1.
Introduction
In our daily life, we deal with problems resulting from indefinite and vague information without using the appropriate modeling tools, this leads to imprecise reasoning and inexact solutions. That is why it is a quite difficult task for the decision-makers (DMs) to make reasonable and logical decisions in handling such problems. So, for such kinds of problems and difficulties, it has become particularly important to address vagueness and uncertainties. Zadeh [1] suggested an innovative idea of fuzzy set, which is an extension of a crisp set. It was eminent attainment and a milestone in the development of fuzzy set theory and fuzzy logic. To address the problems of daily life with vagueness and uncertainties in them, different models and theories have been introduced by the researchers. Later, the concept of the interval-valued fuzzy set (IVFS) was originated by Zadeh [2].
Atanassov [3,4] suggested intuitionistic fuzzy set (IFS) theory and Pythagorean fuzzy sets (PFSs) were suggested by Yager [5,6]. The generalization of PFSs with generalized membership grades was suggested by Yager [7], who named the generalization as follows: q-rung orthopair fuzzy sets (q-ROFSs). The idea of bipolarity was proposed by Zhang [8,9] in terms of bipolar fuzzy set (BFS). A new direct extension of fuzzy set with m degrees of membership grades was suggested by Chen et al. [10] and named m-polar fuzzy sets (mPFSs). Smarandache [11,12] originated the notion of a neutrosophic set, which focuses on truthness, falsity and indeterminacy. The picture fuzzy set (PiFS) was proposed by Cuong [13]. Xu [14] developed IFS based aggregation operators for the information fusion of intuitionistic fuzzy numbers. Garg and Nancy [15] proposed linguistic single-valued neutrosophic prioritized aggregation operators and their applications.
The notion of a soft set was originated by Molodtsov [16]. The structure by merging soft sets with fuzzy sets was introduced by Cagman et al. [17]. Sometimes, it is very difficult for DMs to exactly weigh their certainty in real numbers. So, specifying their degree by the intervals is more appropriate. A hybrid model of a fuzzy set with IVFSs was suggested by Jun et al. [18] and named the cubic set (CS) along with their internal and external modes. A hybrid of the CS and mPFS was proposed by Riaz and Hashmi [19]. They developed aggregation operators for cubic m-polar fuzzy (CmPF) information aggregation. New extensions of fuzzy set, such as the linear Diophantine fuzzy sets (LDFS), linear Diophantine fuzzy soft rough sets, and spherical linear Diophantine sets [20,21,22], have robust features and applications in computational intelligence and information analysis. Liu et al. [23], Liu and Wang [24], and Jain et al. [25] suggested novel concepts of information aggregation for multi-criteria decision making (MCDM) problems. Fuzzy topology takes its motivation from classical analysis, and it has a vast number of applications. Chang [26] proposed the idea of fuzzy topology and the notion of intuitionistic fuzzy topology was introduced by Coker [27]. These ideas ware extended by Olgun et al. [28] to define Pythagorean fuzzy topology. Cagman et al. [29] proposed certain properties of soft topological spaces.
Saha et al. [30,31] developed novel concepts of aggregation operators for information aggregation. Jana et al. [32,33] proposed IFS-based Dombi and bipolar fuzzy Dombi prioritized aggregation operators in MADM. Akram et al. [34] developed MCDM with m-polar fuzzy attributes reduction algorithms. Akram et al. [35] proposed PFS-based extensions of TOPSIS and ELECTRE-I methods. Ashraf and Abdullah [36] introduced fuzzy modeling based on spherical fuzzy sine trigonometric information aggregation methods. Almagrabi [37] proposed a new approach to q-LDFSs and their operational laws with applications.
MCDM is the method that provides the ranking of the objects and also the ranking of feasible objects. The most important problem in decision analysis is how to describe the attribute values in an efficient way. It is very difficult for an individual in various situations to select an option due to the inconsistency in the data that occur because of human error or lacking information.
Many techniques have been used for the fusion of information. The word VIKOR has the abbreviation of "Vlse Kriterijiumska Optimizacija Kompromisno Resenje" and it is a very important technique in decision-making analysis. This process is widely used in decision-making analysis because of its computational comfort. It provides multiple suitable solutions for problems with unequal standards and helps DMs to achieve a neutral ending judgment. Some applications with help of VIKOR technique are discussed in Table 1.
Ali et al. [38,39] proposed the idea of neutrosophic cubic sets and bipolar neutrosophic soft sets with applications in decision making.
The primary objective of this paper was to generate two different types of topological structures on cubic m-polar fuzzy sets (CmPFSs) while keeping in view the two orders of cubic sets. The concepts of CmPF topology with P-order and R-order are defined. The goals of this study are as follows: (i) to define open sets and closed sets in CmPF topology, (ii) to discuss the interior, closure, and exterior of CmPFSs in CmPF topology, (iii) to study the subspace of CmPF topology, (iv) to define the dense set, neighborhood and base of CmPF topology, (v) to develop an extension of the VIKOR method based on CmPFSs, and (vi) to develop a new multi-criteria group decision-making (MCGDM) method based on CmPF topology.
The remaining part of this paper is arranged in the following way. In Section 2, we look back to some elementary concepts like CSs, mPFSs, CmPFSs, and operations on CmPFSs. In Section 3, we describe the notion of a topological structure on CmPFSs under P-order. We also discuss some major results on CmPFs with P-order. In Section 4, we introduce the notion of a topological structure on CmPFSs under R-order. In Section 5, an extension of the VIKOR method for MCGDM with CmPF topology is introduced. An application of the proposed method for chronic kidney disease (CKD) diagnosis is presented, and a comparison analysis of the suggested approach and existing approaches is also given. The conclusion of the study is given in Section 6.
