Research article Special Issues

Averaging aggregation operators under the environment of q-rung orthopair picture fuzzy soft sets and their applications in MADM problems

  • q-Rung orthopair fuzzy soft set handles the uncertainties and vagueness by membership and non-membership degree with attributes, here is no information about the neutral degree so to cover this gap and get a generalized structure, we present hybrid of picture fuzzy set and q-rung orthopair fuzzy soft set and initiate the notion of q-rung orthopair picture fuzzy soft set, which is characterized by positive, neutral and negative membership degree with attributes. The main contribution of this article is to investigate the basic operations and some averaging aggregation operators like q-rung orthopair picture fuzzy soft weighted averaging operator and q-rung orthopair picture fuzzy soft order weighted averaging operator under the environment of q-rung orthopair picture fuzzy soft set. Moreover, some fundamental properties and results of these aggregation operators are studied, and based on these proposed operators we presented a stepwise algorithm for MADM by taking the problem related to medical diagnosis under the environment of q-rung orthopair picture fuzzy soft set and finally, for the superiority we presented comparison analysis of proposed operators with existing operators.

    Citation: Sumbal Ali, Asad Ali, Ahmad Bin Azim, Ahmad ALoqaily, Nabil Mlaiki. Averaging aggregation operators under the environment of q-rung orthopair picture fuzzy soft sets and their applications in MADM problems[J]. AIMS Mathematics, 2023, 8(4): 9027-9053. doi: 10.3934/math.2023452

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  • q-Rung orthopair fuzzy soft set handles the uncertainties and vagueness by membership and non-membership degree with attributes, here is no information about the neutral degree so to cover this gap and get a generalized structure, we present hybrid of picture fuzzy set and q-rung orthopair fuzzy soft set and initiate the notion of q-rung orthopair picture fuzzy soft set, which is characterized by positive, neutral and negative membership degree with attributes. The main contribution of this article is to investigate the basic operations and some averaging aggregation operators like q-rung orthopair picture fuzzy soft weighted averaging operator and q-rung orthopair picture fuzzy soft order weighted averaging operator under the environment of q-rung orthopair picture fuzzy soft set. Moreover, some fundamental properties and results of these aggregation operators are studied, and based on these proposed operators we presented a stepwise algorithm for MADM by taking the problem related to medical diagnosis under the environment of q-rung orthopair picture fuzzy soft set and finally, for the superiority we presented comparison analysis of proposed operators with existing operators.



    Let A be the class of analytic functions in the open unit disk D:={zC:|z|<1} of the form

    f(z)=z+n=2anzn,zD, (1.1)

    and let S be the class of functions fA which are univalent in D.

    Using the principle of subordination Ma and Minda [16] introduced the class S(φ) (so called Ma-Minda-type functions)

    S(φ):={fA:zf(z)f(z)φ(z)}, (1.2)

    where in our paper we suppose that φ is univalent in the unit disk D, it has positive real in D, and satisfies the condition φ(0)=1, while the symbol "" stands for the usual subordination. It is well-known that S(φ)S, and we emphasize that some special subclasses of the class S(φ) play a significant role in Geometric Function Theory because of many interesting geometric aspects.

    For example, taking φ(z):=(1+Az)/(1+Bz), where AC, 1B0 and AB, we get the class S[A,B]. This class with the restriction 1B<A1 reduces to the popular class of Janowski starlike functions.

    Remark 1. (ⅰ) By considering φ(z):=(z)=1+zez in [15], the researchers introduced and studied another Ma-Minda-type function class S() of starlike functions, where maps the unit disk onto a cardioid domain.

    (ⅱ) We emphasize that the class SC of functions fA with φ(z):=ϕC(z)=1+4z3+2z23 such that

    zf(z)f(z)1+4z3+2z23,

    that maps the open unit disk into the cardioid domain

    ϕC(D)={w=u+ivC:(9u2+9v218u+5)216(9u2+9v26u+1)<0},

    was extensively investigated by Sharma et al. [28].

    (ⅲ) In [12], by using the polynomial function φ(z):=ϕcar(z)=1+z+z22 the corresponding class Scar of functions was investigated by different authors (see also, for example [13,22,26]), while the function ϕcar maps the open unit disk into the cardioid domain

    ϕcar(D)={w=u+ivC:(4u2+4v28u1)2+4(4u2+4v212u+1)<0}.

