# Multiple Attribute Decision-Making Problem Using Picture Fuzzy Graph

Authors: Sk. Amanathulla; G. Muhiuddin; D. Al-Kadi; M. Pal
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In a picture fuzzy environment, almost all multiple attribute decision-making (MADM) methods have been discussed a type of problem in which there is no relationship among the attributes. Although the relationship among the attributes should be considered in the actual applications, so we need to pay attention to that important issue. This article applied graph theory to the picture fuzzy set (PFS) and obtained a new method, MADM, to solve complicated problems under a picture fuzzy environment. The developed method can capture the relationship among the attributes that cannot be handled well by any existing methods. This study introduces union, intersection, sum, Cartesian product, the composition of picture fuzzy graphs (PFGs), and their important properties. Finally, by considering the importance of relationships among attributes in the determination process, two algorithms, based on PFG, have developed to solve complicated problems using picture fuzzy information. Also, two numerical examples have introduced to explain how to deal with the MADM problem under picture fuzzy environment.


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## Body

## 1. Introduction

At present, graphs do not disclose all the systems properly because of the uncertainty of the parameters within a system. For instance, a social network can be uttered as the graph, where nodes denote an account (such as institution or person) and edges express the connection between the accounts. If the connections among accounts are mensurable as bad or good according to the recurrence rate of contacts among the accounts, fuzziness should be added to representation. In 1975, Rosenfeld first defined the fuzzy graph considering fuzzy relations on fuzzy sets [1]. A PFS is a generalization of intuitionistic fuzzy set (IFS) [2]. The picture fuzzy model gives more precision, flexibility, and compatibility than the intuitionistic fuzzy model.The concept ofPFS was first introduced by Coung [3] in 2013. In addition to IFS, Coung appended new components which determine the neutral membership degree. IFS gives an element’s membership and nonmembership degree, while PFS gives positive membership degree, neutral membership degree, and negative membership degree of an element. These memberships are almost independent and the sum of these three membership degrees is ≤1. Basically, PFS-based models may be adequate in situations, where we counter several opinions that involve more answers of types: yes, no, abstain, and refusal. If we take voting as an example, human voters may be separated into four possible groups with distinct opinions: vote for, vote against, abstain, and refusal of the voting. Picture fuzzy sets have several interesting applications in system analysis, operation research, economics, medicine, computer science, engineering, mathematics, etc. Some properties of PFS and its operators have been studied in [4, 5].


### 1.1. Review of Literature

After invention of fuzzy graph, it develops with its different branches, such as fuzzy threshold graph [6], balanced interval-valued fuzzy graphs [7], cubic graph [8], m−step fuzzy competition graphs [9], fuzzy planar graphs [10,11], and fuzzy k-competition graph [12]. Pramanik et al. defined interval-valued fuzzy threshold graph and studied several properties [13]. They also have considered planarity in bipolar fuzzy graph, and they extended it to bipolar fuzzy planar graphs [14]. Also, Pramanik et al. have extended fuzzy planar graph to interval-valued fuzzy planar graph [15] and interval-valued fuzzy graph [16]. Voskoglou et al. [17] have discussed and characterized several fuzzy graph theoretic structure and fuzzy hypergraphs. Sahoo et al. [18] have studied the intuitionistic fuzzy competition graph. Balanced intuitionistic fuzzy graphs are discussed by Karunambigai et al. [19]. Also, Sahoo et al. have studied some problems regarding IFG [18, 20, 21]. Recently, some researchers have carried out study regarding picture fuzzy graphs and its applications [22], regular picture fuzzy graph and its application [23], and edge domination in picture fuzzy graphs [24]. Many related problems such as a study on picture Dombi fuzzy graph [25], q-rung picture fuzzy graphs [26], interval-valued picture uncertain linguistic generalized Hamacher aggregation operators and their application in multiple attribute decision-making process [27], multiple attribute decision-making algorithm via picture fuzzy nanotopological spaces [28], decision-making model under complex picture fuzzy Hamacher aggregation operators [29], and fuzzy aggregation operators and their applications to multicriteria decision-making [30] are investigated. In 2018, Ullah et al. [31] have studied similarity measures for T-spherical fuzzy sets with applications in pattern recognition. They also have studied policy decision-making based on some averaging aggregation operators of T-spherical fuzzy sets [32]. The concept of spherical fuzzy set and T-spherical fuzzy set is introduced as a generalization of fuzzy set, intuitionistic fuzzy set, and picture fuzzy set by Mahmood et al. [33].In 2020, Devaraj et al. [34] have studied picture fuzzy labelling graphs, and they also have presented an application of picture fuzzy labelling graphs; also, Mahmood et al. [35] have studied a lot of results regarding the fuzzy cross entropy for picture hesitant fuzzy sets and their application in multicriteria decision-making. Also, T. Mahmood [36] has studied a novel approach towards bipolar soft sets and their applications. Applications of the generalized picture fuzzy soft set in concept selection have been studied by Khan et al. [37]. Exponential operational laws and new aggregation operators for the intuitionistic multiplicative set in the multiple attribute group decision-making process have been studied by Garg [27]. In 2021, Amanathulla et al. have studied a lot of results regarding balanced picture fuzzy graphs [38]. Recently, many researchers have applied various related concepts of the current study on graphs in different aspects (see, for e.g., [39–45]).


### 1.2. Motivation

Most ofMADM methods with picture fuzzy environment are to discuss a type of problem that there is no relationship among attributes. Although this relationship should be considered in the actual applications, so we need to pay attention to that issue. From this point of view, we consider MADM problem using picture fuzzy graph. This article applies graph theory to PFS and obtained a new method, MADM, to solve complicated problems under a picture fuzzy environment. The proposed method can capture the relationship among the attribute that cannot be handled well by any existing technique. Also, we have been given two examples to show that our decision-making algorithm is original. The remaining parts of this article are organized as follows. Some preliminaries are presented in Section 2. In Section 3, PFG and some of its properties are presented. In Section 4, two algorithms based on multiple attribute decision-making for complicated problems are presented. In Section 5, two numerical examples for PFGMADM problem with picture fuzzy information are used to present the applications of the proposed decision-making algorithm. Section 6 is for the brief conclusion.


