# Reliability Analysis of Poll Data with Novel Entropy Information Measure in Multicriteria Decision-Making Based upon Picture Fuzzy Environment

Authors: Ravinder Kumar; Gaurav Gupta; Muhammad Gulzar; Dragan Pamucar; Neeraj Gandotra; Md. Ashraful Alam
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The polling system has a considerable role in the democratic nation. The uncertainty of the people’s participation in polling generally affects the electoral-based system. Therefore, PFS (picture fuzzy set) is the furthermost efficient and useful extension of IFS (intuitionistic fuzzy set) in a fuzzy system capable of precisely handling human perception in the decision-making system. The PFS structure involves the different degrees, i.e., membership, nonmembership, neutral, and hesitancy which are comprehensively applied to such types of complex practical problems in the real-life scenario. This advantage of PFS motivates the author to propose PFSs centred novel entropy measure via this communication, which is comparatively more generalized, reliable, and simplified in place of the existing uncertain measures. The practicability of the proposed present research work is to deal with real-world problems pertaining to the relative importance of the attributes. Therefore, certainly, the proposed novel entropy developed a different approach to handle the uncertainty more precisely as a part of the existing approaches. The validation proof of the proposed entropy measure is proved in an organized manner and practically employed in the perspective of the polling data outcomes about the people’s opinions with the VIKOR-TOPSIS approach.


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## Body

## 1. Introduction

In the modern era, fuzzy sets and their various generalized extensions are predominantly utilized by researchers and decision-makers in the multiple domains of the research field. For the useful life, the different researchers broadly used the concept of fuzzy logic in several applications of modern technology, for instance, washing machine [1], modern cars [2], and traffic signals control system [3]. The author in [4] was the pioneer to present the abstract of fuzzy theory.Later, various authors presented the modified structure ofFss and their efficient use to explore the vagueness and uncertainty in many problems of multicriteria decision-making support systems. Initially, the fuzzy set may be possible to prevail only the knowledge about the membership degree (μ) of the element. It restricts the fuzzy set to the specific domain of MCDM. Further, the author was inspired [5] with the concept of nonmembership function ν and reshaped the existing structure of Fss and added on one more parameter π:0≤μ+ν+π≤1 and IFS comes to light. IFs attracts the great attentions of various researchers, authors, and decision-makers across the globe because of its feature of presenting π. It makes IFs more reliable and concise towards the challenges in real-life scenario. In this perception, several authors around the globe contributed to the research as per their point of view; for instance, the authors in [6] introduced the idea of homomorphism between the subgroup (cyclic) and IFs subgroup. Further, the authors in [7] developed complex IFs approach which relates with group theory and the authors in [8] proposed some important algebraic structures based upon αβcomplex fuzzy−based  subgroup. First time, the author was able to validate various parameters of IFs in context of human perception and related various components of IFs as per the public opinion in the voting system, i.e., μ for Support, ν stands for Unsupport, and π implies neither support nor unsupport. As IFs handles the MCDM problems more precisely and in efficient manner comparatively Fss, however, the researchers analysed that IFs is not applicable comprehensively in all such fuzzy environment-based situations. For instance, in the voting system, the people who are in the mindset of the dilemma are not undertaken in the framework of IFs. To encounter such indeterminacy, the authors [9] were motivated with idea of neutral membership degree and incorporated the new component η to the present structure of IFs:0≤μ+ν+η+π≤1 which is known as PFs. Therefore, whenever IFs restricts its scope, PFs perfectly handles the uncertainty in a better way. Certainly, the parameter η has great importance to capture the logic of the human mind’s ambiguity in the behavioural approach towards complex situations. Figure 1 shows the pretty presentation of various parameters from Fss to PFs.Figure 1
The pretty presentation of various parameters fromFss to PFs.The researchers and decision-makers take advantage of this unique feature ofPFs and used it tremendously in several applications of modern technologies and social issues. The researcher [10] applied the concept of PFs using the distance measure for the health issues. Nowadays, in various research areas, the researchers are using the framework of PFs in the hybrid form. Further, the authors [11] successively utilized PFs-based hybrid distance measure with the concept of rough set theory in selection criteria of the students. In this context, the authors [12] contributed some important correlation coefficients and their application in analysis of clustering and pattern recognition in computing research, and the authors in [13] presented the hybrid form of similarity measures and biparametric distance to handle the uncertainty in medical diagnosis. In the forecasting system, the author [14] significantly implied the PFs feature and proposed the novel method and created the time series model for forecasting. The decision-makers [15] extended the bidirectional projection method for PFs and applied to assess the safety features for construction design.Recently, the researchers have implied the different picture fuzzy uncertain measures with the use of different methods or approaches to cope up the vagueness inMCDM challenges. The authors [16] developed the algorithms based upon interaction aggregation operators defined on PFs for complex multivariable decision-making models. Thus, the researcher [17] contributed the concept of cross-entropy in PFs and its application in MCDM support system. Continuing with the PFs theory, the authors submitted the entropy weighted picture fuzzy information-based clustering-algorithm for segmentation of images. The important entropy similarity measure was proposed by [18], and their application in the decision-making system was developed. Further, the authors [19] worked upon and submitted the divergence information measure of Jenson Tasalli along with its application in decision problems for PFs. The concept of R-S norms was broadly used by the authors in [20, 21], and the information measure for the picture fuzzy situations was proposed. Moreover, the researchers [22] also proposed the entropy information measure based upon linguistic PFs with MCDM application. The author [23] proposed the generalized model based upon the concept of maximum entropy negation to explore the information involved in the field of the artificial intelligent decision-making system. Further, the researcher [24] put forwarded the decision-making models to handle the issues like a cost-aware, fault-tolerant, and reliable strategy in complex problems under fuzzy environment. However, the literature findings help us in problem formulation which prevails across the different areas of the research; fuzzy logic and fuzzy sets are considerably applied to capture the ambiguity available in data as well as in practical problems of the real world.


### 1.1. Research Gap

The human mindsets seem too inefficient to meet the solution of complex multivariable problems in real-time scenarios. Therefore, fuzzy sets are quite proficient and reliable in handling ambiguity and the dilemma approach of human mindsets in complex decision-making issues.PFs can handle the neutral factors involved in the MCDM problems more efficiently; for instance, in the outcomes of the exit poll in the electoral system, the neutral voters have a vital role. So far, several authors proposed a wide range of diverse information measures and approaches defined on picture fuzzy sets to cope with uncertainty as per their mindsets. However, there is still a scarcity of sophisticated, reliable, and simpler measures defined on PFs to deal with complicated MCDM problems in a fuzzy environment. This inspires the authors to work upon the proposed research work. Therefore, the proposed research work certainly overcomes the research gap of the different existing approaches and uncertainty measures of PFs.


### 1.2. Methodology

The different methodologies are proposed to deal withMCDM problems to select the best alternatives in accordance with the choice of the criteria. The purpose of approaches or techniques is to handle the available data with greater accuracy and to reduce uncertainty. The proposed novel entropy is reliable and more applicable with the complex MCDM problems allied with the relative importance of the criteria. Therefore, the practicability of novel entropy is applied to the prediction of the polling data; however, the criteria involved in the present problem have the relative importance of the criteria. Therefore, such type of problems handles with the VIKOR approach which provides the compromise solutions. All through the TOPSIS approach was utilized to select the best alternatives irrespective of the relative importance of the different criteria. Therefore, the application part of the proposed entropy was validated with both approaches.Further, the presentation of the proposed work is classified in various parts as shown in Figure2.Figure 2
Proposed workflow.


