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Dynamic analysis and optimal control of a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response

  • Received: 28 December 2023 Revised: 23 February 2024 Accepted: 27 February 2024 Published: 07 March 2024
  • MSC : 26A33, 92B05

  • In this paper, a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response is established. Two cases are considered: constant control and optimal control. For the constant control system, the existence and uniqueness of the positive solutions are proved, and then the sufficient conditions for the existence and stability of five equilibriums are obtained. For the second case, the Pontryagin's Maximum Principle is used to analyze the optimal control, and the formula of the optimal solution are derived. After that, some numerical simulations are performed to validate the theoretical prediction. Numerical simulations indicate that in the case of HIV/HTLV co-infection, the concentration of CD4+T cells is no longer suitable as an effective reference data for understanding the development process of the disease. On the contrary, the number of HIV virus particles should be used as an important indicator for reference.

    Citation: Ruiqing Shi, Yihong Zhang. Dynamic analysis and optimal control of a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response[J]. AIMS Mathematics, 2024, 9(4): 9455-9493. doi: 10.3934/math.2024462

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  • In this paper, a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response is established. Two cases are considered: constant control and optimal control. For the constant control system, the existence and uniqueness of the positive solutions are proved, and then the sufficient conditions for the existence and stability of five equilibriums are obtained. For the second case, the Pontryagin's Maximum Principle is used to analyze the optimal control, and the formula of the optimal solution are derived. After that, some numerical simulations are performed to validate the theoretical prediction. Numerical simulations indicate that in the case of HIV/HTLV co-infection, the concentration of CD4+T cells is no longer suitable as an effective reference data for understanding the development process of the disease. On the contrary, the number of HIV virus particles should be used as an important indicator for reference.



    In today's culture, scientific and technological advances have resulted in scientific and technological discoveries that have decreased the complications in our daily lives. Yet, despite scientific progress that has made life easier, some concerns, such as decision making (DM), remain complicated. DM, particularly multi-criteria group decision making (MCGDM), has been widely adopted in a variety of sectors where traditional methods have failed in recent years. As in real life, information is usually uncertain, and as information becomes more complex, additional solutions are required. In 1965, Zadeh [35] invented the concept of a fuzzy set (FS). Fuzzy set is a remarkable performance with numerous uses. An FS is defined by its membership degree (MD) for any value between 0 and 1. Atanassov expanded fuzzy set and defined intuitionistic fuzzy set (IFS) in [1]. An IFS is defined by two functions and expressed as and for each element of fixed set on the closed-interval 0 to 1, as well as their sum in the range. Xiao et al. [34] developed a q-rung orthopair fuzzy decision-making model with new score function and best-worst method for manufacturer selection. Xiao et al. [33] suggested an integrated risk assessment method using Z-fuzzy clouds and generalized TODIM.

    In the complex fuzzy set (CFS), an upgraded variety of the classical fuzzy set can be utilized to handle fuzziness information). A complex fuzzy value may deal with information in two ways, since it has both a phase and an amplitude term. Ramot et al. [29] describe the basic operations for CFS. Merigó et al. [23], for example, offer a complicated fuzzy generalized aggregate operator (AO) and describe their applicability in DM. Hu et al. [14,15] shows that CFSs now has approximate paralleling and orthogonality relations. Zhang et al. [37] present the concept of δ-equalities amongst CFSs. Dai et al. [8,16], Alkouri and Salleh [3] provide CFS distance measurements have been defined. Bi et al. [7] proposed new entropy measure classes in CFSs. Liu et al. [21] defined distance between CFSs, their cross-entropy, and their application in DM are all measured. Dai [9] extended rotational in variance to CF operations. Ma et al. [22] provide CFS was defined as a concept for handling problems with many periodic factors. Several CFS applications have been investigated for various intellectuals, such as neighborhood operators Mahmood [24]. Huang et al. [17] developed an assessment and prioritization method of key engineering characteristics for complex products based on cloud rough numbers. Further information on CFSs can be found in [4,5,31].

    The CFSs may only define complex-valued grades for positive membership functions and cannot express complex-valued grades for negative membership grades (NMG), as it is limited in their application. Alkouri et al. [2] proposed the CIFSs structure, which includes mainly two complex membership functions that express an element's positive and NMD. Greenfield et al. [10] introduced the definition of complex interval-valued fuzzy set (CIVFS), which clearly enhanced the paradigm of interval-valued fuzzy sets and helped to conceptualize CFSs. Kumar and Bajaj defined complex intuitionistic fuzzy soft sets with distance measurements and entropy in [20]. Garg and Rani [12] presented some generalized CIF aggregation operations and their application to the MCDM process. Garg and Rani created a strong correlation coefficient measure of CIFSs and its applications in decision-making. Rani and Garg defined CIF power aggregation operators and their applications in MCDM in [30]. Garg and Rani are the first to suggest a power aggregation operator and a ranking method for CIFSs, as well as their use in decision making. Garg and Rani [11] proposed generalized AOs for CIFS based on t-norm methods and discussed some of their DM applications. Huang et al. [18] proposed a design alternative assessment and selection: a novel Z-cloud rough number-based BWM-MABAC model.

    We know that the CFS theory discussed just a supporting grade while leaving out the grade of negative supporting grade, and as a result, various issues have emerged in many cases. For this, Mahmood and Rehman [25] modified the theory of CFS and diagnosed the mathematical form of bipolar CFS (BCFS) with a terminology represented by positive and negative supporting grades in the shape of complex numbers, with real and imaginary parts belonging to the unit intervals [0, 1] and [-1, 0]. Gao et al. [13] developed dual hesitant bipolar fuzzy Hamacher aggregation operators and their applications to MADM. Limited number of researchers produced numerous applications, such as the Hamacher aggregation information examined by Mahmood et al. [26]. Also, Wei et al. [32] defined a bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Additionally, Mahmood and Rehman [27] proposed the core theory of Dombi operators based on BCFS. Jana et al. [19] proposed bipolar fuzzy Dombi prioritized aggregation operators in MADM.

    The main motivation for this analysis is explained below:

    (1) To define new operations for bipolar complex fuzzy credibility numbers (BCFCNs), and to investigate properties these numbers. A BCFCN is superior that a bipolar complex fuzzy number as it carries more comprehensive and reasonable information. Bipolar complex fuzzy credibility numbers is the extension of bipolar complex fuzzy numbers to deal with two-sided contrasting features, which can describe the information with a bipolar complex fuzzy number and an credibility number simultaneously.

    (2) A secondary objective of this paper is to introduce some fundamental operations on BCFCNs, their key properties, and related significant results. Suggested operations are very helpful to strengthen BCFCS theory.

    (3) Since aggregation operators for bipolar complex fuzzy credibility numbers (BCFCNs) have not been established so far, motivated by the above discussion, this paper presents novel averaging and geometric aggregation operators under bipolar complex fuzzy credibility information are proposed.

    (4) An algorithm for new MCDM technique is developed based on proposed aggregation operators using bipolar complex fuzzy credibility information. Proposed technique is also demonstrated by a numerical illustration.

    (5) To demonstrate the validity and capability of the proposed technique, conduct a comparative examination of the developed operators with various current theories.

    The framework of this study is as follows: Section 2 includes certain prevalent ideas such as BCFCS, aggregation operator, and their operational laws. In Section 3, we employed the theory of averaging/geometric aggregation operators to diagnose the well-known operators, such as bipolar complex fuzzy credibility weighted average (BCFCWA), bipolar complex fuzzy credibility ordered weighted average (BCFCOWA), bipolar complex fuzzy credibility hybrid average (BCFCHA), bipolar complex fuzzy credibility weighted geometric (BCFCWG), bipolar complex fuzzy credibility ordered weighted geometric (BCFCOWG), and bipolar complex fuzzy credibility hybrid geometric (BCFCHG) operators, as well as analyses their strategic features and related outcomes. Section 4 proposes an algorithm for multiple criteria group decision making utilizing stated operators. Then, a numerical example of a case study of Hospital selection is discussed. In Section 5, we compared the described operators to existing methodologies to demonstrate the validity and capabilities of the proposed approach. Finally, write the study's conclusion.

