Abstract

Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and the sum of the squares of the degree of membership function and nonmembership function is less than one. We present a novel Pythagorean fuzzy soft Einstein ordered weighted averaging (PFSEOWA) operator based on operational laws for Pythagorean fuzzy soft numbers. Furthermore, some essential properties such as idempotency, boundedness, and homogeneity for the proposed operator have been presented in detail. The choice of a sustainable supplier is also examined as an essential part of sustainable supply chain management (SSCM) and is considered a crucial multiattribute group decision-making (MAGDM) issue. In some MAGDM problems, the relationship between alternatives and uncertain environments will be the main reason for deficient consequences. We have presented a novel aggregation operator for PFSS information to choose sustainable suppliers to cope with those complex issues. The Pythagorean fuzzy soft number (PFSN) helps to represent the obscure information in such real-world perspectives. The priority relationship of PFSS details is beneficial in coping with SSCM. The proposed method’s effectiveness is proved by comparing advantages, effectiveness, and flexibility among the existing studies.

1. Introduction

Decision-making is a preconceived strategy of picking a logical choice between many objects. Decision-making (DM) plays a crucial part in real-life scenarios. Better decision-making will change the process to determine the limits, benefits, and characteristics of the decision-maker. To cope with the designated scenario, Zaheh [1] launched the fuzzy set (FS) paradigm that puts advancement in several fields of science and technology, nominating the membership grade for each object real values among 0 and 1. In conventional set theory, elements of a set can be either 0 or 1, but in FS, the degree of membership ranges from 0 to 1. Atanassov [2] extended the concept of FS and introduced an intuitionistic fuzzy set (IFS) which considered both membership and nonmembership grades. Zeshui Xu [3] presented some novel aggregation operators (AOs) for IFS and utilized their developed operators for DM. Wang and Liu [4] offered intuitionistic fuzzy Einstein weighted geometric and intuitionistic fuzzy Einstein ordered weighted geometric operators with desirable properties. They also constructed multiple attribute decision-making (MADM) techniques based on their developed operators. Atanassov and Gargov [5] prolonged the notion of IFS and established the concept of the interval-valued intuitionistic fuzzy set with some novel operations and their characteristics.

IFS is a powerful concept, which various researchers have studied since its development. However, the dominant concept of IFS has some shortcomings, such as degree of membership and nonmembership taken so that their sum exceeds 1. To cope with these limitations, IFS fails to overcome the scenarios mentioned above. Yager [6] extended the notion of IFS and developed the Pythagorean fuzzy set (PFS) by amending the condition to . Rehman et al. [7] developed the Pythagorean fuzzy weighted averaging aggregation operator with fundamental properties and offered a DM method based on their developed operator. Pamučar and Savin [8] established the best worst method and the compressed proportional assessment models for the assortment of the optimum off-road vehicle. Rehman et al. [9] presented a Pythagorean fuzzy ordered weighted averaging aggregation operator with desirable properties and constructed a MADM approach for the developed operator. Wang and Li [10] planned Bonferroni mean AOs for PFS and built the MADM method utilizing their settled operators. Deveci et al. [11] presented a comprehensive survey to justify the operations and properties for PFS. Garg [12] developed Pythagorean fuzzy geometric interactive AOs based on Einstein operations with their essential properties. Ali et al. [13] proposed the Einstein operational laws utilizing t-norm and t-conorm for complex interval-valued PFS. Alosta et al. [14] utilized the multicriteria decision-making technique to enhance emergency medical service centers’ finest sites. Milosevic et al. [15] constructed a novel model operating fuzzy logic systems to select a route for the transportation of harmful ingredients.

Above mentioned theories and their corresponding DM approaches are acknowledged and utilized by experts in several fields. Still, due to the lack of parametrized values, these approaches cannot solve parametrization problems. Molodtsov [16] presented the solution of vagueness and uncertainty, introduced a soft set (SS), and discussed some basic operations with their properties. Maji et al. [17] extended the concept of SS and defined numerous basic operations with their essential features and operated to solve DM [18] complications. Maji et al. [19] protracted the notion of FSS and introduced the IFSS with some fundamental operations. Zulqarnain et al. [20] established the correlation coefficient (CC) for interval-valued IFSS and operated their settled CC for the structure of the TOPSIS method. Zulqarnain and Dayan [21] employed the intuitionistic fuzzy TOPSIS for the assortment of an autocorporation. Muhammad Zulqarnain et al. [22] protracted the idea of the IFSS to an intuitionistic fuzzy hypersoft set and presented the TOPSIS method based on the CC. Garg and Arora [23] planned the generalized AOs for the IFSS.

