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Research article

Cubic m-polar fuzzy topology with multi-criteria group decision-making

  • The concept of cubic m-polar fuzzy set (CmPFS) is a new approach to fuzzy modeling with multiple membership grades in terms of fuzzy intervals as well as multiple fuzzy numbers. We define some fundamental properties and operations of CmPFSs. We define the topological structure of CmPFSs and the idea of cubic m-polar fuzzy topology (CmPF topology) with P-order (R-order). We extend several concepts of crisp topology to CmPF topology, such as open sets, closed sets, subspaces and dense sets, as well as the interior, exterior, frontier, neighborhood, and basis of CmPF topology with P-order (R-order). A CmPF topology is a robust approach for modeling big data, data analysis, diagnosis, etc. An extension of the VIKOR method for multi-criteria group decision making with CmPF topology is designed. An application of the proposed method is presented for chronic kidney disease diagnosis and a comparative analysis of the proposed approach and existing approaches is also given.

    Citation: Muhammad Riaz, Khadija Akmal, Yahya Almalki, S. A. Alblowi. Cubic m-polar fuzzy topology with multi-criteria group decision-making[J]. AIMS Mathematics, 2022, 7(7): 13019-13052. doi: 10.3934/math.2022721

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  • The concept of cubic m-polar fuzzy set (CmPFS) is a new approach to fuzzy modeling with multiple membership grades in terms of fuzzy intervals as well as multiple fuzzy numbers. We define some fundamental properties and operations of CmPFSs. We define the topological structure of CmPFSs and the idea of cubic m-polar fuzzy topology (CmPF topology) with P-order (R-order). We extend several concepts of crisp topology to CmPF topology, such as open sets, closed sets, subspaces and dense sets, as well as the interior, exterior, frontier, neighborhood, and basis of CmPF topology with P-order (R-order). A CmPF topology is a robust approach for modeling big data, data analysis, diagnosis, etc. An extension of the VIKOR method for multi-criteria group decision making with CmPF topology is designed. An application of the proposed method is presented for chronic kidney disease diagnosis and a comparative analysis of the proposed approach and existing approaches is also given.



    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.

    Table 1.  Some applications of the VIKOR technique.
    Researchers Benchmarks Applications
    Zhao et al. [40] Extended VIKOR Supplier selection
    Joshi and Kumar [41] Extended VIKOR Supplier selection
    Park et al. [42] Extended VIKOR Teacher performance evaluation
    Shouzhen et al. [43] Modified VIKOR Supply chain management
    Arya and Kumar [44] VIKOR-TODIM Management information system
    Devi [45] Extended VIKOR Robot selection
    Luo and Wang [46] Extended VIKOR Distance measure
    Chen [47] PF- VIKOR Evaluating internet stock performance
    Zhou and Chen [48] Extended PF-VIKOR Selection of Blockchain technology
    Bakioglu and Atahan [49] AHP integrated VIKOR Prioritize risk in self-driving vehicle
    Guleria and Bajaj [50] VIKOR Site selection for power plant
    Gul [51] VIKOR Assessment of safety risk
    Kirisci et al. [52] Novel VIKOR Survey on early childhood in quarantine
    Dalapati and Pramanik [53] NC-VIKOR Selection of green supplier for cars
    Pramanik et al. [54] NC-VIKOR Selection of green supplier for cars
    Pramanik et al. [55] VIKOR Best option for money investment
    Wang et al. [56] VIKOR Risk evaluation for construction project
    Arya and Kumar [57] TODIM-VIKOR Selection of team leader in company
    Joshi [58] VIKOR Selection of election bound country
    Arya and Kumar [59] VIKOR-TODIM Selecting opinion polls
    Khan et al. [60] VIKOR Selection of priority area for investment
    Yue [61] Extended VIKOR Software reliability assessment
    Meksavang [62] Extended PiF-VIKOR Supplier management
    Singh and Kumar [63] VIKOR Supplier selection

     | Show Table
    DownLoad: CSV

    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.

    In this section, we discuss some elementary concepts of CmPFSs.

    Definition 2.1. [18] A CS on a universal set k is expressed as

    ={,[A(),A+()],λ():k},

    in which A=[A(),A+()] is an interval-valued fuzzy set and λ() is a fuzzy set on k. For simplicity, the CS ={,[A(),A+()],A():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

    ðp={(,μ1(),,μm()):k},

    where μ1(),,μm() 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

    C={(,[A1(),A+1()],[A2(),A+2()],,[Am(),A+m()],A1(),A2(),,Am()):k}

    Here, [Aj(),A+j()]mj=1 are fuzzy valued intervals and (Aj())mj=1 are fuzzy numbers. For simplicity, we can write the cubic m-polar fuzzy number (CmPFN) as

    Cγ=([Aj,A+j],Aj)mj=1

    Definition 2.4. [18] Let Ia=[ϝa,ϝ+a] and Ib=[b,+b] be any two fuzzy valued intervals. Then

    1. IaIbϝab and ϝ+a+b

    2. IaIbϝab and ϝ+a+b

    3. Ia=Ibϝa=b and ϝ+a=+b

    Definition 2.5. [19] Let us consider two CmPFSs on k given by

    C1={(λ,[Aj,A+j],Aj)mj=1:λk},
    C2={(λ,[Bj,B+j],Bj)mj=1:λk}

    Some basic operations on these sets with P-order are defined as

    1. (C1)c={(λ,[1A+j,1Aj],1Aj)mj=1:λk}

    2. (C2)c={(λ,[1B+j,1Bj],1Bj)mj=1:λk}

    3. C1PC2Aj(λ)Bj(λ),A+j(λ)B+j(λ) and Aj(λ)Bj(λ)

    4. C1PC2={(λ,[Aj(λ)Bj(λ),A+j(λ)B+j(λ)],Aj(λ)Bj(λ))mj=1:λk}

    5. C1PC2={(λ,[Aj(λ)Bj(λ),A+j(λ)B+j(λ)],Aj(λ)Bj(λ))mj=1:λk}

    Similarly, some basic operations on the above two CmPFSs with R-order are defined as

    1. C1RC2Aj(λ)Bj(λ),A+j(λ)B+j(λ) and Aj(λ)Bj(λ)

    2. C1RC2={(λ,[Aj(λ)Bj(λ),A+j(λ)B+j(λ)],Aj(λ)Bj(λ))mj=1:λk}

    3. C1PC2={(λ,[Aj(λ)Bj(λ),A+j(λ)B+j(λ)],Aj(λ)Bj(λ))mj=1:λk}

    Definition 2.6. [19] A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    on a universal set k is an internal CmPFS (ICmPFS) if Aj(λ)Aj(λ)A+j(λ), for all λk and j=1,2,,m.

