# Approach to Multiattribute Decision-Making Problems Based on Neutrality Aggregation Operators of Picture Fuzzy Information

Authors: Mubashar Javed; Shumaila Javeed; Jihad Ahmad; Kifayat Ullah; Lemnaouar Zedam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This manuscript is aimed at developing some novel operational laws named scalar neutrality operation and neutrality addition on picture fuzzy numbers (PFNs). The main focus of this work is to involve the neutral behaviour of the experts towards the priorities of entities where it presents equal degrees to independent membership functions. Moreover, based on these operations, some novel aggregation operators are established to aggregate the different priorities of experts. Some useful relations and characteristics are examined thoroughly. Lastly, the multiattribute group decision-making algorithm in accordance with the suggested operation is illustrated and examined a case study in order to choose a suitable mining company for a mining project along with several numerical examples. The advantages, as well as the superiority of the suggested approach, are exhibited by comparing the results with a few existing methods.


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## Body

## 1. Introduction

Zadeh’s idea of fuzzy set (FS) [1] opened a new horizon for the researchers to deal with real-life scenarios which includes uncertainty. Atanassov [2] generalized the idea of FSs and suggested the framework of intuitionistic fuzzy set (IFS) which provided us a significant tool to handle the imprecision. Atanassov and Gargov [3] and later on Atanassov [4] introduced interval valued intuitionistic fuzzy sets (IVIFSs) which used real numbers in the form of intervals to represent a membership degree (MD), a nonmembership degree (NMD), and hesitancy level. Researchers have frequently applied the concept of FSs to many real-life situations where decision-making is one of the prominent areas among them. Some very interesting fields including Xu et al. [5, 6] developed clustering algorithms, De et al. and Xiao and Ding [7, 8] presented their theories on medical diagnosis, and studies on TOPSIS and TODIM methods were conducted by several authors [9–12]. Pamučar et al. [13] introduced the MABAC method for interval valued fuzzy rough numbers. Mu et al. [14] extended the study by working on Maclaurin symmetric mean based on interval valued Pythagorean fuzzy set (PyFS). As (3,2) FS has larger domain in terms of membership degree as compared to PyFS, so Ibrahim et al. [15] recently investigated the relation of (3,2) FS with other fuzzy sets. Zeng [16] made decision-making more effective by utilizing uncertain intelligence. Muhiuddin et al. [17, 18] recorded their extension in the field of interval valued m polar fuzzy structure. Pan and Deng [19] established a new similarity measure to discuss the clustering problems under IFS. Similarly, Yager [20, 21] adopted the fusion process to aggregate the information during the multiattribute decision-making (MADM) problems. Xu and Yager [22] introduced some innovative aggregation operators (AOs) under IFS environment. Wang and Liu [23] made extension to these AOs by using Einstein operations. Ghani and Isa [24] proposed some interactive AOs based on IFS. Xu et al. [25] established innovative operators by fusing Einstein operator under IFS. Wan et al. [26] have established a decision-making scheme by using IVIFS with the help of incomplete attribute weight. Recently, few authors utilized linear Diophantine fuzzy sets and presented their work [27, 28]. Some useful approaches [29–31] have been adopted to use the fuzzy information more smoothly Furthermore, many authors dealt with MADM problems with more powerful aggregating tools [32].It is clear from above studies that PFSs have the ability to handle the vagueness of data efficiently. Many researchers have taken keen interest in IVIFS and contributed a lot (see [33, 34] for reference). Although IFSs have been utilized to solve many issues, there are so many real-life situations which cannot be represented by using IFSs. One of them is voting which needs three independent functions membership, nonmembership, and neutral membership function to represent the human opinion. To handle such situations, Cuong [35, 36] proposed the concept of picture fuzzy set (PFS) and examined some basic characteristics of PFS. Singh [37] used correlation coefficients to conduct the clustering analysis for PFS. Son [38] presented a new clustering scheme and weather forecasting technique on the basis of PFS. Thong [39] applied PFNs on medical diagnosis to study and support the health care system. Wei and Jana et al. [40–42] presented AOs based on PFSs and solved MADMPs. Due to immense utility of the PFS and SFS, the authors really paid their attention and produced valuable researches [43–46]. Recently, Liu et al. [47], Ullah et al. [48, 49], and Akram et al. [50] made valuable addition in the field of pattern recognition, decision-making, and performance evaluation of solar energy cells by using similarity measures and aggregation operators under picture fuzzy and interval valued T-spherical fuzzy environment.The above-given studies show that different authors have made their contributions to establish different AOs under IFS, IVIFS, PFS, etc. to manage the ambiguities in the data. Yet, it has been observed that all are not unbiased solely. For instance, if a decision-maker has assigned equal degree to MD, NMD, and AD, then the comprehensive values are not equal by utilizing existing tools for aggregation. It shows that the obtained value is not neutral. In order to enhance the utility of the AOs, it is necessary to add the neutrality character to the operational laws and their respective AOs. To achieve this, we construct probability sum (PS) function and the interaction between the MD, NMD, and AD to establish some novel operational laws on PFS. Furthermore, we suggest the picture fuzzy weighted neutral averaging (PFWNA) operator and picture fuzzy ordered neutral averaging (PFOWNA) operator to aggregate the different values of the experts. Finally, we develop a scheme to treat the MAGDMPs and give a numerical example to explore the validity of the scheme.The rest of the paper is set out as follows. In the next section, we review some basic definitions regarding PFS. In Section3, we define some neutral operational laws and its salient features for PFS. In Section 4, we define AOs on the basis of newly proposed operational laws. An innovative algorithm is presented in Section 5 to manage the MAGDMPs. A feasibility study of the scheme and comparison study are conducted in Section 6. Finally, Section 7 gives us final notes to conclude the manuscript.


## 2. Preliminaries

In this section, some fundamental definitions associated with PFS are revised. In this manuscript,X denote a nonempty set and ṧ,ị,ḑ, and r denote membership degree (MD), abstinence degree (AD), nonmembership degree (NMD), and degree of refusal (RD), respectively. The concept of PFS was proposed by Cuong [36] by narrating the fuzzy data by using a MD, NMD, AD, and r.Definition 1 (see [36]).
A PFSÑ on X is defined as
(1)Ñ=x,ṧ,ị,ḑ:ṧ:X⟶0,1,ị:X⟶0,1andḑ:X⟶0,1∀x∈X,0≤ṧx+ịx+ḑx≤1,rx=1−ṧx+ịx+ḑx.Definition 2.
For two PFNsÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2,ị2,ḑ2 and a realλ>0, we have some dcharacteristic axioms defined as
(1)
Ñ1C=ḑ1,ị1,ṧ1(2)
Ñ1⊆Ñ2if ṧ1≤ṧ2, ị1≤ị2 and ḑ1≥ḑ2(3)
Ñ1=Ñ2if Ñ1⊆Ñ2 and Ñ2⊆Ñ1(4)
Ñ1⨁Ñ2=ṧ1+ṧ2,−ṧ1ṧ2,,ị1.ị2,ḑ1.ḑ2(5)
Ñ1⊗Ñ2=ṧ1ṧ2,,ị1+ị2−ị1ị2,ḑ1q+ḑ2q−ḑ1qḑ2q(6)
λÑ1=1−1−ṧ1qλ,ị1λ,ḑ1λ(7)
Ñ1λ=ṧ1λ,1−1−ị1λ,1−1−ḑ1λDefinition 3 (see [51]).
For a PFNÑ=ṧx,ịx,ḑx, the score function and accuracy function for PFNs are defined as under
(2)SCÑ=ṧx−ḑx,andSCÑ∈−1,1(3)ACÑ=ṧx+ịx+ḑx,andACÑ∈0,1Definition 4.
LetÑ1 and Ñ2 be two PFNs; SCÑis the “score function”; and ACÑis “accuracy function”; then, Ñ1>Ñ2, where > refers to “preferred to” if either SCÑ1>SCÑ2 or SCÑ1=SCÑ2 and ACÑ1>ACÑ2 hold.For a collection of PFNsÑj=ṧj,ịj,ḑj,j=1,2,⋯,n, the weighted averaging AOs are defined as follows.Definition 5 (see [52]).
LetÑj=ṧj,ịj,ḑj,j=1,2,⋯,n, be a collection of PFNs and ω=ω1,ω2,⋯,ωn be the weight vector of Ñj with ∑j=1nωj=1; the weighted averaging and ordered weighted averaging AOs are defined as
(4)PFWAÑ1,Ñ2,⋯,Ñn=1−∏j=1n1−ṧjωj,∏j=1nịjωj,∏j=1nḑjωj,PFOWAÑ1,Ñ2,⋯,Ñn=1−∏j=1n1−ṧσjωj,∏j=1nịσjωj,∏j=1nḑσjωj,where σ is a transformation map of 1,2,⋯,n such that Ñσj−1≥Ñσj for all j=2,3,⋯,n.


## 3. New Operational Laws on PFNs

It is not possible with the help of score function given in Definition3 to obtain the accurate ranking for all PFNs. For instance, Ñ1=0.3,0.2,0.25 and Ñ2=0.4.0.2,0.35 are two PFNs and by using score function according to Definition 3, we get SCÑ1=SCÑ2. But we are well aware thatÑ1≠Ñ2. To manage such cases under PFSs, we define a new score function for PFS.


