# Graphical Exploration of Generalized Picture Fuzzy Hypersoft Information with Application in Human Resource Management Multiattribute Decision-Making

Authors: Atiqe Ur Rahman; Muhammad Saeed; Ebenezer Bonyah; Muhammad Arshad
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In this study, the graphical exploration of a novel hybrid, that is, possibility of picture fuzzy hypersoft graph (popfhs-graph) is accomplished. The popfhs-graph is more flexible and reliable in the sense that it has the ability to tackle the limitations of picture fuzzy soft graph regarding the entitlement of multiargument approximate mapping and possibility degree-based setting. Its approximate mapping considers the Cartesian product of subclasses of parameters as domain and then maps it to the power set of universal set. The possibility degree-based setting ensures the assessment of uncertain attitude of approximate elements up to the level of acceptance. First, some of elementary notions and set theoretic operations of popfhs-graph are investigated with the support of numerical examples and pictorial representations. Second, some of its variants, types of products, and composition are also discussed. Lastly, based on aggregation of the popfhs-graph, an algorithm is proposed for multiattribute decision-making problem and validated by resolving daily-life recruitment problem for the best selection of candidate.


---

## Body

## 1. Introduction

The information that are obtained for any kind of computation have some kinds of vagueness and uncertainty. In this regard, fuzzy set (f-set) [1] and intuitionistic fuzzy set (if-set) [2] are initiated to tackle such type of informational uncertainty and vagueness. In f-set, the condition “well defined” of classical set is characterized by a membership function μT defined by a membership grade μTui within 0,1 for all members ui of initial universe U, whereas the if-set characterizes such condition by two functions, that is, membership function μT and nonmembership function μF defined by membership grade μTui and nonmembership grade μFui, respectively, within 0,1 for all ui∈U subject to conditions that both μTui and μFui are dependent and their sum μTui+μFui must lie within 0,1 with hesitancy grade μHui=1−μTui+μFui. Both f-set and if-set are not capable to tackle the situations in which a neutral grade along with refusal grade is required to be emphasized, so picture fuzzy set (pf-set) [3] is conceptualized to manage such scenarios. In pf-set, objective belongingness is further characterized by three membership functions, that is, μT,μI, and μF such that for all ui∈U, their sum μTui+μIui+μFui and refusal grade μRui=1−μTui−μIui−μFui lie within 0,1. Recently, the uncertain complexities in different daily-life problems have been investigated and addressed by the authors [4–8] through employing various multicriteria decision-making techniques based on pf-set. In order to equip f-set, if-set and pf-set with parameterization tool, Molodtsov [9] introduced soft set (s-set) for dealing with uncertainties and vagueness. The fuzzy soft set fs-set [10], intuitionistic fuzzy soft set (ifs-set) [11], and picture fuzzy soft set (pfs-set) [12] are the hybridized structures of f-set, if-set, and pf-set, respectively, with s-set to tackle uncertainties with the support of parameterization tool. The researchers [13–15] discussed the applicability of pfs-set in various fields of study for solving many real-life decision-making based problems. Kamacı et al. [16] discussed multiperiod decision-making problem based on dynamic aggregation operators and Einstein operations of interval-valued pfs-set. Kamacı [17] applied the extended concept of pfs-set with linguistic information in game theory. Akcetin and Kamacı [18] used TOPSIS and ELECTRE decision-making techniques by using three-valued s-set.In various real-world scenarios, the classification of attributes into subattributive values in the form of nonoverlapping sets is necessary. Thes-sets and its hybrids are incompatible with such scenarios, so Smarandache [19] introduced hypersoft sets (hs-sets) to address the insufficiency of s-set-like models. In hs-set, a new approximate function, multiargument approximate function (maa-function), is employed that maps Cartesian product of attribute-valued disjoint sets to the power set of initial universe. The rudiments and elementary axioms of hs-sets have been discussed in [20] and elaborated with numerical examples. Ihsan et al. [21] discussed the validity of hs-sets for the entitlement of multidecisive opinions under expert set environment. Rahman et al. [22–24] investigated the hybridized properties of hs-sets under the environments of convexity and concavity, parameterization, and bijection. They employed decision-making algorithmic approaches to solve real-world problems. The hybridized structures of hs-set with f-set and if-set are fuzzy hypersoft set (fhs-set) [25, 26] and intuitionistic fuzzy hypersoft set (ifhs-set) [27], respectively. Saeed et al. [28] characterized the concept of hypersoft graphs and discussed its some properties.In order to manages-set information in graphs, Thumbakara and George [29] characterized soft graphs in 2014. As its extension, fuzzy soft graph (fs-graph) [30], intuitionistic fuzzy soft graph (ifs-graph) [31], and picture fuzzy soft graph (pfs-graph) [32] are conceptualized. In order to evaluate uncertain nature of approximate elements as a whole in fs-set and ifs-set, Alkhazaleh et al. [33] and Bashir et al. [34] characterized possibility fuzzy soft set (pofs-set) and possibility intuitionistic fuzzy soft set (poifs-set), respectively, by assigning possibility degree to each approximate element collectively in these structures. Akram and Shahzadi [35] developed possibility intuitionistic fuzzy soft graph (poifs-graph) and investigated some of its properties and operations.Inpoifs-graph, neutral membership grade is ignored, and it lacks the consideration of maa-function. Consequently, various daily-life scenarios such as recruitment process, medical diagnosis, optimal product selection, and so on are not tackled by poifs-graph that demands a novel graphical structure to be characterized in literature. We, therefore, present a new graphical model, that is, possibility picture fuzzy hypersoft graph (popfhs-graph) in this study. Its advantageous aspects are as follows: it provides due status to neutral membership degree while dealing uncertainties, and it employs maa-function to cope with the scenarios having further partitioning of attributes into their respective attribute-valued sets. The proposed graphical model is more flexible as it overcomes the shortcomings of existing relevant models for dealing with uncertainties. It assigns a possibility degree to each approximate element of its maa-function to deal with its uncertain behavior. The major contributions of the paper are outlined as follows:(1)
A novel graphical hybridpopfhs-graph is characterized that is capable to cope with the following situations collectively:(a)
The situation in which the categorization of opted parameters into their respective disjoint subclasses having their relevant attributive values is necessary(b)
The situation in which the consideration of multiargument parameterization in the domain of approximate mapping is mandatory to have reliable approximation of alternatives(c)
The situation that demands a mode for the assessment of uncertain nature of approximate elements to assess the level of acceptance(2)
The novel notions ofpfhs-set, popfhs-set, and popfhs-graph are introduced, and then some essential fundamental properties such as subgraph, spanning subgraph, strong subgraph; aggregation operations such as union, intersection, and complement; and products such as Cartesian product, cross product, lexicographic product, strong product, and composition of popfhs-graph are investigated.(3)
A decision-support system is constructed based on the proposal of an algorithm that is implemented in real-life multiattribute decision-making problem for the selection of suitable candidate.The layout of the remaining paper is as follows: some essential definitions relevant to main work are recalled from existing literature for proper understanding of proposed study in Section2. Some fundamentals, that is, properties and set-theoretic operations of popfhs-graph, are characterized with graphical representation-based examples in Section 3. Section 4 presents the analytical cum graphical exploration of some products and compositions of popfhs-graphs. In Section 5, the concept of popfhs-graph is applied in decision-making for HRM-scenario. The comparison analysis is presented in Section 6. Lastly, paper is summarized with more future directions in Section 7.


## 2. Preliminaries

This portion of the paper presents some elementary terms and definitions by reviewing the existing literature for vivid understanding of the proposed study.In literature, Cuong [3] characterized the following concept of pf-set as a generalization of if-set [2] by introducing a new grade called neutral grade μNu^ to furnish the neutrality of decision-makers.Definition 1 (see [3]).
Apf-set Pℱ is defined as Pℱ=u^,<μTu^,μNu^,μF>u^∈u such that μTu^, μNu^,μFu^:U⟶0,1, where μTu^, μNu^, and μFu^ represent positive, neutral, and negative membership grades, respectively, of u^∈U subject to the condition that 0≤μTu^+μNu^+μFu^≤1 with refusal membership grade μRu^=1−μTu^−μNu^−μFu^. The collection of all pf-sets over U is denoted as ℙℱU.
The parameters play a key role in reliable and authentic decision-making process. The existing fuzzy set-like models are inadequate with any kind of parameterization tool; therefore, Molodtsov [9] initiated the following idea of s-set to address the limitations of predefined uncertain models with the provision of parameterization tool.Definition 2 (see [9]).
As-set S over U is a pair ψS,A, where ψS:A⟶ℙU is an approximate function of S and A⊆E (a set of parameters). For any a^∈A,ψSa^ is called an approximate element of S. In this definition, the symbol ℙU is meant for power set of U.
In 2015, Yang et al. [12] developed the following novel model, that is, pfs-set to tackle the insufficiencies of pf-set for parameterization context. In short, they combined the theory of pf-set with Molodtsov’s theory of s-set to carve out a parameterized family of universal set with picture fuzzy setting.Definition 3 (see [12]).
Apfs-set ℙ over U is a pair ψℙ,ℤ, where ψℙ:ℤ⟶ℙℱU and ℤ⊆E. For any z^∈ℤ,ψℙz^ is a pfs-subset and known as approximate element of pfs-set ℙ, that is, ψℙz^ can be represented as pfs-set over U such that ψℙz^=u^,<μTz^u^,μNz^u^,μFz^u^>|u^∈U, where μTz^u^, μNz^u^, and μFz^u^ represent positive, neutral, and negative membership grades, respectively, of u^∈U subject to the condition that 0≤μTz^u^+μNz^u^+μFz^u^≤1 with refusal membership grade μRz^u^=1−μTz^u^−μNz^u^−μFz^u^.
In 2018, Smarandache [19] extended the concept of s-set and developed a novel model, that is, hs-set that utilizes a novel mapping known as maa-function to deal with the shortcomings of s-set regarding the categorization of parameters into their relevant parametric valued subclasses.Definition 4 (see [19]).
Ahs-set ℌ over U is a pair W,D, where D is the Cartesian product of Di,i=1,2,3,…,n,Di∩Dj=∅∀i≠j having attribute values of attributes h^i,i=1,2,3,…,n,h^i≠h^j,i≠j, respectively, and W:D⟶ℙU is called approximate function (so-called maa-function) of ℌ, and for all d^∈D,Wd^ is called approximate element of ℌ. The hs-set ℌ over U is said to be fhs-set and ifhs-set if W:D⟶FU and W:D⟶IFU, respectively, where FU and IFU denote the family of all fuzzy subsets and intuitionistic fuzzy subsets, respectively.
In 2021, Chellamani et al. [32] explored the graphical notations of pfs-sets to handle pfs-information efficiently.Definition 5 (see [32]).
LetM⋆=B,E be a simple graph with B as set of vertices and E as set of edges and E be a nonvoid set of parameters. By a picture fuzzy soft graph (pfs-graph), we mean a four-tuple G=G⋆,Q,L,M with L,E and M,E that are pfs-sets over B and E, respectively; for all e^∈E,Le^,Me^ is a pfs-graph of G if μTMe^b^1b^2≤minμTLe^b^1,μTLe^b^2, μNMe^b^1b^2≤minμNLe^b^1,μNLe^b^2, and μFMe^b^1b^2≤maxμFLe^b^1,μFLe^b^2 such that 0≤μTMe^b^1b^+μNMe^b^1b^+μFMe^b^1b^2≤1 for all b^1,b^2∈B.