2.
Preliminaries
In this section, we discuss some elementary concepts of CmPFSs.
Definition 2.1. [18] A CS ∁ on a universal set k is expressed as
in which A=[A−(ℓ),A+(ℓ)] is an interval-valued fuzzy set and λ(ℓ) is a fuzzy set on k. For simplicity, the CS ∁={ℓ,[A−(ℓ),A+(ℓ)],A(ℓ):ℓ∈k} is denoted as ∁=<A,λ>
Definition 2.2. [10] Let k be a universal set of discourse. An mPFS on k is defined by [0,1]m, and it can be written as
where μ1(ℓ),⋅⋅⋅,μm(ℓ) represents m number of membership grades (MGs) in [0,1].
Definition 2.3. [19] Let k be a universal set. A CmPFS on a universal set k is expressed as
Here, [A−j(ℓ),A+j(ℓ)]mj=1 are fuzzy valued intervals and (Aj(ℓ))mj=1 are fuzzy numbers. For simplicity, we can write the cubic m-polar fuzzy number (CmPFN) as
Definition 2.4. [18] Let Ia=[ϝ−a,ϝ+a] and Ib=[ℸ−b,ℸ+b] be any two fuzzy valued intervals. Then
1. Ia≤Ib⇔ϝ−a≤ℸ−b and ϝ+a≤ℸ+b
2. Ia≥Ib⇔ϝ−a≥ℸ−b and ϝ+a≥ℸ+b
3. Ia=Ib⇔ϝ−a=ℸ−b and ϝ+a=ℸ+b
2.1. Operations on CmPFSs
Definition 2.5. [19] Let us consider two CmPFSs on k given by
Some basic operations on these sets with P-order are defined as
1. (C1)c={(λ,[1−A+j,1−A−j],1−Aj)mj=1:λ∈k}
2. (C2)c={(λ,[1−B+j,1−B−j],1−Bj)mj=1:λ∈k}
3. C1⊆PC2⇔A−j(λ)≤B−j(λ),A+j(λ)≤B+j(λ) and Aj(λ)≤Bj(λ)
4. C1⊔PC2={(λ,[A−j(λ)∨B−j(λ),A+j(λ)∨B+j(λ)],Aj(λ)∨Bj(λ))mj=1:λ∈k}
5. C1⊓PC2={(λ,[A−j(λ)∧B−j(λ),A+j(λ)∧B+j(λ)],Aj(λ)∧Bj(λ))mj=1:λ∈k}
Similarly, some basic operations on the above two CmPFSs with R-order are defined as
1. C1⊆RC2⇔A−j(λ)≤B−j(λ),A+j(λ)≤B+j(λ) and Aj(λ)≥Bj(λ)
2. C1⊔RC2={(λ,[A−j(λ)∨B−j(λ),A+j(λ)∨B+j(λ)],Aj(λ)∧Bj(λ))mj=1:λ∈k}
3. C1⊓PC2={(λ,[A−j(λ)∧B−j(λ),A+j(λ)∧B+j(λ)],Aj(λ)∨Bj(λ))mj=1:λ∈k}
2.2. Some novel concepts of CmPFSs
Definition 2.6. [19] A CmPFS
on a universal set k is an internal CmPFS (ICmPFS) if A−j(λ)≤Aj(λ)≤A+j(λ), for all λ∈k and j=1,2,⋯,m.
Definition 2.7. [19] A CmPFS
on a universal set k is an external CmPFS (ECmPFS) if A−j(λ)≮Aj(λ)≮A+j(λ), for some λ∈k or j=1,2,⋯,m.
For simplicity, the ECmPFS is the inverse of the ICmPFS.
Definition 2.8. A CmPFS
for which [A−j,A+j]=0 and Aj(λ)=0 for all λ∈k and j=1,2,⋯,m is denoted by 0C.
Definition 2.9. A CmPFS
for which [A−j,A+j]=1 and Aj(λ)=1 for all λ∈k and j=1,2,⋯,m is denoted by 1C.
Definition 2.10. A CmPFS
for which [A−j,A+j]=0 and Aj(λ)=1 for all λ∈k and j=1,2,⋯,m is denoted by ¯0C.
Definition 2.11. A CmPFS
for which [A−j,A+j]=1 and Aj(λ)=0 for all λ∈k and j=1,2,⋯,m is denoted by ¯1C.
Definition 2.12. Let Cγ=([A−j,A+j],Aj)mj=1 be a CmPFN. The score function and accuracy functions of a CmPFN are respectively defined as
and
where λ([A−j,A+j] is the length of the fuzzy interval. Clearly, S(Cγ)∈[−1,1] and A(Cγ)∈[0,1].
Let C1γ and C2γ be two CmPFNs. Then the ranking of CmPFNs in association with the proposed score and accuracy functions is defined as follows:
● C1γ<C2γ if S(C1γ)<S(C2γ),
● If S(C1γ)=S(C1γ), then C1γ<Cγ if A(C1γ)<A(C2γ)
● If S(C1γ)=S(C1γ) and A(C1γ)=A(C2γ), then C1γ=C2γ
Definition 2.13. Let C1=⟨[A−1,A+1],[A−2,A+2],⋯,[A−m,A+m],A1,A2,⋯,Am⟩=⟨[A−j,A+j],Aj⟩mj=1 and C2=⟨[B−1,B+1],[B−2,B+2],⋯,[B−m,B+m],B1,ν2,⋯,Bm⟩=⟨[B−j,B+j],Bj⟩mj=1 be two CmPFSs.
