A Novel Approach to Decision-Making with Pythagorean Fuzzy Information

Abstract: A Pythagorean fuzzy set (PFS) is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This paper proposes a new graph, called Pythagorean fuzzy graph (PFG). We investigate some properties of our proposed graphs. We determine the degree and total degree of a vertex of PFGs. Furthermore, we present the concept of Pythagorean fuzzy preference relations (PFPRs). In particular, we solve decision-making problems, including evaluation of hospitals, partner selection in supply chain management, and electronic learning main factors evaluation by using PFGs.


Introduction
Since Zadeh's seminal work [1], the classical logic has been extended to fuzzy logic, which is characterized by a membership function in [0, 1] and provides a powerful alternative to probability theory to characterize imprecision, uncertainty, and obscureness in various fields.Gradually, it has been discovered that sometimes the membership function of the fuzzy set is not enough to reveal the characters of things because of the complexity of data and the ambiguity of the human mind.To overcome this shortcoming of the fuzzy set, Atanassov [2] extended fuzzy set to an intuitionistic fuzzy set (IFS) by adding a non-membership function and a hesitancy function.An IFS is able to describe the things from three aspects of superiority, inferiority and hesitation, which are usually represented by the intuitionistic fuzzy numbers (IFNs) [3].To capture more useful information under imprecise and uncertain circumstances, Yager [4][5][6] recently proposed the concept of Pythagorean fuzzy set (PFS) as a new evaluation format, which is characterized by the membership and the non-membership degree satisfying the condition that their square sum is not greater than 1.Zhang and Xu [7] provided the detailed mathematical expression for PFS and put forward the concept of Pythagorean fuzzy number (PFN).The PFS is more general than the IFS because the space of PFS's membership degree is greater than the space of IFS's membership degree.For instance, when a decision-maker gives the evaluation information whose membership degree is 0.4 and non-membership degree is 0.9, it can be known that the IFN fails to address this issue because 0.4 + 0.9 > 1.However, (0.4) 2 + (0.9) 2 < 1, that is to say, the PFN is capable of representing this evaluation information, as shown in Figure 1.For this case, the PFS shows its wider applicability than the IFS.PFSs as a novel evaluation have been prosperously applied in various fields, such as the internet stocks investment [8], the service quality of domestic airline [7] and the governor selection of the Asian Infrastructure Investment Bank [9].A graph is a convenient way of interpreting information involving the relationship between objects.Fuzzy graphs are designed to represent structures of relationships between objects such that the existence of a concrete object (vertex) and relationship between two objects (edge) are matters of degree.The concept of fuzzy graphs was initiated by Kaufmann [10], based on Zadeh's fuzzy relations [11].Later, another elaborated definition of fuzzy graph with fuzzy vertex and fuzzy edges was introduced by Rosenfeld [12] and obtaining analogs of several graph theoretical concepts such as paths, cycles and connectedness, etc., he developed the structure of fuzzy graphs.Mordeson and Peng [13] defined some operations on fuzzy graphs and investigated their properties.Later, the degrees of the vertices of the resultant graphs, obtained from two given fuzzy graphs using these operations, were determined in [14,15].Parvathi and Karunambigai considered intuitionistic fuzzy graphs (IFGs).Later, IFGs were discussed by Akram and Davvaz [16].After the inception of IFGs, many researchers [17,18] generalized the concept of fuzzy graphs to IFGs.Naz et al. [19,20] discussed some basic notions of single valued neutrosophic graphs along with its application in multi criteria decision-making.More recently, Akram et al. introduced many new concepts related to m-polar fuzzy graph, fuzzy soft graph, rough fuzzy graph, neutrosophic graph and their extensions [21][22][23][24][25].This paper proposes a new graph, called Pythagorean fuzzy graph (PFG).In particular, we solve decision-making problems, including evaluation of hospitals, partner selection in supply chain management, electronic learning main factors evaluation by using PFGs.

Remark 1. •
We call P and Q the Pythagorean fuzzy vertex set and the Pythagorean fuzzy edge set of G, respectively.

•
If Q is a symmetric on P, then G = (P, Q) is called PFG.

•
If Q is not symmetric on P, then D = (P, − → Q ) is called Pythagorean fuzzy digraph.