## 3. Possibility Picture Fuzzy Hypersoft Graphs

This portion presents the characterization ofpopfhs-graph along with some essential properties and results. First, we present the definitions of picture fuzzy hypersoft set pfhs-set and possibility picture fuzzy hypersoft set popfhs-set, which are missing in existing literature.Definition 6.
Apfhs-set ℍ over U is a pair πℍ,ℤ, where πℍ:ℤ⟶ℙℱU and ℤ=Πi=1nℤi where ℤii=1,2,…,n is nonoverlapping attribute-valued sets with respect to distinct attributes. For any z^∈ℤ,πℍz^ is a pfhs-subset and known as approximate element of pfhs-set ℍ, that is, πℍz^ can be represented as pfhs-set over U such that πℍz^=u^,<μTz^u^,μNz^u^,μFz^u^>|u^∈U where μTz^u^, μNz^u^, and μFz^u^ represent positive, neutral, and negative membership grades, respectively, of u^∈U subject to the condition that 0≤μTz^u^+μNz^u^+μFz^u^≤1 with refusal membership grade μRz^u^=1−μTz^u^−μNz^u^−μFz^u^.Definition 7.
A possibility picture fuzzy hypersoft set (popfhs-set) ℍp over U is stated as ℍp=z^,<πℍz^>,δz^|z^∈ℤ, where πℍz^ is a pfhs-set as defined in Definition 6 and δ:U⟶0,1 with δz^ is the possibility degree of z^ to ℍp. The collection of all popfhs-sets over U is represented by ΩPFU.
Throughout the remaining paper,A⋆=B,E is a simple graph with B as set of vertices and E as set of edges; E denotes set of parameters; and Qi denotes the disjoint sets containing subparametric values for distinct parameters e^i,i=1,2,…,n of E. Also, Q=Q1×Q2×Q3×⋯×Qn.Definition 8.
Apopfhs-graph is a four-tuple A=A⋆,Q,J,K, where J:Q⟶ℙFB,K:Q⟶ℙFB×B given by Jσ=Jσ=v^,≺TJσv^,IJσv^,FJσv^≻,μv^,v^∈B and Kσ=Kσ=v^1,v^2,≺TJσv^1,v^2,IJσv^1,v^2,FJσv^1,v^2≻,μJσv^1,v^2,v^1,v^2∈B×B are popf-sets over B and B×B with TJσv^1,v^2≤minTJσv^1,TJσv^2, IJσv^1,v^2≤minIJσv^1,IJσv^2, FJσv^1,v^2≤maxFJσv^1,FJσv^2, μJσv^1,v^2≤minμJσv^1,μJσ−v^2, ∀v^1,v^2∈B×B, and σ∈Q.
Note: The collection of allpopfhs-graphs is represented by ΩPPFHSG.Example 1.
LetA⋆=B,E be a simple graph with B=v^1,v^2,v^3 and Q1=q^11,q^12, Q2=q^21,q^22, and Q3=q^31 such that Q=Q1×Q2×Q3=σ1,σ2,σ3,σ4 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0∀v^i,v^j∈B×B/v^1,v^2,v^1,v^2,v^1,v^2. Table 1 and Figure 1 present its numerical tabulation and geometrical depiction, respectively.Table 1
Numerical computation of Example1 with (a) ℙFσ1, (b) ℙFσ2, (c) ℙFσ3, and (d) ℙFσ4
Jv^1v^2v^3σ10.2,0.1,0.3,0.20.1,0.3,0.2,0.30,0,1,0σ20.1,0.3,0.3,0.20.2,0.4,0.1,0.10,0,1,0σ30.1,0.2,0.2,0.30.3,0.1,0.2,0.20.1,0.2,0.3,0.1σ40.2,0.2,0.1,0.30.3,0.2,0.1,0.40.1,0.3,0.2,0.2Kv^1,v^2v^2,v^3v^1,v^3σ10,0,1,00,0,1,00,0,1,0σ20.1,0.2,0.2,0.40,0,1,00,0,1,0σ30.1,0.3,0.3,0.30.2,0.1,0.3,0.10,0,1,0σ40.1,0.1,0.1,0.60.2,0.2,0.1,0.40.2,0.2,0.1,0.5Figure 1
Geometrical interpretation of Table1.Definition 9.
Apopfhs-graph G=A⋆,Q⋆,J1,K1 is called a popfhs-subgraph of A=A⋆,Q,J,K if(1)
Q⋆⊆Q(2)
Jσ1⊆J implies TJσ1v^≤TJσv^,IJσ1v^≤IJσv^,FJσ1v^≥FJσv^,μJσ1v^≤μJσv^(3)
Kσ1⊆K implies TKσ1v^≤TKσv^,IKσ1v^≤IKσv^,FKσ1v^≥FKσv^,μKσ1v^≤μKσv^∀σ∈Q⋆Example 2.
Repeating Example1 with Q1=α11,α12, Q2=α21, and Q3=α31 and Q⋆=Q1×Q2×Q3=σ1,σ2,σ3, we have a new popfhs-graph A=A⋆,Q⋆,J1,K1 which is popfhs-subgraph of popfhs-graph given in Example 1. Its tabular and geometrical depiction are given in Table 2 and Figure 2, respectively.Table 2
Tabular notation of Example2 with (a) ℙFσ1, (b) ℙFσ2, and (c) ℙFσ3.
Jv^1v^2v^3σ10.1,0.3,0.1,0.10.2,0.2,0.1,0.20,0,1,0σ20.1,0.2,0.1,0.10.2,0.1,0.3,0.10,0,1,0σ30.1,0.4,0.1,0.20.2,0.2,0.1,0.10.2,0.1,0.2,0.1Kv^1,v^2v^2,v^3v^1,v^3σ10,0,1,00,0,1,00,0,1,0σ20.1,0.2,0.1,0.10,0,1,00,0,1,0σ30.1,0.2,0.3,0.20.1,0.2,0.4,0.10,0,1,0Figure 2
Graphical representation of Table2.Definition 10.
Apopfhs-subgraph A⋆,Q⋆,J1,K1 is called a popfhs-spanning subgraph of popfhs-graph A⋆,Q,J,K when Jσ1v^=Jσv^∀v^∈B,σ∈Q.Definition 11.
Apopfhs-subgraph A⋆,Q⋆,J1,K1 is called a strong popfhs-subgraph (SPFHS-subgraph) of popfhs-graph A⋆,Q,J,K when Kσv^1,v^2=Jσv^1∩Jσv^2 for v^1,v^2∈B and σ∈Q.