## 1.1. Research Gap

The human mindsets seem too inefficient to meet the solution of complex multivariable problems in real-time scenarios. Therefore, fuzzy sets are quite proficient and reliable in handling ambiguity and the dilemma approach of human mindsets in complex decision-making issues.PFs can handle the neutral factors involved in the MCDM problems more efficiently; for instance, in the outcomes of the exit poll in the electoral system, the neutral voters have a vital role. So far, several authors proposed a wide range of diverse information measures and approaches defined on picture fuzzy sets to cope with uncertainty as per their mindsets. However, there is still a scarcity of sophisticated, reliable, and simpler measures defined on PFs to deal with complicated MCDM problems in a fuzzy environment. This inspires the authors to work upon the proposed research work. Therefore, the proposed research work certainly overcomes the research gap of the different existing approaches and uncertainty measures of PFs.


## 1.2. Methodology

The different methodologies are proposed to deal withMCDM problems to select the best alternatives in accordance with the choice of the criteria. The purpose of approaches or techniques is to handle the available data with greater accuracy and to reduce uncertainty. The proposed novel entropy is reliable and more applicable with the complex MCDM problems allied with the relative importance of the criteria. Therefore, the practicability of novel entropy is applied to the prediction of the polling data; however, the criteria involved in the present problem have the relative importance of the criteria. Therefore, such type of problems handles with the VIKOR approach which provides the compromise solutions. All through the TOPSIS approach was utilized to select the best alternatives irrespective of the relative importance of the different criteria. Therefore, the application part of the proposed entropy was validated with both approaches.Further, the presentation of the proposed work is classified in various parts as shown in Figure2.Figure 2
Proposed workflow.


## 2. Preliminaries

Definition 1.
. The fuzzy set theory generalized the concept of classical sets, which it perceives as an extension of classical sets. The pioneer of the fuzzy set theory was [4] which defined elements of the fuzzy set Fss in terms of the membership function μ∇ on finite universe of discourse, i.e., Z=Z1,Z2,Z3,……Zn, and defined Fss,∇ as(1)∇=Zi∈Z,where membership function of Fss is μ∇:Z⟶0,1 and μ∇Zi signifies the membership degree μ∇Zi of the Zi∈Z in Fss. The fuzzy set Fss has only explained μ∇Zi and 1−μ∇Zi, i.e., nonmembership function and no idea about hesitancy element. The idea of hesitancy element was further introduced by [5] to the existing framework of Fss called the degree of hesitancy.Definition 2.
The fuzzy sets are restricted to employ in certainMCDM problems, which are unable to handle the uncertainty factor precisely. The idea of IFs intuitionistic fuzzy set was presented by [5] which added on one more important factor, i.e., nonmembership function in the present structure of Fss. Consequently, IFsΔ defined on Zuniverse of discourse is as follows:(2)Δ=Zi∈Z,and satisfies 0≤μΔZi+νΔZi+≤1,∀Zi∈Z, along with hesitancy degree πΔZi=1−μΔZi−νΔZi. Apart of this, the functions μΔ:Z⟶0,1 and νΔ:Z⟶0,1 stated the membership and nonmembership function of IFs, and the terms μΔZi and νΔZi described the degree of membership and nonmembership of the element belongs to IFs. The author in [5] introduced IFs which extend the scope of fuzzy set in decision-making environment. In real-life scenario, the IFs membership, nonmembership, and uncertain degree make the sense of favour, against, and abstain in MCDM problems but are unable to describe refusal degree which restricts its scope of application. Therefore, this inconsistency is vanquished in picture fuzzy set PFS to add on one more component “Neutral membership degree” in the present framework of IFs by [9].Definition 3.
Over the period, the decision-makers analysed thatIFss is not feasible towards certain complex issues. The author in [9] presented the concept of a picture fuzzy set, which is an advanced generalized form of fuzzy sets that are more capable of dealing with complex issues in a fuzzy environment. The picture fuzzy set PFS∇∧ on Z is defined as(3)∇∧=Zi∈Z.with inequality 0≤μ∇∧Zi+η∇∧Zi+ν∇∧Zi≤1,∀Zi∈Z.
This describes the positive membership degreeμ∇∧Zi∈0,1, neutral membership degree η∇∧Zi∈0,, and negative membership degree ν∇∧Zi∈0,1. The component π∇∧Zi=1−μ∇∧Zi−η∇∧Zi−ν∇∧Zi is refusal membership degree for each Zi∈Z. Therefore, the structure of PFS is more generalized and classified form as compared with Fss and IFs to capture the vagueness in MCDM practical problems.Definition 4.
LetQ and R be PFS defined on Z.(4)Q=Zi∈Z,R=Zi∈Z.
The authors in [25] introduced the following important operations and relations for any Q,R∈PFsZ defined as follows:(1)
Q⊆R⇔μQZi≤μRZi,ηQZi≤ηRZi,νRZi≤νQZi∀Zi∈Z(2)
Q=R⇔Q⊆RandR⊆Q(3)
QCthe complement ofQ=Zi∈Z(4)
Q∩R=μQZi∧μRZi,ηQZi∨ηRZi,νQZi∨νRZi|Zi∈Z(5)
Q∪R=μQZi∨μRZi,ηQZi∧ηRZi,νQZi∧νRZi|Zi∈ZDefinition 5.
Assume thatQ and R are two PFs∈Z such that(5)Q=Zi∈Z,R=Zi∈Z.
There are two distance measures, i.e.,Hamming distance measure and Euclidean distance between the two PFsQ and R defined as(6)HDQ,R=0.5∑i=1nμQZi−μRZi+ηQZi−ηRZi+νQZi−νRZi+πQZi−πRZi,(7)EDQ,R=∑i=1nμQZi−μRZi2+ηQZi−ηRZi2+νQZi−νRZi2+πQZi−πRZi2.


## 3. Fuzzy Entropy Information Measure forPFSS

The entropy information measure is the fuzziness degree ofFss. The author [26] introduced the set of postulates based upon probability theory of Shannon for FssDefinition 6.
The entropy is a real functionθ∧:FssZ⟶0,∞ which satisfies the following axioms:(1)
PFSharpness. θ∧Q∧=0⇔Q∧ defined as the crisp set(2)
PFMaximality. θ∧Q∧ acquires maximum value ⇔μQ∧=0.5∀Q∧∈FssZ(3)
PFResolution. Q∧andR∧∈FssZ|Q∧is crisper thanR∧, i.e.,θ∧Q∧≤θ∧R∧⇒μQ∧≤μR∧ for μR∧≤1/2 and μQ∧≥μR∧ifμR∧≥0.5(4)
PFSymmetry. θ∧Q∧=θ∧Q∧c where Q∧c represents the compliment of Q∧Definition 7 (see [27]).
The entropy is a real functionθˇ:IFsZ⟶0,∞ which satisfies the following properties:(1)
PFSharpness. θˇQˇ=0⇔Qˇ∈IFsZ defined as the crisp set(2)
PFMaximality. θˇQˇ is maximum value if μQˇ=νQˇ=πQˇ=1/3∀Q∧∈IFsZ(3)
PFResolution. If QˇandRˇ∈IFsZ|Qˇis crisper thanRˇ,i.e.,θˇQˇ≤θˇRˇ⇒μQˇ≤μRˇandνQˇ≤νRˇ for maximumμRˇ,νRˇ≤1/3and ifμQˇ≥μRˇ,νQˇ≥νRˇ, then minμRˇ,νRˇ≥1/3(4)
PFSymmetry. θˇQˇ=θˇQˇC where QˇC denotes the compliment of θˇ
Moreover,PFSS is the well-known generalized form of IFs with the four components, i.e., μ,ν,η,π which hold the condition such that μ,ν,η,π∈0,1 and μ+ν+η+π=0. These parameters may be presumed as probability measures that characterized the picture fuzzy set. This indicates that entropy measure must have maximum value if all parameters uniformly have the value one-fourth, i.e.,μQ=ηQ=νQ=πQ=1/4 and must be zero if all the components vanish except one such as(8)μQ=1,ηQ=νQ=πQ=0,or,ηQ=1,μQ=νQ=πQ=0,or,νQ=1,νQ=1,μQ=ηQ=πQ=0,or,πQ=1,μQ=ηQ=νQ=0.Definition 8.
The axiomatic definition of entropy measure forPFS proposed in view of the concept of entropy measure of IFs introduced by [27] is as follows.
The entropy measure is a real functionθ:PFSZ⟶0,∞ and agreed with the following axioms:PFS1sharpness. θQ=0⇔PFSQ∈Z which holds Q is a crisp setPFS2maximality. IfμQ=ηQ=νQ=πQ=1/4⇒θQ have the maximum value thenPFS3resolution. This property holds if Q is crisper than R, i.e., θQ≤θR such that μQ≥μR,ηQ≥ηR,νQ≥νRforμR,ηR,νR≥1/4,μQ≤μR,ηQ≤ηR,νQ≤νRforμR,ηR,νR≤1/4PFS4symmetry. If Qc is the complement of Q, symmetry implies θQ=θQc
In fact, several authors proposed different uncertain measures onPFs and utilized them to solve the complex multivariable problems. In this context, the novel entropy measure is proposed to enhance the preciseness and improve the reliability and handle the ambiguity of MCDM problems in a simpler and more logical way. This would certainly overcome such shortcomings existing in uncertain measures defined on PFs. In the next part, a novel entropy information measure has been proposed on PFSS, keeping in view these important fundamental concepts.