    Definition 2.7. [19] A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    on a universal set k is an external CmPFS (ECmPFS) if Aj(λ)Aj(λ)A+j(λ), for some λk or j=1,2,,m.

    For simplicity, the ECmPFS is the inverse of the ICmPFS.

    Definition 2.8. A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    for which [Aj,A+j]=0 and Aj(λ)=0 for all λk and j=1,2,,m is denoted by 0C.

    Definition 2.9. A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    for which [Aj,A+j]=1 and Aj(λ)=1 for all λk and j=1,2,,m is denoted by 1C.

    Definition 2.10. A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    for which [Aj,A+j]=0 and Aj(λ)=1 for all λk and j=1,2,,m is denoted by ¯0C.

    Definition 2.11. A CmPFS

    C={(λ,[Aj,A+j],Aj)mj=1:λk}

    for which [Aj,A+j]=1 and Aj(λ)=0 for all λk and j=1,2,,m is denoted by ¯1C.

    Definition 2.12. Let Cγ=([Aj,A+j],Aj)mj=1 be a CmPFN. The score function and accuracy functions of a CmPFN are respectively defined as

    S(Cγ)=Σmj=1|λ([Aj,A+j])Aj|m (2.1)

    and

    A(Cγ)=Σmj=1(λ([Aj,A+j])+Aj)2m (2.2)

    where λ([Aj,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=[A1,A+1],[A2,A+2],,[Am,A+m],A1,A2,,Am=[Aj,A+j],Ajmj=1 and C2=[B1,B+1],[B2,B+2],,[Bm,B+m],B1,ν2,,Bm=[Bj,B+j],Bjmj=1 be two CmPFSs.

    The distance between two CmPFSs is defined by

    d(C1,C2)=[mj=1|Aj+A+j2Bj+B+j2|m+mj=1|AjBj|m]1/m (2.3)

    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, 1CpτCp

    2. If (Cp)iτCpiΛ then p(Cp)iτCp

    3. If Cp1,Cp2τCp then Cp1pCp2τCp

    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

    Cp1={(k1,[0.23,0.46],[0.31,0.52],[0.47,0.65],0.24,0.32,0.51),(k2,[0.30,0.42],[0.45,0.56],[0.53,0.69],0.20,0.41,0.72),(k3,[0.44,0.63],[0.55,0.78],[0.61,0.83],0.42,0.53,0.60)}Cp2={(k1,[0.19,0.36],[0.24,0.48],[0.39,0.52],0.20,0.31,0.42),(k2,[0.27,0.38],[0.39,0.52],[0.49,0.63],0.15,0.30,0.65),(k3,[0.36,0.61],[0.50,0.68],[0.60,0.81],0.39,0.52,0.58)}

    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.

    Table 2.  Union with P-order.
    p 0Cp 1Cp Cp1 Cp2
    0Cp 0Cp 1Cp Cp1 Cp2
    1Cp 1Cp 1Cp 1Cp 1Cp
    Cp1 Cp1 1Cp Cp1 Cp1
    Cp2 Cp2 1Cp Cp1 Cp2

     | Show Table
    DownLoad: CSV
    Table 3.  Intersection with P-order.
    p 0Cp 1Cp Cp1 Cp2
    0Cp 0Cp 0Cp 0Cp 0Cp
    1Cp 0Cp 1Cp Cp1 Cp2
    Cp1 0Cp Cp1 Cp1 Cp2
    Cp2 0Cp Cp2 Cp2 Cp2

     | Show Table
    DownLoad: CSV

    Then, clearly

    TCp1={0Cp,1Cp,Cp1,Cp2},
    TCp2={0Cp,1Cp,Cp1},
    TCp3={0Cp,1Cp,Cp2},
    TCp4={0Cp,1Cp},

    are cubic 3-polar fuzzy topologies with P-order.

    Definition 3.3. Let kϕ 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 kϕ 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, 1CpτCp. Hence 0Cp and 1Cp are PCmPFOSs.

    2. Let {(Cp)i|iΛ} be PCmPFOSs. Then, Cp)iτCp. By the definition of CmPFTP

    p(Cp)iτCp

    Hence, p(Cp)i represents PCmPFOSs.

    3. Let Cp1,Cp2,...,Cpn be PCmPOSs. Then, by the definition of τCp

    p(Cp)iτCp

    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,1CpτCp. 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|iΛ} be PCmPFCSs. Then,

    ((Cp)i)CτCp

    By the definition of CmPFTP

    p((Cp)i)CτCp

    Hence, p((Cp)i)C represents PCmPFOSs but

    (p((Cp)i)C)=(p(Cp)i)C

    So, p(Cp)i represents PCmPFCSs.

    3. Let Cp1,Cp2,...,Cpn be PCmPCSs. Then, (Cp1)C,(Cp2)C,...,(Cpn)C are PCmPFOSs. So,

    (Cp1)C,(Cp2)C,...,(Cpn)CτCp

    by the definition of τCp

    p((Cp)i)CτCp

    This implies that p((Cp)i)CτCp is PCmPFOSs but

    ((p(Cp)i)C)=(p((Cp)i)C

    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

    τCp1PτCp2

    or

    τCp2PτCp1

    If τCp1PτCp2 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 kϕ; then from Example 3.2,

    τCp1={0Cp,1Cp,Cp1,Cp2}

    and

    τCp2={0Cp,1Cp,Cp1}

    are cubic 3-polar fuzzy topologies on k. Since, τCp2PτCp1. Therefore, τCp2 becomes a P-cubic m-polar coarser than τCp1.

    Definition 3.13. Let (k,TCpk) be a CmPFTPS. Let Yk and TCpY be a CmPFTP on Y with PCmPFOSs that are

    CpY=TCpkp˘Y

    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.,

    TCpY={CpY:CpY=Cpkp˘Y,CpkTCpk}

    Example 3.14. Let k={k1,k2} be a non-empty set.

    Cp1={(k1,[0.14,0.39],[0.28,0.42],0.27,0.33),(k2,[0.26,0.53],[0.43,0.52],0.57,0.61)}Cp2={(k1,[0.61,0.86],[0.58,0.72],0.52,0.64),(k2,[0.37,0.74],[0.48,0.57],0.81,0.72)}

    Then,

    TCpk={0Cp,1Cp,Cp1,Cp2}

    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

    ˘Y={(k1,[1,1],[1,1],1,1)}

    Since

    ˘Yp0Cp=0Cp=´0Cp,
    ˘Yp1Cp=1Cp=˘Y,
    ˘YpCp1=Cp1=´Cp1,
    ˘YpCp2=Cp2=´Cp2,

    we have

    TCpY={´0Cp,˘Y,´Cp1,´Cp2}

    which is a cubic 2-polar fuzzy relative topology of TCpk.