### 3.1. Set-Theoretic Operations ofpopfhs-Graphs

This segment of the paper investigates the basic set theoretic operations ofpopfhs-graphs along with their geometrical interpretations.Definition 12.
The union of twopopfhs-graphs A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2, denoted by A1∪A2, is a popfhs-graph A=A⋆,Q,J,K such that Q=Q1∪Q2. In this graph,(1)TJσv^=TJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2&v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2TJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1maxTJσ1v^,TJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwiseIJσv^=IJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2IJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1maxIJσ1v^,IJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwise&FJσv^=FJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2&v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2FJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1minFJσ1v^,FJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwise.&
Also, thepopfhs-components for K are given as follows:(2)TKσv^1=TKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,TKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxTKσ1v^1,TKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,IKσv^1=IKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,IKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxIKσ1v^1,IKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,FKσv^1=FKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,FKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,minFKσ1v^1,FKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,μKσv^1=μKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,μKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxμKσ1v^1,μKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise.Theorem 1.
IfA1,A2∈ΩPFHSG, then A1∪A2∈ΩPFHSG.Proof.
Its proof is trivial in accordance with Definition12.Example 3.
LetA1=A1⋆,Q1,J1,K1 be a popfhs-graph where A1⋆=B1,E1 with B1=v^1,v^2,v^3 and Q1=α11, Q2=α21, and Q3=α31,α32,α33 such that Q1=Q1×Q2×Q3=σ1,σ2,σ3 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B1×B1/v^1,v^2,v^2,v^3,v^1,v^3. Its tabulation is given in Table 3. Also, let A2=A2⋆,Q2,J2,K2 be a popfhs-graph, where A2⋆=B2,E2 with B2=v^3,v^4,v^5, Q3=α31,α32, and Q4=α41 such that Q5=α51. Q2=Q3×Q4×Q5=σ2,σ4 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B2×B2/v^3,v^4,v^4,v^5,v^3,v^5. Its tabulation is given in Table 4. Now let A=A1∪A2 with Q=Q1∪Q2 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B×B/v^1,v^2,v^1,v^3,v^2,v^3,v^3,v^4,v^3,v^5,v^4,v^5. Its tabulation is given in Table 5.
The graphical interpretations of Tables3–5 are presented in Figures 3–5, respectively.Table 3
Tabulation of Example3.
Jv^1v^2v^3σ10.2,0.1,0.2,0.20.3,0.1,0.1,0.30.2,0.2,0.1,0.3σ20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.1,0.2,0.1,0.5σ30.2,0.1,0.1,0.60.1,0.1,0.4,0.40.1,0.2,0.1,0.7Kv^1,v^2v^2,v^3v^1,v^3σ10.2,0.3,0.2,0.20.2,0.1,0.3,0.20.2,0.1,0.2,0.2σ20.2,0.1,0.3,0.20.2,0.2,0.1,0.20.2,0.3,0.1,0.2σ30,0,1,00.1,0.2,0.1,0.30.1,0.2,0.1,0.2Table 4
Tabulation ofpopfhs-graph A2=A2⋆,Q2,J2,K2 according to Example 3.
Jv^3v^4v^5σ20.2,0.1,0.2,0.30.2,0.1,0.2,0.20.1,0.2,0.1,0.5σ40.2,0.1,0.1,0.60.1,0.2,0.1,0.40.2,0.1,0.2,0.4Kv^3,v^4v^4,v^5v^3,v^5σ20.2,0.3,0.1,0.20.3,0.1,0.2,0.30,0,1,0σ40.2,0.1,0.3,0.20.1,0.3,0.2,0.30.1,0.2,0.2,0.3Table 5
Tabulation ofA=A1∪A2.
Jv^1v^2v^3v^4v^5σ10.2,0.1,0.2,0.20.3,0.1,0.1,0.30.2,0.2,0.1,0.30,0,1,00,0,1,0σ20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.2,0.2,0.1,0.50.2,0.1,0.2,0.20.1,0.2,0.1,0.5σ30.2,0.1,0.1,0.60.1,0.1,0.4,0.40.1,0.2,0.1,0.70,0,1,00,0,1,0σ40,0,1,00,0,1,00.2,0.1,0.1,0.60.1,0.2,0.1,0.40.2,0.1,0.2,0.4Kv^1,v^2v^1,v^3v^2,v^3v^3,v^4v^3,v^5v^4,v^5σ10.2,0.3,0.2,0.20.2,0.1,0.2,0.20.2,0.1,0.3,0.20,0,1,00,0,1,00,0,1,0σ20.2,0.3,0.2,0.20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.2,0.3,0.1,0.20,0,1,00.3,0.1,0.2,0.3σ30,0,1,00.1,0.2,0.1,0.20.1,0.2,0.1,0.30,0,1,00,0,1,00,0,1,0σ40,0,1,00,0,1,00,0,1,00.2,0.1,0.3,0.20.1,0.2,0.2,0.30.1,0.3,0.2,0.3Figure 3
Graphical representation of Table3.Figure 4
Geometrical depiction of Table4.Figure 5
Graphical representation of Table5.Definition 13.
The intersection of twopopfhs-graphs G1=G1⋆,Q1,J1,K1 and G2=G2⋆,Q2,J2,K2, denoted by G1∩G2, is a popfhs-graph G=G⋆,Q,J,K such that Q=Q1∩Q2,B=B1∩B2. In this graph, the uncertain parts for J are as follows:(3)TJσ=TJσ1v^ifσ∈Q1−Q2,TJσ2v^ifσ∈Q2−Q1,minTJσ1v^,TJσ2v^ifσ∈Q1∩Q2,IJσ=IJσ1v^ifσ∈Q1−Q2,IJσ2v^ifσ∈Q2−Q1,minIJσ1v^,IJσ2v^ifσ∈Q1∩Q2,FJσ=FJσ1v^ifσ∈Q1−Q2,FJσ2v^ifσ∈Q2−Q1,maxFJσ1v^,FJσ2v^ifσ∈Q1∩Q2,μJσ=μJσ1v^ifσ∈Q1−Q2,μJσ2v^ifσ∈Q2−Q1,minμJσ1v^,μJσ2v^ifσ∈Q1∩Q2.
The uncertain parts forK are given as follows:(4)TKσ=TKσ1v^ifσ∈Q1−Q2,TKσ2v^ifσ∈Q2−Q1,minTKσ1v^,TKσ2v^ifσ∈Q1∩Q2,IKσ=IKσ1v^ifσ∈Q1−Q2,IKσ2v^ifσ∈Q2−Q1,minIKσ1v^,IKσ2v^ifσ∈Q1∩Q2,FKσ=FKσ1v^ifσ∈Q1−Q2,FKσ2v^ifσ∈Q2−Q1,maxFKσ1v^,FKσ2v^ifσ∈Q1∩Q2,μKσ=μKσ1v^ifσ∈Q1−Q2,μKσ2v^ifσ∈Q2−Q1,minμKσ1v^,μKσ2v^ifσ∈Q1∩Q2.Theorem 2.
IfA1,A2∈ΩPFHSG, then A1∩A2∈ΩPFHSG.Proof.
It can easily be proved with the help of axioms used in Definition13. Therefore, its proof is omitted.Example 4.
LetA1=A1⋆,Q1,J1,K1 be a popfhs-graph where A1⋆=B1,E1 with B1=v^1,v^2,v^3 and Q1,Q2,Q3 are subparametric nonoverlapping sets with respect to distinct attributes α1,α2,α3, where Q1=α11, Q2=α21, and Q3=α31,α32. Q1=Q1×Q2×Q3=σ1,σ2 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1, and∀v^i,v^j∈B1×B1/v^1,v^2,v^2,v^3,v^1,v^3. Table 6 and Figure 6 elaborate its tabulation and geometrical depiction, respectively. Also, let A2=A2⋆,Q2,J2,K2 be a popfhs-graph where A2⋆=B2,E2 with B2=v^2,v^3,v^4 and Q2,Q3,Q4 are subparametric nonoverlapping sets with respect to distinct attributes α2,α3,α4 where Q2=α21, Q3=α31,α32, and Q4=α41. Q2=Q2×Q3×Q4=σ2,σ3 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1∀v^i,v^j∈B2×B2/v^2,v^3,v^3,v^4,v^2,v^4. Its tabulation and pictorial representation are provided in Table 7 and Figure 7, respectively. Consider A=A1∩A2 with Q=Q1∩Q2. Its tabulation and geometrical interpretation are stated in Table 8 and Figure 8, respectively.Table 6
Tabulation ofpopfhs-graph A1=A1⋆,Q1,J1,K1 for Example 4.
Jv^1v^2v^3σ10.2,0.1,0.2,0.20.1,0.2,0.1,0.30.1,0.3,0.2,0.2σ20.1,0.1,0.3,0.30.1,0.2,0.1,0.50.2,0.2,0.4,0.4Kv^1,v^2v^2,v^3v^1,v^3σ10.2,0.2,0.1,0.20.2,0.2,0.1,0.20.2,0.1,0.3,0.2σ20.1,0.3,0.2,0.30.2,0.1,0.3,0.40.3,0.1,0.2,0.3Figure 6
Geometrical interpretation of Table6.Table 7
Tabulation ofpopfhs-graph A2=A2⋆,Q2,J2,K2 for Example 4.
Jv^2v^3v^4σ20.1,0.2,0.1,0.40.2,0.1,0.3,0.50.3,0.1,0.1,0.3σ30.3,0.2,0.1,0.30.1,0.1,0.3,0.20.2,0.1,0.2,0.2Kv^2,v^3v^3,v^4v^2,v^4σ20.2,0.2,0.3,0.20.2,0.1,0.1,0.20.2,0.2,0.3,0.2σ30.1,0.1,0.4,0.30.2,0.1,0.4,0.40.5,0.2,0.1,0.3Figure 7
Graphical representation of Table7.Table 8
Tabulation ofpopfhs-graph A=A1∩A2.
Jv^2v^3σ10.1,0.2,0.1,0.30.1,0.3,0.2,0.2σ20.1,0.2,0.1,0.40.2,0.1,0.4,0.4σ30.3,0.2,0.1,0.30.1,0.1,0.3,0.2Kv^2,v^3σ10.2,0.2,0.1,0.2σ20.2,0.1,0.3,0.2σ30.1,0.1,0.4,0.3Figure 8
Graphical representation of Table8.Definition 14.
The complimentA¯=A⋆¯,Q¯,F¯,G¯ of SPFHS-subgraph A=A⋆,Q,J,K with Kσv^1,v^2=Jσv^1∩Jσv^2 where(1)
Q¯=Q(2)
TJσv^¯=TJσv^,IJσv^¯=IJσv^,FJσv^¯=FJσv^,μJσv^¯=μJσv^∀v^∈B(3)
TJσv^1,v^2¯=minTJσv^1,TJσv^2,ifTKσv^1,v^2=0,0,otherwiseIJσv^1,v^2¯=minIJσv^1,IJσv^2,ifIKσv^1,v^2=0,0,otherwise,FJσv^1,v^2¯=maxFJσv^1,FJσv^2,ifFKσv^1,v^2=0,0,otherwise,μJσv^1,v^2¯=minμJσv^1,μJσv^2,ifμKσv^1,v^2=0,0,otherwise.