## 1.1. Review of Literature

After invention of fuzzy graph, it develops with its different branches, such as fuzzy threshold graph [6], balanced interval-valued fuzzy graphs [7], cubic graph [8], m−step fuzzy competition graphs [9], fuzzy planar graphs [10,11], and fuzzy k-competition graph [12]. Pramanik et al. defined interval-valued fuzzy threshold graph and studied several properties [13]. They also have considered planarity in bipolar fuzzy graph, and they extended it to bipolar fuzzy planar graphs [14]. Also, Pramanik et al. have extended fuzzy planar graph to interval-valued fuzzy planar graph [15] and interval-valued fuzzy graph [16]. Voskoglou et al. [17] have discussed and characterized several fuzzy graph theoretic structure and fuzzy hypergraphs. Sahoo et al. [18] have studied the intuitionistic fuzzy competition graph. Balanced intuitionistic fuzzy graphs are discussed by Karunambigai et al. [19]. Also, Sahoo et al. have studied some problems regarding IFG [18, 20, 21]. Recently, some researchers have carried out study regarding picture fuzzy graphs and its applications [22], regular picture fuzzy graph and its application [23], and edge domination in picture fuzzy graphs [24]. Many related problems such as a study on picture Dombi fuzzy graph [25], q-rung picture fuzzy graphs [26], interval-valued picture uncertain linguistic generalized Hamacher aggregation operators and their application in multiple attribute decision-making process [27], multiple attribute decision-making algorithm via picture fuzzy nanotopological spaces [28], decision-making model under complex picture fuzzy Hamacher aggregation operators [29], and fuzzy aggregation operators and their applications to multicriteria decision-making [30] are investigated. In 2018, Ullah et al. [31] have studied similarity measures for T-spherical fuzzy sets with applications in pattern recognition. They also have studied policy decision-making based on some averaging aggregation operators of T-spherical fuzzy sets [32]. The concept of spherical fuzzy set and T-spherical fuzzy set is introduced as a generalization of fuzzy set, intuitionistic fuzzy set, and picture fuzzy set by Mahmood et al. [33].In 2020, Devaraj et al. [34] have studied picture fuzzy labelling graphs, and they also have presented an application of picture fuzzy labelling graphs; also, Mahmood et al. [35] have studied a lot of results regarding the fuzzy cross entropy for picture hesitant fuzzy sets and their application in multicriteria decision-making. Also, T. Mahmood [36] has studied a novel approach towards bipolar soft sets and their applications. Applications of the generalized picture fuzzy soft set in concept selection have been studied by Khan et al. [37]. Exponential operational laws and new aggregation operators for the intuitionistic multiplicative set in the multiple attribute group decision-making process have been studied by Garg [27]. In 2021, Amanathulla et al. have studied a lot of results regarding balanced picture fuzzy graphs [38]. Recently, many researchers have applied various related concepts of the current study on graphs in different aspects (see, for e.g., [39–45]).


## 1.2. Motivation

Most ofMADM methods with picture fuzzy environment are to discuss a type of problem that there is no relationship among attributes. Although this relationship should be considered in the actual applications, so we need to pay attention to that issue. From this point of view, we consider MADM problem using picture fuzzy graph. This article applies graph theory to PFS and obtained a new method, MADM, to solve complicated problems under a picture fuzzy environment. The proposed method can capture the relationship among the attribute that cannot be handled well by any existing technique. Also, we have been given two examples to show that our decision-making algorithm is original. The remaining parts of this article are organized as follows. Some preliminaries are presented in Section 2. In Section 3, PFG and some of its properties are presented. In Section 4, two algorithms based on multiple attribute decision-making for complicated problems are presented. In Section 5, two numerical examples for PFGMADM problem with picture fuzzy information are used to present the applications of the proposed decision-making algorithm. Section 6 is for the brief conclusion.


## 2. Preliminaries

PFS is an extension of IFS. Some definitions related to PFS are presented below, which we have used later to develop the paper.Definition 1.
(see [4]). Let S=pS,qS,rS and T=pT,qT,rT be two PFSs. Then, the union and the intersection of the PFSs S and T are defined by(i)
S∪Tt=t,pSt∨pTt,qSt∨qTt,rSt∨rTt:t∈X(ii)
S∩Tt=t,pSt∧pTt,qSt∧qTt,rSt∧rTt:t∈XA picture fuzzy number is defined byfn=α,β,γ.Definition 2.
(see [4]). Let fn=α,β,γ be a picture fuzzy number; then, the score function of fn is denoted by scorfn and is defined by scorfn=1+α−2β−γ/2.Observation 1.
Letf1 and f2 be two picture fuzzy numbers; then, scorf1>scorf2⇒f1⇒f2.Definition 3.
(see [4]). A PF relation ρ in a universe A×B is a PFS defined by ρa,b=a,b,pρa,b,qρa,b,rρa,b:a,b∈A×B, where pρ:A×B⟶0,1, qρ:A×B⟶0,1, rρ:A×B⟶0,1, and 0≤pρa,b+qρa,b+rρa,b≤1, for all a,b∈A×B.Definition 4.
(see [4]). Let S=pS,qS,rS and T=pT,qT,rT be two PFSs on a set X. If S be a PFR on X, then S is also a PFR on T if pSa,b≤pTa∧pTb, qSa,b≥qTa∨qTb, and rSa,b≥rTa∨pTb, for all a,b∈X.