The distance between two CmPFSs is defined by
3.
CmPF topology with P-order
Definition 3.1. Let k be a non-empty set and cmp(k) be the collection of all CmPFSs in k. The collection τCp containing the CmPFSs is called CmPF topology with P-order, abbreviated as CmPFTP, if it satisfies the following properties:
1. 0Cp, 1Cp∈τCp
2. If (Cp)i∈τCp∀i∈Λ then ⊔p(Cp)i∈τCp
3. If Cp1,Cp2∈τCp then Cp1⊓pCp2∈τCp
Then, the pair (k,τCp) is called CmPF topological space with P-order, abbreviated as CmPFTPS.
Example 3.2. Let k={k1,k2,k3} be a non-empty set. Then, cmp(k) is the collection of all P-cubic mPFSs (PCmPFSs) in k. We consider two cubic 3-polar fuzzy subsets of cmp(k) given as
The P-union and P-intersection results by applying Definition 2.5 to the CmPFSs Cp1 and Cp1 are given below in Tables 2 and 3, respectively.
Then, clearly
are cubic 3-polar fuzzy topologies with P-order.
Definition 3.3. Let k≠ϕ and τCp={Cpk} where Cpk are all cmPF subsets of k. Then, τCp is a P-cubic m-polar fuzzy topology on k that is also the largest P-cubic m-polar fuzzy topology on k, it is called P-discrete CmPF topology.
Definition 3.4. Let k≠ϕ and τCp={0Cp,1Cp} be a collection of CmPFSs. Then, τCp is a P-cubic m-polar fuzzy topology on k that is also the smallest P-cubic m-polar fuzzy topology on k, it is called P-indiscrete CmPF topology.
Definition 3.5. The members of the P-cubic m-polar fuzzy topology τCp are called P-cubic m-polar fuzzy open sets (PCmPFOSs) in (k,τCp).
Theorem 3.6. Let (k,τCp) be any P-cubic m-polar fuzzy topological space. Then
1. 0Cp and 1Cp are PCmPFOSs.
2. The P-union of any (finite/infinite) number of PCmPFOSs is a PCmPFOS.
3. The P-intersection of finite PCmPFOSs is a PCmPFOS.
Proof. 1. From the definition of the P-cubic m-polar fuzzy topology CmPFTP, 0Cp, 1Cp∈τCp. Hence 0Cp and 1Cp are PCmPFOSs.
2. Let {(Cp)i|i∈Λ} be PCmPFOSs. Then, Cp)i∈τCp. By the definition of CmPFTP
Hence, ⊔p(Cp)i represents PCmPFOSs.
3. Let Cp1,Cp2,...,Cpn be PCmPOSs. Then, by the definition of τCp
Hence, ⊓p(Cp)i represents PCmPFOSs.
Definition 3.7. The complement of the P-cubic m-polar fuzzy open sets are called the P-cubic m-polar fuzzy closed sets (PCmPFCSs) in (k,τCp).
Theorem 3.8. If (k,τCp) is any P-cubic m-polar fuzzy topological space, then
1. 0Cp and 1Cp are PCmPFCSs.
2. The P-intersection of any number of PCmPFCSs is a PCmPFCS.
3. The P-union of finite PCmPFCSs is a PCmPFCS.
Proof. 1. 0Cp and 1Cp are PCmPFCSs. By the definition of CmPFTP, 0Cp,1Cp∈τCp. Since the complement of 0Cp is 1Cp, and the complement of 1Cp is 0Cp. This shows that 0Cp and 1Cp are PCmPFCSs.
2. Let {(Cp)i|i∈Λ} be PCmPFCSs. Then,
By the definition of CmPFTP
Hence, ⊔p((Cp)i)C represents PCmPFOSs but
So, ⊓p(Cp)i represents PCmPFCSs.
3. Let Cp1,Cp2,...,Cpn be PCmPCSs. Then, (Cp1)C,(Cp2)C,...,(Cpn)C are PCmPFOSs. So,
by the definition of τCp
This implies that ⊓p((Cp)i)C∈τCp is PCmPFOSs but
Hence, ⊔p(Cp)i is PCmPFOSs.
Definition 3.9. The PCmPFSs which include both PCmPFOSs and PCmPFCSs are called P-cubic m-polar fuzzy clopen sets in (k,τCp).
Proposition 3.10. 1. In every τCp, 0Cp and 1Cp are P-cubic m-polar fuzzy clopen sets.
2. In discrete P-order CmPF topology, all cubic m-polar subsets of k are P-cubic m-polar fuzzy clopen sets.
3. In in-discrete P-order CmPF topology, only 0Cp and 1Cp are P-cubic m-polar fuzzy clopen sets.
Definition 3.11. Let (k,τCp1) and (k,τCp2) be two CmPFTPSs in k. These CmPFTPSs are said to be comparable if
or
If τCp1⊆PτCp2 then, τCp1 becomes P-cubic m-polar fuzzy coarser than τCp2. Similarly, τCp2 becomes P-cubic m-polar fuzzy finer than τCp1.
Example 3.12. Let k≠ϕ; then from Example 3.2,
and
are cubic 3-polar fuzzy topologies on k. Since, τCp2⊆PτCp1. Therefore, τCp2 becomes a P-cubic m-polar coarser than τCp1.
