Aggregation operators of Pythagorean fuzzy soft sets with their application for green supplier chain management

Abstract

The Pythagorean fuzzy soft sets (PFSS) is a parametrized family and one of the appropriate extensions of the Pythagorean fuzzy sets (PFS). It’s also a generalization of intuitionistic fuzzy soft sets, used to accurately assess deficiencies, uncertainties, and anxiety in evaluation. The most important advantage of PFSS over existing sets is that the PFS family is considered a parametric tool. The PFSS can accommodate more uncertainty comparative to the intuitionistic fuzzy soft sets, this is the most important strategy to explain fuzzy information in the decision-making process. The main objective of the present research is to progress some operational laws along with their corresponding aggregation operators in a Pythagorean fuzzy soft environment. In this article, we introduce Pythagorean fuzzy soft weighted averaging (PFSWA) and Pythagorean fuzzy soft weighted geometric (PFSWG) operators and discuss their desirable characteristics. Also, develop a decision-making technique based on the proposed operators. Through the developed methodology, a technique for solving decision-making concerns is planned. Moreover, an application of the projected methods is presented for green supplier selection in green supply chain management (GSCM). A comparative analysis with the advantages, effectiveness, flexibility, and numerous existing studies demonstrates the effectiveness of this method.

Keywords

Pythagorean fuzzy sets Pythagorean fuzzy soft sets PFSWA operator PFSWG operator GSCM

1 Introduction

Zadeh proposed the idea of fuzzy sets (FS) [1] to solve complex problems that contain uncertainty and ambiguity. In some cases, we must check membership as a non-membership value in the representation of objects that cannot be handled by FS. To overcome this concern, Atanassov anticipated the concept of intuitionistic fuzzy sets (IFS) [2]. Some theories have been developed, such as cubic IFS [3], interval-valued IFS [4], linguistic interval-valued IFS [5], etc. used by investigators. Intimately above-mentioned theories, substance have been well-advised by specialists, and the sum of their two memberships as well as non-membership values cannot overreach one. Atanassov’s IFS only deal with insufficient data due to membership and non-membership values, while IFS cannot deal with inappropriate and vague information. Since the above work is regarded as to visualize our surroundings of linear inequality between the degree of membership (MD) and the degree of non-membership (NMD). However, if the decision-maker goes steady with object MD = 0.8 and NDM = 0.5, then 0.8 + 0.5 ⩽ 1. Clearly, we can see that, it cannot be handled by above studied IFS theories. To overcome the above-mentioned limitations, Yager [6, 7] prolonged the IFS to Pythagorean fuzzy sets (PFSs) by modifying the condition $J + J ⩽ 1 .$ to $J^{2} + J^{2} ⩽ 1 .$ Zhang and Xu [8] defined some operational laws and extended the TOPSIS technique to solve MCDM problems under PFSs environment. Wei and Lu [9] presented several Pythagorean fuzzy power aggregation operators with their properties and proposed the decision-making approaches to solving MADM problems based on developed operators. Wang and Li [10] proposed the Pythagorean fuzzy interaction operational laws and power Bonferroni mean operators. Then, they discussed some specific cases of established operators and considered their properties. In the field of occupational health and safety risk assessment, IIbahar et al. [11] presented the integrated technique Pythagorean fuzzy proportional risk assessment. Zhang [12] established a novel decision-making technique based on Pythagorean fuzzy numbers (PFNs) to solve multiple criteria group decision making (MCGDM) problems. He also developed the accuracy function for the ranking of PFNs and similarity measures under a PFSs environment with some desirable properties. Oztaysi et al. [13] utilized the interval-valued IFS with linguistic information to develop multi experts and MCDM methodologies for the selection of alternative fuel technology.

Garg [14] extended the weighted aggregation operators to PFSs and developed several operators, and presented a decision-making approach based on developed operators. Peng and Yang [15] proposed division and subtraction operators with their several properties under PFSs and developed a ranking method based on developed operators to solve MAGDM problems. Garg [16] introduced the logarithmic operational laws with several weighted averaging and weighted geometric operators under PFSs. Gao et al. [17] established the many Pythagorean fuzzy interaction operators by using arithmetic and geometric operations and proposed some decision-making approaches to solve MADM problems. Wang et al. [18] presented the interactive Hamacher operation under the Pythagorean environment with several aggregation operators and developed the decision-making technique to solve MADM complications by utilizing the developed operators. Wang and Li [19] introduced some novel operators under interval-valued PFSs environment and used them to solve the MAGDM problems. Wang et al. [20] also established the operational laws and aggregation operators for interval-valued q-rung orthopair 2-tuple linguistic to solve the MADM complications. Peng and Yuan [21] presented the Pythagorean fuzzy point operators and established decision-making approaches to solving MADM problems based on developed operators. Cengiz et al. [22] established a unique MCDM methodology according to neutrosophic sets and used it to compare the performance of the firm of a company system composed of one or a lot of legal professionals is solvation of practicing legal professionals. The idea of entropy measure and TOPSIS under the correlation coefficient (CC) has been developed by using complex q-rung orthopair fuzzy information and used the established strategies for decision making [23]. They also developed the operational laws and presented some prioritized aggregation operators under a linguistic IFS environment [24]. Garg [25] developed some improved score functions to analyze the ranking of the normal intuitionistic and interval-valued intuitionistic sets and established the new methodologies to solve multi-attribute decision making (MADM) problems. Ma and Xu [26] improved the score and accuracy functions for PFNs and developed novel averaging and geometric operators under PFSs information.

All the above approaches are widely applied in many areas and fields. However, these theories have limitations due to their incompetence with the parameterization tool. To overcome this kind of complexity, Molodtsov [27] proposed a general mathematical parameterization tool soft sets (SS), which is used to deal with uncertain, ambiguous, and indeterminate components, in which certain specific parameters of the object are evaluated. Maji et al. [28] extended the concept of SS, and proposed some operations with their several properties, and used the established concepts for decision-making [29]. Maji et al. [30] planned the idea of fuzzy soft sets (FSS) by combining FS and SS. Additionally, they projected an intuitionistic fuzzy soft set (IFSS) with fundamental operations along with properties [31]. Garg and Arora [32] progressed the generalized version of the IFSS with weighted averaging and geometric aggregation operators and built a decision-making technique to resolve complications beneath an intuitionistic fuzzy environment. The authors [33] developed the aggregate operators by using dual hesitant fuzzy soft numbers and utilized the proposed operators to solve multi-criteria decision making (MCDM) problems. To measure the relationship among dual hesitant fuzzy soft set Arora and Garg [34] introduced the CC and developed a decision-making approach under the presented environment to solve the MCDM approach, they also used the proposed methodology for decision making. and extended the Maclaurin symmetric mean (MSM) operators to IFSS based on Archimedean T-conorm and T-norm [35]. Garg and Arora introduced the correlation measures on IFSS and constructed the TOPSIS technique on developed correlation measures [36].

In this era, the assumptions and application scenarios of SS and the above-mentioned different research extensions are developing rapidly. Peng et al. completely solved it [37] and proposed a novel idea Pythagorean fuzzy soft sets (PFSSs) by combining two existing theories PFS and SS with some basic operations with ideal characteristics. Athira et al. [38] developed an entropy measure under PFSSs information. They also proposed the Hamming distance and Euclidean distance of PFSSs and used them for decision-making [39]. As far as the author knows, there are few studies on the theory of PFSSs. Therefore, it is a better way to keep PFSSs flexible than IFSS or FSS. This article focuses on developing new aggregation operators on PFSSs. Naeem et al. [40] established some operations with their desirable properties and extended the TOPSIS and VIKOR techniques under linguistic PFSS information. They also performed an application on the consequence of stock exchange investment by utilizing the developed techniques. Riaz et al. [41] established the TOPSIS technique for m polar PFSS and presented an example to solve MCGDM problems under-considered hybrid structure. They also introduced the similarity measure for PFSS [42]. Han et al. [43] improved the TOPSIS approach to solving MAGDM problems by using PFSS information. Hua et al. [44] extended the PFSS to possibility PFSS with fundamental operations and introduced the similarity measure to compare any two possibility PFSS.

