An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

1  Introduction

Green supply chain management (GSCM) is a tactical strategy that implements sustainable practices throughout the supply chain. Due to increasing concerns about the sustainability of the environment, the concept of GSCM acquired severe recognition in the past few years. One of the primary objectives of GSCM is to minimize a supply chain’s emissions and adverse environmental effects. As the world’s economic system grows, more significant resources are used. Still, resource depletion and degradation of the environment have increased indigenous, carrying more attention from researchers, businesses, the general public, and authorities globally [1]. As a consequence, approaches to sustainable development have been established, and many kinds of reinforced environmental regulations, laws, and policies have been placed that combine economic expansion with a sustainable environment [2]. GSCM is being demonstrated to be a viable approach for addressing the problem mentioned earlier. This type of administrative strategy infuses environmentally conscious thinking through the operational phases of the supply chain, including design, which is buying goods manufacturing, transportation, preservation, advertisement, and transaction [3]. The supplier is upstream and in the green supply chain, setting the basis for inter- and intra-organizational efficiency [4]. Consequently, selecting the most capable green supplier is essential for small businesses to contribute to environmental preservation while retaining their fundamental competitive advantage in supply chain management.

Multi-criteria group decision-making is deemed the most suitable method for determining the optimal alternative based on criteria or attributes. However, decisions in real-life situations may involve vague or unclear goals and limitations. In DM, dealing with uncertainties and ambiguities can be a challenge. To address this issue, fuzzy sets (FS) were introduced by Zadeh [5]. Later, interval-valued fuzzy sets (IVFS) were presented by Turksen [6] to handle situations where experts consider membership degrees (MD) in intervals during the DM process. However, FS and IVFS are limited because they do not provide information about alternative non-membership degrees (NMD). Atanassov [7] created the intuitionistic fuzzy set (IFS) to address these limitations. Wang et al. [8] added several operations and AOs for IFS, including the Einstein product and sum. Furthermore, the cubic intuitionistic fuzzy set (CIFS) was proposed by Garg et al. [9] to develop the concept of IFS further. Atanassov [10] extended the notion of IFS to the interval-valued intuitionistic fuzzy set (IVIFS). Torağay et al. [11] utilized the Delphi method based on a 2-Tuple fuzzy Linguistic representation and the TOPSIS model for faculty evaluation. To address these limitations, Yager [12] proposed the Pythagorean fuzzy set (PFS), which resolves issues with the linear relationship between MD and NMD and cases where κ+δ≤1 to κ2+δ2≤1. Rahman et al. [13] introduced an Einstein-weighted geometric operator for PFS and demonstrated a technique for multi-attribute group decision-making (MAGDM) using this operator. Giri et al. [14] developed the decision-making trial and evaluation laboratory method for supplier selection in sustainable supply chain management. Zhang et al. [15] utilized the Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) method to handle multi-criteria decision-making (MCDM) problems using PFS. Wei et al. [16] introduced the Pythagorean fuzzy power AOs and developed a DM technique for multi-attribute decision-making (MADM). Operational laws for interaction with Pythagorean fuzzy numbers were demonstrated by Wang et al. [17], who established power Bonferroni mean operators. Chaurasiya et al. [18] proposed a hybrid MCDM model for PFS. El-Morsy [19] proposed Pythagorean fuzzy numbers (PFN) in the rate of risked return, portfolio risk amount, and expected return rate. Zhang [20] proposed similarity measures for PFS and developed an MCGDM approach to address DM challenges. Anusha et al. [21] developed an interactive group decision-making approach under q-rung probabilistic dual hesitant fuzzy sets. Khan et al. [22] established a MADM model using an Archimedean aggregation operator in a T-spherical fuzzy environment. Peng et al. [23] introduced AOs for interval-valued Pythagorean fuzzy sets (IVPFS) and revealed a DM method using their approach. Yu et al. [24] used modified TOPSIS in an interval-valued Pythagorean fuzzy setting to formulate a distinctive group decision-making sustainable supplier selection strategy. Rahman et al. [25] extended the weighted geometric AOs for IVPFS and provided a DM structure based on their proposed operator. A PFS is a promising approach for handling uncertainties and ambiguities in DM processes. Mu et al. [26] developed the interval-valued Pythagorean fuzzy power Maclaurin symmetric mean operators to resolve MAGDM problems.