### 3.1. A New Score Function

Definition 6.
For a PFNÑ=ṧ,ị,ḑ, an innovative score function is defined as
(5)SÑ=eṧx−ịx−ḑx1+r,where degree of refusal is denoted by r and calculated asr=1−ṧx−ịx−ḑx.Theorem 7.
LetÑ=ṧ,ị,ḑ be a PFN, and S is the score function; then, S monotonically increases for ṧ and decreases forịand ḑ.Proof.
To prove the required result, firstly, we partially differentiateS with respect to ṧ which gives ∂S/∂ṧ=2+reṧ−ị−ḑ/2−ṧ−ị−ḑ2≥0 and then with respect to ị and ḑ gives ∂S/∂ị=−reṧ−ị−ḑ/2−ṧ−ị−ḑ2≤0 and∂S/∂ḑ=−reṧ−ị−ḑ/2−ṧ−ị−ḑ2≤0. Hence, the result is proved.Theorem 8.
For PFNÑ, the score function S holds the following:
(1)
e−1≤S≤e(6)(2)
SÑ=e iff Ñ=1,0,0(3)
SÑ=e−1 iff Ñ=0,1,0 and Ñ=0,0,1
This result has trivial proof.
Proposed score functionS is validated with the help of comparison analysis given in Table 1. Clearly, we observe advantages of newly proposed score functions over the existing function. We justify our claim in Theorem 9.Theorem 9.
For two PFNsÑ1 andÑ2, ifSCÑ1=SCÑ2 and ACÑ1>ACÑ2, then SÑ1>SÑ2. Further, ifSCÑ1=SCÑ2 and ACÑ1=ACÑ2, then SÑ1=SÑ2.Proof.
LetÑ1=ṧ1,ị1,ḑ1and Ñ2=ṧ2.ị2,ḑ2 be two PFNs; then, by using Definition 4, ifSCÑ1=SCÑ2 and ACÑ1>ACÑ2, we haveṧ1−ḑ1=ṧ2,−ḑ2 and ṧ1+ị1+ḑ1>ṧ2,+ị2+ḑ2. Therefore, we can have SÑ1=e,ṧ1−ị1−ḑ1/2−ṧ1+ị1+ḑ12−ṧ1+ị1+ḑ1≥eṧ2,−ị2−ḑ2/2−ṧ2,+ị2+ḑ22−ṧ2,+ị2+ḑ2=SÑ2.Table 1
Comparison of score functions.
ExamplesScore values by using Equation (2)Comparison. By using Equation (5)OrderingÑ1=0.3,0.2,0.25Ñ2=0.4.0.2,0.35SCÑ1=SCÑ2SÑ1<SÑ2Ñ1<Ñ2Ñ1=0.7,0.1,0Ñ2=0.8.0.05,0.1SCÑ1=SCÑ2SÑ1<SÑ2Ñ1<Ñ2Ñ1=0.45,0.15,0.35Ñ2=0.4.0.2,0.3SCÑ1=SCÑ2SÑ1>SÑ2Ñ1>Ñ2In the next subsection, we intend to establish some novel operations on the basis of probability sum (PS) and interaction coefficients. Major advantage of such newly proposed operation for PFS environment can be obtained with a fair decision in the case when all the membership degrees are equal for a PFN.


### 3.2. Meaning of PS and Rules of PFNs

For PFNsÑ1=ṧ1,ị1,ḑ1 andÑ2=ṧ2.ị2,ḑ2, the geometric characteristic of the (PS) is explained in Figure 1. In this figure, ṧ1,ṧ2 represents the MDs of Ñ1 and Ñ2 while ị1,ị2 and ḑ1,ḑ2 represent the ADs and NMDs, respectively. Obviously, ṧ1+ị1+ḑ1 and ṧ2,+ị2+ḑ2 are two such events which do not depend on each other for PSF. Expressṧε+ịε+ḑε to be the PS of occurring at least independent events ofṧ1+ị1+ḑ1 and ṧ2,+ị2+ḑ2; thus,
(6)ṧε+ịε+ḑε=PSṧ1+ị1+ḑ1,ṧ2,+ị2+ḑ2.Figure 1
The geometric interpretation of PS function.From it, we observe that the aggregated values of the PFNsÑ1and Ñ2 can be accomplished by the following:
(1)
Aggregate the MD, NMD, AD, and RD of the PFNsÑ1and Ñ2 by utilizing the algebraic sum operations Tx,y=x+y; as a result, we obtainṧ1+ṧ2,ị1+ị2,ḑ1+ḑ2 and r1+r2 as it is shown with the help of different colours in Figure 2. It is absolutely apparent that their sum exceeds from 1 which is no more a PFN(2)
In order to calculate the overall aggregated values as PFN, we assign a mutual interaction coefficient1−r1r2./ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2 to ṧ1+ṧ2, ị1+ị2, and ḑ1+ḑ2 which can be considered as MD, AD, and NMD. Also, their sum is not greater than 1 and corresponding RD is r1r2.. So(7)ṧε=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ịε=ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ḑε=ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2Figure 2
Geometrical description of the suggested operation.Moreover, PS function used in Equation (6) ist -conorm for PFS and properties like commutativity, associativity, monotonicity, and boundedness which are satisfied. Hence, on the basis of this erudition, we intend to establish some novel operation laws under PFS environment.


### 3.3. Neutral Operational Laws

This section presents novel operational rules for PFS by fusing the neutral attitude of the experts into preferences.Definition 10.
LetÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2 be two PFNs. The neutrality operation of Ñ1and Ñ2 is given as
(8)Ñ1⊝Ñ2=MCSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2,NCSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2ACSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2,,
whereMCSÑ1,Ñ2=ṧ1+ṧ2, NCSÑ1,Ñ2=ḑ1+ḑ2, and ACSÑ1,Ñ2=ị1+ị2 represent the MD, NMD, and AD coefficient sum of Ñ1and Ñ2, respectively. Also, PSx,y=1−1−x1−y is used to represent the PS of the FN Ñ1 and Ñ2.
ForÑ1=ṧ1,ị1,ḑ1 and real τ>1, we get
(9)PSτṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSτ−1ṧ1+ị1+ḑ1.
Hence, we asserted following propositions.Proposition 11.
For PFNÑ1=ṧ1,ị1,ḑ1 and a realτ>0, we get
(10)PSτṧ1+ị1+ḑ1=1−r1τ.Proof.
We apply mathematical induction onτ.
Step 1. Forτ=2 and by using Equation (9), we have
(11)PSτṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSτ−1ṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,1−r1=1−1−ṧ1−ị1−ḑ11−1+r1=1−r12.
The given statement is satisfied forτ=2.
Step 2. Assume Equation (10) is true forτ=n, then, for τ=n+1,(12)PSn+1ṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSnṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,1−r1n=1−1−ṧ1−ị1−ḑ11−1+r1n=1−r1n+1.
Hence, by using induction, Equation (10) is satisfied for allτ.
For a group of “n,” PFNs Ñj=ṧj,ịj,ḑj,j=1,2,⋯,n, such that MCSÑj=ṧj andMCSÑ1,⋯Ñn=MCSÑ1,⋯Ñn−1+ṧn. Thus, we haveMCSÑ1,⋯Ñn=∑j=1nṧj. Likewise, we haveACSÑ1,⋯Ñn=∑j=1nịj andNCSÑ1,⋯Ñn=∑j=1nḑj. By using this, we have the below given proposition.Proposition 12.
For PFNÑ1=ṧ1,ị1,ḑ1 and a real number τ>0, we have MCSτÑ1=τ.MCSÑ1,NCSτÑ1=τ.NCSÑ1, andACSτÑ1=τ.ACSÑ1.Proof.
If we takeÑ1=Ñ2 in MCS, thenMCS2Ñ1=MCSÑ1,Ñ1=ṧ1+ṧ1=2MCSÑ1. Applying induction on τ, we can easily have thatMCSτÑ1=τ.MCSÑ1, NCSτÑ1=τ.NCSÑ1 and ACSτÑ1=τ.ACSÑ1.Definition 13.
For PFNÑ1=ṧ1,ị1,ḑ1 and a realτ≥0, the “scalar neutrality operation” is specified as
(13)τ.Ñ1=MCSτ.Ñ1MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1,NCSτ.Ñ1MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1ACSÑ1,Ñ2MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1,,whereτÑ1=τṧ1,NCSτÑ1=τḑ1, and ACSτÑ1=τị1.
From Equation (8) and Equation (13), we note that decision-makers have neutral behaviour to MD, NMD, and AD which is shown in following preposition.Proposition 14.
For PFNsÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2, if ṧ1=ị1=ḑ1 and ṧ2=ị2=ḑ2, then ṧÑ1⊝Ñ2=ḑÑ1⊝Ñ2=ịÑ1⊝Ñ2.Proof.
Ifṧ1=ị1=ḑ1 andṧ2=ị2=ḑ2, then by using the neutrality operation, we have ṧÑ1⊝Ñ2/ịÑ1⊝Ñ2=MCSÑ1,Ñ2/ACSÑ1,Ñ2=ṧ1+ṧ2/ị1+ị2=1. Also, ṧÑ1⊝Ñ2/ḑÑ1⊝Ñ2=MCSÑ1,Ñ2/NCSÑ1,Ñ2=ṧ1+ṧ2/ḑ1+ḑ2=1. Thus, ṧÑ1⊝Ñ2=ḑÑ1⊝Ñ2=ịÑ1⊝Ñ2. Hence, it makes clear that suggested operations provide us fair decision even if the PFN has equal degrees.Remark 15.
This fact is observed from Definition2 that when ṧ1=ị1=ḑ1 and ṧ2=ị2=ḑ2, we can obtain ṧÑ1⊝Ñ2≠ḑÑ1⊝Ñ2≠ịÑ1⊝Ñ2. Therefore, it shows the neutrality of newly proposed operations.
Now, by utilizing the definition of MCS, NCS, ACS, and PS function, we can write Equation (8) as
(14)Ñ1⊝Ñ2=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2.,
Subsequently, scalar neutrality operation is introduced by using Definition13. Equation (8) gives
(15)Ñ1⊝Ñ1=2Ñ1=MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1,MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1,=ṧ1ṧ1+ị1+ḑ11−r12,ị1ṧ1+ị1+ḑ11−r12,ḑ1ṧ1+ị1+ḑ11−r12.
Similarly,(16)3Ñ1=ṧ1ṧ1+ị1+ḑ11−r13,ị1ṧ1+ị1+ḑ11−r13,ḑ1ṧ1+ị1+ḑ11−r13.
In the same way, for any real positiveτ, we have
(17)τÑ1=ṧ1ṧ1+ị1+ḑ11−r1τ,ị1ṧ1+ị1+ḑ11−r1τ,ḑ1ṧ1+ị1+ḑ11−r1τ.Proposition 16.
The MCS, NCS, and ACS for two PFNsÑ1 and Ñ2 satisfy the following properties andτ,τ1, τ2 are positive real numbers; then,
(1)
MCSÑ1,Ñ2=MCSÑ2,Ñ1(2)
MCSτÑ1,Ñ2=MCSτÑ1,τÑ2(3)
MCSτ1Ñ1,τ2Ñ1=MCSτ1+τ21Ñ1(4)
NCSÑ1,Ñ2=NCSÑ2,Ñ1(5)
NCSτÑ1,Ñ2=NCSτÑ1,τÑ2(6)
NCSτ1Ñ1,τ2Ñ1=NCSτ1+τ2Ñ1(7)
ACSÑ1,Ñ2=ACSÑ2,Ñ1(8)
ACSτÑ1,Ñ2=ACSτÑ1,τÑ2(9)
ACSτ1Ñ1,τ2Ñ1=ACSτ1+τ21Ñ1Proof.
For PFNsÑ1=ṧ1,ị1,ḑ1andÑ2=ṧ2.ị2,ḑ2, we have MCSÑ1,Ñ2=ṧ1+ṧ2, NCSÑ1,Ñ2=ḑ1+ḑ2, and ACSÑ1,Ñ2=ị1+ị2 and MCSτÑ1=τṧ1, NCSτÑ1=τḑ1, and ACSτÑ1=τị1. Thus, MCSτÑ1,τÑ2=τṧ1+τṧ2=τṧ1+ṧ2=τMCSÑ1,Ñ2=MCSτÑ1,Ñ2. Similarly, the remaining parts can be proven