    The aim of this part is to present in a concise manner the preexisting basic definitions for CFS, CIFS, BFC, BCFS and BCFCS.

    Definition 2.1. [28] A CFS C on Q (fixed set) is defined,

    C={ˇn,μC(ˇn)|ˇnQ}, (2.1)

    where μC:U{z:zC,|z|1} and μC(ˇn)=a+ib=χC(ˇn).e2πiΘC(ˇn). Here, χC(ˇn)=a2+K2R and χC(ˇn),ΘC(ˇn)[0,1], where i=1.

    Definition 2.2. [2] A CIFS I on Q (fixed set) is define,

    I={ˇn,μI(ˇn),υI(ˇn)|ˇnQ}, (2.2)

    where μI:U{z1:z1I,|z1|1}, υI:U{z2:z2I,|z2|1}, such as μI(ˇn)=z1=a1+ib1 and υI(ˇn)=z2=a2+ib2, and 0|z1|+|z2|1 or μI(ˇn)=χI(ˇn).e2πiΘχI(ˇn) and υi(ˇn)=ξi(ˇn).e2πiΘξI(ˇn) satisfy the conditions; 0χI(ˇn)+ξI(ˇn)1 and 0ΘχI(ˇn)+ΘξI(ˇn)1. The term HI(ˇn)=R.e2πiΘR, such that R=1(|z1|+|z2|) and ΘR(ˇn)=1(ΘχI(ˇn)+ΘξI(ˇn)) be the hesitancy grade of Q. Furthermore, I=(χ.e2πiΘχ,ξ.e2πiΘξ) indicate the complex intuitionistic fuzzy number (CIFN).

    Definition 2.3. [36] A BFS K on Q (fixed set) is of the form,

    K={ˇn,μ+K(ˇn),μK(ˇn)|ˇnQ}, (2.3)

    where, μ+K:Q[0,1] and μK:Q[1,0].

    Definition 2.4. [25] A BCFS K on Q (fixed set) is of the form,

    K={ˇn,μ+K(ˇn),μK(ˇn)|ˇnQ}, (2.4)

    where, μ+K:Q[0,1]+i[0,1] and μK:Q[1,0]+i[1,0] is called membership degree. μ+K(ˇn)=a+K(ˇn)+ib+K(ˇn) and μK(ˇn)=aK(ˇn)+ibK(ˇn) with a+K(ˇn),ib+K(ˇn)[0,1] and aK(ˇn),ibK(ˇn)[1,0]. The bipolar complex fuzzy number is represented by K=(a+K(ˇn)+ib+K(ˇn),aK(ˇn)+ibK(ˇn)).

    Definition 2.5. [6] A BCFCS K on Q (fixed set) is of the form,

    K={ˇn,(μ+K(ˇn),μK(ˇn)),(υ+K(ˇn),υK(ˇn))|ˇnQ}, (2.5)

    where, μ+K:Q[0,1]+i[0,1],υ+K:Q[0,1]+i[0,1], μK:Q[1,0]+i[1,0] and υK:Q[1,0]+i[1,0] as known as the MG and credibility degree. μ+K(ˇn)=a+K(ˇn)+ib+K(ˇn),υ+K(ˇn)=c+K(ˇn)+id+K(ˇn), μK(ˇn)=aK(ˇn)+ibK(ˇn) and υK=cK+idK with a+K(ˇn),ib+K(ˇn),c+K(ˇn),id+K(ˇn)[0,1] and aK(ˇn),ibK(ˇn),cK(ˇn),idK(ˇn)[1,0]. The bipolar complex fuzzy credibility number is represented as, K=((a+K+ib+K),(aK+ibK),(c+K+id+K),(cK+idK)).

    Definition 2.6. For any two BCFCNs K1=(a+K1+ib+K1,aK1+ibK1,c+K1+id+K1,cK1+idK1) and K2=(a+K2+ib+K2,aK2+ibK2,c+K2+id+K2,cK2+idK2), and for any λ>0. The following operation are defined as

    (1) K1K2={a+K1+a+K2a+K1a+K2+i(b+K1+b+K2b+K1b+K2),(aK1aK2)+i((bK1bK2)),c+K1+c+K2c+K1c+K2+i(c+K1+c+K2c+K1c+K2),(cK1cK2)+i((dK1dK2))};

    (2) K1K2={(a+K1a+K2)+i(b+K1b+K2),aK1+aK2aK1aK2+i(bK1+bK2bK1bK2)(c+K1c+K2)+i(d+K1d+K2),cK1+cK2cK1cK2+i(cK1+cK2cK1cK2)};

    (3) λK1={1(1a+K1)λ+i(1(1b+K1)λ),(aK1)λ+i((bK1)λ)1(1c+K1)λ+i(1(1d+K1)λ),(cK1)λ+i((dK1)λ)};

    (4) Kλ1={(a+K1)λ+i(b+K1)λ,1+(1+a+K1)λ+i(1+(1+b+K1)λ),(c+K1)λ+i(d+K1)λ,1+(1+c+K1)λ+i(1+(1+d+K1)λ)}.

    Definition 2.7. Let K=(a+K+ib+K,aK+ibK,c+K+id+K,cK+idK) be the BCFCN. The score function is defined as

    So(K)=18(4+a+K+ib+K+aK+ibK+c+K+id+K+cK+idK). (2.6)

    Here, we are going to define some aggregation operators like, BCFCWA, BCFCOWA, and BCFCHA operators.

    Definition 3.1. Let Ki=(a+Ki+ib+Ki,aKi+ibKi,c+Ki+id+Ki,cKi+idKi)(i=1,...,n) be a set of BCFCNs with the weights Φ=(Φ1,...,Φn)T, such as ni=1Φi=1 and 0Φi1. Then, the BCFCWA operator is obtain as

    BCFCWA(K1,...,Kn)=nˆȷ=1ΦˆȷKˆȷ, (3.1)

    utilizing Definition 3.1, aggregated value for BCFCWA operator is shown in Theorem 3.2.

    Theorem 3.2. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,c+Kˆȷ+id+Kˆȷ,aKˆȷ+ibKˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, BCFCWA operator is obtained as

    BCFCWA(K1,...,Kn)=nˆȷ=1ΦˆȷKˆȷ={(1nˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b+Kˆȷ)Φˆȷ),nˆȷ=1(aKˆȷ)Φˆȷ+i(nˆȷ=1(bKˆȷ)Φˆȷ)),(1nˆȷ=1(1c+Kˆȷ)Φi+i(1nˆȷ=1(1d+Kˆȷ)Φˆȷ),nˆȷ=1(cKˆȷ)Φˆȷ+i(nˆȷ=1(dKˆȷ)Φˆȷ))}. (3.2)

    Proof. To prove this theorem, we used mathematical induction principle. As we know that

    K1K2=nˆȷ=2ΦˆȷKˆȷ

    and

    Φ1K1={(1(1a+K1)Φ1+i(1(1b+K1)Φ1),(aK1)Φ1+i((bK1)Φ1)),(1(1c+K1)Φ1+i(1(1d+K1)Φ1),(cK1)Φ1+i((dK1)Φ1))}.

    Let Eq (3.2) is true for n=2. Then,

    BCFCWA(K1,K2)=2ˆȷ=1ΦˆȷKˆȷ={(12ˆȷ=1(1a+Kˆȷ)Φˆȷ+i(12ˆȷ=1(1b+Kˆȷ)Φˆȷ),2ˆȷ=1(aKˆȷ)Φˆȷ+i(2ˆȷ=1(bKˆȷ)Φˆȷ)),(12ˆȷ=1(1c+Kˆȷ)Φˆȷ+i(12ˆȷ=1(1d+Kˆȷ)Φˆȷ),2ˆȷ=1(cKˆȷ)Φˆȷ+i(2ˆȷ=1(dKˆȷ)Φˆȷ))}.

