Abstract
In a picture fuzzy environment, almost all multiple attribute decision-making () 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 () and obtained a new method, , 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 (s), and their important properties. Finally, by considering the importance of relationships among attributes in the determination process, two algorithms, based on , have developed to solve complicated problems using picture fuzzy information. Also, two numerical examples have introduced to explain how to deal with the problem under picture fuzzy environment.
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 is a generalization of intuitionistic fuzzy set () [2]. The picture fuzzy model gives more precision, flexibility, and compatibility than the intuitionistic fuzzy model.
The concept of was first introduced by Coung [3] in 2013. In addition to , Coung appended new components which determine the neutral membership degree. gives an element’s membership and nonmembership degree, while 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 . Basically, -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 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], step fuzzy competition graphs [9], fuzzy planar graphs [10,11], and fuzzy -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 [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 of 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 problem using picture fuzzy graph. This article applies graph theory to and obtained a new method, , 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, 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 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
is an extension of . Some definitions related to are presented below, which we have used later to develop the paper.
Definition 1. (see [4]). Let and be two s. Then, the union and the intersection of the s and are defined by(i)(ii)
A picture fuzzy number is defined by .
Definition 2. (see [4]). Let be a picture fuzzy number; then, the score function of is denoted by and is defined by .
Observation 1. Let and be two picture fuzzy numbers; then, .
Definition 3. (see [4]). A relation in a universe is a defined by , where , , , and , for all .
Definition 4. (see [4]). Let and be two s on a set . If be a on , then is also a on if , , and , for all .
3. Picture Fuzzy Graph
In this section, the and some properties and theorems of have been described.
Definition 5. A of a graph is a pair , where is a on and is a on such that , , , and .
Here, is the picture fuzzy node set of and is a picture fuzzy edge set on . Also, , respectively, denote the positive, neutral, and negative membership degree of the node and denote that of edge .
Now, we give some properties of such as composition, Cartesian product, union, and intersection.
Let and be two , where , , , and .
Definition 6. Let be two ; then, the cartesian product of and is defined by , where(i) .(ii) .(iii) .
Theorem 1. Let be two ; then, is a .
Proof. Let . Then, we obtainAgain, let and . Then, we obtainThe above results proves that is a .
Definition 7. Let and be two ; the composition of and is defined by , where(i) , .(ii) , .(iii) , .(iv) , , where .
Theorem 2. Let be two ; then, is a .
Proof. Let . Then, we obtainAgain, let and . Then, we obtainAgain, let , so . Then, we obtainThe proves that is a .
Definition 8. Let be two ; then, the union of and is defined by , where(i)(ii)(iii)(iv)(v)(vi)
Theorem 3. Let be two ; then, is also a .
Proof. If , then we obtainIf , thenIf , then we haveThis shows that is a .
Corollary 1. Let be a family of ; then, is a .
Definition 9. Let be two ; then, the intersection of and is defined by , where(i)(ii)
Theorem 4. Let be two ; then, is also a .
Proof. For , we obtainThis shows that is a .
Corollary 2. Let be a family of ; then is a .
Definition 10. Let be two ; then, the sum of and is defined by , where(i)(ii)(iii)
Theorem 5. Let and be two ; then, is also a .
4. Picture Fuzzy Graph-Based Multiple Attribute Decision-Making
is an important tool to solve real-world problems. deals with inconsistent, incomplete, and indeterminate information or fact. Nowadays, has become an exciting topic for its wide applications. So, can efficiently solve such type of real-world problem.
Here, the concept of the graph is applied to 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 be an arrangement of alternatives and be the arrangement of attribute. be the weight vector of the attribute , , where , for , and .
Let be a picture fuzzy decision matrix, where is the positive membership degree for which alternative satisfies the attribute , which was given by the decision makers, is the neutral membership degree so that alternative does not satisfies the attribute , and is the degree that the alternatives does not fulfill the attribute which was given by the decision maker, where , , , and , . The picture fuzzy relation between two attributes and is defined by , where , , and , , otherwise, .
We proposed two algorithms to develop the graph structure and solve multiattribute decision-making () problems using (Algorithms 1 and 2).
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Let be a decision solution, for . Now, we develop an algorithm that is based on 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.
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5. Numerical Example
In this part, numerical examples for the problem with picture fuzzy information are used to present the application of the proposed algorithms. Here, we consider a 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: : a car company : a food company : a computer company : an arms companyThe investment company makes a decision based on the three attributes: : the risk analysis : the growth analysis : the environmental impact analysisThe growth vector of the attribute is given by .
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 attributes and the evaluation information on the alternative under the factors can be shown in the following picture fuzzy decision matrix :Also, we assume that the relationship among the attribute can be described by a complete graph , where and , 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 asNotice that here is a 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 , , are as follows: Step 2: the attribute of the alternative is calculated below: Step 3: now, we compute the score functions as follows: Step 4: therefore, we rank these alternatives as . From the above numerical observation we have, 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 and set of symptoms as
Let the weight vector of the symptoms be . Also, the performance values of the considered diseases are characterized by , and results are shown in Table 1.
Suppose a patient having all symptoms is represented by the following picture fuzzy information:Let the picture fuzzy edges denote the connection among the symptoms (see Figure 2), which is described as , , , , , , , , , and . The impact coefficient between symptoms is calculated below:Similarly, , , , , , , and .
Now, the associated weighted values of disease are obtained by , where is a .
Therefore,Similarly, , , and .
Again,Similarly, , , , and
Also,Similarly, , , , and
Finally,Similarly, , , , and .
Therefore, the results obtained are shown in Table 2:
The similarity measure between the ideal solution and each diseases , , are calculated below:aTherefore, . The rank of the attributes are .Thus, the patient can be diagnosed with the diseases 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.
6. Conclusion and Future Directions
Graph theory is a needful tool for solving in different areas. is a new dimension of graph theory which is a useful tool for solving real-world problems. Most of 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 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 . Finally, by considering the importance of relationships among attributes in the decision process, two new techniques based on single-valued were developed to solve complicated problems using picture fuzzy information. Also, two numerical examples were presented to explain how to deal with the under a picture fuzzy environment. In the future, we can solve this type of 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
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.