Green supply chain management (GSCM) has grown into a proactive approach to enhancing the overall environmental performance of policies along with products in keeping with the necessities of environmental rules. Supply management and environmental anxieties are becoming progressively related to systematic decision analysis around the world. Urgent needs to incorporate environmental characteristics into the main supply chain management research, because literature investigation shows that orientations to green supply chain management (GSCM) are still not accurate enough. There is also a lack of such management in an administrative organization that reforms policies to address social and environmental issues and promote business and economic processes. Comprehensive classification helps educators, researchers, and interpreters recognize that the GSCM should be linked to a more sensible approach. Numerous corporations have implemented the GSCM procedure to maintain financial support even though those around us remain the same. The selection of green suppliers becomes an integral part of the GSCM and will advise on multi-level group decision making (MCGDM) matters. Lack of consideration for alternative relationships uncertainty will be the main reason for the poor outcome of some MCGDM concerns. By using affluent available content, which comprised prior feedback along with constrained perceptions, related to GSCM is judged rendering circumstantial of the problems mostly effect of the supply chain area. Several analytical and mathematical tools/procedures victimized in the literature for the GSCM environment are going to be express. To enlighten such insufficiencies, we have employed a technique of picking green suppliers with Pythagorean fuzzy soft information, in which the stimulation reassessment is considered by utilizing Pythagorean fuzzy soft numbers (PFSN). The PFSN is intensely worthy to complies with imprecise information that occurs in everyday life complications. Therefore, the main purpose of this work is to propose novel PFSWA and PFSWG operators. An algorithm based on the proposed operators to solve the decision-making problems is proposed, to demonstrate the effectiveness of the proposed decision-making method a numerical example is used to illustrate it. The key benefit of the proposed operators is that under-confident distinct limitations, the proposed operators can reduce IFSS and FSS operators.

The rest of the research organized as follows: in section 2, we present some basic concepts such as SS, FSS, IFSS, and PFSSs which helps us to build the structure of the following research. We define some operational laws and proposed PFSWA and PFSWG operators with their properties based on developed operational laws in section 3. In section 4, develop a decision-making methodology for GSCM by utilizing the proposed operators and present a numerical example to illustrate the proposed technique. Section 5 provides a comparative analysis of some existing techniques to the developed method.

2 Preliminaries

In this section, we recollect some basic definitions which are helpful to build the structure of the following manuscript such as soft sets, FSS, IFSS, and PFSSs.

Definition 2.1 [27] A map $F$ : $E$ → $K^{U}$ is known as a SS, where $K^{U}$ be a collection of all subsets of a universe of discourse $U$ and $E$ is a set of attributes.

Maji et al. explored the theory of FS and SS and planned a more generalized version to handle the uncertainty compared with the existing FS and SS along with its unique features. This is generally known as a fuzzy soft set, which is a combination of FS and SS.

Definition 2.2 [30] $K^{U}$ be a collection of all fuzzy subsets over $U$ and $E$ be a set of attributes. Let $A$ ⊆ $E$ , then a pair ( $F, A$ ) is called FSS over $U$ , where $F$ is a mapping such as $F$ : $A$ → $K^{U}$ .

Definition 2.3 [30] Let $(F, A)$ and $(G, ß)$ be two FSSs over $U$ . Then, some basic operation for FSS defined as follows:

If $A$ ⊆ ß and $F (e)$ ⩽ $G (e)$ , for each $e$ ∈ $A$ , then $(F, A)$ ⊆ $(G, ß)$ .

If $(F, A)$ ⊆ $(G, ß)$ and $(G, ß)$ ⊆ $(F, A)$ , then $(F, A) = (G, ß)$ .

The complement of an FSS $(F, A)$ can be expressed as $(F^{c}, A)$ , and defined as follows:

For each $a \in A$ , $F^{c} (a)$ be an FS over $U$ , whose membership value $F_{a}^{c} (u) = 1 - F_{a} (u)$ for all $u \in U$ .

Let $(F, A)$ and $(G, ß)$ be two FSSs over $U$ . Then, their union is defined as follows:

$(F, A) \cup (G, ß) = {\begin{matrix} F_{e} (u) if e \in A - ß \\ G_{e} (u) if e \in ß - A \\ \max {F_{e} (u), G_{e} (u)} if e \in A \cap ß \end{matrix} .$

Let $(F, A)$ and $(G, ß)$ be two SSs over $U$ . Then, their intersection is defined as follows:

$(F, A) \cap (G, ß) = \min {F_{e} (u), G_{e} (u)}$ for all $e \in A \cap ß$ .

Definition 2.4 [31] A mapping $F$ : $A \to I K^{U}$ is known as an IFSS and defined as $F_{u_{i}} (e) = {(u_{i}, T_{A} (u_{i}), J_{A} (u_{i})) | u_{i} \in U}$ , where $T_{A} (u_{i})$ and $J_{A} (u_{i})$ are the MD and NMD respectively for all $u_{i} \in U$ and 0 ⩽ $T_{A} (u_{i}), J_{A} (u_{i})$ , $T_{A} (u_{i}) + J_{A} (u_{i}) ⩽ 1$ . Where $I K^{U}$ be a collection of all intuitionistic fuzzy subsets of $U$ .

Definition 2.5 [31] Let $(F, A)$ and $(G, ß)$ be two IFSSs over $U$ . Then, some basic operation under IFSS defined as follows:

If $A$ ⊆ ß and $T_{F (e)}$ ⩽ $T_{G (e)}$ and $J_{F (e)}$ ⩾ $J_{G (e)}$ for all $e$ ∈ $A$ , then $(F, A)$ ⊆ $(G, ß)$ .

If $(F, A)$ ⊆ $(G, ß)$ and $(G, ß)$ ⊆ $(F, A)$ , then $(F, A) = (G, ß)$ .

Let $(F, A) = {(u_{i}, T_{A} (u_{i}), J_{A} (u_{i})) | u_{i} \in U}$ be an IFSS over a universe of discourse $U$ . Then, its complement is defined as $(F^{c}, A) = {(u_{i}, J_{A} (u_{i}), T_{A} (u_{i})) | u_{i} \in U}$ .

The above-mentioned IFS cannot deal with the situation when the sum of MD and NMD exceeds one, so to handle such situations Yager [6, 7] presented a most general notion PFS with its characteristics by modifying the condition MD+ NMD ⩽ 1 to MD²+ NMD² ⩽ 1.

Definition 2.6 [6, 7] Let $U$ be a universe of discourse. Then, the PFS over $U$ is stated as $℘ = {(u, T_{℘} (u), J_{℘} (u)) | u \in U}$ , where $T_{℘} (u), J_{℘} (u)$ : $U$ → [0, 1] and represent the MD and NMD respectively, with ${(T_{℘} (u))}^{2} + {(J_{℘} (u))}^{2} ⩽ 1$ for all $u \in U$ . The degree of hesitation is given as $L_{℘} (u) = \sqrt{1 - {(T_{℘} (u))}^{2} + {(J_{℘} (u))}^{2}}$ . Zhang and Xu [8] represented the PFN such as $℘ = (T_{℘}, J_{℘})$ .

Definition 2.7 [7, 15] Let $℘ = (T, J)$ $℘_{1} = (T_{1}, J_{1})$ and $℘_{2} = (T_{2}, J_{2})$ be three PFNs. Then, some elementary operations are defined as follows:

$℘_{1} \cup ℘_{2} = (\max {T_{1}, T_{2}}, \min {J_{1}, J_{2}})$ .

$℘_{1} \cap ℘_{2} = (\min {T_{1}, T_{2}}, \max {J_{1}, J_{2}})$ .

$℘^{c} = (J, T)$ .

Peng et al. [37] established the idea of the Pythagorean fuzzy soft set with its necessary properties, by merging the features of two prevailing structures Pythagorean fuzzy set and parameterization of SS given as follows:

Definition 2.8 [37] Let $U$ and $E$ are a universe of discourse and set of attributes respectively. Then, the PFSS defined over $U$ as follows: $F_{u_{i}} (e_{j}) = {(u_{i}, T_{j} (u_{i}), J_{j} (u_{i})) | u_{i} \in U},$ where $F$ be a mapping such as $F$ : $E \to$ $P K^{U}$ , $P K^{U}$ represent the pythagorean fuzzy subsets of $U$ and for all $u_{i} \in U$ satisfies the $T^{2} + J^{2} ⩽ 1$ .

Simply a PFSNs can be expressed as $F_{u_{i}} (e_{j}) = {(T_{F (e_{j})} (u_{i}), J_{F (e_{j})} (u_{i})) | u_{i} \in U}$ , where $T_{F}$ $(e_{j}), J_{F (e_{j})}$ ∈ [0, 1] and $T_{F (e_{j})}^{2} + J_{F (e_{j})}^{2} ⩽ 1$ .