    The result hold for n=2.

    Now, let Eq (3.2) is true for n=τ. Then, we get

    BCFCWA(K1,...,Kn)=nˆȷ=1ΦˆȷKˆȷ={(1τˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1τˆȷ=1(1b+Kˆȷ)Φˆȷ),τˆȷ=1(aKˆȷ)Φˆȷ+i(τˆȷ=1(bKˆȷ)Φˆȷ)),(1τˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1τˆȷ=1(1d+Kˆȷ)Φˆȷ),τˆȷ=1(cKˆȷ)Φˆȷ+i(τˆȷ=1(dKˆȷ)Φˆȷ))}.

    Next, let Eq (3.2) is true for n=τ+1,

    BCFCWA(K1,...,KτKτ+1)=(τˆȷ=1ΦˆȷKˆȷ)(Φτ+1Kτ+1)={(1τˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1τˆȷ=1(1b+Kˆȷ)Φˆȷ),τˆȷ=1(aKˆȷ)Φˆȷ+i(τˆȷ=1(bKˆȷ)Φˆȷ)),(1τˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1τˆȷ=1(1d+Kˆȷ)Φˆȷ),τˆȷ=1(cKˆȷ)Φˆȷ+i(τˆȷ=1(dKˆȷ)Φˆȷ))}{(1(1a+Kτ+1)Φτ+1+i(1(1b+Kτ+1)Φτ+1),(aKτ+1)Φτ+1+i((bKτ+1)Φτ+1)),(1(1c+Kτ+1)Φτ+1+i(1(1d+Kτ+1)Φτ+1),(cKi)Φτ+1+i((dKτ+1)Φτ+1))}
    ={(1τ+1ˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1τ+1ˆȷ=1(1b+Kˆȷ)Φˆȷ),τ+1ˆȷ=1(aKˆȷ)Φˆȷ+i(τ+1ˆȷ=1(bKˆȷ)Φˆȷ)),(1τ+1ˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1τ+1ˆȷ=1(1d+Kˆȷ)Φˆȷ),τ+1ˆȷ=1(cKˆȷ)Φˆȷ+i(τ+1ˆȷ=1(dKˆȷ)Φˆȷ))},

    which show that Eq (3.2) true for n=τ+1. Hence, the given result is hold for n1.

    The BCFCWA operator satisfied the following properties.

    Theorem 3.3 (Idempotency). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,c+Kˆȷ+id+Kˆȷ,aKˆȷ+ibKˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then

    BCFCWA(K1,...,Kn)=K. (3.3)

    Proof. As we know that;

    BCFCWA(K1,...,Kn)=nˆȷ=1ΦˆȷKˆȷ={(1nˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b+Kˆȷ)Φˆȷ),nˆȷ=1(aKˆȷ)Φˆȷ+i(nˆȷ=1(bKˆȷ)Φˆȷ)),(1nˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d+Kˆȷ)Φˆȷ),nˆȷ=1(cKˆȷ)Φˆȷ+i(nˆȷ=1(dKˆȷ)Φˆȷ))}{(1(1a+K)nˆȷ=1Φˆȷ+i(1(1b+K)nˆȷ=1Φˆȷ),(aK)nˆȷ=1Φˆȷ+i((bK)nˆȷ=1Φˆȷ)),(1(1c+K)nˆȷ=1Φˆȷ+i(1(1d+K)nˆȷ=1Φˆȷ),(cK)nˆȷ=1Φˆȷ+i((dK)nˆȷ=1Φˆȷ))}=(a+K+ib+K,aK+ibK,c+K+id+K,,cK+idK)=K.

    Theorem 3.4 (Monotonicity). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) and K/i=(a/+Kˆȷ+ib/+Kˆȷ,a/Kˆȷ+ib/Kˆȷ,c/+Kˆȷ+id/+Kˆȷ,c/Kˆȷ+id/Kˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if a+Kˆȷa/+Kˆȷ,ib+Kˆȷib/+Kˆȷ,ibKˆȷib/Kˆȷ,aKˆȷa/Kˆȷ, c+Kˆȷc/+Kˆȷ,id+Kˆȷid/+Kˆȷ,cKˆȷc/Kˆȷ, and idKˆȷid/Kˆȷ. Then,

    BCFCWA(K1,...,Kn)BCFCWA(K/1,...,K/n). (3.4)

    Proof. We know that a+Kˆȷa/+Kˆȷ,ib+Kˆȷib/+Kˆȷ,ibKˆȷib/Kˆȷ,aKˆȷa/Kˆȷ, c+Kˆȷc/+Kˆȷ,id+Kˆȷid/+Kˆȷ,cKˆȷc/Kˆȷ, and idKˆȷid/Kˆȷ. Then,

    1a+Kˆȷ1a/+Kˆȷ1nˆȷ=1(1a+Kˆȷ)Φˆȷ1nˆȷ=1(1a/+Kˆȷ)Φˆȷ.

    And

    nˆȷ=1(aKˆȷ)Φˆȷnˆȷ=1(aKˆȷ)Φˆȷ.

    For imaginary part

    (1nˆȷ=1(1ib+Kˆȷ)Φˆȷ)(1nˆȷ=1(1ib+Kˆȷ)Φˆȷ).

    And

    nˆȷ=1(ibKˆȷ)Φˆȷnˆȷ=1(ibKˆȷ)Φˆȷ.

    Similarly, we find for the credibility degree,

    1nˆȷ=1(1c+Kˆȷ)Φˆȷ1nˆȷ=1(1c/+Kˆȷ)Φˆȷ

    and

    nˆȷ=1(cKˆȷ)Φˆȷnˆȷ=1(cKˆȷ)Φˆȷ.

    For imaginary part

    (1nˆȷ=1(1id+Kˆȷ)Φˆȷ)(1nˆȷ=1(1id+Kˆȷ)Φˆȷ).

    And

    nˆȷ=1(idKˆȷ)Φˆȷnˆȷ=1(idKˆȷ)Φˆȷ.

    By the combination of real and imaginary parts, we get

    {(1nˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b+Kˆȷ)Φˆȷ),nˆȷ=1(aKˆȷ)Φˆȷ+i(nˆȷ=1(bKˆȷ)Φˆȷ)),(1nˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d+Kˆȷ)Φˆȷ),nˆȷ=1(cKˆȷ)Φˆȷ+i(nˆȷ=1(dKˆȷ)Φˆȷ))}{(1nˆȷ=1(1a/+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b/+Kˆȷ)Φˆȷ),nˆȷ=1(a/Kˆȷ)Φˆȷ+i(nˆȷ=1(b/Kˆȷ)Φˆȷ)),(1nˆȷ=1(1c/+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d/+Kˆȷ)Φˆȷ),nˆȷ=1(c/Ki)Φˆȷ+i(nˆȷ=1(d/Kˆȷ)Φˆȷ))}.

    We assume that, BCFCWA(K1,...,Kn)=K1 and BCFCWA(K/1,...,K/n)=K/1. So, utilized Eq (2.6), we get

    Sc(K1)Sc(K/1).

    Then, we have two possibility,

    1) When, Sc(K1)Sc(K/1), we get

    BCFCWA(K1,...,Kn)BCFCRWA(K/1,...,K/n).

    2) When, Sc(K1)=Sc(K/1), we get

    {(1nˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b+Kˆȷ)Φˆȷ),nˆȷ=1(aKˆȷ)Φˆȷ+i(nˆȷ=1(bKˆȷ)Φˆȷ)),(1nˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1ni=1(1d+Kˆȷ)Φˆȷ),nˆȷ=1(cKˆȷ)Φˆȷ+i(nˆȷ=1(dKˆȷ)Φˆȷ))}={(1nˆȷ=1(1a/+Kˆȷ)Φˆȷ+i(1ni=1(1b/+Kˆȷ)Φˆȷ),nˆȷ=1(a/Kˆȷ)Φˆȷ+i(nˆȷ=1(b/Kˆȷ)Φˆȷ)),(1nˆȷ=1(1c/+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d/+Kˆȷ)Φˆȷ),nˆȷ=1(c/Kˆȷ)Φˆȷ+i(nˆȷ=1(d/Kˆȷ)Φˆȷ))}.

