Entropy and distance measures of Pythagorean fuzzy soft sets and their applications

Abstract

A Pythagorean fuzzy soft set is a parameterized family of Pythagorean fuzzy sets and a generalization of intuitionistic fuzzy soft sets. In this paper, the notions of entropy and distance measures are defined for the Pythagorean fuzzy soft sets (PFSSs). Since, the already existing techniques for finding entropy and distance measures are not working for PFSSs, it is necessary to introduce these techniques in the contest of PFSSs. This work proposes a characterization of the Pythagorean fuzzy soft entropy. Also, the expressions for the standard distance measures like Hamming distance and Euclidean distance are obtained. Further, the applications of PFSSs in decision making problem and pattern recognition problem are discussed. Finally, comparative studies with other existing equations are also carried out.

Keywords

Pythagorean fuzzy soft sets fuzzy soft sets entropy distance measure decision making problem pattern recognition problem

1. Introduction

Many complicated problems in economics, engineering, management science, medical science, etc. involve uncertain data. These problems, which one comes face to face in our day-to-day life, cannot be solved using classical mathematical methods due to a large number of uncertainties exists. To complete the process, the most important problem is to choose the appropriate decision about the task and to find the optimal results [1, 2]. Decision making is a crucial task for all those professions where experts apply their knowledge in a given zone to take appropriate decision. However, due to various constraints in day-to-day life, decision makers may unable to give their judgments in exactly crisp form. Thus, to handle it, they prefer to give their preferences under the uncertain and imprecise in nature by using the theory of fuzzy set (FS) [3]. In this theory, the measurement of each element is done with the help of the membership degree. But with the growing complexity, there always exists a degree of hesitancy between the preferences of the decision making and hence the analysis conducted under such circumstances is not ideal. To address it, an important extension of FS named as intuitionistic fuzzy set (IFS) introduced by Atanassov [4] by incorporated the degree of non-membership ν into the analysis along with the membership degree μ in such a way that μ + ν ≤ 1. Since it appearance, many researchers are paying more attentions towards it and presents different algorithms for solving the decision-making problems by using aggregation operators (AOs) or information measures.

For example, in the field of AOs, Xu [5] presented the weighted averaging AOs for different intuitionistic fuzzy numbers (IFNs). Garg [6, 7] gave the improved AOs by Einstein operators laws by degree of hesitancy into the analysis for IFNs. Huang Huang [8] and Garg [9] gave the AOs based on the Hamachar operational laws for IFNs. Chen and Chang [10] presented a geometric AOs by using the transformation technique to solve the multiple-attribute group decision-making (MAGDM) problems.

The above considered work is based on the restriction on the decision makers that they have given their preferences only under the environment where the constraint μ + ν ≤ 1 satisfy. However, if an expert gives their preferences towards the object as 0.7 as membership and 0.5 as non-membership then clearly seen that 0.7 + 0.5nleq1 and thus the above IFSs theories cannot handle such problems. To overcome it, Yager [11] introduced the concept of Pythagorean Fuzzy (PF) sets (PFSs), which is a generalization of IFSs, by relaxing the condition μ + ν ≤ 1 to μ² + ν² ≤ 1. Thus the PFS handle much more information than IFS. After its existence, several studies are investigated on it by using AOs, other information measures and graphs which are summarized in [12 –19].

In 1999, Molodtsov introduced soft sets [20] to overcome the lack of parametrization tool when managing vagueness. A soft set is a parameterized family of sets which has been extended into different hybrid structures such as fuzzy soft sets [21], intuitionistic fuzzy soft sets [22] and Pythagorean fuzzy soft sets [23]. Since the Pythagorean fuzzy set is highly capable of dealing with vagueness or uncertainty, the parameterized family of Pythagorean fuzzy sets, that is Pythagorean fuzzy soft set also can perform well. In the case of PFSSs, corresponding to each parameter, a Pythagorean fuzzy set is obtained, and Pythagorean fuzzy sets can manage several situations in real life where the intuitionistic fuzzy sets fail to explain. Suppose there is a case in which someone expresses his degree of satisfaction to particular criteria as 0.6 and degree of dissatisfaction is 0.7. Then their sum exceeds one but the square sum does not. So Pythagorean fuzzy sets can handle this. Thus a Pythagorean fuzzy soft set is an effective parameterizing tool as well as a good medium to represent the vagueness which occurs in many real life situations.

Entropy and distance measures are two important concepts in generalized set theory. The entropy quantifies the degree of vagueness and Zadeh [24] introduced fuzzy entropy. The entropy of a system directly proportional to the irregularity. Thus one can identify that which one is more stable if the entropy of each system is given. The axiomatic Definition of entropy is proposed by De Luca and Termini [25]. They introduced some requirements for entropy known as De Luca and Termini axioms. Majumdar and Samanta [26] obtained the entropy of soft sets by extending the De Luca and Termini’s axioms to measure the uncertainty attached to soft sets. The entropy of intuitionistic fuzzy soft set is explained in [27], and it can be seen that entropy plays an important role in decision making problems. Garg and Arora [28] presented an idea of generalized and group generalized intuitionistic fuzzy soft sets. Also, there are studies for solving decision making problems using many hybrid structures of soft sets like fuzzy soft set [21], intuitionistic fuzzy soft set [29 –31] etc. The authors [21 , 32–34] have put forward some ideas for solving decision making problems in the corresponding hybrid structures.

Since the Pythagorean fuzzy soft set is a good tool to handle more than that of intuitionistic fuzzy soft sets, obtaining the entropy is also relevant. It is identified that entropy can be applied in decision making problem and thus decision making problems manage some bigger domain. Distance measures can be applied to decision making problems, pattern recognition problems, machine learning problems, etc. The most widely used distance measures are Hamming distance and Euclidean distance. The distance measure is inversely proportional to the measure of similarity. So it is helpful to identify the similarity between sets. A distance measure of Pythagorean fuzzy sets are introduced in [35] and demonstrates with a numerical example that the proposed distance measures are practical and adequate. This Definition for PFSs is extended to define the distance measure of PFSSs. Both the proposed entropy and distance measure may help in explain the real life situations adequately. Based on the motivation of the structure of the soft sets and the advantages of Pythagorean fuzzy sets to handle the uncertain and imprecise information, this paper explore the theory of Pythagorean fuzzy soft sets by defining some new information measures named as entropy and distance measures.

The rest of the manuscript is organized as follows. Section 2 briefly overview the basic definitions related to the soft sets and Pythagorean fuzzy sets. In Section 3, we defined the concept of distance and entropy for PFSSs and studied their desirable relations. Section 4 deals with an algorithm to solve the decision making problems based on the proposed measures and demonstrate with a numerical example. Also, a comparative study is conducted with some of the existing approach to show the feasibility of the approach. Finally, a concrete conclusion is drawn in Section 5.

2. Preliminaries

In this section, some basic concepts useful for further discussions are introduced.

2.1. Soft set and fuzzy soft set

Definition 2.1. [20] Let E be set of parameters and U be the universal set. A pair (F, E) is called a soft set over U, where F is a mapping F : E → P (U). In other words, the soft set is a parameterized family of subsets of the set U.

Definition 2.2. [32] Let (F, A) and (G, B) be two soft sets over a common universe U. Then,

(F, A) ⊆ (G, B) if A ⊆ B and F (e) ⊆ G (e) , ∀ e ∈ A.

If $(F, A) \tilde{\subseteq} (G, B)$ and $(G, B) \tilde{\subseteq} (F, A)$ then it is said to be soft equal.

Complement: (F, A) ^c = (F^c, A) , where $F^{c} : A \to P (U)$ is a mapping given by F^c (e) = {F (e)} ^c = X - F (e), ∀e ∈ A.

The union of two soft sets (F, A) and (G, B) over the common universe X is a soft set (H, C), denoted by $(F, A) \tilde{\cup} (G, B) = (H, C)$ , where C = A ∪ B and for each e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B \\ G (e), if e \in B - A \\ F (e) \cup G (e), if e \in A \cap B \end{matrix}$ .