Remark 2.1. If $T^{2} + J^{2} ⩽ 1$ and $T + J ⩽ 1$ both are holds. Then, PFSS reduced to IFSS.

For simplicity, we will express $F_{u_{i}} (e_{j}) = {(T_{F (e_{j})} (u_{i}), J_{F (e_{j})} (u_{i})) | u_{i} \in U}$ as $J_{e_{ij}} = T_{ij},$ $J_{ij}$ is called PFSN. In the process of applying PFSNs in actual problems, it is essential to rank them. For this, the scoring function of $J_{e_{ij}}$ is defined as follows: $S (J_{e_{ij}}) = T_{ij}^{2} - J_{ij}^{2}, S (J_{e_{ij}}) \in [- 1, 1]$ (1)

However, in some cases, the scoring function cannot compare the two PFSNs. Such as $J_{e_{11}} = 〈 . 4, . 7 〉$ and $J_{e_{12}} = 〈 . 5568, . 8 〉$ , it is impossible to know which is greater because $S (J_{e_{11}}) = . 3 = S (J_{e_{12}})$ . For this, an accuracy function is defined as: $H (J_{e_{ij}}) = T_{ij}^{2} + J_{ij}^{2}, H (J_{e_{ij}}) \in [0, 1] .$ (2)

Thus, to compare two PFSNs $J_{e_{ij}}$ and $T_{e_{ij}}$ , the following ranking and comparison laws are defined as follows:

If $S (J_{e_{ij}}) > S (T_{e_{ij}})$ , then $J_{e_{ij}} > T_{e_{ij}}$ .

If $S (J_{e_{ij}}) = S (T_{e_{ij}})$ , then

If $H (J_{e_{ij}}) > H (T_{e_{ij}})$ , then $J_{e_{ij}} > T_{e_{ij}}$

If $H (J_{e_{ij}}) = H (T_{e_{ij}})$ , then $J_{e_{ij}} = T_{e_{ij}}$ .

3 Aggregation operators for pythagorean fuzzy soft numbers

In this section, some aggregation operators, i . e, PFSWA, and PFSWG operators for PFSNs have been presented with their desirable properties.

3.1 Operational laws for PFSNs

For simplicity, we will express $F_{u_{i}} (e_{j}) = {(T_{F (e_{j})} (u_{i}), J_{F (e_{j})} (u_{i})) | u_{i} \in U}$ as $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ and is called PFSN, where $u_{i} \in U$ represent the universe of discourse and $e_{j} \in E$ is a set of attributes.

Definition 3.1. Let $J_{e} = (T, J)$ , $J_{e_{11}} = (T_{11}, J_{11})$ , and $J_{e_{12}} = (T_{12}, J_{12})$ be three PFSNs and α be a positive real number, by algebraic norms, we have

$J_{e_{11}} \oplus J_{e_{12}} = 〈 \sqrt{T_{11}^{2} + T_{12}^{2} - T_{11}^{2} T_{12}^{2}}, J_{11} J_{12} 〉$

$J_{e_{11}} \otimes J_{e_{12}} = 〈 T_{11} T_{12}, \sqrt{J_{11}^{2} + J_{12}^{2} - J_{11}^{2} J_{12}^{2}} 〉$

$α J_{e} = 〈 \sqrt{1 - (1 - T^{2})^{α}}, J^{α} 〉$

$J_{e^{α}} = 〈 T^{α}, \sqrt{1 - (1 - J^{2})^{α}} 〉$

Some averaging and geometric aggregation operators for PFSSs have been defined based on the above laws for the collection of PFSNs Δ.

Definition 3.2. Let $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ be a PFSN, Ω_i and γ_j are weight vector for expert’s and attributes respectively with given conditions Ω_i > 0, $\sum_{i = 1}^{n} Ω_{i} = 1$ , γ_j > 0, $\sum_{j = 1}^{m} γ_{j} = 1$ . Then PFSWA operator is defined as

PFSWA: Δⁿ → Δ defined as follows $PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = \oplus_{j = 1}^{m} γ_{j} (\oplus_{i = 1}^{n} Ω_{i} J_{e_{ij}}) .$ (3)

Theorem 3.1. Let $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ be a collection of PFSNs, where (i = 1, 2, …, n, andj = 1, 2, …, m). Then, the aggregated values obtained by using Equation 3 is also a PFSN and

$\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) \\ = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 . \end{matrix}$ (4) where Ω_i and γ_j are weight vector for expert’s and attributes respectively with given conditions Ω_i > 0, $\sum_{i = 1}^{n} Ω_{i} = 1$ , γ_j > 0, $\sum_{j = 1}^{m} γ_{j} = 1$ .

Proof: By using the principle of mathematical induction PFSWA operator can be proved as follows:

For n = 1, we get Ω₁ = 1. Then, we have $\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{1 m}}) = \oplus_{j = 1}^{m} γ_{j} J_{e_{1 j}} \\ PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) \\ = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(1 - T_{1 j}^{2})}^{γ_{j}}}, \prod_{j = 1}^{m} {(J_{1 j})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 . \end{matrix}$

For m = 1, we get γ₁ = 1. Then, we have $\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{21}}, \dots, J_{e_{n 1}}) = \oplus_{i = 1}^{n} Ω_{i} J_{e_{i 1}} \\ = 〈 \sqrt{1 - \prod_{i = 1}^{n} {(1 - T_{i 1}^{2})}^{Ω_{i}}}, \prod_{i = 1}^{n} {(J_{i 1})}^{Ω_{i}} 〉 \\ = 〈 \sqrt{1 - \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 . \end{matrix}$

This shows that Equation 4 satisfies for n = 1 and m = 1. Consider Equation 4 holds for m = β₁ + 1, n = β₂ and m = β₁, n = β₂ + 1, such as: $\begin{matrix} \oplus_{j = 1}^{β_{1} + 1} γ_{j} (\oplus_{i = 1}^{β_{2}} Ω_{i} J_{e_{ij}}) = 〈 \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \\ \oplus_{j = 1}^{β_{1}} γ_{j} (\oplus_{i = 1}^{β_{2} + 1} Ω_{i} J_{e_{ij}}) = 〈 \sqrt{1 - \prod_{j = 1}^{β_{1}} {(\prod_{i = 1}^{β_{2} + 1} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{β_{1}} {(\prod_{i = 1}^{β_{2} + 1} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \end{matrix}$

For m = β₁ + 1 and n = β₂ + 1, we have $\begin{matrix} \oplus_{j = 1}^{β_{1} + 1} γ_{j} (\oplus_{i = 1}^{β_{2} + 1} Ω_{i} J_{e_{ij}}) = \oplus_{j = 1}^{β_{1} + 1} γ_{j} (\oplus_{i = 1}^{β_{2}} Ω_{i} J_{e_{ij}} \oplus Ω_{β_{2} + 1} J_{e_{(β_{2} + 1) j}}) \\ = \oplus_{j = 1}^{β_{1} + 1} \oplus_{i = 1}^{β_{2}} γ_{j} Ω_{i} J_{e_{ij}} \oplus_{j = 1}^{β_{1} + 1} γ_{j} Ω_{β_{2} + 1} J_{e_{(β_{2} + 1) j}} \\ = 〈 \begin{matrix} \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} \oplus \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {({(1 - T_{(β_{2} + 1) j}^{2})}^{Ω_{β_{2} + 1}})}^{γ_{j}}}, \\ \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} \oplus \prod_{j = 1}^{β_{1} + 1} {({(J_{(β_{2} + 1) j})}^{Ω_{β_{2} + 1}})}^{γ_{j}} \end{matrix} 〉 \\ = 〈 \begin{matrix} \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2} + 1} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2} + 1} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} \end{matrix} 〉 . \end{matrix}$

Hence, it is true for m = β₁ + 1 and n = β₂ + 1.

Remark 3.1.

If $T_{ij}^{2} + J_{ij}^{2} ⩽ 1$ and $T_{ij} + J_{ij} ⩽ 1$ both are holds. Then, the PFSWA operator reduced to the IFSWA operator [46].