    We utilized the accuracy function because the score functions are equal.

    {(1nˆȷ=1(1a+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b+Kˆȷ)Φˆȷ),nˆȷ=1(aKˆȷ)Φˆȷ+i(nˆȷ=1(bKˆȷ)Φˆȷ)),(1nˆȷ=1(1c+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d+Kˆȷ)Φˆȷ),nˆȷ=1(cKˆȷ)Φˆȷ+i(nˆȷ=1(dKˆȷ)Φˆȷ))}{(1ni=1(1a/+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1b/+Kˆȷ)Φˆȷ),ni=1(a/Kˆȷ)Φˆȷ+i(nˆȷ=1(b/Kˆȷ)Φˆȷ)),(1nˆȷ=1(1c/+Kˆȷ)Φˆȷ+i(1nˆȷ=1(1d/+Kˆȷ)Φˆȷ),nˆȷ=1(c/Kˆȷ)Φi+i(nˆȷ=1(d/Kˆȷ)Φˆȷ))}.

    From cases (1) and (2), we have the required proof.

    Theorem 3.5 (Boundedness). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be a set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φi=1 and 0Φˆȷ1, if K+ˆȷ,Kˆȷ are the maximum and minimum BCFCNs. Then,

    K+ˆȷBCFCWA(K1,...,Kn)Kˆȷ. (3.5)

    Proof. We studied two cases (for real and imagined components) separately for MG and credibility degree.

    (1) For membership degree, we have

    (1nˆȷ=1(1min1ˆȷna+Kˆȷ)Φˆȷ)(1nˆȷ=1(1a+Kˆȷ)Φˆȷ)(1nˆȷ=1(1max1ˆȷna+Kˆȷ)Φˆȷ)(1(1min1ˆȷna+Kˆȷ)nˆȷ=1Φˆȷ)(1nˆȷ=1(1a+Kˆȷ)Φˆȷ)(1(1max1ˆȷna+Kˆȷ)nˆȷ=1Φˆȷ).

    As nˆȷ=1Φˆȷ=1, so

    min1ˆȷna+Kˆȷ(1nˆȷ=1(1a+Kˆȷ)Φˆȷ)max1ˆȷna+Kˆȷ.

    Similarly, we can prove for ib+Kˆȷ, and

    nˆȷ=1min1ˆȷn(aKˆȷ)Φˆȷni=1(aKˆȷ)Φˆȷnˆȷ=1max1ˆȷn(aKˆȷ)Φˆȷmin1ˆȷn(aKˆȷ)nˆȷ=1Φˆȷnˆȷ=1(aKˆȷ)Φˆȷmax1ˆȷn(aKˆȷ)nˆȷ=1Φˆȷ.

    As nˆȷ=1Φˆȷ=1. Then

    min1ˆȷnaKˆȷnˆȷ=1(aKˆȷ)Φˆȷmax1ˆȷnaKˆȷ.

    Similarly, we can prove for ibKi.

    (2) For credibility degree, we have

    (1nˆȷ=1(1min1ˆȷnc+Kˆȷ)Φˆȷ)(1nˆȷ=1(1c+Kˆȷ)Φˆȷ)(1nˆȷ=1(1max1ˆȷnc+Kˆȷ)Φˆȷ)(1(1min1ˆȷnc+Kˆȷ)nˆȷ=1Φˆȷ)(1nˆȷ=1(1c+Kˆȷ)Φˆȷ)(1(1max1ˆȷnc+Kˆȷ)nˆȷ=1Φˆȷ).

    As nˆȷ=1Φˆȷ=1, so

    min1ˆȷnc+Kˆȷ(1nˆȷ=1(1c+Kˆȷ)Φˆȷ)max1ˆȷnc+Kˆȷ.

    Similarly, we can prove for id+Kˆȷ. Additionally,

    nˆȷ=1min1ˆȷn(cKˆȷ)Φˆȷnˆȷ=1(cKˆȷ)Φˆȷnˆȷ=1max1ˆȷn(cKˆȷ)Φˆȷmin1ˆȷn(cKˆȷ)nˆȷ=1Φˆȷnˆȷ=1(cKˆȷ)Φˆȷmax1ˆȷn(cKˆȷ)nˆȷ=1Φˆȷ.

    As nˆȷ=1Φˆȷ=1. Then

    min1ˆȷncKˆȷnˆȷ=1(cKˆȷ)Φˆȷmax1ˆȷncKˆȷ.

    Similarly, we have for idKˆȷ.

    Then, combined the above two cases, by the score function, we obtain

    Sc(K+ˆȷ)Sc(Kˆȷ)Sc(Kˆȷ).

    So, based on cases (1) and (2) and the definition of the score function, we get

    K+BCFCWA(K1,...,Kn)K.

    Definition 3.6. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, the BCFCOWA operator is determined as

    BCFCOWA(K1,...,Kn)=nˆȷ=1ΦˆȷKσ(ˆȷ), (3.6)

    and for Kσ(ˆȷ1)Kσ(ˆȷ) the permutation is σ(1),...,σ(n) for all ˆȷ=1,...,n. Utilizing Definition 3.6, aggregated value for BCFCOWA operator is shown in Theorem 3.7.

    Theorem 3.7. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, BCFCOWA operator is obtained as

    BCFCOWA(K1,...,Kn)=nˆȷ=1ΦˆȷKσ(ˆȷ)={(1nˆȷ=1(1a+Kσ(ˆȷ))Φˆȷ+i(1nˆȷ=1(1b+Kσ(ˆȷ))Φˆȷ),nˆȷ=1(aKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(bKσ(ˆȷ))Φˆȷ)),(1nˆȷ=1(1c+Kσ(ˆȷ))Φˆȷ+i(1nˆȷ=1(1d+Kσ(ˆȷ))Φˆȷ),nˆȷ=1(cKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(dKσ(ˆȷ))Φˆȷ))} (3.7)

    where σ(1),...,σ(n) be the permutation of the (ˆȷ=1,...,n), for each Kσ(ˆȷ1)Kσ(ˆȷ) for all (ˆȷ=1,...,n).

    Proof. Proof is follow from Theorem 3.2.

    The BCFCOWA operator satisfied the following properties.

    Theorem 3.8 (Idempotency). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with the weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then

    BCFCOWA(K1,...,Kn)=K. (3.8)

    Proof. Similar to Theorem 4.12.

    Theorem 3.9 (Monotonicity). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) and K/ˆȷ=(a/+Kˆȷ+ib/+Kˆȷ,a/Kˆȷ+ib/Kˆȷ,c/+Kˆȷ+id/+Kˆȷ,c/Kˆȷ+id/Kˆȷ) (ˆȷ=1,...,n) be the set of BCFCNs with the weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if a+Kˆȷa/+Kˆȷ,ib+Kˆȷib/+Kˆȷ,ibKˆȷib/Kˆȷ,aKˆȷa/Kˆȷ, c+Kˆȷc/+Kˆȷ,id+Kˆȷid/+Kˆȷ,cKˆȷc/Kˆȷ, and idKˆȷid/Kˆȷ. Then

    BCFCOWA(K1,...,Kn)BCFCOWA(K/1,...,K/n). (3.9)

    Proof. Similar to Theorem 3.4.

    Theorem 3.10 (Boundedness). Let

    Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)

    (ˆȷ=1,...,n) be the set of BCFCNs with the weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if K+ˆȷ,Kˆȷ are the maximum and minimum BCFCNs. Then

    K+ˆȷBCFCOWA(K1,...,Kn)Kˆȷ. (3.10)

    Proof. Similar to Theorem 3.5.

