Einstein-Ordered Weighted Geometric Operator for Pythagorean Fuzzy Soft Set with Its Application to Solve MAGDM Problem

Abstract

The Pythagorean fuzzy soft set (PFSS) is the most influential and operative tool for maneuvering compared to the Pythagorean fuzzy set (PFS), which can accommodate the parameterization of alternatives. It is also a generalized form of intuitionistic fuzzy soft sets (IFSS), which delivers healthier and more exact valuations in the decision-making (DM) procedure. The primary purpose is to extend and propose ideas related to Einstein’s ordered weighted geometric aggregation operator from fuzzy structure to PFSS structure. The core objective of this work is to present a PFSS aggregation operator, such as the Pythagorean fuzzy soft Einstein-ordered weighted geometric (PFSEOWG) operator. In addition, the basic properties of the proposed operator are introduced, such as idempotency, boundedness, and homogeneity. Moreover, a DM method based on a developed operator has been presented to solve the multiattribute group decision-making (MAGDM) problem. A real-life application of the anticipated method has been offered for a capitalist to choose the most delicate business to finance his money. Finally, a brief comparative analysis with some current methods demonstrates the proposed approach’s effectiveness and reliability.

1. Introduction

MAGDM is considered the most appropriate technique to find the most acceptable alternative from all possible alternatives, following standards or attributes. Traditionally, it is assumed that all information for accessing options based on features and their corresponding weights are expressed in precise numbers. On the other hand, most decisions are made when goals and constraints are usually uncertain or unclear in real life. Zaheh [1] introduced the fuzzy set (FS) model to cope with the specified scenario, making progress in multiple scientific and technical fields. In traditional set theory, the elements of a set can be 0 or 1, but in FS, the degree of membership ranges from 0 to 1. Atanassov [2] extended the perception of FS and developed the notion of the intuitionistic fuzzy set (IFS), which deals with the uncertainty considering the membership (MG) and nonmembership (NMG) grades. Xu [3] protracted the IFS and introduced novel aggregation operators (AOs) for IFS. Wang and Liu [4] proposed some AOs for IFS based on Einstein operations and established a multiple attribute decision-making (MADM) approach using their presented operators. Atanassov [5] established the notion of interval-valued IFS and discussed some essential operations with their properties.

IFS is an influential idea, and various scholars have considered it since its development. However, the leading concept of IFS has some shortcomings; for example, if the sum of membership and nonmembership degree is 1, then IFS cannot deal with such scenarios. To overcome such complications, Yager [6] introduced the PFS, the most generalized form of IFS. Rehman et al. [7] proposed some AOs for PFS and discussed its properties. They also planned a DM approach using their developed AOs. Rehman et al. [8] offered a Pythagorean fuzzy ordered weighted geometric AO with necessary possessions. Wang and Li [9] introduced a MADM approach for PFS using Bonferroni mean AOs. Garg [10] established some novel AOs for PFS, considering the interaction based on Einstein operations.

The abovementioned models and their conforming DM approaches have been familiar and used by specialists in numerous areas. But these models cannot accommodate the parameterized values of the alternatives. Molodtsov [11] developed the concept of soft sets (SSs) and discussed some fundamental operations with their desirable properties. Maji et al. [12] prolonged the notion of SS and stated various elementary operations with its features and utilized it to solve DM complications [13]. Maji et al. [14] combined SS and IFS to grow IFSS and introduced necessary operations with their properties. Zulqarnain et al. [15] introduced the technique for order of preference by similarity to ideal solution (TOPSIS) procedure for interval-valued IFSS employing correlation coefficient (CC). Zulqarnain et al. [16] proposed the robust AOs for the intuitionistic fuzzy hypersoft set. They also constructed an MCDM model using their developed operators to solve DM issues. Zulqarnain et al. [17] proposed the TOPSIS method for intuitionistic fuzzy hypersoft set based on CC to resolve MADM problems. Garg and Arora [18] prolonged the IFSS and projected the generalized AOs for the IFSS.

Several scholars protracted SS ideas by engaging the crucial sorting of FSS. Peng et al. [19] progressed the condition of   1 to of IFSS and established the PFSS with basic operations and possessions. Athira et al. [20] planned the entropy measure for PFSS. Siddique et al. [21] proposed some novel operations for PFSS and established a DM process based on a score matrix. Naeem et al. [22] stretched the concept of PFSS to linguistic PFSS and presented some necessary operations with their possessions. Riaz et al. [23] protracted the PFSS to polar PFSS and developed the TOPSIS technique to resolve the multicriteria group decision-making (MCGDM) problem. Zulqarnain et al. [24] introduced the Pythagorean fuzzy soft Einstein-ordered weighted average operator of PFSS and established the DM technique based on the operator developed by them. Zulqarnain et al. [25] introduced some novel operational laws for PFSS and settled some AOs for PFSS. Zulqarnain et al. [26] developed the TOPSIS method for PFSS based on CC and used their planned approach to resolve the MADM problem. Zulqarnain et al. [27] offered some novel operational laws for PFSS considering the interaction and developed the interaction AOs for PFSS. They also presented an MCDM technique using their proposed interactive AOs. Garg [28, 29] introduced several Einstein AOs under the PFS environment and established the DM techniques based on settled operators to resolve complex difficulties. The existing Einstein AOs and Einstein-weighted ordered AOs are just a weighted Pythagorean fuzzy argument. These PFS Einstein AOs cannot accommodate the parameterized values of alternatives. To overcome the above shortcomings, we focus on developing some novel Einstein AOs for PFSS.

