Let and be two SR-FSs; then (1)(2)

Proof. For the two SR-FSs and , according to Definition 5, we obtain the following: (1)(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. Let , and be three SR-FSs and be a positive real value (). Then their operations are defined as follows: (1)(2)(3)(4)

Example 4. Suppose that and are both SR-FSs for . Then (1)(2)(3), for (4), for

Theorem 11. If and are two SR-FSs, then and are also SR-FSs.

Proof. For SR-FSs and , the following relations are evident: Then, we have which indicates that Since and , then , and we get , and hence, .
Similarly, we get It is obvious that Then we get Therefore and Similarly, we have and These indicate that both of and are SR-FSs.

Theorem 12. Let be an SR-FS and be a positive real value. Then, and are also SR-FSs.

Proof. Since , , and , then It is obvious that ; then we get Similarly, we also get Therefore, and are SR-FSs.

Theorem 13. Let and be two SR-FSs. Then the following properties hold: (1)(2)

Proof. From Definition 10, we obtain the following: (1)(2)

Theorem 14. Let , and be three SR-FSs. Then (1), for (2), for (3), for (4), for

Proof. For the three SR-FSs , and , and , according to Definition 10, we obtain the following: (1).And (2)(3)(4)

Theorem 15. Let and be two SR-FSs and . Then (1)(2)

Proof. For the two SR-FSs and , and , according to Definitions 5 and 10, we obtain the following: (1).And (2)The proof is similar to (1)

Theorem 16. Let , and be three SR-FSs, and . Then (1)(2)(3)(4)

Proof. For the three SR-FSs and , and , according to Definitions 5(3) and 10, we obtain the following: (1)(2)(3)(4)

Theorem 17. Let , , and be three SR-FSs. Then (1)(2)(3)(4)

Proof. By Definitions 5 and 10, we obtain the following: (1).And .
Thus, (2)The proof is similar to (1)(3).And .
Thus, (4)The proof is similar to (3)

Theorem 18. Let , , and be three SR-FSs. Then (1)(2)

Proof. (1)(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-FS can be represented as (2)The accuracy function of an SR-FS can be represented as

Example 5. For an SR-FS , we find that and . In particular, if , then , and if , then .

Theorem 20. The suggested score function of any SR-FS , denoted by lies in .

Proof. Since for any SR-FS , we have , hence and . Therefore, . Hence, .

Remark 21. The suggested accuracy function of any SR-FS , denoted by , lies in .

3.2. Aggregation of SR-Fuzzy Sets

Definition 22. Let be a value of SR-FSs and be the weight vector of with and . Then an SR-Fuzzy (1)Weighted average (SR-FWA) operator is a function -, where(2)Weighted geometric (SR-FWG) operator is a function -, where(3)Weighted power average (SR-FWPA) operator is a function -, where(4)Weighted power geometric (SR-FWPG) operator is a function -, where

Example 6. Suppose that , and are six SR-Fuzzy sets, and let be a weight vector of (). Then (1)(2)(3)(4)

Remark 23. Note that the ordered values induced from the different operators introduced in Definition 22 need not be an SR-FS. To validate this matter, take the ordered values which are given in (4) of the above example. By calculating, we find that which means that - is not an SR-FS.

Theorem 24. Let be a value of SR-FSs, be SR-FS, and be a weight vector of with . Then (1)(2)(3)(4)

Proof. We will display the proof of (1) and (4). The other affirmations are proved in a similar fashion. (1)For any and , we getthat is and By Definition 10 (1) and (2), we have Therefore, from (25), the proof is proved (4)For any and , we getSimilarly Now, by (1) and (2) of Definition 10, we have Hence, --.

Theorem 25. Let be a value of SR-FSs, be SR-FS, and be the weight vector of with ; then (1)--(2)--(3)--(4)--

Proof. We will display the proof of (1). The other affirmations are proved in a similar fashion. (1)For any and , we getthat is Similarly, By (1) and (2) of Definition 10, we have Hence, the desired result is proved.

Theorem 26. Let and be two values of SR-FSs, and be a weight vector of them with . Then (1)(2)(3)(4)

Proof. We will display the proof of (1). The other affirmations are proved in a similar fashion. (1)For any and , we getthat is Similarly By (1) and (2) of Definition 10, we have Thus, --.

Theorem 27. Let be a value of SR-FSs, and be the weight vector of with and ; then (1)(2)(3)(4)

Proof. We will display the proof of (1). The other affirmations are proved in a similar fashion. (1)For any , we haveLet , and we have to show . Using the Newton generalized binomial theorem, we are able to get Thus, , that is Similarly, Therefore .

Theorem 28. Let be a value of SR-FSs, be SR-FS, and be a weight vector of with and . Then (1)(2)(3)(4)

Proof. We will display the proof of (1). The other affirmations are proved in a similar fashion. (1)For any and , we haveLet , and we have to show that . At first we indicate and take the derivative of ; then Therefore, if , then is monotonic increasing, and if , then is monotonic decreasing, so . Since , hence Similarly Hence, .
To prove the following three results, we suppose that the values obtained from the introduced operators are an SR-FS (see Remark 23).

Theorem 29. Let be a value of SR-FSs, be SR-FS, and be the weight vector of with ; then (1)--(2)--(3)--(4)--

Proof. We will display the proof of (1). The other affirmations are proved in a similar fashion. (1)For any and , we getSimilarly By (1) and (2) of Definition 10, we obtain Hence, the desired result is proved.

Theorem 30. Let and be two values of SR-FSs and be a weight vector of them with . Then (1)(2)(3)(4)

Theorem 31. Let be a value of SR-FSs and be the weight vector of with and ; then (1)(2)(3)(4)

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 Remark 23, 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. Let be a set of alternatives (universities) and be a set of seven attributes for the selection of universities, where Suppose that the weight vector of the attributes given by the decision-maker is . Obviously, . The SR-F values of the alternatives according to different attributes are given in Table 1, where is the positive membership degree for which alternative obeys the given attribute and is the membership degree for which alternative does not obey the given attribute such that and .

Applying the proposed aggregation operators given in Definition 22, score and accuracy functions, we find, as demonstrated in Table 2, that the optimal ranking order of the three universities is , and thus, the top alternative is .

Note that the score functions for the data given in Table 2 are unequal, so they are enough to determine the optimal alternative.

The method adopted in this application is illustrated in Figure 3.

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Copyright © 2022 Tareq M. Al-shami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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