    Definition 3.11. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCN with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, and ϑ=(ϑ1,...,ϑn)T, such that nˆȷ=1ϑˆȷ=1 and 0ϑˆȷ1 be the associated weights of BCFCNs. Then, the BCFCHA operator is obtained as

    BCFCHA(K1,...,Kn)=nˆȷ=1ΦˆȷKσ(ˆȷ), (3.11)

    and for Kσ(ˆȷ1)Kσ(ˆȷ) the permutation is σ(1),...,σ(n) for all (ˆȷ=1,...,n). Utilizing Definition 4.11, aggregated value for BCFCHA operator is shown in Theorem 4.12.

    Theorem 3.12. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, and ϑ=(ϑ1,...,ϑn)T, such as nˆȷ=1ϑˆȷ=1 and 0ϑˆȷ1 be the associated weight vector of the given set of BCFCNs. Then, BCFCHA operator is obtained as

    BCFCHA(K1,...,Kn)=nˆȷ=1ΦˆȷKσ(ˆȷ)={(1nˆȷ=1(1a+Kσ(ˆȷ))Φˆȷ+i(1nˆȷ=1(1b+Kσ(ˆȷ))Φˆȷ),nˆȷ=1(aKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(bKσ(ˆȷ))Φˆȷ)),(1nˆȷ=1(1c+Kσ(ˆȷ))Φˆȷ+i(1nˆȷ=1(1d+Kσ(ˆȷ))Φˆȷ),nˆȷ=1(cKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(dKσ(ˆȷ))Φˆȷ))} (3.12)

    where σ(1),...,σ(n) be the permutation of the (ˆȷ=1,...,n), for each Kσ(ˆȷ1)Kσ(ˆȷ) for all (ˆȷ=1,...,n). The largest permutation value from the family of BCFCNs is represented by Kσ(ˆȷ)=nϑˆȷKˆȷ, and n stands for the balancing coefficient.

    Proof. Proof is follow from Theorem 3.2.

    Here, we are going to defined some aggregation operators like as, BCFCWG, BCFCOWG, and BCFCHG operators.

    Definition 4.1. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+K1+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be a set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, the BCFCWG operator is determined as

    BCFCWG(K1,...,Kn)=nˆȷ=1(Kˆȷ)Φˆȷ, (4.1)

    utilizing Definition 4.1, aggregated value for BCFCWG operator is shown in Theorem refthE1.

    Theorem 4.2. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,c+Kˆȷ+id+Kˆȷ,aKˆȷ+ibKˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, BCFCWG operator is obtained as

    BCFCWG(K1,...,Kn)=nˆȷ=1(Kˆȷ)Φˆȷ={(nˆȷ=1(a+Kˆȷ)Φˆȷ+i(nˆȷ=1(b+Kˆȷ)Φˆȷ),1+nˆȷ=1(1+aKˆȷ)Φˆȷ+i(1+nˆȷ=1(1+bKˆȷ)Φˆȷ)),(nˆȷ=1(c+Kˆȷ)Φˆȷ+i(nˆȷ=1(d+Kˆȷ)Φˆȷ),1+nˆȷ=1(1+cKˆȷ)Φˆȷ+i(1+nˆȷ=1(1+dKˆȷ)Φˆȷ))}. (4.2)

    Proof. Proof is same as Theorem 3.2.

    The BCFCWG operator satisfied the following properties.

    Theorem 4.3 (Idempotency). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with the weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then,

    BCFCWG(K1,...,Kn)=K. (4.3)

    Proof. Proof is same as Theorem 3.3.

    Theorem 4.4 (Monotonicity). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) and K/ˆȷ=(a/+Kˆȷ+ib/+Kˆȷ,a/Kˆȷ+ib/Kˆȷ,c/+Kˆȷ+id/+Kˆȷ,c/Kˆȷ+id/Kˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if a+Kˆȷa/+Kˆȷ,ib+Kˆȷib/+Kˆȷ,ibKˆȷib/Kˆȷ,aKˆȷa/Kˆȷ, c+Kˆȷc/+Kˆȷ,id+Kˆȷid/+Kˆȷ,cKˆȷc/Kˆȷ, and idKˆȷid/Kˆȷ. Then,

    BCFCWG(K1,...,Kn)BCFCWG(K1,...,Kn). (4.4)

    Proof. Proof is same as (3.4).

    Theorem 4.5 (Boundedness). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) (ˆȷ=1,...,n) be a set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if K+ˆȷ,Kˆȷ are the maximum and minimum BCFCNs. Then

    K+ˆȷBCFCWG(K1,...,Kn)Kˆȷ. (4.5)

    Proof. Proof is same as Theorem 3.5.

    Definition 4.6. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weight vector Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, the BCFCOWG operator is determined as

    BCIFOWG(K1,...,Kn)=nˆȷ=1(Kσ(ˆȷ))Φˆȷ, (4.6)

    and for Kσ(ˆȷ1)Kσ(ˆȷ) the permutation is σ(1),...,σ(n) for all (ˆȷ=1,...,n). Utilizing Definition 4.6, aggregated value for BCFCOWG operator is shown in Theorem 4.7.

    Theorem 4.7. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then, BCFCOWG operator is obtained as

    BCFCOWG(K1,...,Kn)=nˆȷ=1(Kσ(ˆȷ))Φˆȷ={(nˆȷ=1(aKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(bKσ(ˆȷ))Φˆȷ),1+nˆȷ=1(1+a+Kσ(ˆȷ))Φˆȷ+i(1+nˆȷ=1(1+b+Kσ(ˆȷ))Φˆȷ)),(nˆȷ=1(cKσ(ˆȷ))Φˆȷ+i(nˆȷ=1(dKσ(ˆȷ))Φˆȷ),1+nˆȷ=1(1+c+Kσ(ˆȷ))Φˆȷ+i(1+nˆȷ=1(1+d+Kσ(ˆȷ))Φˆȷ))} (4.7)

    where σ(1),...,σ(n) be the permutation of the (ˆȷ=1,...,n), for each Kσ(ˆȷ1)Kσ(ˆȷ) for all (ˆȷ=1,...,n).

    Proof. Proof follows from Theorem 3.2.

    The BCFCOWG operator satisfied the following properties.

    Theorem 4.8 (Idempotency). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with the weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1. Then

    BCFCOWG(K1,...,Kn)=K. (4.8)

    Proof. Similar to Theorem 3.3.

    Theorem 4.9 (Monotonicity). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) and K/ˆȷ=(a/+Kˆȷ+ib/+Kˆȷ,a/Kˆȷ+ib/Kˆȷ,c/+Kˆȷ+id/+Kˆȷ,c/Kˆȷ+id/Kˆȷ) (ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if a+Kˆȷa/+Kˆȷ,ib+Kˆȷib/+Kˆȷ,ibKˆȷib/Kˆȷ,aKˆȷa/Kˆȷ, c+Kˆȷc/+Kˆȷ,id+Kˆȷid/+Kˆȷ,cKˆȷc/Kˆȷ, and ˆȷdKˆȷˆȷd/Kˆȷ. Then

    BCFCOWG(K1,...,Kn)BCFCOWG(K/1,...,K/n). (4.9)

    Proof. Similar to Theorem 3.4.

    Theorem 4.10 (Boundedness). Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ) (ˆȷ=1,...,n) be the set of BCFCNs with weight are Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, if K+ˆȷ,Kˆȷ are the maximum and minimum BCFCNs. Then

    K+ˆȷBCFCOWG(K1,...,Kn)Kˆȷ. (4.10)

    Proof. Similar to Theorem 3.5.