    The logarithmic coefficients γn of the function fS are defined with the aid of the following power series expansion

    Ff(z):=logf(z)z=2n=1γn(f)zn,zD,wherelog1=0. (1.3)

    These coefficients play an important role for different estimates in the theory of univalent functions, and note that we use γn instead of γn(f); in this regard see [17, Chapter 2] and [18,19]. In [6], authors determined bounds on the difference of the moduli of successive coefficients for some classes defined by subordination using the logarithmic coefficients.

    The logarithmic coefficients γn of an arbitrary function fS (see [10, Theorem 4]) satisfy the inequality

    n=1|γn|2π26,

    and the equality is obtained for the Koebe function. For fS, the inequality |γn|1/n holds but it is not true for the whole class S (see [9, Theorem 8.4]). However, the problem of the best upper bounds for the logarithmic coefficients of univalent functions for n3 is presumably still a concern.

    Recently, Ponnusamy et al. [24] studied the logarithmic coefficients problems in families related to starlike and convex functions and obtained the sharp upper bound for |γn| when n=1,2,3 and f belongs to the families. Some first logarithmic coefficients γn were obtained for certain subclasses of close-to-convex functions by Ali and Vasudevarao [5] and Pranav Kumar and Vasudevarao [25]. In [14], Kowalczyk and Lecko obtained related bounds with these coefficients for strongly starlike and strongly convex functions.

    Due to the major importance of the study of the logarithmic coefficients, in recent years several authors have investigated the issues regarding the logarithmic coefficients and the related problems for some subclasses of analytic functions (for example, see [2,3,4,7,8,11,21,23,25,30,31,32].

    In [1] the authors obtained the bounds for the logarithmic coefficients γn (nN) of the general class S(φ), while the given bounds would generalize many previous results.

    Theorem A. [1, Theorem 1(ⅰ)] Let the function fS(φ). If φ is convex (univalent), then the logarithmic coefficients of f satisfy the inequalities:

    |γn||B1|2n,nN:={1,2,3,}, (1.4)

    and

    n=1|γn|214n=1|Bn|2n2. (1.5)

    All inequalities in (1.4) and (1.5) are sharp for the function fn given by zfn(z)/fn(z)=φ(zn) for any nN and the function f given by zf(z)/f(z)=φ(z), respectively.

    We correct that the next inequality (and some other results in [1]) is sharp only for n=1 as it is shown in the following result:

    Theorem B. [1, Theorem 1(ⅱ)] Let the function fS(φ). If φ is starlike (univalent) with respect to 1, then the logarithmic coefficients of f satisfy the inequality

    |γn||B1|2,nN.

    The above inequality is sharp for n=1 for the function f given by zf(z)/f(z)=φ(z).

    If we do direct calculations to get the upper bound of |γ1| we will also get the same sharp bound |γ1||B1|2. Therefore, this shows this theorem is sharp for n = 1. But there are some examples like the mentioned classes in Remark 1(ⅱ) and Remark 1(ⅲ) and using Theorem 3 and Theorem 2, respectively, that show Theorem B is not sharp for n>1.

    A possible sharper version of Theorem B could be conjectured as follows:

    Conjecture. Let the function fS(φ) for some φ. If φ is starlike (univalent) with respect to 1, then the logarithmic coefficients of f satisfy the inequality

    |γn||B1|2n,nN.

    The above inequality is sharp for each n.

    Lemma 1. [9,27] (Theorem 6.3, p. 192; Rogosinski's Theorem Ⅱ (ⅰ)) Let f(z)=n=1anzn and g(z)=n=1bnzn be analytic in D, and suppose that f(z)g(z) where g is univalent in D. Then,

    nk=1|ak|2nk=1|bk|2,nN.

    Using Theorem 3.1d of [20] (see also [29]), we have the following result.

    Lemma 2. Let h be starlike in D, with h(0)=0. If F is analytic in D, with F(0)=0, and satisfies

    F(z)h(z),

    then

    z0F(t)tdtz0h(t)tdt=:q(z). (1.6)

    Moreover, the function q is convex and is the best dominant.

    The main purpose of this paper is to get the sharp bounds for some relations associated with the logarithmic coefficients of the functions belonging to the class S(φ) of Ma-Minda type functions and of other subclasses. Some applications of our results are given here as special cases.