## 3.1. Set-Theoretic Operations ofpopfhs-Graphs

This segment of the paper investigates the basic set theoretic operations ofpopfhs-graphs along with their geometrical interpretations.Definition 12.
The union of twopopfhs-graphs A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2, denoted by A1∪A2, is a popfhs-graph A=A⋆,Q,J,K such that Q=Q1∪Q2. In this graph,(1)TJσv^=TJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2&v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2TJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1maxTJσ1v^,TJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwiseIJσv^=IJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2IJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1maxIJσ1v^,IJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwise&FJσv^=FJσ1v^ifσ∈Q1−Q2&v^∈B1−B2or ifσ∈Q1−Q2&v^∈B1∩B2or ifσ∈Q1∩Q2&v^∈B1−B2FJσ2v^ifσ∈Q2−Q1&v^∈B2−B1or ifσ∈Q2−Q1&v^∈B2∩B1or ifσ∈Q1∩Q2&v^∈B2−B1minFJσ1v^,FJσ2v^ifσ∈Q1∩Q2&v^∈B1∩B20, otherwise.&
Also, thepopfhs-components for K are given as follows:(2)TKσv^1=TKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,TKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxTKσ1v^1,TKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,IKσv^1=IKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,IKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxIKσ1v^1,IKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,FKσv^1=FKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,FKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,minFKσ1v^1,FKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise,μKσv^1=μKσ1v^1ifσ∈Q1−Q2&v^1,v^2∈B1×B1−B2×B2or ifσ∈Q1−Q2&v^1,v^2∈B1×B1∩B2×B2or ifσ∈Q1∩Q2&v^1,v^2∈B1×B1−B2×B2,μKσ2v^1ifσ∈Q2−Q1&v^1,v^2∈B2×B2−B1×B1or ifσ∈Q2−Q1&v^1,v^2∈B2×B2∩B1×B1or ifσ∈Q1∩Q2&v^1,v^2∈B2×B2−B1×B1,maxμKσ1v^1,μKσ2v^1ifσ∈Q1∩Q2&v^1,v^2∈B1×B1∩B2×B2,0,otherwise.Theorem 1.
IfA1,A2∈ΩPFHSG, then A1∪A2∈ΩPFHSG.Proof.
Its proof is trivial in accordance with Definition12.Example 3.
LetA1=A1⋆,Q1,J1,K1 be a popfhs-graph where A1⋆=B1,E1 with B1=v^1,v^2,v^3 and Q1=α11, Q2=α21, and Q3=α31,α32,α33 such that Q1=Q1×Q2×Q3=σ1,σ2,σ3 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B1×B1/v^1,v^2,v^2,v^3,v^1,v^3. Its tabulation is given in Table 3. Also, let A2=A2⋆,Q2,J2,K2 be a popfhs-graph, where A2⋆=B2,E2 with B2=v^3,v^4,v^5, Q3=α31,α32, and Q4=α41 such that Q5=α51. Q2=Q3×Q4×Q5=σ2,σ4 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B2×B2/v^3,v^4,v^4,v^5,v^3,v^5. Its tabulation is given in Table 4. Now let A=A1∪A2 with Q=Q1∪Q2 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1,μKσv^i,v^j=0, and ∀v^i,v^j∈B×B/v^1,v^2,v^1,v^3,v^2,v^3,v^3,v^4,v^3,v^5,v^4,v^5. Its tabulation is given in Table 5.
The graphical interpretations of Tables3–5 are presented in Figures 3–5, respectively.Table 3
Tabulation of Example3.
Jv^1v^2v^3σ10.2,0.1,0.2,0.20.3,0.1,0.1,0.30.2,0.2,0.1,0.3σ20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.1,0.2,0.1,0.5σ30.2,0.1,0.1,0.60.1,0.1,0.4,0.40.1,0.2,0.1,0.7Kv^1,v^2v^2,v^3v^1,v^3σ10.2,0.3,0.2,0.20.2,0.1,0.3,0.20.2,0.1,0.2,0.2σ20.2,0.1,0.3,0.20.2,0.2,0.1,0.20.2,0.3,0.1,0.2σ30,0,1,00.1,0.2,0.1,0.30.1,0.2,0.1,0.2Table 4
Tabulation ofpopfhs-graph A2=A2⋆,Q2,J2,K2 according to Example 3.
Jv^3v^4v^5σ20.2,0.1,0.2,0.30.2,0.1,0.2,0.20.1,0.2,0.1,0.5σ40.2,0.1,0.1,0.60.1,0.2,0.1,0.40.2,0.1,0.2,0.4Kv^3,v^4v^4,v^5v^3,v^5σ20.2,0.3,0.1,0.20.3,0.1,0.2,0.30,0,1,0σ40.2,0.1,0.3,0.20.1,0.3,0.2,0.30.1,0.2,0.2,0.3Table 5
Tabulation ofA=A1∪A2.
Jv^1v^2v^3v^4v^5σ10.2,0.1,0.2,0.20.3,0.1,0.1,0.30.2,0.2,0.1,0.30,0,1,00,0,1,0σ20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.2,0.2,0.1,0.50.2,0.1,0.2,0.20.1,0.2,0.1,0.5σ30.2,0.1,0.1,0.60.1,0.1,0.4,0.40.1,0.2,0.1,0.70,0,1,00,0,1,0σ40,0,1,00,0,1,00.2,0.1,0.1,0.60.1,0.2,0.1,0.40.2,0.1,0.2,0.4Kv^1,v^2v^1,v^3v^2,v^3v^3,v^4v^3,v^5v^4,v^5σ10.2,0.3,0.2,0.20.2,0.1,0.2,0.20.2,0.1,0.3,0.20,0,1,00,0,1,00,0,1,0σ20.2,0.3,0.2,0.20.2,0.3,0.1,0.20.2,0.2,0.1,0.20.2,0.3,0.1,0.20,0,1,00.3,0.1,0.2,0.3σ30,0,1,00.1,0.2,0.1,0.20.1,0.2,0.1,0.30,0,1,00,0,1,00,0,1,0σ40,0,1,00,0,1,00,0,1,00.2,0.1,0.3,0.20.1,0.2,0.2,0.30.1,0.3,0.2,0.3Figure 3
Graphical representation of Table3.Figure 4
Geometrical depiction of Table4.Figure 5
Graphical representation of Table5.Definition 13.
The intersection of twopopfhs-graphs G1=G1⋆,Q1,J1,K1 and G2=G2⋆,Q2,J2,K2, denoted by G1∩G2, is a popfhs-graph G=G⋆,Q,J,K such that Q=Q1∩Q2,B=B1∩B2. In this graph, the uncertain parts for J are as follows:(3)TJσ=TJσ1v^ifσ∈Q1−Q2,TJσ2v^ifσ∈Q2−Q1,minTJσ1v^,TJσ2v^ifσ∈Q1∩Q2,IJσ=IJσ1v^ifσ∈Q1−Q2,IJσ2v^ifσ∈Q2−Q1,minIJσ1v^,IJσ2v^ifσ∈Q1∩Q2,FJσ=FJσ1v^ifσ∈Q1−Q2,FJσ2v^ifσ∈Q2−Q1,maxFJσ1v^,FJσ2v^ifσ∈Q1∩Q2,μJσ=μJσ1v^ifσ∈Q1−Q2,μJσ2v^ifσ∈Q2−Q1,minμJσ1v^,μJσ2v^ifσ∈Q1∩Q2.
The uncertain parts forK are given as follows:(4)TKσ=TKσ1v^ifσ∈Q1−Q2,TKσ2v^ifσ∈Q2−Q1,minTKσ1v^,TKσ2v^ifσ∈Q1∩Q2,IKσ=IKσ1v^ifσ∈Q1−Q2,IKσ2v^ifσ∈Q2−Q1,minIKσ1v^,IKσ2v^ifσ∈Q1∩Q2,FKσ=FKσ1v^ifσ∈Q1−Q2,FKσ2v^ifσ∈Q2−Q1,maxFKσ1v^,FKσ2v^ifσ∈Q1∩Q2,μKσ=μKσ1v^ifσ∈Q1−Q2,μKσ2v^ifσ∈Q2−Q1,minμKσ1v^,μKσ2v^ifσ∈Q1∩Q2.Theorem 2.
IfA1,A2∈ΩPFHSG, then A1∩A2∈ΩPFHSG.Proof.
It can easily be proved with the help of axioms used in Definition13. Therefore, its proof is omitted.Example 4.
LetA1=A1⋆,Q1,J1,K1 be a popfhs-graph where A1⋆=B1,E1 with B1=v^1,v^2,v^3 and Q1,Q2,Q3 are subparametric nonoverlapping sets with respect to distinct attributes α1,α2,α3, where Q1=α11, Q2=α21, and Q3=α31,α32. Q1=Q1×Q2×Q3=σ1,σ2 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1, and∀v^i,v^j∈B1×B1/v^1,v^2,v^2,v^3,v^1,v^3. Table 6 and Figure 6 elaborate its tabulation and geometrical depiction, respectively. Also, let A2=A2⋆,Q2,J2,K2 be a popfhs-graph where A2⋆=B2,E2 with B2=v^2,v^3,v^4 and Q2,Q3,Q4 are subparametric nonoverlapping sets with respect to distinct attributes α2,α3,α4 where Q2=α21, Q3=α31,α32, and Q4=α41. Q2=Q2×Q3×Q4=σ2,σ3 and TKσv^i,v^j=0,IKσv^i,v^j=0,FKσv^i,v^j=1∀v^i,v^j∈B2×B2/v^2,v^3,v^3,v^4,v^2,v^4. Its tabulation and pictorial representation are provided in Table 7 and Figure 7, respectively. Consider A=A1∩A2 with Q=Q1∩Q2. Its tabulation and geometrical interpretation are stated in Table 8 and Figure 8, respectively.Table 6
Tabulation ofpopfhs-graph A1=A1⋆,Q1,J1,K1 for Example 4.
Jv^1v^2v^3σ10.2,0.1,0.2,0.20.1,0.2,0.1,0.30.1,0.3,0.2,0.2σ20.1,0.1,0.3,0.30.1,0.2,0.1,0.50.2,0.2,0.4,0.4Kv^1,v^2v^2,v^3v^1,v^3σ10.2,0.2,0.1,0.20.2,0.2,0.1,0.20.2,0.1,0.3,0.2σ20.1,0.3,0.2,0.30.2,0.1,0.3,0.40.3,0.1,0.2,0.3Figure 6
Geometrical interpretation of Table6.Table 7
Tabulation ofpopfhs-graph A2=A2⋆,Q2,J2,K2 for Example 4.
Jv^2v^3v^4σ20.1,0.2,0.1,0.40.2,0.1,0.3,0.50.3,0.1,0.1,0.3σ30.3,0.2,0.1,0.30.1,0.1,0.3,0.20.2,0.1,0.2,0.2Kv^2,v^3v^3,v^4v^2,v^4σ20.2,0.2,0.3,0.20.2,0.1,0.1,0.20.2,0.2,0.3,0.2σ30.1,0.1,0.4,0.30.2,0.1,0.4,0.40.5,0.2,0.1,0.3Figure 7
Graphical representation of Table7.Table 8
Tabulation ofpopfhs-graph A=A1∩A2.
Jv^2v^3σ10.1,0.2,0.1,0.30.1,0.3,0.2,0.2σ20.1,0.2,0.1,0.40.2,0.1,0.4,0.4σ30.3,0.2,0.1,0.30.1,0.1,0.3,0.2Kv^2,v^3σ10.2,0.2,0.1,0.2σ20.2,0.1,0.3,0.2σ30.1,0.1,0.4,0.3Figure 8
Graphical representation of Table8.Definition 14.
The complimentA¯=A⋆¯,Q¯,F¯,G¯ of SPFHS-subgraph A=A⋆,Q,J,K with Kσv^1,v^2=Jσv^1∩Jσv^2 where(1)
Q¯=Q(2)
TJσv^¯=TJσv^,IJσv^¯=IJσv^,FJσv^¯=FJσv^,μJσv^¯=μJσv^∀v^∈B(3)
TJσv^1,v^2¯=minTJσv^1,TJσv^2,ifTKσv^1,v^2=0,0,otherwiseIJσv^1,v^2¯=minIJσv^1,IJσv^2,ifIKσv^1,v^2=0,0,otherwise,FJσv^1,v^2¯=maxFJσv^1,FJσv^2,ifFKσv^1,v^2=0,0,otherwise,μJσv^1,v^2¯=minμJσv^1,μJσv^2,ifμKσv^1,v^2=0,0,otherwise.