## 4. A Novel Entropy Information Measure forPFSS

Let us assume the finite set of comprehensive probability distribution signified asLn=〈℧=″a1,a2………an″:ai≥0and∑i=1nai=1〉;n≥2for any℧∈Ln. Firstly, Shannon [28] initiated to define the information measure explained which is widely known as Shannon entropy(9)R℧=−∑i=1nailogai.In the passage of time, Zadeh (1965) put forward the concept of fuzzy set which develops the new approach to quantify the uncertainty/fuzziness. Further, in 1968, Zadeh prolonged the concept of Shannon entropy; however, this information was not capable to meet the purpose. The continued efforts of researchers with information measure in 1972 [26] were capable to extend the axioms of Shannon entropy and proposed the new fuzzy information postulates which are accepted widely across the world and set as criteria for fuzzy entropy measure mentioned in Definition 6 This also motivated the authors [26] to propose the fuzzy entropy measure consistent with Shannon entropy presented as(10)RLT∇=−1n∑i=1nμΔZilog logμΔZi+1−μΔZilog log1−μΔZi,where∇andZi∈FssZ.This developed the author’s insight more intoFss and presented different information measures according to their point of view. The authors [29] contributed exponential entropy measure for IFs given as follows:(11)RRBΔ=1nexp−1∑i=1nμΔZi+1−νΔZi2expμΔZi+1−νΔZi2+μΔZi+1−νΔZi2expμΔZi+1−νΔZi2−1,whereΔandZi∈IFs.Further, the researchers [30] have explored the idea of entropy measure based on exponential function to PPFS (Pythagorean fuzzy set) given by(12)RRGΔ∧=1n1−exp32∑i=1n1−expμΔ∧2Zi+νΔ∧2Zi−πΔ∧2ZiwhereΔ∧andZi∈PPFS.Over the period, several authors have introduced the different work on entropy information measure. The authors [31] contributed residual entropy information measure in the application of the group decision-making. The geometric aggregation operators were introduced [32] based upon IFs, respectively, with undefined criteria weights and their application in MCDM. Further, the researcher [33] presented information entropy measure with the generalized form of IFs and utilized in decision-making problems.With this information as background, in the next part, new entropy measure is proposed to the picture fuzzy setPFS as the generalized form of IFs.Definition 9.
The real functionθ is defined on Q∈PFSZ. The new probability type picture information measure is proposed as follows:(13)θQ=1.582nexp∑i=1nexp−expμQZi+ηQZi−νQZi−πQZi:QandZi∈PFSZ,whereμQZi⟶positive membership degree,ηQZi⟶neutral membership degree,νQZi⟶negative membership degree,πQZi⟶degree of the refusal membership1.582=ExpExp−1.The validity of information measure is proved if it satisfied the fundamental postulates of entropy measure listed in Definition8.


### 4.1. Validation of Proposed Information Measure

Firstly, the following inequalities for anyQandR∈PFSZ:Qcrisper thanRlisted in axiomsPFS3 are proved:(14)μQZi−14+ηQZi−14+νQZi−14+πQZi−14≥μRZi−14+ηRZi−14+νRZi−14+πRZi−14,and(15)μQZi−142+ηQZi−142+νQZi−142+πQZi−142≥μRZi−142+ηRZi−142+νRZi−142+πRZi−142.Proof.
IfμQZi≤μRZi,ηQZi≤ηRZi,νQZi≤νRZiandπQZi≥πRZiwith maxμRZi,ηRZi,νRZi≤1/4⇒μQZi≤μRZi≤1/4 and ηQZi≤ηRZi≤1/4,νQZi≤νRZi≤1/4andπQZi≥πRZi≥1/4This proves that equations (14) and (15) are correct.
IfifμQZi≥μRZi,ηQZi≥ηRZi,νQZi≥νRZiandπQZi≤πRZiwithminμRZi,ηRZi,νRZi≥1/4⇒μQZi≥μRZi≥1/4 and ηQZi≥ηRZi≥1/4,νQZi≥νRZi≥1/4andπQZi≤πRZi≤1/4 which hold (16) and (20).
There are two distance measures, i.e., Hamming and Euclidean distance which are widely used to determine the distance between the two differentIFss. The authors in [34] introduced the concept of distance between the two different IFss specifically in terms of their parameters μ,ν,π. Subsequently, PFSS is the generalized form of IFss which have four components, i.e., μ,ν,η,π. Therefore, this concept of distance measure is significantly used from IFss to PFSS. Thus, inequalities (14) and (15) concluded that PFSR is nearest to the maximum value 1/41/41/41/4 than Q.Theorem 1.
To validate the proof of the novel entropy information measure forPFS.Proof.
The proof of the validity of the proposed measure will be established if it agreed with the axioms listed in Definition8.
PFS1sharpness. Initially, we assume θQ=0.(16)θQ=1n∑i=1n1.582expexp−expμQZi+ηQZi−νQZi−πQZi:QandZi∈PFSZ.
This will hold if following cases are possible:(17)IfμQZi=1,ηQZi=νQZi=πQZi=0,or,ηQ=1μQZi=νQZi=πQZi=0,or,νQZi=1,μQZi=ηQZi=πQZi=0,or,πQZi=1,μQZi=ηQZi=νQZi=0.
Hence, all the above cases show thatθQ=0. This implies that θ is the crisp set. Therefore conversely, assume that θ is the crisp set. This provides(18)eitherμQZi=1ηQZi=νQZi=πQZi=0,orηQ=1μQZi=νQZi=πQZi=0,orνQZi=1μQZi=ηQZi=πQZi=0,orπQZi=1μQZi=ηQZi=νQZi=0.
Thus,θQ=0⇔Qis a crisp set.
PFS2maximality. Equation (16) has the maximum value for(19)θQ=1n∑i=1n1.582expexp−expμQZi+ηQZi−νQZi−πQZi:Q,Zi∈PFSZ.
It is obvious that ifμQ=ηQ=νQ=πQ=1/4, this implies that entropy measure has the maximum value.
PFS3resolution. The property PFS2 shows that θQ attains the maximum value if μQ=ηQ=νQ=πQ=1/4. Therefore, if max μQZi,ηQZi,νQZi≤1/4, and, μQ≤μR,ηQ≤ηR, andνQ≤νR. This concludes πQZi≥πRZi≥1/4.
Subsequently, if minimumμQZi,ηQZi,νQZi≥1/4, Then, μQ≥μR,ηQ≥ηR,andνQ≥νR. Thus, πQZi≤πRZi≤1/4. Therefore, use of equations (14) and (15). This is concluded that θQ holds the property of resolution.
PFS4symmetry. This property is satisfied directly from the definition, i.e., θQ=θQc.
This is the complete proof of the theorem.Theorem 2.
. If any twoQ,R∈PFSZ:Q⊆RorR⊆Q, then(20)θQ∪R+θQ∩R=θQ+θR.Proof.
LetZ1andZ2be the two parts of∈Zuniversal set such that(21)Z1=Zi∈Z:Q⊆R,Z2=Zi∈Z:R⊆Q,(22)μQ≤μRηQ≤ηR,νQ≥νRfor allZi∈Z1,μQ≥μR,ηQ≥ηR,νQ≤νRfor allZi∈Z2,(23)θQ∪R=1.582nexp∑i=1nexp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi,θQ∪R=1.582nexp∑Z1exp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi+1.582nexp∑Z2exp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi,θQ∪R=1.582nexp∑Z1exp−expμRZi+ηRZi−νRZi−πRZi+1.582nexp∑Z2exp−expμQZi+ηQZi−νQZi−πQZi.
As accordingly,(24)θQ∩R=1.582nexp∑Z1exp−expμQ∩RZi+ηQ∩RZi−νQ∩RZi−πQ∩RZi+1.582nexp∑Z2exp−expμQ∩RZi+ηQ∩RZi−νQ∩RZi−πQ∩RZi,θQ∩R=1.582nexp∑Z1exp−expμQZi+ηQZi−νQZi−πQZi+1.582nexp∑Z2exp−expμRZi+ηRZi−νRZi−πRZi.
Therefore, equations (23) and (24) prove that(25)θQ∪R+θQ∩R=θQ+θR.


