Pythagorean fuzzy soft rough sets and their applications in decision-making

ABSTRACT This paper aims to originate two new notions that are soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set, and investigate some important properties of soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set in detail. Furthermore, classical representations of Pythagorean fuzzy soft rough approximation operators are presented. Then the proposed operators are applied on decision-making problem in which the experts provide their preferences in Pythagorean fuzzy soft rough environment. Finally through an illustrative example, it is shown that how the proposed operators work in decision-making problems.


Introduction
In last few years, many complicated models in engineering, economics, medical sciences, social sciences and many other fields including uncertainties and many other well-known theories have been introduced for the exploration of uncertainties. Among these theories, the most popular theories are fuzzy set [1], soft set (in short SS) [2] and rough set [3]. Different complications are pointed out in this existing literature as mentioned in [2]. To tackle these complications the new approach of SS was originated by Moldstove [2], which is different from the other existing theories due to its parameterization tools. SS theory is free from the inherent complications and handles the vague and uncertain knowledge easily. Due to these characteristics, the theory of SS is famous among scholars in the recent era. This theory has strong concepts in various directions such as Riemann integration, game theory, perron integration, smoothness of functions, operational research, the theory of measure and probability theory [2,4]. In the recent scenario, SS is one of the most significant areas of research. Maji et al. [5] initiated various operations on SS. Through these operations, the theoretical study of SS becomes improved and emerges very rapidly in research area. On the basis of [5], Ali et al. [6] improved the existing literature and presented some new operations on SS. They developed the concept of complement and asserted that certain De Morgans's laws are satisfied on the basis of these newly developed operations on SS. The generalization of SS is improving very rapidly in the research area to hybridize the different mathematical structures with SS. The combined structure of fuzzy set and SS were presented by Maji et al. [7] to initiate the new concept of fuzzy SS (FSS). Yang et al. [8] further generalized the concept of FSS and originated the new idea of multi FSS and under an imprecise situation, they provided their applications in decision-making (in short DM).
The fundamental approach of rough sets theory (RST) attracted the attention of scholars towards this topic, and had been applied in various directions like machine learning, data analysis, knowledge discovery, pattern recognition and information process, etc. with successful results. RST by Pawlak is based on indiscernibility relation. Actually, indiscernibility relation is also known as an equivalence relation and we do not have enough knowledge about the universe of discourse, which makes it difficult for us to define the target. So, different authors defined various structures of RST with less restrictions. Feng et al. [9] defined the hybrid theory of fuzzy set, rough set and SS to get several important theories like, soft rough set, rough SS and soft rough fuzzy set. Mahmood et al. [10] presented the concept of generalized roughness with the hepl of isotone and monotone mappings in ordered semigroups and for details see [11,12]. The detail of combining the study of rough set and SS are presented in [9,13,14]. Ma [15] defined two new types of fuzzy covering-based rough set models by the new concepts of fuzzy β-covering and fuzzy β -neighbourhood, gave the properties of the two models, and revealed the relationships between the two models and some others. For the fuzzy β-covering approximation space proposed in [15], Yang et al. [16] introduced the concept of fuzzy β-neighbourhood family, fuzzy β-cover and consistent fuzzy β-covering, and gave some propositions about the fuzzy β-cover and the consistent fuzzy βcovering. Zhang [17] presented the novel fuzzy rough set models and corresponding applications to multicriteria decision-making (MCDM). Zhang and Zhan [18] proposed the concepts of fuzzy soft β-coverings, fuzzy soft β-neighbourhoods and fuzzy soft complement βneighbourhoods and some related properties are studied. Zhan and Sun [19] initiated the notion of coveringbased soft fuzzy rough theory and its application to MCDM.
Intuitionistic fuzzy set (IF set) [20] consist of two functions which are the generalization of fuzzy set and like a fuzzy set, these two functions also map on unit closed interval. Here, the first function represents the membership grade and the second function represents non-membership grade. IF set has a limitation that is the sum of both these functions does not exceed 1. IF set plays a vital role for tackling vagueness and uncertainty. Many authors developed the new notions by combining rough set and SS with IF set and proposed the new results, such as Maji et al. [21,22] originated the notion of intuitionistic fuzzy SS (IFSS). Karaaslan [23] presented the concept of IF parameterized IFSS and validated the proposed results with the help of applications in DM. Geng et al. [24] initiated the theory of generalized IFSS and studied their applications. Recently Feng et al. [25] clarified and reformulated the existing concepts of generalized IFSS and studied their applications in MCDM. Furthermore, they improved some existing concepts and results in the literature through these new notions. Jiang et al. [26] presented the applications of DM in IF soft environment. The concept of IF rough set was originated by Zhang et al. [27], which depends on two universes and general IF relation. The notions of rough IF set and IF rough set were initiated by Zhou and Wu [28], and they studied the constructive and axiomatic approaches in detail. Zhang et al. in [29] investigated the notion of IF soft rough set (IFSRS) and proposed its application in DM. The concept of generalized IFSRS was presented by Zhang et al. [30] and studied their applications in DM. Zhan and Sun [31] introduced three classes of coverings based IF rough set model via IFβ-neighbourhoods and IF complementary β-neighbourhood and studied their application in MCDM. Zhang et al. [32] proposed the concept of covering-based general multigranulation IF rough set and presented their corresponding application to MCDM. Yager [33] provided more comfortability and relaxation in limitation of values for the decision makers that is, the square sum of membership and nonmembership grades must not exceed 1, and he named this notion as Pythagorean fuzzy set (PFS), which got more attention of scholars in the recent era. Hussain et al. [34] proposed the concept of rough Pythagorean fuzzy ideals in semigroups. By viewing existing literature, it is clear that there does not exist any concepts of soft rough Pythagorean fuzzy set (SRPFS) and Pythagorean fuzzy soft rough set (PFSRS). From the investigation of existing study, it appears a lack of application in DM problems with the evaluation of PF information by means of PFSRS. This motivates the present research to present the novel approach of SRPFS and PFSRS and presenting its application in DM problems with the evaluation of PF information.
The arrangement of the rest of portion of the manuscript is given as Section 2 consists of the brief review of IF sets, Pythagorean fuzzy sets (PFSs), (α, β)level cut sets, rough sets and SSs. Furthermore, in Section 3 the new structure of soft rough Pythagorean fuzzy set (SRPFS) is presented and the related properties of SRPF approximation operators are studied. Section 4 consists of the study of Pythagorean fuzzy soft rough set (PFSRS) and investigates their related properties in detail. Section 5 consists of the technique for DM process and its algorithm. In the final Section 6, we presented an illustrative example on the proposed methods SRPFS and PFSRS, and show that how the proposed operators work in DM problems.