    First, we give a similar result to the inequality (1.5) of Theorem A for the case that φ is starlike with respect to 1 and univalent in D, that is under weaker assumption than those of Theorem A.

    Theorem 1. Let the function fS(φ), with φ(z)=1+n=1Bnzn be a starlike function with respect to 1, univalent in D, and Reφ(z)>0 for all zD. Then, the logarithmic coefficients of f satisfy the following inequalities:

    n=1|γn|214n=1|Bn|2n2, (2.1)

    and

    n=1n2|γn|214n=1|Bn|2. (2.2)

    Both inequalities are sharp since there are attained for the function fS(φ) given by zf(z)/f(z)=φ(z), that is

    f(z)=zexp(z0φ(t)1tdt). (2.3)

    Proof. Supposing that fS(φ), let us define the function H(z):=f(z)z, which is an analytic function in D, H(0)=1. From the relation (1.2) it satisfies

    zH(z)H(z)=zf(z)f(z)1φ(z)1=:ϕ(z), (2.4)

    where ϕ is starlike univalent in D.

    Now, let's take in Lemma 2 the functions

    F(z):=zH(z)H(z),h(z):=ϕ(z). (2.5)

    Then, since ϕ is starlike in D with ϕ(0)=0, and F(0)=0 (because H(0)=1), we should only to prove that F is analytic in D. Since the function fS(φ), with Reφ(z)>0 for all zD, fS(φ)SS, where S represents the class of starlike functions in D. Thus, f(z)0 for zD{0} and z0=0 is a simple zero for f. Hence, H(z)=f(z)z0 for all zD, and therefore F is analytic in D. Since all conditions of Lemma 2 are satisfied, from (1.6) and (2.5) it follows that

    z0H(t)H(t)dtz0ϕ(t)tdt, (2.6)

    and the relation (2.6) results in

    logH(z)logH(0)z0ϕ(t)tdt, (2.7)

    where (2.7) is equal to

    logf(z)zz0ϕ(t)tdt. (2.8)

    In addition, we know that if ϕ is starlike in D, then z0ϕ(t)tdt is convex (univalent) in D, and conversely. Denoting with γn the logarithmic coefficients of f given by (1.3), the subordination (2.8) is equivalent to

    n=12γnznn=1Bnznn. (2.9)

    Since the function z0ϕ(t)tdt is univalent in D, by using Lemma 1 the subordination (2.9) implies

    4kn=1|γn|2kn=1|Bn|2n2n=1|Bn|2n2,kN, (2.10)

    and taking k in (2.10) we conclude that

    4n=1|γn|2n=1|Bn|2n2. (2.11)

    Now, the relation (2.11) shows that the inequality (2.1) is proved.

    To prove the second inequality of our theorem, let fS(φ). Then, using the power series expansion formula (1.3) we get

    n=12nγnzn=zddz(logf(z)z)=zf(z)f(z)1φ(z)1=:ϕ(z). (2.12)

    Now, according to Lemma 1 the subordination (2.12) leads to

    kn=14n2|γn|2kn=1|Bn|2n=1|Bn|2,kN, (2.13)

    and letting k in (2.13), the assertion (2.2) is proved.

    For proving the sharpness of these bounds it is sufficient to use the equality

    n=12nγnzn=zf(z)f(z)1=n=1Bnzn. (2.14)

    The relation (2.14) shows that the upper bound of the inequalities (2.1) and (2.2) is the best possible and it is attained for the function f given by zf(z)f(z)=φ(z).

    The following results represent two special cases of the above theorem connected with the logarithmic coefficients γn for the subclasses S(z+1+z2) and S(1+sinz) defined in [26] and [8], respectively.

    Corollary 1. If the function fS(z+1+z2), then the logarithmic coefficients of f satisfy the inequalities

    n=1|γn|214(1+n=1|(12n)|2(2n)2),

    and

    n=1n2|γn|214(1+n=1|(12n)|2).

    These results are sharp for the function fS(z+1+z2) given by zf(z)/f(z)=z+1+z2.