    Let (k,TCpk) be a CmPFTPS. Let Yk and TCpY be a CmPFTP on Y with PCmPFOSs that are

    CpY=TCpkpY

    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.,

    TCpY={CpY:CpY=CpkpY,CpkTCpk}

    Example 3.15. From Example 3.14

    TCpk={0Cp,1Cp,Cp1,Cp2}

    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

    Y={(k1,[0.33,0.43],[0.49,0.68],0.42,0.51)}

    Since

    ˘Yp0Cp=0Cp=´0Cp,
    ˘Yp1Cp=1Cp=Y,
    ˘YpCp1=Cp1=´Cp1,
    ˘YpCp2=Y,

    we have

    TCpY={´0Cp,Y,´Cp1}

    which is a cubic 2-polar fuzzy relative topology of TCpk.

    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 Cp3cmp(k) given as

    Cp3={(k1,[0.37,0.49],[0.38,0.56],[0.54,0.69],0.39,0.41,0.60),(k2,[0.41,0.52],[0.48,0.61],[0.57,0.83],0.38,0.43,0.83),(k3,[0.53,0.72],[0.59,0.91],[0.64,0.89],0.61,0.82,0.73)}.

    Then,

    Cp30=0CppCp1pCp2=Cp1.

    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

    (Cp)0=Cp

    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

    Cp1pCp2(Cp1)0p(Cp2)0

    (Cp1pCp2)0=(Cp1)0p(Cp2)0

    (Cp1pCp2)0p(Cp1)0p(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

    (Cp1)c={(k1,[0.54,0.77],[0.48,0.69],[0.35,0.53],0.76,0.68,0.49),(k2,[0.58,0.70],[0.44,0.55],[0.31,0.47],0.80,0.59,0.28),(k3,[0.37,0.56],[0.22,0.45],[0.17,0.39],0.58,0.47,0.40)}(Cp2)c={(k1,[0.64,0.81],[0.52,0.76],[0.48,0.61],0.80,0.69,0.58),(k2,[0.62,0.73],[0.48,0.61],[0.37,0.51],0.85,0.70,0.35),(k3,[0.39,0.64],[0.32,0.50],[0.19,0.40],0.61,0.48,0.42)}

    Let Cp4cmp(k) be given as

    Cp4={(k1,[0.43,0.69],[0.39,0.54],[0.28,0.49],0.52,0.49,0.36),(k2,[0.31,0.67],[0.40,0.52],[0.26,0.35],0.48,0.49,0.31),(k3,[0.35,0.43],[0.21,0.38],[0.12,0.36],0.36,0.32,0.21)}

    Then,

    ¯Cp4=1Cpp(Cp2)c=(Cp2)c

    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

    ¯Cp=Cp

    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

    Fr(Cp)=¯Cpp¯(Cp)c

    Definition 3.24. Let Cp be a P-cubic m-polar fuzzy subset of (k,TCp); then, its exterior is denoted by

    Ext(Cp)=(¯Cp)c=(Cpc)0

    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 Cpp(Cp)c=0Cp and law of the excluded middle Cpp(Cp)c=1Cp do not hold in general. From Example 3.17,

    Cp3p(Cp3)c0Cp

    Cp3p(Cp3)c1Cp

    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)pCp01Cp

    6. Fr(Cp)=Fr(Cpc)

    7. Fr(Cp)pCp00Cp

    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)pCp01Cp. Given Example 3.25, we can see that

    Ext(Cp3)pFr(Cp3)pCp01Cp.

    6. Fr(Cpc)=¯(Cpc)p¯((Cpc)c) Fr(Cpc)=¯(Cpc)p¯(Cp)=Fr(Cp)

    7. Fr(Cp)pCp00Cp. From Example 3.25, we can see that,

    Fr(Cp3)pCp00Cp.

    Definition 3.27. A PCmPFS is said to be dense in a universal set k if

    ¯Cp=1Cp

    From Example 3.17 ¯Cp3=1Cp. So, ¯Cp3 is dense in k.

    Definition 3.28. A P-cubic m-polar number Cγ=([Aj,A+j],Aj)mj=1 belong PCmPFS if, and only if, Ai(λ)Aj(λ),A+i(λ)A+j(λ) and Ai(λ)Aj(λ) j=1,2,...,m and lk

    Let (k,TCp) be a CmPFTPS. A PCmPFS ´Cp of k which contains a PCmPF number Cγk is said to be a neighborhood of Cγ if, there exists a PCmPFOS Cp such that

    CγCpp´Cp.

    Example 3.29. From Example 3.2, consider a PCmPF number

    Cγ={[0.14,0.39],[0.28,0.42],0.27,0.33}

    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.

    Definition 3.30. Let (k,TCp) be a CmPFTPS. Then BTCp is said to be the P-cubic m-polar fuzzy basis for TCp if, for each CpTCp, BB such that

    Cp=pB

    Example 3.31. From Example 3.2,

    τCp={0Cp,1Cp,Cp1,Cp2}

    is a P-cubic 3-polar fuzzy topology on k. Then,

    B={1Cp,Cp1,Cp2}

    is a P-cubic 3-polar fuzzy basis for τCp.

    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, ¯1CrTCr

    2. If (Cr)iTCriΛ then r(Cr)iTCr

    3. If Cr1,Cr2TCr then Cr1rCr2TCr

    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

    Cr1={(k1,[0.19,0.39],[0.31,0.48],[0.38,0.61],0.8,0.7,0.6),(k2,[0.21,0.40],[0.34,0.53],[0.40,0.70],0.7,0.6,0.5),(k3,[0.24,0.52],[0.40,0.62],[0.47,0.81],0.6,0.5,0.4)}Cr2={(k1,[0.27,0.41],[0.32,0.50],[0.39,0.65],0.7,0.6,0.5),(k2,[0.24,0.49],[0.39,0.58],[0.42,0.73],0.6,0.5,0.4),(k3,[0.26,0.58],[0.44,0.70],[0.51,0.89],0.5,0.4,0.3)}.

    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.