    Definition 4.11. Let Kˆȷ=(a+Kˆȷ+ib+Kˆȷ,aKˆȷ+ibKˆȷ,c+Kˆȷ+id+Kˆȷ,cKˆȷ+idKˆȷ)(ˆȷ=1,...,n) be the set of BCFCNs with weights Φ=(Φ1,...,Φn)T, such as nˆȷ=1Φˆȷ=1 and 0Φˆȷ1, and ϑ=(ϑ1,...,ϑn)T, such as nˆȷ=1ϑˆȷ=1 and 0ϑˆȷ1 be the associated weights of BCFCNs. Then, the BCFCHG operator is obtained as;

    BCFCHG(K1,...,Kn)=nˆȷ=1(Kσ(ˆȷ))Φˆȷ, (4.11)

    and \mathcal{K}_{\sigma (\hat{\jmath}-1)}\geq \mathcal{K}_{\sigma \left(\hat{\jmath}\right) } the permutation is \sigma (1), ..., \sigma (n) for all \left(\hat{\jmath} = 1, ..., n\right). Utilizing Definition 4.11, aggregated value of BCFCHG operator is given in Theorem 4.12.

    Theorem 4.12. Let \mathcal{K}_{\hat{\jmath}} = \left(\langle a_{\mathcal{K} _{\hat{\jmath}}}^{+}+ib_{\mathcal{K}_{\hat{\jmath}}}^{+}, a_{\mathcal{K}_{ \hat{\jmath}}}^{-}+ib_{\mathcal{K}_{\hat{\jmath}}}^{-}\rangle, \langle c_{\mathcal{K}_{\hat{\jmath}}}^{+}+id_{\mathcal{K}_{\hat{\jmath }}}^{+}, c_{\mathcal{K}_{\hat{\jmath}}}^{-}+id_{\mathcal{K}_{\hat{\jmath} }}^{-}\rangle \right) \left(\hat{\jmath} = 1, ..., n\right) be the set of BCFCNs with weights \Phi = \left(\Phi _{1}, ..., \Phi _{n}\right) ^{T}, such as \sum_{\hat{\jmath} = 1}^{n}\Phi _{\hat{\jmath}} = 1 and 0\leq \Phi _{ \hat{\jmath}}\leq 1, and \vartheta = \left(\vartheta _{1}, ..., \vartheta _{n}\right) ^{T}, such that \sum_{\hat{\jmath} = 1}^{n}\vartheta _{\hat{ \jmath}} = 1 and 0\leq \vartheta _{\hat{\jmath}}\leq 1 be the associated weights of BCFCNs. Then, BCFCHG operator is obtained as

    \begin{eqnarray} &&BCFCHG\left( \mathcal{K}_{1},...,\mathcal{K}_{n}\right) = \bigotimes\limits_{\hat{\jmath} = 1}^{n}\left( \mathcal{K}_{\sigma \left( \hat{\jmath}\right) }^{\ast }\right) ^{\Phi _{\hat{\jmath}}} \\ & = &\left\{ \begin{array}{c} \left( \prod\limits_{\hat{\jmath} = 1}^{n}\left( a_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast -}\right) ^{\Phi _{\hat{\jmath}}}+i\left( \prod\limits_{\hat{\jmath} = 1}^{n}\left( b_{\mathcal{K}_{\sigma \left( \hat{ \jmath}\right) }}^{\ast -}\right) ^{\Phi _{\hat{\jmath}}}\right) ,-1+\prod\limits_{\hat{\jmath} = 1}^{n}\left( 1+a_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast +}\right) ^{\Phi _{\hat{\jmath}}}+i\left( -1+\prod\limits_{\hat{\jmath} = 1}^{n}\left( 1+b_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast +}\right) ^{\Phi _{\hat{\jmath}}}\right) \right) , \\ \left( \prod\limits_{\hat{\jmath} = 1}^{n}\left( c_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast -}\right) ^{\Phi _{\hat{\jmath}}}+i\left( \prod\limits_{\hat{\jmath} = 1}^{n}\left( d_{\mathcal{K}_{\sigma \left( \hat{ \jmath}\right) }}^{\ast -}\right) ^{\Phi _{\hat{\jmath}}}\right) ,-1+\prod\limits_{\hat{\jmath} = 1}^{n}\left( 1+c_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast +}\right) ^{\Phi _{\hat{\jmath}}}+i\left( -1+\prod\limits_{\hat{\jmath} = 1}^{n}\left( 1+d_{\mathcal{K}_{\sigma \left( \hat{\jmath}\right) }}^{\ast +}\right) ^{\Phi _{\hat{\jmath}}}\right) \right) \end{array} \right\} \end{eqnarray} (4.12)

    where \sigma (1), ..., \sigma (n) be the permutation of the (\hat{\jmath} = 1, ..., n), for each \mathcal{K}_{\sigma (\hat{\jmath}-1)}\geq \mathcal{K} _{\sigma (\hat{\jmath})} for all \left(\hat{\jmath} = 1, ..., n\right). The largest permutation value from the set of BCFCNs is represented by \mathcal{ K}_{\sigma \left(\hat{\jmath}\right) }^{\ast } = \left(\mathcal{K}_{\hat{ \jmath}}\right) ^{n\vartheta _{\hat{\jmath}}} , and n stands for the balancing coefficient.

    Proof. Proof is follow from Theorem 3.2.

    In this section, we construct an approach to tackle the MCGDM problem using the proposed bipolar complex intuitionistic fuzzy set. Assume that \acute{E} = \left\{ \acute{E}_{1}, ..., \acute{E}_{n}\right\} is the collection of n criteria and \wp = \left\{ \wp _{1}, ..., \wp _{m}\right\} is the set of m alternatives for a MCGDM problem. Let the weights for the criterion \acute{E }_{\hat{\jmath}} as \Phi = (\Phi _{1}, ..., \Phi _{n})^{T}, such as \sum_{ \hat{\jmath} = 1}^{n}\Phi _{\hat{\jmath}} and 0\leq \Phi _{\hat{\jmath}} . The following are the key steps:

    Step 1: Create a decision matrix using the assessment data collected in accordance with the criteria \acute{E}_{\hat{\jmath}} for qualified experts for each alternative \wp ;

    \begin{equation*} {\bf{M}} = \left[ \begin{array}{c} \Im _{11} & \Im _{12} & . & . & \Im _{1n} \\ \Im _{21} & \Im _{22} & . & . & \Im _{2n} \\ \Im _{31} & \Im _{32} & . & . & \Im _{3n} \\ . & . & . & . & . \\ \Im _{m1} & \Im _{m2} & . & . & \Im _{mn} \end{array} \right]. \end{equation*}

    Step 2:Evaluate the aggregate information given by experts with the help of BCFCWA operators.

    Step 3: Evaluate the aggregate information using BCFCOWA operators.

    Step 4: Evaluate the score value of the information obtained with the help of operators.

    Step 5: Give alternatives a ranking based on the score value.

    As per predictions, chronic diseases (CDs) can cause one-third of all deaths in the world and are one of the main causes of death and disability in the world. There are numerous diseases related with CDs, like diabetes, hypertension heart disease and cardiovascular diseases (CVDs), some of them present higher risks than others (in particular, CVD). It is one of the main causes of disability and presents threats to population vitality. Because it causes diseases like hypertension, arrhythmia, stroke, and heart attacks, as well as deaths, CVD is a global crisis. The diagnosis, monitoring, and treatment of CVD are necessary since it is a life-threatening disease. However, there are also other problems that can make diagnosis and treatment more difficult, like the lack of qualified cardiologists or patients who live in remote areas far from hospitals. Modern technologies, such as the Internet of Things (IoT), are utilized to monitor patients with CD in order to address these issues. IoT has set the stage for a variety of uses, and it has played a remarkable role in telemedicine in the health care field and in managing patients who are located elsewhere.

    In this real life example we have to select the best hospital using our proposed work. There are four alternatives (Hospitals) and six criteria with the weights are \Phi = \left(0.20, 0.30, 0.15, 0.25, 0.10\right) ^{T} , which is discussed as follows:

    (1) Surgical Doctors (SD): In this type of criteria, we have discussed a special groups of doctors, which have the ability to treat all the surgical patient.