## 4. Some Products and Composition ofpopfhs-Graphs

In this section, we discussed some products and composition ofpopfhs-graphs with the help of graphical representation and numerical examples.Definition 15.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E1 and A2⋆=B2,E2. Let A=A1×ℙA2 where A=J,K,Q1×Q2 and J=J1×J2,K=K1×K2 is popfhs-set over B=B1×B2, K=K1×K2,Q1×Q2 is popfhs-set over E=π^,τ^1,π^,τ^2|π^∈B1,τ^1,τ^2∈E2\\∪π^1,τ^,π^2,τ^|τ^∈B2,π^1,π^2∈E1, and K=J,K,Q1×Q2 is popfhs-graphs where (1)
TJγ,ζπ^,τ^=TJ1γπ^∧TJ2tτ^IJγ,ζπ^,τ^=IJ1γπ^∧IJ2tτ^FJγ,ζπ^,τ^=FJ1γπ^∨FJ2tτ^μJγ,ζπ^,τ^=μJ1γπ^∧μJ2tτ^∀π^,τ^∈B,s,t∈Q1×Q2(2)
TKγ,ζπ^,τ^1,π^,τ^2=TJ1γπ^∧TK2ζτ^1,τ^2IKγ,ζπ^,τ^1,π^,τ^2=IJ1γπ^∧IK2ζτ^1,τ^2FKγ,ζπ^,τ^1,π^,τ^2=FJ1γπ^∨FK2ζτ^1,τ^2μKγ,ζπ^,τ^1,π^,τ^2=μJ1γπ^∧μK2ζτ^1,τ^2∀π^∈B1,τ^1,τ^2∈E2,(3)
TKγ,ζπ^1,τ^,π^2,τ^=TJ2ζτ^∧TK1γπ^1,π^2IKγ,ζπ^1,τ^,π^2,τ^=IJ2ζτ^∧IK1γπ^1,π^2FKγ,ζπ^1,τ^,π^2,τ^=FJ2ζτ^∧FK1γπ^1,π^2μKγ,ζπ^1,τ^,π^2,τ^=μJ2ζτ^∧μK1γπ^1,π^2∀τ^∈B2,π^1,π^2∈E1,∀γ,ζ∈Q1×Q2 and Wγ,ζ=W1γ×W2ζ are popfhs-graphs of A.Example 5.
LetA1⋆=B1,E1 be a simple graph with B1=π^1,π^2,π^3 and E1=π^1π^2,π^1π^3,π^2π^3 and Q1,Q2,Q3 are subparametric nonoverlapping sets with respect to distinct attributes α1,α2,α3 where Q1=α11, Q2=α21,α22, and Q3=α31. Q1=Q1×Q2×Q3=ω^1,ω^2. A1=W1,Q1=W1ω^1,W1ω^2 is popfhs -graph, which is stated in Table 9.
LetA2⋆=B2,E2 be a simple graph with B2=τ^1,τ^2,τ^3,τ^4 and E2=τ^1τ^2,τ^1τ^3,τ^1τ^4,τ^3τ^4, and Q1,Q2,Q3 are subparametric nonoverlapping sets with respect to distinct attributes α1,α2,α3, where Q1=α11, Q2=α21,α22, and Q3=α31. Q2=Q1×Q2×Q3=ω^3,ω^4. A2=W2,Q2=W2ω^3,W2ω^4 is popfhs-graph that is depicted in Table 10.
A=A1×ℙA2=W,Q1×Q2, where Q1×Q2=ω^1,ω^3,ω^2,ω^3,ω^1,ω^4,ω^2,ω^4. Here, Wω^1,ω^3=W1ω^1×ℙW2ω^3, Wω^2,ω^3=W1ω^2×ℙW2ω^3, Wω^1,ω^4=W1ω^1×ℙW2ω^4, and Wω^2,ω^4=W1ω^2×ℙW2ω^4; for convenience we will write π^p,τ^q=η^pq for p=1,2,3 and q=1,2,3,4 also TKω^η^i,η^j=0,IKω^η^i,η^j=0,FKω^η^i,η^j=1∀η^pq∈B×B.(5)η^11,η^12,η^11,η^13,η^11,η^21,η^11,η^31,η^12,η^22,η^12,η^32,η^13,η^23,η^13,η^33,η^13,η^14,η^14,η^24,η^14,η^34,η^21,η^22,η^21,η^23,η^21,η^31,η^22,η^32,η^23,η^24,η^23,η^33,η^24,η^34,η^31,η^32,η^31,η^33,η^33,η^34,
popfhs-graph of Wω^1,ω^3=W1ω^1×ℙW2ω^3 is given in Table 11.
The graphical representation of Table9 is provided in Figure 9.
The graphical explanations of Tables10 and 11 are presented in Figures 10 and 11, respectively.Table 9
popfhs-graph A1=A⋆,Q1,J1,K1 (Example 5).
Jπ^1π^2π^3ω^10.3,0.2,0.2,0.30.2,0.1,0.3,0.50.2,0.3,0.3,0.5ω^20.2,0.1,0.1,0.40.2,0.2,0.3,0.50.2,0.3,0.1,0.6Kπ^1,π^2π^1,π^3π^2,π^3ω^10.2,0.1,0.4,0.30.2,0.3,0.2,0.20.2,0.4,0.1,0.3ω^20.3,0.2,0.2,0.30.2,0.4,0.2,0.30,0,1,0Table 10
popfhs-graph A2=A⋆,Q2,J2,K2 (Example 5).
Jτ^1τ^2τ^3τ^4ω^30.3,0.2,0.2,0.50.2,0.1,0.2,0.40.3,0.1,0.1,0.40.3,0.1,0.2,0.6ω^40.1,0.1,0.3,0.50.3,0.2,0.2,0.70.1,0.2,0.1,0.50.2,0.3,0.1,0.8Kτ^1,τ^2τ^1,τ^3τ^1,τ^4τ^2,τ^3τ^2,τ^4τ^3,τ^4ω^30.1,0.2,0.1,0.30.2,0.1,0.2,0.30,0,1,00,0,1,00,0,1,00.1,0.3,0.1,0.3ω^40.2,0.1,0.2,0.40.3,0.1,0.1,0.30.1,0.2,0.2,0.40,0,1,00,0,1,00,0,1,0Table 11
popfhs-graph Wω^1,ω^3=W1ω^1×ℙW2ω^3 of A=A1×ℙA2 (Example 5).
Jη^11η^12η^13η^14η^21η^22ω^1,ω^30.3,0.2,0.2,0.30.2,0.1,0.2,0.30.3,0.1,0.2,0.30.3,0.1,0.2,0.30.2,0.1,0.3,0.50.2,0.1,0.3,0.4Jη^23η^24η^31η^32η^33η^34ω^1,ω^30.2,0.1,0.3,0.40.2,0.1,0.3,0.50.2,0.2,0.3,0.50.2,0.1,0.3,0.40.2,0.1,0.3,0.40.2,0.1,0.3,0.5Kη^11,η^12η^11,η^13η^11,η^21η^11,η^31η^12,η^22η^12,η^32ω^1,ω^30.1,0.2,0.2,0.30.2,0.1,0.2,0.30.2,0.1,0.4,0.30.2,0.2,0.2,0.20.2,0.1,0.4,0.30.2,0.1,0.2,0.2Kη^13,η^33η^13,η^14η^14,η^24η^14,η^34η^21,η^22η^21,η^23ω^1,ω^30.2,0.1,0.2,0.20.1,0.2,0.2,0.30.2,0.1,0.4,0.30.2,0.1,0.2,0.20.1,0.1,0.3,0.30.2,0.1,0.3,0.3Kη^22,η^32η^23,η^24η^23,η^33η^24,η^34η^31,η^32η^31,η^33ω^1,ω^30.2,0.1,0.2,0.30.1,0.1,0.3,0.30.2,0.1,0.1,0.30.2,0.1,0.2,0.30.1,0.2,0.3,0.30.2,0.1,0.3,0.3Kη^13,η^23η^21,η^31η^33,η^34ω^1,ω^30.2,0.1,0.4,0.30.2,0.2,0.2,0.30.1,0.3,0.3,0.3Figure 9
Graphical representation of Table9 with (a) Wω^1 and (b) Wω^2.Figure 10
Geometrical depiction of Table10 with (a) Wω^3 and (b) Wω^4.
(a)(b)Figure 11
Geometrical interpretation ofWω^1,ω^3=W1ω^1×ℙW2ω^3.Theorem 3.
The Cartesian product of twopopfhs-graphs is popfhs-graph.Proof.