If a set of attributes contains only one parameter, then the PFSWA operator reduced to PFWA operator [12] such as

$PFWA (J_{e_{11}}, J_{e_{21}}, \dots, J_{e_{n 1}}) = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(1 - T_{ij}^{2})}^{γ_{j}}}, \prod_{j = 1}^{m} {(J_{ij})}^{γ_{j}} 〉 .$

If $T_{ij}^{2} + J_{ij}^{2} ⩽ 1$ and $T_{ij} + J_{ij} ⩽ 1$ both are holds and a set of attributes contains only one parameter. Then, the PFSWA operator reduced to the IFWA operator [47].

Example 3.1. Let $U = {u_{1} u_{2}, u_{3}}$ be a set of experts with weights Ω_i = (.243, . 514, . 343) ^T, who are going to describe the attractiveness of a house under the defined set of attributes $L^{'} = {e_{1} = lawn, e_{2} = security system, e_{3} = rooms envoirnment, e_{4} = infrastucture, e_{5} = electricity}$ with weights γ_j = (. 25, . 15, . 2, . 1, . 3) ^T. The assumed rating values of the experts for each attribute in the form of PFSNs $(J, L^{'}) = {〈 T_{ij}, J_{ij} 〉}_{3 \times 5}$ given as follows: $(J, L^{'}) = [\begin{matrix} (. 3, . 8) & (. 4, . 6) & (. 3, . 6) & (. 5, . 6) & (. 4, . 5) \\ (. 8, . 3) & (. 7, . 4) & (. 7, . 3) & (. 4, . 8) & (. 6, . 4) \\ (. 3.6) & (. 5, . 7) & (. 6, . 5) & (. 5, . 4) & (. 4, . 5) \end{matrix}]$

By using Equation 4, $\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{44}}) = 〈 \sqrt{1 - \prod_{j = 1}^{5} {(\prod_{i = 1}^{3} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{5} {(\prod_{i = 1}^{3} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - (\begin{matrix} {(. 91)^{. 243} (. 36)^{. 514} (. 91)^{. 343}}^{. 25} {(. 84)^{. 243} (. 51)^{. 514} (. 75)^{. 343}}^{. 15} {(. 91)^{. 243} (. 51)^{. 514} (. 64)^{. 343}}^{. 2} \\ {(. 75)^{. 243} (. 84)^{. 514} (. 75)^{. 343}}^{. 1} {(. 84)^{. 243} (. 64)^{. 514} (. 84)^{. 343}}^{. 3} \end{matrix}),} \\ (\begin{matrix} {(. 8)^{. 243} (. 3)^{. 514} (. 6)^{. 343}}^{. 25} {(. 6)^{. 243} (. 4)^{. 514} (. 7)^{. 343}}^{. 15} {(. 6)^{. 243} (. 3)^{. 514} (. 5)^{. 343}}^{. 2} \\ {(. 6)^{. 243} (. 8)^{. 514} (. 4)^{. 343}}^{. 1} {(. 5)^{. 243} (. 4)^{. 514} (. 5)^{. 343}}^{. 3} \end{matrix}) 〉 \\ = 〈 . 6009, . 4342 〉 . \end{matrix}$

By utilizing the proposed PFSWA operator, we establish some properties for the collection of PFSNs based on Theorem 3.1.

3.2 Properties of PFSWA operator

3.2.1 (Idempotency)

If $J_{e_{ij}} = J_{e} = 〈 T_{ij}, J_{ij} 〉 \forall i$ , j, then, $PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = J_{e} .$

Proof: As we know that all $J_{e_{ij}} = J_{e} = 〈 T_{ij}, J_{ij} 〉$ , then by Equation 4 $\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - {({(1 - T_{ij}^{2})}^{\sum_{i = 1}^{n} Ω_{i}})}^{\sum_{j = 1}^{m} γ_{j}}}, {({(J_{ij})}^{\sum_{i = 1}^{n} Ω_{i}})}^{\sum_{j = 1}^{m} γ_{j}} 〉 \\ = 〈 \sqrt{1 - (1 - T_{ij}^{2})}, J_{ij} 〉 = 〈 T_{ij}, J_{ij} 〉 = J_{e} . \end{matrix}$

Which completes the proof.

3.2.2 (Boundedness)

Let $J_{e_{ij}}$ be a collection of PFSNs and $J_{e_{ij}}^{-} = 〈 \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}}, \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {J_{ij}} 〉$ and $J_{e_{ij}}^{+} = 〈 \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}}, \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {J_{ij}} 〉$ , then $J_{e_{ij}}^{-} ⩽ PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) ⩽ J_{e_{ij}}^{+} .$

Proof: As we know that $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ be a PFSN, then $\begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}} ⩽ T_{ij}^{2} ⩽ \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}}$ $\Rightarrow 1 - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}} ⩽ 1 - T_{ij}^{2} ⩽ 1 - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}}$ $\Leftrightarrow {(1 - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}})}^{Ω_{i}} ⩽ {(1 - T_{ij}^{2})}^{Ω_{i}} ⩽ {(1 - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}})}^{Ω_{i}}$ $\Leftrightarrow {(1 - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}})}^{\sum_{i = 1}^{n} Ω_{i}} ⩽ \prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}} ⩽ {(1 - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}})}^{\sum_{i = 1}^{n} Ω_{i}}$ $\Leftrightarrow {(1 - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}})}^{\sum_{j = 1}^{m} γ_{j}} ⩽ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}} ⩽ {(1 - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}})}^{\sum_{j = 1}^{m} γ_{j}}$ $\Leftrightarrow 1 - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}} ⩽ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}} ⩽ 1 - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}}$ $\Leftrightarrow \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}^{2}} ⩽ 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}} ⩽ \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}^{2}}$ $\Leftrightarrow \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}} ⩽ \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} ⩽ \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}} (a)$

Similarly, $\begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {J_{ij}} ⩽ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} ⩽ \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {J_{ij}}$ (b)

Let $PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = 〈 T_{δ}, J_{δ} 〉 = J_{δ}$ , then inequalities (a) and (b) can be transformed into the following form:

$\begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}} ⩽$ $T_{δ}$ ⩽ $\begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}}$ and $\begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {J_{ij}}$ ⩽ $J_{δ}$ ⩽ $\begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {J_{ij}}$ respectively.

So, by utilizing Equation 1, we have: $S (J_{δ}) = T_{δ}^{2} - J_{δ}^{2} ⩽ \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}} - \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {J_{ij}} = S (J_{e_{ij}}^{+}),$ $S (J_{δ}) = T_{δ}^{2} - J_{δ}^{2} ⩾ \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}} - \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {J_{ij}} = S (J_{e_{ij}}^{-}) .$

Then, by order relation among two PFSNs, we have $J_{e_{ij}}^{-} ⩽ PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) ⩽ J_{e_{ij}}^{+} .$

3.2.3 (Shift Invariance)

If $J_{e} = 〈 T, J 〉$ be an PFSN. Then, $PFSWA (J_{e_{11}} \oplus J_{e}, J_{e_{12}} \oplus J_{e}, \dots, J_{e_{nm}} \oplus J_{e}) = PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) \oplus J_{e} .$

Proof: Consider $J_{e}$ and $J_{e_{ij}}$ be two PFSNs. Then by using operational laws defined under PFSNs in Definition 3.1 (1), we have:

$J_{e} \oplus J_{e_{ij}} = 〈 \sqrt{T^{2} + T_{ij}^{2} - T^{2} T_{ij}^{2}}, J J_{ij} 〉$ , therefore $\begin{matrix} PFSWA (J_{e_{11}} \oplus J_{e}, J_{e_{12}} \oplus J_{e}, \dots, J_{e_{nm}} \oplus J_{e}) = \oplus_{j = 1}^{m} γ_{j} (\oplus_{i = 1}^{n} Ω_{i} (J_{e_{ij}} \oplus J_{e})) \\ = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}} {(1 - T^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}} {(J)}^{Ω_{i}})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - (1 - T^{2}) \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, J \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}} 〉 \oplus T, J \\ = PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) \oplus J_{e} . \end{matrix}$

Which completes the proof.

3.2.4 (Homogeneity)

Prove that $PFSWA (α J_{e_{11}}, α J_{e_{12}}, \dots, α J_{e_{nm}}) = α$ $PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}})$ for any positive real number α.