    Proof. Taking

    φ(z)=z+1+z2=1+z+n=1(12n)z2n=1+z+n=1B2nz2n=1+z+z22z48+,zD,

    and using Theorem 2.1 of [26] it follows that Reφ(z)>0 for all zD, and considering the main branch of the square root function we have φ(0)=1. According to the Figure 1 made with MAPLE™ software we get that the function Φ defined by

    Φ(z):=Rezφ(z)φ(z)1,zD.

    is positive in D, and hence φ(z)=z+1+z2 is a starlike function in D with respect to 1. Since, in addition φ(0)=10, the function φ is univalent in D. Thus our result follows immediately from Theorem 1.

    Figure 1.  The image of Φ(eit), t[0,2π].

    Corollary 2. If the function fS(1+sinz), then the logarithmic coefficients of f satisfy the inequalities

    n=1|γn|214(1+n=11[(2n+1)!(2n+1)]2),

    and

    n=1n2|γn|214(1+n=11[(2n+1)!])2).

    These results are sharp for the function fS(1+sinz) given by zf(z)/f(z)=1+sinz.

    Proof. Considering

    φ(z)=1+sinz=1+z+n=1(1)n(2n+1)!z2n+1=1+z+n=1B2n+1z2n+1,zD,

    and using the Figure 2 (a) made with MAPLE™ software it follows that Reφ(z)>0 for all zD. Also, from the Figure 2 (b) made with the same computer software, we get that the function Φ defined by

    Ψ(z):=Rezφ(z)φ(z)1=zcotz,zD,

    is positive in D, and hence φ(z)=1+sinz is starlike with respect to 1. From here, since φ(0)=10, the function φ is univalent in D. Therefore using Theorem 1 we get our result.

    Figure 2.  Figures for the proof of Corollary 2.

    Remark 2. Corollary 2 is an improvement, without the convexity condition in D for the function φ(z)=1+sin(r0z) if r00.345, of the result given by [1, Corollary 3].

    Corollary 3. If the function fS() where S() was defined in the Remark 1 (i), then the logarithmic coefficients of f satisfy the inequalities

    n=1|γn|214n=11n2[(n1)!]2,

    and

    n=1n2|γn|214n=11[(n1)!]2.

    These results are sharp for the function fS() given by

    zf(z)f(z)=1+zez=1+n=1zn(n1)!.

    Proof. For φ(z):=(z)=1+zez it is easy to check that

    Rez(z)(z)1=Re(1+z)>0,zD,

    and this implies that the function φ is starlike with respect to 1 and univalent in D because φ(0)=10. Also, from Lemma 2.1 (ⅰ) of [15] we obtain that Reφ(z)0.136038 for all z¯D, hence φ has real positive part in D. Then, according to Theorem 1 we obtain the desired result.

    Corollary 4. If the function fSC where SC was defined in the Remark 1 (ii), then the logarithmic coefficients of f satisfy the inequalities

    n=1|γn|21736,

    and

    n=1n2|γn|259.

    These results are sharp for the function f given by

    zf(z)f(z)=1+4z3+2z23.

    Proof. For the function φ(z):=ϕC(z)=1+4z3+2z23 we have

    Rezφ(z)φ(z)1=Re1+z1+0.5z=1+Re0.5z1+0.5z>11=0,zD,

    and φ(0)=4/30. These show that the function φ is starlike with respect to 1 and univalent in D. Also, from the right hand side of the relation (3.5) in [28] we obtain that φ has real positive part in D, and using Theorem 1 we get our result.

    Corollary 5. If the function fScar where Scar was defined in the Remark 1 (iii), then the logarithmic coefficients of f satisfy the inequalities

    n=1|γn|21764,

    and

    n=1n2|γn|2516.

    These results are sharp for the function f given by zf(z)f(z)=1+z+z22.

    Proof. By using simple computations it is easy to check that the function φ(z):=ϕcar(z)=1+z+z22 has positive real part, and it is starlike with respect to 1 in D [12, p. 1148]. Since φ(0)=10, it follows that φ is univalent in D. Hence, the result follows from Theorem 1.

    The following two results give the best upper bounds of the logarithmic coefficients γn for two subclasses Scar and SC.

    Theorem 2. If fScar (see the Remark 1 (iii)), then

    |γn|12n,nN.

    This inequality is sharp for each nN.