## 3. Picture Fuzzy Graph

In this section, thePFG and some properties and theorems of PFG have been described.Definition 5.
APFG of a graph G∗=V,E is a pair G=S,T, where S=pS,qS,rS is a PFS on V and T=pT,qT,rT is a PFR on E such that pTa,b≤PSa∧pb, qTa,b≥qSa∨qb, rTa,b≥rSa∨rb, and 0≤pTa,b+qTa,b+rTa,b≤1.Here,S is the picture fuzzy node set of G and T is a picture fuzzy edge set on G. Also, pSa,qSa,andrSa, respectively, denote the positive, neutral, and negative membership degree of the node a and pTa,b,qTa,b,andrTa,b denote that of edge a,b.Now, we give some properties ofPFG such as composition, Cartesian product, union, and intersection.LetG1=S1,T1 and G2=S2,T2 be two PFGs, where S1=pS1,qS1,rS1, S2=pS2,qS2,rS2, T1=pT1,qT1,rT1, and T2=pT2,qT2,rT2.Definition 6.
LetG1andG2 be two PFGs; then, the cartesian product G1×G2 of G1 and G2 is defined by G1×G2=S1×S2,T1×T2, where(i)
pS1×pS2a1,a2=pS1a1∧pS2a2qS1×qS2a1,a2=qS1a1∨qS2a2rS1×rS2a1,a2=rS1a1∨rS2a2,a1,a2∈V1×V2=V.(ii)
pT1×pT2a,a2,a,b2=pS1a∧pT2a2,b2qT1×qT2a,a2,a,b2=qS1a∨qT2a2,b2rT1×rT2a,a2,a,b2=rS1a∨rT2a2,b2,a∈V1,a2,b2∈E2.(iii)
pT1×pT2a1,c,b1,c=pT1a1,b1∧pS2cqT1×qT2a1,c,b1,c=qT1a1,b1∨qS2crT1×rT2a1,c,b1,c=rT1a1,b1∨pS2c,c∈V2,a1,b1∈E1.Theorem 1.
LetG1andG2 be two PFGs; then, G1×G2 is a PFG.Proof.
Leta∈V1,a2,b2∈E2. Then, we obtain(1)pT1×pT2a,a2,a,b2=pS1a∧pT2a2,b2≤pS1a∧pS2a2∧pS2b2=pS1a∧pS2a2∧pS1a∧pS2b2=PS1×PS2a,a2∧PS1×PS2a,b2,qT1×qT2a,a2,a,b2=qS1a∨qT2a2,b2≥qS1a∨qS2a2∨qS2b2=qS1a∨qS2a2∨qS1a∨qS2b2=qS1×qS2a,a2∨qS1×qS2a,b2,rT1×rT2a,a2,a,b2=rS1a∨rT2a2,b2≥rS1a∨rS2a2∨rS2b2=rS1a∨rS2a2∨rS1a∨rS2b2=rS1×rS2a,a2∨rS1×rS2a,b2.
Again, letc∈V2 and a1,b1∈E1. Then, we obtain(2)pT1×pT2a1,c,b1,c=pT1a1,b1∧pS2c≤pS2c∧pS1a1∧pS1b1=pS1a1∧pS2c∧pS1b1∧pS2c=PS1×PS2a1,c∧PS1×PS2b1,c,qT1×qT2a1,c,b1,c=qT1a1,b1∨qS2c≥qS2c∨qS1a1∨qS1b1=qS1a1∨qS2c∨qS1b1∨qS2c=qS1×qS2a1,c∨qS1×qS2b1,c,rT1×rT2a1,c,b1,c=rT1a1,b1∨rS2c≥rS2c∨rS1a1∨rS1b1=rS1a1∨rS2c∨rS1b1∨rS2c=rS1×rS2a1,c∨rS1×rS2b1,c.
The above results proves thatG1×G2 is a PFG.Definition 7.
LetG1 and G2 be two PFGs; the composition of G1 and G2 is defined by G1G2=S1⋅S2,T1⋅T2, where(i)
pS1⋅pS2a1,a2=pS1a1∧pS2a2qS1⋅qS2a1,a2=qS1a1∨qS2a2rS1⋅rS2a1,a2=rS1a1∨rS2a2, a1,a2∈V1×V2=V.(ii)
pT1⋅pT2a,a2,a,b2=pS1a∧pT2a2,b2qT1⋅qT2a,a2,a,b2=qS1a∨qT2a2,b2rT1⋅rT2a,a2,a,b2=rS1a∨rT2a2,b2, a∈V1,a2,b2∈E2.(iii)
pT1⋅pT2a1,c,b1,c=pT1a1,b1∧pS2cqT1⋅qT2a1,c,b1,c=qT1a1,b1∨qS2crT1⋅rT2a1,c,b1,c=rT1a1,b1∨pS2c, c∈V2,a1,b1∈E1.(iv)
pT1⋅pT2a1,a2,b1,b2=pS2a2∧pS2b2∧pT1a1,b1qT1⋅qT2a1,a2,b1,b2=qS2a2∨qS2b2∨qT1a1,b1rT1⋅rT2a1,a2,b1,b2=rS2a2∨rS2b2∨rT1a1,b1, a1,a2,b1,b2∈E⋅−E, where E⋅=E∪a1,a2,b1,b2:a1,b1∈E1,a2≠b2.Theorem 2.
LetG1andG2 be two PFGs; then, G1G2 is a PFG.Proof.
Leta∈V1,a2,b2∈E2. Then, we obtain(3)pT1⋅pT2a,a2,a,b2=pS1a∧pT2a2,b2≤pS1a∧pS2a2∧pS2b2=pS1a∧pS2a2∧pS1a∧pS2b2=PS1⋅PS2a,a2∧PS1⋅PS2a,b2,qT1⋅qT2a,a2,a,b2=qS1a∨qT2a2,b2≥qS1a∨qS2a2∨qS2b2=qS1a∨qS2a2∨qS1a∨qS2b2=qS1⋅qS2a,a2∨qS1⋅qS2a,b2,rT1⋅rT2a,a2,a,b2=rS1a∨rT2a2,b2≥rS1a∨rS2a2∨rS2b2=rS1a∨rS2a2∨rS1a∨rS2b2=rS1⋅rS2a,a2∨rS1⋅rS2a,b2.
Again, letc∈V2 and a1,b1∈E1. Then, we obtain(4)pT1⋅pT2a1,c,b1,c=pT1a1,b1∧pS2c≤pS2c∧pS1a1∧pS1b1=pS1a1∧pS2c∧pS1b1∧pS2c=PS1⋅PS2a1,c∧PS1⋅PS2b1,c,qT1⋅qT2a1,c,b1,c=qT1a1,b1∨qS2c≥qS2c∨qS1a1∨qS1b1=qS1a1∨qS2c∨qS1b1∨qS2c=qS1⋅qS2a1,c∨qS1⋅qS2b1,c,rT1⋅rT2a1,c,b1,c=rT1a1,b1∨rS2c≥rS2c∨rS1a1∨rS1b1=rS1a1∨rS2c∨rS1b1∨rS2c=rS1⋅rS2a1,c∨rS1⋅rS2b1,c.
