Pythagorean Fuzzy Digraphs and Its Application in Healthcare Center

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Introduction
Fuzzy set is unlike ordinary set whose elements of a set have membership value. It is found by Zadeh [1] in the year 1965. Graph theory is a mathematical tool to solve network problem and study the relation between objects (node). Rosenfeld [2] explained the notion of graph theory under fuzzy environment. Later, researchers defined the various operations of fuzzy graph and types of fuzzy graph such as complement of a fuzzy graph and regular fuzzy graph [3,4].
Definition of IFS is a generalized fuzzy set whose elements of a set have both membership and nonmembership value and it is introduced by Atanassov [5][6][7]. e concepts of intuitionistic fuzzy relations (IFR) and intuitionistic fuzzy graph (IFG) are the generalizations of fuzzy graphs (FG) and it is developed by Shannon and Atanassov [8]. Operations on IFG and shortest path problem under intuitionistic fuzzy environment were developed by Parvathi et al. [9][10][11] and Karunambigai et al. [12]. Intuitionistic fuzzy graphs of nth type were developed by Davvaz et al. [13]. Intuitionistic fuzzy graphs of nth type are a generalization of intuitionistic fuzzy graph and intuitionistic fuzzy graph of second type. In research, several extensions of IFG can be seen in recent years [14][15][16][17][18][19][20][21][22][23][24].
However, the use of IFSs is identified in many fields; it has some limitations. Limitation of IFSs confines the truth and false membership value, that is, sum of truth and false membership does not exceed 1. Pythagorean fuzzy set is one of the extensions of intuitionistic fuzzy set. PyFS was developed to overcome the limitations in IFS and this theory is introduced by Yager [25][26][27]where a membership grade (μ) and a nonmembership grade (]) with the condition μ 2 + ] 2 ≤ 1. Pythagorean fuzzy number (PyFN) is developed by Zhang and Xu [28] to interpret the dual aspects of an element.
But Zhang and Xu theory failed to address the decisionmaking problem where the membership grade and the nonmembership grade are, respectively, 0.9 and 0.3; but 0.9 2 + 0.3 2 ≤ 1. PyFG was originally studied by Naz et al. [29] as a generalized notion of IFG and application of the proposed notion was also investigated. Akram et al. [30] recently studied the operations of PyFGs and properties of PyFGs. e concept of planar graph under Pythagorean fuzzy environment was developed by Akram et al. [31]. Notion of maximal product of two PyFGs, residue product of two PyFGs, and its properties have been introduced and studied by Akram et al. [32]. He et al. [33] combined Pythagorean 2-tuple linguistic fuzzy set and QUALIFLEX method. is combination is used to evaluate the full quality of operation personnel in engineering field. Zhang et al. [17] combined the novel TODIM along with cumulative prospect theory under 2-tuple linguistic Pythagorean fuzzy sets (2-TLPFS) and also basic definitions and operators of 2-TLPFSs introduced by Zhang et al. [34]. Li et al. [35] proposed a new similarity measure under Pythagorean fuzzy environment and investigated multiple criteria group decision-making problem to prove the feasibility of the proposed method. e objective of our work in this paper is to introduce some operations on Pythagorean fuzzy digraph, decisionmaking algorithm for solving problem using the notion of Pythagorean fuzzy digraph. Finally, we explore the proposed algorithm with a real-life example. is paper may motivate to study various real-life problems using the proposed algorithm.
Basic definitions which are used in our work are presented in Section 2. Definition of Pythagorean fuzzy diagraph and its operations are presented in Section 3. Algorithm for solving decision-making problem using the proposed concept is developed in Section 4. Also, a decisionmaking problem is considered and solved using the developed algorithm in Section 5. Comparative study is presented in Section 6. In Section 7, conclusion is presented and also discussed the future work.