## 4.1. Validation of Proposed Information Measure

Firstly, the following inequalities for anyQandR∈PFSZ:Qcrisper thanRlisted in axiomsPFS3 are proved:(14)μQZi−14+ηQZi−14+νQZi−14+πQZi−14≥μRZi−14+ηRZi−14+νRZi−14+πRZi−14,and(15)μQZi−142+ηQZi−142+νQZi−142+πQZi−142≥μRZi−142+ηRZi−142+νRZi−142+πRZi−142.Proof.
IfμQZi≤μRZi,ηQZi≤ηRZi,νQZi≤νRZiandπQZi≥πRZiwith maxμRZi,ηRZi,νRZi≤1/4⇒μQZi≤μRZi≤1/4 and ηQZi≤ηRZi≤1/4,νQZi≤νRZi≤1/4andπQZi≥πRZi≥1/4This proves that equations (14) and (15) are correct.
IfifμQZi≥μRZi,ηQZi≥ηRZi,νQZi≥νRZiandπQZi≤πRZiwithminμRZi,ηRZi,νRZi≥1/4⇒μQZi≥μRZi≥1/4 and ηQZi≥ηRZi≥1/4,νQZi≥νRZi≥1/4andπQZi≤πRZi≤1/4 which hold (16) and (20).
There are two distance measures, i.e., Hamming and Euclidean distance which are widely used to determine the distance between the two differentIFss. The authors in [34] introduced the concept of distance between the two different IFss specifically in terms of their parameters μ,ν,π. Subsequently, PFSS is the generalized form of IFss which have four components, i.e., μ,ν,η,π. Therefore, this concept of distance measure is significantly used from IFss to PFSS. Thus, inequalities (14) and (15) concluded that PFSR is nearest to the maximum value 1/41/41/41/4 than Q.Theorem 1.
To validate the proof of the novel entropy information measure forPFS.Proof.
The proof of the validity of the proposed measure will be established if it agreed with the axioms listed in Definition8.
PFS1sharpness. Initially, we assume θQ=0.(16)θQ=1n∑i=1n1.582expexp−expμQZi+ηQZi−νQZi−πQZi:QandZi∈PFSZ.
This will hold if following cases are possible:(17)IfμQZi=1,ηQZi=νQZi=πQZi=0,or,ηQ=1μQZi=νQZi=πQZi=0,or,νQZi=1,μQZi=ηQZi=πQZi=0,or,πQZi=1,μQZi=ηQZi=νQZi=0.
Hence, all the above cases show thatθQ=0. This implies that θ is the crisp set. Therefore conversely, assume that θ is the crisp set. This provides(18)eitherμQZi=1ηQZi=νQZi=πQZi=0,orηQ=1μQZi=νQZi=πQZi=0,orνQZi=1μQZi=ηQZi=πQZi=0,orπQZi=1μQZi=ηQZi=νQZi=0.
Thus,θQ=0⇔Qis a crisp set.
PFS2maximality. Equation (16) has the maximum value for(19)θQ=1n∑i=1n1.582expexp−expμQZi+ηQZi−νQZi−πQZi:Q,Zi∈PFSZ.
It is obvious that ifμQ=ηQ=νQ=πQ=1/4, this implies that entropy measure has the maximum value.
PFS3resolution. The property PFS2 shows that θQ attains the maximum value if μQ=ηQ=νQ=πQ=1/4. Therefore, if max μQZi,ηQZi,νQZi≤1/4, and, μQ≤μR,ηQ≤ηR, andνQ≤νR. This concludes πQZi≥πRZi≥1/4.
Subsequently, if minimumμQZi,ηQZi,νQZi≥1/4, Then, μQ≥μR,ηQ≥ηR,andνQ≥νR. Thus, πQZi≤πRZi≤1/4. Therefore, use of equations (14) and (15). This is concluded that θQ holds the property of resolution.
PFS4symmetry. This property is satisfied directly from the definition, i.e., θQ=θQc.
This is the complete proof of the theorem.Theorem 2.
. If any twoQ,R∈PFSZ:Q⊆RorR⊆Q, then(20)θQ∪R+θQ∩R=θQ+θR.Proof.
LetZ1andZ2be the two parts of∈Zuniversal set such that(21)Z1=Zi∈Z:Q⊆R,Z2=Zi∈Z:R⊆Q,(22)μQ≤μRηQ≤ηR,νQ≥νRfor allZi∈Z1,μQ≥μR,ηQ≥ηR,νQ≤νRfor allZi∈Z2,(23)θQ∪R=1.582nexp∑i=1nexp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi,θQ∪R=1.582nexp∑Z1exp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi+1.582nexp∑Z2exp−expμQ∪RZi+ηQ∪RZi−νQ∪RZi−πQ∪RZi,θQ∪R=1.582nexp∑Z1exp−expμRZi+ηRZi−νRZi−πRZi+1.582nexp∑Z2exp−expμQZi+ηQZi−νQZi−πQZi.
As accordingly,(24)θQ∩R=1.582nexp∑Z1exp−expμQ∩RZi+ηQ∩RZi−νQ∩RZi−πQ∩RZi+1.582nexp∑Z2exp−expμQ∩RZi+ηQ∩RZi−νQ∩RZi−πQ∩RZi,θQ∩R=1.582nexp∑Z1exp−expμQZi+ηQZi−νQZi−πQZi+1.582nexp∑Z2exp−expμRZi+ηRZi−νRZi−πRZi.
Therefore, equations (23) and (24) prove that(25)θQ∪R+θQ∩R=θQ+θR.