3.1. Sub spaces of CmPFTP
Definition 3.13. Let (k,TCpk) be a CmPFTPS. Let Y⊆k and TCpY be a CmPFTP on Y with PCmPFOSs that are
where Cpk are PCmPFOSs of TCpk, TCpY are PCmPFOSs of TCpY and ˘Y is an absolute PCmPFS on Y. Then, TCpY is the P-cubic m-polar fuzzy subspace of TCpk i.e.,
Example 3.14. Let k={k1,k2} be a non-empty set.
Then,
is a cubic 2-polar fuzzy topology with P-order on k. Now, the absolute cubic 2-polar fuzzy set on Y={k1}⊆k is
Since
we have
which is a cubic 2-polar fuzzy relative topology of TCpk.
Let (k,TCpk) be a CmPFTPS. Let Y⊆k and TCpY be a CmPFTP on Y with PCmPFOSs that are
where Cpk denotes the PCmPFOSs of TCpk, TCpY are PCmPFOSs of a TCpY and Y is any subset of PCmPFS on Y. Then, TCpY is the P-cubic m-polar fuzzy subspace of TCpk i.e.,
Example 3.15. From Example 3.14
is a cubic 2-polar fuzzy topology with P-order on k.
Now, take any cubic 2-polar fuzzy subset on k such that Y={k1}⊆k is
Since
we have
which is a cubic 2-polar fuzzy relative topology of TCpk.
3.2. Interior, closure, frontier and exterior of the PCmPFS
Definition 3.16. Let (k,TCp) be a CmPFTPS and Cp ∈cmp(k) then the interior of Cp is denoted as Cp0 and defined as the union of all open CmPF subsets contained in Cp. It is the greatest open cubic m-polar fuzzy set contained in Cp.
Example 3.17. Consider the cubic 3-polar topological space as presented in Example 3.2. Let Cp3∈cmp(k) given as
Then,
Theorem 3.18. Let (k,TCp) be a CmPFTPS and Cp ∈cmp(k). Then Cp is an open CmPFS if, and only if, (Cp)0=Cp.
Proof. If Cp is an open CmPFS then the greatest open CmPFS contained in Cp is itself Cp. Thus
Conversely, if (Cp)0=Cp then (Cp)0 is an open CmPFS. This means that Cp is an open CmPFS.
Theorem 3.19. Let (k,TCp) be a CmPFTPS and Cp1, Cp2 ∈cmp(k). Then
● ((Cp)0)0=(Cp)0
● Cp1⊆pCp2⇒(Cp1)0⊆p(Cp2)0
● (Cp1⊓pCp2)0=(Cp1)0⊆p(Cp2)0
● (Cp1⊔pCp2)0⊇p(Cp1)0⊔p(Cp2)0
Proof. The proof is obvious.
Definition 3.20. Let (k,TCp) be a CmPFTPS and Cp ∈cmp(k) then the closure of Cp is denoted as ¯Cp and is defined as the intersection of all closed cubic m-polar fuzzy supersets of Cp. It is the smallest closed cubic m-polar fuzzy superset of Cp.
Example 3.21. Consider the cubic 3-polar topological space given in Example 3.2. Then, the closed CmPFSs are (0Cp)c=1Cp and (1Cp)c=0Cp
Let Cp4∈cmp(k) be given as
Then,
is a closed CmPFS.
Theorem 3.22. Let (k,TCp) be a CmPFTPS and Cp ∈cmp(k). Then, Cp is a closed CmPFS if, and only if, ¯Cp=Cp.
Proof. If Cp is a closed CmPFS then the smallest closed CmPFS superset of Cp is itself Cp. Thus
Conversely, if ¯Cp=Cp then ¯Cp is a closed CmPFS. This means that Cp is a closed CmPFS.
Definition 3.23. Let Cp be a P-cubic m-polar fuzzy subset of (k,TCp); then, its frontier or boundary is denoted by
Definition 3.24. Let Cp be a P-cubic m-polar fuzzy subset of (k,TCp); then, its exterior is denoted by
Example 3.25. Consider the cubic 3-polar topological space as constructed in Example 3.2 and Cp3 and Cp4 from Example 3.17 and Example 3.21, respectively. Then,
(Cp3)0=Cp1, ¯Cp3=1Cp
Fr(Cp3)=Cp1, Ext(Cp3)=0Cp
(Cp4)0=0Cp, ¯Cp4=(Cp2)c
Fr(Cp4)=(Cp2)c, Ext(Cp4)=0Cp
Remark. In the case of the CmPFTPS, the law of contradiction Cp⊓p(Cp)c=0Cp and law of the excluded middle Cp⊔p(Cp)c=1Cp do not hold in general. From Example 3.17,
Cp3⊓p(Cp3)c≠0Cp
Cp3⊔p(Cp3)c≠1Cp
Theorem 3.26. Let (k,TCp) be a CmPFTPS and Cp ∈cmp(k). Then
1. (Cp0)c=¯(Cpc)
2. (¯Cp)c=(Cpc)0
3. Ext(Cpc)=Cp0
4. Ext(Cp)=(Cpc)0
5. Ext(Cp)⊔pFr(Cp)⊔pCp0≠1Cp
6. Fr(Cp)=Fr(Cpc)
7. Fr(Cp)⊓pCp0≠0Cp
Proof. 1. The proof is obvious.
