1. Introduction
Fuzzy set theory and rough set theory are two effective tools for dealing with fuzzy and uncertain information. In 1965, L.A. Zadeh proposed the fuzzy set theory [
1], marking the birth of fuzzy mathematics. The core of fuzzy mathematics is the fuzzy sets, which aims to imitate the fuzzy thinking of the human brain and provide effective ideas and methods for solving various practical problems, especially the processing problems of complex systems with human intervention. It has found extensive applications in fields such as healthcare diagnosis [
2], system evaluation [
3], decision making [
4], and other domains. In 1982, Pawlak first gave the concept of rough set theory [
5], where the objective is to classify known knowledge through the equivalence relations, construct upper and lower approximate operators, and use known knowledge to describe unknown knowledge. It has been effectively utilized in fields such as machine learning [
6], pattern analysis [
7], knowledge discovery [
8], and various other domains. In 1990, based on fuzzy sets and rough sets, the scholars Dubois et al. proposed fuzzy rough sets (FRSs) [
9], where they introduced fuzzy relations into rough sets, enriched the rough set models, and expanded the scope of application of rough sets. Currently, there are many research results on fuzzy rough sets [
10,
11,
12,
13,
14].
In many practical fuzzy problems, the membership and non-membership of classical fuzzy sets are often insufficient to explain the relationship between objects and sets. For example, in simulated voting problems, in addition to approval and disapproval, there is also the operation of abstention to remain neutral. Obviously, it is inaccurate to use classical fuzzy sets to deal with such problems. Therefore, in 1986, Atanassov introduced the hesitation degree to solve such problems and proposed IF sets [
15]. IF sets are characterized by three parameters: membership, non-membership, and hesitation degree. Various generalized IFRS models have been studied thus far [
16,
17,
18,
19,
20,
21]. With the continuous in-depth study of many problems, researchers have found that the condition in which the sum of membership and non-membership degrees is less than or equal to one has become increasingly restrictive in solving certain problems. In 2013, Yager et al. proposed PF sets [
22], which relaxed the constraint from the sum of membership and non-membership being less than one to the sum of their squares being less than or equal to one, greatly expanding the applicability of fuzzy set theory. Subsequently, the PF numbers [
23] and the PF relations [
24] were proposed. Naturally, PF sets have been introduced into rough set theory. Zhang et al. [
25] first gave the definition of PFRSs. Since then, various generalized PFRS models have been proposed [
26,
27,
28,
29]. However, generalized PFRS models based on overlapping functions remain unexplored.
The process of combining a large amount of data from many different sources into a representative value is called information aggregation, and the mathematical models used to perform information aggregation are called aggregation functions. Bustince et al. proposed overlap functions in 2010 [
30]. As a binary continuous aggregation function, this function emerged from addressing problems in image processing and classification tasks. Compared with the classical aggregation operators, namely t-norms and t-conorms, the overlap function is free from the constraints of “one as the unit element” and “associativity”. It has made rapid progress in both theory and applications since its proposal. There are many generalized overlap functions, such as interval-valued overlap functions [
31] and IFOFs. However, existing research results have limited applicability in the PF domain.
This paper is mainly driven by two research motivations. First, since overlap functions are difficult to work with in the PF environment, they are extended to the PF domain, and a special lattice-valued overlap function, the PFOF, is proposed. Second, generalized PFRS models based on overlap functions have not been studied; thus, PFOFs are combined with rough set theory, and PF rough approximation operators are constructed using PFOFs and their induced fuzzy implications, leading to the proposal of a generalized PFRS model. The relationship between the research content of this paper and existing models is shown in
Figure 1 (The blue part in the
Figure 1 is the research content of this paper). Compared with classical PFRSs, PFRSs based on PFOFs offer greater flexibility by allowing different PFOFs to be used. (This flexibility is specifically demonstrated in
Section 6.2.4 through a comparison between generalized PFRSs and the classical case).
The rest of this paper is organized as follows.
Section 2 introduces some fundamental definitions which have been studied.
Section 3 proposes the concept of PFOFs, discusses their representable forms, and presents a general construction method for PFOFs.
Section 4 introduces a similarity measure for PF sets based on fuzzy scoring functions and verifies its effectiveness and rationality through examples.
Section 5 proposes a novel PFRS based on PFOFs and the associated fuzzy residual implications, discusses its relevant properties, generalizes it to the dual-domain case, and constructs an approximation accuracy measure for PFRSs.
