# A New Idea to Evaluate Networking Problem and MCGDM Problem in Parametric Interval Valued Pythagorean Arena

Authors: Avishek Chakraborty; Sankar Prasad Mondal; Shariful Alam; Dragan Pamucar; Dragan Marikovic
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this article, the concept of parametric interval valued Pythagorean number (PIVPN) has been introduced, which is an extended version of Pythagorean number (PN). Here, a new score and accuracy function have been innovated in the PIVPN environment along with the De-Pythagorean value concept. The new tool and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision making problem (MCGDM) problem. In order to solve the MCGDM problem, we have prepared Parametric Interval valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO) and Parametric Interval valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO) operator in PIVPN environment. Finally, sensitivity analysis and industrious comprehensive numerical simulations have been performed to identify the reliability, efficiency, and usefulness of this novel work. In this article, we have shown that PIVPNs are a more well-organized representation to grip a real-life problem, and they can handle inconsistent conditions in a better compatible way in comparison to the other existing methods.


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## Body

## 1. Introduction

Fuzzy set theory [1] plays a quintessential role in modern science, medical diagnoses, engineering, and technical problem, but there is a basic note of interrogation as to how we can relate or use the impreciseness concept in our computational mathematical modeling. Myriad researchers from the distinct field have defined many approaches to define it and have given numerous recommendations like triangular [2], trapezoidal [3], pentagonal [4] fuzzy number, and their perspectives to use the theory of uncertainty. Further, researchers developed the concept of Intuitionistic fuzzy number (IFN) [5], introducing the perception of both belongingness and nonbelongingness membership functions. After that, triangular [6] and trapezoidal [7] IFN’s are formulated and applied in the distinct domain of the mathematical field. Further, the idea of interval-valued IFN [8] is manifested, which is the logical extension of IFN. After that, the concept of neutrosophic set (NS) [9] is established in the research domain which is the extension of IFN. In the case of NS theory, it actually deals with three components, namely, truth, false, and hesitation, whereas IFN contains two components, namely membership and nonmembership functions. Apart from this literature survey, researchers observed one important point. For instance, someone considered their liking towards any item is 0.7 whereas malaise is 0.4, and then the definition of intuitionistic fuzzy number has been violated. Thus, researchers developed the idea of Pythagorean number [10], which is the modified version of the intuitionistic fuzzy number considering the nonlinearity portions of the membership functions. Further, aggregation operators [11], generalized averaging aggregation operators [12], and confidence level-based aggregation operators [13] are developed by researchers day by day. Also, strategic decision-making approach [14], interval-valued Pythagorean number [15], some new results on Pythagorean set [16], decision-making approach on Pythagorean set [17], new novel score and accuracy function development [18], and linear programming method based on score value [19] have been developed by several researchers from different aspects. Apart from it, several researchers are worked in this field till now, and they proposed lots of concepts [20–25] related to multicriteria decision-making, score value, correlation coefficient, accuracy function of Pythagorean or intervalued Pythagorean number. There is plentiful literary evidence to classify some basic uncertain parameters. It should be adumbrated that there is no such unique representation of this uncertainty parameter. It can be optimized according to the choice of decision-makers for solving problems & can be varied as well as presented as different applications. Here, we present some information about the parameter of uncertainty showing how they vary from other concepts maneuvering the idea of uncertainty by utilizing some definitions, flowcharts, and diagrams. In this paper, we recommend that the researchers take the uncertainty parameter as PIVPN.In this current era, the MCGDM problem is a predominant topic in the advance science and engineering field. The information of several parameters is precisely unspecified to create a model in real-life circumstances. Involving such cases, there is an occurrence of the notion of impreciseness. Uncertainty too forms the chief theme of the MCGDM problem, but the complication of such a method is how we can measure the uncertainty. Buyukozkan and Guleryuz [26] have adopted MCGDM approach to select better smart phones; Wang et al. [27] manifested intuitionistic linguistic operators on MCGDM problems; You et. al. [28] focused on an entropy based MCGDM method in interval-valued IFN; interval valued intuitionistic work in the MCGDM zone has been surveyed by Wang et al. [29]; Chakraborty A. [30–33] introduced MCGDM based problems in distinct domain, Chaio [34] derived MCGDM methodology based on type 2 fuzzy linguistic judgments; Wibowo [35] introduced MCGDM model in human resources management information projects; Chuu [36] established MCGDM model in manufacturing selection problem; Liu and Liu [37] proposed MCGDM model using Power Bonferroni Mean operator in IFN domain; Liu and Jiang [38] incorporated MCDM problem in interval valued intuitionistic set; Kumar and Garg [39] introduced TOPSIS in interval valued domain; Garg and Kumar [40] proposed advanced TOPSIS method in using SPA theory in IFS domain; Du et al. [41] focused on MADM problem in IVPN environment; Li et al. [42] described MADM model in IVPN environment. Apart from this literature survey, the concept of the Pythagorean set has also been fruitfully applied in several domains like operation research, graph theory, matrix evolution, etc. Luqman et al. [43] incorporated the Pythagorean concept in risk management problems using Digraph and matrix concept; Akram et al. [44] proposed LP model in the Pythagorean arena with equality constraints; Akram et al. [44, 45] manifested L-R type Pythagorean LP approaches with mixed and equality constraints; Enayattabar et al. [46] introduced the concept of shortest path problem in IVPN environment; Di Caprio et al. [47] focused on ant colony algorithm based on Pythagorean arena; Ebrahimnejad et al. [48] developed advanced bee colony algorithm in IVPN domain; Akram et al. [49] incorporated the concept of a planar graph in the Pythagorean environment which plays an essential role in graph theory problem.From the available literature, it is observed that all the works which had been done earlier are mainly based on a general Pythagorean set or in an IVPN environment. Researchers have not explored the parametric representation of IVPN and its utility and usefulness in the domain of decision-making. In this article, we focused on the parametric representation of IVPN and constructed a few operators, de-Pythagorean skills in this environment that have been successfully and effectively applied to some real-life problems in the field of decision making. In this research article, we have developed two different operators, namely PIVPWAMO and PIVPWGMO, in an uncertain environment and introduced a new de-Pythagorean technique along with a new score and accuracy function for the crispification of PIVPN. The newly developed tools and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision-making (MCGDM) problem. Finally, sensitivity analysis and industrious comprehensive numerical simulations has been performed to identify the reliability, efficiency, and usefulness of this novel work.


### 1.1. Motivation

Although a good number of research works have already been toiled in the area of Pythagorean number that has been effectively and successfully applied to enrich MCDM/MADM/MCGDM techniques but only a few work is available in the area of IVPN. It is fascinating to note that many researchers have written Pythagorean numbers where they deploy the extension concept of FN and IFN. But why can we not structure the concept of the Pythagorean number using a simple parametric interval valued concept? Also, what will be the mathematical structure of PIVPN, and how can we relate it to real-life problems? Moreover, can we tackle the networking problem in PIVPN environment? If all the entities of decision matrices are considered as PIVPN, then how can we find out the best alternative! How can we perform a rigorous sensitivity and numerical analysis of an MCGDM model in a PIVPN environment? In this research article, an attempt is made to address all these issues and we proposed an efficient layout of MCGDM and networking when the decision-maker scored the feasible alternatives in the form of PIVPN. Different classifications of uncertain parameters and the conceptual workflow have been presents in Figure1.Figure 1
Classification of uncertain parameters and the workflow.


### 1.2. Novelties

The novelties of the work are described as follows:(i)
Formulation of PIVPN based on the concept of IVPN(ii)
Establishment of De-Pythagorean value using a noble technique to crispify a PIVPN into a real number.(iii)
Realistic application of PIVPN into MCGDM and Networking model.(iv)
Apply PIVPWAMO and PIVPWGMO in decision matrices to evaluate ranking.(v)
Well planned sensitivity analysis and comprehensive numerical simulations have been performed in order to understand the effect of weights on the PIVPN in the MCGDM problem.