Several investigators prolonged the SS concept utilizing the fundamental definition of FSS. Peng et al. [24] proposed the impression of IFSS to PFSS by modifying the condition to with some desirable operations. Athira et al. [25] utilized the Hamming distance and Euclidean distance to propose the entropy measure for PFSS. Athira et al. [26] established a DM technique using distance-based entropy measures to resolve DM complications for PFSS. Naeem et al. [27] considered the linguistic PFSS and introduced some basic operations with their properties for PFSS. They also proposed the TOPSIS and VIKOR methods under considered environment to solve DM issues. Riaz et al. [28] prolonged the notion of m-polar PFSS and offered the TOPSIS technique for polar PFSS to resolve multicriteria group decision-making (MCGDM) problems. Riaz et al. [29] anticipated the similarity measures for PFSS and constructed a DM method for PFSS using their developed similarity measures. Zulqarnain et al. [30] settled the AOs for PFSS and projected a DM procedure based on their developed operators. Zulqarnain et al. [31] prolonged the TOPSIS method for PFSS based on CC and employed their progressive approach for MADM problems. Zulqarnain et al. [32] introduced some novel operational laws considering the interaction and proposed interactive AOs for PFSS. They also developed the MCDM approach utilizing their established interactive AOs. Siddique et al. [33] acquired some algebraic operations for PFSS and built a DM technique for PFSS based on a score matrix. It has been observed that fuzzy numbers can only measure uncertainty, and intuitionistic fuzzy numbers can measure true and false membership values. The sum of true and false membership values must be less than 1. However, in our developed methodology, we can measure the values of truth and false membership by modifying the intuitionistic fuzzy numbers condition, such as the sum of the square of true and false values must be less or equal to 1.

Selection and evaluation of suppliers are essential features in professional activities. The fluctuations of the current government strategy use supplier classification as measured by multiple theories with environmental and social needs. Therefore, in the literature, the issue is called sustainable supplier selection, a reference issue of MCGDM. At the same time, multiple credentials [3437] point to the need for further study through the MCDM approach in supplier selection, focusing on appropriate glossary considerations on environmental realities and expert predictions. To solve such shortcomings, we have implemented a method of choosing sustainable suppliers with Pythagorean fuzzy soft information. The stimulation reassessment is considered by utilizing Pythagorean fuzzy soft numbers. The PFSN is helpful to comply with imprecise information in everyday life complications. In the prevailing literature [38, 39], numerous Einstein AOs have been familiarized, such as Pythagorean fuzzy Einstein weighted average, Einstein weighted geometric, Einstein ordered weighted average, and Einstein ordered weighted geometric operators, to solve the complex problems of DM. The Einstein weighted AOs only weight the Pythagorean fuzzy argument. At the same time, the Einstein weighted ordered AOs only weight the orderly position of the Pythagorean fuzzy argument not the Pythagorean fuzzy argument itself. These Einstein ordered operators for PFS are unable to accommodate the parametrization values of the alternatives. To overcome the drawbacks mentioned above, we focus on developing some novel Einstein AOs for PFSS.

Thus, the current work intends to offer a novel PFSEOWA operator. It is expected to follow the proposed operator’s algorithm rules to solve the DM problem and numerical examples used to prove the effectiveness of the introduced DM method. The proposed operators’ key benefit is that the proposed operators can reduce IFSS and FSS operators under specific confident limitations. The rest of the research is organized as follows: some fundamental concepts like FS, IFS, PFS, SS, FSS, IFSS, and PFSS are discussed in Section 2. In Section 3, we developed the PFSEOWA operator. Some desired properties of proposed operators also have been discussed in Section 3. Section 4 developed the MAGDM approach based on proposed operators and presented a numerical example of SSCM. In Section 5, a comparison with some existing methodologies has been provided.