The intersection of two soft sets (F, A) and (G, B) over the common universe X is a soft set (H, C), denoted by $(F, A) \tilde{\cap} (G, B) = (H, C)$ where C = A ∩ B, and H (e) = F (e) ∩ G (e) , ∀ e ∈ C.

Definition 2.3. [33] Let E = {e₁, e₂, . . . , e_m} and U = {x₁, x₂, . . . x_n} be parameter set, universal set respectively. For any two soft sets (F, E) and (G, E), the normalized Hamming distance between them is defined as, $d_{H} (F, G) = \frac{1}{mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} | F (e_{j}) (x_{i}) - G (e_{j}) (x_{i}) |$ (1) The normalized distance between (F, E) and (G, E) is defined as, $d_{E} (F, G) = {[\frac{1}{mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} (F (e_{j}) (x_{i}) - G (e_{j}) (x_{i}))^{2}]}^{1 / 2} .$ (2)

Maji et al. [21] initiated the study of hybrid structures involving fuzzy sets and soft sets. They proposed the concept of fuzzy soft sets.

Definition 2.4. [21] Let E be set of parameters and U be the universal set. A pair $(F, A)$ is called a fuzzy soft set over U, if $F : A \to f (U)$ is a mapping from A into set of all fuzzy sets in U, where f (U) is set of all fuzzy subset of U.

Definition 2.5. [21] Let (F, A) and (G, B) be any two fuzzy soft sets over a common universe U then,

$(F, A) ⊑ (G, B)$ if, A ⊂ B and $F (e) ⪯ G (e)$ for each e ∈ A.

$(F, A) = (G, B)$ are equal if $(F, A) ⊑$ $(G, B)$ and $(G, B) ⊑ (F, A)$ .

Complement of a fuzzy soft set $(F, A)$ is a fuzzy soft set $(F^{c}, A)$ defined by for each a ∈ A, $F^{c} (a)$ is a fuzzy set in U whose membership function $F_{a}^{c} (x) = 1 - F_{a} (x)$ for all x ∈ U.

Union of $(F, A)$ and $(G, B)$ is a fuzzy soft set $(H, C)$ , denoted as $(F, A) ⊔ (G, B) = (H, C)$ where C = AUB and for each c ∈ C, $ℋ (e) = {\begin{array}{l} ℱ_{c} (x), if c \in A \ B \\ G_{c} (x), if c \in B \ A \\ ℱ_{c} (x) \lor G_{c} (x), if c \in A \cap^{} B \end{array}$ .

Intersection of $(F, A)$ and $(G, B)$ is the fuzzy soft set denoted as $(H, C) = (F, A) ⊓ (G, B)$ where C = A ∩ B and for each c ∈ C, $H_{c} (x) = F_{c} (x) \land G_{c} (x)$ .

The following is the definition of intuitionistic fuzzy soft sets introduced by Maji et al.[22].

Definition 2.6. [22] Let U be an initial universe E be a set of parameters and IFS (U) denotes the intuitionistic fuzzy power set of U and A ⊂ E. A pair (F, A) is called an intuitionistic fuzzy soft set over U, where F is a mapping given by, F : A → IFS (U).

An intuitionistic fuzzy soft set is a parameterized family of intuitionistic fuzzy subsets of U. A fuzzy soft set is a special case of an intuitionistic fuzzy soft set, because when all the intuitionistic fuzzy subsets of U degenerate into fuzzy subsets, the corresponding intuitionistic fuzzy soft sets degenerate into fuzzy soft set.

Definition 2.7. [22] Let U be an initial universe and E be a set of parameters. Suppose that A, B ⊆ E, (F, A) and (G, B) are any two intuitionistic fuzzy soft sets over U.

(F, A) is a subset of (G, B) if A ⊆ B and F (e) ≤ G (e) for all e ∈ A.

(F, A) and (G, B) are equal if (F, A) is a subset of (G, B) and (G, B) is a subset of (F, A).

Union of (F, A) and (G, B) is a intuitionistic fuzzy soft set (H, C), where C = A ∪ B and for each e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B \\ G (e), if e \in B - A \\ F (e) \underline{\cup} G (e), if e \in A \cap B \end{matrix}$ .

The intersection of (F, A) and (G, B) is a wo intuitionistic fuzzy soft set (H, C), where C = A ∩ B, and $H (e) = F (e) \underline{\cap} G (e), \forall e \in C$ .

Definition 2.8. [27] Let E = {e₁, e₂, . . . , e_m} and U = {x₁, x₂, . . . x_n} be parameter set, universal set respectively. For any two intuitionistic fuzzy soft sets (F, E) and (G, E), the normalized Hamming distance between them is defined as, $\begin{matrix} d_{H} (F, G) = \frac{1}{2 mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} | μ_{F (e_{j})} (x_{i}) \\ - μ_{G (e_{j})} (x_{i}) | + | ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}) | \end{matrix}$ (3)

The normalized distance between (F, E) and (G, E) is defined as, $\begin{matrix} d_{E} (F, G) = [\frac{1}{2 mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} (μ_{F (e_{j})} (x_{i}) \\ {- μ_{G (e_{j})} (x_{i}))^{2} + (ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}))^{2}]}^{1 / 2} \end{matrix}$ (4)

2.2. Pythagorean fuzzy soft set

Definition 2.9. [11] Let U be a universe of discourse. A Pythagorean fuzzy set (PFS) in U is given by, P = {(x, μ_p (x) , ν_p (x)) : x ∈ U} where, μ_p : U → [0, 1] denotes the degree of membership and ν_p : U → [0, 1] denotes the degree of nonmembership of the element x ∈ U to the set P with the condition that $0 \leq μ_{p}^{2} (x) + ν_{p}^{2} (x) \leq 1$ .

For convenience, Zhang and Xu [36] called (μ_p (x) , ν_p (x)) a Pythagorean fuzzy number (PFN) denoted by p = (μ_p, ν_p). Yager [11] proposed another representation of Pythagorean fuzzy number which is p = (r_p, k_p). The relation between p = (μ_p, ν_p) and p = (r_p, k_p) is that μ_p = r_p cos θ_p, ν_p = r_p sin θ_p where, k_p = 1 -2θ_p/π.

Definition 2.10. [11] Let p = (x, μ_p (x) , ν_p (x)) be a Pythagorean fuzzy set. Then, the degree of indeterminacy is denoted by π_p and defined by $π_{p} (x) = \sqrt{1 - μ_{p}^{2} (x) - ν_{p}^{2} (x)}$ .

The following definition gives various operations of Pythagorean fuzzy sets.

Definition 2.11. [11, 12] Let p₁ = (μ₁, ν₁) and p₂ = (μ₂, ν₂) be two PFNs. Then their operations are defined as follows;

p₁ ∪ p₂ = (max {μ₁, μ₂} , min {ν₁, ν₂})

p₁ ∩ p₂ = (min {μ₁, μ₂} , max {ν₁, ν₂})

p^c = (ν, μ)

$p_{1}^{c} \cup p_{2}^{c} = (p_{1} \cap p_{2})^{c}$

$p_{1}^{c} \cap p_{2}^{c} = (p_{1} \cup p_{2})^{c}$

Definition 2.12. [35] If p₁ = (μ₁, ν₁) and p₂ = (μ₂, ν₂) are two Pythagorean fuzzy numbers, then the normalized Hamming distance between p₁ and p₂ is defined as follows;

$\begin{matrix} d_{H} (p_{1}, p_{2}) = \frac{1}{4} (| μ_{p_{1}} - μ_{p_{2}} | + | ν_{p_{1}} - ν_{p_{2}} | \\ + | r_{p_{1}} - r_{p_{2}} | + | k_{p_{1}} - k_{p_{2}} |) \end{matrix}$ (5) The normalized Euclidean distance between p₁ and p₂ is defined as follows;

$\begin{matrix} d_{E} (p_{1}, p_{2}) = [\frac{1}{4} ((μ_{p_{1}} - μ_{p_{2}})^{2} + (ν_{p_{1}} - ν_{p_{2}})^{2} \\ {+ (r_{p_{1}} - r_{p_{2}})^{2} + (k_{p_{1}} - k_{p_{2}})^{2})]}^{1 / 2} \end{matrix}$ (6)

Definition 2.13. [23] Let U be the universal set and E be a set of parameters. The PFSSs is defined as the pair (F, E) where, F : E → PFS (U) and PFS (U) is the set of all Pythagorean fuzzy subsets of U.