The earlier theories have various applications but may be ineffective in parametric chemistry. Molodtsov [27] developed the theory of soft sets (SS) to address uncertainty and misperceptions. Fu et al. [28] then developed some essential operations for interval-valued soft sets and demonstrated a technique for DM. Khalil et al. [29] established a novel DM methodology based on FSS. Other researchers have also proposed different variations of FSS, such as intuitionistic fuzzy soft sets (IFSS) [30]. Arora et al. [31] proposed a method that uses AOs for IFSS. Jiang et al. [32] developed the interval-valued IFSS (IVIFSS) with basic operations and their possessions. Zulqarnain et al. [33] proposed the TOPSIS technique for multiple attribute decision-making (MADM) problems in IVIFSS using the correlation coefficient (CC). Peng et al. [34] combined PFS and SS to create PFSS with some basic operations. Zulqarnain et al. [35,36] developed AOs and interaction AOs for PFSS and used their developed operators in green supply chain management (GSCM). Kirişci et al. [37] proposed a new DM approach that uses PFSS. Zulqarnain et al. [38,39] prolonged the Einstein-ordered weighted AOs for PFSS. Siddique et al. [40] developed a novel DM method based on a score matrix for PFSS to solve daily life hurdles. Zulqarnain et al. [41] extended the AOs for IVPFSS and presented a DM technique to solve the MAGDM problem. Selecting sustainable suppliers is vital to managing a sustainable supply chain, ensuring all stakeholders adhere to environmental, social, and ethical standards. Qiu et al. [42] chose the best private partner using a single-valued neutrosophic set. Chatterjee et al. [43] suggested that to create a healthier environment, it is important for practitioners to identify the key criteria necessary for implementing sustainable policies, particularly in the rapidly growing electronics sector. Stevic et al. [44] utilized the DM trial and evaluation laboratory method based on rough numbers to select suppliers for a construction company. Stojic et al. [45] established a new MCDM model to select suppliers for a polyvinyl chloride carpentry company. Stevic et al. [46] introduced a new method for measuring alternatives and ranking them according to compromise solutions for sustainable supplier selection in the healthcare industry.

1.1 Motivation

The IVPFSS is a type of structure that combines two different systems, IVPFS and SS. These systems are used to handle information that is both certain and uncertain. When making decisions, it is important to be able to combine information from different sources to get a comprehensive understanding of the situation. However, the current tools for IVPFSS are not very effective when dealing with imprecise or uncertain data. This can make it difficult to determine the best course of action. Researchers have been exploring using interaction AOs for IVPFSS to address this issue. These AOs, such as IVPFSIWA and IVPFSIWG, can be compared to other fusion extensions of FS. The models that determine the overall NMD are based on the compatible NMD interval values. However, this approach can lead to adverse outcomes and make it difficult to assess alternatives accurately. To improve the effectiveness of these models, researchers are exploring the use of interval-valued Pythagorean fuzzy soft numbers (IVPFSNs) and their interactions. The current AOs used to check data concepts are ineffective and can lead to imprecise outcomes. By incorporating IVPFSNs and their interactions, researchers hope to develop better tools for decision-making that can handle uncertain and imprecise data more effectively. Let U = {u1,u2} be a set of two experts with weights ωi = (.6,.4)T and e1,e2 be the selected factors with their compatible parameters with weights νj = (.4,.6)T. Let I be an alternate, then preferences of experts can be precise as I = [([.7,.8],[.0,.0])([.2,.6],[.3,.5])([.3,.6],[.5,.7])([.5,.7],[.1,.6])] longsighted the parameters of the planned aspects in terms of IVPFSNs. Then, we conquered the collected value by the IVPFSWA [41] operator is ⟨[.8333,.9487],[.0,.0]⟩. Similarly, we engaged the IVPFSWG [41] operator and achieved a collected value ⟨[.3584,.6505],[.0,.0]⟩. This shows that there is no effect on the collective consequence. δij. As δij = δ11 = [0.0,0.0], δ12 = [0.5,0.7], δ21 = [0.3,0.5], and δ22 = [0.1,0.6], which is arbitrary. An improved approach to analyzing complex and perplexing data is important for investigators. The current methods used to determine the significance of certain AOs are not always reliable and may not provide enough information to make decisions about alternatives. As a result, researchers are exploring the use of IVPFSNs and their interactions to improve the accuracy of these methods. However, the existing methodologies for analyzing IVPFSNs are ineffective in scrutinizing data or providing clear implications for decision-making. To address this issue, researchers have developed new interaction AOs, such as IVPFSIWA and IVPFSIWG, to evaluate better the significance of different factors in the decision-making process. The development of these new interaction AOs is an important step forward in the field of IVPFSS, as it allows investigators to analyze complex data and make better decisions based on the available information.