•
The proposed concept of PFGs is a generalization of the notion of Akram and Davvaz's IFGs [16].
Example 1.Consider a graph G = (V, E), where V = {l, m, n, o, p, q, r} and E = {lm, mn, no, nr, pq, qr}.Let P and Q be the Pythagorean fuzzy vertex set and the Pythagorean fuzzy edge set defined on V and E, respectively: By direct calculations, it is easy to see from Figure 2 that G = (P, Q) is a PFG.Definition 3. The degree and total degree of a vertex x ∈ V in a PFG G is defined as d G (x) = (d µ (x), d ν (x)) and td G (x) = (td µ (x), td ν (x)), respectively, where 2) and the total degree of a vertex n in Figure 2 and defined as: Proposition 1.Let G 1 and G 2 be the PFGs of the graphs G 1 and G 2 , respectively.The direct product Definition 5. Let G 1 = (P 1 , Q 1 ) and G 2 = (P 2 , Q 2 ) be two PFGs.Then, for any vertex, (x 1 , Proof.By definition of vertex degree of G 1 × G 2 , we have Proof.The proof is straightforward using Definition 6 and Theorem 1. Example 2. Consider two PFGs G 1 = (P 1 , Q 1 ) and G 2 = (P 2 , Q 2 ) on V 1 = {l, m, n} and V 2 = {o, p}, respectively, as shown in Figure 3. Their direct product G 1 × G 2 is shown in Figure 4.

Since µ
In addition, by Theorem 2, we have Similarly, it is easy to find the degree and total degree of all the vertices in G 1 × G 2 .
respectively.The Cartesian product of G 1 and G 2 is denoted by G 1 2 G 2 = (P 1 2 P 2 , Q 1 2 Q 2 ) and defined as: Proposition 2. Let G 1 and G 2 be the PFGs of the graphs G 1 and G 2 , respectively.The Cartesian product Proof.By definition of vertex degree of G 1 2 G 2 , we have (by (by Proof.By definition of vertex total degree of G 1 2 G 2 , Example 3. Consider two PFGs G 1 and G 2 as in Example 2, where µ Then, by Theorem 3, we have In addition, by Theorem 4, we have Similarly, we can find the degree and total degree of all the vertices in G 1 2 G 2 .
, is defined as: Proposition 3. Let G 1 and G 2 be the PFGs of the graphs G 1 and G 2 , respectively.The semi-strong product Proof.By definition of vertex degree of G 1 • G 2 , we have .

Proof. By definition of vertex total degree
Analogously, we can prove (ii).
and their semi-strong product G 1 • G 2 is shown in Figure 6.Thus, by Theorem 5, we have

1).
In addition, by Theorem 6, we have Similarly, we can find the degree and total degree of all the vertices in respectively.The strong product of these two PFGs is denoted by G 1 G 2 = (P 1 P 2 , Q 1 Q 2 ) and defined as: Proposition 4. Let G 1 and G 2 be the PFGs of the graphs G 1 and G 2 , respectively.The strong product Proof.By definition of vertex degree of G 1 G 2 , we have Analogously, it is easy to show that Proof.For any vertex (x 1 , Analogously, we can prove (ii).
Example 5. Consider two PFGs G 1 and G 2 as in Example 2, where µ shown in Figure 7.Then, by Theorem 7, we must have In addition, by Theorem 8, we must have Similarly, we can find the degree and total degree of all the vertices in G 1 G 2 .
respectively.The lexicographic product of these two PFGs is denoted by and defined as follows: Proof.For any vertex (x 1 , Analogously, we can show that Proof.For any vertex (x 1 , Analogously, we can prove (ii).
Example 6.Consider two PFGs G 1 and G 2 as in Example 2, where µ Then, by Theorem 9, we must have In addition, by Theorem 10, we must have Similarly, we can find the degree and total degree of all the vertices in G

Definition 19. The union
, respectively, is defined as follows: In addition, respectively, as shown in Figure 9.In addition, their union G 1 ∪ G 2 is shown in Figure 10.
, respectively, is defined as follows: In both cases: , is defined as: where E is the set of all edges joining the vertices of and only if G 1 and G 2 are PFGs of G 1 and G 2 , respectively, where P 1 , P 2 , Q 1 and Q 2 are the Pythagorean fuzzy subsets of V 1 , V 2 , E 1 and E 2 , respectively, and