If $μ_{A}^{2} (x) + ν_{A}^{2} (x) \leq 1$ and μ_A (x) + ν_A (x) ≤1, then PFSSs degenerate into intuitionistic fuzzy soft sets. The following definitions show various set operations of PFSSs.

Definition 2.14. [23] Suppose A, B ⊆ E and (F, A) , (G, B) are two PFSSs over U. (F, A) is said to be Pythagorean fuzzy soft subset of (G, B) denoted as $(F, A) \hat{\subset} (G, B)$ if,

A ⊆ B

∀e ∈ A, F (e) is a Pythagorean fuzzy subset of G (e) that is, ∀x ∈ U and ∀e ∈ A, μ_F(e) (x) ≤ μ_G(e) (x) and ν_F(e) (x) ≥ ν_G(e) (x).

(F, A) \hat{\subset} (G, B)

and

(G, B) \hat{\subset} (F, A)

then (F, A) and (G, B) are said to be equal.

Definition 2.15. [23] Intersection of two PFSSs (F, A) and (G, B) over a common universe U is the Pythagorean fuzzy soft set (H, C) where, C = A ∩ B and H (e) = F (e) ∩ G (e), ∀e ∈ E and denoted as, $(F, A) \hat{\cap} (G, B) = (H, C)$ , while union of two PFSSs over a common universe U is (H, C) where, C = A ∪ B and

$H (e) = {\begin{matrix} F (e) & if e \in A \ B \\ G (e) & if e \in B \ A \\ F (e) \cup G (e) & if e \in A \cap B \end{matrix}$ (7) This can be represented as, $(F, A) \hat{\cap} (G, B) = (H, C)$ .

3. Entropy and distance measures of PFSSs

Entropy is an essential tool to measure uncertain information. If the entropy is less the uncertainty is also less thus one can quickly identify that which one is the more stable information. PFSS being a more generalized structure, it is able to represent an information in which other existing structures fail. Thus introducing the measure for entropy is important in the current scenario. In this section, by introducing some definitions and results, the expression for entropy and distance measure for PFSSs are obtained and illustrated with examples.

Definition 3.1. Let (F, A) and (G, B) are two PFSSs over U with parameter set E. Define a relation (F, A) is less than or equal to (G, B), denoted as (F, A) ⪯ (G, B) if, for all x ∈ U and e ∈ E, μ_F(e) (x) ≤ μ_G(e) (x) and ν_F(e) (x) ≤ ν_G(e) (x).

The following definition is about a mapping which maps every Pythagorean fuzzy soft set to a fuzzy soft set. It is also shown that the collection of images of PFSSs with α ∈ [0, 1] and with the relation ⊆ is a totally ordered family of fuzzy soft sets.

Definition 3.2. Let α ∈ [0, 1]. The mapping f_α : PFSS (U) → FSS (U) is defined as, f_α ((F, E)) = (F_α, E), for every Pythagorean fuzzy soft set (F, E) with membership value μ_F(e) and nonmembership value ν_F(e) and F_α (e) = f_α (F (e)) and,

$\begin{matrix} f_{α} (F (e)) = {(x, μ_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x), \\ 1 - μ_{F (e)}^{2} (x)) - α π_{F (e)}^{2} (x) : x \in U} . \end{matrix}$ (8) Thus the map f_α assign every PFSSs to a fuzzy soft set.

To demonstrate the Definition 3, we provide a numerical example as follows:

Example 3.1. Let α = 0.6 and $(F, E) = [a_{ij}] = [\begin{matrix} (0.6, 0.6) & (0.5, 0.3) \\ (0.2, 0.9) & (0.6, 0.4) \end{matrix}]$ . Then, we can compuite $\begin{matrix} F_{α} (e_{1}) & = & f_{α} ((u_{1}, 0.6, 0.6), (u_{2}, 0.5, 0.3)) \\ = & {(u_{1}, 0.58, 0.472), (u_{2}, 0.646, 0.354)}, \\ and \\ F_{α} (e_{2}) & = & f_{α} ((u_{1}, 0.2, 0.9), (u_{2}, 0.6, 0.4)) \\ = & {(u_{1}, 0.13, 0.87), (u_{2}, 0.648, 0.352)} \end{matrix}$ Thus fuzzy soft set is obtained and is represented by matrix $[\begin{matrix} (0.528, 0.472) & (0.646, 0.354) \\ (0.13, 0.87) & (0.648, 0.352) \end{matrix}]$ .

The following Theorem establishes some properties of the map f_α and as a corollary of this Theorem it is obtained that ({p_α} _α∈[0,1], ⊆) is a totally ordered family of fuzzy soft sets.

Theorem 3.1. For any α, β ∈ [0, 1] and $p, \bar{p} \in PFSS (U)$ , the following statements are true.

If α ≤ β then f_α (p) ⊂ f_β (p).

If $p \tilde{\subset} \bar{p}$ then $f_{α} (p) \subset f_{α} (\bar{p})$ .

f_α (f_β (p)) = f_β (p) if and only if α = 0.5.

(f_α (p^c)) ^c = f_1-α (p).

Proof. Let p = (F, E), F (e) = {(x, μ_F(e) (x), ν_F(e) (x)) : x ∈ U} ∀ e ∈ U, $\bar{p} = (G, E)$ , G (e) = {(x, μ_G(e) (x), ν_G(e) (x)) : x ∈ U} ∀ e ∈ U and f_α (p) = (F_α, E), where f_α (e) = {(x, $μ_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x)$ , $1 - μ_{F (e)}^{2} (x) - α π_{F (e)}^{2} (x)) : x \in U} \forall e \in U .$

Given that α ≤ β. Then, for all x ∈ U and e ∈ E, $μ_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x) \leq μ_{F (e)}^{2} (x) + β π_{F (e)}^{2} (x)$ . Thus, μ_Fα(e) (x) ≤ μ_Fβ(e) (x) ∀ x ∈ Uande ∈ E. Therefore, f_α (p) ⊂ f_β (p) .

Given $p \tilde{\subset} \bar{p}$ which implies that μ_F(e) (x) ≤ μ_G(e) (x) and ν_F(e) (x) ≥ ν_G(e) (x) ∀ x ∈ U, e ∈ E. Then, $\begin{matrix} μ_{F α (e)} (x) & = & μ_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x) \\ = & μ_{F (e)}^{2} (x) + α (1 - μ_{F (e)}^{2} (x) - ν_{F (e)}^{2} (x)) \\ = & μ_{F (e)}^{2} (x) (1 - α) + α - α ν_{F (e)}^{2} (x) \\ \leq & μ_{G (e)}^{2} (x) (1 - α) + α - α ν_{G (e)}^{2} (x) \\ = & μ_{G (e)}^{2} (x) + α π_{G (e)}^{2} (x) = μ_{G α (e) (x)} \end{matrix}$ Thus, μ_Fα(e) (x) ≤ μ_Gα(e) (x). Therefore, $f_{α} (p) \subset f_{α} (\bar{p})$ .