    Proof. If fScar, then by the definition of Scar we obtain

    zddz(logf(z)z)=zf(z)f(z)1z+z22. (2.15)

    Using the definition of the logarithmic coefficients γn of the function f given by (1.3) and the relation (2.15) we get

    n=12nγnznz+z22. (2.16)

    Setting in [27, Theorem Ⅵ (ⅰ)] the sequence A1=1, A2=12, An=0 for all n3, and Bk=0 for all kn+1, the function F1 given by [27, (1.10.1) p. 62] becomes

    F1(z)=12+12z.

    The function F1 is analytic in D and satisfy ReF1(z)>0, zD, hence all assumptions of [27, Theorem Ⅵ (ⅰ)] are satisfied. Therefore, from (2.16) we get

    2n|γn|A1=1,nN,

    that represents our result.

    For a fixed n0N, suppose that fn0Scar satisfies the relation

    zfn0(z)fn0(z)=ϕcar(zn0)ϕcar(z). (2.17)

    Then, according to (2.17) the function fn0 is of the form

    fn0(z)=zexp(z0ϕcar(tn0)1tdt)=z+1n0zn0+1+,zD, (2.18)

    hence from (2.18) we get

    logfn0(z)z=2k=n0γk(fn0)zk=1n0zn0+,zD. (2.19)

    The relation (2.19) concludes that the bound given by our theorem is sharp for the fixed value n0N if f=fn0, that is |γn0|=12n0 for f=fn0, and this completes the proof.

    Theorem 3. If fSC (see the Remark 1 (ii)), then

    |γn|23n,nN.

    This inequality is sharp for each nN.

    Proof. If fSC, then by the definition of SC we have

    zddz(logf(z)z)=zf(z)f(z)14z3+2z23, (2.20)

    and using the logarithmic coefficients γn of the function f given by (1.3) and (2.20) we get

    n=12nγnzn4z3+2z23. (2.21)

    Setting now in [27, Theorem Ⅵ (ⅰ)] the sequence A1=43, A2=23, An=0 for all n3, and Bk=0 for all kn+1, the function F1 given by [27, (1.10.1) p. 62] becomes

    F1(z)=23+23z.

    The function F1 is analytic in D and satisfy ReF1(z)>0, zD, hence all assumptions of [27, Theorem Ⅵ (ⅰ)] are satisfied, therefore (2.21) results in

    2n|γn|A1=43,nN.

    For a fixed n0N, suppose that fn0Scar such that

    zfn0(z)fn0(z)=ϕC(zn0)ϕC(z). (2.22)

    According to (2.22) the function fn0 has the form

    fn0(z)=zexp(z0ϕC(tn0)1tdt)=z+43n0zn0+1+,zD, (2.23)

    thus (2.23) implies

    logfn0(z)z=2k=1γk(fn0)zk=43n0zn0+,zD. (2.24)

    Regrading (2.24) the bound given in our result is sharp for the fixed n0N if f=fn0, that is |γn0|=23n0 for f=fn0, which completes the proof.

    In the current paper we obtained the upper bounds for some expressions associated with the logarithmic coefficients γn (nN) of functions that belong to the well-known class S(φ). Furthermore, we took some other particular functions φ in main theorem to obtain the corresponding special cases. Moreover, we gave the best upper bounds of the logarithmic coefficients γn for two subclasses Scar and SC.

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

    Prof. Dr. Nak Eun Cho is the Guest Editor of special issue "Geometric Function Theory and Special Functions" for AIMS Mathematics. Prof. Dr. Nak Eun Cho was not involved in the editorial review and the decision to publish this article.

    The authors declare that they have no conflicts of interest.