## 1.1. Motivation

Although a good number of research works have already been toiled in the area of Pythagorean number that has been effectively and successfully applied to enrich MCDM/MADM/MCGDM techniques but only a few work is available in the area of IVPN. It is fascinating to note that many researchers have written Pythagorean numbers where they deploy the extension concept of FN and IFN. But why can we not structure the concept of the Pythagorean number using a simple parametric interval valued concept? Also, what will be the mathematical structure of PIVPN, and how can we relate it to real-life problems? Moreover, can we tackle the networking problem in PIVPN environment? If all the entities of decision matrices are considered as PIVPN, then how can we find out the best alternative! How can we perform a rigorous sensitivity and numerical analysis of an MCGDM model in a PIVPN environment? In this research article, an attempt is made to address all these issues and we proposed an efficient layout of MCGDM and networking when the decision-maker scored the feasible alternatives in the form of PIVPN. Different classifications of uncertain parameters and the conceptual workflow have been presents in Figure1.Figure 1
Classification of uncertain parameters and the workflow.


## 1.2. Novelties

The novelties of the work are described as follows:(i)
Formulation of PIVPN based on the concept of IVPN(ii)
Establishment of De-Pythagorean value using a noble technique to crispify a PIVPN into a real number.(iii)
Realistic application of PIVPN into MCGDM and Networking model.(iv)
Apply PIVPWAMO and PIVPWGMO in decision matrices to evaluate ranking.(v)
Well planned sensitivity analysis and comprehensive numerical simulations have been performed in order to understand the effect of weights on the PIVPN in the MCGDM problem.


## 2. Mathematical Preliminaries

Definition 1.
Interval Number: An interval numberX is denoted by XL,XR and defined as X=XL,XR=x:XL≤x≤XR,x∈R, where R real number set and XL and XR generally denoted the left and right range of the interval, respectively.Lemma 1.
The intervalXL,XR can also be represented as Pα=XL1−αXRα for α∈0,1.Definition 2.
Fuzzy Set: A setS˜, generally defined as S˜=α,μS˜α:α∈S,μS˜α∈0,1, denoted by the pair α,μS˜α, where αϵS and μS˜αϵ0,1, then set S˜ is called a fuzzy set.Definition 3.
Intuitionistic Fuzzy Set (IFS): [2] S˜ in the universal discourse X which is denoted generically by x is said to be a IFS if S˜=x;τx,φx⋮x∈X, where τx:X⟶0,1 is called the truth membership function, φx:X⟶0,1 is called the indeterminacy membership function. τx,φx exhibits the following relation:(1)0≤τx+φx≤1.
The degree of hesitation is between membership function and nonmembership function is defined asπx=1−τx−φx.Definition 4.
Pythagorean Fuzzy Set (PFS): [4] P˜ in the universal discourse X which is denoted generically by x is said to be a Pythagorean fuzzy set if P˜=x;τx,φx⋮x∈X, where τx:X⟶0,1 is called the truth membership function which represents the degree of confidence, φx:X⟶0,1 is called the indeterminacy membership function which represents the degree of falsity. Geometrical representation of PFN is shown in Figure 2. Here, τx,φx exhibits the following relation:(2)0≤τx2+φx2≤1.
The degree of hesitation is between the membership function and the nonmembership function is defined asπx=1−τx2−φx2. Zhang and Xu [6] defined the score function as SPˇ=τx2−φx2, where SPˇ∈−1,1 for two Pythagorean fuzzy numbers P1ˇ and P2ˇ if(i)
SP1ˇ>SP2ˇ, then P1ˇ>P2ˇ(ii)
SP1ˇ<SP2ˇ, then P1ˇ<P2ˇ(iii)
SP1ˇ=SP2ˇ, then P1ˇ∼P2ˇ
and accuracy function is defined asHPˇ=τx2+φx2, where HPˇ∈0,1Figure 2
Graphical representation of IFN and PFN.Definition 5.
Interval Valued Pythagorean fuzzy Set (IVPFS): A set “A” in X is defined as follows:(3)A=<X;τLx,τUx,φLx,φUx>|x∈X,where 0≤τLx≤τUx≤1 and 0≤φLx≤φUx≤1
Also,τx2+φx2≤1.
For each elementx∈X its hesitation interval relative to A is defined as follows:(4)πAx=πALx,πAUx=1−τLx2−φLx2,1−τUx2−φUx2.Definition 6.
Triangular Interval Valued Pythagorean fuzzy Set (TIVPFS): A TIVPFS is denoted asA˜invi=a1,b,c1;λ,a,b,c;ω,d1,b,e1;δ,d,b,e;μ where 0<ω≤λ≤1 , 0<δ≤μ≤1 and a1<a<b<c<c1 also 0<ω2+μ2≤1 and 0<λ2+δ2≤1. Here λ,ω are the maximum degree of the membership function of the triangular fuzzy number a1,b,c1 and a,b,c, respectively, and δ,μ are the minimum degree of the nonmembership function of the triangular fuzzy number d1,b,e1 and d,b,e, respectively.
The upper and lower membership function is defined by the following:(5)μAˇinviUx=λx−a1b−a1.,a1≤x≤b,λc1−xc1−b.,b≤x≤c1,0,otherwise,μAˇinviLx=ωx−ab−a.,a≤x≤b,ωc−xc−b.,b≤x≤c,0,otherwise,and non membership functions are defined as follows:(6)ϑAˇinviUx=δb−xb−d1.,d1≤x≤b,δx−be1−b.,b≤x≤e1,1,otherwise,ϑAˇinviLx=μb−xb−d.,d≤x≤b,μx−be−b.,b≤x≤e,1,otherwise.


## 3. Parametric Interval Valued Pythagorean Number and Its Properties

Definition 7.
Parametric Interval Valued Pythagorean Number: PIVPN is defined as follows:(7)Sα=τLxα1,τUx1−α1,φLxα2,φUx1−α2,where 0≤τLx≤τUx≤1;0≤φLx≤φUx≤1;α1,α2∈0,1and0≤α1+α2≤1.
Also,τx2+φx2≤1.


### 3.1. De-Pythagorean of Parametric Interval Valued Pythagorean Number

A PIVPNA=τLxα1,τUx1−α1,φLxα2,φUx1−α2. can be converted into a crisp number C as follows:(8)C=ρτLxα1+τUx1−α1/2+1−ρφLxα2+φUx1−α2/22,where ρ is an index parameter of the decision-maker and ρ∈0,1. That is, the crisp value of an IVPN fully depends on the parameter ρ.Example 1.
There is an IVPN written in the parametric form as1α12.51−α1,3.5α24.51−α2. Then the value of the crisp number when index λ=0.5 is 1.48.
LetA1,A2 be two IVPNs whose De-Pythagorean values are C1andC2; then the relation between them is defined as follows:(9)1.C1>C2thenA1>A2,2.C1<C2thenA1<A2,3.C1=C2thenA1=A2.
We have computed some ranking results using the De-Pythagorean value in Table1 for different IVPNs.Table 1
Numerical example.
Interval valued Pythagorean numberValue ofρDe-Pythagorean valueRankingA=<1.5,2.5,2,3>0.41.15B>AB= <[3.5, 5.5], [4, 6]>2.4C= <[3.5, 6.5], [5, 7]>0.32.85D>CD= <[7.5, 9.5], [8, 11]>9.2


### 3.2. Properties and Operators in Parametric Interval Valued Pythagorean Number

Let us consider two IVPNsS1=<τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>andS2=<τL2xα1,τU2x1−α1,φL2xα2,φU2x1−α2> where α1,α2∈0,1 and 0 ≤α1+α2≤1 then, we have the following:


#### 3.2.1. Addition

(10)S=S1+S2=<τL1x+τL2xα1,τU1x+τU2x1−α1,φL1x+φL2xα1,φU1x+φU2x1−α1>.


#### 3.2.2. Subtraction

(11)S=S1−S2−<τL1x−τL2xα1,τU1x−τU2x1−α1,φL1x−φL2xα1,φU1x−φU2x1−α1>.


#### 3.2.3. Multiplication

(12)S=S1×S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1,MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2>.