Again, leta1,a2,b1,b2∈E⋅−E, so a1,b1∈E1,a2≠b2. Then, we obtain(5)pT1⋅pT2a1,a2,b1,b2=pS2a2∧pS2b2∧pT1a1,b1≤pS2a2∧pS2b2∧pS1a1∧pS1b1=pS1a1∧pS2a2∧pS1b1∧pS2b2=pS1⋅pS2a1,a2∧pS1⋅pS2b1,b2,qT1⋅qT2a1,a2,b1,b2=qS2a2∨qS2b2∨qT1a1,b1≤pS2a2∧pS2b2∧pS1a1∧pS1b1≥qS2a2∨qS2b2∨qS1a1∨qS1b1=qS1a1∨qS2a2∨qS1b1∨qS2b2=qS1⋅qS2a1,a2∨qS1⋅qS2b1,b2,rT1⋅rT2a1,a2,b1,b2=rS2a2∨rS2b2∨rT1a1,b1≤pS2a2∧pS2b2∧pS1a1∧pS1b1≥rS2a2∨rS2b2∨rS1a1∨rS1b1=rS1a1∨rS2a2∨rS1b1∨rS2b2=rS1⋅rS2a1,a2∨rS1⋅rS2b1,b2.
The proves thatG1G2 is a PFG.Definition 8.
LetG1andG2 be two PFGs; then, the union of G1 and G2 is defined by G=G1∪G2=S1∪S2,T1∪T2, where(i)
(6)pS1∪pS2a=pS1a,ifa∈V1∩V2c,pS2a,ifa∈V2∩V1c,pS1a∨pS2a,ifa∈V1∩V2.(ii)
(7)qS1∪qS2a=qS1a,ifa∈V1∩V2c,qS2a,ifa∈V2∩V1c,qS1a∧qS2a,ifa∈V1∩V2.(iii)
(8)rS1∪rS2a=rS1a,ifa∈V1∩V2c,rS2a,ifa∈V2∩V1c,rS1a∧rS2a,ifa∈V1∩V2.(iv)
(9)pT1∪pT2a,b=pT1a,b,ifa,b∈E1∩E2c,pT2a,b,ifa,b∈E2∩E1c,pT1a,b∨pT2a,b,ifa,b∈E1∩E2.(v)
(10)qT1∪qT2a,b=qT1a,b,ifa,b∈E1∩E2c,qT2a,b,ifa,b∈E2∩E1c,qT1a,b∧qT2a,b,ifa,b∈E1∩E2.(vi)
(11)rT1∪rT2a,b=rT1a,b,ifa,b∈E1∩E2c,rT2a,b,ifa,b∈E2∩E1c,rT1a,b∧rT2a,b,ifa,b∈E1∩E2.Theorem 3.
LetG1andG2 be two PFGs; then, G1∪G2 is also a PFG.Proof.
Ifa,b∈E1∩E2, then we obtain(12)pT1∪pT2a,b=pT1a,b∨pT2a,b≤pS1a∧pS1b∨pS2a∧pS2b=pS1a∨pS2a∧pS1b∨pS2b=pS1∪pS2a∧pS1∪pS2b,qT1∪qT2a,b=qT1a,b∧qT2a,b≥qS1a∨qS1b∧qS2a∨qS2b=qS1a∧qS2a∨qS1b∧qS2b=qS1∪qS2a∧qS1∪qS2b,rT1∪rT2a,b=rT1a,b∧rT2a,b≥rS1a∨rS1b∧rS2a∨rS2b=rS1a∧rS2a∨rS1b∧rS2b=rS1∪rS2a∧rS1∪rS2b.
Ifa,b∈E1∩E2c, then(13)pT1∪pT2a,b=pT1a,b≤pS1a∧pS1b=pS1∪pS2a∧pS1∪pS2b,qT1∪qT2a,b=qT1a,b≥qS1a∨qS1b=qS1∪qS2a∨qS1∪qS2b,rT1∪rT2a,b=rT1a,b≥rS1a∨rS1b=rS1∪rS2a∨rS1∪rS2b.
Ifa,b∈E2∩E1c, then we have(14)pT1∪pT2a,b=pT2a,b≤pS2a∧pS2b=pS1∪pS2a∧pS1∪pS2b,qT1∪qT2a,b=qT2a,b≥qS2a∨qS2b=qS1∪qS2a∨qS1∪qS2b,rT1∪rT2a,b=rT2a,b≥rS2a∨rS2b=rS1∪rS2a∨rS1∪rS2b.
This shows thatG1∪G2 is a PFG.Corollary 1.
LetGλ:λ∈Λ be a family of PFGs; then, ∪λ∈ΛGλ is a PFG.Definition 9.
LetG1andG2 be two PFGs; then, the intersection of G1 and G2 is defined by G=G1∩G2=S1∩S2,T1∩T2, where(i)
(15)pS1∩pS2a=pS1a∧pS2a,a∈V1∩V2,qS1∩qS2a=qS1a∨qS2a,a∈V1∩V2,rS1∩rS2a=rS1a∨rS2a,a∈V1∩V2.(ii)
(16)pS1∩pS2a,b=pT1a,b∧pT2a,b,a,b∈E1∩E2,qT1∩qT2a,b=qT1a,b∨qT2a,b,a,b∈E1∩E2,rT1∩rT2a,b=rT1a,b∨rT2a,b,a,b∈E1∩E2.Theorem 4.
LetG1andG2 be two PFGs; then, G1∩G2 is also a PFG.Proof.
Foru,v∈V, we obtain(17)pT1∩pT2a,b=pT1a,b∧pT2a,b≤pS1a∧pS1b∧pS2a∧pS2b=pS1a∧pS2a∧pS1b∧pS2b=pS1∩pS2a∧pS1∩pS2b,qT1∩qT2a,b=qT1a,b∨qT2a,b≥qS1a∨qS1b∨pS2a∨qS2b=qS1a∨qS2a∨qS1b∨qS2b=qS1∩qS2a∨qS1∩qS2b,rT1∩rT2a,b=rT1a,b∨rT2a,b≥rS1a∨rS1b∨rS2a∨rS2b=rS1a∨rS2a∨rS1b∨rS2b=rS1∩rS2a∨rS1∩rS2b.
This shows thatG1∩G2 is a PFG.Corollary 2.
LetGλ:λ∈Λ be a family of PFGs; then ∩λ∈ΛGλ is a PFG.Definition 10.
LetG1andG2 be two PFGs; then, the sum of G1 and G2 is defined by G=G1+G2=S1+S2,T1+T2, where(i)
(18)pS1+pS2a=pS1∪pS2a,qS1+qS2a=qS1∩qS2a,rS1+rS2a=rS1∩rS2a,ifa∈V1∪V2.(ii)
(19)pT1+pT2a,b=pT1∪pT2a,b=pT1a,b,qT1+qT2a,b=qT1∩qT2a,b=qT1a,b,rT1+rT2a,b=rT1∩rT2a,b=rT1a,b,ifa,b∈E1∩E2.(iii)
(20)pT1+pT2a,b=pS1a∨pS2a,qT1+qT2a,b=qS1a∧qS2a,rT1+rT2a,b=rS1a∧rS2aifa,b∈E′,whereE′is the set of edges connecting the nodes ofV1.Theorem 5.
LetG1 and G2 be two PFGs′; then, G1+G2 is also a PFG.