## 5. The Proposed Picture Fuzzy Entropy Measure with Application inMCDM Process

TheMCDM process involves the different techniques to solve the decision and planning problems, which involve multiple criteria. The objective is to support decision-maker in tackling such problems. To attain the unique optimal solution of such problems, the decision-makers fixed a set of alternatives and criteria. The fuzzy set and its various extended forms were included in MCDM techniques because of the vagueness and ambiguity in the data analysis. Therefore, fuzzy sets have a great importance in the decision-making process. The MCDM problems may be symbolized as a matrix say PFDMmˇ×nˇ which represents the mˇ alternatives "ALT1,ALT2,……………ALTm″ and nˇ criteria ″CR1,CR2…………CRn″ in the shape of rows and columns or vice versa. The entries in PFMmˇ×nˇ refer the degrees in accordance with PFS described in Definition 3. In practical problems where the human opinions are there, PFS is comparatively useful. Let the group of decision-makers Nˇ expressed in PFS describe the (+) membership degree for support nˇposij, neutral membership degree nˇneuij signify the undecided or neutral, and (-) membership nˇnegij represent unfavour of alternative for specific criteria. Consequently, the following statistical tools are used to express different degrees pertaining to PFS.(26)μˇij=nˇposijNˇ,ηˇij=nˇneuijNˇ,νˇij=nˇnegijNˇ.The statistical tool mentioned in (26)is significantly used if the available data are in the form of numerical quantity. In decision-making analysis, the different degrees of PFS are entirely reliant upon the single opinion or choice of decision-makers apart of any kind of information to render a decision. Such practical problems in PFS decision-making process are solved more effectively with greater accuracy.


### 5.1. Picture Fuzzy Sets with Proposed VIKOR Approach

The author [35] was the pioneer of VIKOR approach to obtain the most desired alternatives based upon the closeness index to the (+) ideal solution and farthest from the (−) ideal solution in decision-making problems. The key salient feature of present approach helps to provide the compromise solution. Initially, the idea of compromise solution was developed by [36] in1973. The following procedural steps are widely used to solve the fuzzy decision-making problems.Step 1.
Formation of picture fuzzy decision matrixPFDMmˇ×nˇ.
PFDMmˇ×nˇ is constructed in such a way that entries of the matrix elements are computed in accordance with different membership degrees of PFS briefed in equation (26).Step 2.
Normalization ofPFDMmˇ×nˇ.
The fabricated matrixPFDMmˇ×nˇ is normalized as shown in Table 1.(27)Where,μˇij,ηˇij,νˇij=μij,ηij,νijfor benefit criteriaνij,ηij,μijfor cost criteria.Table 1
The fabricated matrixPFDMmˇ×nˇ.
CR1CR2…CRnALT1μˇ11,ηˇ11,νˇ11μˇ12,ηˇ12,νˇ12…μˇ1n,ηˇ1n,νˇ1nALT2μˇ21,ηˇ21,νˇ21μˇ22,ηˇ22,νˇ22…μˇ2n,ηˇ2n,νˇ2n⋮⋮⋮⋱⋮ALTmμˇm1,ηˇm1,νˇm1μˇm2,ηˇm2,νˇm2…μˇmn,ηˇmn,νˇmnNote.μˇij,ηˇij,νˇij=μij,ηij,νijfor benefit criteriaνij,ηij,μijfor cost criteria.Step 3.
Assessment of criteria weights
The choice of selecting the weights of criteria or attributes has a substantial role in the way out of the solution ofMCDM problems. The appropriate and justified assessment of criteria weights might lead to the desired result of a suitable alternative. Considering the classified the criteria weights in two categories, i.e., subjective and objective weights [37], the objective weight apprehends the criteria information in a quantified manner; therefore, entropy practice is the best reliable approach to capture and handle such information. Sometimes, in MCDM process, the decision-makers have the criteria weights information in separate manner, i.e., entirely known or unknown and may be partially known according to problem scenario.
As a result, the following approaches are used to compute the criteria weights for both the cases:(a)
Methodology for Unknown Criteria Weights. To compute the unspecified criteria-weight, the author [37] proposed the method. (28)w∧j=I−θQJn−∑j=1nθQJ:j=1,2,3……n,where(29)θQ=1n∑i=1m1.582expexp−expμij+ηij−νij−πij.(b)
Methodology for Partially Known Criteria Weights.  There are many complexities which are cited by the decision-makers to solve the problems in the decision-making system such as paucity of time, information, and lack of expertise around problem domain. Therefore, the decision-makers are enabled to give their judgment or decision precisely in the form of numeral value. They often express their opinions of the judgment in intervals to overcome such incongruities. In this framework, to explore the attribute weights from the set of partial information, the decision-makers generally use the method of minimum entropy proposed by [38]. Hence, the alternatives entropy across the distinct criteria CRj is presented as follows:(30)α∧ALTi=∑j=1nα∧μij,ηij,νij,πij,such thatα∧μij,ηij,νij,πij=1n∑i=1m1.582expexp−expμij+ηij−νij−πij.
Thus, the following programming model is frequently used by the decision-makers for the optimum criteria evaluation:(31)E∧=∑I=1MWj.α∧ALTi=1.582nexp∑i=1m∑j=1nwjexp−expμij+ηij−νij−πij,wherew∈β∑j=1nwj=1wj≥1:j=1,2…n.
β denotes the set of partial known informations to evaluate the weights of different criteria.Step 4.
Determination of the relative ideal solution forPFS.
InMCDM problems, the ideal solution provides the best alternatives which maximize the most benefited attribute or criteria and minimize the least cost criteria. The objective of the VIKOR method is to opt the best alternative nearest to the (+) ideal solution along with farthest away from the (−) ideal solution. Firstly, the picture fuzzy relative (+) ideal solution and (−) ideal solution are constructed to determine the alternatives.
For benefit criteria,(32)ALT+=ALT1+,ALT2+,………ALTn+,ALT−=ALT1−,ALT2−,………ALTn−,where(33)ALTJ+=μij+,ηij+,νij+=μˇij,ηˇij,νˇij,πij+=1−μij+−ηij+−νij+;j=1,2…n,ALTJ−=μij−,ηij−,νij−=μˇij,ηˇij,νˇij,πij+=1−μˇij−ηˇij−νˇij;j=1,2…n.
For cost criteria,(34)ALT+=ALT1+,ALT2+,………ALTn+,ALT−=ALT1−,ALT2−,………ALTn−.where(35)ALTJ+=μij+,ηij+,νij+=μˇij,ηˇij,νˇij,πij+=1−μij+−ηij+−νij+;j=1,2,…n,ALTJ−=μij−,ηij−,νij−=μˇij,ηˇij,νˇij,πij+=1−μˇij−ηˇij−νˇij;j=1,2…n.Step 5.
Evaluation of compromise solution
In decision sciences, the VIKOR method is very useful for the compromise solution, when the preference of decision-makers about alternatives is conflicting from the set of criteria information. In the VIKOR method, the compromise solution is obtained with the use ofL∧p metric which was earlier introduced by [36]. The metric L∧p is an aggregate function used as follows:(36)L∧p,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−p1/p,p∈1,∞,i∈1,2,……m,where ALT+ and ALT− are described in equations (33) and (35).(37)Φ∧ij=μˇij,ηˇij,νˇij,wjare the weight of the criteriajth;j=1……n.
The parameterp has the distinct values varying between 1,∞ which imply the different meaning in perspective of group utility and individual regret. As the increased value of p, the individual regret is substantially increased and the group utility or efficacy is decreased Hence, the VIKOR method gives the compromise solution with maximum group efficacy L∧1,i and reduces the individual regret L∧∞,i as follows:(38)G∧ui=L∧1,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−,(39)I∧Ri=L∧∞,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−.Step 6.
VIKOR index coefficient.
The index coefficientI∧c in the VIKOR method is computed as(40)I∧c=Θ˜G∧ui−G∧u⋇G∧u⋇−G∧u⋇+1−Θ˜I∧Ri−I∧R⋇I∧R⋇−I∧R⋇,whereG∧u⋇=G∧ui,G∧u⋇=G∧ui,I∧R⋇=I∧Ri,I∧R⋇=I∧Ri.
In addition, the parametersΘ˜ and 1−Θ˜ represent the group utility weight and individual regret as per the choice of its value by decision-makers as follows:(41)IFΘ˜=0.5⇒consenus>0.5⇒voting by majority<0.5⇒Veto.Step 7.
Finally, the ascending order of the determined values ofG∧ui,I∧Ri, andI∧c is arranged, and for the most preferred alternatives, the CONI and CONII are two respective conditions satisfied as(42)CONI⇒Aceptable Advantage:IF the value,I∧ALT2−I∧ALT1≥1n−1;ALT1andALT2are cataloged at1stand2ndplace in distinct value ofI∧c,CONII⇒AceptableStablity:ALT1must be cata loged at1stplace in the distinct value of eitherG∧uiandI∧Ri.
If the stated condition does not satisfy together, then we look upon a compromise solution proceed as follows.Case 1.
ALT1andALT2 are the compromise solutions if Acceptable Stability condition does not satisfy.Case 2.
If Acceptable Advantage is not accepted, then alternativesALT1,ALT2,……………ALTm have the compromise solution along with ALTm if it does hold the following condition:(43)I∧ALTm−I∧ALT1<1n−1;m⇒for some maximum andnimplies the number of alternatives.