#### 3.2.4. Multiplication by a Constant

(13)S=k×S1=<kτL1xα1,kτU1x1−α1,kφL1xα2,kφU1x1−α2>whenk>0=<kτU1xα1,kτL1x1−α1,kφU1xα2,kφL1x1−α2>whenk>0.


#### 3.2.5. Division

(14)S=S1S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1=MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2.


#### 3.2.6. Inverse

(15)hp−1=1τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2=τL1x−α1,τU1xα1−1,φL1x−α2,φU1xα2−1.


#### 3.2.7. Example

IfI=2.5α16.31−α1,1α221−α2andJ=10α1201−α1,5α27.51−α2forα1,α2,α3∈0,1FindI+J,I−J,IJ,kI,I/Jfork=0.5,−0.2.Solution:(16)I+J=12.5α126.31−α1,6α29.51−α2,I−J=−17.5α1−3.71−α1,−6.5α2−31−α2,IJ=25α11261−α1,5α2151−α2,kI=1.25α13.151−α1,0.5α211−α2k=0.5,−1.26α1−0.51−α1,−0.4α2−0.21−α2k=−0.2,IJ=0.125α10.631−α1,0.133α20.41−α2.


### 3.3. Operators

We consider some operators used for interval valued parametric Pythagorean fuzzy numbers.


#### 3.3.1. Parametric Interval Valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO)

AssumingSi=<τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>, where i=1,…n is family of PIVPN, λ is a set of PIVPN. A mapping PIVPWAMO such that λn⟶λ, then PIVPWAMO S1,….SN=<∑i=1nwiτiL1α1,wiτiU11−α1,∑i=1nwiφiL1α2,wiφiU11−α2. where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


#### 3.3.2. Parametric Interval Valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO)

AssumingSi=<τL1xα1,τU1x1−α1,>φL1xα2,φU1x1−α2,τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>. where i=1,2,3,…..n is family of PIVPN and λ is a set of PIVPN. A mapping PIVPWGMO such that λn⟶λ, thenPIVPWGMOS1,S2,S3,………SN=<∏i=1nτiL1α1wi,τiU11−α1wi,∏i=1nφiL1α1wi,φiU11−α1wi>, where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


## 3.1. De-Pythagorean of Parametric Interval Valued Pythagorean Number

A PIVPNA=τLxα1,τUx1−α1,φLxα2,φUx1−α2. can be converted into a crisp number C as follows:(8)C=ρτLxα1+τUx1−α1/2+1−ρφLxα2+φUx1−α2/22,where ρ is an index parameter of the decision-maker and ρ∈0,1. That is, the crisp value of an IVPN fully depends on the parameter ρ.Example 1.
There is an IVPN written in the parametric form as1α12.51−α1,3.5α24.51−α2. Then the value of the crisp number when index λ=0.5 is 1.48.
LetA1,A2 be two IVPNs whose De-Pythagorean values are C1andC2; then the relation between them is defined as follows:(9)1.C1>C2thenA1>A2,2.C1<C2thenA1<A2,3.C1=C2thenA1=A2.
We have computed some ranking results using the De-Pythagorean value in Table1 for different IVPNs.Table 1
Numerical example.
Interval valued Pythagorean numberValue ofρDe-Pythagorean valueRankingA=<1.5,2.5,2,3>0.41.15B>AB= <[3.5, 5.5], [4, 6]>2.4C= <[3.5, 6.5], [5, 7]>0.32.85D>CD= <[7.5, 9.5], [8, 11]>9.2


## 3.2. Properties and Operators in Parametric Interval Valued Pythagorean Number

Let us consider two IVPNsS1=<τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>andS2=<τL2xα1,τU2x1−α1,φL2xα2,φU2x1−α2> where α1,α2∈0,1 and 0 ≤α1+α2≤1 then, we have the following:


### 3.2.1. Addition

(10)S=S1+S2=<τL1x+τL2xα1,τU1x+τU2x1−α1,φL1x+φL2xα1,φU1x+φU2x1−α1>.


### 3.2.2. Subtraction

(11)S=S1−S2−<τL1x−τL2xα1,τU1x−τU2x1−α1,φL1x−φL2xα1,φU1x−φU2x1−α1>.


### 3.2.3. Multiplication

(12)S=S1×S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1,MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2>.


### 3.2.4. Multiplication by a Constant

(13)S=k×S1=<kτL1xα1,kτU1x1−α1,kφL1xα2,kφU1x1−α2>whenk>0=<kτU1xα1,kτL1x1−α1,kφU1xα2,kφL1x1−α2>whenk>0.


### 3.2.5. Division

(14)S=S1S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1=MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2.


### 3.2.6. Inverse

(15)hp−1=1τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2=τL1x−α1,τU1xα1−1,φL1x−α2,φU1xα2−1.


### 3.2.7. Example

IfI=2.5α16.31−α1,1α221−α2andJ=10α1201−α1,5α27.51−α2forα1,α2,α3∈0,1FindI+J,I−J,IJ,kI,I/Jfork=0.5,−0.2.Solution:(16)I+J=12.5α126.31−α1,6α29.51−α2,I−J=−17.5α1−3.71−α1,−6.5α2−31−α2,IJ=25α11261−α1,5α2151−α2,kI=1.25α13.151−α1,0.5α211−α2k=0.5,−1.26α1−0.51−α1,−0.4α2−0.21−α2k=−0.2,IJ=0.125α10.631−α1,0.133α20.41−α2.


## 3.2.1. Addition

(10)S=S1+S2=<τL1x+τL2xα1,τU1x+τU2x1−α1,φL1x+φL2xα1,φU1x+φU2x1−α1>.


## 3.2.2. Subtraction

(11)S=S1−S2−<τL1x−τL2xα1,τU1x−τU2x1−α1,φL1x−φL2xα1,φU1x−φU2x1−α1>.


## 3.2.3. Multiplication

(12)S=S1×S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1,MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2>.


## 3.2.4. Multiplication by a Constant

(13)S=k×S1=<kτL1xα1,kτU1x1−α1,kφL1xα2,kφU1x1−α2>whenk>0=<kτU1xα1,kτL1x1−α1,kφU1xα2,kφL1x1−α2>whenk>0.


## 3.2.5. Division

(14)S=S1S2=<MinτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1xα1,MaxτL1xτL2x,τU1xτU2x,τL1xτU2x,τL2xτU1x1−α1=MinφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1xα2,MaxφL1xφL2x,φU1xφU2x,φL1xφU2x,φL2xφU1x1−α2.


## 3.2.6. Inverse

(15)hp−1=1τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2=τL1x−α1,τU1xα1−1,φL1x−α2,φU1xα2−1.


## 3.2.7. Example

IfI=2.5α16.31−α1,1α221−α2andJ=10α1201−α1,5α27.51−α2forα1,α2,α3∈0,1FindI+J,I−J,IJ,kI,I/Jfork=0.5,−0.2.Solution:(16)I+J=12.5α126.31−α1,6α29.51−α2,I−J=−17.5α1−3.71−α1,−6.5α2−31−α2,IJ=25α11261−α1,5α2151−α2,kI=1.25α13.151−α1,0.5α211−α2k=0.5,−1.26α1−0.51−α1,−0.4α2−0.21−α2k=−0.2,IJ=0.125α10.631−α1,0.133α20.41−α2.


## 3.3. Operators

We consider some operators used for interval valued parametric Pythagorean fuzzy numbers.


