Research article Special Issues

An innovative fuzzy parameterized MADM approach to site selection for dam construction based on sv-complex neutrosophic hypersoft set

  • Received: 22 August 2022 Revised: 30 October 2022 Accepted: 08 November 2022 Published: 09 December 2022
  • MSC : 03B52, 03E72, 90B50

  • Dams are water reservoirs that provide adequate freshwater to residential, industrial, and mining sites. They are widely used to generate electricity, control flooding, and irrigate agricultural lands. Due to recent urbanization trends, industrialization, and climatic changes, the construction of dams is in dire need, which is planning intensive, quite expensive, and time-consuming. Moreover, finding an appropriate site to construct dams is also considered a challenging task for decision-makers. The dam site selection problem (DSSP) has already been considered a multi-criteria decision-making (MCDM) problem under uncertain (fuzzy set) environments by several researchers. However, they ignored some essential evaluating features (e.g., (a) fuzzy parameterized grades, which assess the vague nature of parameters and sub-parameters, (b) the hypersoft setting, which provides multi-argument-based domains for the approximation of alternatives, (c) the complex setting which tackles the periodicity of data, and (d) the single-valued neutrosophic setting which facilitates the decision makers to provide their opinions in three-dimensional aspects) that can be used in DSSP to make it more reliable and trustworthy. Thus this study aims to employ a robust fuzzy parameterized algebraic approach which starts with the characterization of a novel structure "fuzzy parameterized single valued complex neutrosophic hypersoft set (˜λ-set)" that is competent to deal with the above-mentioned features jointly. After that, it integrates the concept of fuzzy parameterization, decision-makers opinions in terms of single-valued complex neutrosophic numbers, and the classical matrix theory to compute the score values for evaluating alternatives. Based on the stages of the proposed approach, an algorithm is proposed, which is further explained by an illustrative example in which DSSP is considered a multiple attributes decision-making (MADM) scenario. The computed score values are then used to evaluate some suitable sites (regions) for dam construction. The computational results of the proposed algorithm are found to be precise and consistent through their comparison with some already developed approaches.

    Citation: Atiqe Ur Rahman, Muhammad Saeed, Mazin Abed Mohammed, Alaa S Al-Waisy, Seifedine Kadry, Jungeun Kim. An innovative fuzzy parameterized MADM approach to site selection for dam construction based on sv-complex neutrosophic hypersoft set[J]. AIMS Mathematics, 2023, 8(2): 4907-4929. doi: 10.3934/math.2023245

    Related Papers:

    [1] Mohammad Faisal Khan . Certain new applications of Faber polynomial expansion for some new subclasses of υ-fold symmetric bi-univalent functions associated with q-calculus. AIMS Mathematics, 2023, 8(5): 10283-10302. doi: 10.3934/math.2023521
    [2] Ekram E. Ali, Miguel Vivas-Cortez, Rabha M. El-Ashwah . New results about fuzzy γ-convex functions connected with the q-analogue multiplier-Noor integral operator. AIMS Mathematics, 2024, 9(3): 5451-5465. doi: 10.3934/math.2024263
    [3] Ebrahim Amini, Mojtaba Fardi, Shrideh Al-Omari, Rania Saadeh . Certain differential subordination results for univalent functions associated with q-Salagean operators. AIMS Mathematics, 2023, 8(7): 15892-15906. doi: 10.3934/math.2023811
    [4] Shuhai Li, Lina Ma, Huo Tang . Meromorphic harmonic univalent functions related with generalized (p, q)-post quantum calculus operators. AIMS Mathematics, 2021, 6(1): 223-234. doi: 10.3934/math.2021015
    [5] Shujaat Ali Shah, Ekram Elsayed Ali, Adriana Cătaș, Abeer M. Albalahi . On fuzzy differential subordination associated with q-difference operator. AIMS Mathematics, 2023, 8(3): 6642-6650. doi: 10.3934/math.2023336
    [6] Luminiţa-Ioana Cotîrlǎ . New classes of analytic and bi-univalent functions. AIMS Mathematics, 2021, 6(10): 10642-10651. doi: 10.3934/math.2021618
    [7] Murugusundaramoorthy Gangadharan, Vijaya Kaliyappan, Hijaz Ahmad, K. H. Mahmoud, E. M. Khalil . Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function. AIMS Mathematics, 2021, 6(12): 13235-13246. doi: 10.3934/math.2021765
    [8] Pinhong Long, Jinlin Liu, Murugusundaramoorthy Gangadharan, Wenshuai Wang . Certain subclass of analytic functions based on q-derivative operator associated with the generalized Pascal snail and its applications. AIMS Mathematics, 2022, 7(7): 13423-13441. doi: 10.3934/math.2022742
    [9] Haiyan Zhou, K. A. Selvakumaran, S. Sivasubramanian, S. D. Purohit, Huo Tang . Subordination problems for a new class of Bazilevič functions associated with k-symmetric points and fractional q-calculus operators. AIMS Mathematics, 2021, 6(8): 8642-8653. doi: 10.3934/math.2021502
    [10] Thongchai Botmart, Soubhagya Kumar Sahoo, Bibhakar Kodamasingh, Muhammad Amer Latif, Fahd Jarad, Artion Kashuri . Certain midpoint-type Fejér and Hermite-Hadamard inclusions involving fractional integrals with an exponential function in kernel. AIMS Mathematics, 2023, 8(3): 5616-5638. doi: 10.3934/math.2023283
  • Dams are water reservoirs that provide adequate freshwater to residential, industrial, and mining sites. They are widely used to generate electricity, control flooding, and irrigate agricultural lands. Due to recent urbanization trends, industrialization, and climatic changes, the construction of dams is in dire need, which is planning intensive, quite expensive, and time-consuming. Moreover, finding an appropriate site to construct dams is also considered a challenging task for decision-makers. The dam site selection problem (DSSP) has already been considered a multi-criteria decision-making (MCDM) problem under uncertain (fuzzy set) environments by several researchers. However, they ignored some essential evaluating features (e.g., (a) fuzzy parameterized grades, which assess the vague nature of parameters and sub-parameters, (b) the hypersoft setting, which provides multi-argument-based domains for the approximation of alternatives, (c) the complex setting which tackles the periodicity of data, and (d) the single-valued neutrosophic setting which facilitates the decision makers to provide their opinions in three-dimensional aspects) that can be used in DSSP to make it more reliable and trustworthy. Thus this study aims to employ a robust fuzzy parameterized algebraic approach which starts with the characterization of a novel structure "fuzzy parameterized single valued complex neutrosophic hypersoft set (˜λ-set)" that is competent to deal with the above-mentioned features jointly. After that, it integrates the concept of fuzzy parameterization, decision-makers opinions in terms of single-valued complex neutrosophic numbers, and the classical matrix theory to compute the score values for evaluating alternatives. Based on the stages of the proposed approach, an algorithm is proposed, which is further explained by an illustrative example in which DSSP is considered a multiple attributes decision-making (MADM) scenario. The computed score values are then used to evaluate some suitable sites (regions) for dam construction. The computational results of the proposed algorithm are found to be precise and consistent through their comparison with some already developed approaches.



    Let A denote the class of functions that are analytic in the open unit disc U={zC:|z|<1} and let A0 be the subclass of A consisting of functions h with the normalization h(0)=h(0)1=0 and has the following series expansion,

    h(z)=z+n=2anzn. (1.1)

    A continuous function f=u+iv is a complex valued harmonic function defined in U, where u and v are real harmonic functions in U. We can write f(z)=h+¯g where h and g are analytic in U (see [2]). We call h the analytic part and g the co-analytic part of f, where hA0 is given by (1.1) and gA has the following power series expansion (see, for details, [4,5,10]):

    g(z)=n=1bnzn, |b1|<1. (1.2)

    A necessary and sufficient condition for f to be locally univalent and sense preserving in U is that |h(z)|>|g(z)| in U.

    We denote by SH the class of functions f=h+¯g, that are harmonic univalent and sense preserving in U and satisfies the normalization conditions. For f=h+¯gSH, where h(z) and g(z) are given in (1.1) and (1.2). Note that SH reduces to S, the class of normalized analytic univalent functions if the co-analytic part of f=h+¯g is identically zero. Also, SH is the subclass of SH consisting of functions f that map U={z:|z|<1} onto starlike domain (see [27]).

    The fractional q-calculus is the q-extension of the ordinary fractional calculus and dates back to early 20th century. The theory of q-calculus operators are used in various areas of science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, and also in the Geometric Function Theory of complex analysis. Initially in 1908, Jackson [7] defined the q-analogue of derivative and integral operator as well as provided some of their applications. Further in [6] Ismail et al. gave the idea of\ q-extension of\ class of q-starlike functions after that Srivastava [28] studied q-calculus in the context of univalent functions theory. Kanas and Raducanu [14] introduced the q-analogue of Ruscheweyh differential operator and Srivastava in [30] studied q-starlike functions related with generalized conic domain. By using the concept of convolution Srivastava et al. [31] introduced q-analogue of Noor integral operator and studied some of its applications, also Srivastava et al. also published a set of articles in which they concentrated class of q-starlike functions from different aspects (see [32,33,35,36]). Additionally, a recently published survey-cum-expository review article by Srivastava [29] is potentially useful for researchers and scholars working on these topics. For some more recent investigation about q-calculus we may refer to [20,22,23,37,38,39].