To solve these shortcomings, we indicated the finest alternate with PFSS. Pythagorean fuzzy soft numbers support to conform with inexact statistics in the difficulties of everyday life. In this study, the PFSS operator and steering mechanism of PFSEOWG are based on the assumption of Pythagorean fuzzy soft number (PFSN). Therefore, compared to IFSS, IFS, and FS, it is better to maintain inaccurate and imprecise information flexibility. The core objective of this article is to focus on the development of the new AO for PFSS. It is expected that the operational laws of the proposed operations will be followed to solve the DM problem, and numerical example will be used to prove the effectiveness of the introduced DM method. The main benefit of the proposed operator is that the proposed operator can reduce the number of IFSS and fuzzy soft set (FSS) operators under certain confidence limits. The rest of the research is ordered as follows: Section 2 discusses fundamental concepts such as FS, IFS, PFS, SS, FSS, IFSS, and PFSS. In Section 3, we settled the PFSEOWG operator. Section 3 also discusses some desirable properties of the suggested operator. Section 4 develops the MAGDM method based on the proposed operator and provides a numerical example for selecting the most suitable vehicle. In Section 5, a comparison with some popular methods has been given. Section 6 gives the conclusion.

2. Preliminaries

This section contains some basic definitions, such as SS, IFS, PFS, IFSS, and PFSS, which will form the following manuscript’s structure.

Definition 1 (see [11]). Let be a universal set and be the set of attributes and is a mapping such as , where represents the subsets collection of . Then, is called a SS over .

Definition 2 (see [30]). Let be a collection of substances, then a PFS over is defined aswhere represents the MG and NMG functions, respectively. Furthermore, and is called degree of indeterminacy.
From the above definition, we can see that the only difference lies in the condition, that is, in IFS, what we deal with is the state and , whereas in PFS, we have condition and .

Definition 3 (see [14]). Let be a universal set and be set of attributes and is a mapping such as , where is a collection of intuitionistic fuzzy subsets. Then, is named an IFSS over .where are MG and NMG functions respectively with and

Definition 4 (see [14]). 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 , for all .(2)Complement of is denoted by and is defined as(3)Union of two IFSS is defined as follows:(4)The intersection of and can be defined as follows:

Definition 5 (see [19]). Let be a universal set and be set of attributes and be a mapping such as , where is a collection of Pythagorean fuzzy subsets. Then, is called a PFSS over .where represents the MG and NMG functions, respectively, with , degree of independency , and
The PFSN can be expressed as for readers’ convenience. Zulqarnain et al. [25] presented the score and accuracy functions for PFSN such aswhere   . Sometimes, score function cannot differentiate the PFSNs. For example, let  =  and  = , then equation (7); we have  =  and  = . So, it is difficult to decide which alternative is most suitable in this case. An accuracy function has been developed to overcome the limitations mentioned above.where . The following comparison laws have been presented for PFSNs.(1)If , then .(2)If , then(i)If , then .(ii)If , then .

3. Einstein-Ordered Weighted Geometric Operator for Pythagorean Fuzzy Soft Set

The subsequent section will develop the Einstein-ordered weighted geometric operator for PFSS with some fundamental properties.

Definition 6. Let be a collection of PFSNs, where ( = 1, 2, …, n) and ( = 1, 2, …, m), then the Pythagorean fuzzy soft Einstein-ordered weighted geometric (PFSEOWG) operator is defined aswhere and represent the weight vectors such that , and , and and are permutations of ( = 1, 2, …, n) and ( = 1, 2, …, m) such that and .

Theorem 1. Let be a collection of PFSNs, where ( = 1, 2, …, n) and ( = 1, 2, …, m), then the aggregated value obtained by equation (9) is given aswhere and represent the weight vectors such that , and , and and are permutations of ( = 1, 2, …, n) and ( = 1, 2, …, m) such that and .

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

Example 1. Let be a set of decision makers with weigh vector , who want to decide a bike under the set of attributes with weight vector . Then, is given asFirst, we find the associated ordered position matrix by using the score function, which isas we know that

3.1. Properties of PFSEOWG Operator

3.1.1. Idempotency

Let be a collection of PFSNs, where ( = 1, 2, …, n) and (  = 1, 2, …, m). If are identical, thenwhere and represent the weight vectors such that , and , .

Proof. We know that PFSEOWGAs , so

3.1.2. Boundedness

Let be a collection of PFSNs, where ( = 1, 2, …, n) and ( = 1, 2, …, m), where and represent the weight vectors such that , and , . If  =  and  = , then  ≤ PFSEOWG (, …, ) ≤ .

Proof. Let , , then . So, is decreasing function on .
As  ≤  ≤ , , so  ≤  ≤  and  ≤  ≤ .
Let and represent the weight vectors such that , and , , then we have