### 3.4. Characteristics of the Proposed Operations

The next subsection discusses some remarkable properties of the suggested operational laws as under.Theorem 17.
IfÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2 are two distinct PFNs, then operations of Ñ1 andÑ2, Ñ1⊝Ñ2, and τÑ1 are also PFNs forτ>0.Proof.
For PFNsÑ1=ṧ1,ị1,ḑ1 andÑ2=ṧ2.ị2,ḑ2, we haveṧ1,ị1,ḑ1ṧ2,ị2,ḑ2∈0,1, r1=1−ṧ1−ị1−ḑ1∈0,1 andr2=1−ṧ2−ị2−ḑ2∈0,1. Let Ñ1⊝Ñ2=ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2. To obtain the required result Ñ1⊝Ñ2∈ PFN, we will just show that ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2∈0,1 and ṧÑ1⊝Ñ2+ịÑ1⊝Ñ2+ḑÑ1⊝Ñ2≤1. Sinceṧ1+ị1+ḑ1≤1, ṧ2+ị2+ḑ2≤1 and hence, ṧ1+ṧ2/ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2≤1, providedṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2≠0. Also, r1,r2∈0,1 which implies that1−r1r2∈0,1. Hence, it can be obtained that ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2∈0,1. Further, ṧÑ1⊝Ñ2+ịÑ1⊝Ñ2+ḑÑ1⊝Ñ2=1−r1r2∈0,1. Hence, Ñ1⊝Ñ2 is a PFN. Similarly, τÑ1 can also be proven as a PFN.Theorem 18.
LetÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2be two PFN and τ,τ1τ2≥0 be a real number; then,
(1)
Ñ1⊝Ñ2=Ñ2⊝Ñ1(2)
τÑ1⊝Ñ2=τÑ1⊝τβ2(3)
τ1Ñ1⊝τ2Ñ1=τ1+τ2Ñ1Proof.
(1)
Easily can be followed from Equation (14)(2)
For PFNsÑ1and Ñ2,we get(18)τÑ1⊝τÑ2=MCSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2,NCSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2,ACSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2=τṧ1+ṧ2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ,τị1+ị2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ,τḑ1+ḑ2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ,ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2,ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2=τÑ1⊝Ñ2.
Forτ1,τ2>0,
(19)τ1Ñ1=ṧ1ṧ1+ị1+ḑ11−r1τ1,ị1ṧ1+ị1+ḑ11−r1τ1,ḑ1ṧ1+ị1+ḑ11−r1τ1,τ2Ñ1=ṧ1ṧ1+ị1+ḑ11−r1τ2,ị1ṧ1+ị1+ḑ11−r1τ2,ḑ1ṧ1+ị1+ḑ11−r1τ2.
Thus, by using Equation (8), we get
(20)τ1Ñ1⊝τ2Ñ1=MCSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2,NCSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2ACSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2,=ṧ1ṧ1+ị1+ḑ11−r1τ1+τ2,ị1ṧ1+ị1+ḑ11−r1τ1+τ2,ḑ2ṧ1+ị1+ḑ11−r1τ1+τ2=τ1+τ2Ñ1.


## 3.1. A New Score Function

Definition 6.
For a PFNÑ=ṧ,ị,ḑ, an innovative score function is defined as
(5)SÑ=eṧx−ịx−ḑx1+r,where degree of refusal is denoted by r and calculated asr=1−ṧx−ịx−ḑx.Theorem 7.
LetÑ=ṧ,ị,ḑ be a PFN, and S is the score function; then, S monotonically increases for ṧ and decreases forịand ḑ.Proof.
To prove the required result, firstly, we partially differentiateS with respect to ṧ which gives ∂S/∂ṧ=2+reṧ−ị−ḑ/2−ṧ−ị−ḑ2≥0 and then with respect to ị and ḑ gives ∂S/∂ị=−reṧ−ị−ḑ/2−ṧ−ị−ḑ2≤0 and∂S/∂ḑ=−reṧ−ị−ḑ/2−ṧ−ị−ḑ2≤0. Hence, the result is proved.Theorem 8.
For PFNÑ, the score function S holds the following:
(1)
e−1≤S≤e(6)(2)
SÑ=e iff Ñ=1,0,0(3)
SÑ=e−1 iff Ñ=0,1,0 and Ñ=0,0,1
This result has trivial proof.
Proposed score functionS is validated with the help of comparison analysis given in Table 1. Clearly, we observe advantages of newly proposed score functions over the existing function. We justify our claim in Theorem 9.Theorem 9.
For two PFNsÑ1 andÑ2, ifSCÑ1=SCÑ2 and ACÑ1>ACÑ2, then SÑ1>SÑ2. Further, ifSCÑ1=SCÑ2 and ACÑ1=ACÑ2, then SÑ1=SÑ2.Proof.
LetÑ1=ṧ1,ị1,ḑ1and Ñ2=ṧ2.ị2,ḑ2 be two PFNs; then, by using Definition 4, ifSCÑ1=SCÑ2 and ACÑ1>ACÑ2, we haveṧ1−ḑ1=ṧ2,−ḑ2 and ṧ1+ị1+ḑ1>ṧ2,+ị2+ḑ2. Therefore, we can have SÑ1=e,ṧ1−ị1−ḑ1/2−ṧ1+ị1+ḑ12−ṧ1+ị1+ḑ1≥eṧ2,−ị2−ḑ2/2−ṧ2,+ị2+ḑ22−ṧ2,+ị2+ḑ2=SÑ2.Table 1
Comparison of score functions.
ExamplesScore values by using Equation (2)Comparison. By using Equation (5)OrderingÑ1=0.3,0.2,0.25Ñ2=0.4.0.2,0.35SCÑ1=SCÑ2SÑ1<SÑ2Ñ1<Ñ2Ñ1=0.7,0.1,0Ñ2=0.8.0.05,0.1SCÑ1=SCÑ2SÑ1<SÑ2Ñ1<Ñ2Ñ1=0.45,0.15,0.35Ñ2=0.4.0.2,0.3SCÑ1=SCÑ2SÑ1>SÑ2Ñ1>Ñ2In the next subsection, we intend to establish some novel operations on the basis of probability sum (PS) and interaction coefficients. Major advantage of such newly proposed operation for PFS environment can be obtained with a fair decision in the case when all the membership degrees are equal for a PFN.


## 3.2. Meaning of PS and Rules of PFNs

For PFNsÑ1=ṧ1,ị1,ḑ1 andÑ2=ṧ2.ị2,ḑ2, the geometric characteristic of the (PS) is explained in Figure 1. In this figure, ṧ1,ṧ2 represents the MDs of Ñ1 and Ñ2 while ị1,ị2 and ḑ1,ḑ2 represent the ADs and NMDs, respectively. Obviously, ṧ1+ị1+ḑ1 and ṧ2,+ị2+ḑ2 are two such events which do not depend on each other for PSF. Expressṧε+ịε+ḑε to be the PS of occurring at least independent events ofṧ1+ị1+ḑ1 and ṧ2,+ị2+ḑ2; thus,
(6)ṧε+ịε+ḑε=PSṧ1+ị1+ḑ1,ṧ2,+ị2+ḑ2.Figure 1
The geometric interpretation of PS function.From it, we observe that the aggregated values of the PFNsÑ1and Ñ2 can be accomplished by the following:
(1)
Aggregate the MD, NMD, AD, and RD of the PFNsÑ1and Ñ2 by utilizing the algebraic sum operations Tx,y=x+y; as a result, we obtainṧ1+ṧ2,ị1+ị2,ḑ1+ḑ2 and r1+r2 as it is shown with the help of different colours in Figure 2. It is absolutely apparent that their sum exceeds from 1 which is no more a PFN(2)
In order to calculate the overall aggregated values as PFN, we assign a mutual interaction coefficient1−r1r2./ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2 to ṧ1+ṧ2, ị1+ị2, and ḑ1+ḑ2 which can be considered as MD, AD, and NMD. Also, their sum is not greater than 1 and corresponding RD is r1r2.. So(7)ṧε=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ịε=ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ḑε=ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2Figure 2
Geometrical description of the suggested operation.Moreover, PS function used in Equation (6) ist -conorm for PFS and properties like commutativity, associativity, monotonicity, and boundedness which are satisfied. Hence, on the basis of this erudition, we intend to establish some novel operation laws under PFS environment.