### 3.3.1. Parametric Interval Valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO)

AssumingSi=<τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>, where i=1,…n is family of PIVPN, λ is a set of PIVPN. A mapping PIVPWAMO such that λn⟶λ, then PIVPWAMO S1,….SN=<∑i=1nwiτiL1α1,wiτiU11−α1,∑i=1nwiφiL1α2,wiφiU11−α2. where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


### 3.3.2. Parametric Interval Valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO)

AssumingSi=<τL1xα1,τU1x1−α1,>φL1xα2,φU1x1−α2,τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>. where i=1,2,3,…..n is family of PIVPN and λ is a set of PIVPN. A mapping PIVPWGMO such that λn⟶λ, thenPIVPWGMOS1,S2,S3,………SN=<∏i=1nτiL1α1wi,τiU11−α1wi,∏i=1nφiL1α1wi,φiU11−α1wi>, where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


## 3.3.1. Parametric Interval Valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO)

AssumingSi=<τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>, where i=1,…n is family of PIVPN, λ is a set of PIVPN. A mapping PIVPWAMO such that λn⟶λ, then PIVPWAMO S1,….SN=<∑i=1nwiτiL1α1,wiτiU11−α1,∑i=1nwiφiL1α2,wiφiU11−α2. where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


## 3.3.2. Parametric Interval Valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO)

AssumingSi=<τL1xα1,τU1x1−α1,>φL1xα2,φU1x1−α2,τL1xα1,τU1x1−α1,φL1xα2,φU1x1−α2>. where i=1,2,3,…..n is family of PIVPN and λ is a set of PIVPN. A mapping PIVPWGMO such that λn⟶λ, thenPIVPWGMOS1,S2,S3,………SN=<∏i=1nτiL1α1wi,τiU11−α1wi,∏i=1nφiL1α1wi,φiU11−α1wi>, where wi is the weight vector assigned with the PIVPN and ∑i=1nwi=1, also each wi∈0,1.


## 4. Score and Accuracy Function IVPFN

A score function of IVPFS is mainly depends on hesitation indicator and score indicator. Letσ=<a,b,c,d> be an IVPFN, and a new score function Sˇσ depends on the following:(i)
1−b2−d2,1−a2−c2 are called hesitation indicators.(ii)
a21−a2−c2 , b21−b2−d2 are called benefit degrees.(iii)
c21−a2−c2 , d21−b2−d2 are called nonbeneficiary degrees.So(17)Sˇσ=a2−c21−a2−c2+b2−d21−b2−d22,is the score function where Sˇσϵ−1,1Ifσ=<0,0,1,1> then Sˇσ=−1 and if σ=<1,1,0,0>thenSˇσ=1The new accuracy function is defined as follows:(18)Hˇσ=a2+c21−a2−c2+b2+d21−b2−d22,where Hˇσϵ0,1.Letγ,δ are two IVPFN then If(1)
sˇγ>sˇδthenγ>δ,(2)
sˇγ<sˇδthenγ<δ.(3)
sˇγ=sˇδthen.(i)
Hˇγ>Hˇδthenγ>δ,(ii)
Hˇγ<Hˇδthenγ<δ(iii)
Hˇγ=Hˇδthenγ∼δ


## 5. Critical Path Problem in Pythagorean Fuzzy Environment

The full form of CPM method is the critical path method which is a project management technique for progress planning that defines critical and noncritical tasks with the goal of preventing time frame problems. The critical path is defined as the (time) longest possible path from the start point to endpoint of the chart. Each project has at least one critical path. Each critical path consists of a list of activities on which the project manager should focus the most if he wants to ensure timely completion of the project. The date of the completion of the last task on the critical path is also the date of the completion of the project. For critical tasks to be applied, the total time reserve and thus free time reserve is equal to zero, i.e., that the start delay of this task or to extend its duration will affect the final date of the project. The critical path is reflected in the schedule and project management in all phases of the project lifecycle.Activity: It is any portion of a project that has a definite beginning and ending and may use some resources such as time, labor, material, equipment, etc.Event or Node: Beginning and ending points of activities denoted by circles are called nodes or events.Critical Path: The sequence of activities in a network is called the critical path. It is the longest path in the network, from the starting event to the ending event, and defines the minimum time required to complete the project.Forward pass method: Based on fixed occurrence time of the initial network event, the forward pass computation yields the earliest start and earliest finish times for each activity and, indirectly, the earliest expected occurrence time for each event.(1)
The computations begin from the start node and move to the end node. To accomplish this, the forward pass computations start with an assumed earliest occurrence time of zero for the initial project event.(2)
For any activityi,j, let ESi denote the earliest time of event i, then Sj=ESi+tij .(3)
If more than one activity enters an event, the earliest start time for that event is computed asESj=maxESi+tij for all activities emanating from node i entering into j.Backward pass method: The latest occurrence event time specifies the time by which all activities entering into that event must be completed without delaying the total project. These are computed by reversing the method of calculation used for the earliest event times.(1)
To compute backward pass, starting from the final node, the computation proceeds from right to left, up to the initial event.(2)
For any activityi,j, let LFi denote the latest finished time of event i, then Fi=LFj−tij .(3)
If more than one activity enters an event the latest finish time for that event is computed asLFi=minLFj−tij for all activities emanating from node j entering into i.In our developed technique, we consider all the activities are IVPN to calculate theESi and LFj; we utilize the concept of a new score function Sˇσ=a2−c21−a2−c2+b2−d21−b2−d2/2, which can compare the finite number of distinct IVPFN to estimate the minimum and maximum value. If the score values are equal, then estimate the IVPFN using the new accuracy function defined as Hˇσ=a2+c21−a2−c2+b2+d21−b2−d2/2.


### 5.1. Flowchart

Flowchart of critical path problem in Pythagorean fuzzy environment is presented in Figure3.Figure 3
Critical path problem in pythagorean fuzzy environment.


### 5.2. Numerical Problem

Table2 and Figure 4 present a numerical problem implemented in this study.(1)
Sketch the project network(2)
Evaluate theESi and LFi(3)
Compute the critical path and total project duration.Step-1Step-2 After computationESi and LFi we have a network as follows (Figure 5):Table 2
Numerical example.
NodesDescriptionPredecessorsActivity1Choice of vehicles—a=0.4,0.7;0.3,0.52Choice of destination—b=0.3,0.8;0.2,0.63Choice of driver1e=0.3,0.7;0.2,0.64Final plan and blueprint2n=0.6,0.8;0.3,0.55Route map creation2O=0.7,0.9;0.2,0.36Fuel consumption for different places1d=0.6,0.8;0.5,0.67Instruction and training3h=0.4,0.7;0.3,0.68Initial startup time4m=0.4,0.7;0.2,0.69Vehicle testing timing1c=0.4,0.6;0.2,0.410Run the system5, 7, 8p=0.8,0.9;0.1,0.2k=0.4,0.7;0.3,0.5l=0.5,0.7;0.3,0.611Final destination6, 9, 10g=0.3,0.6;0.2,0.5f=0.1,0.8;0.05,0.3j=0.4,0.8;0.3,0.6Figure 4
Network diagram.In step-1, we draw the network diagram of the given problem in Figure3, and further after computation of forward pass and backward pass, we finally get Figure 4, which is the final diagram with node allocation. Hence, the expected project duration −0.879 and Critical Path- 1 ⟶ 3 ⟶ 7 ⟶ 11Figure 5
Final computational network diagram. line denote the critical path.


### 5.3. Comparative Analysis

In Table3 we compare our work with other previous score function and accuracy function described by the authors [15, 18, 19] for finding the alternatives.Table 3
Comparison of work.
Score functionCritical pathProject durationGarg [18]Can’t be determined since negativity arise at the earliest and latest time counting—Garg [19]2⟶5⟶10⟶112.281Zhang [15]Can’t determined since negativity arise in earliest and latest time counting—Our proposed1⟶ 3 ⟶ 7 ⟶ 110.879


#### 5.3.1. Discussion

We have compared the outcome of our proposed method with the outcome of other similar research work that has been presented in Table2. Here, we observed that some score and accuracy functions fail to calculate the critical path as negativity arises in calculating ESi and LFi which is impossible for a networking problem. It is to be further noted that our proposed approach can find out a critical path that gives us a minimum project time duration in comparison to the approach of Garg [19]. Hence, we can conclude that it is a better and modified approach which can solve critical path problem in networking domain.


## 5.1. Flowchart

Flowchart of critical path problem in Pythagorean fuzzy environment is presented in Figure3.Figure 3
Critical path problem in pythagorean fuzzy environment.