2. The proof is obvious.
3. Ext(Cpc)=(¯Cpc)c
Ext(Cpc)=((Cpc)c)0
Ext(Cpc)=Cp0
4. Ext(Cp)=(¯Cp)c
Ext(Cp)=(Cpc)0
5. Ext(Cp)⊔pFr(Cp)⊔pCp0≠1Cp. Given Example 3.25, we can see that
Ext(Cp3)⊔pFr(Cp3)⊔pCp0≠1Cp.
6. Fr(Cpc)=¯(Cpc)⊓p¯((Cpc)c) Fr(Cpc)=¯(Cpc)⊓p¯(Cp)=Fr(Cp)
7. Fr(Cp)⊓pCp0≠0Cp. From Example 3.25, we can see that,
Fr(Cp3)⊓pCp0≠0Cp.
Definition 3.27. A PCmPFS is said to be dense in a universal set k if
From Example 3.17 ¯Cp3=1Cp. So, ¯Cp3 is dense in k.
Definition 3.28. A P-cubic m-polar number Cγ=([A−j,A+j],Aj)mj=1 belong PCmPFS if, and only if, A−i(λ)≤A−j(λ),A+i(λ)≤A+j(λ) and Ai(λ)≤Aj(λ) j=1,2,...,m and l∈k
Let (k,TCp) be a CmPFTPS. A PCmPFS ´Cp of k which contains a PCmPF number Cγ∈k is said to be a neighborhood of Cγ if, there exists a PCmPFOS Cp such that
Example 3.29. From Example 3.2, consider a PCmPF number
which belongs to the PCmPFOS Cp1, which is a PCmPF subset of Cp2. From this, we can say that Cp2 is a neighborhood of Cγ.
3.3. P-cubic m-polar fuzzy basis
Definition 3.30. Let (k,TCp) be a CmPFTPS. Then B⊆TCp is said to be the P-cubic m-polar fuzzy basis for TCp if, for each Cp∈TCp, ∃ B∈B such that
Example 3.31. From Example 3.2,
is a P-cubic 3-polar fuzzy topology on k. Then,
is a P-cubic 3-polar fuzzy basis for τCp.
4.
CmPF topology with R-order
Definition 4.1. Let k be a non-empty set and cmp(k) be the assemblage of all CmPFSs in k. The collection TCr containing CmPFSs is called a CmPF topology with R-order, abbreviated as CmPFTr, if it satisfies the following properties:
1. ¯0Cr, ¯1Cr∈TCr
2. If (Cr)i∈TCr∀i∈Λ then ⊔r(Cr)i∈TCr
3. If Cr1,Cr2∈TCr then Cr1⊓rCr2∈TCr
Then, the pair (k,TCr) is called the CmPF topological space with R-order, abbreviated as CmPFTrS.
Example 4.2. Let k={k1,k2,k3} be a non-empty set. Then, cmp(k) is the collection of all R-cubic m-polar fuzzy sets (RCmPFSs) in k. We consider two cubic 3-polar fuzzy subsets of cmp(k), given as
The R-union and R-intersection results obtained by applying Definition 2.5 to the CmPFSs Cr1 and C21 are given below in Tables 4 and 5, respectively.
Then, clearly
are cubic 3-polar fuzzy topologies with R-order.
Example 4.3. Let k={k1,k2} be a non-empty set. Then, cmp(k) is the collection of all RCmPFSs in k. We consider five cubic 3-polar fuzzy subsets of cmp(k), given as
The R-union and R-intersection results obtained by applying Definition 2.5 to these CmPFSs are given below in Tables 6 and 7, respectively.
Then clearly,
are cubic 3-polar fuzzy topologies with R-order.
Definition 4.4. Let k≠ϕ and TCr={Crk}, where Crk denotes all of the CmPFSs of k. Then, TCr is a R-cubic m-polar fuzzy topology on k that is also the largest R-cubic m-polar fuzzy topology on k; it is called an R-discrete CmPF topology.
Definition 4.5. Let k≠ϕ and TCr={¯0Cr,¯1Cr} be the collection of CmPFSs. Then, TCr is a R-cubic m-polar fuzzy topology on k that is also the smallest R-cubic m-polar fuzzy topology on k; it is called an R-indiscrete cubic m-polar fuzzy topology.
Definition 4.6. The members of the RCmPF topology TCr are called RCmPFOSs in (k,TCr).
Theorem 4.7. If (k,TCr) is any R-cubic m-polar fuzzy topological space, then
1. ¯0Cr and ¯1Cr are RCmPFOSs.
2. The R-union of any (finite/infinite) number of RCmPFOSs is an RCmPFOS.
3. The R-intersection of finite RCmPFOSs is an RCmPFOS.
Proof. The proof is the same as that for PCmPFOSs.
Definition 4.8. The complement of RCmPFOSs are called RCmPFCSs in (k,TCr).
Theorem 4.9. If (k,TCr) is any R-cubic m-polar fuzzy topological space, then
1. ¯0Cr and ¯1Cr are RCmPFCSs.
2. The R-intersection of any (finite/infinite) number of RCmPFCSs is an RCmPFCS.
3. The R-union of finite RCmPFCSs is an RCmPFCS.
Proof. The proof is the same as that for PCmPFCSs.
Definition 4.10. Let (k,TCr1) and (k,TCr2) be two CmPFTrSs in k. Two CmPFTrSs are said to be comparable if,
Furthermore if, TCr1⊆RTCr2, then TCr1 becomes R-cubic m-polar fuzzy coarser than TCr2. Similarly, TCr2 becomes R-cubic m-polar fuzzy finer than TCr1.