Preliminaries
This section consists of a brief review of IF set, Pythagorean fuzzy set (PFS), RST, SS and soft rough set. These concepts will help us in next sections.

Definition 2.3:
with the ordered relation denoted as For any two (κ 1 , (1) does not hold.

Lemma 2.1:
The ordered set S * with respect to the ordered relation is a complete lattice.

Definition 2.4 ([2]):
Suppose a universal setÛ and E be the initial set of parameters. Suppose that a function F : E → P(Û), then the pair (F, E) is known to be a SS onÛ, where the family of all subsets of a universal set is denoted by P(Û).

Definition 2.5 ([37]):
Consider a SS (F, E) onÛ. Then a relation fromÛ × E is known to be a crisp soft relation from a setÛ to E,which is given by  on a universal setÛ. Then a relation fromÛ × E is known to be a fuzzy soft relation (FSR) from a setÛ to E and is given as: Consider if setÛ = {κ 1 , κ 2 , . . . , κ m } and E = {x 1 , x 2 , . . . , x n }, then the FSR from setÛ to E is given in the following table: The relation is known to be serial if for all κ ∈Û, (κ) = ϕ. Then (Û, E, ) is known to be a crisp soft approximation space. Consider for a nonempty subset ⊆ E, then the lower and upper soft approximations of with respect to (Û, E, ) are represented by ( ) and ( ) are defined as:

Soft rough Pythagorean fuzzy set (SRPFS)
In this section, we will present the concept of SRPFS by combining the crisp soft relation fromÛ to E with the rough Pythagorean fuzzy set. Furthermore, the concept of soft rough Pythagorean fuzzy (SRPF) approximation operators are investigated, and some basic properties of proposed operators are also discussed.

Definition 3.1: Consider a crisp soft approximation
Then the lower and upper soft approximations of with respect to (Û, E, ) are represented by ( ) and ( ) are defined as; Then the pair ( ( ), ( )) is known to be an SRPFS of with respect to (Û, E, ) where ( ) = ( ). Thus, the operators ( ), ( ) : PFS(E) → PFS(Û) are known to be lower and upper SRPF approximation operators with respect to (Û, E, ).

Remark 3.1: Suppose a crisp soft approximation space
is the family of fuzzy sets. Then the defined SRPF approximation operators ( ) and ( ) degenerate into the soft rough fuzzy set, that is: Hence the pair ( ( ), ( )) is known to be soft rough fuzzy set.