Basic Definitions
is section contributes to present the basic definitions of [1,2,10,28,30] which we used in our work. roughout the paper, let U be the universal set.
An object A � (a, μ A (a)): a ∈ X is called FS over the universe X, where the mapping μ A : X ⟶ [0, 1] is called the membership function of A for each element a ∈ X.
Let V be a nonempty vertex set. Fuzzy graph is denoted by G � (P, E) where the mappings of a fuzzy set P : V ⟶ [0, 1] on V and the relation E : e notation ∧ represents the minimum operator.
Definition 3 (see [5]). An object A � (a, μ A (a), η A (a)): a ∈ X is called IFS over the universe X, where the mapping μ A : X ⟶ [0, 1] and η A : X ⟶ [0, 1] are called the membership function and nonmembership function of A for each element a ∈ X.
Pythagorean fuzzy set is an order pair (μ p (e), η P (e)) whose first element is positive grade value and second element is negative grade value for each element e in U. Positive and negative grade mapping, respectively, is called the refusal membership.
Let V be a nonempty vertex set. PyFG is denoted by G � (P, E) where the mappings of Pythagorean fuzzy set e notation ∧ represents the minimum operator; ∨ represents the maximum operator.

Pythagorean Fuzzy Directed Graphs
Pythagorean fuzzy digraph is defined in this section and also operations of Pythagorean fuzzy number, score function is presented in this section.
Remark 1. As the name implies, PyFDG does not hold a symmetric relation on V, like a PyFG holding on V.
e operations PyFN are defined as follows: Definition 8. Let P � (μ, η) be PyFN. Its score function and accuracy function can be derived by using the following formulas: Let P 1 � (μ 1 , η 1 ), P 2 � (μ 2 , η 2 ) be two PyFNs. Comparisons of two PyFNs are defined as follows: Journal of Mathematics (1) If score function of P 1 is less than the score function P 2 , then P 1 < P 2 (2) If score function of P 1 is equal to the score function P 2 , then P 1 � P 2 (a) If accuracy function of P 1 is less than the accuracy function P 2 , then P 1 < P 2 (b) If accuracy function of P 1 is equal to the accuracy function P 2 , then P 1 � P 2

Algorithm for Pythagorean Fuzzy Directed Graphs
Shortest path and its length of a graph were calculated using many algorithms such as Dijkstra's algorithm and Bellman-Ford algorithm. We proposed a new algorithm for calculating shortest path from node i to node j and its length of a PyFDG. Vertex and edges of a PyFDG are assumed as a PyFN. e proposed algorithm in this section motivates to investigate the real-life problem.
� source node, 2, 3, . . .n � terminus node} and edge set which connect two vertices by an arrow V � {1, 2, 3,. . .n}. e path of the PyFDG is denoted by P ij and it is defined as P ij � weight of the arc which connects node i to node j . e existence of a minimum of one path P 1i in the digraph is supposed for every i in V-{1}. Pythagorean fuzzy distance along the path is defined as d(p) � d ij .

Algorithm.
Step 1: label the source node as p 1 � (0, 1) Step 2: compute Pythagorean fuzzy value p j , where p j is the minimum of direct sum of p i and p ij for j � 2, 3,. . .,n Step 3: identify the minimum p j in step 2 and then label [p j , i] if p j is reached from node i Step 4: find the shortest Pythagorean fuzzy path from source node to j � 2, 3, 4,. . .,n by combining the label [p j , i] calculated in step 3 and its corresponding p i Step 5: choose the path from source node to destination node and its corresponding p i is the shortest Pythagorean fuzzy length