## 3.3. Neutral Operational Laws

This section presents novel operational rules for PFS by fusing the neutral attitude of the experts into preferences.Definition 10.
LetÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2 be two PFNs. The neutrality operation of Ñ1and Ñ2 is given as
(8)Ñ1⊝Ñ2=MCSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2,NCSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2ACSÑ1,Ñ2MCSÑ1,Ñ2+NCSÑ1,Ñ2+ACSÑ1,Ñ2.PSṧ1+ị1+ḑ1,ṧ2+ị2+ḑ2,,
whereMCSÑ1,Ñ2=ṧ1+ṧ2, NCSÑ1,Ñ2=ḑ1+ḑ2, and ACSÑ1,Ñ2=ị1+ị2 represent the MD, NMD, and AD coefficient sum of Ñ1and Ñ2, respectively. Also, PSx,y=1−1−x1−y is used to represent the PS of the FN Ñ1 and Ñ2.
ForÑ1=ṧ1,ị1,ḑ1 and real τ>1, we get
(9)PSτṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSτ−1ṧ1+ị1+ḑ1.
Hence, we asserted following propositions.Proposition 11.
For PFNÑ1=ṧ1,ị1,ḑ1 and a realτ>0, we get
(10)PSτṧ1+ị1+ḑ1=1−r1τ.Proof.
We apply mathematical induction onτ.
Step 1. Forτ=2 and by using Equation (9), we have
(11)PSτṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSτ−1ṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,1−r1=1−1−ṧ1−ị1−ḑ11−1+r1=1−r12.
The given statement is satisfied forτ=2.
Step 2. Assume Equation (10) is true forτ=n, then, for τ=n+1,(12)PSn+1ṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,PSnṧ1+ị1+ḑ1=PSṧ1+ị1+ḑ1,1−r1n=1−1−ṧ1−ị1−ḑ11−1+r1n=1−r1n+1.
Hence, by using induction, Equation (10) is satisfied for allτ.
For a group of “n,” PFNs Ñj=ṧj,ịj,ḑj,j=1,2,⋯,n, such that MCSÑj=ṧj andMCSÑ1,⋯Ñn=MCSÑ1,⋯Ñn−1+ṧn. Thus, we haveMCSÑ1,⋯Ñn=∑j=1nṧj. Likewise, we haveACSÑ1,⋯Ñn=∑j=1nịj andNCSÑ1,⋯Ñn=∑j=1nḑj. By using this, we have the below given proposition.Proposition 12.
For PFNÑ1=ṧ1,ị1,ḑ1 and a real number τ>0, we have MCSτÑ1=τ.MCSÑ1,NCSτÑ1=τ.NCSÑ1, andACSτÑ1=τ.ACSÑ1.Proof.
If we takeÑ1=Ñ2 in MCS, thenMCS2Ñ1=MCSÑ1,Ñ1=ṧ1+ṧ1=2MCSÑ1. Applying induction on τ, we can easily have thatMCSτÑ1=τ.MCSÑ1, NCSτÑ1=τ.NCSÑ1 and ACSτÑ1=τ.ACSÑ1.Definition 13.
For PFNÑ1=ṧ1,ị1,ḑ1 and a realτ≥0, the “scalar neutrality operation” is specified as
(13)τ.Ñ1=MCSτ.Ñ1MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1,NCSτ.Ñ1MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1ACSÑ1,Ñ2MCSτ.Ñ1+NCSτ.Ñ1+ACSτ.Ñ1.PSτ.ṧ1+ị1+ḑ1,,whereτÑ1=τṧ1,NCSτÑ1=τḑ1, and ACSτÑ1=τị1.
From Equation (8) and Equation (13), we note that decision-makers have neutral behaviour to MD, NMD, and AD which is shown in following preposition.Proposition 14.
For PFNsÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2, if ṧ1=ị1=ḑ1 and ṧ2=ị2=ḑ2, then ṧÑ1⊝Ñ2=ḑÑ1⊝Ñ2=ịÑ1⊝Ñ2.Proof.
Ifṧ1=ị1=ḑ1 andṧ2=ị2=ḑ2, then by using the neutrality operation, we have ṧÑ1⊝Ñ2/ịÑ1⊝Ñ2=MCSÑ1,Ñ2/ACSÑ1,Ñ2=ṧ1+ṧ2/ị1+ị2=1. Also, ṧÑ1⊝Ñ2/ḑÑ1⊝Ñ2=MCSÑ1,Ñ2/NCSÑ1,Ñ2=ṧ1+ṧ2/ḑ1+ḑ2=1. Thus, ṧÑ1⊝Ñ2=ḑÑ1⊝Ñ2=ịÑ1⊝Ñ2. Hence, it makes clear that suggested operations provide us fair decision even if the PFN has equal degrees.Remark 15.
This fact is observed from Definition2 that when ṧ1=ị1=ḑ1 and ṧ2=ị2=ḑ2, we can obtain ṧÑ1⊝Ñ2≠ḑÑ1⊝Ñ2≠ịÑ1⊝Ñ2. Therefore, it shows the neutrality of newly proposed operations.
Now, by utilizing the definition of MCS, NCS, ACS, and PS function, we can write Equation (8) as
(14)Ñ1⊝Ñ2=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1r2.,
Subsequently, scalar neutrality operation is introduced by using Definition13. Equation (8) gives
(15)Ñ1⊝Ñ1=2Ñ1=MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1,MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1MCSÑ1,Ñ1MCSÑ1,Ñ1+NCSÑ1,Ñ1+ACSÑ1,Ñ1.PSṧ1+ị1+ḑ1,ṧ1+ị1+ḑ1,=ṧ1ṧ1+ị1+ḑ11−r12,ị1ṧ1+ị1+ḑ11−r12,ḑ1ṧ1+ị1+ḑ11−r12.
Similarly,(16)3Ñ1=ṧ1ṧ1+ị1+ḑ11−r13,ị1ṧ1+ị1+ḑ11−r13,ḑ1ṧ1+ị1+ḑ11−r13.
In the same way, for any real positiveτ, we have
(17)τÑ1=ṧ1ṧ1+ị1+ḑ11−r1τ,ị1ṧ1+ị1+ḑ11−r1τ,ḑ1ṧ1+ị1+ḑ11−r1τ.Proposition 16.
The MCS, NCS, and ACS for two PFNsÑ1 and Ñ2 satisfy the following properties andτ,τ1, τ2 are positive real numbers; then,
(1)
MCSÑ1,Ñ2=MCSÑ2,Ñ1(2)
MCSτÑ1,Ñ2=MCSτÑ1,τÑ2(3)
MCSτ1Ñ1,τ2Ñ1=MCSτ1+τ21Ñ1(4)
NCSÑ1,Ñ2=NCSÑ2,Ñ1(5)
NCSτÑ1,Ñ2=NCSτÑ1,τÑ2(6)
NCSτ1Ñ1,τ2Ñ1=NCSτ1+τ2Ñ1(7)
ACSÑ1,Ñ2=ACSÑ2,Ñ1(8)
ACSτÑ1,Ñ2=ACSτÑ1,τÑ2(9)
ACSτ1Ñ1,τ2Ñ1=ACSτ1+τ21Ñ1Proof.
For PFNsÑ1=ṧ1,ị1,ḑ1andÑ2=ṧ2.ị2,ḑ2, we have MCSÑ1,Ñ2=ṧ1+ṧ2, NCSÑ1,Ñ2=ḑ1+ḑ2, and ACSÑ1,Ñ2=ị1+ị2 and MCSτÑ1=τṧ1, NCSτÑ1=τḑ1, and ACSτÑ1=τị1. Thus, MCSτÑ1,τÑ2=τṧ1+τṧ2=τṧ1+ṧ2=τMCSÑ1,Ñ2=MCSτÑ1,Ñ2. Similarly, the remaining parts can be proven