Remark 3.2:
Suppose a crisp soft approximation space (Û, E, ) and for a crisp set ∈ P(E) of E. Then the defined SRPF approximation operators ( ) and ( ) degenerate into soft rough approximation operators as defined in Definition 2.9. Hence, it is clear that Definition 3.1 is the generalization of Definition 2.9.
Now to define crisp soft relation onÛ×E, that is Now to define a Pythagorean fuzzy set ∈ PFS(E) as follows Proof: (i) By Definition 3.1, we have Proofs (v) to (viii) are straightforward and follows the above results and Definition 3.1.
Here, by counter example we will show that the equality does not hold in parts (iv) and (viii) From Definition 2.9 to get the set-valued functions * , that are * (κ 1  Therefore, it is clear that implies κ 1 , 0.7, 0.3 κ 1 , 0.7, 0.5 ⇒ 0.7 ≤ 0.7 but 0.3 0.5. Similarly, we can show that

Pythagorean fuzzy soft rough set (PFSRS)
In this section, on the bases of IF soft rough set [29] we originate the new notion of PFSRS. Moreover, some fundamental properties of Pythagorean fuzzy soft rough (PFSR) approximation operators are also discussed in detail.

Definition 4.1:
Let E be a set of parameter andÛ be a universal set. Then (F, E) is known to be a Pythagorean fuzzy SS overÛ, if F : represents the membership grade and non-membership grade of κ ∈Û to F(x) respectively, which satisfies that 0 Then the pair ( ( ), ( )) is known to be a PFSRS of with respect to. (Û, E, ) where ( ) = ( ).

Remark 4.2: By taking fuzzy soft approximation space (Û, E, ) and let ∈ F(E) where F(E) is the collection of fuzzy sets. Then the PFSR approximation operators
( ) and ( ) in Definition 4.2, degenerate into soft fuzzy rough approximation operators defined by Sun and Ma [41]. Hence, it is clear that PFSR approximation operators in Definition 4.2, is the generalization of soft fuzzy rough approximation operators defined by Sun and Ma [41].
Proof: Proofs are straightforward as the proof of Theorem 3.1.

Theorem 4.3: Consider a fuzzy soft approximation space (Û, E, ) and ∈ PFS(E). Then the upper PFSR approximation operator can be shown as follows, for all κ∈Û;
(i) and more over for any α ∈ [0, 1], (iii) On the same way we can prove that (ii) The upper crisp soft rough approximation operator according to Definition 2.9, we have On the same way we can prove that thus as a result we get α + (κ) ∩ α + = ϕ. Therefore by Definition 2.9, we have Next for any κ ∈ α ( α ), we have α ( α )(κ) = 1. Since (i) and more over for any α ∈ [0, 1], Proof: The proofs of (i) and (ii) according to Theorems 4.2 and 4.3. Now for any κ ∈Û, consider Therefore, by the duality of upper and lower PFSR approximation operators (see Theorem 4.1), we can get Similarly, the proof of the above result, we can get The proofs of (iii) and (vi) are straightforward to the proofs of (iii) and (vi) of Theorem 4.3.

Application of Pythagorean fuzzy soft rough set (PFSRS) in decision-making
Here in this section, the technique for the DM process is constructed on the approach of PFSRS. For this, we will define the ring sum and ring product operations on PFSs. By the operation, the basic concept of this method and approach to DM is given, which is based on the PFSRS approach.  (μ 1 (κ), ψ 1 (κ)), 2 = (μ 2 (κ), ψ 2 (κ)) ∈PFS(Û). Then the ring sum for PFSs 1 and 2 can be defined as follows:

Algorithm
This subsection is devoted for the step wise algorithm of the proposed model and consists of the following steps: step i First to find the FSR fromÛ×E or fuzzy SS (F, E) overÛ, accordance to the interests of decision maker. step ii For the evaluation of certain decision input each person has different point of views on the attribute of the same parameter, so to find the optimum normal decision object in accordance with the demand of expert/decision maker. step iii From Definition 4.2, calculate the PFSR approximation operators ( ) and ( ). step iv By ring sum operation or ring product operation calculate the choice set.
It is clear that in choice set H the Pythagorean fuzzy value λ is the maximum choice value. If μ H (κ j ) ≥ S * μ and ψ H (κ j ) ≥ S * ψ, then the optimum decision value is κ j .
The final decision is only one, one may go back to the second step and change the optimum decision object in the final step of the given algorithm, when there exist too many "optimal choices" to be chosen.
The concept of the proposed algorithm is illustrated with the help of the following example.