1.2 Significant Contribution

The interactional AOs for IVPFSS are aimed at addressing the limitations of existing AOs. The traditional AOs do not consider the interaction between different attributes or experts, which can lead to biased or inaccurate DM. The interactional AOs allow for viewing interaction between different attributes or experts, which can lead to more comprehensive and accurate DM. These operators consider the weights assigned to different attributes or experts and their agreement and disagreement levels. Therefore, by introducing interactional AOs for IVPFSS, we can improve the accuracy and reliability of DM in various fields such as engineering, finance, medical diagnosis, sustainable supplier selection, and other areas where uncertainty and ambiguity are common. Sustainable supplier selection minimizes negative environmental and societal impacts while maximizing positive outcomes, such as increased efficiency and profitability. This is accomplished by integrating sustainability criteria into the supplier evaluation process. Evaluating potential suppliers involves considering quality, price, delivery time, and reliability factors to determine the best fit for a particular business or organization. The selection and assessment of suppliers are critical for ensuring the success of supply chain management and achieving the desired results. The literature highlights the need for an MCGDM method for sustainable supplier selection that considers environmental concerns and expert opinions. To address these limitations, we propose a strategy that uses IVPFSS information. IVPFSS is a hybrid intellectual structure that incorporates both IVPFSS and fuzzy soft information to improve the sorting process and help decision-makers deal with complex and incomplete information. Based on research findings, IVPFSS plays a crucial role in DM by integrating multiple sources into a single value.

•   However, when dealing with MD and NMD, existing AOs may not yield accurate results, making an improved sorting process essential. Based on research findings, IVPFSS plays a crucial role in DM by consolidating multiple sources into a single value. Although combining IVPFS and SS is a widely recognized concept, integration with the IVPFSS background has been lacking. To address this, we propose extending IVPFSS by introducing interaction AOs with their fundamental properties. Our investigation aims to expand the concept of IVPFSS by introducing interaction operators based on rough data, with the following objectives:

•   Extend the PFSS to IVPFSS and develop interaction AOs to effectively handle multiple attributes in DM processes.

•   While interaction AOs for IVPFSS are a reputable and effective method for evaluating attributes in the DM process, they may not always provide accurate results. To address this, we present new operational laws for IVPFSNs.

•   Introduce the IVPFSIWA and IVPFSIWG operators and their essential properties, utilizing the established operational laws.

•   Design a new method using the established operators to address MCGDM problems in the IVPFSS environment.

•   Sustainable supplier selection is crucial for sustainable supply chain management and a significant step in professional development. It aims to ensure that all aspects meet concrete conditions.

•   Compare the newly proposed MCGDM technique and existing methods to evaluate the effectiveness and superiority of the new approach.