If α = 0.5 then f_α (f_β ((F, E))) = f_α (F_β, E) = ((F_β) _α, E) where, ((F_β) _α) (e) = f_α (F_β (e)) = f_α (f_β (F (e))) ∀ e ∈ E. So, $((F_{β})_{α}) (e) = f_{α} ((μ_{F (e)}^{2} (x) + β π_{F (e)}^{2} (x)$ , $1 - μ_{F (e)}^{2} (x) - β π_{F (e)}^{2} (x)))$ . Since α = 0.5, $[μ_{F (e)}^{2} (x) + β π_{F (e)}^{2} (x)]^{2} + 0.5 [1 - (μ_{F (e)}^{2} (x) + β$ $π_{F (e)}^{2} (x))^{2} - (1 - μ_{F (e)}^{2} (x) - β π_{F (e)}^{2} (x))^{2}]$ $= (μ_{F (e)}^{2} (x) + β π_{F (e)}^{2} (x)$ , $1 - μ_{F (e)}^{2} (x) - β π_{F (e)}^{2} (x)) = f_{β} (p)$ . Now, f_α (f_β (p)) = f_β (p) which implies f_α ((F, A)) = (F, A). Since (F, A) is a fuzzy soft set, F (e) = {(x, μ_F(e) (x) , 1 - μ_F(e) (x) : x ∈ U} ∀ e ∈ E. Thus, $μ_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x) = μ_{F (e)} (x)$ and $1 - μ_{F (e)}^{2} (x) - α π_{F (e)}^{2} (x) = 1 - μ_{F (e)} (x)$ i.e., $α (1 - μ_{F (e)}^{2} (x) - 1 - μ_{F (e)}^{2} (x) + 2 μ_{F (e)} (x)) = μ_{F (e)} (x) - μ_{F (e)}^{2} (x)$ . Thus α = 0.5.

Let p^c = (F, E) ^c = (F^c, ¬ E) = {(x, ν_F(e) (x) , μ_F(e) (x)) : x ∈ U} ∀ e ∈ ¬ E. f_α (p^c) = f_α (F^c, ¬ E) = ((F^c) _α, ¬ E) where, $(F^{c} (α)) (e) = {(x, ν_{F (e)}^{2} (\neg e) (x) + α π_{F (\neg e)}^{2} (x), 1 - ν_{F (\neg e)}^{2} (x) - α π_{F (\neg e)}^{2} (x) : x \in U} \forall e \in \neg E$ . $(F_{α}^{c})^{c} (e) = {(x, 1 - ν_{F (e)}^{2} (x) - α π_{F (e)}^{2} (x), ν_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x)) : x \in U} \forall e \in E$ . $f_{(1 - α)} (p) = f_{(1 - α)} ((F, E)) = f_{(1 - α)} ({(x, μ_{F (e)} (x), ν_{F (e)} (x)) : x \in U}) = (μ_{F (e)}^{2} (x) + (1 - α) (π_{F (e)}^{2} (x), 1 - μ_{F (e)}^{2} (x) - (1 - α) (π_{F (e)}^{2} (x)) = (1 - ν_{F (e)}^{2} (x) - α (1 - μ_{F (e)}^{2} (x) - ν_{F (e)}^{2} (x)), ν_{F (e)}^{2} (x) + α π_{F (e)}^{2} (x)) = F_{α^{c}}^{2} (e)$ . Thus, (f_α (p^c)) ^c = f_1-α (p).

Corollary 3.1. For all p = (F, E) ∈ PFSS (U), ({p_α} _α∈[0,1], ⊆) is a totally ordered family of fuzzy soft sets.

Proof. Since {p_α} _α∈[0,1] is a collection of PFSSs with common universe U and parameter set E, it satisfies reflexivity, antisymmetry and transitivity under the relation ⊆. Thus ({p_α} _α∈[0,1], ⊆) is a partial ordered family.

Now ∀p_β, p_γ ∈ {p_α} _α∈[0,1]. Without loss of generality, assume that β ≤ γ. Then property 1 of Theorem 3 implies that p_β ⊆ p_γ. Thus any two elements of {p_α} _α∈[0,1] is comparable.

Next, we have defined the axiomatic definition of the Pythagorean fuzzy soft entropy.

Definition 3.3. A real function $E : PFSS (U) \to ℝ^{+}$ is called a Pythagorean fuzzy soft entropy on PFSS (U), if E has following properties;

E (p) =0 if and only if p ∈ S (U)

Let p = (F, E) = [a_ij] _m×n, E (p) = mn if and only if μ_F(e) (x) =0 = ν_F(e) (x) ∀ e ∈ E, ∀ x ∈ U

E (p) = E (p^c) p ∈ PFSS (U)

If $p ⪯ \bar{p}$ then $E (p) \geq E (\bar{p})$ where (F, E) = p and $(G, E) = \bar{p}$ .

From the definition it is clear that entropy is minimum (zero) when the Pythagorean fuzzy soft set degenerates into soft set. The following Theorem discussed the case when the entropy is maximum.

Theorem 3.2. Entropy of a Pythagorean fuzzy soft set p is maximum if and only if p = (F, E) = [a_ij] _m×n = [0] _m×n i.e., μ_{F(e_j)} (u_i) = ν_{F(e_j)} (u_i) =0∀ e_j ∈ E, u_i ∈ U where 0 ≤ i ≤ m and 0 ≤ j ≤ n and p∈ PFSS (U).

Proof. Let p = (F, E) = [0] _m×n. Let $\bar{p} = (G, E)$ be any Pythagorean fuzzy soft set. Since μ_{G(e_j)} (u_i) ≥0 and ν_{G(e_j)} (u_i) ≥0 for all e_j ∈ E, u_i ∈ U, where 0 ≤ i ≤ m and 0 ≤ j ≤ n, by Definition 3, $p ⪯ \bar{p}$ . So, by property 4 of Definition 3 $E (p) \geq E (\bar{p})$ for all $\bar{p}$ . Thus E (p) is maximum.

Conversely, let E (p) be maximum. Assume that p = (F, E) ≠ [0] _m×n, then there exist e_j ∈ E and x_i ∈ U such that μ_{F(e_j)} (u_i) ≠0 or ν_{F(e_j)} (u_i) ≠0 . Now construct the Pythagorean fuzzy soft set as $\bar{p} = (G, E)$ with μ_{G(e_j)} (u_i) = μ_{F(e_j)} (u_i)/2 and ν_{G(e_j)} (u_i) = ν_{F(e_j)} (u_i)/2 for all e_j ∈ E and x_i ∈ U. Then by Definition 3, $p \leq \bar{p}$ . This implies that $E (\bar{p}) \geq E (p)$ which contradicts the hypothesis E (p) is maximum. Thus p = [0] _m×n.

In order to obtain an expression for entropy, a function Φ_D is constructed as, Φ_D : D → [0, 1] where the domain D is given by {(x, y) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1} and Φ_D satisfies following conditions.

Φ_D (x, y) =1 if and only if (x, y) = (0, 1) or (1, 0).

Φ_D (x, y) =0 if and only if x = y = 0 .

Φ_D (x, y) = Φ_D (y, x).

If x ≤ x′ and y ≤ y′ then Φ_D (x, y) ≤ Φ_D (x′, y′).

The following Theorem gives an expression to construct different Pythagorean fuzzy soft entropies.

Theorem 3.3. Let E: $PFSS (U) \to ℝ^{+}$ and p = (F, E) = [a_ij] _m×n ∈ PFSS (U). If $E (p) = \sum_{j = 1}^{n}$ $\sum_{i = 1}^{m} (1 - (Φ_{D} (μ_{F (e_{j})} (u_{i})$ , ν_{F(e_j)} (u_i)))) where Φ_D satisfies the conditions (1) - (4) given above then E is a PFSS entropy.

Proof. E (p) =0 if and only if $\sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (Φ_{D} (μ_{F (e_{j})} (u_{i})$ , ν_{F(e_j)} (u_i)))) =0 if and only if Φ_D (μ_{F(e_j)} (u_i), ν_{F(e_j)} (u_i)) =1 ∀ e_j ∈ E and u_i ∈ U if and only if μ_{F(e_j)} (u_i) =1, ν_{F(e_j)} (u_i) =0or μ_{F(e_j)} (u_i) =0, ν_{F(e_j)} (u_i) =1 if and only if p is a soft set. Thus E satisfies property 1 of Definition 3.3.

E (p) = mn if and only if $\sum_{j = 1}^{n} \sum_{i = 1}^{m} 1 - ((Φ_{D} (μ_{F (e_{j})} (u_{i})$ , ν_{F(e_j)} (u_i)))) = mn if and only if Φ_D (μ_{F(e_j)} (u_i), ν_{F(e_j)} (u_i)) =0 ∀e_j ∈ E and u_i ∈ U if and only if μ_{F(e_j)} (u_i) =0 = ν_{F(e_j)} (u_i) ∀e_j ∈ E and u_i ∈ U. Therefore, E satisfies property 2 of Definition 3.3.