#### 5.1.1. Picture Fuzzy Sets with Proposed TOPSIS Approach

The concept of TOSIS approach initiates and incorporates inMCDM decision sciences by [39]. Since then, this method is prominently used to many complex practical problems in a fuzzy situation. The selection process of the best attributes in the proposed approach is to focus on the alternatives that one closest to the ideal solution and farthest, respectively, from the (−) ideal solution.Further, to complete the calculative part of theMCDM problem, we carry forward the first four steps stated above in the first approach.Step 8.
This step involves the picture fuzzy discriminant measureED+ and ED− for distinct attributes from ALT+ and ALT−, respectively, with use of equation (7).(44)ED+=∑j=1nwjΦ∧ij−ALT+20.5,ED−=∑j=1nwjΦ∧ij−ALT−20.5.Step 9.
The relative closeness coefficientRcc is computed as follows:(45)Rcc=ED−ED−+ED+.Step 10.
At the end, alternatives ranking have been done according to the ordering the distinct values ofRcc. The larger the value is, the most suitable the alternatives are. Thus, we have successfully presented the procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS and for the sake of understanding the flow chart shown in Figure 3.Figure 3
The procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS.


## 5.1. Picture Fuzzy Sets with Proposed VIKOR Approach

The author [35] was the pioneer of VIKOR approach to obtain the most desired alternatives based upon the closeness index to the (+) ideal solution and farthest from the (−) ideal solution in decision-making problems. The key salient feature of present approach helps to provide the compromise solution. Initially, the idea of compromise solution was developed by [36] in1973. The following procedural steps are widely used to solve the fuzzy decision-making problems.Step 1.
Formation of picture fuzzy decision matrixPFDMmˇ×nˇ.
PFDMmˇ×nˇ is constructed in such a way that entries of the matrix elements are computed in accordance with different membership degrees of PFS briefed in equation (26).Step 2.
Normalization ofPFDMmˇ×nˇ.
The fabricated matrixPFDMmˇ×nˇ is normalized as shown in Table 1.(27)Where,μˇij,ηˇij,νˇij=μij,ηij,νijfor benefit criteriaνij,ηij,μijfor cost criteria.Table 1
The fabricated matrixPFDMmˇ×nˇ.
CR1CR2…CRnALT1μˇ11,ηˇ11,νˇ11μˇ12,ηˇ12,νˇ12…μˇ1n,ηˇ1n,νˇ1nALT2μˇ21,ηˇ21,νˇ21μˇ22,ηˇ22,νˇ22…μˇ2n,ηˇ2n,νˇ2n⋮⋮⋮⋱⋮ALTmμˇm1,ηˇm1,νˇm1μˇm2,ηˇm2,νˇm2…μˇmn,ηˇmn,νˇmnNote.μˇij,ηˇij,νˇij=μij,ηij,νijfor benefit criteriaνij,ηij,μijfor cost criteria.Step 3.
Assessment of criteria weights
The choice of selecting the weights of criteria or attributes has a substantial role in the way out of the solution ofMCDM problems. The appropriate and justified assessment of criteria weights might lead to the desired result of a suitable alternative. Considering the classified the criteria weights in two categories, i.e., subjective and objective weights [37], the objective weight apprehends the criteria information in a quantified manner; therefore, entropy practice is the best reliable approach to capture and handle such information. Sometimes, in MCDM process, the decision-makers have the criteria weights information in separate manner, i.e., entirely known or unknown and may be partially known according to problem scenario.
As a result, the following approaches are used to compute the criteria weights for both the cases:(a)
Methodology for Unknown Criteria Weights. To compute the unspecified criteria-weight, the author [37] proposed the method. (28)w∧j=I−θQJn−∑j=1nθQJ:j=1,2,3……n,where(29)θQ=1n∑i=1m1.582expexp−expμij+ηij−νij−πij.(b)
Methodology for Partially Known Criteria Weights.  There are many complexities which are cited by the decision-makers to solve the problems in the decision-making system such as paucity of time, information, and lack of expertise around problem domain. Therefore, the decision-makers are enabled to give their judgment or decision precisely in the form of numeral value. They often express their opinions of the judgment in intervals to overcome such incongruities. In this framework, to explore the attribute weights from the set of partial information, the decision-makers generally use the method of minimum entropy proposed by [38]. Hence, the alternatives entropy across the distinct criteria CRj is presented as follows:(30)α∧ALTi=∑j=1nα∧μij,ηij,νij,πij,such thatα∧μij,ηij,νij,πij=1n∑i=1m1.582expexp−expμij+ηij−νij−πij.
Thus, the following programming model is frequently used by the decision-makers for the optimum criteria evaluation:(31)E∧=∑I=1MWj.α∧ALTi=1.582nexp∑i=1m∑j=1nwjexp−expμij+ηij−νij−πij,wherew∈β∑j=1nwj=1wj≥1:j=1,2…n.
β denotes the set of partial known informations to evaluate the weights of different criteria.Step 4.
Determination of the relative ideal solution forPFS.
InMCDM problems, the ideal solution provides the best alternatives which maximize the most benefited attribute or criteria and minimize the least cost criteria. The objective of the VIKOR method is to opt the best alternative nearest to the (+) ideal solution along with farthest away from the (−) ideal solution. Firstly, the picture fuzzy relative (+) ideal solution and (−) ideal solution are constructed to determine the alternatives.
For benefit criteria,(32)ALT+=ALT1+,ALT2+,………ALTn+,ALT−=ALT1−,ALT2−,………ALTn−,where(33)ALTJ+=μij+,ηij+,νij+=μˇij,ηˇij,νˇij,πij+=1−μij+−ηij+−νij+;j=1,2…n,ALTJ−=μij−,ηij−,νij−=μˇij,ηˇij,νˇij,πij+=1−μˇij−ηˇij−νˇij;j=1,2…n.
For cost criteria,(34)ALT+=ALT1+,ALT2+,………ALTn+,ALT−=ALT1−,ALT2−,………ALTn−.where(35)ALTJ+=μij+,ηij+,νij+=μˇij,ηˇij,νˇij,πij+=1−μij+−ηij+−νij+;j=1,2,…n,ALTJ−=μij−,ηij−,νij−=μˇij,ηˇij,νˇij,πij+=1−μˇij−ηˇij−νˇij;j=1,2…n.Step 5.
Evaluation of compromise solution
In decision sciences, the VIKOR method is very useful for the compromise solution, when the preference of decision-makers about alternatives is conflicting from the set of criteria information. In the VIKOR method, the compromise solution is obtained with the use ofL∧p metric which was earlier introduced by [36]. The metric L∧p is an aggregate function used as follows:(36)L∧p,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−p1/p,p∈1,∞,i∈1,2,……m,where ALT+ and ALT− are described in equations (33) and (35).(37)Φ∧ij=μˇij,ηˇij,νˇij,wjare the weight of the criteriajth;j=1……n.
The parameterp has the distinct values varying between 1,∞ which imply the different meaning in perspective of group utility and individual regret. As the increased value of p, the individual regret is substantially increased and the group utility or efficacy is decreased Hence, the VIKOR method gives the compromise solution with maximum group efficacy L∧1,i and reduces the individual regret L∧∞,i as follows:(38)G∧ui=L∧1,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−,(39)I∧Ri=L∧∞,i=∑j=1nwjALT+−Φ∧ijALT+−ALT−.Step 6.
VIKOR index coefficient.
The index coefficientI∧c in the VIKOR method is computed as(40)I∧c=Θ˜G∧ui−G∧u⋇G∧u⋇−G∧u⋇+1−Θ˜I∧Ri−I∧R⋇I∧R⋇−I∧R⋇,whereG∧u⋇=G∧ui,G∧u⋇=G∧ui,I∧R⋇=I∧Ri,I∧R⋇=I∧Ri.
In addition, the parametersΘ˜ and 1−Θ˜ represent the group utility weight and individual regret as per the choice of its value by decision-makers as follows:(41)IFΘ˜=0.5⇒consenus>0.5⇒voting by majority<0.5⇒Veto.Step 7.
Finally, the ascending order of the determined values ofG∧ui,I∧Ri, andI∧c is arranged, and for the most preferred alternatives, the CONI and CONII are two respective conditions satisfied as(42)CONI⇒Aceptable Advantage:IF the value,I∧ALT2−I∧ALT1≥1n−1;ALT1andALT2are cataloged at1stand2ndplace in distinct value ofI∧c,CONII⇒AceptableStablity:ALT1must be cata loged at1stplace in the distinct value of eitherG∧uiandI∧Ri.
If the stated condition does not satisfy together, then we look upon a compromise solution proceed as follows.Case 1.
ALT1andALT2 are the compromise solutions if Acceptable Stability condition does not satisfy.Case 2.
If Acceptable Advantage is not accepted, then alternativesALT1,ALT2,……………ALTm have the compromise solution along with ALTm if it does hold the following condition:(43)I∧ALTm−I∧ALT1<1n−1;m⇒for some maximum andnimplies the number of alternatives.