    Table 4.  Union with R-order.
    r ¯0Cr ¯1Cr Cr1 Cr2
    ¯0Cr ¯0Cr ¯1Cr Cr1 Cr2
    ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr
    Cr1 Cr1 ¯1Cr Cr1 Cr2
    Cr2 Cr2 ¯1Cr Cr2 Cr2

     | Show Table
    DownLoad: CSV
    Table 5.  Intersection with R-order.
    r ¯0Cr ¯1Cr Cr1 Cr2
    ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr
    ¯1Cr ¯0Cr ¯1Cr C11 Cr2
    Cr1 ¯0Cr Cr1 Cr1 Cr1
    Cr2 ¯0Cr Cr2 Cr1 Cr2

     | Show Table
    DownLoad: CSV

    Then, clearly

    TCr1={¯0Cr,¯1Cr,Cr1,Cr2},
    TCr2={¯0Cr,¯1Cr,Cr1},
    TCr3={¯0Cr,¯1Cr,Cr2},
    TCr4={¯0Cr,¯1Cr},

    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

    Cr1={(k1,[0.20,0.41],[0.33,0.50],[0.44,0.60],0.9,0.8,0.7),(k2,[0.24,0.42],[0.35,0.60],[0.45,0.65],0.8,0.7,0.6)},Cr2={(k1,[0,0],[0,0],[0,0],0.9,0.8,0.7),(k2,[0,0],[0,0],[0,0],0.8,0.7,0.6)},Cr3={(k1,[1,1],[1,1],[1,1],0.9,0.8,0.7),(k2,[1,1],[1,1],[1,1],0.8,0.7,0.6)},Cr4={(k1,[0.20,0.41],[0.33,0.50],[0.44,0.60]0,0,0),(k2,[0.24,0.42],[0.35,0.60],[0.45,0.65],0,0,0)},Cr5={(k1,[0.20,0.41],[0.33,0.50],[0.44,0.60]1,1,1),(k2,[0.24,0.42],[0.35,0.60],[0.45,0.65],1,1,1)}.

    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.

    Table 6.  Union with R-order.
    r 0Cr 1Cr ¯0Cr ¯1Cr Cr1 Crr2 Cr3 Cr4 Cr5
    0Cr 0Cr ¯1Cr 0Cr ¯1Cr Cr4 0Cr ¯1Cr Cr4 Cr4
    1Cr ¯1Cr 1Cr 1Cr ¯1Cr Cr3 Cr3 Cr3 ¯1Cr 1Cr
    ¯0Cr 0Cr 1Cr ¯0Cr ¯1Cr Cr1 Cr2 Cr3 Cr4 Cr5
    ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr ¯1Cr
    Cr1 Cr4 Cr3 Cr1 ¯1Cr Cr1 Cr1 Cr3 Cr4 Cr1
    Cr2 0Cr Cr3 Crr2 ¯1Cr Cr1 Cr2 Cr3 ¯1Cr Cr1
    Cr3 ¯1Cr Cr3 Cr3 ¯1Cr Cr3 Cr3 Cr3 ¯1Cr Cr3
    Cr4 Cr4 ¯1Cr Cr4 ¯1Cr Cr4 Cr4 ¯1Cr Cr4 Cr4
    Cr5 Cr4 1Cr Cr5 ¯1Cr Cr1 Cr1 Cr3 Cr4 Cr5

     | Show Table
    DownLoad: CSV
    Table 7.  Intersection with R-order.
    r 0Cr 1Cr ¯0Cr ¯1Cr Cr1 Cpr2 Cr3 Cr4 Cr5
    0Cr 0Cr ¯0Cr ¯0Cr 0Cr Cr2 Cr2 Cr2 0Cr ¯0Cr
    1Cr ¯0Cr 1Cr ¯0Cr 1Cr Cr5 ¯0Cr 1Cr Cr5 Cr5
    ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr ¯0Cr
    ¯1Cr 0Cr 1Cr ¯0Cr ¯1Cr Cr1 Cr2 Cr3 Cr4 Cr5
    Cr1 Cr2 Cr5 ¯0Cr Cr1 Cr1 Cr2 Cr1 Cr1 Cr5
    Cr2 Cr2 ¯0Cr ¯0Cr Cr2 Cr2 Cr2 Cr2 Cr2 ¯0Cr
    Cr3 Cr2 1Cr ¯0Cr Cr3 Cr1 Cr2 Cr3 Cr1 Cr5
    Cr4 0Cr Cr5 ¯0Cr Cr4 Cr1 Cr2 Cr1 Cr4 Cr5
    Cr5 ¯0Cr Cr5 ¯0Cr Cr5 Cr5 ¯0Cr Cr5 Cr5 Cr5

     | Show Table
    DownLoad: CSV

    Then clearly,

    TCr={0Cr,1Cr,¯0Cr,¯1Cr,Cr1,Cpr2,Cr3,Cr4,Cr5},
    TCr={0Cr,1Cr,¯0Cr,¯1Cr},

    are cubic 3-polar fuzzy topologies with R-order.

    Definition 4.4. Let kϕ 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 kϕ 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,

    TCr1RTCr2orTCr2RTCr1.

    Furthermore if, TCr1RTCr2, 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 kϕ; then from Example 4.3,

    TCr1={0Cr,1Cr,¯0Cr,¯1Cr,Cr1,Cpr2,Cr3,Cr4,Cr5},
    TCr2={0Cr,1Cr,¯0Cr,¯1Cr},

    are cubic 3-polar fuzzy topologies on k. Since TCr2RTCr1. So that, TCr2 is R-cubic m-polar coarser then TCr1.

    Definition 4.12. Let (k,TCrk) be a CmPFTrS. Let Yk and TCrY be a CmPFTr on Y with the following RCmPFOSs:

    CrY=TCrkR˘Y

    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.,

    TCrY={CrY:CrY=CrkR˘Y,CrkTCrk}

    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

    ˘Y={(k1,[1,1],[1,1],[1,1],1,1,1)}

    Since

    ˘YR0Cr=¯0Cr=´¯0Cr,
    ˘YR1Cr=1Cr=˘Y,
    ˘YR¯0Cr=¯0Cr=´¯0Cr,
    ˘YR¯1Cr=1Cr=˘Y,
    ˘YRCr1=Cr5=´Cr5,
    ˘YRCr2=¯0Cr=´¯0Cr,
    ˘YRCr3=1Cr=˘Y,
    ˘YRCr4=Cr5=´Cr5,
    ˘YRCr5=Cr5=´Cr5,

    it follows that

    TCrY={´¯0Cr,˘Y,´Cr5}

    is a cubic 3-polar fuzzy relative topology of TCrk.