Illustrative example
For a certain senior position of a doctor in the Pakistan Institute of Medical Sciences (PIMS) Cardiac Centre, the appointment of new faculty has to face a very complex evaluation and DM process. The skill and ability of a candidate may be judged with respect to various attributes like "physical and surgical productivity" "managerial skill" "ability to work under pressure" "research productivity", etc. In order to take the right decision about the candidate, the professional experts' opinions are needed for these criteria.
Consider thatÛ = {κ 1 , κ 2 , κ 3 , κ 4 , κ 5 } be set of five candidates who fulfil the requirements for the senior faculty position in PIMS. In order to appoint the most qualified and suitable candidates for the required position, a team of experts is organized and chaired by Prof. Z as a director. The team of experts will judge the candidates upon the criteria in the parameter set According to the background and experience, the team of experts wants to appoint the candidate which qualifies with the parameters of E who deserves extremely from candidate inÛ.

For PFSRS
step i Consider that the experts explain the gorgeous/attractiveness of the candidates by calculating an FSR fromÛ × E which is given in the following Table 2. step ii Now consider the team of experts present the optimum normal decision object which is a Pythagorean fuzzy subset over the a set E as follows: If μ H (κ j ) ≥ S * μ and ψ H (κ j ) ≥ S * ψ, then the optimum decision value is κ j .Hence, the optimal decision is λ = κ 5 = (0.8544, 0.08).

Ring Product Operator
Now to calculate the optimal decision through ring product operator, we have   Table 3. Comparitive study of the proposed method with existing literature.

Ring Product Operator
Now to calculate the optimal decision through ring product operator, we have Hence, the optimal decision is λ = κ 5 = (0.4, 0.44). Therefore, the most qualified and suitable candidate for the required position is κ 5 .

Comparative study
From the above analysis, it is clear, that the proposed approach is better than an intuitionistic fuzzy rough set (IFRS) [43], soft rough intuitionistic fuzzy set (SRIFS) and intuitionistic fuzzy soft rough set (IFSRS) [29]. The advantages of the proposed method with existing literature are given below. Advantages is the family of fuzzy sets. Then the defined SRPF approximation operators ( ) and ( ) degenerate into the soft rough fuzzy set. (b) Suppose a crisp soft approximation space (Û, E, ) and for a crisp set ∈ P(E) of E. Then the defined SRPF approximation operators ( ) and ( ) degenerate into soft rough approximation operators as defined in Definition 2.9. (c) By taking crisp soft approximation space (Û, E, ) and let ∈ PFS(E). Then the PFSR approximation operators ( ) and ( ) in Definition 4.2, degenerate into SRPF approximation operators ( ) and ( ) in Definition 3.1. (d) By taking fuzzy soft approximation space (Û, E, ) and let ∈ F(E) where F(E)is the collection of fuzzy sets. Then the PFSR approximation operators ( ) and ( ) in Definition 4.2, degenerate into soft fuzzy rough approximation operators defined by Sun and Ma [41]. Now to verify the effectiveness of the developed approach with some existing methods are presented in Table 3 by considering the above Illustrative Example of Section 6. IFRS [43] having no information about parameterization tools, so due to lake of this information the method developed in [43] failed to handle the proposed example. On the other hand, if the sum of PF value (μ(κ), ψ(κ)) is greater than 1, that is μ(κ) + ψ(κ) > 1 in optimum normal decision object of Step ii. So, in this case, the method presented in [29] failed to tackle the situation. Thus from the comparative study, it is clear that the proposed method is more superior and provides more freedom to the decision makers for the selection of membership and non-membership degrees as compared to existing literature.

Conclusion
The theories of rough set, Soft set, IF set and PFS all are important mathematical tools for dealing with uncertainties. In this manuscript, we have presented two new concepts: SRPFS and PFSRS, which can be seen as two new generalizations of soft rough set models. Then we investigated some important properties of SRPFS and PFSRS in detail. Moreover, classical representations of PFSR approximation operators are presented. In addition, the validity and effactiveness of the proposed operators are checked by applying them to the problems of DM in which the experts provide their preferences in the PFSR environment. Finally, through a numerical example, it is demonstrated that how the proposed operators work in DM problems. By comparative analysis, we find that it is more effective to deal with DM problem with the evaluation of PF information based on SRPFS and PFSRS models than DM problems with the evaluation of SRIFS and IFSRS models.