### 5.1.1. Picture Fuzzy Sets with Proposed TOPSIS Approach

The concept of TOSIS approach initiates and incorporates inMCDM decision sciences by [39]. Since then, this method is prominently used to many complex practical problems in a fuzzy situation. The selection process of the best attributes in the proposed approach is to focus on the alternatives that one closest to the ideal solution and farthest, respectively, from the (−) ideal solution.Further, to complete the calculative part of theMCDM problem, we carry forward the first four steps stated above in the first approach.Step 8.
This step involves the picture fuzzy discriminant measureED+ and ED− for distinct attributes from ALT+ and ALT−, respectively, with use of equation (7).(44)ED+=∑j=1nwjΦ∧ij−ALT+20.5,ED−=∑j=1nwjΦ∧ij−ALT−20.5.Step 9.
The relative closeness coefficientRcc is computed as follows:(45)Rcc=ED−ED−+ED+.Step 10.
At the end, alternatives ranking have been done according to the ordering the distinct values ofRcc. The larger the value is, the most suitable the alternatives are. Thus, we have successfully presented the procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS and for the sake of understanding the flow chart shown in Figure 3.Figure 3
The procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS.


## 5.1.1. Picture Fuzzy Sets with Proposed TOPSIS Approach

The concept of TOSIS approach initiates and incorporates inMCDM decision sciences by [39]. Since then, this method is prominently used to many complex practical problems in a fuzzy situation. The selection process of the best attributes in the proposed approach is to focus on the alternatives that one closest to the ideal solution and farthest, respectively, from the (−) ideal solution.Further, to complete the calculative part of theMCDM problem, we carry forward the first four steps stated above in the first approach.Step 8.
This step involves the picture fuzzy discriminant measureED+ and ED− for distinct attributes from ALT+ and ALT−, respectively, with use of equation (7).(44)ED+=∑j=1nwjΦ∧ij−ALT+20.5,ED−=∑j=1nwjΦ∧ij−ALT−20.5.Step 9.
The relative closeness coefficientRcc is computed as follows:(45)Rcc=ED−ED−+ED+.Step 10.
At the end, alternatives ranking have been done according to the ordering the distinct values ofRcc. The larger the value is, the most suitable the alternatives are. Thus, we have successfully presented the procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS and for the sake of understanding the flow chart shown in Figure 3.Figure 3
The procedural steps to solve the picture fuzzy MCDM problems by employing the proposed picture fuzzy entropy measure with the modified form of VIKOR and TOPSIS.