    [1] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
    [2] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, 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] Q. Song, A. Kandel, M. Schneider, Parameterized fuzzy operators in fuzzy decision-making, Int. J. Intell. Syst., 18 (2003), 971–987. https://doi.org/10.1002/int.10124 doi: 10.1002/int.10124
    [4] H. Zhao, Z. Xu, M. Ni, S. Liu, Generalized aggregation operators for intuitionistic fuzzy sets, Int. J. Intell. Syst., 25 (2010), 1–30. https://doi.org/10.1002/int.20386 doi: 10.1002/int.20386
    [5] C. Tan, Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision-making, Soft Comput., 15 (2011), 867–876. doi: 10.1007/s00500-010-0554-6
    [6] C. Tan, W. Yi, X. Chen, Generalized intuitionistic fuzzy geometric aggregation operators and their application to multi-criteria decision making, J. Oper. Res. Soc., 66 (2015), 1919–19. https://doi.org/10.1057/jors.2014.104 doi: 10.1057/jors.2014.104
    [7] B. C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybern., 30 (2014), 409. https://doi.org/10.15625/1813-9663/30/4/5032 doi: 10.15625/1813-9663/30/4/5032
    [8] H. Garg, Some picture fuzzy aggregation operators and their applications to multi-criteria decision-making, Arab. J. Sci. Eng., 42 (2017), 5275–5290. https://doi.org/10.1007/s13369-017-2625-9 doi: 10.1007/s13369-017-2625-9
    [9] G. Wei, Picture fuzzy aggregation operators and their application to multiple attribute decision-making, J. Intell. Fuzzy Syst., 33 (2017), 713–724. https://doi.org/10.3233/JIFS-161798 doi: 10.3233/JIFS-161798
    [10] S. Khan, S. Abdullah, S. Ashraf, Picture fuzzy aggregation information based on Einstein operations and their application in decision-making, Math. Sci., 13 (2019), 213–229. https://doi.org/10.1007/s40096-019-0291-7 doi: 10.1007/s40096-019-0291-7
    [11] C. Jana, T. Senapati, M. Pal, R. R. Yager, Picture fuzzy Dombi aggregation operators: Application to MADM process, Appl. Soft Comput., 74 (2019), 99–109. https://doi.org/10.1016/j.asoc.2018.10.021 doi: 10.1016/j.asoc.2018.10.021
    [12] R. R. Yager, Pythagorean fuzzy subsets, In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, Edmonton, Canada, 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
    [13] R. R. Yager, Pythagorean membership grades in multi-criteria decision making, IEEE Trans. Fuzzy Syst., 22 (2014), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
    [14] R. R. Yager, A. M. Abbasov, Pythagorean membership grades, complex numbers, and decision making, Int. J. Intell. Syst., 2 (2014), 436–452. https://doi.org/10.1002/int.21584 doi: 10.1002/int.21584
    [15] P. Liu, P. Wang, Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision-making, Int. J. Intell. Syst., 33 (2018), 259–280. https://doi.org/10.1002/int.21927 doi: 10.1002/int.21927
    [16] P. Liu, J. Liu, some q-rung orthopair fuzzy Bonferroni mean operators and their application to multi-attribute group decision-making, Int. J. Intell. Syst., 33 (2018), 315–347. https://doi.org/10.1002/int.21933 doi: 10.1002/int.21933
    [17] P. Liu, S. M. Chen, P. Wang, Multiple-attribute group decision-making based on q-rung orthopair fuzzy power maclurin symmetric mean operators, IEEE Trans. Syst. Man Cybern. Syst., 2018, 1–16. https://doi.org/10.1109/TSMC.2018.2852948 doi: 10.1109/TSMC.2018.2852948
    [18] C. Jana, G. Muhiuddin, M. Pal, Some Dombi aggregation of q-rung orthopair fuzzy numbers in multiple-attribute decision-making, Int. J. Intell. Syst., 34 (2019), 3220–3240. https://doi.org/10.1002/int.22191 doi: 10.1002/int.22191
    [19] H. Garg, S. M. Chen, Multi-attribute group decision-making based on neutrality aggregation operators of q-rung orthopair fuzzy sets, Inf. Sci., 517 (2020), 427–447. https://doi.org/10.1016/j.ins.2019.11.035 doi: 10.1016/j.ins.2019.11.035
    [20] R. R. Yager, Generalized orthopair fuzzy sets, IEEE T. Fuzzy Syst., 25 (2016), 1222–1230. https://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
    [21] 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
    [22] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9 (2001), 589–602.