## 4. Picture Fuzzy Graph-Based Multiple Attribute Decision-Making

PFS is an important tool to solve real-world problems. PFS deals with inconsistent, incomplete, and indeterminate information or fact. Nowadays, PFS has become an exciting topic for its wide applications. So, PFG can efficiently solve such type of real-world problem.Here, the concept of the graph is applied toMADMP with a picture fuzzy environment, and we proposed two algorithms. Also, to illustrate our proposed decision-making algorithm, we have been given two examples. Let A=A1,A2,A2,…,Am be an arrangement of alternatives and C=C1,C2,C3,…,Cn be the arrangement of attribute. w=w1,w2,w3,…,wn be the weight vector of the attribute Ci, i=1,2,…,n, where wi≥0, for i=1,2,…,n, and ∑i=1nwi=1.LetM=bkjm×n=ppkj,qkj,rkjm×n be a picture fuzzy decision matrix, where pkj is the positive membership degree for which alternative Aj satisfies the attribute Cj, which was given by the decision makers, qkj is the neutral membership degree so that alternative Ak does not satisfies the attribute Cj, and rkj is the degree that the alternatives Ak does not fulfill the attribute Cj which was given by the decision maker, where pkj∈0,1, qkj∈0,1, rkj∈0,1, and 0≤pkj+qkj+rkj≤1, k=1,2,…,m. The picture fuzzy relation between two attributes Ci=pi,qi,ri and Cj=pj,qj,rj is defined by fij=pij,qij,rij, where pij≤pi∧pj, qij≥qi∨qj, and rij≥ri∨rj, i,j=1,2,…,m, otherwise, fij=0,0,1.We proposed two algorithms to develop the graph structure and solve multiattribute decision-making (MADM) problems using PFG (Algorithms 1 and 2).Algorithm 1: Computation of best alternative.
Step 1: calculate the impact coefficient between the attributesCi and Cj by ηij=pij+1−qij1−rij/3 for i,j=1,2,…,n, where ηij=pij,qij,rij is the picture fuzzy edge between the nodes Ci and Cj, for i,j=1,2,…,n. We have ηij=1 and ηij=ηji if i=j.Step 2: find the attribute of the alternativeAk by Ak˜=pk˜,qk˜,rk˜=1/3∑j=1nwj∑s=1nηsjbks, where fsj=psj,qsj,rsj.Step 3: calculate the score function of the alternativeAk˜ by scorAk˜=1/21+pk˜−2qk˜−rk˜.Step 4: rank all the alternativeAk depending on scorAk˜ and then select the best alternative.Step 5: stop.LetA=pj,qj,rj be a decision solution, for j=1,2,…,n. Now, we develop an algorithm that is based on PFG and the similarity measure between picture fuzzy numbers. Here, the main advantage is that it can compute the relationship among multiple-input arguments through the graph theory approach.Algorithm 2: Computation of best alternative using similarity measure.
Step 1: calculate the impact coefficient between the attributesCi and Cj by ηij=pij+1−qij1−rij/3 for i,j=1,2,3,…,n, where ηij=pij,qij,rij is the picture fuzzy edge between the nodes Ci and Cj for i,j=1,2,…,n. We have ηij=1 and ηij=ηji if i=j.Step 2: compute the associated weighted value of attributeCj, for j=1,2,3,…,n, over the other criteria by bkj˜=pkj˜,qkj˜,rkj˜=1/3wj∑s=1nηsjbks.Step 3: find the similarity measure between the decision solutionA=pj,qj,rj, j=1,2,3,…,n, and every alternative Ak, k=1,2,3,…,m, by scorA,Ak=1−1/3n∑j=1npj−pkj˜+qj−qkj˜+rj−rkj˜.Step 4: rank all the alternativeAk according to scorA,Ak, k=1,2,3,…,m.Step 5: stop.