    (2) Surgical Team (ST): In this criteria, there are one head doctor and having surgical doctors to handle the surgical problems.

    (3) Surgical Room (SR): This criteria is about surgical room which is use for special purpose for any surgical patient.

    (4) Oxygen Supplier (OS): In this criteria the surgical team can provide oxygen to any surgical patient.

    (5) Send Ambulance (SA): This type of criteria is important to carry any surgical patient to the surgical team.

    (6) Prove Medications (PM): In this types of criteria the surgical team provide specific medicine to any surgical patient.

    Let, there are their experts with the weight vector \kappa = (0.3, 0.4, 0.3)^{T}, provided their individual assessment for each option and the corresponding assessments are presented in Tables 13.

    Table 1.  BCF evaluation information given by expert one.
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{2}
    \wp _{1} \left(\begin{array}{c} \langle 0.3+i0.4, -0.5-i0.4\rangle, \\ \langle 0.7+i0.5, -0.2-i0.5\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.1+i0.8, -0.1-i0.8\rangle, \\ \langle 0.8+i0.1, -0.6-i0.1\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.2+i0.3, -0.8-i0.3\rangle, \\ \langle 0.8+i0.6, -0.2-i0.6\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.3+i0.4, -0.5-i0.4\rangle, \\ \langle 0.7+i0.5, -0.2-i0.5\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.5+i0.7, -0.4-i0.3\rangle, \\ \langle 0.3+i0.3, -0.6-i0.6\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.6+i0.7, -0.6-i0.2\rangle, \\ \langle 0.4+i0.2, -0.3-i0.7\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.6+i0.5, -0.3-i0.2\rangle, \\ \langle 0.4+i0.3, -0.7-i0.8\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.2+i0.5, -0.3-i0.5\rangle, \\ \langle 0.5+i0.4, -0.7-i0.5\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{4}
    \wp _{1} \left(\begin{array}{c} \langle 0.3+i0.3, -0.6-i0.6\rangle, \\ \langle 0.5+i0.7, -0.4-i0.3\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.2+i0.4, -0.3-i0.2\rangle, \\ \langle 0.7+i0.4, -0.6-i0.7\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.4+i0.2, -0.3-i0.7\rangle, \\ \langle 0.6+i0.7, -0.6-i0.2\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.5+i0.4, -0.7-i0.5\rangle, \\ \langle 0.2+i0.5, -0.3-i0.5\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.3+i0.4, -0.5-i0.4\rangle, \\ \langle 0.7+i0.5, -0.2-i0.5\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.4+i0.2, -0.3-i0.7\rangle, \\ \langle 0.6+i0.7, -0.6-i0.2\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.8+i0.6, -0.2-i0.6\rangle, \\ \langle 0.2+i0.3, -0.8-i0.3\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.7+i0.4, -0.6-i0.7\rangle, \\ \langle 0.2+i0.4, -0.3-i0.2\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{5} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{6}
    \wp _{1} \left(\begin{array}{c} \langle 0.4+i0.3, -0.7-i0.8\rangle, \\ \langle 0.6+i0.5, -0.3-i0.2\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.8+i0.1, -0.6-i0.1\rangle, \\ \langle 0.1+i0.8, -0.1-i0.8\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.3+i0.3, -0.6-i0.6\rangle, \\ \langle 0.5+i0.7, -0.4-i0.3\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.2+i0.4, -0.3-i0.2\rangle, \\ \langle 0.7+i0.4, -0.6-i0.7\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.8+i0.1, -0.6-i0.1\rangle, \\ \langle 0.1+i0.8, -0.1-i0.8\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.2+i0.3, -0.8-i0.3\rangle, \\ \langle 0.8+i0.6, -0.2-i0.6\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.5+i0.4, -0.7-i0.5\rangle, \\ \langle 0.2+i0.5, -0.3-i0.5\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.6+i0.5, -0.3-i0.2\rangle, \\ \langle 0.4+i0.3, -0.7-i0.8\rangle \end{array} \right)

     | Show Table
    DownLoad: CSV
    Table 2.  BCF evaluation information given by expert two.
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{2}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle ,\\ \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle ,\\ \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.2+i0.5,-0.3-i0.5\right\rangle ,\\ \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle ,\\ \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle ,\\ \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle ,\\ \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle ,\\ \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle ,\\ \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{4}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle ,\\ \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle ,\\ \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle ,\\ \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.5,-0.3-i0.5\right\rangle\\ \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle , \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle ,\\ \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle ,\\ \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle ,\\ \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle ,\\ \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{5} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{6}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle\\ \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle ,\\ \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle ,\\ \left\langle 0.2+i0.5,-0.3-i0.5\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle\\ \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle , \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle ,\\ \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle\\ \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle , \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle ,\\ \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle ,\\ \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle \end{array} \right)

     | Show Table
    DownLoad: CSV
    Table 3.  BCF evaluation information given by expert three.
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{2}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle\\ \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle\\ \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle , \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle ,\\ \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle\\ \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle , \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle ,\\ \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle ,\\ \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle\\ \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.2,-0.3-i0.5\right\rangle ,\\ \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{4}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle\\ \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle ,\\ \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.6+i0.7,-0.6-i0.2\right\rangle ,\\ \left\langle 0.4+i0.2,-0.3-i0.7\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.5,-0.3-i0.5\right\rangle\\ \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle , \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle ,\\ \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle\\ \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle , \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle\\ \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.4,-0.3-i0.2\right\rangle\\ \left\langle 0.7+i0.4,-0.6-i0.7\right\rangle , \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{5} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{6}
    \wp _{1} \left( \begin{array}{c} \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle ,\\ \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.1+i0.8,-0.1-i0.8\right\rangle\\ \left\langle 0.8+i0.1,-0.6-i0.1\right\rangle , \end{array} \right)
    \wp _{2} \left( \begin{array}{c} \left\langle 0.5+i0.4,-0.7-i0.5\right\rangle ,\\ \left\langle 0.2+i0.5,-0.3-i0.5\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle ,\\ \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle \end{array} \right)
    \wp _{3} \left( \begin{array}{c} \left\langle 0.6+i0.5,-0.3-i0.2\right\rangle ,\\ \left\langle 0.4+i0.3,-0.7-i0.8\right\rangle \end{array} \right) \left( \begin{array}{c} \left\langle 0.7+i0.5,-0.2-i0.5\right\rangle\\ \left\langle 0.3+i0.4,-0.5-i0.4\right\rangle , \end{array} \right)
    \wp _{4} \left( \begin{array}{c} \left\langle 0.5+i0.7,-0.4-i0.3\right\rangle\\ \left\langle 0.3+i0.3,-0.6-i0.6\right\rangle , \end{array} \right) \left( \begin{array}{c} \left\langle 0.2+i0.3,-0.8-i0.3\right\rangle\\ \left\langle 0.8+i0.6,-0.2-i0.6\right\rangle , \end{array} \right)

     | Show Table
    DownLoad: CSV

    Step 1:The total information given by experts for each alternative \wp _{i} under the criteria \acute{E}_{\hat{\jmath}} in Tables 13.

    Step 2:Using the BCFCWA operators and expert-provided data against their weights.