Since p = (F, E) ^c = (F^c, ¬ E) where F^c (e) = {(u_i, ν_{F(e_j)} (u_i), μ_{F(e_j)} (u_i)) : u_i ∈ U} ∀ ¬ e_j ∈ ¬ E, then, $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (Φ_{D} (μ_{F (e_{j})} (u_{i})$ , ν_{F(e_j)} $(u_{i})))) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (Φ_{D} (ν_{F (e_{j})} (u_{i}), μ_{F (e_{j})} (u_{i})))) = E (p^{c})$ . Thus E satisfies property 3 of Definition 3.

Let $\bar{p} = (G, E) = [b_{ij}]_{m \times n}$ . If $p \leq \bar{p}$ then, μ_{F(e_j)} (u_i) ≤ μ_{G(e_j)} (u_i) & ν_{F(e_j)} (u_i) ≤ ν_{G(e_j)} (u_i) which implies Φ_D (μ_{F(e_j)} (u_i) , ν_{F(e_j)} (u_i)) ≤ Φ_D (μ_{G(e_j)} (u_i) , ν_{G(e_j)} (u_i)).

Hence $E (p) \geq E (\bar{p})$ . Therefore E satisfies property 4 of Definition 3.3. Thus E is a PFSS entropy. ◻

Example 3.2. $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e)}^{3} + ν_{F (e)}^{3}))$ .

To verify that above expression is an entropy of PFSSs it is enough to verify that $μ_{F (e)}^{3} + ν_{F (e)}^{3}$ satisfy the four conditions of Φ_D defined above. $Φ_{D} (x, y) = μ_{F (e)}^{3} + ν_{F (e)}^{3}$ is a function from D = {(μ_F(e),) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1} to [0, 1]. Also, $μ_{F (e)}^{3} + ν_{F (e)}^{3} = 1$ if and only if μ_F(e) = 1, ν_F(e) = 0 or μ_F(e) = 0, ν_F(e) = 1 in the domain D. All other conditions are trivial.

Definition 3.4. Let φ, φ′ : [0, 1] → [0, 1] such that if x² + y² ≤ 1 then φ (x²) + φ′ (y²) ≤1 with x, y ∈ [0, 1]. Define E_φ,φ′ function of the Pythagorean fuzzy soft set p = (F, E) = [a_ij] _m×n to $ℝ^{+}$ as, $E_{φ, φ^{'}} (p) = mn - \sum_{j = 1}^{n} \sum_{i = 1}^{m} φ (μ_{F (e_{j})} (u_{i}) + φ^{'} (ν_{F (e_{j})} (u_{i}))$ (9)

Obviously 0 ≤ E_φ,φ′ (p) ≤ mn and ∀p = [a_ij] _m×n belonging to PFSS (U).

Theorem 3.4. If φ : [0, 1] → [0, 1] satisfies,

φ is increasing,

φ (x) =0 if and only if x = 0,

φ (x) + φ (y) =1 if and only if (x, y) = (0, 1) or (1, 0).

Then φ (x) + φ (y) satisfies the conditions 1 - 4 of the Φ_D function defined previously.

Proof. Consider Φ_D (x, y) = φ (x) + φ (y) then the property 3 of this Theorem is same as the condition 1 of Φ_D. Now, Φ_D (x, y) = φ (x) + φ (y) =0 if and only if φ (x) =0 = φ (x) from property 2, x = y = 0. Thus second condition of Φ_D is obtained. Also, φ (x) + φ (y) = φ (y) + φ (x) thus Φ_D (x, y) = Φ_D (y, x). Since φ is increasing, condition 4 of Φ_D is obtained. Hence φ (x) + φ (y) satisfies the conditions 1 - 4 of the Φ_D function.

The following Theorem characterizes the entropies of PFSSs in a general way.

Theorem 3.5. Let $E : PFSS (U) \to ℝ^{+}$ , φ : [0, 1] → [0, 1] and p = (F, E) = [a_ij] _m×n ∈ PFSS (U). E is a Pythagorean fuzzy soft entropy and a E_φ,φ-function if and only if $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i})))$ where φ satisfies the conditions 1 - 3 of Theorem 3.

Proof. Consider Φ : [0, 1] × [0, 1] → [0, 1] such that Φ (x, y) = φ (x) + φ (y) and take the following set {(x, y) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1}. Restrict the function Φ_D : D → [0, 1], which is simply the notation.

If $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) +$ φ (ν_{F(e_j)} (u_i))) where φ satisfies the conditions 1 - 3 of Theorem 3. Then E (p) is a Pythagorean fuzzy soft entropy by Theorem 3. Now it is enough to prove that E is an E_φ,φ function.

Let α, β ∈ [0, 1] and α² + β² ≤ 1 to prove φ (α²) +φ (β²) ≤1, construct the following Pythagorean fuzzy soft set:

$[a_{ij}]_{m \times n} = [\begin{matrix} (α^{2}, β^{2}) & (1, 0) & . . . & (1, 0) \\ (α^{2}, β^{2}) & (1, 0) & . . . & (1, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (α^{2}, β^{2}) & (1, 0) & . . . & (1, 0) \end{matrix}]$ (10) Thus, $\begin{matrix} E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i}))) \\ = mn - m (φ (α^{2}) + φ (β^{2})) - m (n - 1) (φ (1) + φ (0)) \end{matrix}$ By property 3 of Theorem 3, φ (0) + φ (1) =1. Then, E (p) = mn - m (φ (α²) + φ (β²)) - m (n - 1) = m (1 - (φ (α²) + φ (β²))). Since E is an entropy, E (p) ≥0. Therefore m (1 - (φ (α²) + φ (β²))) ≥0, which implies that φ (α²) + φ (β²) ≤1. Therefore E is an entropy and E_φ,φ function.

Conversely let E be an entropy and E_φ,φ function, then E has the form, $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i})))$ . It is enough to prove that φ satisfies conditions 1 - 4 of Theorem 3.

(1) Let α ≤ β, α, β ∈ [0, 1] construct the following PFSSs: $p = (F, E) = [a_{ij}]_{m \times n} = [\begin{matrix} (α, 0) & (0, 0) & . . . & (0, 0) \\ (α, 0) & (0, 0) & . . . & (0, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (α, 0) & (0, 0) & . . . & (0, 0) \end{matrix}]$ and, $\bar{p} = (G, E) = [b_{ij}]_{m \times n} = [\begin{matrix} (β, 0) & (0, 0) & . . . & (0, 0) \\ (β, 0) & (0, 0) & . . . & (0, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (β, 0) & (0, 0) & . . . & (0, 0) \end{matrix}]$

Thus, $\begin{matrix} E (p) & = & \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i}))) \\ = & mn - m (φ (α) + φ (0)) - m (n - 1) 2 φ (0), \\ and \\ E (\bar{p}) & = & \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i}))) \\ = & mn - m (φ (β) + φ (0)) - m (n - 1) 2 φ (0) . \end{matrix}$ Since α ≤ β, $p ⪯ \bar{p}$ so, $E (p) \geq E (\bar{p})$ . Hence, $mn - m (φ (α) + φ (0)) - m (n - 1) 2 φ (0) \geq mn - m (φ (β) + φ (0)) - m (n - 1) 2 φ (0)$ implies φ (α) ≤ φ (β). Thus φ is increasing.

(2) To prove φ (α) =0 if and only if α = 0;

If α = 0, construct the following Pythagorean fuzzy soft set: $p = (F, E) = [a_{ij}]_{m \times n} = [\begin{matrix} (0, 0) & (0, 0) & . . . & (0, 0) \\ (0, 0) & (0, 0) & . . . & (0, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (0, 0) & (0, 0) & . . . & (0, 0) \end{matrix}]$

Thus,

E (p) = mn - mn (φ (0)) + φ (0)) then φ (0) =0 i.e, φ (α) =0.