This paper’s organization content is divided into five sections. Section 2 overviews some crucial concepts that will facilitate the subsequent study. Section 3 introduces new operational laws for IVPFSNs and presents the IVPFSIWA and IVPFSIWG operators based on these laws. These operators are developed to improve the boundaries of communicating indefinite and unreliable data in real-life surroundings. Section 4 proposes an MCGDM method using the newly developed interaction AOs. This section describes how the proposed method can be used to select sustainable suppliers effectively. To demonstrate the practicality of the approach, we provide a numerical example of sustainable supplier selection in this section. Finally, Section 5 compares the proposed technique with existing approaches, showing its efficiency and effectiveness. The comparison will help to understand the advantages of the proposed method over other existing methods.

2  Preliminaries

This section lays the foundation for the subsequent work by providing essential definitions.

2.1 Definition [5]

A is a fuzzy set over a universe of discourse U, where U is a collection of objects, and A is a subset of U.

A={(t,κA(t))|t∈U}

where κA(t)∈[0,1] is a membership grade function.

2.2 Definition [6]

A is an interval-valued fuzzy set over a universe of discourse U, where U is a collection of objects, and A is a subset of U.

A={(t,[κAl(t),κAu(t)]) | t∈U}

where κAl(t),κAu(t)∈[0,1] and expressed the lower and upper values of the membership grade.

2.3 Definition [10]

A is an interval-valued intuitionistic fuzzy set over a universe of discourse U, where U is a collection of objects, and A is a subset of U.

A={(t,([κAl(t),κAu(t)],[δAl(t),δAu(t)]))|t∈U}

where [κAl(t),κAu(t)] and [δAl(x),δAu(x)] are intervals for MD and NMD, respectively, whereas [κAl(t),κAu(t)]and[δAl(t),δAu(t)]⊆[0,1] and

0≤κAl(t),κAu(t),δAl(t),δAu(t)≤1

Furthermore, 0≤κAu(x)+δAu(x)≤1.

2.4 Definition [23]

A is an interval-valued Pythagorean fuzzy set (IVPFS) over a universe of discourse U, where U is a collection of objects, and A is a subset of U.

A={(x,([κAl(t),κAu(t)],[δAl(t),δAu(t)]))|t∈U}

where [κAl(t),κAu(t)] and [δAl(t),δAu(t)] represents the MD and NMD intervals, respectively. Also, κAl(t),κAu(t), δAl(t),δAu(t) ∈[0,1] and satisfied the subsequent condition 0≤(κAu(t))2+(δAu(t))2≤1.

2.5 Definition [27]

A soft set is defined as a pair (Ω,A), where U is a universal set, and ζ is a set of parameters such as A⊆ζ. Its mapping can be defined as:

Ω:A→𝒫(U)

Also, it can be defined as follows:

(Ω,A)={Ω(t)∈𝒫(U):t∈ζ,Ω(t)=∅ift∉A}

2.6 Definition [32]

An interval-valued intuitionistic fuzzy soft set is defined as a pair (Ω,A), where U is a universal set, and ζ is a set of parameters, where Ω:ζ→IKU is a mapping and IKU is known as a collection of all intuitionistic fuzzy soft subsets of the universal set U, and A⊂ζ.

(Ω,A)={t,([κAl(t),κAu(t)],[δAl(t),δAu(t)])|t∈A}

where [κAl(t),κAu(t)]and[δAl(t),δAu(t)] are intervals for MD and NMD, respectively, with 0≤κAu(t)+δAu(t)≤1.

2.7 Definition [41]

An interval-valued Pythagorean fuzzy soft set is defined as a pair (Ω,A), where U is a universal set, and ζ is a set of parameters, where Ω:ζ→℘KU is a mapping and ℘KU is identified as a collection of all Pythagorean fuzzy soft subsets of universal set U and A⊂ζ.