## 3.4. Characteristics of the Proposed Operations

The next subsection discusses some remarkable properties of the suggested operational laws as under.Theorem 17.
IfÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2 are two distinct PFNs, then operations of Ñ1 andÑ2, Ñ1⊝Ñ2, and τÑ1 are also PFNs forτ>0.Proof.
For PFNsÑ1=ṧ1,ị1,ḑ1 andÑ2=ṧ2.ị2,ḑ2, we haveṧ1,ị1,ḑ1ṧ2,ị2,ḑ2∈0,1, r1=1−ṧ1−ị1−ḑ1∈0,1 andr2=1−ṧ2−ị2−ḑ2∈0,1. Let Ñ1⊝Ñ2=ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2. To obtain the required result Ñ1⊝Ñ2∈ PFN, we will just show that ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2∈0,1 and ṧÑ1⊝Ñ2+ịÑ1⊝Ñ2+ḑÑ1⊝Ñ2≤1. Sinceṧ1+ị1+ḑ1≤1, ṧ2+ị2+ḑ2≤1 and hence, ṧ1+ṧ2/ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2≤1, providedṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2≠0. Also, r1,r2∈0,1 which implies that1−r1r2∈0,1. Hence, it can be obtained that ṧÑ1⊝Ñ2,ịÑ1⊝Ñ2,ḑÑ1⊝Ñ2∈0,1. Further, ṧÑ1⊝Ñ2+ịÑ1⊝Ñ2+ḑÑ1⊝Ñ2=1−r1r2∈0,1. Hence, Ñ1⊝Ñ2 is a PFN. Similarly, τÑ1 can also be proven as a PFN.Theorem 18.
LetÑ1=ṧ1,ị1,ḑ1 and Ñ2=ṧ2.ị2,ḑ2be two PFN and τ,τ1τ2≥0 be a real number; then,
(1)
Ñ1⊝Ñ2=Ñ2⊝Ñ1(2)
τÑ1⊝Ñ2=τÑ1⊝τβ2(3)
τ1Ñ1⊝τ2Ñ1=τ1+τ2Ñ1Proof.
(1)
Easily can be followed from Equation (14)(2)
For PFNsÑ1and Ñ2,we get(18)τÑ1⊝τÑ2=MCSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2,NCSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2,ACSτÑ1,τÑ2MCSτÑ1,τÑ2+NCSτÑ1,τÑ2+ACSτÑ1,τÑ2.PSPSτṧ1+ị1+ḑ1,PSτṧ2+ị2+ḑ2=τṧ1+ṧ2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ,τị1+ị2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ,τḑ1+ḑ2τṧ1+ṧ2+τị1+ị2+τḑ1+ḑ2.PS1−r1τ,1−r2τ=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ,ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ21−r1τr2τ=ṧ1+ṧ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2,ị1+ị2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2,ḑ1+ḑ2ṧ1+ṧ2+ị1+ị2+ḑ1+ḑ2PSτ1−r1r2=τÑ1⊝Ñ2.
Forτ1,τ2>0,
(19)τ1Ñ1=ṧ1ṧ1+ị1+ḑ11−r1τ1,ị1ṧ1+ị1+ḑ11−r1τ1,ḑ1ṧ1+ị1+ḑ11−r1τ1,τ2Ñ1=ṧ1ṧ1+ị1+ḑ11−r1τ2,ị1ṧ1+ị1+ḑ11−r1τ2,ḑ1ṧ1+ị1+ḑ11−r1τ2.
Thus, by using Equation (8), we get
(20)τ1Ñ1⊝τ2Ñ1=MCSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2,NCSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2ACSτ1Ñ1,τ2Ñ1MCSτ1Ñ1,τ2Ñ1+NCSτ1Ñ1,τ2Ñ1+ACSτ1Ñ1,τ2Ñ1.PSPSτ1ṧ1+ị1+ḑ1,PSτ2ṧ2+ị2+ḑ2,=ṧ1ṧ1+ị1+ḑ11−r1τ1+τ2,ị1ṧ1+ị1+ḑ11−r1τ1+τ2,ḑ2ṧ1+ị1+ḑ11−r1τ1+τ2=τ1+τ2Ñ1.


## 4. Neutral Aggregation Operators for PFNs

In the next section, the PFWN and PFOWNA operators of PFNs are discussed, including their related characteristics. To get this, we assumeΩ to be the group of PFNs.Definition 19.
LetÑi be a collection of “n” PFNs. The PFWNA of Ñ1,Ñ2,⋯,Ñn is given by
(21)PFWNAÑ1,Ñ2,⋯,Ñn=⊝i=1nωiÑi,where ωi>0 is the weighting vector of Ñiand∑i=1nωi=1.Theorem 20.
The aggregated result of PFNs with the help of Definition19 is also a PFN and given by
(22)PFWNAÑ1,Ñ2,⋯,Ñn=∑i=1nωiṧi∑i=1nωiṧi+ịi+ḑi.1−∏i=1nriωi,∑i=1nωiịi∑i=1nωiṧi+ịi+ḑi.1−∏i=1nriωi,∑i=1nωiḑi∑i=1nωiṧi+ịi+ḑi.1−∏i=1nriωi.Proof.
For “n” PFNs Ñi and realωi>0, by utilizing Theorem 7, the first result is trivially satisfied. For the existence of Equation (22), we apply mathematical induction on “n” which is compiled as follows:
Step 1. Forn=1, we have Ñi=ṧi,ịi,ḑi andωi=1. Therefore, it can be written that
(23)PFWNAÑ1=ω1Ñ1=ṧ1,ị1,ḑ1=ω1ṧ1ω1ṧ1+ị1+ḑ11−r1ω1,ω1ị1ω1ṧ1+ị1+ḑ11−r1ω1,ω1ḑ1ω1ṧ1+ị1+ḑ11−r1ω1.
Thus, Equation (22) is satisfied.
Step 2. Assume Equation (22) holds forn=k, that is,
(24)PFWNAÑ1,Ñ2,⋯,Ñk=∑i=1kωiṧi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi,∑i=1kωiịi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi,∑i=1kωiḑi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi.
Now, forn=k+1, we get
(25)PFWNAÑ1,Ñ2,⋯,Ñk+1=TSFWNAÑ1,Ñ2,⋯,Ñk⊝ωk+1Ñk+1=MCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1MCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+NCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+ACSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1.PS1−∏i=1kriωi,1−rk+1ωk+1,ACSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1MCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+NCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+ACSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1.PS1−∏i=1kriωi,1−rk+1ωk+1NCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1MCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+NCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1+ACSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1.PS1−∏i=1kriωi,1−rk+1ωk+1.,
After substituting the expressions of MCS, ACS, and NCS, we get(26)MCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1=MCSPFWNAÑ1,Ñ2,⋯,Ñk+MCSωk+1Ñk+1=∑i=1kωiṧi+ωk+1ṧk+1=∑i=1k+1ωiṧi.
Similarly, we have(27)ACSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1=ACSTSFWNAÑ1,Ñ2,⋯,Ñk+ACSωk+1Ñk+1=∑i=1kωiịi+ωk+1ịk+1=∑i=1k+1ωiịi,NCSPFWNAÑ1,Ñ2,⋯,Ñk,ωk+1Ñk+1=NCSPFWNAÑ1,Ñ2,⋯,Ñk+NCSωk+1Ñk+1=∑i=1kωiḑi+ωk+1ḑk+1=∑i=1k+1ωiḑi.
By using the definition of PS, we get(28)PS1−∏i=1kriωi,1−rk+1ωk+1=1−1−1+∏i=1kriωi1−1+rk+1ωk+1=1−∏i=1k+1riωi.
Thus,(29)PFWNAÑ1,Ñ2,⋯,Ñk+1=∑i=1k+1ωiṧi∑i=1k+1ωiṧi+ịi+ḑi.1−∏i=1k+1riωi,∑i=1k+1ωiịi∑i=1k+1ωiṧi+ịi+ḑi.1−∏i=1k+1riωi,∑i=1k+1ωiḑi∑i=1k+1ωiṧi+ịi+ḑi.1−∏i=1k+1riωi.
Equation (22) holds for n=k+1. Therefore, by applying induction, Equation (22) is true for alln.Example 1.
Let us consider four PFNsÑ1=0.4,0.3,0.1,Ñ2=0.5,0.2,0.2,Ñ3=0.1,0.2,0.4, andÑ4=0.3,0.3,0.2; the weigh vector associated with them isω=0.2,0.4,0.3,0.1; we use here PFWNA to calculate the four PFNs, since
(30)PFWNAÑ1,Ñ2,Ñ3,Ñ4=∑i=14ωiṧi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi,∑i=14ωiịi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi,∑i=14ωiḑi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi.
Further,(31)∑i=14ωiṧi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi=0.3479,∑i=14ωiịi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi=0.2353,∑i=14ωiḑi∑i=14ωiṧi+ịi+ḑi.1−∏i=1k+1riωi=0.2456.
Finally, we get(32)PFWNAÑ1,Ñ2,Ñ3,Ñ4=0.3479,0.2553,0.2456,which is also a PFN.Theorem 21 (Idempotency).
LetÑi=ṧi,ịi,ḑii=1,2,3...,n be a set of PFNs. If Ñi=ṧ0,ị0,ḑ0 for alli, thenPFWNAÑ1,Ñ2,⋯,Ñn=ṧ0,ị0,ḑ0.Proof.
For “n” PFNs Ñi=ṧi,ịi,ḑi and Ñ0=ṧ0,ị0,ḑ0 such that Ñi=Ñ0, we have ṧi=ṧ0,ịi=ị0 andḑi=ḑ0 for all i. Then, by Equation (22) and ωi>0 with ∑i=1nωi=1, we get
(33)TSFWNAÑ1,Ñ2,⋯,Ñn=ṧ0ṧ0+ị0+ḑ0.1−r0∑i=1nωi,ị0ṧ0+ị0+ḑ0.1−r0∑i=1nωi,ḑ0ṧ0+ị0+ḑ0.1−r0∑i=1nωi=ṧ0ṧ0+ị0+ḑ0.ṧ0+ị0+ḑ0,ị0ṧ0+ị0+ḑ0.ṧ0+ị0+ḑ0,ḑ0ṧ0+ị0+ḑ0.ṧ0+ị0+ḑ0=Ñ0.Theorem 22.
Consider a collection of “n” PFNsÑi; we have
(1)
minṧi+ịi+ḑi≤ṧp+ịp+ḑp≤maxṧi+ịi+ḑi(2)
minṧi+ịi+ḑi.minṧi/maxṧi+ịi+ḑi≤ṧp≤maxṧi+ịi+ḑi.maxṧi/minṧi+ịi+ḑi(3)
minṧi+ịi+ḑi.minịi/maxṧi+ịi+ḑi≤ịp≤maxṧi+ịi+ḑi.maxịi/minṧi+ịi+ḑi(4)
minṧi+ịi+ḑi.minḑi/maxṧi+ịi+ḑi≤ḑp≤maxṧi+ịi+ḑi.maxḑi/minṧi+ịi+ḑiwhere PFWNAÑ1,Ñ2,⋯,Ñn=ṧp,ịp,ḑp. This result is known as boundedness.Proof.
For a collection of “n” PFNsÑi=ṧi,ịi,ḑi, we have
(34)minṧi+ịi+ḑi=1−1−minṧi+ịi+ḑi∑i=1nωi=1−∏i=1n1−minṧi+ịi+ḑiωi≤1−∏i=1n1−ṧi−ịi−ḑiωi≤1−∏i=1n1−maxṧi+ịi+ḑiωi=1−1−maxṧi+ịi+ḑi∑i=1nωi=maxṧi+ịi+ḑi
Thus, we have(35)minṧi+ịi+ḑi≤1−∏i=1n1−ṧi−ịi−ḑiωi≤maxṧi+ịi+ḑi.
By using Theorem20,
(36)ṧp=∑i=1kωiṧi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi,ịp=∑i=1kωiịi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi,ḑp=∑i=1kωiḑi∑i=1kωiṧi+ịi+ḑi.1−∏i=1kriωi.
Therefore,(37)ṧp+ịp+ḑp=1−∏i=1n1−ṧi−ịi−ḑiωi.
Hence, we getminṧi+ịi+ḑi≤ṧp+ịp+ḑp≤maxṧi+ịi+ḑi.(1)
Since,ṧi≥minṧi,so by expression ṧp, we have(38)ṧp≥∑i=1nωiminṧi∑i=1nωimaxṧi+ịi+ḑi1−∏i=1n1−minṧi+ịi+ḑiωi=minṧimaxṧi+ịi+ḑi1−1−minṧi+ịi+ḑi∑i=1nωi=minṧi+ịi+ḑiminṧimaxṧi+ịi+ḑi
Further,(39)ṧp≤∑i=1nωimaxṧi∑i=1nωiminṧi+ịi+ḑi1−∏i=1n1−maxṧi+ịi+ḑiωi=maxṧiminṧi+ịi+ḑi1−1−maxṧi+ịi+ḑi∑i=1nωi=maxṧi+ịi+ḑimaxṧiminṧi+ịi+ḑi=minṧi+ịi+ḑiminṧimaxṧi+ịi+ḑi≤ṧp≤maxṧi+ịi+ḑi.maxṧiminṧi+ịi+ḑi.(2)
Similar to part (2)(3)
Similar to part (2)Theorem 23 (Monotonicity).
LetÑi=ṧÑi,ịÑi,ḑÑi and Ḿi=ṧḾi,ịḾi,ḑḾi be collection of “n” PFNs. Then,
(1)
ṧpÑ+ịpÑ+ḑpÑ≤ṧpḾ+ịpḾ+ḑpḾif ṧÑi+ịÑi+ḑÑi≤ṧḾi+ịḾi+ḑḾi(2)
ṧpÑ≤ṧpḾ, ịpÑ≥ịpḾ,ḑpÑ≥ḑpḾ if ṧÑi+ịÑi+ḑÑi=ṧḾi+ịḾi+ḑḾi and ṧÑi≤ṧḾi(3)
PFWNAÑ1,Ñ2,⋯,Ñn≤PFWNAḾ1,Ḿ2,⋯,Ḿn if ṧÑi+ịÑi+ḑÑi=ṧḾi+ịḾi+ḑḾi and ṧÑi≤Proof.
For “n” PFNs Ñ1,Ñ2,⋯,Ñnand Ḿ1,Ḿ2,⋯,Ḿnand by using Theorem 20, we get PFWNA Ñ1,Ñ2,⋯,Ñn=ṧpÑ,ịpÑ,ḑpÑand PFWNAḾ1,Ḿ2,⋯,Ḿn=ṧpḾ,ịpḾ,ḑpḾ, where
(40)ṧpÑ=∑i=1kωiṧβi∑i=1kωiṧpÑ+ịpÑ+ḑpÑ1−∏i=1n1−ṧpÑ−ịpÑ−ḑpÑωi,ịpÑ=∑i=1kωiịÑi∑i=1kωiṧpÑ+ịpÑ+ḑpÑ1−∏i=1n1−ṧpÑ−ịpÑ−ḑpÑωi,ḑpÑ=∑i=1kωiḑβi∑i=1kωiṧpÑ+ịpÑ+ḑpÑ1−∏i=1n1−ṧpÑ−ịpÑ−ḑpÑωi,ṧpḾ=∑i=1kωiṧḾi∑i=1kωiṧpḾ+ịpḾ+ḑpḾ1−∏i=1n1−ṧpḾ−ịpḾ−ḑpḾωi,ịpḾ=∑i=1kωiịḾi∑i=1kωiṧpḾ+ịpḾ+ḑpḾ1−∏i=1n1−ṧpḾ−ịpḾ−ḑpḾωi,ḑpḾ=∑i=1kωiḑḾi∑i=1kωiṧpḾ+ịpḾ+ḑpḾ1−∏i=1n1−ṧpḾ−ịpḾ−ḑpḾωi.
On the basis of above information, we have(1)
IfṧÑi+ịÑi+ḑÑi=ṧḾi+ịḾi+ḑḾi, then we get(41)ṧpÑ+ịpÑ+ḑpÑ≤1−∏i=1n1−ṧÑi−ịÑi−ḑÑiωi≤1−∏i=1n1−ṧḾi−ịḾi−ḑḾiωi=ṧpḾ+ịpḾ+ḑpḾ(2)
IfṧÑi+ịÑi+ḑÑi=ṧḾi+ịḾi+ḑḾi, andṧÑi≤ṧḾi, then we get(42)ṧpÑ=∑i=1kωiṧÑi∑i=1kωiṧÑi+ịÑi+ḑÑi1−∏i=1n1−ṧÑi−ịÑi−ḑÑiωi≤∑i=1kωiṧḾi∑i=1kωiṧḾi+ịḾi+ḑḾi1−∏i=1n1−ṧḾi−ịḾi−ḑḾiωi=ṧpḾ,ịpÑ=∑i=1kωiịÑi∑i=1kωiṧÑi+ịÑi+ḑÑi1−∏i=1n1−ṧÑi−ịÑi−ḑÑiωi≤∑i=1kωiịḾi∑i=1kωiṧḾi+ịḾi+ḑḾi1−∏i=1n1−ṧḾi−ịḾi−ḑḾiωi=ịpḾ,ḑpÑ=∑i=1kωiḑÑi∑i=1kωiṧÑi+ịÑi+ḑÑi1−∏i=1n1−ṧÑi−ịÑi−ḑÑiωi≤∑i=1kωiḑḾi∑i=1kωiṧḾi+ịḾi+ḑḾi1−∏i=1n1−ṧḾi−ịḾi−ḑḾiωi=ḑpḾ(3)
From part (2), we obtainṧpÑ≤ṧpḾ, ịpÑ≥ịpḾ,ḑpÑ≥ḑpḾ. Hence, by utilizing the score function of Equation (2), we getṧpÑ−ịpÑ−ḑpÑ≤ṧpḾ−ịpḾ−ḑpḾ. So, by an order relation, we get PFWNAÑ1,Ñ2,⋯,Ñn≤PFWNAḾ1,Ḿ2,⋯,ḾnDefinition 24.
LetÑi=ṧi,ịi,ḑii=1,2,3...,n be a set of PFNs. The PFOWNA operator of Ñi is defined by
(43)PFOWNAÑ1,Ñ2,⋯,Ñn=⊝i=1nωiÑσi,where σ is the permutation map of 1,2,3...,n such that Ñσi−1≥Ñσi and Ñi∈Ω and ω=ω1,ω2,⋯,ωnT is the weight vector associated with PFOWNA operator, satisfying ωi>0 and ∑i=1nωi=1.Theorem 25.
The aggregated value by using Definition24 for “n” PFNs Ñi=ṧi,ịi,ḑi is also a PFN, where
(44)PFOWNAÑ1,Ñ2,⋯,Ñn=∑i=1nωiṧσi∑i=1nωiṧσi+ịσi+ḑσi.1−∏i=1nrσiωi,∑i=1nωiịσi∑i=1nωiṧσi+ịσi+ḑσi.1−∏i=1nrσiωi,∑i=1nωiḑσi∑i=1nωiṧσi+ịσi+ḑσi.1−∏i=1nrσiωi.
Theproof of this Theorem 25 is omitted because it is similar to Theorem 20.Remark 26.
The PFOWNA operator satisfies all the properties which are satisfied by the PFWNA operator mentioned in Theorems20, 21, 22, and 23. Also, their proofs are omitted due to similarity.