    Let (k,TCrk) be a CmPFTrS. Let Yk and TCrY be a CmPFTr on Y with the following RCmPFOSs:

    CrY=TCrkRY

    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.,

    TCrY={CrY:CrY=CrkRY,CrkTCrk}

    Example 4.14. 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

    Y={(k1,[0.31,0.52],[0.43,0.62],[0.54,0.70],0.8,0.7,0.6)}

    Since

    ˘YR0Cr={(k1,[0,0],[0,0],[0,0],0.8,0.7,0.6)}=´Cr6,YR1Cr={(k1,[0.31,0.52],[0.43,0.62],[0.54,0.70],1,1,1)}=´Cr7,YR¯0Cr={(k1,[0,0],[0,0],[0,0],1,1,1)}=´¯0Cr,YR¯1Cr={(k1,[0.31,0.52],[0.43,0.62],[0.54,0.70],0.8,0.7,0.6)}=Y,YRCr1={(k1,[0.2,0.41],[0.33,0.50],[0.44,0.60],0.9,0.8,0.7)}=´Cr1,YRCr2={(k1,[0,0],[0,0],[0,0],0.9,0.8,0.7)}=´Cr2,YRCr3={(k1,[0.31,0.52],[0.43,0.62],[0.54,0.70],0.9,0.8,0.7)}=´Cr3,YRCr4={(k1,[0.2,0.41],[0.33,0.50],[0.44,0.60],0.8,0.7,0.6)}=´Cr4,YRCr5={(k1,[0.2,0.41],[0.33,0.50],[0.44,0.60],1,1,1)}=´Cr5,

    it follows that

    TCrY={´¯0Cr,Y,´Cr1,´Cr2,´Cr3,´Cr4,´Cr5,´Cr6,´Cr7}

    is a cubic 3-polar fuzzy relative topology of TCrk.

    Definition 4.15. Let (k,TCr) be a CmPFTrS and Crcmp(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 Cr6cmp(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,

    Cr06=0CrRCr1RCr2RCr5=Cr1

    is open CmPFS.

    Theorem 4.17. Let (k,TCr) be a CmPFTrS and Crcmp(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

    (Cr1)c={(k1,[0.59,0.8],[0.0.50,0.67],[0.40,0.56],0.1,0.2,0.3),(k2,[0.58,0.76],[0.40,0.65],[0.35,0.55],0.2,0.3,0.4)}(Cr2)c={(k1,[1,1],[1,1],[1,1],0.1,0.2,0.3),(k2,[1,1],[1,1],[1,1],0.2,0.3,0.4)}(Cr3)c={(k1,[0,0],[0,0],[0,0],0.1,0.2,0.3),(k2,[0,0],[0,0],[0,0],0.2,0.3,0.4)}(Cr4)c={(k1,[0.59,0.8],[0.0.50,0.67],[0.40,0.56],1,1,1),(k2,[0.58,0.76],[0.40,0.65],[0.35,0.55],1,1,1)}(Cr4)c={(k1,[0.59,0.8],[0.50,0.67],[0.40,0.56],0,0,0),(k2,[0.58,0.76],[0.40,0.65],[0.35,0.55],0,0,0)}

    Let Cr7cmp(k) be given as

    Cr7={(k1,[0.43,0.76],[0.47,0.53],[0.29,0.42],0.2,0.3,0.4),(k2,[0.49,0.72],[0.38,0.58],[0.30,0.49],0.3,0.4,0.5)}

    Then,

    ¯Cr7=(¯0Cr)cR(Cr1)cR(Cr2)cR(Cr5)c=(Cr1)c

    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

    Fr(Cr)=¯CrR¯(Cr)c

    Definition 4.22. Let Cr be an R-cubic m-polar fuzzy subset of (k,TCr); then, its exterior is denoted by

    Ext(Cr)=(¯Cr)c=(Crc)0

    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, Crp(Cr)c=0Cr, and the law of excluded middle, CrR(Cr)c=1Cr do not hold in general. From Example 4.19

    Cr7R(Cr7)c0Cr

    Cr7R(Cr7)c1Cr

    Definition 4.24. An RCmPFS is said to be dense in a universal set k if

    ¯Cr=1Cr

    Definition 4.25. An RCmPF number Cγ=([Aj,A+j],Aj)mj=1 belong RCmPFS if, and only if,

    Ai(λ)Aj(λ),A+i(λ)A+j(λ) and Ai(λ)Aj(λ) j=1,2,...,m and lk

    Let (k,TCr) be a CmPFTrS. An RCmPFS ´Cr of k which contains an RCmPF number Cγk is said to be a neighborhood of Cγ if, there exists an RCmPFOS Cr containing Cγ, such that

    CγCrR´Cr

    Example 4.26. From Example 4.3, an R-cubic m-polar fuzzy number

    Cγ={k1,[0.20,0.41],[0.33,0.50],[0.44,0.60],0.9,0.8,0.7}

    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.

    Definition 4.27. Let (k,TCr) be a CmPFTrS. Then, BTCr is said to be the R-cubic m-polar fuzzy basis for TCr if, for each CrTCr, BB such that

    Cr=RB

    Example 4.28. From Example 4.3,

    TCr={0Cr,1Cr,¯0Cr,¯1Cr,Cr1,Cpr2,Cr3,Cr4,Cr5}

    is an R-cubic 3-polar fuzzy topology on k. Then,

    B={1Cr,¯1Cr,Cr1,Cpr2,Cr3,Cr4,Cr5}

    is the R-cubic 3-polar fuzzy basis for TCr.

    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.

    Table 8.  Linguistic terms for weighing choices.
    Linguistic terms Fuzzy weights
    Stage 0: Healthy kidney (S0) 0.10
    Stage 1: Beginning of kidney damage (S1) 0.30
    Stage 2: Moderate kidney damage (S2) 0.50
    Stage 3: Severe kidney damage (S3) 0.70
    Stage 4: Kidney failure (S4) 0.90

     | Show Table
    DownLoad: CSV

    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=ωjkiω2jk. Then, the weights are gathered as W=(ω1,ω2,,ωı), where ωi=1ȷiˉωjkjˉωjk.

    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: ζk=minjPςjk or ζk=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

    Si=Σmj=1ωj(d(¨ζj+,¨ζj)d(¨ζj+,¨ζj))Ri=mmaxj=1ωj(d(¨ζj+,¨ζj)d(¨ζj+,¨ζj))Qi=χ(SiSS+S)+(1χ)(RiRR+R)

    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=[A1,A+1],[A2,A+2],,[Am,A+m],A1,A2,,Am=[Aj,A+j],Ajmj=1, and

    C2=[B1,B+1],[B2,B+2],,[Bm,B+m],B1,ν2,,Bm=[Bj,B+j],Bjmj=1, is defined as

    d(C1,C2)=[mj=1|Aj+A+j2Bj+B+j2|m+mj=1|AjBj|m]1/m

    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)1ı1

    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

    Q(ϱak)Q(ϱa1)<1ı1

    then, ϱa1,ϱa2,...,ϱak should act as multiple compromise solutions.