Proof: Let $J_{e_{ij}}$ be a PFSN and α > 0, then by using Definition 3.1 (3), we have $α J_{e_{ij}} = 〈 \sqrt{1 - {(1 - T_{ij}^{2})}^{α}}, J^{α} 〉$ . So, $\begin{matrix} PFSWA (α J_{e_{11}}, α J_{e_{12}}, \dots, α J_{e_{nm}}) = 〈 \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{α Ω_{i}})}^{γ_{j}}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{α Ω_{i}})}^{γ_{j}} 〉 \\ = 〈 \sqrt{1 - {(\prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}})}^{α}}, {(\prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(J_{ij})}^{Ω_{i}})}^{γ_{j}})}^{α} 〉 \\ = α PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) . \end{matrix}$

Completes the proof.

Definition 3.3. Let $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ be a PFSN, Ω_i and γ_j are weight vector for expert’s and attributes respectively with given conditions Ω_i > 0, $\sum_{i = 1}^{n} Ω_{i} = 1$ , γ_j > 0, $\sum_{j = 1}^{m} γ_{j} = 1$ . Then PFSWG operator is defined as

PFSWG: Δⁿ → Δ defined as follows $PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = \otimes_{j = 1}^{m} {(\otimes_{i = 1}^{n} J_{e_{nm}}^{Ω_{i}})}^{γ_{j}} .$ (5)

Theorem 3.2. Let $J_{e_{ij}} = 〈 T_{ij}, J_{ij} 〉$ be a collection of PFSNs. Then, the aggregated values obtained by using Equation 5 is also a PFSN and $PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = 〈 \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉$ (6) where Ω_i and γ_j are weight vector for expert’s and attributes respectively with given conditions Ω_i > 0, $\sum_{i = 1}^{n} Ω_{i} = 1$ , γ_j > 0, $\sum_{j = 1}^{m} γ_{j} = 1$ .

Proof: By using the principle of mathematical induction PFSWG operator can be proved as follows:

For n = 1, we get Ω₁ = 1. Then, we have $\begin{matrix} PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{1 m}}) = \otimes_{j = 1}^{m} J_{e_{1 j}}^{γ_{j}} \\ PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = 〈 \prod_{j = 1}^{m} {(T_{1 j})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{m} {(1 - J_{1 j}^{2})}^{γ_{j}}} 〉 \\ = 〈 \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 . \end{matrix}$

For m = 1, we get γ₁ = 1. Then, we have $\begin{matrix} PFSWG (J_{e_{11}}, J_{e_{21}}, \dots, J_{e_{n 1}}) = \otimes_{i = 1}^{n} {(J_{e_{i 1}})}^{Ω_{i}} \\ = 〈 \prod_{i = 1}^{n} {(T_{i 1})}^{Ω_{i}}, \sqrt{1 - \prod_{i = 1}^{n} {(1 - J_{i 1}^{2})}^{Ω_{i}}} 〉 \\ = 〈 \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 . \end{matrix}$ which shows that Equation 6 satisfies for n = 1 and m = 1. Consider Equation 6 holds for m = β₁ + 1, n = β₂ and m = β₁, n = β₂ + 1, such as: $\begin{matrix} \otimes_{j = 1}^{β_{1} + 1} {(\otimes_{i = 1}^{β_{2}} {(J_{e_{ij}})}^{Ω_{i}})}^{γ_{j}} = 〈 \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 \\ \otimes_{j = 1}^{β_{1}} {(\otimes_{i = 1}^{β_{2}} {(J_{e_{ij}})}^{Ω_{i}})}^{γ_{j}} = 〈 \prod_{j = 1}^{β_{1}} {(\prod_{i = 1}^{β_{2} + 1} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{β_{1}} {(\prod_{i = 1}^{β_{2} + 1} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 \end{matrix}$

For m = β₁ + 1 and n = β₂ + 1, we have $\begin{matrix} \otimes_{j = 1}^{β_{1} + 1} {(\otimes_{i = 1}^{β_{2} + 1} {(J_{e_{ij}})}^{Ω_{i}})}^{γ_{j}} = \otimes_{j = 1}^{β_{1} + 1} {(\otimes_{i = 1}^{β_{2}} {(J_{e_{ij}})}^{Ω_{i}} \otimes {(J_{e_{(β_{2} + 1) j}})}^{Ω_{β_{2} + 1}})}^{γ_{j}} \\ = \otimes_{j = 1}^{β_{1} + 1} {(\otimes_{i = 1}^{β_{2}} {(J_{e_{ij}})}^{Ω_{i}})}^{γ_{j}} \otimes_{j = 1}^{β_{1} + 1} {({(J_{e_{(β_{2} + 1) j}})}^{Ω_{β_{2} + 1}})}^{γ_{j}} \\ = 〈 \begin{matrix} \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(T_{ij})}^{Ω_{i}})}^{γ_{j}} \otimes \prod_{j = 1}^{β_{1} + 1} {({(T_{(β_{2} + 1) j})}^{Ω_{β_{2} + 1}})}^{γ_{j}}, \\ \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2}} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} \otimes \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {({(1 - J_{(β_{2} + 1) j}^{2})}^{Ω_{β_{2} + 1}})}^{γ_{j}}} \end{matrix} 〉 \\ = 〈 \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2} + 1} {(T_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{β_{1} + 1} {(\prod_{i = 1}^{β_{2} + 1} {(1 - J_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 . \end{matrix}$

Hence, it is true for m = β₁ + 1 and n = β₂ + 1.

Remark 3.2.

If $T_{ij}^{2} + J_{ij}^{2} ⩽ 1$ and $T_{ij} + J_{ij} ⩽ 1$ both are holds. Then, the PFSWG operator was reduced to the IFSWG operator [46].

If $T_{ij}^{2} + J_{ij}^{2} ⩽ 1$ and $T_{ij} + J_{ij} ⩽ 1$ both are holds and a set of attributes contains only one parameter. Then, the PFSWG operator was reduced to the IFWG operator [47].

Example 3.2. Reconsider Example 3.1 $(J, L^{'}) = [\begin{matrix} (. 3, . 8) & (. 4, . 6) & (. 3, . 6) & (. 5, . 6) & (. 4, . 5) \\ (. 8, . 3) & (. 7, . 4) & (. 7, . 3) & (. 4, . 8) & (. 6, . 4) \\ (. 3.6) & (. 5, . 7) & (. 6, . 5) & (. 5, . 4) & (. 4, . 5) \end{matrix}]$

By using Equation 6, $\begin{matrix} PFSWA (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{44}}) = 〈 \prod_{j = 1}^{5} {(\prod_{i = 1}^{3} {(J_{ij})}^{Ω_{i}})}^{γ_{j}}, \sqrt{1 - \prod_{j = 1}^{5} {(\prod_{i = 1}^{3} {(1 - T_{ij}^{2})}^{Ω_{i}})}^{γ_{j}}} 〉 \\ 〈 \begin{matrix} (\begin{matrix} {(. 3)^{. 243} (. 8)^{. 514} (. 3)^{. 343}}^{. 25} {(. 4)^{. 243} (. 7)^{. 514} (. 5)^{. 343}}^{. 15} {(. 3)^{. 243} (. 7)^{. 514} (. 6)^{. 343}}^{. 2} \\ {(. 5)^{. 243} (. 4)^{. 514} (. 5)^{. 343}}^{. 1} {(. 4)^{. 243} (. 6)^{. 514} (. 4)^{. 343}}^{. 3} \end{matrix}), \\ \sqrt{1 - (\begin{matrix} {(. 36)^{. 243} (. 91)^{. 514} (. 64)^{. 343}}^{. 25} {(. 64)^{. 243} (. 84)^{. 514} (. 51)^{. 343}}^{. 15} {(. 64)^{. 243} (. 91)^{. 514} (. 75)^{. 343}}^{. 2} \\ {(. 64)^{. 243} (. 36)^{. 514} (. 84)^{. 343}}^{. 1} {(. 75)^{. 243} (. 84)^{. 514} (. 75)^{. 343}}^{. 3} \end{matrix})} \end{matrix} 〉 \\ = 〈 . 4679, . 5590 〉 . \end{matrix}$

We establish some properties for the collection of PFSNs based on Theorem 3.2, by utilizing the proposed PFSWG operator.