Example 4.11. Let k≠ϕ; then from Example 4.3,
are cubic 3-polar fuzzy topologies on k. Since TCr2⊆RTCr1. So that, TCr2 is R-cubic m-polar coarser then TCr1.
4.1. Sub spaces of CmPFTr
Definition 4.12. Let (k,TCrk) be a CmPFTrS. Let Y⊆k and TCrY be a CmPFTr on Y with the following RCmPFOSs:
where Crk are RCmPFOSs of TCrk, TCrY represents the RCmPFOSs of TCrY, and ˘Y is an absolute RCmPFS on Y. Then, TCrY is the R-cubic m-polar fuzzy subspace of TCrk i.e.,
Example 4.13. Let k={k1} be a non-empty set. From Example 4.3,
TCr={0Cr,1Cr,¯0Cr,¯1Cr,Cr1,Crr2,Cr3,Cr4,Cr5} is a cubic 3-polar fuzzy topology with R-order on k.
Now, the absolute cubic 3-polar fuzzy set on Y={k1}⊆k is
Since
it follows that
is a cubic 3-polar fuzzy relative topology of TCrk.
Let (k,TCrk) be a CmPFTrS. Let Y⊆k and TCrY be a CmPFTr on Y with the following RCmPFOSs:
where Crk represents the RCmPFOSs of TCrk, TCrY are RCmPFOSs of TCRY and Y is any subset of an RCmPFS on Y. Then, TCrY is an R-cubic m-polar fuzzy subspace of TCrk i.e.,
Example 4.14. Let k={k1} be a non-empty set. From Example 4.3
is a cubic 3-polar fuzzy topology with R-order on k.
Now, the absolute cubic 3-polar fuzzy set on Y={k1}⊆k is
Since
it follows that
is a cubic 3-polar fuzzy relative topology of TCrk.
4.2. Interior, closure, frontier and exterior of RCmPFSs
Definition 4.15. Let (k,TCr) be a CmPFTrS and Cr∈cmp(k) then the interior of Cr is denoted as Cr0 and is defined as the union of all open cubic m-polar fuzzy subsets contained in Cr. It is the greatest open CmPFS contained in Cr.
Example 4.16. Consider the cubic 3-polar topological space as constructed in Example 4.3. Let Cr6∈cmp(k): given as
Cr6={(k1,[0.33,0.50],[0.42,0.69],[0.52,0.73],0.8,0.7,0.6),(k2,[0.38,0.49],[0.51,0.64],[0.63,0.82],0.7,0.6,0.5)}.
So,
is open CmPFS.
Theorem 4.17. Let (k,TCr) be a CmPFTrS and Cr∈cmp(k). Then, Cr is an open CmPFS if, and only if, (Cr)0=Cr.
Proof. The proof is obvious.
Definition 4.18. Let (k,TCr) be a CmPFTrS and Cr ∈cmp(k) then the closure of Cr is denoted as ¯Cr and is defined as the intersection of all closed cubic m-polar fuzzy supersets of Cr. It is the smallest closed cubic m-polar fuzzy superset of Cr.
Example 4.19. Consider the cubic 3-polar topological space constructed in Example 4.3. Then, the closed CmPFSs are
(0Cp)c=1Cp, (1Cp)c=0Cp, (¯0Cr)c=¯1Cr, (¯0Cr)c=¯0Cr
Let Cr7∈cmp(k) be given as
Then,
is closed a CmPFS.
Theorem 4.20. Let (k,TCr) be a CmPFTrS and Cr ∈cmp(k). Then, Cr is a closed CmPFS ⇔ ¯Cr=Cr.
Proof. The proof is obvious.
Definition 4.21. Let Cr be an R-cubic m-polar fuzzy subset of (k,TCr); then, its frontier or boundary is denoted by
Definition 4.22. Let Cr be an R-cubic m-polar fuzzy subset of (k,TCr); then, its exterior is denoted by
Example 4.23. Consider the cubic 3-polar topological space given in Example 4.3 and Cr7 from Example 4.19. Then
Cr07=(Cr1)c, ¯Cr7=(Cr1)c
Fr(Cr7)=(Cr1)c, Ext(Cr7)=(Cr1)c
Remark. For a CmPFTrS, the law of contradiction, Cr⊓p(Cr)c=0Cr, and the law of excluded middle, Cr⊔R(Cr)c=1Cr do not hold in general. From Example 4.19
Cr7⊓R(Cr7)c≠0Cr
Cr7⊔R(Cr7)c≠1Cr
Definition 4.24. An RCmPFS is said to be dense in a universal set k if
Definition 4.25. An RCmPF number Cγ=([A−j,A+j],Aj)mj=1 belong RCmPFS if, and only if,
A−i(λ)≤A−j(λ),A+i(λ)≤A+j(λ) and Ai(λ)≥Aj(λ) j=1,2,...,m and l∈k
Let (k,TCr) be a CmPFTrS. An RCmPFS ´Cr of k which contains an RCmPF number Cγ∈k is said to be a neighborhood of Cγ if, there exists an RCmPFOS Cr containing Cγ, such that
Example 4.26. From Example 4.3, an R-cubic m-polar fuzzy number
belongs to the RPCmPFOS Cr1 which is an R-cubic m-polar fuzzy subset of Cr3. From this, we can say that Cr3 is a neighborhood of Cγ.