2. Preliminaries

This section comprises some basic definitions such as SS, IFS, PFS, IFSS, and PFSS, which will provide a foundation to form the structure of the following manuscript.

Definition 1 (see [16]). Let be a universal set and be the set of attributes, then a pair is called a soft set (SS) over where is a mapping and is known as a collection of all subsets of universal set .

Definition 2 (see [6]). Let be a collection of objects, then a PFS over is defined as where represent the membership and nonmembership grade functions, respectively. Furthermore, and is called degree of indeterminacy.
We can see from the above definitions that the only difference is in the conditions, i.e., in IFS, we deal with the state and whereas in PFS, we have condition and . We can say that a PFS is the general case of IFS.

Definition 3 (see [24]). Let be a universal set and be set of attributes, then a pair is called an IFSS over where is a mapping and is known as a collection of all IFS subsets of universal set . where are membership grade and nonmembership functions, respectively, with and .

Definition 4 (see [24]). Let and be two IFSS. Then, some basic operations for IFSS are defined as follows: (1) is said to be an intuitionistic fuzzy soft subset of . If and and , for all .(2)Complement of is denoted by and is defined as(3)Union of two IFSSs is defined as follows:(4)The intersection of and can be defined as follows:

Definition 5 (see [24]). Let be a universal set and be set of attributes, then a pair is called a PFSS over where is a mapping and is known as the collection of all PFS subsets of universal set . where represent the membership grade and nonmembership functions, respectively, with , degree of independency , and .
For the sake of readers convenience, we express the PFSN as . For calculating the ranking of alternatives, Zulqarnain et al. [30] introduced the score and accuracy functions for as follows:where . It is informed that the score function is unable to differentiate the PFSNs in some cases. For example, let and , then according to the definition of score function, we have and . So, it is impossible to find the most acceptable alternative utilizing the score function in this case. To handle this drawback, an accuracy function has been developed.where .
Thus, to compare two PFSNs and , following comparison laws are defined:(1)If , then (2)If , then(a)If , then (b)If , then

3. Pythagorean Fuzzy Soft Einstein Ordered Weighted Average Operator

The following section will develop the Einstein ordered weighted average operator for PFSS with some fundamental properties.

Definition 6. Let be a collection of PFSNs, where and , then the Pythagorean fuzzy soft Einstein ordered weighted averaging (PFSEOWA) operator is defined as follows: where and represent the weight vectors such that , , and , , and are permutations of and such that and .

Theorem 1. Let be a collection of PFSNs, where and , then the aggregated value obtained by equation (9) is given as where and represent the weight vectors such that , , and , , and are permutations of and such that and .

Proof. We will prove it by using mathematical induction.For n = 1, we get .For , we get .So, equation (9) is true for n = 1 and m = 1.
Suppose that equation holds for , , and for , .Now, we prove the equation for and :So, it is valid for and .

Example 1. Let be a set of decision-makers with weight vector , who want to decide a bike under the set of attributes with weight vector . The assumed rating values for each attribute in the form of PFSNs are given as follows: First, we find the associated ordered position matrix by using the score function, which is as follows:As we know,

3.1. Properties of PFSEOWA Operator
3.1.1. Idempotency

Let be a collection of PFSNs, where and .

If are mathematically identical, thenwhere and represent the weight vectors such that , , and , .

Proof. As we know,

3.1.2. Boundedness

Let be a collection of PFSNs, where and .

If and , thenwhere and represent the weight vectors such that , , and , .

Proof. Let , , then , which shows that is decreasing function on . So, .
Hence, :Let and represent the weight vectors such that , , and , . Then, we haveLet , , then . So, is decreasing function on .
As , , so and .
Let and represent the weight vectors such that , , and , , then we haveLetThen, equations (22) and (23) can be written as . Thus, and If and , then we haveIf , then we have and . Thus, . Therefore, If , then we have . and .
Thus, . Therefore,So, using (26)–(28), we get

3.1.3. Homogeneity

Prove that PFSEOWA , PFSEOWA for any positive real number .

Proof. Let be a PFSN and . Then, we know thatSo,