## 5.2. Numerical Problem

Table2 and Figure 4 present a numerical problem implemented in this study.(1)
Sketch the project network(2)
Evaluate theESi and LFi(3)
Compute the critical path and total project duration.Step-1Step-2 After computationESi and LFi we have a network as follows (Figure 5):Table 2
Numerical example.
NodesDescriptionPredecessorsActivity1Choice of vehicles—a=0.4,0.7;0.3,0.52Choice of destination—b=0.3,0.8;0.2,0.63Choice of driver1e=0.3,0.7;0.2,0.64Final plan and blueprint2n=0.6,0.8;0.3,0.55Route map creation2O=0.7,0.9;0.2,0.36Fuel consumption for different places1d=0.6,0.8;0.5,0.67Instruction and training3h=0.4,0.7;0.3,0.68Initial startup time4m=0.4,0.7;0.2,0.69Vehicle testing timing1c=0.4,0.6;0.2,0.410Run the system5, 7, 8p=0.8,0.9;0.1,0.2k=0.4,0.7;0.3,0.5l=0.5,0.7;0.3,0.611Final destination6, 9, 10g=0.3,0.6;0.2,0.5f=0.1,0.8;0.05,0.3j=0.4,0.8;0.3,0.6Figure 4
Network diagram.In step-1, we draw the network diagram of the given problem in Figure3, and further after computation of forward pass and backward pass, we finally get Figure 4, which is the final diagram with node allocation. Hence, the expected project duration −0.879 and Critical Path- 1 ⟶ 3 ⟶ 7 ⟶ 11Figure 5
Final computational network diagram. line denote the critical path.


## 5.3. Comparative Analysis

In Table3 we compare our work with other previous score function and accuracy function described by the authors [15, 18, 19] for finding the alternatives.Table 3
Comparison of work.
Score functionCritical pathProject durationGarg [18]Can’t be determined since negativity arise at the earliest and latest time counting—Garg [19]2⟶5⟶10⟶112.281Zhang [15]Can’t determined since negativity arise in earliest and latest time counting—Our proposed1⟶ 3 ⟶ 7 ⟶ 110.879


### 5.3.1. Discussion

We have compared the outcome of our proposed method with the outcome of other similar research work that has been presented in Table2. Here, we observed that some score and accuracy functions fail to calculate the critical path as negativity arises in calculating ESi and LFi which is impossible for a networking problem. It is to be further noted that our proposed approach can find out a critical path that gives us a minimum project time duration in comparison to the approach of Garg [19]. Hence, we can conclude that it is a better and modified approach which can solve critical path problem in networking domain.


## 5.3.1. Discussion

We have compared the outcome of our proposed method with the outcome of other similar research work that has been presented in Table2. Here, we observed that some score and accuracy functions fail to calculate the critical path as negativity arises in calculating ESi and LFi which is impossible for a networking problem. It is to be further noted that our proposed approach can find out a critical path that gives us a minimum project time duration in comparison to the approach of Garg [19]. Hence, we can conclude that it is a better and modified approach which can solve critical path problem in networking domain.


## 6. Proposed MCGDM Problem in Pythagorean Environment

In this MCGDM arena, we shall always focus on finding out the best feasible alternatives by disjunctive kind of factors, which is basically called the criteria. Though, it is not a very easy job to find out the attribute value in the format of crisp numbers due to the presence of impreciseness. All the information of the decision matrix is of the interval valued Pythagorean parametric form in nature. In this article, we actually consider an MCGDM, where there will be finite number of distinctive decision-makers available and according to their viewpoint we need to find out the best alternative. To do so, we construct an algorithm based on IVPWAMO and the normalization approach using the conception of IVPWGMO such that we can tackle the impreciseness problem very easily and using the conception of De-Pythagorean value, we shall choose the best alternative among all of them.


### 6.1. Illustration of the MCGDM Problem

We consider the problem as follows:Letπ=π1,π2,π3…πm. be “m” number of distinctive alternatives and σ=σ1,σ2,σ3………..σn. be “n” number of distinctive attribute values, respectively. Let δ=δe1,δe2,δe3………..δen. be the weight set associated with the attributes σ where each δe≥ 0 and also satisfies the relation ∑i=1nδei=1. We also consider the set of decision-maker μ=μ1,μ2,μ3………..μK associated with alternatives whose weight vector is defined as φ=φ1,φ2,φ3………..φk where each φi≥ 0 and also satisfies the relation ∑i=1kφi=1. This weight vector will be selected according to the decision-makers quality of judgment, knowledge, thinking power, etc. The flowchart of the proposed methodology is presented in Figure 6.Step-1 Creation of Decision MatricesFirst, we formulate the decision matrices for each decision maker’s choice related to alternatives versus attribute functions. We consider the elements of the matrices in IVPN in parametric form, so allaij's are members of the Pythagorean fuzzy number set. The related matrix is described as follows:(19)CK=.σ1σ2σ3...σnπ1a11ka12ka13k....a1nkπ2a21ka22ka23k...a2nkπ3.πm...am1k..am2k..am3k...........amnk.Step-2 Creation of Single decision matrixTo create the single group decision matrix, we utilize the operator PIVPWAMO for each decision matrixCi . Thus, we get the modified matrix as follows:(20)C=.σ1σ2σ3...σnπ1a11′a12′a13′....a1n′π2a21′a22′a23′...a2n′π3.πm..am1′..am2′..am3′...........amn′.Step-3 Creation of Final matrix according to weight PriorityTo obtain a single Column decision matrix, we all use the operator PIVPWGMO for each Column, and hence we get the decision matrix as follows:(21)C=.σ1π1a11″π2a21″....πmam1″.Step-4 RankingNow, we consider the De-Pythagorean value and convert the matrix (3) into a crisp one such that we can evaluate the best alternative corresponding to the best attributes.Figure 6
Flowchart of the MCGDM methodology.