Section 6 presents three MCDM methods based on PF information and verifies the effectiveness and rationality of the algorithm through examples. Finally,
Section 7 summarizes the entire paper.
3. PF Overlap Functions
In this section, we first give the definition of PFOFs, introduce the concept of representable PFOFs, and finally present a general method for generating PFOFs.
Definition 11. Let be the set of all pairs , where and . We define a partial order on as follows. For any , if and only if and .
Theorem 1. The partially ordered set is a complete lattice, that is, for any subset , the supremum and infimum exist in .
Proof. Let
be an arbitrary subset. We define
Since , let . We claim that and that it serves as the lowest upper bound of S in the order .
Claim 1. Take any . By the definition of A, there exists such that . Since , we know , and hence . But is the infimum of all such , and thus for every , we have . In particular, since x can be arbitrarily chosen to be close to , letting yields . When squaring both sides, , and thus .
Claim 2. is an upper bound of S.
Take any . With the construction of A and B, we have , and thus . Recall that is defined by if and only if and , and hence . Thus, is an upper bound of S under .
Claim 3. is the lowest upper bound of S under .
Suppose is any other upper bound of S. Then, for every , we have which means and . In particular, . Thus, we have , showing that is the lowest upper bound of S in .
Under Claims 1–3, the element is precisely in .
A completely analogous argument (reversing the roles of ≤ and ≥ or with a “dual” proof) shows that S also has an infimum in . Consequently, is a complete lattice. □
Definition 12. A binary function is called a PFOF if for any , it satisfies the following:
- a.
Commutativity: ;
- b.
Boundary condition: or ;
- c.
Boundary condition: ;
- d.
Monotonicity: ;
- e.
Continuity: where are the supremum and infimum operators of , respectively.
If satisfies for all , then is called one-section deflation, and if satisfies for all , then is called one-section inflation.
Proposition 1. For any , we define the function as follows:where O is an overlap function. Then, we call a PFOF, satisfies one-section deflation if O satisfies one-section deflation, and satisfies one-section inflation if O satisfies one-section inflation. Proof. - (a)
Because of the commutativity of
O, for all
, we have
- (b)
According to Definition 11 and the boundary condition of O, we can obtain or
- (c)
According to the boundary condition of O, we have
- (d)
If , then according to Definition 11, we can obtain , and thus we have . By letting , we have and . From the monotonicity of the overlap function, we know that , and thus there is ; that is, ;
- (e)
Firstly, we prove
. Because of the continuity of the overlap function, we have
Similarly, we can obtain . Therefore, is continuous.
- (f)
For , if O satisfies one-section deflation, then , and thus , satisfies one-section deflation. Similarly, if O satisfies one-section inflation, then satisfies one-section inflation.
□
Example 1. - (1)
Given , then is a PFOF;
- (2)
Given , then is a PFOF.
Definition 13. PFOF is representable if , and it can be expressed as , where O is an overlap function.
Remark 4. We can break down the expressible form of the PFOF into two parts for clarity. The first part, , represents the aggregation of the membership part of the PF fuzzy numbers, where O is the overlap function, and and represent the membership values of the two PF fuzzy numbers. The second part, , is the dual of the first part, except that and represent the non-membership values of the two PF fuzzy numbers. This part aggregates the non-membership values of the two PF fuzzy numbers.
Example 2. - (1)
The PFOFs in Example 1 are all representable PFOFs;
- (2)
is an unrepresentable PFOF.
We can easily find that is a PFOF (it satisfies the conditions of Definition 12). Let . There are four variables in the function , namely , , , and , where if , we have . The value of the function also depends on . If , then we can find , and does not satisfy the condition of the overlap function. It is proven that is an unrepresentable PFOF.
Proposition 2. A binary function is a PFOF if there are two functions and such that :where satisfies the following: - a.
and are symmetric;
- b.
u is non-decreasing and v is non-increasing;
- c.
;
- d.
;
- e.
u and v are continuous.
Example 3. Let . Then, is a PFOF.
5. PFRSs Based on PFOFs
In this section, we introduce a novel PFRS model derived from PFOFs, discuss its related properties, and extend it to the dual domain. Finally, we propose a new approximation accuracy measure for PFRSs.
Definition 17. We define the function aswhere is a PFOF. Then, we call the function a residual implication induced by . Example 5. - (1)
Given , we can find the induced residual implication as follows: - (2)
Given , we can find the induced residual implication as follows:
Definition 18 ([
42]).