3.3 Properties of PFSWG operator

3.3.1 (Idempotency)

If $J_{e_{ij}} = J_{e} = 〈 T_{ij}, J_{ij} 〉 \forall$ i, j, then, $PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) = J_{e} .$

3.3.2 (Boundedness)

Let $J_{e_{ij}}$ be a collection of PFSNs and $J_{e_{ij}}^{-} = 〈 \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {T_{ij}}, \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {J_{ij}} 〉$ and $J_{e_{ij}}^{+} = 〈 \begin{matrix} \max \\ j \end{matrix} \begin{matrix} \max \\ i \end{matrix} {T_{ij}}, \begin{matrix} \min \\ j \end{matrix} \begin{matrix} \min \\ i \end{matrix} {J_{ij}} 〉$ , then $J_{e_{ij}}^{-}$ ⩽ $PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}})$ ⩽ $J_{e_{ij}}^{+}$ .

3.3.3 (Shift Invariance)

If $J_{e} = 〈 T, J 〉$ be an PFSN. Then, $PFSWG (J_{e_{11}} \oplus J_{e}, J_{e_{12}} \oplus J_{e}, \dots, J_{e_{nm}} \oplus J_{e}) = PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}}) \oplus J_{e}$ .

3.3.4 (Homogeneity)

Prove that $PFSWG (α J_{e_{11}}, α J_{e_{12}}, \dots, α J_{e_{nm}}) = α$ $PFSWG (J_{e_{11}}, J_{e_{12}}, \dots, J_{e_{nm}})$ for any positive real number α.

4 Decision-making approach based on proposed operators

An MCDM approach is presented here based on the proposed operators and described the numerical examples for showing their efficiency.

4.1 Proposed decision-making approach

Consider ℵ ={ ℵ ¹, ℵ ², ℵ ³, …, ℵ ^s } and $U = {δ_{1}, δ_{2}, δ_{3}, \dots, δ_{n}}$ are set of s alternatives n experts respectively, the weights of experts is given as Ω = (Ω₁, Ω₁, …, Ω_n) ^T and Ω_i > 0, $\sum_{i = 1}^{n} Ω_{i} = 1$ . Let $L^{'} = {e_{1} e_{2}, e_{3}, \dots e_{m}}$ be a set of attributes with weights γ = (γ₁, γ₂, γ₃, …, γ_m) ^T such as γ_j > 0, $\sum_{j = 1}^{m} γ_{j} = 1$ . Experts provide the decision matrix $D^{(z)} = {(J_{e_{ij}})}_{n \times m} = {(T_{ij}, J_{ij})}_{n \times m}$ for each alternative in term of PFSNs with the condition that 0 ⩽ $T_{ij}, J_{ij}$ ⩽ 1 and $T_{ij}^{2} + J_{ij}^{2}$ ⩽ 1 for all i, j are given in Table1. Construct the aggregated PFSNs $L_{k}$ , based on the expert’s preference values for each alternative by using PFSWA and PFSWG operators. Finally, utilize the ranking of the alternatives based on the score function by using Equation 1. The above technique is summarized in the following steps:

Table 1
Standards for assessing the superlative alternative waste electrical and electronic equipment; constraint of hazardous ingredients

Criteria Definition

$e_{1}$ Quality Reject rate, upgrading procedure, guarantee coverage and rights, and superiority assurance.

$e_{2}$ Distribution Percentage to achieve order, principal period, as well as order regularity

$e_{3}$ Services Accountability, organization record, and readiness.

$e_{4}$ Atmosphere Eco-design stipulations, compounds containing ozone-depleting

$e_{5}$ Corporate societal concern Worker freedoms and constitutional rights, investor constitutional rights, information revelation, and admiration for the rule.

	Criteria	Definition
$e_{1}$	Quality	Reject rate, upgrading procedure, guarantee coverage and rights, and superiority assurance.
$e_{2}$	Distribution	Percentage to achieve order, principal period, as well as order regularity
$e_{3}$	Services	Accountability, organization record, and readiness.
$e_{4}$	Atmosphere	Eco-design stipulations, compounds containing ozone-depleting
$e_{5}$	Corporate societal concern	Worker freedoms and constitutional rights, investor constitutional rights, information revelation, and admiration for the rule.

Step 1. Acquire decision matrix $D^{(z)} = {(T_{ij}, J_{ij})}_{n \times m}$ in form of PFSNs for alternatives in accordance with experts.

Step 2. Normalize the collective information decision matrix by using the normalization formula by converting the rating value of the cost type parameters into benefit type parameters. $H_{ij} = {\begin{matrix} J_{e_{ij}}^{c} = (J_{ij}, T_{ij}); cost type parameter \\ J_{e_{ij}} = (T_{ij}, J_{ij}); benefit type parameter \end{matrix}$

Step 3. Aggregate the PFSNs $J_{e_{ij}}$ for each alternative ℵ ={ ℵ ¹, ℵ ², ℵ ³, …, ℵ ^s } into a collective decision matrix $L_{k}$ by using the developed PFSWA or PFSWG operators.

Step 4. Compute the score values of $L_{k}$ for each alternative by using Equation 1.

Step 5. Choose the alternative with the maximum score value.

Step 6. Analyze the ranking.

The graphical representation of the above-presented algorithm can be seen in Fig.1.

Fig. 1

Flow chart of presented PFSWA and PFSWG operators.

4.2 Application of proposed technique for decision making

4.2.1 Case study

The supply chain is a series of activities to buy acquire substances, process them into immovable goods as well as produce them and deliver them to buyers, including correspondence from suppliers to consumers. In recent decades, such as forest fires, respect for states and worldwide territories, which are the major aspects of weather fluctuations, and global warming. Besides, environmental scarcity and air as well as water pollution also serious effects on plants, wildlife, and human life, including ischemic diseases vascular disease, lung cancer, chronic obstructive lung disease, stroke, guinea worm disease, cholera, tuberculosis, and typhoid fever. Interpretation of green supply chain objectives to minimize ecological deprivation and adjust air, water, and waste contamination by green activities. The fundamental idea behindhand the green conception is of course perfect environmental fortification. However, the green perception followed by the firms serves as “kill twosome birds with one bird stone” meanwhile the green supply chain can diminish ecological affluence as well as manufacturing cost so that encouraging financial development. Sustainability or green supply chain is correlated to the idea that supportable apparatuses need to be unified into a mostly well-ordered supply chain [50]. One involves procurement, assembling, strategy, manufacture, gathering, delivery, and scrap administration, provider. Supplier selection concerns include testing and measuring the performance of a group of suppliers so that they can be ranked and choose to be able to accelerate the excellence of their existing supply chain. Because different conflicting factors need to be considered in the analysis, multi-stability models and methods are frequently used to solve this problem. In recent times, the promotion of environmental awareness has helped the emergence of a new model of green supply. Therefore, green quality has been added to the issue of supplier selection.

All over the world, the consciousness of environmental protection, firms have been facing tremendous pressure the harmful effect of various contributors, including governments as well as consumers, on the surroundings. after all, the industrial sector should deliberate combining its operational practices with sustainable development the cost of entering services as well as manufacturing and the end-to-end supply chain is rising, possess a competitive advantage. In the past, some decades, global warming, climate change, waste material, and smog have motivated professionals all over the world to think over more environmental protection [51]. Rath [52] represented GSCM as a catalyst for the sustainable development of the organization’s rise. due to environmental problems, GSCM needs to have unalterable desires in underdeveloped nations to continue to grow. Furthermore, underdeveloped nations have currently become part of the green environment campaign. managers of the green supply chain must be integrated with the management supply chain, which included design, procurement as well as variety, mechanism, the redistribution of the final product to consumers, and the end of product life. From the extraction of resources to accumulation, use, and reproduction, to final use and disposal, every aspect of life will affect the supply chain environment. As a result, GSCM is defined as green purchase, green accumulation, green distribution as well as reverse logistics. GSCM purposes to eliminate or cut off the waste material within the supply chain (waste, emissions, chemical hazardous materials, solid waste)

Reverse logistics activity model performs an essential part in modifying green supply chain environmental, social as well as economic performance. However, including reverse logistics, the GSCM strategy is not prosperous. The feature of reverse logistics is the procedure of possessions allocation used for things like reuse, capture, or suitable refinement omitting its conventional destination. Product flow from the supplier to the end-user in the supply chain network. The efficiency of the flow is resolute by time delivery indicators for supply chain staff. This is a prominent index for the supply chain to ensure that you do distribute quickly and efficiently from the moment he/she areas a terminal order. Reverse logistics serves as more advanced compared to progressive logistics and needs a lot of attention. It will be more systematic as well as also part of the booking approach to any company in the company product assembling, gross revenue, warehousing, parceling along with service. Institutionally, reverse logistics is a region that is often confused and overlooked by any assembling company. However, this is no longer the case. Because companies without projected reverse logistics secret plans, the trend could be a gloomy picture in terms of financial performance and market share. The eco-management value should be carried out inside the entire customer order cycles, such as design, procurement, assembling as well as assembly, publicity, hauling along with delivery [53]. GSCM incorporates environmental protection suggestions within supply chain management to enhance the environment achieve sustainability by way of numerous green procedures, which included green procurement, green circulation and storing, green transportation with biofuels, green mechanisms, and end-of-life products [54]. Over the years, [55] does have some modifications in its classifications as well as nomenclature, since we have a definition of GSCM. The subsequent detailed list contains some words that characterize this definition. Sustainable management of the supply network [56]. Sustainability of supply and demand across corporate social responsibility networks [57], Environmental Supply Chain Management [58], Green procurement [59], Green logistics [60].