Applications to Decision-Making
Decision making is a common activity in daily life, aiming to select the best alternative from a given finite set of alternatives.In actual decision-making problems, decision makers usually rely on their intuition and prior expertise to make decisions.Owing to the complexity of decision-making problems, the precondition is to represent the fuzzy and vague information appropriately in the process of decision-making.To express the decision makers' or experts' preferences over the given alternatives (criteria), preference relation is one of the useful techniques by which the ranking of criteria can be obtained.For a set of criteria X = {x 1 , x 2 , . . ., x n }, the experts compare each pair of criteria and construct preference relations, respectively.If every element in the preference relations is a PFN, then the concept of the Pythagorean fuzzy preference relation (PFPR) can be put forward as follows: Definition 25.A PFPR on the set X = {x 1 , x 2 , . . . ,x n } is represented by a matrix R = (r ij ) n×n , where r ij = (x i x j , µ(x i x j ), ν(x i x j )) for all i, j = 1, 2, ..., n.For convenience, let r ij = (µ ij , ν ij ) where µ ij indicates the degree to which the object x i is preferred to the object x j , ν ij denotes the degree to which the object x i is not preferred to the object x j , and ij is interpreted as a hesitancy degree, with the conditions: 5, for all i, j = 1, 2, . . ., n.
Therefore, the best hospital is x 1 .Now, using Pythagorean fuzzy weighted geometric (PFWG) operator [6], we aggregate all r ij , j = 1, 2, . . ., 5 corresponding to the hospital x i , and then get the complex PFN r i of the hospital x i , over all the other hospitals: According to s(r i ), i = 1, 2, . . ., 5, we get the ranking of the hospitals x i , i = 1, 2, . . ., 5 as: Therefore, the best hospital is x 1 again.

Comparison with IFSs
In this sub-section, the application example concerning the evaluation of hospitals is compared with IFSs, in order to present the novelty of the introduced approach.
To present a comparison with IFS methods, firstly, we check the constraint condition 0 ≤ µ α + ν α ≤ 1 for the decision matrix of preference relations, as shown in Table 1.
Table 1.Checking intuitionistic fuzzy sets (IFSs) requirement to address the problem.The bold values represent values that do not satisfy the mentioned constraint.From Table 1, it can be observed that the space in which a number can be defined in IFSs is reduced in comparison with PFSs.Since the values of experts' judgments are not in the space of IFS, it is not possible to use the exact values to IFSs methods.Therefore, it is concluded that using PFS increases the flexibility and power of the experts in expressing their judgments and, while considering the limited space of IFSs in comparison with PFSs, it is not possible to solve the case presented in this paper by using IFSs.Similarly, the other application examples concerning the partner selection in supply chain management and the electronic learning main factors evaluation can be compared with IFSs to show the superiorities of the introduced approach.

Conclusions
A fuzzy graph can well describe the uncertainty of all kinds of networks.Pythagorean fuzzy models give more precision, flexibility and compatibility to the system as compared to the classical, fuzzy and intuitionistic fuzzy models.In this paper, we have introduced a new concept of PFGs.We have developed a series of operational laws of PFGs and investigated their desirable properties in detail.We have determined the degree and total degree of a vertex in PFGs formed by these operations in terms of the degree of vertices in the given PFGs in some particular cases.We have proposed the concept of PFPRs.Finally, applications of PFG theory in decision-making based on PFPRs are presented to illustrate the applicability of the proposed generalization of fuzzy graph theory.We are extending our research work to (1) Interval-valued Pythagorean fuzzy graphs; (2) Simplified interval-valued Pythagorean fuzzy graphs; and (3) Hesitant Pythagorean fuzzy graphs.

Figure 1 .
Figure 1.Comparison of spaces of the intuitionistic fuzzy numbers (IFNs) and the Pythagorean fuzzy numbers (PFNs).

Example 4 .
Consider two PFGs G 1 and G 2 as given in Example 2, where µ

C 1 :
Response time and supply capacity; C 2 : Quality and technical skills; C 3 : Price and cost; C 4 : Service level.In order to rank the above four factors C i (i = 1, 2, 3, 4), a committee of three decision makers e k (k = 1, 2, 3) is invited.The decision makers compare each pair of these factors and provide Pythagorean fuzzy preferences contained in the PFPRs R k = (r (k) ij ) 4×4 (k = 1, 2, 3), respectively:

Figure 12 .
Figure 12.Directed network of the fused PFPR.

Figure 13 .
Figure 13.Partial directed network of the fused PFPR.

Figure 17 .
Figure 17.Partial directed network of the fused PFPR.