## 5. Proposed MAGDM Approach Based on Aggregation Operators

This section introduces an inventive scheme to work out the MAGDM problem under the PFS conditions. Consider that a collection of alternativesƑ=Ƒ1,Ƒ2,⋯Ƒm and objective is to choose the best possible one under the evaluation of a set of attributes H=H1,H2,⋯Hn with weight vector ω>0such that ∑j=1nωj=1. The set H containing attributes is split into two mutually exclusive sets, that is, cost type and benefit type attributes, respectively F1 andF2. To rank the given alternatives, in this event, a team of experts E^=E∧1,E∧2,⋯E∧l is selected with weight vectorwk>0 such that∑k=1lwk=1, with a task to propose the best possible solution. For the sake of smooth assessment, decision-makers will utilize PFS environment as δijk=ṧijk,ịijk,ḑijk with 0≤ṧijk,ịijk,ḑijk≤1 s.t. 0≤ṧijk+ịijk+ḑijk≤1 for i=1,2⋯,m,j=1,2⋯n, and k=1,2,⋯l. Henceforth, a PF decision matrix can be considered as
(45)Nk=δ11kδ12k⋯δ21kδ22k⋯⋮δm1k⋮δm2k⋱⋯δ1nkδ2nk⋮δmnk.Now, the detailed strategy of new technique is presented below:Step 1.
A decision matrixN is formulated under PFS environment corresponding to each expertE^.Step 2.
By using a weight vector and suitable AOs, that is, PFWNA or PFOWNA aggregate all the decision-makers’ preferencesδijk,k=1,2,⋯,lintoδij=ṧij,ịij,ḑij.Step 3.
As there are two types of attribute indicators, namely, the cost (F1) and the benefit (F2), so we normalize δij into rij if it is essential.
(46)rij=ḑij,ịij,ṧij,forF1criteria,ṧij,ịij,ḑij,forF2criteria.Step 4.
In the case if weights of the attributes are already given, then use them as they are. Contrarily, if the detail about the attributes is partially known, which is signified in setG, then an optimization model is formulated to calculate an unknown weight vector for each attribute.
(47)maxf=∑i=1m∑j=1nωijSij(48)s.t.∑j=1nωj=1,ωj≥0,ω∈G,where Sij is the score of each alternative, Ƒi; the above model givesω=ω1,ω2,⋯ωn.Step 5.
Using weights and utilizing suggested PF averaging AOs to compute the comprehensive valuesri=ṧi,ịi,ḑi of alternatives Ƒii=1,2,⋯,m.Step 6.
Equation (49) is used to calculate the score values ri=ṧi,ịi,ḑi,i=1,2,⋯,m.
(49)Sri=eṧi−ịi−ḑi2−ṧi−ịi−ḑi.Step 7.
By using Definition4, the alternatives Ƒii=1,2,⋯,m are ranked.