    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

    =[ωij]4×4
    =(S2S3S0S1S3S4S1S2S1S0S2S3S4S2S0S1)
    =(0.500.700.100.300.700.900.300.500.300.100.500.700.900.500.100.30)

    where ωij represents the weights assigned by the DMs to criteria.

    Step 3. The normalized weighted matrix appears to be

    =(0.390.560.160.310.540.720.500.520.230.080.830.720.700.400.160.31)

    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

    PCmPPIS={˙ζ+1,˙ζ+2,˙ζ+3,˙ζ+4}PCmPPIS={[0.13,0.22],[0.15,0.28],0.13,0.15,[0.14,0.17],[0.18,0.21],0.14,0.18,=[0.34,0.15],[0.15,0.17],0.13,0.16,[0.15,0.16],[0.12,0.18],0.16,0.18}
    PCmPNIS={˙ζ1,˙ζ2,˙ζ+,˙ζ4}PCmPNIS={[0.09,0.12],[0.11,0.12],0.01,0.11,[0.10,0.13],[0.14,0.16],0.11,0.14,=[0.11,0.10],[0.10,0.11],0.09,0.11,[0.10,0.11],[0.10,0.13],0.11,0.12}

    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

    Si=Σ4j=1ωj(d(¨ζj+,¨ζj)d(¨ζj+,¨ζj))Ri=4maxj=1ωj(d(¨ζj+,¨ζj)d(¨ζj+,¨ζj))Qi=χ(SiSS+S)+1χ(RiRR+R)

    They are given in Table 9.

    Table 9.  Values of Si, Ri and Qi for each alternative.
    Alternatives Si Ri Qi
    ϱ1 0.6217 0.2487 0.6894
    ϱ1 0.8013 0.2258 0.8271
    ϱ1 0.5164 0.1836 0.000
    ϱ1 0.5350 0.2209 0.3192

     | Show Table
    DownLoad: CSV

    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.

    Table 10.  Advantages of the CmPFS.
    Fuzzy models Advantages and limitations
    Cubic set (CS) It describes information in terms of a fuzzy interval and a
    (Jun et. al. [18]) fuzzy number. It can not handel multi-polarity.
    m-Polar fuzzy set (mPFS) It describes the multi-polarity of objects with m grades.
    (Chen et al. [10]) It can not deal with fuzzy intervals.
    Cubic m-polar fuzzy set A strong hybrid model for the CS and mPFS to address the cubic
    (Riaz and Hashmi [19]) environment and multi-polarity of objects.

     | Show Table
    DownLoad: CSV

    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.

    Table 11.  Distance and coefficient of closeness of each patient.
    Alternatives d+i di Ci
    ϱ1 0.1114 0.0781 0.4123
    ϱ2 0.1274 0.0445 0.2590
    ϱ3 0.0958 0.0788 0.4566
    ϱ4 0.0917 0.0907 0.4973

     | Show Table
    DownLoad: CSV

    Step 9. Thus, the ranking of each patient is

    ϱ4>ϱ3>ϱ1>ϱ2

    This ranking shows that Patient ϱ4 is in a more critical situation. So this is optimal decision for the VIKOR and TOPSIS approaches.

    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.

    The authors declare that they have no conflict of interest.

    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.