(Ω,A)={x,([κAl(t),κAu(t)],[δAl(t),δAu(t)])|t∈A}

where [κAl(t),κAu(t)],[δAl(t),δAu(t)] represents the MD and NMD intervals, respectively.

Also, κAl(t),κAu(t), δAl(t),δAu(t) ∈[0,1] and satisfied the subsequent condition 0≤(κAu(t))2+(δAu(t))2≤1 and A⊂ζ.

Simply IVPFSN can be expressed as ℳ=([κl,κu],[δl,δu]). The score and accuracy functions can be defined as follows to compute the ranking of the alternatives:

S(ℳe)=(κl)2+(κu)2−(δl)2−(δu)22

A(ℳe)=(κl)2+(κu)2+(δl)2+(δu)22

2.8 Definition [41]

Let ℳe=([κl,κu],[δl,δu]), ℳe11=([κ11l,κ11u],[δ11l,δ11u]), and ℳe12=([κ12l,κ12u],[δ12l,δ12u]) be three IVPFSNs and β be a positive real number, then the algebraic operational norms are defined as follows:

1.    ℳe11⊕ℳe12=([κ11l2+κ12l2−κ11l2κ12l2,κ11u2+κ12u2−κ11u2κ12u2],[δ11lδ12l,δ11uδ12u])

2.    ℳe11⊗ℳe12=([κ11lκ12l,κ11uκ12u],[δ11l2+δ12l2−δ11l2δ12l2,δ11u2+δ12u2−δ11u2δ11u2])

3.    βℳe=([1−(1−κl2)β,1−(1−κu2)β],[δlβ,δuβ])=(1−(1−[κl,κu]2)β,[δlβ,δuβ])

4.    ℳeβ=([κlβ,κuβ],[1−(1−δl2)β,1−(1−δu2)β])=([κlβ,κuβ],1−(1−[δl,δu]2)β).

For the collection of IVPFSNs ℳeij, where ωi and νj are weight vectors for experts and attributes correspondingly, with assumed circumstances ωi>0,∑i=1nωi=1;νj>0,∑j=1mνj=1. Zulqarnain et al. [41] planned AOs for IVPFSNs as follows:

IVPFSWA(ℳe11,ℳe12,………,ℳenm)=(1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=1m(∏i=1n([δijl,δiju])ωi)νj)

IVPFSWG (ℳe11,ℳe12,………,ℳenm)=(∏j=1m(∏i=1n([κijl,κiju])ωi)νj,1−∏j=1m(∏i=1n(1−[δijl,δiju]2)ωi)νj)

It is noticed that the IVPFSWA and IVPFSWG operators provide some unattractive outcomes in certain conditions, as mentioned in 1.1. To overwhelm such consequences, we present the interaction AOs for IVPFSNs.

3  Interaction Aggregation Operators for Interval Valued Pythagorean Fuzzy Soft Sets

This section will describe the concept of interaction operational laws in the context of IVPFSNs. Utilizing these operational laws, we will introduce the IVPFSIWA and IVPFSIWG operators.

3.1 Interaction Operational Laws for IVPFSNs

Let ℳe=([κl,κu],[δl,δu]), ℳe11=([κ11l,κ11u],[δ11l,δ11u]), and ℳe12=([κ12l,κ12u],[δ12l,δ12u]) be three IVPFSNs, and β be a positive real number. Then the algebraic operational laws for IVPFSNs are given as follows:

1.    ℳe11⊕ℳe12=([κ11l2+κ12l2−κ11l2κ12l2,κ11u2+κ12u2−κ11u2κ12u2],[δ11l2+δ12l2−δ11l2δ12l2−κ11l2δ12l2−δ11l2κ12l2,δ11u2+δ12u2−δ11u2δ12u2−κ11u2δ12u2−δ11u2κ12u2])

2.    ℳe11⊗ℳe12=([k11l2+k12l2−k11l2k12l2−κ11l2δ12l2−δ11l2κ12l2,k11u2+k12u2−k11u2k12u2−κ11u2δ12u2−δ11u2κ12u2],[δ11l2+δ12l2−δ11l2δ12l2,δ11u2+δ12u2−δ11u2δ11u2])

3.    βℳe=(1−(1−[κl,κu]2)β,(1−[κl,κu]2)β−[1−([κl,κu]2+[δl,δu]2)]β)

4.    ℳeβ=((1−[δl,δu]2)β−[1−([κl,κu]2+[δl,δu]2)]β,1−(1−[δl,δu]2)β).