### 6.2. Illustrative Example

Let us consider a problem associated with four distinctive products and their three different attributes. Generally, lots of commodities are available in the market, having different types of components with different quality and facilities. Now in this problem, there are four different kinds of decision-makers available. We choose the problem as follows:C1= Comodity 1, C2= Comodity 2, C3= Comodity 3, C4= Comodity 4 are the different kinds of alternatives. S1= Quality, S2= longibity, S3= Price, S4= Service are the four different attributes. There are three distinct types of decision-makers classified according to their ages D1= Young aged people,D2= Middle-aged people,D3= Old aged people having corresponding weight set D=0.30,0.33,0.37 and we also consider the weight function related to the attribute function σ=0.35,0.33,0.32. We consider a verbal matrix for the designer to assist the decision-maker in the creation of his decision matrix. The verbal phrases are listed in Table 4 and the verbal matrix of this MCGDM problem is shown in Table 5.Step-1 All the members of the matrices are PIVPN in nature. So, the decision matrices are as follows:(22)M1=⋅S1S2S3C15.5α16.51−α1,1.3α22.81−α25.3α16.51−α1,6.7α28.61−α25.8α16.31−α1,0.9α22.51−α2C26.3α17.21−α1,1.5α22.31−α25.0α16.01−α1,1.9α22.51−α24.2α16.01−α1,1.5α22.21−α2C31.2α12.41−α1,5.9α26.41−α21.9α12.51−α1,9.0α29.91−α20.9α12.51−α1,7.2α28.31−α2C44.0α15.31−α1,5.5α26.51−α22.5α13.01−α1,5.2α26.51−α23.2α14.61−α1,2.5α23.01−α2.For decision makerD1(23)M2=⋅S1S2S3C14.5α17.51−α1,1.3α22.81−α25.3α16.51−α1,6.7α28.61−α25.8α16.31−α1,0.9α22.51−α2C25.3α17.21−α1,1.2α22.81−α25.0α16.01−α1,1.9α22.51−α24.2α16.01−α1,1.5α22.21−α2C32.2α25.41−α2,7.9α38.41−α31.9α22.51−α2,9.0α39.91−α30.9α22.51−α2,7.2α38.31−α3C43.0α25.81−α2,4.5α36.81−α32.5α23.01−α2,5.2α36.51−α33.2α14.61−α1,2.5α23.01−α2.For decision makerD2(24)M3=⋅S1S2S3C12.5α14.51−α1,2.3α23.31−α24.3α16.81−α1,3.7α25.61−α25.2α17.31−α1,1.9α24.51−α2C26.0α18.01−α1,2.5α26.31−α25.2α16.81−α1,2.9α24.51−α23.2α16.51−α1,2.5α24.21−α2C37.2α19.41−α1,2.9α25.41−α23.9α14.51−α1,7.0α28.91−α22.9α15.51−α1,4.2α26.31−α2C42.0α16.31−α1,4.5α27.51−α21.5α14.01−α1,3.2α24.51−α21.2α13.61−α1,1.5α23.51−α2.For decision makerD3Step-2Now, we utilized our parametric interval valued Pythagorean weighted arithmetic mean operator (PIVPWAMO)to calculate all individual decision matrices and we get the final weighted normalized matrix as follows:(25)M=⋅S1S2S3C14.06α16.091−α1,1.67α22.991−α24.93α16.611−α1,5.59α27.491−α25.58α16.671−α1,1.27α23.241−α2C25.86α17.501−α1,1.77α23.951−α25.08α16.301−α1,2.27α23.241−α23.83α16.191−α1,1.87α22.941−α2C33.75α15.981−α1,5.45α26.691−α22.64α13.241−α1,8.26α29.531−α21.64α13.611−α1,6.09α27.561−α2C43.75α15.841−α1,4.80α26.971−α22.13α13.371−α1,4.46α2.5.761−α22.46α14.231−α1,2.13α23.191−α2.Step-3 Now, we utilized our parametric interval valued Pythagorean weighted geometric mean operator (PIVPWGMO) considering the weight factors of attributes to calculate the matrixM in step 2, and we get the following:(26)M=.SC14.9346α17.49171−α1,0.8581α22.95461−α2C13.8891α17.35141−α1,0.9580α22.24181−α2C11.1482α13.18411−α1,7.0812α29.90081−α2C11.4718α13.74731−α1,1.9331α23.58351−α2.Step-4 Now, we apply our developed De-Pythagorean value to convert the fuzzy number into a crisp one considering=0.5,α1=0.5,α2=0.5 , and then we have the final decision matrix M as follows:(27)M=.SC10.9505C20.8949C31.0829C40.8040.Table 4
List of verbal phrase.
Sl no.AttributeVerbal phraseQuantitative attributes1Quantity of the commodityVery high (VH), high (L), intermediate (I), small (S), very small (VS)2Legibility of the commodityVery high (VH), high (H), mid (M), low (L), very low (VL)3Price rate of the commodityVery high (VH), high (H), mid (M), low (l), very low (VL)4Service life of the commodityVery high (VH), high (H), mid (M), low (l), very low (VL)Table 5
The verbal matrix.
S1S2S3S4C1LILVHC2VLMHVHC3LIIHC4VLLMHFrom the final decision matrix we can say that the ranking will beC3>C1>C2>C4.


### 6.3. Comparison of Work

We compare our work with other previously established work described by the authors 31,8,32, for finding the best alternatives wheneverρ=0.5,α1=0.5,α2=0.5. The comparison result has been presented in Table 6.Table 6
Comparison of work.
ApproachRankingYe [7]C3>C1>C2>C4Garg [18]C3>C1>C2>C4Li et al. [42]C3>C1>C4>C2Our proposed modelC3>C1>C2>C4From Table5, we notice that C3 has been declared the best alternative by our model, which matches with the outcomes of other researchers, namely Ye [7], Garg [18], Li et al. [42], although we have taken the parametric interval valued form of Pythagorean number which is more generalized from in compare to the established forms. Thus, we may conclude that PIVPN is a more robust structure that is reliable in the decision-making process.


### 6.4. Sensitivity Analysis

The main target of performing the sensitivity analysis is to check the efficiency, effectiveness, and stability of the proposed work. Here, we performed a well-organized sensitivity analysis by exchanging the weights of the attributes and values of the parameterρ keeping values of other terms unchanged. For different values of α1,α2,ρ the change of ranking have been shown in Table 7.Table 7
Sensitivity analysis.
Value ofα1Value ofα2Value ofρFinal decisionMatrix-ranking0.60.30.1.SC10.8163C20.7300C31.5915C40.8979C3>C4>C1>C20.2.SC10.8602C20.7734C31.4890C40.8803C3>C4>C1>C20.3.SC10.9040C20.8167C31.3865C40.8627C3>C1>C4>C20.4.SC10.94479C20.8600C31.2840C40.8450C3>C1>C2>C40.5.SC10.9917C20.9033C31.1815C40.8274C3>C1>C2>C40.6.SC11.0355C20.9467C31.0790C40.8098C3>C1>C2>C40.7.SC11.0794C20.9900C30.9765C40.7922C1>C2>C3>C40.8.SC11.1232C21.0333C30.8739C40.7746C1>C2>C3>C40.9.SC11.1671C21.0767C30.7714C40.7569C1>C2>C4>C30.50.50.1.SC10.7191C20.6742C31.3781C40.8175C3>C4>C1>C20.2.SC10.7770C20.7294C31.3043C40.8141C3>C4>C1>C20.3.SC10.8348C20.7846C31.2305C40.8108C3>C1>C4>C20.4.SC10.8926C20.8397C31.1567C40.8074C3>C1>C2>C40.5.SC10.9505C20.8949C31.0829C40.8040C3>C1>C2>C40.6.SC11.0083C20.9501C31.0092C40.8007C3>C1>C2>C40.7.SC11.0661C21.0053C30.9354C40.7973C1>C2>C3>C40.8.SC11.1240C21.0605C30.8616C40.7940C1>C2>C3>C40.9.SC11.1818C21.1157C30.7878C40.7906C1>C2>C3>C40.30.40.1.SC10.7854C20.7250C31.4650C40.8679C3>C4>C1>C20.2.SC10.8567C20.7984C31.3937C40.8727C3>C4>C1>C20.3.SC10.9280C20.8718C31.3223C40.8775C3>C1>C4>C20.4.SC10.9993C20.9453C31.2510C40.8823C3>C1>C2>C40.5.SC11.0706C21.0187C31.1796C40.8871C3>C1>C2>C40.6.SC11.1419C21.0922C31.1083C40.8919C1>C3>C2>C40.7.SC11.2133C21.1656C31.0370C40.8967C1>C2>C3>C40.8.SC11.2846C21.2390C30.9656C40.9014C1>C2>C3>C40.9.SC11.3559C21.3125C30.8943C40.9062C1>C2>C4>C3


#### 6.4.1. Graphical Chart

Here, we computed the graphical chart of comparison of work in Figures7–9 which shows the ranking of the alternatives for different values of α1,α2 and ρ.Here, Series-1, Series-2, Series-3, and Series-4 denote the alternatives Commodity 1, Commodity 2, Commodity 3, and Commodity 4, respectively.Figure 7
Chart for comparison of ranking whenα1=0.6, α2=0.3, row (ρ) varies from 0.1 to 0.9.Figure 8
Chart for comparison of ranking whenα1=0.3, α2=0.4, row (ρ) varies from 0.1 to 0.9.Figure 9
Chart for comparison of ranking whenα1=0.5, α2=0.5, row (ρ) varies from 0.1 to 0.9.