If φ (α) =0 construct the following Pythagorean fuzzy soft set: $p = (F, E) = [a_{ij}]_{m \times n} = [\begin{matrix} (α, 0) & (0, 0) & . . . & (0, 0) \\ (α, 0) & (0, 0) & . . . & (0, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (α, 0) & (0, 0) & . . . & (0, 0) \end{matrix}]$

Thus, $\begin{matrix} E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) + φ (ν_{F (e_{j})} (u_{i}))) \\ = mn - m (φ (α) + φ (0)) - m (n - 1) (φ (0) + φ (0)) . \end{matrix}$ By previous part, φ (0) =0 and given that φ (α) =0 So E (p) = mn. Thus property 2 of Definition 3 implies α = 0.

(3) To prove φ (α) + φ (β) =1 if and only if (α, β) = (0, 1) or (1, 0);

If (α, β) = (1, 0) or (0, 1) take the following Pythagorean fuzzy soft set: $p = (F, E) = [a_{ij}]_{m \times n} = [\begin{matrix} (α, β) & (α, β) & . . . & (α, β) \\ (α, β) & (α, β) & . . . & (α, β) \\ . & . & . \\ . & . & . \\ . & . & . \\ (α, β) & (α, β) & . . . & (α, β) \end{matrix}]$ .

Then p ∈ S (U). By property 1 of Definition 3, E (p) =0.

Thus $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - φ (μ_{F (e_{j})} (u_{i})) +$ φ (ν_{F(e_j)} (u_i))) =0. i.e, φ (α) + φ (β) =1.

If φ (α) + φ (β) =1 construct the following Pythagorean fuzzy soft set: $p = [a_{ij}]_{m \times n} = [\begin{matrix} (α, β) & (1, 0) & . . . & (1, 0) \\ (α, β) & (1, 0) & . . . & (1, 0) \\ . & . & . \\ . & . & . \\ . & . & . \\ (α, β) & (1, 0) & . . . & (1, 0) \end{matrix}]$

Thus, $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m}$ (1 - φ (μ_{F(e_j)} (u_i)) + φ (ν_{F(e_j)} (u_i))) = mn - m (φ (α) + φ (β)) - m (n - 1) (φ (1) + φ (0)). Given φ (α) + φ (β) =1 and by previous result φ (0) + φ (1) =1. Thus E (p) =0. By property 1 of Definition 3 p ∈ S (U) i.e, (α, β) = (1, 0) or (0, 1).

Remark 3.1. Let p = (F, E) = [a_ij] _m×n ∈ PFSS (U) then entropy of p is given by, $\begin{matrix} E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}^{q} (u_{i})) + (ν_{F (e_{j})}^{q} (u_{i}))), \\ q = 3, 4, 5, \dots \end{matrix}$ (11)

The following part of this section includes definition of distance measure of PFSSs. According to Li and Zeng [35], distance measure of PFSSs are characterized by four parameters μ_p, ν_p, r_p and k_p, which was introduced by Yager [11]. The Hamming and Euclidean distance measures are obtained through generalizing the corresponding measures of distances for Pythagorean fuzzy sets.

Axiomatic Definition of distance measure between PFSSs is given below.

Definition 3.5. Let p = (F, A) and $\bar{p} = (G, A)$ be two PFSSs over U. Let d be a mapping given by $d : PFSS (U) \times PFSS (U) \to ℝ^{+} \cup {0}$ and $d (p, \bar{p})$ satisfies the following axioms.

$d (p, \bar{p}) \geq 0$ .

$d (p, \bar{p}) = d (\bar{p}, p)$ .

$d (p, \bar{p}) = 0$ if and only if $p = \bar{p}$ .

For any q = (H, A) ∈ PFSS (U), $d (p, \bar{p}) + d (\bar{p}, q) \geq d (p, q)$

Then $d (p, \bar{p})$ is a distance measure between PFSSs p and $\bar{p}$ .

Definition 3.6. Let U = {u₁, u₂, . . . u_n}, E = {e₁, e₂, . . . e_m} & p₁ = (F, E), p₂ = (G, E) be two PFSSs over U. Then normalized Hamming distance between p₁ and p₂ is defined as follows;

$\begin{matrix} d_{H} (p_{1}, p_{2}) = \frac{1}{4 mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} \\ (\begin{matrix} | μ_{F (e_{j})} (x_{i}) - μ_{G (e_{j})} (x_{i}) | + | ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}) | \\ + | r_{F (e_{j})} (x_{i}) - r_{G (e_{j})} (x_{i}) | + | k_{F (e_{j})} (x_{i}) - k_{G (e_{j})} (x_{i}) | \end{matrix}) . \end{matrix}$ (12) The normalized Euclidean distance between p₁ and p₂ is defined as follows;

$\begin{matrix} d_{E} (p_{1}, p_{2}) = [\frac{1}{4 mn} \sum_{j = 1}^{m} \sum_{i = 1}^{n} \\ {(\begin{matrix} {(μ_{F (e_{j})} (x_{i}) - μ_{G (e_{j})} (x_{i}))}^{2} \\ + {(ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}))}^{2} \\ + {(r_{F (e_{j})} (x_{i}) - r_{G (e_{j})} (x_{i}))}^{2} \\ + {(k_{F (e_{j})} (x_{i}) - k_{G (e_{j})} (x_{i}))}^{2} \end{matrix})]}^{\frac{1}{2}} . \end{matrix}$ (13)

Remark 3.2. It is easy to verify that given Definitions satisfy four conditions given in Definition 3.5.

To illustrate the above Definition 3.5, we provide a numerical example as follows:

Example 3.3. Let $p_{1} = [\begin{matrix} (0.9, 0.2) & (0.4, 0.6) & (0.2, 0.8) \\ (0.3, 0.6) & (0.4, 0.8) & (0.6, 0.9) \\ (0.1, 0.8) & (0.5, 0.8) & (0.6, 0.3) \end{matrix}]$ and $p_{2} = [\begin{matrix} (0.2, 0.7) & (0.2, 0.9) & (0.5, 0.3) \\ (0.9, 0.2) & (0.7, 0.3) & (0.6, 0.2) \\ (0.6, 0.5) & (0.8, 0.2) & (0.4, 0.8) \end{matrix}]$

The matrix representation of PFSSs p₁ = (r_{F(e_j)} (u_i), k_{F(e_j)} (u_i)) and p₂ = (r_{G(e_j)} (u_i), k_{G(e_j)} (u_i)) are given below, $p_{1} = [\begin{matrix} (0.922, 0.861) & (0.721, 0.374) & (0.825, 0.156) \\ (0.671, 0.295) & (0.894, 0.705) & (1.08, 0.374) \\ (0.806, 0.079) & (0.943, 0.356) & (0.671, 0.705) \end{matrix}]$ and, $p_{2} = [\begin{matrix} (0.729, 0.177) & (0.922, 0.861) & (0.583, 0.656) \\ (0.922, 0.861) & (0.762, 0.742) & (0.632, 0.795) \\ (0.781, 0.558) & (0.825, 0.844) & (0.894, 0.295) \end{matrix}]$

Thus, the Hamming distance between p₁ and p₂ is given by, $\begin{matrix} d_{H} (p_{1}, p_{2}) = \frac{1}{4 \times 3 \times 3} \sum_{j = 1}^{3} \sum_{i = 1}^{3} \\ (\begin{matrix} | μ_{F (e_{j})} (x_{i}) - μ_{G (e_{j})} (x_{i}) | + | ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}) | \\ + | r_{F (e_{j})} (x_{i}) - r_{G (e_{j})} (x_{i}) | + | k_{F (e_{j})} (x_{i}) - k_{G (e_{j})} (x_{i}) | \end{matrix}) \\ = & 0.185 . \end{matrix}$ and, $\begin{matrix} d_{E} (p_{1}, p_{2}) = [\frac{1}{4 \times 9} \sum_{j = 1}^{3} \sum_{i = 1}^{3} \\ {(\begin{matrix} (μ_{F (e_{j})} (x_{i}) - μ_{G (e_{j})} (x_{i}))^{2} \\ + (ν_{F (e_{j})} (x_{i}) - ν_{G (e_{j})} (x_{i}))^{2} \\ + (r_{F (e_{j})} (x_{i}) - r_{G (e_{j})} (x_{i}))^{2} \\ + (k_{F (e_{j})} (x_{i}) - k_{G (e_{j})} (x_{i}))^{2} \end{matrix})]}^{\frac{1}{2}} \\ = & 0.416 . \end{matrix}$

4. Algorithms based on the proposed measures for PFSSs

This section explains applications of the proposed measures. An algorithm for decision making problems using Pythagorean fuzzy soft entropy as well as an algorithm for pattern recognition problem using proposed distance measures are explained. Then illustrative examples for the proposed algorithms followed by comparative studies are given.