It can easily be proved by following the axiomatic concepts provided in Definition15 and Example 5. Therefore, we have omitted its proof.Definition 16.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1⊚ℙA2 be cross product A1 and A2 where A=J,K,Q1×Q2 is popfhs-set over B=B1×B2, K=K1⊚ℙK2,Q1×Q2 is popfhs-set over E=π^1,τ^1,π^2,τ^2|π^1,π^2∈E1,τ^1,τ^2∈E2, and K=K1⊚ℙK2,Q1×Q2 is popfhs-graph where(1)
TJγ,ζπ^,τ^=TJ1γπ^∧TJ2tτ^IJγ,ζπ^,τ^=IJ1γπ^∧IJ2tτ^FJγ,ζπ^,τ^=FJ1γπ^∨FJ2tτ^μJγ,ζπ^,τ^=μJ1γπ^∧μJ2tτ^∀π^,τ^∈B,s,t∈Q1×Q2(2)
TKγ,ζπ^1,τ^1,π^2,τ^2=TK1γπ^1,π^2∧TK2ζτ^1,τ^2IKγ,ζπ^1,τ^1,π^2,τ^2=IK1γπ^1,π^2∧IK2ζτ^1,τ^2FKγ,ζπ^1,τ^1,π^2,τ^2=FK1γπ^1,π^2∨FK2ζτ^1,τ^2μKγ,ζπ^1,τ^1,π^2,τ^2=μK1γπ^1,π^2∧μK2ζτ^1,τ^2∀π^1,π^2∈E1,τ^1,τ^2∈E2∀π^,τ^∈Q1×Q2 and Wγ,ζ=W1γ⊚ℙW2ζ are popfhs-graphs of A.Theorem 4.
The cross product of twopopfhs-graphs is popfhs-graph.Proof.
It can easily be proved by following the axiomatic concepts provided in Definition16. Therefore, we have omitted its proof.Definition 17.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1⊙ℙA2 be lexicographic product A1 and A2 where A=J,K,Q1×Q2 is popfhs-set over B=B1×B2, K=K1⊙ℙK2,Q1×Q2 is popfhs-set over(6)E=π^,τ^1,π^,τ^2|π^∈B1,τ^1,τ^2∈E2∪π^1,τ^1,π^2,τ^2|π^1,π^2∈E1,τ^1,τ^2∈E2,and K=K1⊙ℙK2,Q1×Q2 is popfhs-graphs where (1)
TJγ,ζπ^,τ^=TJ1γπ^∧TJ2tτ^IJγ,ζπ^,τ^=IJ1γπ^∧IJ2tτ^FJγ,ζπ^,τ^=FJ1γπ^∨FJ2tτ^μJγ,ζπ^,τ^=μJ1γπ^∧μJ2tτ^∀π^,τ^∈B,s,t∈Q1×Q2(2)
TKγ,ζπ^,τ^1,π^,τ^2=TJ1γπ^∧TK2ζτ^1,τ^2IKγ,ζπ^,τ^1,π^,τ^2=IJ1γπ^∧IK2ζτ^1,τ^2FKγ,ζπ^,τ^1,π^,τ^2=FJ1γπ^∨FK2ζτ^1,τ^2μKγ,ζπ^,τ^1,π^,τ^2=μJ1γπ^∧μK2ζτ^1,τ^2∀π^∈B1,τ^1,τ^2∈E2(3)
TKγ,ζπ^1,τ^1,π^2,τ^2=TK1γπ^1,π^2∧TK2ζτ^1,τ^2IKγ,ζπ^1,τ^1,π^2,τ^2=IK1γπ^1,π^2∧IK2ζτ^1,τ^2FKγ,ζπ^1,τ^1,π^2,τ^2=FK1γπ^1,π^2∨FK2ζτ^1,τ^2μKγ,ζπ^1,τ^1,π^2,τ^2=μK1γπ^1,π^2∧μK2ζτ^1,τ^2∀π^1,π^2∈E1,τ^1,τ^2∈E2
Here,∀π^,τ^∈Q1×Q2 and Wγ,ζ=W1γ⊙ℙW2ζ are popfhs-graphs of A.Theorem 5.
The lexicographical product of twopopfhs-graphs is popfhs-graph.Proof.
It can easily be proved by following the axiomatic concepts provided in Definition17. Therefore, we have omitted its proof.Definition 18.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1⊗ℙA2 be strong product of A1 and A2, where A=J,K,Q1×Q2 is popfhs-set over B=B1×B2, K=K1⊗ℙK2,Q1×Q2 is popfhs-set over(7)E=π^,τ^1,π^,τ^2|π^∈B1,τ^1,τ^2∈E2∪π^1,τ^,π^2,τ^|π^1,π^2∈E1,τ^∈B2∪π^1,τ^1,π^2,τ^2π^1,π^2∈E1,τ^1,τ^2∈E2,and K=K1⊗ℙK2,Q1×Q2 are popfhs-graphs where(1)
TJγ,ζπ^,τ^=TJ1γπ^∧TJ2tτ^IJγ,ζπ^,τ^=IJ1γπ^∧IJ2tτ^FJγ,ζπ^,τ^=FJ1γπ^∨FJ2tτ^μJγ,ζπ^,τ^=μJ1γπ^∧μJ2tτ^∀π^,τ^∈B,s,t∈Q1×Q2(2)
TKγ,ζπ^,τ^1,π^,τ^2=TJ1γπ^∧TK2ζτ^1,τ^2IKγ,ζπ^,τ^1,π^,τ^2=IJ1γπ^∧IK2ζτ^1,τ^2FKγ,ζπ^,τ^1,π^,τ^2=FJ1γπ^∨FK2ζτ^1,τ^2μKγ,ζπ^,τ^1,π^,τ^2=μJ1γπ^∧μK2ζτ^1,τ^2∀π^∈B1,τ^1,τ^2∈E2(3)
TKγ,ζπ^1,τ^,π^2,τ^=TK1γπ^1,π^2∧TJ2ζτ^IKγ,ζπ^1,τ^,π^2,τ^=IK1γπ^1,π^2∧IJ2ζτ^FKγ,ζπ^1,τ^,π^2,τ^=FK1γπ^1,π^2∨FJ2ζτ^μKγ,ζπ^1,τ^,π^2,τ^=μK1γπ^1,π^2∧μJ2ζτ^∀π^1,π^2∈E1,τ^∈B2(4)
TKγ,ζπ^1,τ^1,π^2,τ^2=TK1γπ^1,π^2∧TK2ζτ^1,τ^2IKγ,ζπ^1,τ^1,π^2,τ^2=IK1γπ^1,π^2∧IK2ζτ^1,τ^2FKγ,ζπ^1,τ^1,π^2,τ^2=FK1γπ^1,π^2∨FK2ζτ^1,τ^2μKγ,ζπ^1,τ^1,π^2,τ^2=μK1γπ^1,π^2∧μK2ζτ^1,τ^2∀π^1,π^2∈E1,τ^1,τ^2∈E2
Here,∀π^,τ^∈Q1×Q2 and Wγ,ζ=W1γ⊗ℙW2ζ are popfhs-graphs of A.Theorem 6.
The strong product of twopopfhs-graphs is popfhs-graph.Definition 19.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1A2 be composition of A1 and A2, where A=J,K,Q1×Q2 is popfhs-set over B=B1×B2, K=K1×K2,Q1×Q2 is popfhs-set over(8)E=π^,τ^1,π^,τ^2|π^∈B1,τ^1,τ^2∈E2∪π^1,τ^,π^2,τ^|π^1,π^2∈E1,τ^∈B2∪π^1,τ^1,π^2,τ^2|π^1,π^2∈E1,τ^1≠τ^2,and K=K1×K2,Q1×Q2 is popfhs-graphs where(1)
TJγ,ζπ^,τ^=TJ1γπ^∧TJ2tτ^IJγ,ζπ^,τ^=IJ1γπ^∧IJ2tτ^FJγ,ζπ^,τ^=FJ1γπ^∨FJ2tτ^μJγ,ζπ^,τ^=μJ1γπ^∧μJ2tτ^∀π^,τ^∈B,s,t∈Q1×Q2(2)
TKγ,ζπ^,τ^1,π^,τ^2=TJ1γπ^∧TK2ζτ^1,τ^2IKγ,ζπ^,τ^1,π^,τ^2=IJ1γπ^∧IK2ζτ^1,τ^2FKγ,ζπ^,τ^1,π^,τ^2=FJ1γπ^∨FK2ζτ^1,τ^2μKγ,ζπ^,τ^1,π^,τ^2=μJ1γπ^∧μK2ζτ^1,τ^2∀π^∈B1,τ^1,τ^2∈E2(3)
TKγ,ζπ^1,τ^,π^2,τ^=TK1γπ^1,π^2∧TJ2ζτ^IKγ,ζπ^1,τ^,π^2,τ^=IK1γπ^1,π^2∧IJ2ζτ^FKγ,ζπ^1,τ^,π^2,τ^=FK1γπ^1,π^2∨FJ2ζτ^μKγ,ζπ^1,τ^,π^2,τ^=μK1γπ^1,π^2∧μJ2ζτ^∀π^1,π^2∈E1,τ^∈B2(4)
TKγ,ζπ^1,τ^1,π^2,τ^2=TK1γπ^1,π^2∧TJ1ζτ^2∧TJ2ζτ^1IKγ,ζπ^1,τ^1,π^2,τ^2=IK1γπ^1,π^2∧IJ1ζτ^2∧IJ2ζτ^1FKγ,ζπ^1,τ^1,π^2,τ^2=FK1γπ^1,π^2∧FJ1ζτ^2∧FJ2ζτ^1μKγ,ζπ^1,τ^1,π^2,τ^2=μK1γπ^1,π^2∧μJ1ζτ^2∧μJ2ζτ^1∀π^1,π^2∈E1τ^1≠τ^2
Here,∀γ,ζ∈Q1×Q2 and Wγ,ζ=W1γW2ζ are popfhs-graphs of A.Theorem 7.