Let U and W be two domains. A PF set is called a PF relation on the dual domain . For every , represents the degree to which the elements and have the relation , represents the degree to which the elements and do not have the relation , and the set of all PF relations on is denoted by . If , then is called a PF relation on U.The PF relation on the domain U is called reflexive if , it satisfies ; is called symmetric if , it satisfies ; and is called transitive if , it satisfies .
Definition 19. Let be a PF approximation space, U be a domain, and be a PF relation on the domain U. . We denote the PF rough approximation operators of B by as follows:where is a PFOF and is the residual implication induced by . If , then B is called undefinable, and the pair is called a PFRS. Example 6. Let be a PF approximation space and the domain . Table 4 is the PF relation matrix, where . When taking and in Example 5(2), according to Definition 19, we can obtain Theorem 3. Let be the PF approximation space . The PF rough approximation operators in Definition 19 satisfy the following:
- (1)
;
- (2)
;
- (3)
;
- (4)
, where is the intersection operation of PF sets;
- (5)
, where is the union operation of PF sets;
- (6)
;
- (7)
.
Proof. (3) Since
,
, we have
and
, that is,
. Due to the monotonicity of
, for any
, we have
Therefore, we can obtain . Similarly, we can prove that ;
(6, 7) These are direct consequences of (3). □
Theorem 4. Let U be a domain . If , then for any , we have the following:
- (1)
;
- (2)
.
Proof. - (1)
Since
,
, we have
, and because of the monotonicity of
, we have
Therefore, we can find ;
- (2)
The proof method is similar to that in (1). We can easily obtain .
□
We can also prove that the idempotence law of this model does not hold; that is, .
Example 7. Let B be the PF set in Example 6, and let it be known that . Then, we can obtain . We can also obtain .
Definition 20. Let be a dual-domain PF approximation space, U and V be two domains, and be a PF relation on . Then, , denote the PF rough approximation operators of B in by as follows:where is a PFOF, is the residual implication induced by , and the pair is called a dual-domain PFRS of B in . Theorem 5. Let be the PF approximation space. Then, , the PF rough approximation operators in Definition 20 satisfies the following:
- (1)
;
- (2)
;
- (3)
;
- (4)
;
- (5)
.
Proof. The proof method is consistent with Theorem 3 and is omitted. □
Theorem 6. Let U and V be two domains . If , then for any , we have the following:
- (1)
;
- (2)
.
Proof. The proof method is consistent with Theorem 4 and is omitted. □
Next, the approximate accuracy and roughness of classical rough sets are extended to PFRSs.
Definition 21. Let be a fuzzy approximation space, be a fuzzy relation on U, , and the two fuzzy sets and be the fuzzy rough approximation operators of M:
- (1)
The approximate accuracy of the fuzzy set M is defined as follows: It is stipulated that ;
- (2)
The roughness of the fuzzy set M is defined as follows:
From Definition 8, we observe that the approximation accuracy of classical rough sets is given by , where and denote the cardinalities of the sets and , respectively; that is, they represent the number of elements belonging to and . Since the membership degree of elements in classical sets is either zero or one, the cardinality of a set can also be interpreted as the sum of the membership degrees of its elements. Therefore, the approximation accuracy of fuzzy rough sets (FRSs) extends that of classical rough sets and remains applicable in the classical case.
For , a higher membership degree of elements in leads to a greater value of . Since, in classical fuzzy sets, the sum of the membership and non-membership degrees of each element equals one, an increase in the non-membership degree results in a decrease in , thereby reducing the value of . The same principle applies to .
However, as intuitionistic fuzzy (IF) sets and Pythagorean fuzzy (PF) sets introduce hesitation, these rules no longer hold.
Example 8. Two PF sets are given as follows: However, , and the approximate accuracy of classical FRSs is no longer applicable to PFRSs.
Definition 22. Let be a PF approximation space, be a PF relation on U, , and the two PF sets and be the PF rough approximation operators of B.
- (1)
The approximate accuracy of the PF set B is defined as follows:where is the fuzzy score functions in Definition 14. It is stipulated that ; - (2)
The roughness of the PF set B is defined as follows:
Example 9. is taken from Example 4(3), and the approximate accuracy of the PFRS B in Example 6 is calculated to be 0.3719. Figure 4 shows the values of different elements in B and its upper and lower approximations, which more intuitively reflects the approximate accuracy of B. Since is a special case of , the approximate accuracy of PFRSs is an extension of FRS and rough sets, and it is also applicable to the classical cases.