## 5. Numerical Example

In this part, numerical examples for thePFGMADM problem with picture fuzzy information are used to present the application of the proposed algorithms. Here, we consider a MADMP taken from S. Ashraf et al. [46].Example 1.
An investment company wants to invest money in the best choice. There are four measurable alternatives:A1: a car companyA2: a food companyA3: a computer companyA4: an arms company
The investment company makes a decision based on the three attributes:C1: the risk analysisC2: the growth analysisC3: the environmental impact analysis
The growth vector of the attribute is given byw=0.35,0.25,0.40.
The four possible alternatives are to be measured under the three attributes and are given in the form of picture fuzzy information by decision-making according to three attributesC1,C2,andC3 and the evaluation information on the alternative A1,A2,A3,andA4 under the factors C1,C2,andC3 can be shown in the following picture fuzzy decision matrix M:(21)M=0.6,0.2,0.20.8,0.1,0.10.6,0.1,0.30.5,0.3,0.20.5,0.2,0.30.8,0.1,0.10.4,0.2,0.40.6,0.3,0.10.4,0.2,0.40.3,0.1,0.60.7,0.1,0.20.7,0.1,0.2.
Also, we assume that the relationship among the attributeC1,C2,andC3 can be described by a complete graph G=V,E, where V=C1,C2,C3 and E=C1,C2,C1,C3,C2,C3, see Figure 1.
From equation (1), we get all the impact coefficient to find out the relationships among the attribute. Now, the picture fuzzy edges denoting the connections among the attributes are described as(22)f12=p12,q12,r12=0.3,0.4,0.5,f13=p13,q13,r13=0.3,0.3,0.4,f23=p23,q23,r23=0.2,0.4,0.4.
Notice that hereG=V,E is a PFG according to the relationship among the attribute for every alternatives. To find the best alternatives, we perform the following steps:Step 1: the impact coefficient between the attributeCj, j=1,2,3, are as follows:(23)η12=p12+1−q121−r123=0.3+1−0.41−0.43=0.22,η13=p13+1−q131−r133=0.3+1−0.31−0.43=0.24,η23=p23+1−q231−r233=0.2+1−0.41−0.43=0.187.Step 2: the attribute of the alternativeA1 is calculated below:(24)A1˜=13w1η11b11+η21b12+η31b13+w2η12b11+η22b12+η32b13+w3η13b11+η23b12+η33b13=130.351×0.6,0.2,0.2+0.22×0.8,0.1,0.1+0.24×0.6,0.1,0.3+0.250.22×0.6,0.2,0.2+1×0.8,0.1,0.1+0.187×0.6,0.1,0.3+0.40.24×0.6,0.2,0.2+0.187×0.8,0.1,0.1+1×0.6,0.1,0.3=0.314,0.065,0.1,A2˜=13w1η11b21+η21b22+η31b23+w2η12b21+η22b22+η32b23+w3η13b21+η23b22+η33b23=130.351×0.5,0.3,0.2+0.22×0.5,0.2,0.3+0.24×0.8,0.1,0.1+0.250.22×0.5,0.3,0.2+1×0.5,0.2,0.3+0.187×0.8,0.1,0.1+0.40.24×0.5,0.3,0.2+0.187×0.5,0.2,0.3+1×0.8,0.1,0.1=0.292,0.088,0.078,A3˜=13w1η11b31+η21b32+η31b33+w2η12b31+η22b32+η32b23+w3η13b31+η23b32+η33b33=130.351×0.4,0.2,0.4+0.22×0.6,0.3,0.1+0.24×0.4,0.2,0.4+0.250.22×0.4,0.2,0.4+1×0.6,0.3,0.1+0.187×0.4,0.2,0.4+0.40.24×0.4,0.2,0.4+0.187×0.6,0.3,0.1+1×0.4,0.2,0.4=0.218,0.109,0.151,A4˜=13w1η11b41+η21b42+η31b43+w2η12b41+η22b42+η32b43+w3η13b41+η23b42+η33b43=130.351×0.3,0.1,0.6+0.22×0.7,0.1,0.2+0.24×0.7,0.1,0.2+0.250.22×0.3,0.1,0.6+1×0.7,0.1,0.2+0.187×0.7,0.1,0.2+0.40.24×0.3,0.1,0.6+0.187×0.7,0.1,0.2+1×0.7,0.1,0.2=0.268,0.048,0.162.Step 3: now, we compute the score functions as follows:(25)scorA1˜=121+p1˜−2q1˜−r1˜=121+0.314−2×0.065−0.1=0.542,scorA2˜=121+p2˜−2q2˜−r2˜=121+0.292−2×0.088−0.078=0.519,scorA3˜=121+p3˜−2q3˜−r3˜=121+0.218−2×0.109−0.151=0.425,scorA4˜=121+p4˜−2q4˜−r4˜=121+0.264−2×0.048−0.162=0.505.Step 4: therefore, we rank these alternatives asA1>A2>A4>A3. From the above numerical observation we have, A1 is the best choice in the decision-making problem.Figure 1
The graph relationship among the three attributes.Example 2.