4.3. R-Cubic m-polar fuzzy basis
Definition 4.27. Let (k,TCr) be a CmPFTrS. Then, B⊆TCr is said to be the R-cubic m-polar fuzzy basis for TCr if, for each Cr∈TCr, ∃ B∈B such that
Example 4.28. From Example 4.3,
is an R-cubic 3-polar fuzzy topology on k. Then,
is the R-cubic 3-polar fuzzy basis for TCr.
5.
Extension of VIKOR method to CmPFSs
As a sample, in this section, we first discuss several types of CKD by providing a brief but comprehensive overview of this fatal disease, including its types and symptoms, and then use the established VIKOR methodology to diagnose those who are affected.
Case Study
Kidneys have very important positions for human beings. They work as channels for your blood, eliminating waste, poisons, and surplus liquids. They also help to manage circulatory strain and synthetic compounds in the blood, keep bones sound and animate red platelet creation. If you have CKD, your kidneys have been damaged for more than a few months. Diseased kidneys also do not channel blood properly, which can prompt an assortment of genuine human concerns. There are five stages to CKD. The phases are determined by the results of an eGFR test and how successfully your kidneys filter waste and excess fluid from your blood. Kidney disease worsens as the stages progress and your kidneys become less effective. The stages of CKD*, †, ‡ are summarized below.
*https://www.freseniuskidneycare.com/kidney-disease/stages
†https://www.healthline.com/health/ckd-stages
‡https://www.cdc.gov/kidneydisease/prevention-risk.html
Stage 1 of CKD
An eGFR of 90 or higher indicates that your kidneys are healthy, but you might have other symptoms of kidney damage. Protein in your urine or physical harm to your kidneys could be signs of kidney damage. When the kidneys function at a 90 or higher eGFR, there are usually no symptoms. Here are some things one can do to help slow the harm to the kidneys in Stage 1. In the case of diabetes, keep the blood sugar under control and maintain the blood pressure. Consume a nutritious diet. Do not use tobacco or smoke. Engage in physical activity for 30 minutes five days a week and maintain a healthy weight.
Stage 2 of CKD
If you have Stage 2 CKD kidney problem, it is slight, and corresponds to an eGFR between 60 and 89. Almost all of the time, an eGFR between 60 and 89 indicates that your kidneys are healthy and operating correctly. However, if you do have Stage 2 kidney disease, you have some other symptoms of kidney damage even if your eGFR is normal. You may still be symptom-free at this point. Or the symptoms are general, such as fatigue, appetite loss, sleep issues and weakness.
Stage 3 of CKD
If your eGFR is between 30 and 59, you have Stage 3 CKD. An eGFR of 30 to 59 implies that your kidneys have been harmed and are not operating correctly. There are two stages in Stage 3; if your eGFR is between 45 and 59, you are at Stage 3a and if your eGFR is between 30 and 44, you are in Stage 3b. The kidneys are not filtering waste, poisons, or fluids efficiently, and they are starting to pile up. Many persons with kidney disease in Stage 3 do not show any noticeable symptoms. However, if symptoms exist, they may include hand and foot swelling, pain in the back, and more or less urination than usual.
Stage 4 of CKD
You have Stage 4 CKD if your eGFR is between 15 and 29. An eGFR between 15 and 30 indicates that your kidneys are relatively or badly damaged and are not operating normally. Stage 4 kidney disease must be taken seriously because it is the final step before kidney failure. Many people with Stage 4 CKD experience symptoms such as swelling of the hands and feet, back pain and more or less urination than usual. At Stage 4, you will most likely experience medical complications as waste piles up in your kidneys and your body fails to function properly, such as excessively high blood pressure, anemia and a disease of the bones.
Stage 5 of CKD
You have Stage 5 CKD if your eGFR is below 15. An eGFR less than 15 indicates that the kidneys are nearing failure or have failed completely. When your kidneys fail, waste accumulates in your blood, making you very sick. The following are some of the symptoms of kidney failure: itching, cramps in the muscles, throwing up, swelling of the hands and feet, back pain, more or less urination than usual, breathing difficulties and sleeping problems. Once your kidneys fail, you will need dialysis or a kidney transplant to survive.
We show, in this part, how the VIKOR might be applied to CmPFSs. Right away, we will apply VIKOR to the CmPFS and later apply it to deal with an issue from life sciences. We start by expounding the recommended strategy stage by stage as described below. The linguistic terms for weighing choices are given in Table 8.
Suppose there are ȷ number of DMs, that is, D1,D2,⋯,Dȷ, subject to 'ı' number of criteria μ1,μ2,⋯,μı and 'λ' number of alternatives ν1,ν2,⋯,νλ.
Step 1. In the first step: the DMs have to allocate the preference weights to the criteria. Let ωjk be the weight allocated by jth DM to the kth criteria. We set the weights in the matrix form ℘=[ωjk]ȷ×ı for convenience.
Step 2. The weights assigned by DMs must be normalized. Suppose that the weights ωjk for the criteria are not normal, so they must be normalized by utilizing the formula ˉωjk=ωjk√∑iω2jk. Then, the weights are gathered as W=(ω1,ω2,⋯,ωı), where ωi=1ȷ∑iˉωjk∑jˉωjk.
Step 3. Every model of a DM is a PCmPF matrix Dx=(ζxjk)λ×ı, x=1,2,⋯,ȷ, where ζxjk is the value that is allotted by the DM X to the criteria K corresponding the alternative J.