Srivastava [61] gave a wide-ranging clarification of GSCM, GSCM is environmental intellection in supply chain management, which included product design as well as materials procurement, selection, manufacturing process, product parceling, and scrap management products for customers. For innovativeness, GSCM has many compensations. The allegory that greening will cause sales to fall and higher operating costs have disappeared because many companies have now realized that they will not able to satisfy the client’s desire to incorporate environmental protection plans into their needs supply chain and turn it into higher profits. There is a connection between a better environment many companies develop sustainability and financial incentives. Business got insights into the supply chain and discovering areas that can change the way of working to increase income. Green logistics can help reduce various emissions such as carbon dioxide and CO. Consumption containing non-fossil sources of energy (such as electrical vehicles) alleviates smog that has effects on our surroundings trying to inhale is good for humanoid health. Several procedures of fossil fuels have been devastation the environment because of affluence, for instance, for marine life, air travel will also impact contamination because of diesel engine consumption. In the literature analysis, we investigated the factors for choosing green suppliers consistent with different researchers. In this article, the selection standards for green suppliers have been considered are given in the following Table1.

4.3 Numerical example

Let {ℵ⁽¹⁾, ℵ⁽²⁾, ℵ⁽³⁾, ℵ⁽⁴⁾, ℵ⁽⁵⁾} be a set of alternatives and $e_{1}$ =Quality, $e_{2}$ =Distribution, $e_{3}$ =Services, $e_{4}$ =Atmosphere, and $e_{5}$ =Corporate societal concern are considered attributes with weights (. 25, . 15, . 20, . 10, . 30) ^T. Consider {u₁, u₂, u₃} be a set of experts to evaluate the best alternative having weight vector (. 243, . 514, . 343) ^T. The team of decision-makers evaluates the alternatives according to the above-mentioned parameters. Every decision-maker evaluates the ratings for alternatives in PFSNs form under the considered parameters. The developed method to find the best alternative is given in 4.1.

4.3.1 By using PFSWA operator

Step 1. The decision-makers will evaluate the condition in the case of PFSNs, there are just five alternatives; parameters, and a summary of their scores given in Table2 –4.

Table 2
PFS Decision Matrix for u₁

$e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $e_{5}$

ℵ⁽¹⁾ (. 3, . 8) (. 4, . 6) (. 3, . 6) (. 5, . 6) (. 4, . 5)

ℵ⁽²⁾ (. 7, . 3) (. 6, . 5) (. 3, . 6) (. 7, . 4) (. 6, . 4)

ℵ⁽³⁾ (. 7, . 4) (. 7, . 2) (. 9, . 2) (. 5, . 4) (. 4, . 5)

ℵ⁽⁴⁾ (. 8, . 3) (. 5, . 5) (. 5, . 4) (. 5, . 6) (. 4, . 7)

ℵ⁽⁵⁾ (. 6, . 3) (. 3, . 8) (. 5, . 6) (. 4, . 2) (. 5, . 5)

	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$
ℵ⁽¹⁾	(. 3, . 8)	(. 4, . 6)	(. 3, . 6)	(. 5, . 6)	(. 4, . 5)
ℵ⁽²⁾	(. 7, . 3)	(. 6, . 5)	(. 3, . 6)	(. 7, . 4)	(. 6, . 4)
ℵ⁽³⁾	(. 7, . 4)	(. 7, . 2)	(. 9, . 2)	(. 5, . 4)	(. 4, . 5)
ℵ⁽⁴⁾	(. 8, . 3)	(. 5, . 5)	(. 5, . 4)	(. 5, . 6)	(. 4, . 7)
ℵ⁽⁵⁾	(. 6, . 3)	(. 3, . 8)	(. 5, . 6)	(. 4, . 2)	(. 5, . 5)

Table 3

PFS Decision Matrix for u₂

	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$
ℵ⁽¹⁾	(. 8, . 3)	(. 7, . 4)	(. 7, . 3)	(. 4, . 8)	(. 6, . 4)
ℵ⁽²⁾	(. 8, . 5)	(. 6, . 5)	(. 6, . 3)	(. 4, . 8)	(. 7, . 5)
ℵ⁽³⁾	(. 5, . 4)	(. 4, . 3)	(. 7, . 5)	(. 7, . 5)	(. 6, . 4)
ℵ⁽⁴⁾	(. 3, . 4)	(. 4, . 6)	(. 7, . 3)	(. 6, . 5)	(. 7, . 4)
ℵ⁽⁵⁾	(. 5, . 5)	(. 5, . 5)	(. 5, . 1)	(. 5, . 8)	(. 4, . 8)

Table 4

PFS Decision Matrix for u₃

	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$
ℵ⁽¹⁾	(. 3, . 6)	(. 5, . 7)	(. 6, . 5)	(. 5, . 4)	(. 4, . 5)
ℵ⁽²⁾	(. 3, . 5)	(. 5, . 3)	(. 7, . 2)	(. 8, . 4)	(. 3, . 7)
ℵ⁽³⁾	(. 6, . 2)	(. 2, . 5)	(. 3, . 5)	(. 6, . 4)	(. 5, . 5)
ℵ⁽⁴⁾	(. 6, . 5)	(. 6, . 2)	(. 6, . 1)	(. 7, . 4)	(. 4, . 5)
ℵ⁽⁵⁾	(. 4, . 7)	(. 3, . 6)	(. 6, . 7)	(. 3, . 8)	(. 8, . 3)

Step 2. No need for normalization because all parameters are of the same type (benefit type).

Step 3. The opinion of the decision-makers by using Equation 4, are summarized and we get:

$L_{1} = 〈 . 6009, . 4342 〉$ , $L_{2} = 〈 . 6499, . 4078 〉$ , $L_{3} = 〈 . 6179, . 3506 〉$ , $L_{4} = 〈 . 6076, . 3527 〉$ , and $L_{5} = 〈 . 5493, . 4345 〉$ .

Step 4. By using Equation 1, compute the score values of $L_{k}$ for each alternative. $\begin{matrix} S (L_{1}) = . 1667, S (L_{2}) = . 2421, S (L_{3}) \\ = . 2673, S (L_{4}) = . 2549, and S (L_{5}) = . 1148 . \end{matrix}$

Step 5. So, by utilizing the score values ℵ⁽³⁾ be the best alternative.

Step 6. Therefore, the ranking of the alternatives is as follows $S (L_{3}) > S (L_{4}) > S (L_{2}) > S (L_{1}) > S (L_{5})$ . So, ℵ⁽³⁾ > ℵ ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ ⁽¹⁾ > ℵ ⁽⁵⁾, hence, the alternative ℵ⁽³⁾ is the most suitable alternative.

4.3.2 By using PFSWG operator

Step 1. Similar to 4.3.1.

Step 2. All parameters are of the same type (benefit type), so no need to normalize them.

Step 3. Decision-makers’ opinions summarized as follows by using Equation 6.

$L_{1} = 〈 . 4679, . 5590 〉$ , $L_{2} = 〈 . 5157, . 5289 〉$ , $L_{3} = 〈 . 4892, . 4387 〉$ , $L_{4} = 〈 . 4910, . 4751 〉$ , and $L_{5} = 〈 . 4440, . 6407 〉$ .

Step 4. By using Equation 1, compute the score values of $L_{k}$ for each alternative.

$S (L_{1}) = - 0.0911$ , $S (L_{2}) = - 0.0132$ , $S (L_{3}) = 0.0505$ , $S (L_{4}) = 0.0159$ , and $S (L_{5}) = - 0.1967$ .