## 6. Illustrative Examples

To validate the suggested method, we assume a MAGDM problem and examine the performance of the proposed method further; for the sake of comparison, some existing methods are compared with the suggested method.


### 6.1. Case Study

Mineral resources are the most vital among the category of natural resources. Minerals provide raw material to all kind of industries which is considered as a backbone of economic growth. Mining is the process through which different minerals are extracted from the earth. Mining sector has made great contributions for human development. Especially in developing countries, it has really helped to raise the standards of health and education. Pakistan is an Asian country and full of natural resources. Balochistan is a resource rich province of Pakistan. A well-known geological survey team has reported that Balochistan has round about ninety mineral resources with sufficient layers. Most of them have not been discovered yet. Recently, there are three main projects working in Balochistan named as Reko-diq, Dudder, and Saindak. All of them are being managed by the Chinese companies. The management of Balochistan province wants to explore more mineral resources. In order to speed up the mining work, the government has invited four different companies with four required attributes for selection named as technical capabilityH1, financial status H2, company back ground (H3), and employability H4. There are three panels E∧1,E∧2, and E∧3 of decision-makers which are appraising the four companies: Ƒ1 (“China National Coal Group”), Ƒ2 (“China Northern Rare Earth Group”), Ƒ3 (“High-Tech Mining Associates”), and Ƒ4 (“HBIS Group”) with the abovementioned four attributes. Suppose that w=0.39,0.27,0.34 is the weight vector of the experts and their assessment matricesN1,N2 and N3 under PFNs are presented in Table 2. The purpose of the study is to choose the appropriate company for new mining project. The steps of the proposed MAGDM process are performed as follows:Step 1.
All the values evaluated by the experts are summarized in Table2.Step 2.
We assume the corresponding weight vectorw=0.39,0.27,0.34 and PFWNA operator to aggregate the information of each alternative. The values are listed in Table 3.Step 3.
All of the given attributes are the same type, so it is not essential to normalize the information.Step 4.
The partial weight information regarding attribute importance as proposed by decision-makers isG=0.2≤ω1≤0.3,0.25≤ω2≤0.4,0.12≤ω3≤0.4,0.2≤ω4≤0.35,ω1+ω3≤ω2,ω1−ω4≤ω3; an optimization model has been formulated by using Equation (48), and after some calculations, we obtained ω=0.26.0.4,0.14,0.2.Step 5.
In accordance with the PFWNA operator and weightsω, we obtain the overall value ri of each alternative; values are given as r1=0.4018,0.2740,0.2917,r2=0.3509,0.4021,0.3335,r3=0.4130,0.3376,0.2223, and r4=0.3911,0.3117,0.2661.Step 6.
By using Equation (5), score values are calculated asSr1=0.8221,Sr2=0.7452,Sr3=0.8407, and Sr4=0.8046.Step 7.
AsSr3>Sr1>Sr4>Sr2, therefore the preferences are ordered asƑ3>Ƒ1>Ƒ4>Ƒ2. So Ƒ3 is the optimal choice.Table 2
The decision matrix from experts.
ExpertsAlternativesH1H2H3H4E∧1Ƒ10.5,0.19,0.30.4,0.25,0.30.2,0.4,0.30.3,0.3,0.3Ƒ20.4,0.39,0.20.45,0.24,0.30.37,0.5,0.10.27,0.32,0.4Ƒ30.5,0.19,0.30.34,0.49,0.10.29,0.4,0.20.4,0.35,0.21Ƒ40.21,0.31,0.330.51,0.27,0.20.32,0.27,0.40.42,0.25,0.3E∧2Ƒ10.5,0.19,0.30.5,0.22,0.20.3,0.23,0.20.28,0.28,0.28Ƒ20.34,0.42,0.2024,0.3,0.0.390.25,0.3,0.390.41,0.28,0.3Ƒ30.35,0.35,0.250.44,0.35,0.180.35,0.34,0.30.5,0.29,0.2Ƒ40.34,0.42,0.20.39,0.3,0.30.3,0.35,0.290.4,0.37,0.2E∧3Ƒ10.5,0.38,0.10.4,0.28,0.30.2,0.39,0.40.33,0.33,0.33Ƒ20.37,0.31,0.30.29,0.33,0.320.32,0.25,0.40.4,0.55,0.04Ƒ30.4,0.3,0.290.54,0.22,0.20.34,0.34,0.30.31,0.38,0.3Ƒ40.51,0.23,0.20.39,0.32,0.270.32,0.42,0.210.34,0.31,0.3Table 3
Aggregated values of the experts by the PFWNA operator.
AlternativesH1H2H3H4Ƒ10.5004,0.2548,0.23220.4350,0.2376,0.27810.2313,0.3436,0.39450.3160,0.3160,0.3160Ƒ20.3748,0.3721,0.23470.3432,0.2904,0.33530.3212,0.3617,0.28090.3543,0.3899,0.2385Ƒ30.4278,0.2721,0.28460.4363,0.3615,0.15610.3301,0.13711,0.26650.3982,0.3456,0.2390Ƒ40.3520,0.3169,0.25420.44371,0.2953,0.25100.3176,0.2458,0.30860.3878,0.3031,0.2733


### 6.2. Alteration of AOs

We can choose a different pair of proposed AOs except from the above-given analysis. Here, we examine the ranking pattern of the alternatives if experts select distinct AOs in Step 2 and Step 5, respectively. As it is quite clear that every AO has its own importance and specific features as per given situation. For instance, PFWNAO gives more weight to PFNs; on the other hand, PFWONAO gives more importance to the position of PFNs after ordering them. In this way, decision-makers can alter the AOs as per requirement. Table4 shows different values under alteration of AOs. It shows the stable behaviour of ranking pattern, and Ƒ3 remains the best choice throughout the ranking calculations. Figure 3 clearly demonstrates the stability of newly proposed AOs. Use of PFOWNA in Step 2 and PFWNA in step 5 generates relatively low score values of alternatives. However, the overall ranking pattern and best possible alternatives remain unchanged.Table 4
Influence of alteration of AOs and ranking order.
Operators usedScore valuesRankingStep 2Step 5Ƒ1Ƒ2Ƒ3Ƒ4PFWNAPFWNA0.82210.74520.84070.8046Ƒ3>Ƒ4>Ƒ2>Ƒ1PFOWNAPFWNA0.81350.73680.81440.7742Ƒ3>Ƒ4>Ƒ2>Ƒ1PFWNAPFOWNA0.39610.33320.39780.3728Ƒ3>Ƒ4>Ƒ2>Ƒ1PFOWNAPFOWNA0.83510.74360.85060.8107Ƒ3>Ƒ4>Ƒ2>Ƒ1Figure 3
Geometrical interpretation of alteration of AOs.


### 6.3. Comparative Analysis

Example 2.
Again, utilizing the data from Table2, here, we will calculate the aggregation values and validate the proposed AOs with the help of different tools which already exist in the literature. It is necessary to mention here that the already available work by Garg et al. [52, 53] on neutrality operators is not sufficient enough to manage the information recorded in Table 2. Here, we apply the operators proposed by Garg [52], Wei [40], and Jana at el. [42] on the information given in Table 2 in order to conduct a comparative study among different aggregating tools. The ranking patterns obtained by using the aforementioned existing AOs are enlisted in Table 5. Results obtained using the proposed PFWNA and PFOWNA operators have consistency. Ranking patterns are quite similar with results obtained by Garg [52], Wei [40], and Jana at el. [42]. Moreover, neutrality towards the selection of alternatives plays an important part during the overall decision-making process so, in this way, the proposed operators provide better environment for decision-making. The validity of suggested AOs is evident from Figure 4. Throughout the analysis, there is no change in ranking patterns which shows that results obtained using novel operators agree with the results that exist in the literature. Furthermore, the stability of the results makes it clear from Figure 4 that the approach prescribed in this manuscript is much effective than the remaining methods recorded in Table 5.Table 5
Comparative analysis by using existing and proposed methods.
OperatorReferenceƑ1Ƒ2Ƒ3Ƒ4RankingPFDWAJana at el. [42]0.46570.43670.50450.4809Ƒ3>Ƒ4>Ƒ1>Ƒ2PFDWGJana at el. [42]0.22170.23460.27880.2477Ƒ3>Ƒ4>Ƒ2>Ƒ1PFWAGarg [52]−0.1245−0.2272−0.1136−0.1668Ƒ3>Ƒ1>Ƒ4>Ƒ2PFWGWei [40]−0.1824−0.2946−0.1621−0.2003Ƒ3>Ƒ1>Ƒ4>Ƒ2PFWNAThis paper0.82200.74520.84070.8046Ƒ3>Ƒ1>Ƒ4>Ƒ2PFOWNAThis paper0.81350.73640.81440.7742Ƒ3>Ƒ1>Ƒ4>Ƒ2Figure 4
Pictorial analysis of validity and comparison.