## 6. Validation of Proposed Entropy Measure with Example

The application of the PFS is to solve the MCDM problems, where simultaneously several attributes are likely to exist. In such type of environment, the human mindsets are always in dilemma; therefore,PFSS is considerably used. In that point of view, a national welfare society conducted the social survey to get the public opinion for the functioning and implementation of the government policies for the national importance. They fix the framework of six different criteria, i.e.,CR1: unemploymentCR2: foreign relationsCR3: national education policyCR4: financial servicesCR5: strategy for COVID−19CR6: national policies for social development and empowermentAnd the national parties are classified into five categories,i.e.,″ALT1,ALT2,ALT3,ALT4,ALT5.″ The survey was conducted on a sample of 1000=Nˇ peoples. The results show 450 peoples in support with CR1, 320 peoples unsupport with CR1, 120 peoples are neutral, and 110 peoples are not participated, i.e., abstain. Then, with the use of equation (26), the entries of the PFS matrix can be determined as follows:(46)μˇ11=4501000,ηˇ11=1201000,νˇ11=3201000.Further, entries ofPFDM5×6 are computed in the same manner.VIKOR approach for proposed entropy measure is as follows:(1)
The accumulated information as per opinion of peoples constitutes the normalizedPFDM5×6 listed in Table 2.(2)
The notionPFDMW used for the calculated values of different criteria with information measure mentioned in (14) is listed in Table 3.(3)
Utilizing equation (28), the weights are determined for the defined criteria throughout the alternatives with use of values PFDMW catalogued in Table 4.(4)
The values forALT+ and ALT− are computed with use of equation (34) as listed in Table 5.(5)
Table6 briefs the computed values for G∧ui,I∧Ri, andI∧c by using equations (39), (40), and (41).(6)
In accordance with the computed values forG∧ui,I∧Ri, andI∧c, the ranking of the alternatives is shown in Table 7.(7)
The indexing showsALT1andALT3 located at 2nd and 1st position in Table 7.TheCONI condition I∧ALT1−I∧ALT3=0.1219−0.016=0.1059≥≠/6−1=0.2 does not hold.Therefore, we proceed for the compromise solution, i.e.,(47)I∧ALT1−I∧ALT3=0.1219−0.016=0.1059<16−1=0.2,I∧ALT2−I∧ALT3=0.5596−0.016=0.5436>16−1=0.2,I∧ALT4−I∧ALT3=1−0.016=0.984>16−1=0.2,I∧ALT5−I∧ALT3=0.1757−0.016=0.1597<16−1=0.2.Thus,ALT1, ALT3, and ALT5 hold the CONII condition, and the alternatives ALT1, ALT3, and ALT5 are the compromise solution.TOPSIS approachProceed with the same manner up to Step 4.(8)
The picture fuzzy discriminant measuresED+ and ED− are computed for distinct attributes from ALT+ and ALT−, respectively, with use of equation (46) and listed in Table 8.(9)
The value of relative closeness coefficientRcc degree with use of (47) is mentioned in Table 9.Table 2
Opinion of peoples constitutes the normalizedPFDM5×6.
CR1CR2CR3CR4CR5CR6ALT1 0.45, 0.32, 0.120.34, 0.15, 0.070.69, 0.06, 0.170.78, 0.02, 0.120.84, 0.04, 0.080.13, 0.53, 0.05ALT2 0.46, 0.05, 0.120.18, 0.13, 0.310.81, 0.11, 0.040.69, 0.12, 0.110.36, 0.32, 0.310.05, 0.66, 0.16ALT3 0.24, 0.27, 0.120.57, 0.10, 0.270.81, 03, 0.100.48, 0.24, 0.230.17, 0.58, 0.160.77, 0.04, 0.12ALT4 0.36, 0.24, 0.080.4, 0.23, 0.340.08, 0.72, 0.060.23, 0.14, 0.150.28, 0.28, 0.280.59, 0.10, 0.20ALT5 0.40, 0.30, 0.140.56, 0.33, 0.070.64, 0.12, 0.210.56, 0.13, 0.110.51, 0.10, 0.360.35, 0.39, 0.12Table 3
Calculated values of different criteria with information measure.
CR1CR2CR3CR4CR5CR6PFDMV0.828950.724630.482850.651640.698090.66986Table 4
The weights for the defined criteria throughout the alternatives with use of valuesPFDMW.
W^1W^2W^3W^4W^5W^60.087980.141650.266020.179200.155300.16982Table 5
The values forALT+ and ALT−.
ALT+0.46, 0.05, 0.080.57, 0.10, 0.070.81, 0.03, 0.040.78, 0.02, 0.110.84, 0.04, 0.080.77, 0.04, 0.05ALT−0.24, 0.32, 0.140.18, 0.33, 0.340.08, 0.72, 0.210.23, 0.24, 0.230.17, 0.58, 0.360.05, 0.66, 0.20Table 6
The computed values forG^ui,I^Ri,andI^c.
ALT1ALT2ALT3ALT4ALT5G^ui0.30380.49650.31540.66520.4024I^Ri0.1340.18310.0990.24250.1103I^c0.12190.55960.01610.1757Table 7
The computed values forG^ui,I^Ri,andI^c.
ALT1ALT2ALT3ALT4ALT5G^uiIIVIIVIIII^RiIIIIVIVIII^cIIIVIVIIITable 8
The picture fuzzy discriminant measuresED+ and ED− are computed for distinct attributes from ALT+ and ALT−.
ALT1ALT2ALT3ALT4ALT5ED+0.420050.483850.401900.635090.33341ED−0.599310.579400.726700.417470.61122Table 9
The value of relative closeness coefficientRcc.
ALT1ALT2ALT3ALT4ALT50.58790.54490.64380.39660.6470Finally, the indexing of the alternatives in descending order list is shown in Table10.Table 10
The indexing of the alternatives in descending order.
ALT1ALT2ALT3ALT4ALT5IIIIVIIVIFrom the indexing of the alternatives, it is concluded thatALT5 is the most desirable alternative.


## 7. Comparative Evaluation

To verify the effectiveness and practicability of the proposed information measure with TOSIS and VIKOR approaches, the result prevails in such a way that the TOPSIS method provides the best alternativesALT5 whereas VIKOR has a compromise solution, i.e., ALT1,ALT3, and ALT5. Thus, it is quite natural to think about the most benefited alternative. The authors [40] have briefly explained the relative difference between these two approaches for selection of best alternative. The feasible compromise solution in the VIKOR approach formed a consent of mutual compromises focussed on optimising G∧ui and minimising I∧Ri; however, the TOPSIS approach adopts the feasible solution without considering the relative importance of the distances which are nearest and farthest from an ideal solution. In the perspective of the proposed example, if the results are interpreted in the broader sense, then it shows that national party ALT5 is the choice of the peoples in majority and would be able to form the government. In compromise solution, it reveals that the parties ALT1,ALT3, and ALT5 are also the choice of the peoples which leads the alliance situation to establish the government. There is also the possibility of an alliance between the parties either two or three to have a majority criterion to form the government.


## 8. Concluded Remarks

Till now, several mindsets have developed different approaches or methods of information measure to explore the information in picture fuzzy environment, and still the researchers are working hard to develop such information measures which are more reliable, precise, and informative reasonably with existing measure techniques. The proposed picture novel entropy was validated with the two different approaches. The results reveal that the best alternative is analogous in both approaches and proved that information measure is quite appropriate to solve the complex issues onMCDM system. The trustworthiness and practical applicability of the proposed work proved to improve the voting system and the opinion of the peoples about the parties to the national importance. The proposed model could be used to predict the future outcomes of the polling. The novel research work is significantly suitable to the MCDM complex multivariable problems, relating to the neutral factors or variables. The role of neutral variables in complex issues affects the result of decision-makers. This advantage of the proposed work facilitates the decision-makers to handle the ambiguity that arises due to neutral variables in complex issues. Therefore, it concluded that novel picture information measure is more reliable, sophisticated, and informative as well as simpler, to tackle the uncertainty or vagueness more precisely and effectively in a picture fuzzy environment.


### 8.1. The Advantage and Future Scope of the Current Research

The present research work could be extensively applied as a model to predict the forthcoming results in the field of data analysis, i.e., healthcare, agriculture, smart cities, and many more approaches in relation to sustainable development. Further, it can be explored toPFSS in terms of divergence measure, intervalued, and many more. The proposed research work could be bridged between the complex evidence theory as well as quantum mechanics theory to envisage the human decision behaviours in decision theory. Moreover, the presented research work model is also utilized in intelligent generalized quality-based approach which is capable enough to fuse the complex multivariable informations. In fact, decision-makers are comfortable to express the human judgments or suggestions in the linguistic form in picture fuzzy environment. Therefore, present research works can be utilized in fuzzy linguistic models.


## 8.1. The Advantage and Future Scope of the Current Research

The present research work could be extensively applied as a model to predict the forthcoming results in the field of data analysis, i.e., healthcare, agriculture, smart cities, and many more approaches in relation to sustainable development. Further, it can be explored toPFSS in terms of divergence measure, intervalued, and many more. The proposed research work could be bridged between the complex evidence theory as well as quantum mechanics theory to envisage the human decision behaviours in decision theory. Moreover, the presented research work model is also utilized in intelligent generalized quality-based approach which is capable enough to fuse the complex multivariable informations. In fact, decision-makers are comfortable to express the human judgments or suggestions in the linguistic form in picture fuzzy environment. Therefore, present research works can be utilized in fuzzy linguistic models.



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*Source: 2505397-2022-02-27.xml*