In this example, we consider medical diagnosis problem adapted from Ye [47]. Let us consider a set of diagnosis as A=A1viral fever,A2malaria,A3typhoid,A4gastritis,A5stenocardia and set of symptoms as C=C1temperature,C2headache,C3stomach pain,C4cough,C5sttenocardia
Let the weight vector of the symptoms bew=0.25,0.15,0.10,0.20,0.30. Also, the performance values of the considered diseases are characterized by PFS, and results are shown in Table 1.
Suppose a patientA having all symptoms is represented by the following picture fuzzy information:(26)Apatient=C1,0.8,0.1,0.1,C2,0.6,0.2,0.1,C3,0.2,0.0,0.8,C4,0.6,0.1,0.1,C5,0.1,0.1,0.6
Let the picture fuzzy edges denote the connection among the symptoms (see Figure2), which is described as f12=0.2,0.3,0.1, f13=0.1,0.1,0.8, f14=0.3,0.4,0.2, f15=0.1,0.2,0.6, f23=0.0,0.2,0.8, f24=0.4,0.3,0.1, f25=0.1,0.2,0.6, f34=0.1,0.1,0.8, f35=0.1,0.1,0.8, and f45=0.1,0.2,0.6. The impact coefficient between symptoms is calculated below:(27)η12=13p12+1−q121−r12=130.2+1−0.31−0.1=0.277,η13=13p13+1−q131−r13=130.1+1−0.11−0.8=0.273,η14=13p14+1−q141−r14=130.3+1−0.41−0.2=0.260.
Similarly,η15=0.140, η23=0.053, η24=0.343, η25=0.140, η34=0.093, η35=0.280, and η45=0.140.
Now, the associated weighted values of disease are obtained bybkj˜=wj/3∑s=15ηsjbks, where bkj˜=pkj˜,qkj˜,rkj˜ is a PFN.
Therefore,(28)b11˜=w13∑s=15ηs1b1s=w13η11b11+η21b12+η31b13+η41b14+η51b15=0.2531×0.4,0.3,0.0+0.277×0.3,0.1,0.5+0.273×0.1,0.2,0.7+0.260×0.4,0.2,0.3+0.140×0.1,0.1,0.7=0.052,0.037,0.042,b12˜=w23∑s=15ηs2b1s=w23η12b11+η22b12+η32b13+η42b14+η52b15=0.1530.277×0.4,0.3,0.0+1×0.3,0.1,0.5+0.053×0.1,0.2,0.7+0.343×0.4,0.2,0.3+0.140×0.1,0.1,0.7=0.028,0.014,0.037,b13˜=w13∑s=15ηs3b1s=w33η13b11+η23b12+η33b13+η43b14+η53b15=0.1030.273×0.4,0.3,0.0+0.053×0.3,0.1,0.5+1×0.1,0.2,0.7+0.093×0.4,0.2,0.3+0.280×0.1,0.1,0.7=0.010,0.011,0.031,b14˜=w43∑s=15ηs4b1s=w43η14b11+η24b12+η34b13+η44b14+η54b15=0.2030.26×0.4,0.3,0.0+0.034×0.3,0.1,0.5+0.093×0.1,0.2,0.7+1×0.4,0.2,0.3+0.140×0.1,0.1,0.7=0.042,0.023,0.042,b15˜=w53∑s=15ηs5b1s=w53η15b11+η25b12+η35b13+η45b14+η55b15=0.3030.140×0.4,0.3,0.0+0.140×0.3,0.1,0.5+0.280×0.1,0.2,0.7+0.140×0.4,0.2,0.3+1×0.1,0.1,0.7=0.028,0.024,0.101,b21˜=w13∑s=15ηs1b2s=w13η11b21+η21b22+η31b23+η41b24+η51b25=0.2531×0.7,0.1,0.0+0.277×0.2,0.1,0.6+0.273×0.0,0.1,0.9+0.260×0.7,0.2,0.0+0.140×0.1,0.0,0.8=0.079,0.016,0.038,b22˜=w23∑s=15ηs2b2s=w23η12b21+η22b22+η32b23+η42b24+η52b25=0.1530.277×0.7,0.1,0.0+1×0.2,0.1,0.6+0.053×0.0,0.1,0.9+0.343×0.7,0.2,0.0+0.140×0.1,0.0,0.8=0.032,0.010,0.038.
Similarly,b23˜=0.010,0.005,0.039, b24˜=0.062,0.027,0.027, and b25˜=0.032,0.008,0.114.
Again,(29)b31˜=w13∑s=15ηs1b3s=w13η11b31+η21b32+η31b33+η41b34+η51b35=0.2531×0.3,0.2,0.3+0.277×0.6,0.2,0.1+0.273×0.2,0.1,0.7+0.260×0.2,0.1,0.6+0.140×0.1,0.0,0.9=0.049,0.026,0.067.
Similarly,b32˜=0.039,0.015,0.028, b33˜=0.012,0.006,0.037, b34˜=0.034,0.015,0.060, and b35˜=0.031,0.010,0.124
Also,(30)b41˜=w13∑s=15ηs1b4s=w13η11b41+η21b42+η31b43+η41b44+η51b45=0.2531×0.1,0.1,0.7+0.277×0.2,0.3,0.4+0.273×0.8,0.1,0.0+0.260×0.2,0.1,0.7+0.140×0.2,0.1,0.7=0.038,0.021,0.091.
Similarly,b42˜=0.018,0.019,0.047, b43˜=0.030,0.006,0.016, b44˜=0.026,0.017,0.074, and b45˜=0.049,0.020,0.095
Finally,(31)b51˜=w13∑s=15ηs1b5s=w13η11b51+η21b52+η31b53+η41b54+η51b55=0.2531×0.1,0.1,0.8+0.277×0.0,0.1,0.8+0.273×0.2,0.0,0.8+0.260×0.2,0.0,0.8+0.140×0.8,0.1,0.1=0.027,0.012,0.0122.
Similarly,b52˜=0.011,0.007,0.068, b53˜=0.016,0.002,0.039, b54˜=0.024,0.005,0.091, and b55˜=0.090,0.013,0.066.
Therefore, the results obtained are shown in Table2:
The similarity measure between the ideal solutionA and each diseases Ak, k=1,2,3,4,5, are calculated below:(32)scorA,A1=1−115p1−p11+q1−q11+r1−r11+p2−p12+q2−q12+r2−r12+p3−p13+q3−q13+r3−r13+p4−p14+q4−q14+r4−r14+p5−p15+q5−q15+r5−r15=1−1150.869+0.821+0.691+0.693+0.647=0.7519,scorA,A2=1−115p1−p21+q1−q21+r1−r21+p2−p22+q2−q22+r2−r22+p3−p23+q3−q23+r3−r23+p4−p24+q4−q24+r4−r24+p5−p25+q5−q25+r5−r25=1−1150.867+0.820+0.956+0.684+0.646=0.7351,scorA,A3=d1−115p1−p31+q1−q31+r1−r31+p2−p32+q2−q32+r2−r32+p3−p33+q3−q33+r3−r33+p4−p34+q4−q34+r4−r34+p5−p35+q5−q35+r5−r35=1−1150.858+0.818+0.957+0.691+0.635=0.7361,scorA,A4=1−115p1−p41+q1−q41+r1−r41+p2−p42+q2−q42+r2−r42+p3−p43+q3−q43+r3−r43+p4−p44+q4−q44+r4−r44+p5−p45+q5−q45+r5−r45=1−1150.850+0.816+0.960+0.717+0.636=0.7347,scorA,A5=1−115p1−p51+q1−q51+r1−r51+p2−p52+q2−q52+r2−r52+p3−p53+q3−q53+r3−r53+p4−p54+q4−q54+r4−r54+p5−p55+q5−q55+r5−r55=1−1150.883+0.815+0.947+0.680+0.631=0.7363.a
Therefore,scorA,A1>scorA,A5>scorA,A3>scorA,A2>scorA,A4. The rank of the attributes are A1>A5>A3>A2>A4.
Thus, the patientA can be diagnosed with the diseases A1viral fever according to the recognition principle. The ranking is the same as J.Ye [2011]. The above example indicates that this type of decision-making algorithm is well suitable for picture fuzzy environment and is a useful technique that provides a different respective than others for picture fuzzy environment.Table 1
Performance values of the disease.
C1temperatureC2headacheC3stomach painC4(cough)C5chestpainA1viral fever0.052,0.037,0.0420.028,0.014,0.0370.010,0.011,0.3100.042,0.023,0.0420.028,0.024,0.101A2malaria0.079,0.016,0.0380.032,0.010,0.0380.010,0.005,0.0390.062,0.027,0.0270.032,0.008,0.114A3typhoid0.049,0.026,0.0670.039,0.015,0.0280.012,0.006,0.0370.034,0.015,0.0600.031,0.010,0.124A4gastritis0.038,0.021,0.0910.018,0.019,0.0470.030,0.006,0.0160.026,0.017,0.0740.049,0.020,0.095A5stenocardia0.027,0.012,0.1220.011,0.007,0.0680.016,0.002,0.0390.024,0.005,0.0910.090,0.013,0.066Figure 2
The graph relationship among the five attributes.Table 2
The associated weighted values of the disease.
C1C2C3C4C5A10.052,0.037,0.0420.028,0.014,0.0370.010,0.011,0.3100.042,0.023,0.0420.028,0.024,0.101A20.079,0.016,0.0380.032,0.010,0.0380.010,0.005,0.0390.062,0.027,0.0270.032,0.008,0.114A30.049,0.026,0.0670.039,0.015,0.0280.012,0.006,0.0370.034,0.015,0.0600.031,0.010,0.124A40.038,0.021,0.0910.018,0.019,0.0470.030,0.006,0.0160.026,0.017,0.0740.049,0.020,0.095A50.027,0.012,0.1220.011,0.007,0.0680.016,0.002,0.0390.024,0.005,0.0910.090,0.013,0.066


## 6. Conclusion and Future Directions

Graph theory is a needful tool for solvingMADMP in different areas. PFG is a new dimension of graph theory which is a useful tool for solving real-world problems. Most of MADM algorithms with picture fuzzy environment discuss a type of problem with no relationship among attributes. Although this relationship should be considered in the actual applications, so we need to pay attention to that issue. This article applies graph theory to PFS and obtained a new method for solving complicated problems under picture fuzzy information. The proposed method can capture the relationship among the attributes that cannot be handled well by any available methods. In this study, we introduce union, intersection, sum, Cartesian product, and the composition of PFG. Finally, by considering the importance of relationships among attributes in the decision process, two new techniques based on single-valued PFG were developed to solve complicated problems using picture fuzzy information. Also, two numerical examples were presented to explain how to deal with the MADMP under a picture fuzzy environment. In the future, we can solve this type of MADM problem using soft sets, picture fuzzy hesitant fuzzy sets, and spherical and T-spherical fuzzy sets.



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