### 6.4. Advantages of the Suggested Approach

Some salient features and advantages of the suggested approach are elaborated in the following:(1)
Suggested AOs can handle all the human aspects; in this way, decision-makers can handle various real-life situations more efficiently(2)
Degree of refusal plays an important role to choose the best alternative; during the information aggregation, by using AOs presented in this, work we can control refusal degree(3)
The main characteristic of this novel extension is the inclusion of decision-makers’ attitude(4)
The ability to handle neutral behaviour of the experts makes this work more effective as compared to other studies presented in Table5


## 6.1. Case Study

Mineral resources are the most vital among the category of natural resources. Minerals provide raw material to all kind of industries which is considered as a backbone of economic growth. Mining is the process through which different minerals are extracted from the earth. Mining sector has made great contributions for human development. Especially in developing countries, it has really helped to raise the standards of health and education. Pakistan is an Asian country and full of natural resources. Balochistan is a resource rich province of Pakistan. A well-known geological survey team has reported that Balochistan has round about ninety mineral resources with sufficient layers. Most of them have not been discovered yet. Recently, there are three main projects working in Balochistan named as Reko-diq, Dudder, and Saindak. All of them are being managed by the Chinese companies. The management of Balochistan province wants to explore more mineral resources. In order to speed up the mining work, the government has invited four different companies with four required attributes for selection named as technical capabilityH1, financial status H2, company back ground (H3), and employability H4. There are three panels E∧1,E∧2, and E∧3 of decision-makers which are appraising the four companies: Ƒ1 (“China National Coal Group”), Ƒ2 (“China Northern Rare Earth Group”), Ƒ3 (“High-Tech Mining Associates”), and Ƒ4 (“HBIS Group”) with the abovementioned four attributes. Suppose that w=0.39,0.27,0.34 is the weight vector of the experts and their assessment matricesN1,N2 and N3 under PFNs are presented in Table 2. The purpose of the study is to choose the appropriate company for new mining project. The steps of the proposed MAGDM process are performed as follows:Step 1.
All the values evaluated by the experts are summarized in Table2.Step 2.
We assume the corresponding weight vectorw=0.39,0.27,0.34 and PFWNA operator to aggregate the information of each alternative. The values are listed in Table 3.Step 3.
All of the given attributes are the same type, so it is not essential to normalize the information.Step 4.
The partial weight information regarding attribute importance as proposed by decision-makers isG=0.2≤ω1≤0.3,0.25≤ω2≤0.4,0.12≤ω3≤0.4,0.2≤ω4≤0.35,ω1+ω3≤ω2,ω1−ω4≤ω3; an optimization model has been formulated by using Equation (48), and after some calculations, we obtained ω=0.26.0.4,0.14,0.2.Step 5.
In accordance with the PFWNA operator and weightsω, we obtain the overall value ri of each alternative; values are given as r1=0.4018,0.2740,0.2917,r2=0.3509,0.4021,0.3335,r3=0.4130,0.3376,0.2223, and r4=0.3911,0.3117,0.2661.Step 6.
By using Equation (5), score values are calculated asSr1=0.8221,Sr2=0.7452,Sr3=0.8407, and Sr4=0.8046.Step 7.
AsSr3>Sr1>Sr4>Sr2, therefore the preferences are ordered asƑ3>Ƒ1>Ƒ4>Ƒ2. So Ƒ3 is the optimal choice.Table 2
The decision matrix from experts.
ExpertsAlternativesH1H2H3H4E∧1Ƒ10.5,0.19,0.30.4,0.25,0.30.2,0.4,0.30.3,0.3,0.3Ƒ20.4,0.39,0.20.45,0.24,0.30.37,0.5,0.10.27,0.32,0.4Ƒ30.5,0.19,0.30.34,0.49,0.10.29,0.4,0.20.4,0.35,0.21Ƒ40.21,0.31,0.330.51,0.27,0.20.32,0.27,0.40.42,0.25,0.3E∧2Ƒ10.5,0.19,0.30.5,0.22,0.20.3,0.23,0.20.28,0.28,0.28Ƒ20.34,0.42,0.2024,0.3,0.0.390.25,0.3,0.390.41,0.28,0.3Ƒ30.35,0.35,0.250.44,0.35,0.180.35,0.34,0.30.5,0.29,0.2Ƒ40.34,0.42,0.20.39,0.3,0.30.3,0.35,0.290.4,0.37,0.2E∧3Ƒ10.5,0.38,0.10.4,0.28,0.30.2,0.39,0.40.33,0.33,0.33Ƒ20.37,0.31,0.30.29,0.33,0.320.32,0.25,0.40.4,0.55,0.04Ƒ30.4,0.3,0.290.54,0.22,0.20.34,0.34,0.30.31,0.38,0.3Ƒ40.51,0.23,0.20.39,0.32,0.270.32,0.42,0.210.34,0.31,0.3Table 3
Aggregated values of the experts by the PFWNA operator.
AlternativesH1H2H3H4Ƒ10.5004,0.2548,0.23220.4350,0.2376,0.27810.2313,0.3436,0.39450.3160,0.3160,0.3160Ƒ20.3748,0.3721,0.23470.3432,0.2904,0.33530.3212,0.3617,0.28090.3543,0.3899,0.2385Ƒ30.4278,0.2721,0.28460.4363,0.3615,0.15610.3301,0.13711,0.26650.3982,0.3456,0.2390Ƒ40.3520,0.3169,0.25420.44371,0.2953,0.25100.3176,0.2458,0.30860.3878,0.3031,0.2733


## 6.2. Alteration of AOs

We can choose a different pair of proposed AOs except from the above-given analysis. Here, we examine the ranking pattern of the alternatives if experts select distinct AOs in Step 2 and Step 5, respectively. As it is quite clear that every AO has its own importance and specific features as per given situation. For instance, PFWNAO gives more weight to PFNs; on the other hand, PFWONAO gives more importance to the position of PFNs after ordering them. In this way, decision-makers can alter the AOs as per requirement. Table4 shows different values under alteration of AOs. It shows the stable behaviour of ranking pattern, and Ƒ3 remains the best choice throughout the ranking calculations. Figure 3 clearly demonstrates the stability of newly proposed AOs. Use of PFOWNA in Step 2 and PFWNA in step 5 generates relatively low score values of alternatives. However, the overall ranking pattern and best possible alternatives remain unchanged.Table 4
Influence of alteration of AOs and ranking order.
Operators usedScore valuesRankingStep 2Step 5Ƒ1Ƒ2Ƒ3Ƒ4PFWNAPFWNA0.82210.74520.84070.8046Ƒ3>Ƒ4>Ƒ2>Ƒ1PFOWNAPFWNA0.81350.73680.81440.7742Ƒ3>Ƒ4>Ƒ2>Ƒ1PFWNAPFOWNA0.39610.33320.39780.3728Ƒ3>Ƒ4>Ƒ2>Ƒ1PFOWNAPFOWNA0.83510.74360.85060.8107Ƒ3>Ƒ4>Ƒ2>Ƒ1Figure 3
Geometrical interpretation of alteration of AOs.


## 6.3. Comparative Analysis

Example 2.
Again, utilizing the data from Table2, here, we will calculate the aggregation values and validate the proposed AOs with the help of different tools which already exist in the literature. It is necessary to mention here that the already available work by Garg et al. [52, 53] on neutrality operators is not sufficient enough to manage the information recorded in Table 2. Here, we apply the operators proposed by Garg [52], Wei [40], and Jana at el. [42] on the information given in Table 2 in order to conduct a comparative study among different aggregating tools. The ranking patterns obtained by using the aforementioned existing AOs are enlisted in Table 5. Results obtained using the proposed PFWNA and PFOWNA operators have consistency. Ranking patterns are quite similar with results obtained by Garg [52], Wei [40], and Jana at el. [42]. Moreover, neutrality towards the selection of alternatives plays an important part during the overall decision-making process so, in this way, the proposed operators provide better environment for decision-making. The validity of suggested AOs is evident from Figure 4. Throughout the analysis, there is no change in ranking patterns which shows that results obtained using novel operators agree with the results that exist in the literature. Furthermore, the stability of the results makes it clear from Figure 4 that the approach prescribed in this manuscript is much effective than the remaining methods recorded in Table 5.Table 5
Comparative analysis by using existing and proposed methods.
OperatorReferenceƑ1Ƒ2Ƒ3Ƒ4RankingPFDWAJana at el. [42]0.46570.43670.50450.4809Ƒ3>Ƒ4>Ƒ1>Ƒ2PFDWGJana at el. [42]0.22170.23460.27880.2477Ƒ3>Ƒ4>Ƒ2>Ƒ1PFWAGarg [52]−0.1245−0.2272−0.1136−0.1668Ƒ3>Ƒ1>Ƒ4>Ƒ2PFWGWei [40]−0.1824−0.2946−0.1621−0.2003Ƒ3>Ƒ1>Ƒ4>Ƒ2PFWNAThis paper0.82200.74520.84070.8046Ƒ3>Ƒ1>Ƒ4>Ƒ2PFOWNAThis paper0.81350.73640.81440.7742Ƒ3>Ƒ1>Ƒ4>Ƒ2Figure 4
Pictorial analysis of validity and comparison.


## 6.4. Advantages of the Suggested Approach

Some salient features and advantages of the suggested approach are elaborated in the following:(1)
Suggested AOs can handle all the human aspects; in this way, decision-makers can handle various real-life situations more efficiently(2)
Degree of refusal plays an important role to choose the best alternative; during the information aggregation, by using AOs presented in this, work we can control refusal degree(3)
The main characteristic of this novel extension is the inclusion of decision-makers’ attitude(4)
The ability to handle neutral behaviour of the experts makes this work more effective as compared to other studies presented in Table5


## 7. Conclusion

Aggregation operators have a pivotal role during the MADM problems. Hence, we suggest some novel AOs in this paper named as PFWNA and PFOWNA on the basis of PFSs. Many researchers have made enormous contributions for IFSs which consider only MD and NMD. It has been noticed that some real-life scenarios cannot be represented clearly by using IFSs. During this work, we represented our information under PFS environment which extends the idea of IFS. Also, a novel MAGDM scheme is proposed on the basis of newly suggested AOs for PFSs. Some numerical examples have been demonstrated to prove the effectiveness of this approach. Finally, a comparative analysis is presented which shows the supremacy and advantages of this scheme. The concept can be further extended to develop neutrality AOs for spherical and T-spherical fuzzy sets [54, 55].



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*Source: 2762067-2022-04-01.xml*