3.2 Interval Valued Pythagorean Fuzzy Soft Interaction Weighted Average Operator

Suppose we have a collection of IVPFSNs denoted by ℳeij=([κijl,κiju],[δijl,δiju]). Let ωi and νj be the weight vectors for experts and parameters, respectively, subject to the assumptions that ωi>0,∑i=1nωi=1;νj>0,∑j=1mνj=1. Then, the IVPFSIWA operator can be defined as:

IVPFSIWA:Ψn⟶Ψ

IVPFSIWA(ℳe11,ℳe12,………,ℳenm)=⊕j=1mνj(⊕i=1nωiℳeij)

3.2.1 Theorem

Let ℳeij=([κijl,κiju],[δijl,δiju]) be a collection of IVPFSNs, where (i=1,2,3,……,n and j=1,2,3,………,m). The resulting aggregated value is also an IVPFSN, denoted as:

IVPFSIWA(ℳe11,ℳe12,………,ℳenm)

=(1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj−∏j=1m(∏i=1n(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

The weight vectors for experts and attributes are denoted by ωi and νj, respectively, where ωi>0,∑i=1nωi=1;νj>0,∑j=1mνj=1.

Proof:

To demonstrate the IVPFSIWA operator, we will employ the principle of mathematical induction.

For n=1, we get ω1=1. Then, we have

IVPFSIWA(ℳe11,ℳe12,………,ℳe1m)=⊕j=1mνjℳe1j=(1−∏j=1m⁡(1−[κ1jl,κ1ju]2)νj,∏j=1m⁡(1−[κ1jl,κ1ju]2)νj−∏j=1m⁡(1−([κ1jl,κ1ju]2+[δ1jl,δ1ju]2))νj)

=(1−∏j=1m⁡(∏i=11⁡(1−[κijl,κiju]2)ωi)νj,∏j=1m⁡(∏i=11⁡(1−[κijl,κiju]2)ωi)νj−∏j=1m⁡(∏i=11⁡(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj).

For m=1, we get ν1=1. Then, we have

IVPFSIWA(ℳe11,ℳe21,………,ℳen1)=⊕i=1nωiℳei1

=(1−∏i=1n(1−[κi1l,κi1u]2)ωi,∏i=1n(1−[κi1l,κi1u]2)ωi−∏i=1n(1−([κi1l,κi1u]2+[δi1l,δi1u]2))ωi)

=(1−∏j=11(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=11(∏i=1n(1−[κijl,κiju]2)ωi)νj−∏j=11(∏i=1n(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

This demonstrates that the theorem given above is true for n=1 and m=1. To further prove the theorem, let us assume that the theorem holds for m=α1+1,n=α2 and m=α1,n=α2+1, such as

⊕j=1α1+1νj(⊕i=1α2ωiℳeij)=(1−∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj,∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj−∏j=1α1+1(∏i=1α2(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

⊕j=1α1νj(⊕i=1α2+1ωiℳeij)=(1−∏j=1α1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj,∏j=1α1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj−∏j=1α1(∏i=1α2+1(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

For m=α1+1 and n=α2+1, we have

⊕j=1α1+1νj(⊕i=1α2+1ωiℳeij)=⊕j=1α1+1νj(⊕i=1α2ωiℳeij⊕ωα2+1ℳe(α2+1)j)