4.1. Decision making using PFSS entropy

Decision making can be regarded as a problem-solving activity terminated by a solution deemed to be optimal, or at least satisfactory. Soft set [20] is a good tool to handle vagueness in decision making. Also, there are studies for solving decision making problems using many hybrid structures of soft sets like fuzzy soft set [21], intuitionistic fuzzy soft set [29 –31] etc. In this section, based on the proposed distance and entropy measures, we have presented an algorithm for solving the decision making problems under the PFSS environment.

An algorithm for decision making problems

Let X = {x₁, x₂, . . . x_m} be the universal set under consideration and E = {e₁, e₂, . . . e_n} be the parameter set which consists some qualities of universal set. The algorithm given below explains how to solve the decision making problem using the expression for entropy.

Input each of the PFSSs p₁, p₂, . . . p_k.

Compute entropy of each PFSSs using the expression, $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}^{3} (x_{i})) + (ν_{F (e_{j})}^{3} (x_{i}))) .$ (14)

Find p_r such that $E (p_{r}) = min_{i = 1, 2, . . k} E (p_{i})$ .

Optimal decision is to select p_r obtained from Step 3.

If more than one optimal solution is obtained, any one of them may be chosen.

The following example illustrates the algorithm for decision making problems using Pythagorean fuzzy soft entropy.

Example 4.1. Suppose that a real estate dealer who has three customers wants to know which customer would possibly to buy an apartment from him. Suppose he has customers Mr.A, Mr.B, and Mr.C looking for an apartment with qualities beautiful, near the city and luxurious. Let U = {x₁, x₂, x₃} be collection of apartments and E = {e₁ = beautiful, e₂ = nearthecity, e₃ = luxurious} be a set of parameters. The attractiveness of the set the apartment to Mr.A, Mr.B and Mr.C represented by $(F, A)$ , $(G, A)$ and $(H, A)$ respectively.

Construct the PFSSs $(F, A)$ , $(G, A)$ and, $(H, A)$ representing attractiveness of apartment of Mr.A, Mr.B and, Mr.C respectively. This can be prepared with the help of customers. $\begin{array}{l} (ℱ, A) = \\ {\begin{matrix} e_{1} = (x_{1} / < 0.7, 0.4 >), (x_{2} / < 0.5, 0 >), (x_{3} / < 0.2, 0.8 >), \\ e_{2} = (x_{1} / < 0.9, 0.3 >), (x_{2} / < 0.6, 0.6 >), (x_{3} / < 0.3, 0.6 >), \\ e_{3} = (x_{1} / < 0.4, 0.7 >), (x_{2} / < 0.7, 0.3 >), (x_{3} / < 0.9, 0.1 >) \end{matrix}} \\ (G, A) = \\ {\begin{matrix} e_{1} = (x_{1} / < 0.3, 0.8 >), (x_{2} / < 0.5, 0.5 >), (x_{3} / < 0.8, 0.2 >), \\ e_{2} = (x_{1} / < 0.1, 0.8 >), (x_{2} / < 0.6, 0.3 >), (x_{3} / < 0.3, 0.6 >), \\ e_{3} = (x_{1} / < 0.9, 0.1 >), (x_{2} / < 0.5, 0.3 >), (x_{3} / < 0.5, 0.1 >) \end{matrix}} \\ (ℋ, A) = \\ {\begin{matrix} e_{1} = (x_{1} / < 0.8, 0.1 >), (x_{2} / < 0.9, 0.3 >), (x_{3} / < 0.2, 0.2 >), \\ e_{2} = (x_{1} / < 0.9, 0 >), (x_{2} / < 0.6, 0.2 >), (x_{3} / < 0.7, 0.1 >), \\ e_{3} = (x_{1} / < 0.4, 0.5 >), (x_{2} / < 0.1, 0.3 >), (x_{3} / < 0.2, 0.1 >) \end{matrix}} \end{array}$

Compute the entropies of $(F, A)$ , $(G, A)$ and, $(H, A)$ using the equation $E (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}^{3} (x_{i})) + (ν_{F (e_{j})}^{3} (x_{i})))$ .

Thus, $E (F, A) = 5.01$ , $E (G, A) = 5.68$ , $E (H, A) = 6.19$ .

Find the Pythagorean fuzzy soft set which has minimum value of entropy, which is $(F, A)$ .

Optimal decision is to select $(F, A)$ .

Since there is only one optimal decision, Mr.A has higher possibility to buy apartment from real estate dealer.

Thus the conclusion is Mr.A has higher possibility to buy the apartment from the dealer.

4.2. Pattern recognition using PFSS distance measure

Pattern recognition may be defined as a procedure by which one can search for structures in data and classify these structures such that the degree of association is high among the structures of the same category and low between the structures of different categories. Fuzzy sets, soft sets, fuzzy soft sets, etc. are good tools to represent patterns. Since the measure of similarity is considered as a dual concept of distance measure, sets with smaller distance are taken as similar. Pythagorean fuzzy soft set also can represent patterns with a more clear representation of vagueness.

An algorithm for pattern recognition problems

At first, the given pattern is represented as PFSSs A₁, A₂, . . . A_k in the feature space X = {x₁, x₂, . . . x_m} and the parameters E = {e₁, e₂, . . . e_n}, where E represents some qualities of patterns to be considered. Also, the pattern is to be represented as Pythagorean fuzzy soft set B which should be recognized with one of A_i’s. Then the distance between each A_i, i = 1, 2, . . . k and B is calculated, and the pattern A_i which has least distance with B is identified. This A_i is the most similar to the pattern B. The algorithm for pattern recognition is given below.

Input the patterns A₁, A₂ . . . , A_k.

Input the pattern B which should be recognized.

Find both Hamming distance and Euclidean distance between each set in step 1 and the set in step 2.

Final decision is to select those A_i which has least Hamming distance and Euclidean distance.

The following example illustrates the proposed algorithm for pattern recognition problems.

Example 4.2. There are three kinds of common hybrid minerals and expressed them by PFSSs A₁, A₂, A₃ in the feature space X = {x₁, x₂, x₃} and the parameters E = {e₁, e₂} which represent any qualities of the minerals. Given another kind of hybrid mineral B, it is to be identified which category do B belong to. Distance measure can help to answer this question.