The composition of twopopfhs-graphs is popfhs-graph.Proof.
It can easily be proved by following the axiomatic concepts provided in Definition19. Therefore, we have omitted its proof.Definition 20.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1∪A2 be the union of A1 and A2 where A=J,K,Q1∪Q2 is popfhs-set over B=B1∪B2 and K=K1∪K2,Q1∪Q2 is popfhs-set over E=E1∪E2, where for π^,τ^∈B, popf-components are stated as follows:(1)
TJω^τ^=TJ1ω^τ^TJ2ω^τ^TJ1ω^τ^∨TJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2IJω^τ^=IJ1ω^τ^IJ2ω^τ^IJ1ω^τ^∨IJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2FJω^τ^=FJ1ω^τ^FJ2ω^τ^FJ1ω^τ^∧FJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2μJω^τ^=μJ1ω^τ^μJ2ω^τ^μJ1ω^τ^∨μJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2(2)
TKω^π^τ^=TK1ω^π^τ^;ω^∈Q1−Q2TK2ω^π^τ^;ω^∈Q2−Q1,TK1ω^π^τ^∨TK2ω^π^τ^;ω^∈Q1∩Q2,IKω^π^τ^=IK1ω^π^τ^;ω^∈Q1−Q2,IK2ω^π^τ^;ω^∈Q2−Q1,IK1ω^π^τ^∨IK2ω^π^τ^;ω^∈Q1∩Q2,FKω^π^τ^=FK1ω^π^τ^;ω^∈Q1−Q2,FK2ω^π^τ^;ω^∈Q2−Q1,FK1ω^π^τ^∨IK2ω^π^τ^;ω^∈Q1∩Q2,μKω^π^τ^=μK1ω^π^τ^;ω^∈Q1−Q2,μK2ω^π^τ^;ω^∈Q2−Q1,μK1ω^π^τ^∨IK2ω^π^τ^;ω^∈Q1∩Q2.Definition 21.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1∩A2 be the intersection of A1 and A2, where A=J,K,Q1∪Q2 is popfhs-set over B=B1∩B2 and K=K1∪K2,Q1∪Q2 is popfhs-set over E=E1∩E2, where for π^,τ^∈B, popf-components can be given by(1)
ɛTJω^τ^=TJ1ω^τ^TJω^τ^TJ1ω^τ^∧TJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2IJω^τ^=IJ1ω^τ^IJ2ω^τ^IJ1ω^τ^∧IJω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2FJω^τ^=FJ1ω^τ^FJω^τ^FJ1ω^π^τ^∨FJ1ω^π^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2μJω^τ^=μJ1ω^τ^μJω^τ^μJ1ω^τ^∧μJ2ω^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2(2)
TKω^π^τ^=TK1ω^π^τ^TK2ω^π^τ^TK1ω^π^τ^∧TK2ω^π^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2IKω^π^τ^=IK1ω^π^τ^IK2ω^π^τ^IK1ω^π^τ^∧IK2ω^π^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2FKω^π^τ^=FK1ω^π^τ^FK2ω^π^τ^FK1ω^π^τ^∨FK2ω^π^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2μKω^π^τ^=μK1ω^π^τ^μK2ω^π^τ^μK1ω^π^τ^∧μK2ω^π^τ^;ω^∈Q1−Q2;ω^∈Q2−Q1;ω^∈Q1∩Q2Definition 22.
For twopopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2. Let A=A1⊎A2 be the join of A1 and A2 where A=J1⊎J2,K1⊎K2,Q1∪Q2 is popfhs-set over B=B1∪B2 and K=K1⊎K2,Q1∪Q2 is popfhs-set over E=E1∪E2 where(1)
J1⊎J2,Q1∪Q2=J1,Q1∪J2,Q2(2)
K1⊎K2,Q1∪Q2=K1,Q1∪K2,Q2,ifπ^τ^∈E1∪E2when ω^∈Q1∩Q2 and π^τ^∈E, and uncertain parts are as follows:(9)TK1⊎K2ω^π^τ^=minTJ1ω^π^τ^,TJ2ω^π^τ^,IK1⊎K2ω^π^τ^=minIJ1ω^π^τ^,IJ2ω^π^τ^,FK1⊎K2ω^π^τ^=minFJ1ω^π^τ^,FJ2ω^π^τ^,μK1⊎K2ω^π^τ^=minμJ1ω^π^τ^,μJ2ω^π^τ^.Definition 23.
The complementAc=A⋆c,Qc,Jc,Kc of popfhs-graph A=A⋆,Q,J,K is a popfhs-graph for which π^,τ^∈B and ω^∈Q, and it satisfies the following conditions:(1)
Qc=Q(2)
Jcω^=Jω^(3)
TKcω^π^,τ^=TJω^π^∧TJω^τ^−TKω^π^,τ^(4)
IKcω^π^,τ^=IJω^π^∧IJω^τ^−IKω^π^,τ^(5)
FKcω^π^,τ^=FJω^π^∧FJω^τ^−FKω^π^,τ^(6)
μKcω^π^,τ^=μJω^π^∧μJω^τ^−μKω^π^,τ^Definition 24.
IfAc=A where A=A⋆,Q,J,K is a popfhs-graph, then A is self-complementary.Definition 25.
IfKω^ is popfhs-graph of A, ∀ω^∈Q, then A is complete, and one can write(10)TKω^π^τ^=minTJω^π^,TJω^τ^,IKω^π^τ^=minIJω^π^,IJω^τ^,IKω^π^τ^=minIJω^π^,IJω^τ^,μKω^π^τ^=minμJω^π^,μJω^τ^.Definition 26.
Apopfhs-graph A=A⋆,Q,J,K is strong popfhs-graph if Kω^ is popfhs-graph of A and ∀ω^∈Q.Theorem 8.
For two strongpopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2; then A1A2; the composition of A1 and A2 is strong popfhs-graph.Theorem 9.
For two strongpopfhs-graphs, A1=A1⋆,Q1,J1,K1 and A2=A2⋆,Q2,J2,K2 with respect to A1⋆=B1,E2 and A2⋆=B2,E2; then A1×ℙA2; the Cartesian product of A1 and A2 is strong popfhs-graph.Definition 27.
The complementAc=A⋆c,Qc,Jc,Kc of strong popfhs-graph A=A⋆,Q,J,K∀ω^∈J,π^,τ^∈B is given by(1)
Qc=Q(2)
Jcω^π^=Jω^π^(3)
TKcω^π^,τ^=minTJω^π^,TJω^τ^0;TKω^π^,τ^=0;TKω^π^,τ^>0(4)
IKcω^π^,τ^=minIJω^π^,IJω^τ^0;IKω^π^,τ^=0;IKω^π^,τ^>0(5)
FKcω^π^,τ^=minFJω^π^,FJω^τ^0;FKω^π^,τ^=0;FKω^π^,τ^>0(6)
μKcω^π^,τ^=minμJω^π^,μJω^τ^0;μKω^π^,τ^=0;μKω^π^,τ^>0Theorem 10.
The complementAc=A⋆c,Qc,Jc,Kc of strong popfhs-graph A=A⋆,Q,J,K∀ω^∈J,π^,τ^∈B is strong popfhs-graph.Theorem 11.
IfA=A⋆,Q,J,K and its complement Ac=A⋆c,Qc,Jc,Kc are strong popfhs-graphs ∀ω^∈J,π^,τ^∈B, then A∪Ac is complete popfhs-graph.