    Step 3: Using the BCFCOWA operator and the aggregated value of Table 4, with the weight vector \Phi = \left(0.20, 0.30, 0.15, 0.25, 0.10\right) ^{T} . The total values of the alternatives \wp _{i}(i = 1, ..., 4) as

    \begin{equation*} \begin{array}{l} \mathbb{R} _{1} & \left( \langle 0.432+i0.405,-0.228-i0.321\rangle ,\langle 0.443+i0.407,-0.327-i0.332\rangle \right). \\ \mathbb{R} _{2} & \left( \langle 0.377+i0.185,-0.328-i0.326\rangle ,\langle 0.486+i0.701,-0.325-i0.341\rangle \right). \\ \mathbb{R} _{3} & \left( \langle 0.571+i0.342,-0.690-i0.325\rangle ,\langle 0.436+i0.332,-0.170-i0.438\rangle \right). \\ \mathbb{R} _{4} & \left( \langle 0.589+i0.327,-0.489-i0.211\rangle ,\langle 0.357+i0.376,-0.329-i0.462\rangle \right). \end{array} \end{equation*}
    Table 4.  Aggregated values using BCFCWA operator.
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{2}
    \wp _{1} \left(\begin{array}{c} \langle 0.361+i0.491, -0.198-i0.376\rangle, \\ \langle 0.475+i0.403, -0.531-i0.616\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.498+i0.379, -0.291-i0.436\rangle, \\ \langle 0.316+i0.372, -0.187-i0.439\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.338+i0.364, -0.229-i0.375\rangle, \\ \langle 0.218+i0.287, -0.251-i0.336\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.382+i0.385, -0.421-i0.333\rangle, \\ \langle 0.498+i0.405, -0.627-i0.209\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.223+i0.590, -0.327-i0.369\rangle, \\ \langle 0.319+i0.497, -0.418-i0.232\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.625+i0.221, -0.339-i0.195\rangle, \\ \langle 0.252+i0.590, -0.393-i0.284\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.377+i0.471, -0.370-i0.305\rangle, \\ \langle 0.516+i0.264, -0.409-i0.260\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.442+i0.408, -0.275-i0.196\rangle, \\ \langle 0.610+i0.384, -0.224-i0.398\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{4}
    \wp _{1} \left(\begin{array}{c} \langle 0.443+i0.357, -0.509-i0.287\rangle, \\ \langle 0.369+i0.361, -0.432-i0.339\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.542+i0.215, -0.537-i0.332\rangle, \\ \langle 0.437+i0.118, -0.503-i0.437\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.400+i0.325, -0.326-i0.642\rangle, \\ \langle 0.455+i0.452, -0.509-i0.351\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.542+i0.225, -0.417-i0.499\rangle, \\ \langle 0.378+i0.245, -0.352-i0.350\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.542+i0.590, -0.431-i0.376\rangle, \\ \langle 0.423+i0.480, -0.548-i0.327\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.246+i0.324, -0.562-i0.325\rangle, \\ \langle 0.411+i0.325, -0.503-i0.653\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.480+i0.531, -0.762-i0.390\rangle, \\ \langle 0.437+i0.531, -0.664-i0.328\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.434+i0.476, -0.438-i0.332\rangle, \\ \langle 0.368+i0.432, -0.666-i0.598\rangle \end{array} \right)
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{5} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \acute{E}_{6}
    \wp _{1} \left(\begin{array}{c} \langle 0.243+i0.437, -0.359-i0.127\rangle, \\ \langle 0.392+i0.315, -0.420-i0.391\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.244+i0.346, -0.248-i0.432\rangle, \\ \langle 0.382+i0.142, -0.266-i0.258\rangle \end{array} \right)
    \wp _{2} \left(\begin{array}{c} \langle 0.452+i0.205, -0.474-i0.349\rangle, \\ \langle 0.383+i0.125, -0.532-i0.302\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.240+i0.351, -0.472-i0.230\rangle, \\ \langle 0.147+i0.501, -0.564-i0.338\rangle \end{array} \right)
    \wp _{3} \left(\begin{array}{c} \langle 0.252+i0.325, -0.227-i0.424\rangle, \\ \langle 0.419+i0.189, -0.531-i0.270\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.252+i0.250, -0.241-i0.360\rangle, \\ \langle 0.343+i0.418, -0.358-i0.137\rangle \end{array} \right)
    \wp _{4} \left(\begin{array}{c} \langle 0.261+i0.234, -0.252-i0.235\rangle, \\ \langle 0.418+i0.135, -0.363-i0.363\rangle \end{array} \right) \left(\begin{array}{c} \langle 0.404+i0.495, -0.365-i0.462\rangle, \\ \langle 0.450+i0.262, -0.259-i0.316\rangle \end{array} \right)

     | Show Table
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    Table 5.  Score values of the alternatives.
    Operators Sc\left(\acute{E}_{1}\right) Sc\left(\acute{E} _{2}\right) Sc\left(\acute{E}_{3}\right) Sc\left(\acute{E} _{4}\right)
    BCFCWA 0.328 0.437 0.492 0.290

     | Show Table
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    Step 4: Analyze the score value of the data we collected using various operators.

    Step 5: Utilize the ranking values shown in Table 6 to determine which choice is the best.

    Table 6.  Alternative ranking.
    BCFCWA \acute{E}_{3} > \acute{E}_{2} > \acute{E}_{1} > \acute{E}_{4}

     | Show Table
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    Comparative study

    The comparative study of determined techniques was discussed in this section, along with certain common operators based on accepted concept like as, BCFSs.

    For this, we choose a few famous theories that are, Mahmood et al. [25], BCFSs and their applications in generalized similarity measures; Mahmood et al. [26], BCFS-based Hamacher aggregation information; Mahmood et al. [27], Dombi AOs under bipolar complex fuzzy information. Jana et al. [19], Bipolar fuzzy Dombi prioritized aggregation operators; Gao et al. [13], Dual hesitant bipolar fuzzy Hamacher aggregation operators.

    The method described in [13,19,25,26,27] contains bipolar fuzzy set details, but the given model cannot be solved using this method. Reviewing Table 5 reveals that the methods now in use lack basic information and are unable to solve or rank the case that has been provided. Compared to other methods already in use, the strategy suggested in this study is more capable and dependable. The main analysis of the identified and proposed hypotheses is presented in Table 7.

    Table 7.  Ranking of the existing methods.
    Methods Score value Ranking
    Sc\left(\acute{E}_{1}\right) Sc\left(\acute{E}_{2}\right) Sc\left(\acute{E}_{3}\right) Sc\left(\acute{E}_{4}\right)
    Mahmood et al. [25] 0.760 \ \ 0.792 \ \ \ \ 0.826 \ \ 0.731 \acute{E}_{3} > \acute{E}_{2} > \acute{E}_{1} > \acute{E}_{4}
    Mahmood et al. [26] 0.529 \ \ \ 0.553 \ \ \ 0.587 \ \ 0.502 \acute{E}_{3} > \acute{E}_{2} > \acute{E}_{1} > \acute{E}_{4}
    Mahmood et al. [27] 0.831 \ \ \ 0.842 \ \ \ 0.889 \ \ 0.782 \acute{E}_{3} > \acute{E}_{1} > \acute{E}_{4} > \acute{E}_{2}
    Jana et al. [19] 0.453 \ \ \ 0.441 \ \ \ 0.488 \ \ 0.427 \acute{E}_{3} > \acute{E}_{1} > \acute{E}_{2} > \acute{E}_{4}
    Gao et al. [13] 0.663 \ \ \ 0.684 \ \ \ 0.699 \ \ 0.650 \acute{E}_{3} > \acute{E}_{2} > \acute{E}_{1} > \acute{E}_{4}

     | Show Table
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    We define a number of operations, the scoring function, and the accuracy function for BCFCS in this paper. We also established several aggregation operators based on BCFC operational laws, such as BCFCWA, BCFCOWA, BCFCHA, BCFCWG, BCFCOWG, and BCFCHG operators. We explored the essential properties of the aforementioned operators' specific situations, such as idempotency, boundedness, and monotonous. Next, utilizing these operators, we solved the bipolar complex fuzzy MCGDM problem. To validate the interpreted techniques, we provided a numerical example of selecting fire extinguishers. Finally, we compared our findings to those of existing operators to establish the usefulness and applicability of our method.

    In the future, we will use our proposed operators in different domains, like as, complex Pythagorean fuzzy set, complex picture fuzzy set, complex Spherical fuzzy set, and complex fractional orthotriple fuzzy set.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number: IFP22UQU4310396DSR080. This research was also supported by office of Research Management, Universiti Malaysia Terengganu, Malaysia.

    The authors declare that they have no conflicts of interest.



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