Step 5. So, by utilizing the score values ℵ⁽³⁾ be the best alternative.

Step 6. Therefore, the ranking of the alternatives is as follows $S (L_{3}) > S (L_{4}) > S (L_{2}) > S (L_{1}) > S (L_{5})$ . So, ℵ⁽³⁾ > ℵ ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ ⁽¹⁾ > ℵ ⁽⁵⁾. Hence, the alternative ℵ⁽³⁾ is the most suitable alternative.

5 Discussion and comparative analysis

In the following section, we will debate the effectiveness, naivety, flexibility, and advantages of the planned method. We also conducted an ephemeral comparative analysis of the following: the proposed method and some existing methods.

5.1 Advantages and flexibility of proposed approach

The recommended technique is effective and applicable to all forms of input data. Here, we introduce a novel algorithm based on PFSSs, by using the PFSWA or PFSWG operator. Compared with the existing techniques, our proposed technique is more effective and can provide the best results in MCDM problems. The recommended algorithm is simple and easy to understand, can deepen the understanding, and will apply to many types of choices and measures. The developed algorithm is flexible and easy to change to adapt to different situations, inputs, and outputs. There are subtle differences between the rankings of the suggested methods because different techniques have different ranking strategies, so they can be affordable according to their considerations.

5.2 Superiority of the proposed method

Through this research and comparative analysis, we have concluded that the results obtained by the proposed method are more common than existing techniques. However, in the decision-making process, compared with the existing decision-making methods, it contains more information to deal with the uncertainty in the data. Moreover, many hybrid structures of FS have become special cases of PFSSs, and some suitable conditions have been added. Among them, the information related to the object can be expressed more accurately and empirically see Table5, so it is a convenient tool to combine inaccurate and uncertain information in the decision-making process. Therefore, our proposed method is effective, flexible, simple, and superior to other hybrid structures of fuzzy sets.

Table 5
Comparison of PFSSs with some existing theories

Set Truth information Falsity Loss of information Parametrization Advantages Limitations

Zadeh [1] FS ✓ × ✓ × Deals uncertainty by using fuzzy interval Cannot deal with NMD

Zhang et al. [48] IFS ✓ ✓ ✓ × Deals uncertainty by using MD and NMD Cannot deal with problems that satisfies 1 < MD+ NMD ⩽ 0

Xu et al. [49] IFS ✓ ✓ × × Deals uncertainty by using MD and NMD Cannot deal with problems that satisfies 1 < MD+ NMD ⩽ 0

Yager [6, 7] PFS ✓ ✓ × × Deals more uncertainty comparative to IFS Cannot deal with problems that satisfy 1 < MD²+ NMD² ⩽ 0

Proposed approach PFSSs ✓ ✓ × ✓ Deals more uncertainty comparative to IFS Cannot deal with problems that satisfy 1 < MD²+ NMD² ⩽ 0

	Set	Truth information	Falsity	Loss of information	Parametrization	Advantages	Limitations
Zadeh [1]	FS	✓	×	✓	×	Deals uncertainty by using fuzzy interval	Cannot deal with NMD
Zhang et al. [48]	IFS	✓	✓	✓	×	Deals uncertainty by using MD and NMD	Cannot deal with problems that satisfies 1 < MD+ NMD ⩽ 0
Xu et al. [49]	IFS	✓	✓	×	×	Deals uncertainty by using MD and NMD	Cannot deal with problems that satisfies 1 < MD+ NMD ⩽ 0
Yager [6, 7]	PFS	✓	✓	×	×	Deals more uncertainty comparative to IFS	Cannot deal with problems that satisfy 1 < MD²+ NMD² ⩽ 0
Proposed approach	PFSSs	✓	✓	×	✓	Deals more uncertainty comparative to IFS	Cannot deal with problems that satisfy 1 < MD²+ NMD² ⩽ 0

5.3 Comparative analysis

Through the prevailing exploration as well as comparative studies given above, it can be concluded that the outcomes through the suggested approach overlap with the accessible techniques, so the established PFSS methodology is conservative. However, associated with accessible decision-making techniques, the main benefit of the planned approach is that it accommodates more information to deal with uncertainty in the data. Among them, the information associated with the objects could be expressed more accurately and objectively, it is also a useful tool for solving imprecise and incomprehensible information in the decision-making process. Besides, it’s noted from the comparative analysis that the calculation process of the planned approach isn’t the same as the available techniques. In this article, we propose a new algorithm based on PFSSs by using the developed PFSWA and PFSWG operators. Next, the proposed algorithm is applied for green supplier selection in GSCM. The ranks of all alternatives using the existing operators gave the same final decision, that is, ℵ⁽³⁾ is the best choice for GSCM.

The results show that the algorithm is effective and practical, all rankings are also calculated by applying existing methods. The proposed method is also compared with other existing methods, such as Zadeh [1], FS only deals the uncertain problems by using MD, while our proposed method deals with the uncertainty by utilizing MD and NMD. Zhang et al. [48] and Xu et al. [49] presented IFS handles the vagueness by using MD and NMD. But these theories cannot deal with the situation when their sum exceeds one, on the other hand, our presented technique can accommodate more vagueness comparative to the above-mentioned theories by upgrading the MD+ NMD ⩽ 1 to MD²+ NMD² ⩽ 1. Yager [6, 7] PFS accommodates more uncertainty comparative to IFS, but it cannot deal with the parametrization. But our proposed method can easily solve these obstacles and provide more effective results. A comparison can be seen in the above-listed Table6, which is the final ranking of the top 5 alternatives for GSCM. It can be observed that the best choice made by the proposed method is compared with the established method of expressing itself, thus proving the reliability and effectiveness of the proposed method. Therefore, the technique we developed is more efficient and can provide better results for decision-makers through a variety of information.

Table 6
Comparative analysis with existing operators

Method Ranking order Best alternative

Zhang [48] PFSWA ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽²⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Zhang [48] PFSWG ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽²⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Garg [14] PFEWA ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Ma and Xu [26] SPFWA ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽⁵⁾> ℵ⁽²⁾ ℵ⁽³⁾

Garg [45] PFEWG ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Proposed PFSWA operator ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Proposed PFSWG operator ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾ ℵ⁽³⁾

Method	Ranking order	Best alternative
Zhang [48] PFSWA	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽²⁾> ℵ⁽⁵⁾	ℵ⁽³⁾
Zhang [48] PFSWG	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽²⁾> ℵ⁽⁵⁾	ℵ⁽³⁾
Garg [14] PFEWA	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾	ℵ⁽³⁾
Ma and Xu [26] SPFWA	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽¹⁾ > ℵ⁽⁵⁾> ℵ⁽²⁾	ℵ⁽³⁾
Garg [45] PFEWG	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾	ℵ⁽³⁾
Proposed PFSWA operator	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾	ℵ⁽³⁾
Proposed PFSWG operator	ℵ⁽³⁾ > ℵ⁽⁴⁾ > ℵ ⁽²⁾ > ℵ⁽¹⁾> ℵ⁽⁵⁾	ℵ⁽³⁾

6 Conclusion

In this article, two novel aggregation operators have been proposed such as PFSWA and PFSWG operators in the PFSSs environment. For it, the algebraic structure of the PFSNs is proposed and in accordance with its operational laws, the aggregation operators are developed. We also studied some of their essential possessions in detail. A decision-making method based on PFSWA and PFSWG operators is presented, in which preferences are adopted associated with different alternatives are considered in terms of PFSNs. The demonstration of the presented aggregation operators described by a numerical illustration for green supplier selection in GSCM. Furthermore, A comparative analysis is presented to authenticate the strength and demonstration of the planned method. Based on the consequences attained, it is determined that the recommended technique showed higher solidity and practicality for decision-makers in the decision-making process. In the future, the correlation coefficient, the TOPSIS method based on correlation coefficient under PFSS can be presented. Future research will concentration on presenting numerous other operators under the PFSS environment to solve decision-making issues. Many other structures such as topological, algebraic, ordered structures, etc. can be developed and discussed under-considered environment. The strategies outlined in this research can be extended to specific mixed structures of fuzzy sets. The suggested idea can be applied to quite a lot of issues in real life, including the medical profession, robotics, artificial intelligence, pattern recognition, economics, etc. We hope this article will open new ways for researchers in this field.

Footnotes

Acknowledgments

This research is partially supported by a grant of National Natural Science Foundation of China (11971384).

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