## 5. Application in Decision-Making

In order to validate the proposed study, an algorithm-based application is discussed for reliable decision-making process (The brief graphical description of steps involved in Algorithm 1 is presented in Figure12).Figure 12
Step-wise brief graphical description of proposed algorithm.Example 6.
Suppose an organization intends to recruit a candidate to fill a vacant post of assistant manager. Six candidates, that is,B=ℭ1,ℭ2,ℭ3,ℭ4,ℭ5,ℭ6, have been scrutinized by recruitment committee. The committee further requires evaluation to select one of these candidates. The evaluation indicators are qualification β1, relevant experience β2, and computer skill β3. Their subparametric disjoint sets are Q1=β11, Q2=β21,β22,β23, and Q3=β31, respectively, such that Q=Q1×Q2×Q3=ω^1,ω^2,ω^3 and A=W,Q=Wω^1,Wω^2,Wω^3 are popfhs-graph. This selection is accomlished by proposing an algorithm (i.e. Algorithm 1 which is presented in Figure 1)
The graphical explanation of Table12 is provided in Figure 13.
Thepopfhs-graphs Wω^1,Wω^2, and Wω^3 with respect to subparametric values are given in Table 12 and stated in Figure 2. The I-Matrices of popfhs-graphs are(11)Wω^1=0,0,0,00.2,0.2,0.1,0.20,0,0,00.3,0.2,0.3,0.30,0,0,00,0,0,00.2,0.2,0.1,0.20,0,0,00.3,0.1,0.1,0.30.2,0.2,0.2,0.20.3,0.1,0.1,0.30,0,0,00,0,0,00.3,0.1,0.1,0.30,0,0,00,0,0,00.2,0.1,0.2,0.40.2,0.2,0.3,0.20.3,0.2,0.3,0.30.2,0.2,0.2,0.20,0,0,00,0,0,00.1,0.2,0.3,0.40,0,0,00,0,0,00.3,0.1,0.1,0.30.2,0.1,0.2,0.40.1,0.2,0.3,0.40,0,0,00.3,0.1,0.1,0.30,0,0,00,0,0,00.2,0.2,0.3,0.20,0,0,00.3,0.1,0.1,0.30,0,0,0,Wω^2=0,0,0,00.2,0.2,0.3,0.60.2,0.1,0.1,0.20.2,0.2,0.1,0.50,0,0,00,0,0,00.2,0.2,0.3,0.60,0,0,00,0,0,00.1,0.2,0.3,0.60,0,0,00.5,0.1,0.1,0.50.2,0.1,0.1,0.20,0,0,00,0,0,00.2,0.1,0.4,0.20,0,0,00.3,0.2,0.1,0.30.2,0.2,0.1,0.50.1,0.2,0.3,0.60.2,0.1,0.4,0.20,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00,0,0,00.5,0.1,0.1,0.50.3,0.2,0.1,0.30,0,0,00,0,0,00,0,0,0,Wω^3=0,0,0,00.2,0.1,0.1,0.50,0,0,00,0,0,00.1,0.3,0.2,0.50,0,0,00.2,0.1,0.1,0.50,0,0,00.1,0.2,0.2,0.40.2,0.2,0.1,0.40,0,0,00,0,0,00,0,0,00.1,0.2,0.2,0.40,0,0,00,0,0,00.2,0.3,0.1,0.40.2,0.3,0.3,0.40,0,0,00.2,0.2,0.1,0.40,0,0,00,0,0,00.2,0.1,0.2,0.50,0,0,00.1,0.3,0.2,0.50,0,0,00.2,0.3,0.1,0.40.2,0.1,0.2,0.50,0,0,00.3,0.1,0.4,0.30,0,0,00,0,0,00.2,0.3,0.3,0.40,0,0,00.3,0.1,0.4,0.30,0,0,0.
The incidence matrix for resultantpopfhs-graph can be calculated by ω^=ω^1∧ω^2∧ω^3, which is given by(12)Wω^=0,0,0,00.2,0.1,0.3,0.20,0,0.1,00,0,0.3,00,0,0.2,00,0,0,00.2,0.1,0.3,0.20,0,0,00,0,0.2,00.1,0.2,0.2,0.20,0,0.1,00,0,0.1,00,0,0.1,00,0,0.2,20,0,0,00,0,0.4,00,0,0.2,00.2,0.2,0.3,0.20,0,0.3,00.1,0.2,0.2,0.20,0,0.4,00,0,0,00,0,0.3,00,0,0,00,0,0.2,00,0,0.1,00,0,0.2,00,0,0.3,00,0,0,00,0,0.4,00,0,0,00,0,0.1,00.2,0.2,0.3,0.20,0,0,00,0,0.4,00,0,0,0.
Score and its average values are given in Table13.
Since according to Table13, the maximum average score value for ℭ2 is 1.525, so the candidate ℭ2 is selected.Algorithm 1: Optimal selection of candidate by using popfhs-graph.
StartInput(1)
InputB and Q as universal set and set of parametric tuples, respectively.(2)
Inputpopfhs-sets J,Q and K,Q.Construction(3)
Constructpopfhs-graph A=A⋆,Q,J,K.(4)
Construct resultantpopfhs-graph for ω^=∧κω^κ∀ values of κ.(5)
Constructpopfhs-graph Wω^ along with I-Matrix.Computation(6)
Calculate scoreSκ of ℭκ∀ values of κ and its average by the formula Sκ=Tκ+Iκ−Fκ+μκ+1/4.Decision(7)
Optℭκ if ℭκ=maxiℭi.(8)
Opt any one ofℭκ if κ bears multiple values.EndTable 12
popfhs-graph A=A⋆,Q,J,K.
Jℭ1ℭ2ℭ3ℭ4ℭ5ℭ6ω^10.2,0.1,0.3,0.40.3,0.1,0.1,0.30.2,0.1,0.3,0.50.1,0.2,0.1,0.60.2,0.1,0.1,0.50.2,0.1,0.2,0.3ω^20.1,0.1,0.3,0.70.2,0.3,0.2,0.70.1,0.3,0.2,0.30.3,0.1,0.3,0.80,0,0,00.2,0.3,0.1,0.7ω^30.3,0.4,0.1,0.70.2,0.3,0.1,0.60.1,0.1,0.3,0.50.2,0.1,0.2,0.60.3,0.2,0.1,0.70.2,0.1,0.3,0.5Kℭ1,ℭ2ℭ1,ℭ3ℭ1,ℭ4ℭ1,ℭ5ℭ2,ℭ3ℭ2,ℭ4ℭ2,ℭ5ω^10.2,0.2,0.1,0.20,0,1,00.3,0.2,0.3,0.30,0,1,00.3,0.1,0.1,0.30.2,0.2,0.2,0.20.3,0.1,0.1,0.3ω^20.2,0.2,0.3,0.60.2,0.1,0.1,0.20.2,0.2,0.1,0.50,0,1,00,0,1,00.1,0.2,0.3,0.60,0,1,0ω^30.2,0.1,0.1,0.50,0,1,00,0,1,00.1,0.3,0.2,0.50.1,0.2,0.2,0.40.2,0.2,0.1,0.40,0,1,0Kℭ2,ℭ6ℭ3,ℭ4ℭ3,ℭ5ℭ3,ℭ6ℭ4,ℭ5ℭ5,ℭ6ω^10,0,1,00,0,1,00.2,0.1,0.2,0.40.2,0.2,0.3,0.20.1,0.2,0.3,0.40.3,0.1,0.1,0.3ω^20.5,0.1,0.1,0.50.2,0.1,0.4,0.20,0,1,00.3,0.2,0.1,0.30,0,1,00,0,1,0ω^30,0,1,00,0,1,00.2,0.3,0.1,0.40.2,0.3,0.3,0.40.2,0.1,0.2,0.50.3,0.1,0.4,0.3Figure 13
Pictorial depiction of Table1 with (a) Wω^1, (b) Wω^2, and (c) Wω^3.
(a)(b)(c)Table 13
Score values with choice values.
ℭ1ℭ2ℭ3ℭ4ℭ5ℭ6ℭκℭ10.2500.3000.2250.1750.2000.2501.400ℭ20.3000.2500.2000.3250.2250.2251.525ℭ30.2250.2000.2500.1500.2000.3251.400ℭ40.1750.3250.1500.2500.1750.2501.325ℭ50.2000.2250.2000.1750.2500.1501.200ℭ60.2500.2250.3250.2500.1500.2501.450


## 6. Comparison Analysis

The problem of project selection for investment selection is of great importance for investors. A very few literature exists in this regard under uncertain environments (i.e., fuzzy set-like environments). But there exists no literature regarding this project under soft set-like and possibility soft set-like environments. It is an advantageous aspect of the proposed study over the existing relevant models that an important problem has been discussed through this study. Due to lack of relevant literature, the numerical comparison is not possible, but we present the comparison of the proposed model with some existing relevant models on structural basis to depict its advantageous aspects and flexibility. Table14 presents this structural comparison.Table 14
Advantageous aspects of proposed model over relevant existing models.
ModelsGrade of true membershipGrade of false membershipGrade of neutralityMapping with single argumentMapping with multiargumentEntitlement of possibility degreeProject selection rankingfs-graph [30]✓××✓×××ifs-graph [31]✓✓×✓×××pfs-graph [32]✓✓✓✓×××pofs-set [33]✓××✓×✓×poifs-set [34]✓✓×✓×✓×poifs-graph [35]✓✓×✓×✓×Proposed model✓✓✓✓✓✓✓


## 7. Conclusions

In this research, authors have managed the real-world decision-making situation that demands: (i) the categorization of opted parameters into their respective disjoint subclasses having their relevant attributive values, (ii) the consideration of multiargument parameterization in the domain of approximate mapping to have reliable approximation of alternatives, and (iii) a mode for the assessment of uncertain nature of approximate elements to have level of acceptance, collectively with the development of novel context ofpopfhs-graph. As a conceptual framework, essential rudiments, operations, and products are characterized with the help of elaborated instances. A real-world decision-making problem from human resource management for the optimal selection of candidate is resolved with the proposal of an intelligent algorithm. Since the sum of values of uncertain components in proposed model is considered within [0,1], but it is not sufficient for the situations where decision-makers provide their opinions as uncertain values whose sum exceeds 1; therefore, future work may include the extension of this work to possibility neutrosophic hypersoft setting to deal with above describe limitation. Moreover, many other notions of classical graph may also be characterized by utilizing the proposed context.



---

*Source: 6435368-2022-09-15.xml*