=⊕j=1α1+1⊕i=1α2νjωiℳeij⊕j=1α1+1νjωα2+1ℳe(α2+1)j=(1−∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj⨁1−∏j=1α1+1((1−[κ(α2+1)jl,κ(α2+1)ju]2)ωα2+1)νj,∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj−∏j=1α1+1(∏i=1α2(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj⨁∏j=1α1+1((1−[κ(α2+1)jl,κ(α2+1)ju]2)ωα2+1)νj−∏j=1α1+1((1−([κ(α2+1)jl,κ(α2+1)ju]2+[δ(α2+1)jl,δ(α2+1)ju]2)ωα2+1)νj)

=(1−∏j=1α1+1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj,∏j=1α1+1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj−∏j=1α1+1(∏i=1α2+1(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

Therefore, it holds for m=α1+1 and n=α2+1. Therefore, based on the principle of mathematical induction, we can conclude that the theorem holds ∀ m and n.

3.2.2 Example

Suppose χ={x1,x2,x3} be set of experts and φ={e1=Resale Value,e2=Mileage,e3=Cost of bike} be a set of attributes. The weights of experts and attributes are ωi=(0.38,0.45,0.17)T and νj=(0.25,0.45,0.3)T, respectively. The team of experts wants to buy a motorbike under their deliberated parameters. First of all, experts deliver their evaluations in terms of IVPFSNs for each alternate (ℳ,φ)=([κijl,κiju],[δijl,δiju])3×3 is given as

(ℳ,φ)=[([.3,.8],[.4,.5])([.4,.6],[.3,.7])([.5,.8],[.5,.5])([.1,.5],[.2,.3])([.3,.8],[.5,.5])([.2,.4],[.2,.3])([.2,.9],[.2,.3])([.5,.7],[.2,.6])([.2,.4],[.2,.8])]

Applying the theorem mentioned above

IVPFSIWA(ℳe11,ℳe12,………,ℳe33)=(1−∏j=13(∏i=13(1−[κijl,κiju]2)ωi)νj,∏j=13(∏i=13(1−[κijl,κiju]2)ωi)νj−∏j=13(∏i=13(1−([κijl,κiju]2+[δijl,δiju]2))ωi)νj)

=(1−({[0.36,0.91]0.38[0.75,0.99]0.45[0.19,0.96]0.17}0.25{[0.64,0.84]0.38[0.36,0.91]0.45[0.51,0.75]0.17}0.45{[0.36,0.75]0.38[0.84,0.96]0.45[0.84,0.96]0.17}0.3),(({[0.36,0.91]0.38[0.75,0.99]0.45[0.19,0.96]0.17}0.25{[0.64,0.84]0.38[0.36,0.91]0.45[0.51,0.75]0.17}0.45{[0.36,0.75]0.38[0.84,0.96]0.45[0.84,0.96]0.17}0.3)−({(1−[0.25,0.89])0.38(1−[0.05,0.34])0.45(1−[0.08,0.9])0.17}0.25{(1−[0.25,0.85])0.38(1−[0.34,0.89])0.45(1−[0.29,0.85])0.17}0.45{(1−[0.5,0.89])0.38(1−[0.08,0.25])0.45(1−[0.08,0.8])0.17}0.3)))

=(1−({[0.6783,0.9648][0.8786,0.9955][0.7540,0.9931]}0.25{[0.8440,0.9359][0.6314,0.9584][0.8918,0.9523]}0.45{[0.6783,0.8964][0.9245,0.9818][0.9708,0.9931]}0.3),(({[0.6783,0.9648][0.8786,0.9955][0.7540,0.9931]}0.25{[0.8440,0.9359][0.6314,0.9584][0.8918,0.9523]}0.45{[0.6783,0.8964][0.9245,0.9818][0.9708,0.9931]}0.3)−({([0.11,0.75])0.38([0.66,0.95])0.45([0.1,0.92])0.17}0.25{([0.15,0.75])0.38([0.11,0.66])0.45([0.15,0.71])0.17}0.45{([0.11,0.5])0.38([0.25,0.92])0.45([0.2,0.92])0.17}0.3)))