    [1] L. A. Zadeh, Fuzzy sets, Inform. Control., 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
    [2] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci., 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90036-5 doi: 10.1016/0020-0255(75)90036-5
    [3] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3
    [4] K. T. Atanassov, Intuitionistic fuzzy sets: Theory and applications, Springer-Verlag Berlin Heidelberg GmbH, 283 (2012), 1–322. https://doi.org/10.1007/978-3-7908-1870-3 doi: 10.1007/978-3-7908-1870-3
    [5] R. R. Yager, Pythagorean fuzzy subsets, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
    [6] R. R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE Trans. Fuzzy Syst., 22 (2014), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
    [7] R. R. Yager, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst., 25 (2017), 1220–1230. https://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
    [8] W. R. Zhang, Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis, NAFIPS/IFIS/NASA94. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intellige., (1994), 305–309. https://doi.org/10.1109/IJCF.1994.375115
    [9] W. R. Zhang, (Yin)(Yang) bipolar fuzzy sets, IEEE International Conference on Fuzzy Systems Proceedings, IEEE World Congress Comput. Intell., 1 (1998), 835–840. https://doi.org/10.1109/FUZZY.1998.687599 doi: 10.1109/FUZZY.1998.687599
    [10] J. Chen, S. Li, S. Ma, X. Wang, m-Polar fuzzy sets: An extension of bipolar fuzzy sets, Sci. World J., 2014 (2014), 1–8. https://doi.org/10.1155/2014/416530 doi: 10.1155/2014/416530
    [11] F. Smarandache, A unifying field in logics, neutrosophy: Neutrosophic probability, set and logic, Amer. Res. Press: Rehoboth, DE, USA., (1999). 1–141.
    [12] F. Smarandache, Neutrosophic set-a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math., 24 (2005), 287–297. https://doi.org/10.1089/blr.2005.24.297 doi: 10.1089/blr.2005.24.297
    [13] B. C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybern., 30 (2014), 409–420. https://doi.org/10.15625/1813-9663/30/4/5032
    [14] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Trans. Fuzzy Syst., 15 (2007), 1179–1187. https://doi.org/10.1109/TFUZZ.2006.890678 doi: 10.1109/TFUZZ.2006.890678
    [15] H. Garg, Nancy, Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making, J. Ambient. Intell. Human. Comput., 9 (2018), 1975–1997. https://doi.org/10.1007/s12652-018-0723-5 doi: 10.1007/s12652-018-0723-5
    [16] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
    [17] N. agman, S. Enginoglu, Soft set theory and uniint decision making, Eur. J. Oper. Res., 207 (2010), 848–855. https://doi.org/10.1016/j.ejor.2010.05.004 doi: 10.1016/j.ejor.2010.05.004
    [18] Y. B. Jun, C. S. Kim, K. O. Yang, Cubic Sets, Annal. Fuzzy Math. Inform., 4 (2012), 83–98.
    [19] M. Riaz, M. R. Hashmi, MAGDM for agribusiness in the environment of various cubic m-polar fuzzy averaging aggregation operators, J. Intell. Fuzzy Syst., 37 (2019), 3671–3691. https://doi.org/10.3233/JIFS-182809 doi: 10.3233/JIFS-182809
    [20] M. Riaz, M. R. Hashmi, Linear Diophantine fuzzy set and its applications towards multi-attribute decision making problems, J. Intell. Fuzzy Syst., 37 (2019), 5417–5439. https://doi.org/10.3233/JIFS-190550 doi: 10.3233/JIFS-190550
    [21] M. Riaz, M. R. Hashmi. H. Kalsoom, D. Pamucar, Y. M. Chu, Linear Diophantine fuzzy soft rough sets for the selection of sustainable material handling equipment, Symmetry., 12 (2020), 1–39. https://doi.org/10.3390/sym12081215 doi: 10.3390/sym12081215
    [22] M. Riaz, M. R. Hashmi, D. Pamucar, Y. M. Chu, Spherical linear Diophantine fuzzy sets with modeling uncertainties in MCDM, Comput. Model. Eng. Sci., 126 (2021), 1125–1164. https://doi.org/10.32604/cmes.2021.013699 doi: 10.32604/cmes.2021.013699
    [23] P. Liu, Z. Ali, T. Mahmood, N. Hassan, Group decision-making using complex q-rung orthopair fuzzy Bonferroni mean, Int. J. Comput. Intell. Syst., 13 (2020), 822–851. https://doi.org/10.2991/ijcis.d.200514.001 doi: 10.2991/ijcis.d.200514.001
    [24] P. Liu, P. Wang, Multiple attribute group decision making method based on intuitionistic fuzzy Einstein interactive operations, Int. J. Fuzzy Syst., 22 (2020), 790–809. https://doi.org/10.1007/s40815-020-00809-w doi: 10.1007/s40815-020-00809-w
    [25] A. Jain, J. Darbari, A. Kaul, P. C. Jha, Selection of a green marketing strategy using MCDM under fuzzy environment, In: Soft Computing for Problem Solving, (2020), https://doi.org/10.1007/978-981-15-0184-5_43
    [26] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–190. https://doi.org/10.1016/0022-247X(68)90057-7
    [27] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst., 88 (1997), 81–89. https://doi.org/10.1016/S0165-0114(96)00076-0 doi: 10.1016/S0165-0114(96)00076-0
    [28] M. Olgun, M. Unver, Yardimci, Pythagorean fuzzy topological spaces, Complex Intell. Syst., 5 (2019), 177–183. https://doi.org/10.1007/s40747-019-0095-2 doi: 10.1007/s40747-019-0095-2
    [29] N. Cagman, S. Karatas, S. Enginoglu, Soft topology, Comput. Math. Applic., 62 (2011), 351–358. https://doi.org/10.1016/j.camwa.2011.05.016
    [30] A. Saha, T. Senapati, R. R. Yager, Hybridizations of generalized Dombi operators and Bonferroni mean operators under dual probabilistic linguistic environment for group decision-making, Int. J. Intell. Syst., 11 (2021), 6645–6679. https://doi.org/10.1002/int.22563 doi: 10.1002/int.22563
    [31] A. Saha, H. Garg, D. Dutta, Probabilistic linguistic q-rung orthopair fuzzy generalized Dombi and Bonferroni mean operators for group decision-making with unknown weights of experts, Int. J. Intell. Syst., 12 (2021), 7770–7804. https://doi.org/10.1002/int.22607 doi: 10.1002/int.22607
    [32] C. Jana, G. Muhiuddin, M. Pal, D. Al-Kadi, Intuitionistic fuzzy Dombi hybrid decision-making method and their applications to enterprise financial performance evaluation, Math. Prob. Eng., 2021 (2021), 1–14. https://doi.org/10.1155/2021/3218133 doi: 10.1155/2021/3218133
    [33] C. Jana, M. Pal, J. Wang, Bipolar fuzzy Dombi prioritized aggregation operators in multiple attribute decision making, Soft Comput., 24 (2020), 3631–3646. https://doi.org/10.1007/s00500-019-04130-z. doi: 10.1007/s00500-019-04130-z
    [34] M. Akram, G. Ali, J. C. R. Alcantud, Attributes reduction algorithms for m-polar fuzzy relation decision systems, Int. J. Approx. Reas., 140 (2022), 232–254. https://doi.org/10.1016/j.ijar.2021.10.005 doi: 10.1016/j.ijar.2021.10.005
    [35] M. Akram, A. Luqman, J. C. R. Alcantud, Risk evaluation in failure modes and effects analysis: Hybrid TOPSIS and ELECTRE I solutions with Pythagorean fuzzy information, Neural Comput. Applic., 33 (2021), 5675–5703. https://doi.org/10.1007/s00521-020-05350-3 doi: 10.1007/s00521-020-05350-3
    [36] S. Ashraf, S. Abdullah, Decision aid modeling based on sine trigonometric spherical fuzzy aggregation information, Soft Comput., 25 (2021), 8549–8572. https://doi.org/10.1007/s00500-021-05712-6 doi: 10.1007/s00500-021-05712-6