Step 4. Compute the PCmPFS decision matrix by taking the average. The matrix that is obtained can be named A=(ζjk)λ×ı.
Step 5. Construct the PCmPF weighted matrix to be B=(ςjk)λ×ı, where ςjk=ωkζjk.
Step 6. Positive ideal solutions (PIS) and negative ideal solutions (NIS) with P-order (R-order) for CmPFSs are respectively obtained by using the relations given as
PCmpF-PIS: ζ+k=maxjPςjk or ζ+k=maxjRςjk
PCmPF-NIS: ζ−k=minjPςjk or ζ−k=minjRςjk.
Step 7. To find the strategic value of VIKOR, the utility value Si, regret value Ri and compromise value Qi are calculated by using following formula
Here, S+=∨iSi,S−=∧iSi,R+=∨iRi and R−=∧iR. The parameter χ is coefficient of decision analysis. If, in decision making, the majority selects the compromise solution, then we take χ>0.5, where χ>0.5 denotes a veto and, for agreement, χ=0.5
Note that the distance between two CmPFNs,
C1=⟨[A−1,A+1],[A−2,A+2],⋯,[A−m,A+m],A1,A2,⋯,Am⟩=⟨[A−j,A+j],Aj⟩mj=1, and
C2=⟨[B−1,B+1],[B−2,B+2],⋯,[B−m,B+m],B1,ν2,⋯,Bm⟩=⟨[B−j,B+j],Bj⟩mj=1, is defined as
Step 8. We rank Si, Ri and Qi by arranging them in ascending order. The alternative ϱa considered as a compromise solution if it holds the highest ranking (minimum value) and satisfies the following two conditions at the same time.
C-1 If ϱa1 and ϱa1 are the top two alternatives having minimum values in Qi, then
Q(ϱa2)−Q(ϱa1)≥1ı−1
where ı is the number of criteria.
C-2 The alternative ϱa1 must also be supreme ranked by at least one of the Si or Ri.
If the above two conditions are not satisfied at a time, then we have multiple compromise solutions. In this case, the conditions are given as follows:
● If C-1 is satisfied, then ϱa1 and ϱa2 are both compromise solutions.
● If C-1 is not satisfied and
then, ϱa1,ϱa2,...,ϱak should act as multiple compromise solutions.
5.1. Numerical example
Step 1. Suppose D={Di:i=1,⋅⋅⋅,4} is the set of medical experts, P={ϱj:j=1,⋅⋅⋅,4} is the set of patients and C={ξk:k=1,⋅⋅⋅,4} for a set of criteria, where ξ1=vomitting, ξ2= nausea, ξ3= loss of appetite and ξ4= fatigue and weakness.
Step 2. The matrix ℘ of the weighted criteria is
where ωij represents the weights assigned by the DMs to criteria.
Step 3. The normalized weighted matrix appears to be
and thus the weight vectors comes out to be W={0.26,0.24,0.23,0.26}
Step 4. We consider that the four experts give the following four assessment PCmPFS matrices.
The aggregated matrix is
Step 5. The weighted PCmPFS matrix is
Step 6. Thus, PCmP-PIS and PCmP-NIS, respectively are
Step 7. By selecting χ=0.5, we calculated the values of Si, Ri and Qi for each alternative by making use of following formula
They are given in Table 9.
Step 8. The ranking of choices are as follows:
By Qi : ϱ3<ϱ4<ϱ1<ϱ2
By Si : ϱ3<ϱ4<ϱ1<ϱ2
By Ri : ϱ3<ϱ4<ϱ2<ϱ1
Since, Q(ϱ4)−Q(ϱ3)≱13
So, by Q(ϱ4)−Q(ϱ3)>13, we infer that both ϱ4 and ϱ3 serve as multiple compromise solutions.
Comparative analysis:
The advantages of using a CmPFS are described in Table 10.
To comparatively analyze the method of the ranking of alternatives and an optimal alternative, we solve the above problem by applying the TOPSIS approach to the same data. The first six steps of TOPSIS is the same as VIKOR. In Step 7, we find the the closeness of each alternative from the PCmP-PIS and PCmP-NIS. In Step 8, we find the relative closeness of each alternative. In Step 9, we rank the alternatives.
Step 7 and 8. The distance of each alternative from the PCmP-PIS and PCmP-NIS and their relative closeness are given in Table 11.
Step 9. Thus, the ranking of each patient is
This ranking shows that Patient ϱ4 is in a more critical situation. So this is optimal decision for the VIKOR and TOPSIS approaches.
6.
Conclusions
When applied in MCDM techniques, a CmPFS is an effective model for coping with uncertain information. A CS is a two-component system that can describe information in terms of a fuzzy interval and a fuzzy number. Alternatively, an mPFS describes multi-polarity with m degrees. To take advantages of a CS and mPFS, we focused on a hybrid model of CmPFS and introduced the idea of a topological structure of CmPFSs and CmPF topology with P-order (R-order). We defined certain concepts of a CmPF topology such as, open sets, closed sets, subspaces and dense sets, as well as the interior, exterior, frontier, neighborhood, and basis with P-order (R-order). A CmPF topology is a robust approach for modeling big data, data analysis, and diagnosis, etc. An extension of the VIKOR method for MCGDM with a CmPF topology was designed and its application to CKD diagnosis is presented. A comparative analysis of the proposed approach and TOPSIS method was also performed to seek the optimal decision.
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgement
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia for funding this work through the Research Group Program under grant number RGP.2/211/43.