    [23] P. Maji, R. Biswas, A. Roy, Intuitionistic fuzzy soft sets, J. Fuzzy Math., 9 (2001), 677–692.
    [24] A. Hussain, M. I. Ali, T. Mahmood, M. Munir, q-Rung orthopair fuzzy soft average aggregation operators and their application in multicriteria decision-making, Int. J. Intell. Syst., 35 (2020), 571–599. https://doi.org/10.1002/int.22217 doi: 10.1002/int.22217
    [25] F. Smarandache, A unifying field in logics neutrosophy: Neutrosophic probability, set and logic, American Research Press, Rehoboth, 1999.
    [26] 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
    [27] R. Arora, H. Garg, A robust aggregation operators for multi-criteria decision-making with intuitionistic fuzzy soft set environment, Sci. Iran., 25 (2018), 913–942. https://doi.org/10.24200/sci.2017.4433 doi: 10.24200/sci.2017.4433
    [28] R. M. Zulqarnain, X. L. Xin, H. Garg, W. A. Khan, Aggregation operators of Pythagorean fuzzy soft sets with their application for green supplier chain management, J. Intell. Fuzzy Syst., 40 (2021), 5545–5563. https://doi.org/10.3233/JIFS-202781 doi: 10.3233/JIFS-202781
    [29] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
    [30] B. P. Joshi, A. Singh, P. K. Bhatt, K. S. Vaisla, Interval valued q-rung orthopair fuzzy sets and their properties, J. Intell. Fuzzy Syst., 35 (2018), 5225–5230. https://doi.org/10.3233/JIFS-169806 doi: 10.3233/JIFS-169806
    [31] K. Hayat, M. S. Raja, E. Lughofer, N. Yaqoob, New group-based generalized interval-valued q-rung orthopair fuzzy soft aggregation operators and their applications in sports decision-making problems, Comput. Appl. Math., 42 (2023), 1–28. https://doi.org/10.1007/s40314-022-02130-8 doi: 10.1007/s40314-022-02135-3
    [32] X. Yang, K. Hayat, M. S. Raja, N. Yaqoob, C. Jana, Aggregation and interaction aggregation soft operators on interval-valued q-rung orthopair fuzzy soft environment and application in automation company evaluation, IEEE Access, 10 (2022), 91424–91444. https://doi.org/10.1109/ACCESS.2022.3202211 doi: 10.1109/ACCESS.2022.3202211
    [33] K. Hayat, R. A. Shamim, H. Al Salman, A. Gumaei, X. P. Yang, M. A. Akbar, Group Generalized q-Rung orthopair fuzzy soft sets: New aggregation operators and their applications, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/5672097 doi: 10.1155/2021/5672097
    [34] I. Deli, N. Çağman, Intuitionistic fuzzy parameterized soft set theory and its decision making, Appl. Soft Comput., 28 (2015), 109–113. https://doi.org/10.1016/j.asoc.2014.11.053 doi: 10.1016/j.asoc.2014.11.053
    [35] I. Deli, A TOPSIS method by using generalized trapezoidal hesitant fuzzy numbers and application to a robot selection problem, J. Intell. Fuzzy Syst., 38 (2020), 779–793. https://doi.org/10.3233/JIFS-179448 doi: 10.3233/JIFS-179448
    [36] I. Deli, S. Broumi, Neutrosophic soft matrices and NSM-decision making, J. Intell. Fuzzy Syst., 28 (2015), 2233–2241. https://doi.org/10.3233/IFS-141505 doi: 10.3233/IFS-141505
    [37] M. Akram, G. Shahzadi, J. C. R. Alcantud, Multi-attribute decision-making with q-rung picture fuzzy information, Granular Comput., 7 (2022), 197–215. https://doi.org/10.1007/s41066-021-00260-8 doi: 10.1007/s41066-021-00260-8
    [38] M. Akram, M. Shabir, A. N. Al-Kenani, J. C. R. Alcantud, Hybrid decision-making frameworks under complex spherical fuzzy N-soft sets, J. Math., 2021 (2021), 1–46. https://doi.org/10.1155/2021/5563215 doi: 10.1155/2021/5563215
    [39] 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. Appl., 33 (2021), 5675–5703. https://doi.org/10.1007/s00521-020-05350-3 doi: 10.1007/s00521-020-05350-3
    [40] M. Akram, F. Wasim, J. C. R. Alcantud, A. N. Al-Kenani, Multi-criteria optimization technique with complex Pythagorean fuzzy n-soft information, Int. J. Comput. Intel. Syst., 14 (2021), 1–24. https://doi.org/10.1007/s44196-021-00008-x doi: 10.1007/s44196-021-00008-x
    [41] M. Akram, M. Amjad, J. C. R. Alcantud, G. Santos-García, Complex Fermatean fuzzy N-soft sets: A new hybrid model with applications, J. Amb. Intel. Hum. Comp., 14 (2022), 1–34. https://doi.org/10.1007/s12652-021-03629-4 doi: 10.1007/s12652-021-03629-4
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