Multiple Attribute Decision-Making Problem Using Picture Fuzzy Graph

Department of Mathematics, Raghunathpur College, Raghunathpur 723101, India Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia Department of Mathematics and Statistic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India


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]. e picture fuzzy model gives more precision, flexibility, and compatibility than the intuitionistic fuzzy model. e concept of PFS 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. ese 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].
ey 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], decisionmaking 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. ey also have studied policy decision-making based on some averaging aggregation operators of T-spherical fuzzy sets [32]. e 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][40][41][42][43][44][45]).

Motivation.
Most of MADM 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. is article applies graph theory to PFS and obtained a new method, MADM, to solve complicated problems under a picture fuzzy environment. e 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. e 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 decisionmaking algorithm. Section 6 is for the brief conclusion.

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 � (p S , q S , r S ) and T � (p T , q T , r T ) be two PFSs. en, the union and the intersection of the PFSs S and T are defined by A picture fuzzy number is defined by f n � (α, β, c).
Definition 2 (see [4]). Let f n � (α, β, c) be a picture fuzzy number; then, the score function of f n is denoted by scor(f n ) and is defined by Observation 1. Let f 1 and f 2 be two picture fuzzy numbers; then, scor(f 1 ) > scor(f 2 )⇒f 1 ⇒f 2 .

Definition 3 (see [4]). A PF relation ρ in a universe
Definition 4 (see [4]). Let S � (p S , q S , r S ) and T � (p T , q T , r T ) be two PFSs on a set X. If S be a PFR on X, then S is also a PFR on

Picture Fuzzy Graph
In this section, the PFG and some properties and theorems of PFG have been described.
Here, S is the picture fuzzy node set of G and T is a picture fuzzy edge set on G. Also, p S (a), q S (a), and r S (a), respectively, denote the positive, neutral, and negative membership degree of the node a and p T (a, b), q T (a, b), and r T (a, b) denote that of edge (a, b). Now, we give some properties of PFG such as composition, Cartesian product, union, and intersection. Let Definition 6. Let G 1 and G 2 be two PFGs; then, the cartesian product G 1 × G 2 of G 1 and G 2 is defined by

Mathematical Problems in Engineering
e above results proves that G 1 × G 2 is a PFG. □ Definition 7. Let G 1 and G 2 be two PFGs; the composition of Theorem 2. Let G 1 and G 2 be two PFGs; then,

Mathematical Problems in Engineering
Again, let c ∈ V 2 and (a 1 , b 1 ) ∈ E 1 . en, we obtain Mathematical Problems in Engineering 5 e proves that G 1 [G 2 ] is a PFG.
□ Definition 8. Let G 1 and G 2 be two PFGs; then, the union of Theorem 3. Let G 1 and G 2 be two PFGs; then, is shows that G 1 ∪ G 2 is a PFG. □ Corollary 1. Let G λ : λ ∈ Λ be a family of PFGs; then, ∪ λ∈Λ G λ is a PFG. Definition 9. Let G 1 and G 2 be two PFGs; then, the intersection of G 1 and G 2 is defined by Theorem 4. Let G 1 and G 2 be two PFGs; then, G 1 ∩ G 2 is also a PFG.
Proof. For u, v ∈ V, we obtain is shows that G 1 ∩ G 2 is a PFG.
Definition 10. Let G 1 and G 2 be two PFGs; then, the sum of (ii) (iii) where E ′ is the set of edges connecting the nodes of V 1 .
Mathematical Problems in Engineering Theorem 5. Let G 1 and G 2 be two PFGs ′ ; then, G 1 + G 2 is also a PFG.

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 to MADMP 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 � A 1 , A 2 , A 2 , . . . , A m } be an arrangement of alternatives and C � C 1 , C 2 , C 3 , . . . , C n be the arrangement of attribute. w � w 1 , w 2 , w 3 , . . . , w n be the weight vector of the attribute C i , i � 1, 2, . . . , n, where w i ≥ 0, for i � 1, 2, . . . , n, and n i�1 w i � 1.
Let M � (b kj ) m×n � (p p kj ,q kj ,r kj ) m×n be a picture fuzzy decision matrix, where p kj is the positive membership degree for which alternative A j satisfies the attribute C j , which was given by the decision makers, q kj is the neutral membership degree so that alternative A k does not satisfies the attribute C j , and r kj is the degree that the alternatives A k does not fulfill the attribute C j which was given by the decision maker, where p kj ∈ [0, 1], q kj ∈ [0, 1], r kj ∈ [0, 1], and 0 ≤ p kj + q kj + r kj ≤ 1, k � 1, 2, . . . , m. e picture fuzzy relation between two attributes C i � (p i , q i , r i ) and C j � (p j , q j , r j ) is defined by f ij � (p ij , q ij , r ij ), where p ij ≤ p i ∧p j , q ij ≥ q i ∨q j , and r ij ≥ r i ∨r j , i, j � 1, 2, . . . , m, otherwise, f ij � (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).
Let A � (p j , q j , r j ) 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.

Numerical Example
In this part, numerical examples for the PFGMADM 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].
Also, we assume that the relationship among the attribute C 1 , C 2 , and C 3 can be described by a complete graph 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)
Notice that here G � (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 attribute C j , j � 1, 2, 3, are as follows: Mathematical Problems in Engineering Step 2: the attribute of the alternative A 1 is calculated below: Step 1: calculate the impact coefficient between the attributes C i and C j by η ij � ((p ij + (1 − q ij )(1 − r ij ))/3) for i, j � 1, 2, . . . , n, where η ij � (p ij , q ij , r ij ) is the picture fuzzy edge between the nodes C i and C j , for i, j � 1, 2, . . . , n. We have η ij � 1 and η ij � η ji if i � j.
Step 2: find the attribute of the alternative A k by A k � (p k , q k , r k ) � (1/3) n j�1 w j ( n s�1 η sj b ks ), where f sj � (p sj , q sj , r sj ).
Step 4: rank all the alternative A k depending on scor(A k ) and then select the best alternative.
Step 5: stop. ALGORITHM 2: Computation of best alternative using similarity measure.
Step 3: now, we compute the score functions as follows: Step 4: therefore, we rank these alternatives as A 1 > A 2 > A 4 > A 3 . From the above numerical observation we have, A 1 is the best choice in the decision-making problem.
Example 2. In this example, we consider medical diagnosis problem adapted from Ye [47]. Let us consider a set of diagnosis as A � A 1 (viral fever), A 2 (malaria), A 3 (typhoid), A 4 (gastritis), A 5 (stenocardia)} and set of symptoms as C � C 1 (temperature), C 2 (headache), C 3 (stomach pain), C 4 (cough), C 5 (sttenocardia)} Let the weight vector of the symptoms be w � (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.
Again,    erefore, the results obtained are shown in Table 2: e similarity measure between the ideal solution A and each diseases A k , k � 1, 2, 3, 4, 5, are calculated below: Mathematical Problems in Engineering 13  us, the patient A can be diagnosed with the diseases A 1 (viral fever) according to the recognition principle. e ranking is the same as J. Ye [2011]. e 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.

Conclusion and Future Directions
Graph theory is a needful tool for solving MADMP 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. is article applies graph theory to PFS and obtained a new method for solving complicated problems under picture fuzzy information. e 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.