## 6.1. Illustration of the MCGDM Problem

We consider the problem as follows:Letπ=π1,π2,π3…πm. be “m” number of distinctive alternatives and σ=σ1,σ2,σ3………..σn. be “n” number of distinctive attribute values, respectively. Let δ=δe1,δe2,δe3………..δen. be the weight set associated with the attributes σ where each δe≥ 0 and also satisfies the relation ∑i=1nδei=1. We also consider the set of decision-maker μ=μ1,μ2,μ3………..μK associated with alternatives whose weight vector is defined as φ=φ1,φ2,φ3………..φk where each φi≥ 0 and also satisfies the relation ∑i=1kφi=1. This weight vector will be selected according to the decision-makers quality of judgment, knowledge, thinking power, etc. The flowchart of the proposed methodology is presented in Figure 6.Step-1 Creation of Decision MatricesFirst, we formulate the decision matrices for each decision maker’s choice related to alternatives versus attribute functions. We consider the elements of the matrices in IVPN in parametric form, so allaij's are members of the Pythagorean fuzzy number set. The related matrix is described as follows:(19)CK=.σ1σ2σ3...σnπ1a11ka12ka13k....a1nkπ2a21ka22ka23k...a2nkπ3.πm...am1k..am2k..am3k...........amnk.Step-2 Creation of Single decision matrixTo create the single group decision matrix, we utilize the operator PIVPWAMO for each decision matrixCi . Thus, we get the modified matrix as follows:(20)C=.σ1σ2σ3...σnπ1a11′a12′a13′....a1n′π2a21′a22′a23′...a2n′π3.πm..am1′..am2′..am3′...........amn′.Step-3 Creation of Final matrix according to weight PriorityTo obtain a single Column decision matrix, we all use the operator PIVPWGMO for each Column, and hence we get the decision matrix as follows:(21)C=.σ1π1a11″π2a21″....πmam1″.Step-4 RankingNow, we consider the De-Pythagorean value and convert the matrix (3) into a crisp one such that we can evaluate the best alternative corresponding to the best attributes.Figure 6
Flowchart of the MCGDM methodology.


## 6.2. Illustrative Example

Let us consider a problem associated with four distinctive products and their three different attributes. Generally, lots of commodities are available in the market, having different types of components with different quality and facilities. Now in this problem, there are four different kinds of decision-makers available. We choose the problem as follows:C1= Comodity 1, C2= Comodity 2, C3= Comodity 3, C4= Comodity 4 are the different kinds of alternatives. S1= Quality, S2= longibity, S3= Price, S4= Service are the four different attributes. There are three distinct types of decision-makers classified according to their ages D1= Young aged people,D2= Middle-aged people,D3= Old aged people having corresponding weight set D=0.30,0.33,0.37 and we also consider the weight function related to the attribute function σ=0.35,0.33,0.32. We consider a verbal matrix for the designer to assist the decision-maker in the creation of his decision matrix. The verbal phrases are listed in Table 4 and the verbal matrix of this MCGDM problem is shown in Table 5.Step-1 All the members of the matrices are PIVPN in nature. So, the decision matrices are as follows:(22)M1=⋅S1S2S3C15.5α16.51−α1,1.3α22.81−α25.3α16.51−α1,6.7α28.61−α25.8α16.31−α1,0.9α22.51−α2C26.3α17.21−α1,1.5α22.31−α25.0α16.01−α1,1.9α22.51−α24.2α16.01−α1,1.5α22.21−α2C31.2α12.41−α1,5.9α26.41−α21.9α12.51−α1,9.0α29.91−α20.9α12.51−α1,7.2α28.31−α2C44.0α15.31−α1,5.5α26.51−α22.5α13.01−α1,5.2α26.51−α23.2α14.61−α1,2.5α23.01−α2.For decision makerD1(23)M2=⋅S1S2S3C14.5α17.51−α1,1.3α22.81−α25.3α16.51−α1,6.7α28.61−α25.8α16.31−α1,0.9α22.51−α2C25.3α17.21−α1,1.2α22.81−α25.0α16.01−α1,1.9α22.51−α24.2α16.01−α1,1.5α22.21−α2C32.2α25.41−α2,7.9α38.41−α31.9α22.51−α2,9.0α39.91−α30.9α22.51−α2,7.2α38.31−α3C43.0α25.81−α2,4.5α36.81−α32.5α23.01−α2,5.2α36.51−α33.2α14.61−α1,2.5α23.01−α2.For decision makerD2(24)M3=⋅S1S2S3C12.5α14.51−α1,2.3α23.31−α24.3α16.81−α1,3.7α25.61−α25.2α17.31−α1,1.9α24.51−α2C26.0α18.01−α1,2.5α26.31−α25.2α16.81−α1,2.9α24.51−α23.2α16.51−α1,2.5α24.21−α2C37.2α19.41−α1,2.9α25.41−α23.9α14.51−α1,7.0α28.91−α22.9α15.51−α1,4.2α26.31−α2C42.0α16.31−α1,4.5α27.51−α21.5α14.01−α1,3.2α24.51−α21.2α13.61−α1,1.5α23.51−α2.For decision makerD3Step-2Now, we utilized our parametric interval valued Pythagorean weighted arithmetic mean operator (PIVPWAMO)to calculate all individual decision matrices and we get the final weighted normalized matrix as follows:(25)M=⋅S1S2S3C14.06α16.091−α1,1.67α22.991−α24.93α16.611−α1,5.59α27.491−α25.58α16.671−α1,1.27α23.241−α2C25.86α17.501−α1,1.77α23.951−α25.08α16.301−α1,2.27α23.241−α23.83α16.191−α1,1.87α22.941−α2C33.75α15.981−α1,5.45α26.691−α22.64α13.241−α1,8.26α29.531−α21.64α13.611−α1,6.09α27.561−α2C43.75α15.841−α1,4.80α26.971−α22.13α13.371−α1,4.46α2.5.761−α22.46α14.231−α1,2.13α23.191−α2.Step-3 Now, we utilized our parametric interval valued Pythagorean weighted geometric mean operator (PIVPWGMO) considering the weight factors of attributes to calculate the matrixM in step 2, and we get the following:(26)M=.SC14.9346α17.49171−α1,0.8581α22.95461−α2C13.8891α17.35141−α1,0.9580α22.24181−α2C11.1482α13.18411−α1,7.0812α29.90081−α2C11.4718α13.74731−α1,1.9331α23.58351−α2.Step-4 Now, we apply our developed De-Pythagorean value to convert the fuzzy number into a crisp one considering=0.5,α1=0.5,α2=0.5 , and then we have the final decision matrix M as follows:(27)M=.SC10.9505C20.8949C31.0829C40.8040.Table 4
List of verbal phrase.
Sl no.AttributeVerbal phraseQuantitative attributes1Quantity of the commodityVery high (VH), high (L), intermediate (I), small (S), very small (VS)2Legibility of the commodityVery high (VH), high (H), mid (M), low (L), very low (VL)3Price rate of the commodityVery high (VH), high (H), mid (M), low (l), very low (VL)4Service life of the commodityVery high (VH), high (H), mid (M), low (l), very low (VL)Table 5
The verbal matrix.
S1S2S3S4C1LILVHC2VLMHVHC3LIIHC4VLLMHFrom the final decision matrix we can say that the ranking will beC3>C1>C2>C4.


## 6.3. Comparison of Work

We compare our work with other previously established work described by the authors 31,8,32, for finding the best alternatives wheneverρ=0.5,α1=0.5,α2=0.5. The comparison result has been presented in Table 6.Table 6
Comparison of work.
ApproachRankingYe [7]C3>C1>C2>C4Garg [18]C3>C1>C2>C4Li et al. [42]C3>C1>C4>C2Our proposed modelC3>C1>C2>C4From Table5, we notice that C3 has been declared the best alternative by our model, which matches with the outcomes of other researchers, namely Ye [7], Garg [18], Li et al. [42], although we have taken the parametric interval valued form of Pythagorean number which is more generalized from in compare to the established forms. Thus, we may conclude that PIVPN is a more robust structure that is reliable in the decision-making process.