Application of Pythagorean Fuzzy Digraph
Healthcare center is a center for maintaining, improving, and helping individual's health through diagnosis and treatment by professionals. Healthcare centers are depending on medical professionals, psychiatrists, physiotherapists, dentists, and nurses. Healthcare centers are classified into four types, namely, primary healthcare center, secondary healthcare center, tertiary healthcare center, and quaternary healthcare center. Primary healthcare center is a first point of contact by all patients within the region. General practitioner provides treatment to the patients in this center. If the problem is serious then the practitioner in the primary healthcare center recommends visiting secondary healthcare center. Secondary healthcare center is found in emergency unit in the hospital. In this center, professionals provide treatment for severe injury and emergency medical condition and during child birth also. Secondary healthcare center service is for a short-period of time while the primary healthcare center service is for a day. Tertiary healthcare center provides advanced medical treatment. e patients admitted in this center are mostly referred by primary or secondary healthcare center. Professionals working in this center are specialists for cardiac, cancer, plastic surgery, neuro surgery, and more complex illnesses. Quaternary healthcare center is a national health center because these centers are found only in limited regions. is center provides advanced treatment compared to tertiary healthcare center. Medical practitioner in the primary healthcare center attends the patients; if the health issues of the patients are severe, then he will recommend visiting secondary healthcare center. Also, if the problem is in advanced level, then the medical practitioner recommends the patients to visit tertiary healthcare center. If the health issue of a patient became severe and is in advanced level, then the patient may visit quaternary healthcare center. ere are two possibilities to each patient who visits these healthcare centers either he/she gets recovered or is forwarded to next healthcare center. Positive membership refers to the patient if he/she recovers in that center, and negative membership refers to the patient if his/her health worsens in that center; and then the patient is forwarded to another center. e node 1 and 2 represent the primary healthcare centers in two regions. Node 3, 4, and 5 represent secondary healthcare centers. Node 6 and 7 represent the tertiary healthcare centers. Node 8 is the quaternary healthcare center. Patient in one region visits a particular center; if more patients visit a center, then the medical practitioner in  that region recommends visiting the another center. After diagnosing the patients, the practitioner recommends to visit another center depending on the severity of the issue. We constructed a Pythagorean fuzzy network for this case which is shown in Figure 2 and arc weight of this network is shown in Table 2. Presume that a patient visits center 1 with an ailment. e proposed algorithm shows the way to visit center 1 to center 8. e nodes in the above PyFDG (see Figure 2) are the healthcare centers and connection between the healthcare centers is denoted by edges whose weights are PyFN. Weight of each edge is given in Table 2.
We applied the proposed algorithm to find the shortest path from healthcare center 1 to center 8.
Healthcare center 1 is assumed as source node p 1 � (0, 1) and distance is labelled as p 1 � [(0, 1), 1] and center 8 is the terminus node. So, Pythagorean fuzzy value of p j , where j � 2, 3, 4, 5, 6, 7, 8, can be obtained in the following iterations.    Shortest path of PyFDG is obtained by working backward from healthcare center 8 and including the permanently labelled healthcare centers from which the subsequent label arose. e shortest path of PyFDG is 1⟶3⟶5⟶8, with the length (0.88944, 0.064).
SP from the healthcare center 1 to healthcare center j given in Table 3 and the thick lines in PyFDG indicate the SP from the healthcare center 1 to healthcare center 8 shown in Figure 3.

Comparative Analysis
Advantages and limitations of existing digraph and also Pythagorean digraph are shown in Table 4.

Conclusion
Pythagorean fuzzy set has been applied in many fields to deal with uncertainty. is set has been applied in graph structure to find the shortest path. But Pythagorean fuzzy set is not discussed for digraph. So, Pythagorean fuzzy digraph is defined and operations on PyFDG are studied. Crisp values are Pythagorean fuzzified for calculation and score function is used for Pythagorean defuzzification. A real-life problem is investigated with the help of the proposed algorithm. e advantage of this work is to handle imprecise edge weight when sum of membership and nonmembership of an edge exceeds one. Future work will be investigating the various complex problems using Pythagorean fuzzy diagraph.

Data Availability
No data were used to support this study.

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

Type of digraph Advantages Limitations
Classical digraph [33] is is applicable when arc weights are precise is method is not applicable when arc weights are imprecise Fuzzy digraph [34] is concept can be applied for imprecise arc weights Membership degree in an arc is discussed but the nonmembership degree in the same arc is discussed Intuitionistic fuzzy digraph [35] is notion can be applied to the imprecise edge weight involving membership and nonmembership degree is concept fails when sum of membership and nonmembership degree of an edge weight exceeds 1 Pythagorean fuzzy digraph (proposed model) is environment can deal with imprecise edge weight when sum of membership and nonmembership degree of an edge weight exceeds 1 Hesitancy degree of an edge weight is not discussed in this concept Journal of Mathematics 5