    [37] A. O. Almagrabi, S. Abdullah, M. Shams, Y. D. Al-Otaibi, S. Ashraf, A new approach to q-linear Diophantine fuzzy emergency decision support system for COVID19, J. Ambient. Intell. Human. Comput., 13 (2021), 1687–1713. https://doi.org/10.1007/s12652-021-03130-y doi: 10.1007/s12652-021-03130-y
    [38] M. Ali, I. Deli, F. Smarandache, The theory of neutrosophic cubic sets and their applications in pattern recognition, J. Intell. Fuzzy Syst., 30 (2016), 1957–1963. https://doi.org/10.3233/IFS-151906 doi: 10.3233/IFS-151906
    [39] M. Ali, L. H. Son, I. Deli, N. D. Tien, Bipolar neutrosophic soft sets and applications in decision making, J. Intell. Fuzzy Syst., 33 (2017), 4077–4087. https://doi.org/10.3233/JIFS-17999 doi: 10.3233/JIFS-17999
    [40] J. Zhao, X. Y. You, H. C. Liu, S. M. Wu, An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection, Symmetry, 9 (2017), 1–16. https://doi.org/10.3390/sym9090169 doi: 10.3390/sym9090169
    [41] R. Joshi, S. Kumar, An intuitionistic fuzzy information measure of order-(α,β) with a new approach in supplier selection problems using an extended VIKOR method, J. Appl. Math. Comput., 60 (2019), 27–50. https://doi.org/10.1007/s12190-018-1202-z doi: 10.1007/s12190-018-1202-z
    [42] J. H. Park, H. J. Cho, J. S. Hwang, Y. C. Kwun, Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making, Third International Workshop on Advanced Computational Intelligence, (2010), 189–195. https://doi.org/10.1109/IWACI.2010.5585223
    [43] Z. Shouzhen, C. S. Ming, K. L. Wei, Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method, Inform. Sci., 488 (2019), 76–92. https://doi.org/10.1016/j.ins.2019.03.018 doi: 10.1016/j.ins.2019.03.018
    [44] V. Arya, S. Kumar, A novel VIKOR-TODIM Approach based on Havrda-Charvat-Tsallis entropy of intuitionistic fuzzy sets to evaluate management information system, Fuzzy Inform. Eng., 11 (2019), 357–384. https://doi.org/10.1080/16168658.2020.1840317 doi: 10.1080/16168658.2020.1840317
    [45] K. Devi, Extension of VIKOR method in intuitionistic fuzzy environment for robot selection, Expert Syst. Applic., 38 (2011), 14163–14168. https://doi.org/10.1016/j.eswa.2011.04.227 doi: 10.1016/j.eswa.2011.04.227
    [46] X. Luo, X. Wang, Extended VIKOR method for intuitionistic fuzzy multiattribute decision-making based on a new distance measure, Math. Prob. Eng., 2017 (2017), 1–16. https://doi.org/10.1155/2017/4072486 doi: 10.1155/2017/4072486
    [47] T. Y. Chen, Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis, Inf. Fus., 41 (2018), 129–150. https://doi.org/10.1016/j.inffus.2017.09.003 doi: 10.1016/j.inffus.2017.09.003
    [48] F. Zhou, T. Y. Chen, An extended Pythagorean fuzzy VIKOR method with risk preference and a novel generalized distance measure for multicriteria decision-making problems, Neural Comput. Applic., 33 (2021), 11821–11844. https://doi.org/10.1007/s00521-021-05829-7 doi: 10.1007/s00521-021-05829-7
    [49] G. Bakioglu, A. O. Atahan, AHP integrated TOPSIS and VIKOR methods with Pythagorean fuzzy sets to prioritize risks in self-driving vehicles, Appl. Soft Comput., 99 (2021), 1–19. https://doi.org/10.1016/j.asoc.2020.106948 doi: 10.1016/j.asoc.2020.106948
    [50] A. Guleria, R. K. Bajaj, A robust decision making approach for hydrogen power plant site selection utilizing (R, S)-Norm Pythagorean Fuzzy information measures based on VIKOR and TOPSIS method, Int. J. Hydr. Energy., 45 (2020), 18802–18816. https://doi.org/10.1016/j.ijhydene.2020.05.091 doi: 10.1016/j.ijhydene.2020.05.091
    [51] M. Gul, Application of Pythagorean fuzzy AHP and VIKOR methods in occupational health and safety risk assessment: the case of a gun and rifle barrel external surface oxidation and colouring unit, Int. J. Occup. Safety Ergon., 26 (2020), 705–718. https://doi.org/10.1080/10803548.2018.1492251 doi: 10.1080/10803548.2018.1492251
    [52] M. Kirisci, I. Demir, N. Simsek, N. Topa, M. Bardak, The novel VIKOR methods for generalized Pythagorean fuzzy soft sets and its application to children of early childhood in COVID-19 quarantine, Neural Comput. Applic., 34 (2021), 1877–1903. https://doi.org/10.1007/s00521-021-06427-3 doi: 10.1007/s00521-021-06427-3
    [53] S. Dalapati, S. Pramanik, A revisit to NC-VIKOR based MAGDM strategy in neutrosophic cubic set environment, Neutrosophic Sets Sy., 21 (2018), 131–141. https://doi.org/10.20944/preprints201803.0230.v1 doi: 10.20944/preprints201803.0230.v1
    [54] S. Pramanik, S. Dalapati, S. Alam, T. K. Roy, NC-VIKOR based MAGDM strategy under neutrosophic cubic set environment, Neutrosophic Sets Sy., 20 (2018), 95–108. https://doi.org/10.20944/preprints201803.0230.v1 doi: 10.20944/preprints201803.0230.v1
    [55] S. Pramanik, S. Dalapati, S. Alam, T. K. Roy, VIKOR based MAGDM strategy under bipolar neutrosophic set environment, Neutrosophic Sets Sy., 19 (2018), 57–69. https://doi.org/10.20944/preprints201801.0006.v1 doi: 10.20944/preprints201801.0006.v1
    [56] L. Wang, H. Y. Zhang, J. Q. Wang, L. Li, Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project, Appl. Soft Comput., 64 (2018), 216–226. https://doi.org/10.1016/j.asoc.2017.12.014 doi: 10.1016/j.asoc.2017.12.014
    [57] V. Arya, S. Kumar, A picture fuzzy multiple criteria decision-making approach based on the combined TODIM-VIKOR and entropy weighted method, Cogn. Comput., 13 (2021), 1172–1184. https://doi.org/10.1007/s12559-021-09892-z doi: 10.1007/s12559-021-09892-z
    [58] R. Joshi, A novel decision-making method using R-Norm concept and VIKOR approach under picture fuzzy environment, Expert Syst. Applic., 147 (2020), 1–12. https://doi.org/10.1016/j.eswa.2020.113228 doi: 10.1016/j.eswa.2020.113228
    [59] V. Arya, S. Kumar, A new picture fuzzy information measure based on shannon entropy with applications in opinion polls using extended VIKOR-TODIM approach, Comp. Appl. Math., 39 (2020), 1–24. https://doi.org/10.1007/s40314-020-01228-1 doi: 10.1007/s40314-020-01228-1
    [60] M. J. Khan, P. Kumam, W. Kumam, A. N. A. Kenani, Picture fuzzy soft robust VIKOR method and its applications in decision-making, Fuzzy Inf. Eng., 13 (2021), 296–322. https://doi.org/10.1080/16168658.2021.1939632. doi: 10.1080/16168658.2021.1939632
    [61] C. Yue, Picture fuzzy normalized projection and extended VIKOR approach to software reliability assessment, Appl. Soft Comput., 88 (2020), 1–13. https://doi.org/10.1016/j.asoc.2019.106056. doi: 10.1016/j.asoc.2019.106056
    [62] P. Meksavang, H. Shi, S. M. Lin, H. C. Liu, An extended picture fuzzy VIKOR approach for sustainable supplier management and its application in the beef industry, Symmetry., 11 (2019), 1–19. https://doi.org/10.3390/sym11040468. doi: 10.3390/sym11040468
    [63] A. Singh, S. Kumar, Picture fuzzy Choquet integral-based VIKOR for multicriteria group decision-making problems, Gran. Comput., 6 (2021), 587–601. https://doi.org/10.1007/s41066-020-00218-2. doi: 10.1007/s41066-020-00218-2
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