## 6.4. Sensitivity Analysis

The main target of performing the sensitivity analysis is to check the efficiency, effectiveness, and stability of the proposed work. Here, we performed a well-organized sensitivity analysis by exchanging the weights of the attributes and values of the parameterρ keeping values of other terms unchanged. For different values of α1,α2,ρ the change of ranking have been shown in Table 7.Table 7
Sensitivity analysis.
Value ofα1Value ofα2Value ofρFinal decisionMatrix-ranking0.60.30.1.SC10.8163C20.7300C31.5915C40.8979C3>C4>C1>C20.2.SC10.8602C20.7734C31.4890C40.8803C3>C4>C1>C20.3.SC10.9040C20.8167C31.3865C40.8627C3>C1>C4>C20.4.SC10.94479C20.8600C31.2840C40.8450C3>C1>C2>C40.5.SC10.9917C20.9033C31.1815C40.8274C3>C1>C2>C40.6.SC11.0355C20.9467C31.0790C40.8098C3>C1>C2>C40.7.SC11.0794C20.9900C30.9765C40.7922C1>C2>C3>C40.8.SC11.1232C21.0333C30.8739C40.7746C1>C2>C3>C40.9.SC11.1671C21.0767C30.7714C40.7569C1>C2>C4>C30.50.50.1.SC10.7191C20.6742C31.3781C40.8175C3>C4>C1>C20.2.SC10.7770C20.7294C31.3043C40.8141C3>C4>C1>C20.3.SC10.8348C20.7846C31.2305C40.8108C3>C1>C4>C20.4.SC10.8926C20.8397C31.1567C40.8074C3>C1>C2>C40.5.SC10.9505C20.8949C31.0829C40.8040C3>C1>C2>C40.6.SC11.0083C20.9501C31.0092C40.8007C3>C1>C2>C40.7.SC11.0661C21.0053C30.9354C40.7973C1>C2>C3>C40.8.SC11.1240C21.0605C30.8616C40.7940C1>C2>C3>C40.9.SC11.1818C21.1157C30.7878C40.7906C1>C2>C3>C40.30.40.1.SC10.7854C20.7250C31.4650C40.8679C3>C4>C1>C20.2.SC10.8567C20.7984C31.3937C40.8727C3>C4>C1>C20.3.SC10.9280C20.8718C31.3223C40.8775C3>C1>C4>C20.4.SC10.9993C20.9453C31.2510C40.8823C3>C1>C2>C40.5.SC11.0706C21.0187C31.1796C40.8871C3>C1>C2>C40.6.SC11.1419C21.0922C31.1083C40.8919C1>C3>C2>C40.7.SC11.2133C21.1656C31.0370C40.8967C1>C2>C3>C40.8.SC11.2846C21.2390C30.9656C40.9014C1>C2>C3>C40.9.SC11.3559C21.3125C30.8943C40.9062C1>C2>C4>C3


### 6.4.1. Graphical Chart

Here, we computed the graphical chart of comparison of work in Figures7–9 which shows the ranking of the alternatives for different values of α1,α2 and ρ.Here, Series-1, Series-2, Series-3, and Series-4 denote the alternatives Commodity 1, Commodity 2, Commodity 3, and Commodity 4, respectively.Figure 7
Chart for comparison of ranking whenα1=0.6, α2=0.3, row (ρ) varies from 0.1 to 0.9.Figure 8
Chart for comparison of ranking whenα1=0.3, α2=0.4, row (ρ) varies from 0.1 to 0.9.Figure 9
Chart for comparison of ranking whenα1=0.5, α2=0.5, row (ρ) varies from 0.1 to 0.9.


## 6.4.1. Graphical Chart

Here, we computed the graphical chart of comparison of work in Figures7–9 which shows the ranking of the alternatives for different values of α1,α2 and ρ.Here, Series-1, Series-2, Series-3, and Series-4 denote the alternatives Commodity 1, Commodity 2, Commodity 3, and Commodity 4, respectively.Figure 7
Chart for comparison of ranking whenα1=0.6, α2=0.3, row (ρ) varies from 0.1 to 0.9.Figure 8
Chart for comparison of ranking whenα1=0.3, α2=0.4, row (ρ) varies from 0.1 to 0.9.Figure 9
Chart for comparison of ranking whenα1=0.5, α2=0.5, row (ρ) varies from 0.1 to 0.9.


## 7. Numerical Analysis

In this section, we consider our sensitivity results and we did a numerical analysis whenever the value of Row (ρ actually varies from 0.05 to 0.95. For different three sets of values of α1,α2, we got the following figures numerically (Figures 10–12),Figure 10
Graph of de-Pythagorean value of different alternatives forα1=0.3, α2=0.4.Figure 11
Graph of de-Pythagorean value of different alternatives forα1=0.5, α2=0.5.Figure 12
Graph of de-Pythagorean value of different alternatives forα1=0.6, α2=0.3.


### 7.1. Discussion

After the rigorous sensitivity analysis, numerical simulations, and graphical analysis, we have observed that in Figure8, alternatives 3, 1 change their position in a range of (0.55, 0.6) and become the best one. Now, if we observe Figure 9, then we can conclude that alternatives 3, 1 change their position in a range of (0.6, 0.62) and become the best one. Lastly, if we observe Figure 10, then we can conclude that the alternatives 3, 1 change their position in a range of (0.6, 0.68) and becomes the best one. Finally, from the sensitivity analysis, we can conclude that the ranking of the alternatives depends on the range of the parameter, which plays an essential role in this proposed method.


### 7.2. Advantages and Limitations of the Proposed Method

(1)
After this scientific discussion mentioned above, we have noticed that the existing uncertain parameters such as FS, IFS, PFS can be easily obtained from PIVPNs, and hence PIVPNs are a more well-organized representation to grip a problem more effectively where the existing theories fail. Moreover, PIVPNs can grab the inconsistent condition in a more compatible way rather than other existing methods. Henceforth, the complete studies under the PIVPN environment are effective in solving MCGDM problems.(2)
Through sensitivity analysis, numerical simulation, and graphical representation of the result, we have noticed that if we vary the weights of the attributes within a certain range, then the best alternatives may change their positions within the given range. That is, it is basically case dependent which shows the limitation of this proposed technique.


## 7.1. Discussion

After the rigorous sensitivity analysis, numerical simulations, and graphical analysis, we have observed that in Figure8, alternatives 3, 1 change their position in a range of (0.55, 0.6) and become the best one. Now, if we observe Figure 9, then we can conclude that alternatives 3, 1 change their position in a range of (0.6, 0.62) and become the best one. Lastly, if we observe Figure 10, then we can conclude that the alternatives 3, 1 change their position in a range of (0.6, 0.68) and becomes the best one. Finally, from the sensitivity analysis, we can conclude that the ranking of the alternatives depends on the range of the parameter, which plays an essential role in this proposed method.


## 7.2. Advantages and Limitations of the Proposed Method

(1)
After this scientific discussion mentioned above, we have noticed that the existing uncertain parameters such as FS, IFS, PFS can be easily obtained from PIVPNs, and hence PIVPNs are a more well-organized representation to grip a problem more effectively where the existing theories fail. Moreover, PIVPNs can grab the inconsistent condition in a more compatible way rather than other existing methods. Henceforth, the complete studies under the PIVPN environment are effective in solving MCGDM problems.(2)
Through sensitivity analysis, numerical simulation, and graphical representation of the result, we have noticed that if we vary the weights of the attributes within a certain range, then the best alternatives may change their positions within the given range. That is, it is basically case dependent which shows the limitation of this proposed technique.


## 8. Conclusion

The concept of PIVPN is logical and pragmatic and it has practical usefulness in the current research arena. In this research article, we have been extended the conception of the Pythagorean fuzzy set into PIVPN and developed some logical operators related to it, namely PIVPWAMO and PIVPWGMO. We also developed a score and accuracy function in the PIVPN environment and solved a critical path problem and performed a comparison with the other existing method to observe the advantage of this proposed model. Also, marketing based real life MCGDM problem is described in this article to find out the best alternatives whenever all the decision matrices members are in the PIVPN form. Applying the concept of developed operators, we solved this problem and we utilized the concept of De-Pythagorean value to short out the ranking of the alternatives. Finally, we performed a rigorous sensitivity analysis, numerical simulation, and comparison of work to check the utility, reliability, usefulness, and advantages of this proposed novel work.In a future study, we can extend some theoretical ideas of PIVPN and can form the structure of exponential operational law, and logarithmic operational law in the PIVPN environment. Further, we can apply it to different fields like cloud computing, image segmentation, pattern recognition, medical diagnosis, banking policy making problems, economic and social science based several real-life problems, etc.



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