Pythagorean fuzzy soft set representation of A₁, A₂, A₃ and B is given below. This can be prepared with the help of experts. $\begin{array}{l} A_{1} = \\ {\begin{matrix} e_{1} = (x_{1} / < 0.6, 0.5 >), (x_{2} / < 0.5, 0.5 >), (x_{3} / < 0.1, 0.6 >), \\ e_{2} = (x_{1} / < 0.9, 0.1 >), (x_{2} / < 0.7, 0.4 >), (x_{3} / < 0.4, 0.7) \end{matrix}} \\ A_{2} = \\ {\begin{matrix} e_{1} = (x_{1} / < 0.9, 0.3 >), (x_{2} / < 0.9, 0.3 >), (x_{3} / < 0.7, 0.5 >), \\ e_{2} = (x_{1} / < 0.1, 0.4 >), (x_{2} / < 0.2, 0.4 >), (x_{3} / < 0.1, 0.5) \end{matrix}} \\ and \\ A_{3} = \\ {\begin{cases} e_{1} = (x_{1} / < 0.9, 0.1 >), (x_{2} / < 0.5, 0.1 >), (x_{3} < 0.8, 0.2 >) \\ e_{2} = (x_{1} / < 0.3, 0.7 >), (x_{2} / < 0.4, 0.8 >), (x_{3} / < 0.7, 0.1) \end{cases}} \end{array}$

Construct PFSSs of pattern B where $B = {\begin{matrix} e_{1} = (x_{1} / < 0.9, 0.1 >), (x_{2} / < 0.9, 0.2 >), \\ (x_{3} / < 0.7, 0.6 >) \\ e_{2} = (x_{1} / < 0.2, 0.3 >), (x_{2} / < 0.3, 0.1 >), \\ (x_{3} / < 0.2, 0.4 >) \end{matrix}}$

Hamming distance and Euclidean distance of A₁, A₂, A₃ with B is given by,

d_H (A₁, B) =0.335, d_E (A₁, B) =0.370;

d_H (A₂, B) =0.113, d_E (A₂, B) =0.255;

d_H (A₃, B) =0.265, d_E (A₃, B) =0.340

Hamming distance and Euclidean distance between A₂ and B is the least. So the pattern A₂ is more similar to pattern B.

It can be concluded that hybrid mineral B should be produced by the mineral field A₂.

4.3. Comparative studies

The comparative analysis of both entropy and distance measures for PFSSs with the existing ones is illustrated with examples.

To compare the proposed entropy measure for PFSSs, the intuitionistic fuzzy soft entropy given in [27] is taken. Since every intuitionistic fuzzy soft set is a Pythagorean fuzzy soft set and there is no formula available for entropy of PFSSs other than the introduced one, a comparison will be done with the definition given by [27] for intuitionistic fuzzy soft sets.

Example 4.3. Consider the PFSSs A₁, A₂ and A₃ given as, $A_{1} = [\begin{matrix} (0.3, 0.2) & (0.6, 0) & (0.5, 0.4) \\ (0.6, 0.3) & (0.7, 0.2) & (0.4, 0.3) \\ (0.8, 0.1) & (0.8, 0.1) & (0.6, 0.1) \end{matrix}],$ $A_{2} = [\begin{matrix} (0.6, 0.2) & (0.8, 0.1) & (0.8, 0.1) \\ (0.5, 0.5) & (0.7, 0.2) & (0.5, 0.4) \\ (0.7, 0.1) & (0.6, 0.3) & (0.6, 0.3) \end{matrix}], and$ $A_{3} = [\begin{matrix} (0.5, 0.4) & (0.4, 0.1) & (0.6, 0.2) \\ (0.6, 0.2) & (0.7, 0.1) & (0.8, 0.1) \\ (0.9, 0) & (0.5, 0.1) & (0.6, 0.3) \end{matrix}] .$

It can be seen that A₁, A₂ and A₃ are intuitionistic fuzzy soft sets also. Thus different entropies given in [27] as well as proposed entropy are obtained for A₁, A₂ and A₃. The Table 1 represents the comparative analysis for A₁, A₂ and A₃ where, $E_{1} (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})} (x_{i})) +$ (ν_{F(e_j)} (x_i))), E₂ (p) = $\sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}$ (x_i)) + (ν_{F(e_j)} (x_i))) e^{1-(μ_{F(e_j)}(x_i))+(ν_{F(e_j)}(x_i))}, $E_{3} (p) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}^{3}$ $(x_{i})) + (ν_{F (e_{j})}^{3} (x_{i})))$ and $E_{4} (p) = \sum_{j = 1}^{n}$ $\sum_{i = 1}^{m} (1 - (μ_{F (e_{j})}^{4} (x_{i})) + (ν_{F (e_{j})}^{4} (x_{i})))$ . Here E₁, E₂ are entropies taken from [27] and E₃, E₄ are proposed entropies. Thus corresponding to E₁, E₂, E₃ and E₄, we get A₂ has minimum entropy and A₁ has maximum. So it can be concluded that proposed equations for entropy are consistent.

Table 1
Table showing comparison of entropies

Intuitionistic fuzzy soft entropy Pythagorean fuzzy soft entropy

E ₁ E ₂ E ₃ E ₄

A ₁ 2.0 2.77 6.63 7.37

A ₂ 1.0 1.15 6.13 7.08

A ₃ 1.9 2.59 6.34 7.11

	Intuitionistic fuzzy soft entropy	Pythagorean fuzzy soft entropy
A ₁	2.0	2.77	6.63	7.37
A ₂	1.0	1.15	6.13	7.08
A ₃	1.9	2.59	6.34	7.11

Example 4.4. Consider the same PFSSs A₁, A₂, A₃ given in Example 4.3. The comparison of normalized Euclidean distance and Hamming distance given in Jiang et al. [27] with proposed distance measures are given in Table 2, where,

D_{1}^{E}

D_{1}^{H}

D_{2}^{E}

and

D_{2}^{H}

Euclidean distance, Hamming distance (obtained from[27]), proposed Euclidean distance and Hamming distance, respectively. From Table 2, it can be concluded that the distance between A₁, A₂ is the least and A₂, A₃ is the highest.

Table 2

Table showing comparison of distance measures

	Intuitionistic fuzzy soft distance measures		Pythagorean fuzzy soft distance measures
	$D_{1}^{E}$	$D_{1}^{H}$	$D_{2}^{E}$	$D_{2}^{H}$
(A₁, A₂)	0.14	0.13	0.15	0.13
(A₂, A₃)	0.19	0.15	0.18	0.15
(A₁, A₃)	0.17	0.14	0.17	0.14

5. Conclusions

In this paper, axiomatic definitions of the entropy and distance measures of PFSSs are introduced. Some formulas to calculate the entropy and distance measures of PFSSs have also been put forward. These concepts can be effectively applied in many fields such as decision making and pattern recognition. The study of proposed measures is obtained at the first time for PFSSs. Even though the entropy and distance measures are defined for other set generalizations, the same definition cannot use for PFSSs directly. Thus the definitions for entropy and distance measures are extended to PFSSs. Since every IFSS is a PFSS, the proposed measures can be applied for IFSSs and thus for fuzzy soft sets, soft sets, etc. The advantages of this work are the following, it can be used to know the amount of uncertainty involved with a PFSS, identify the similarity between any two PFSSs through the proposed distance measures, and also it is comparable with other existing structures in the literature. Entropy and distance measures in another type of generalized structures and their comparisons can be seen in future work. Also, topological, algebraic, and order theoretical structure can be introduced and studied for PFSSs. Future research will focus on introducing the various other operators under the intuitionistic fuzzy soft sets [37, 38] and uncertain linguistic information [39 –41] to solve the decision making problems.

Footnotes

Acknowledgments

The authors would like to thank the reviewers for their insightful and constructive comments and suggestions that led to an improved version of this paper. The authors also acknowledge the MHRD, India for providing financial support.

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	Intuitionistic fuzzy soft entropy		Pythagorean fuzzy soft entropy
	E ₁	E ₂	E ₃	E ₄
A ₁	2.0	2.77	6.63	7.37
A ₂	1.0	1.15	6.13	7.08
A ₃	1.9	2.59	6.34	7.11

Entropy and distance measures of Pythagorean fuzzy soft sets and their applications

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Soft set and fuzzy soft set

4.1. Decision making using PFSS entropy

4.3. Comparative studies

Table 1 Table showing comparison of entropies Intuitionistic fuzzy soft entropy Pythagorean fuzzy soft entropy E 1 E 2 E 3 E 4 A 1 2.0 2.77 6.63 7.37 A 2 1.0 1.15 6.13 7.08 A 3 1.9 2.59 6.34 7.11

Footnotes

Acknowledgments

References

Table 1
Table showing comparison of entropies

Intuitionistic fuzzy soft entropy Pythagorean fuzzy soft entropy

E ₁ E ₂ E ₃ E ₄

A ₁ 2.0 2.77 6.63 7.37

A ₂ 1.0 1.15 6.13 7.08

A ₃ 1.9 2.59 6.34 7.11