    The theory of symmetric q-calculus has been applied to many areas of mathematics and physics such as fractional calculus and quantum physics. The symmetric q-calculus has proven to be valuable in a few areas, specially in quantum mechanics [3,24]. Recently in [13] Kanas et al. defined a symmetric operator of q-derivative and introduced a new family of univalent functions. Furthermore Shahid et al. [21] used the concepts of symmetric q-calculus operator theory and defined symmetric conic domains. But research on symmetric q-calculus in connection with Geometric Function Theory and especially harmonic univalent functions is fairly new and not much is published on this topic.

    Jahangiri in [9] applied certain q-calculus operators to complex harmonic functions, while Porwal and Gupta discussed applications of q -calculus to harmonic univalent functions in [25]. Srivastava [34] defined the q-analogue of derivative operator as well as provided some of its applications to complex harmonic functions. In this article, first time we apply symmetric q-calculus theory in order to define symmetric Salagean q-differential operator to complex harmonic functions and introduce a new class of harmonic univalent functions.

    For better understanding of the article, we recall some concept details and definitions of the symmetric q-difference calculus. We suppose throughout in this paper that 0<q<1 and that

    N={1,2,3,...}=N0{0}  (N0={0,1,2,3,...}).

    Definition 1.1. For nN, the symmetric q-number is defined by:

    ~[n]q=qnqnq1q,   ~[0]q=0.

    We note that the symmetric q-number do not reduce to the q-number, and frequently occurs in the study of q-deformed quantum mechanical simple harmonic oscillator (see [1]).

    Definition 1.2. For any nZ+{0}, the symmetric q-number shift factorial is defined by:

    ~[n]q!={~[n]q~[n1]q~[n2]q...~[2]q~[1]q,          n1,1                                              n=0.

    Note That

    limq1~[n]q!=n!.

    Definition 1.3. The symmetric q-derivative (q-difference) operator of symmetric q -calculus operated on the function h([12]) is defined by

    ˜qh(z)=1z(h(qz)h(q1z)qq1), zU,=1+n=1~[n]qanzn1, (z0, q1), (1.3)

    and

    ˜qzn=~[n]qzn1,  ˜q{n=1anzn}=n=1~[n]qanzn1.

    We can observe that

    limq1˜qh(z)=h(z).

    A successive application of the symmetric q-derivative (q-difference) operator of symmetric q-calculus as defined in (1.3) leads to symmetric Salagean q-differential operator which is define as:

    Definition 1.4. The symmetric Salagean q-differential operator of h is defined by

    ˜D0qh(z)=h(z), ˜D1qh(z)=z˜qh(z)=h(qz)h(q1z)qq1,...,˜Dmqh(z)=z˜q˜Dm1qh(z)=h(z)(z+n=2~[n]mqzn)=z+n=2~[n]mqanzn,

    where m is a positive integer and the operator stands for the Hadamard product or convolution of two analytic power series. The operator ˜Dmqh(z) is called symmetric Salagean q-differential operator.

    Note that

    limq1˜Dmqh(z)=z+n=2nmanzn,

    which is the famous Salagean operator defined in [26].

    It is the aim of this article to define the symmetric q-derivative (q -difference) operator by using symmetric q-calculus on the complex functions that are harmonic in U and obtain sharp coefficient bounds, distortion theorems and covering results.

    Definition 1.5. We define the symmetric Salagean q-differential operator for harmonic function f=h+¯g as follows:

    ˜Dmqf(z)=˜Dmqh(z)+(1)m˜Dmq¯g(z), (1.4)

    where

    ˜Dmqh(z)=z+n=2~[n]mqanzn,˜Dmqg(z)=n=1~[n]mqbnzn.

    Remark 1.6. It is easy to see that, for q1, we obtain symmetric Salagean differential operator for harmonic functions f=h+¯g.

    Definition 1.7. For 0α<1, let ˜Hmq(α) denote the family of harmonic functions f=h+¯g which satisfy the condition

    {˜Dm+1qf(z)˜Dmqf(z)}α,  (1.5)

    where~ Dmqf(z) is given by (1.4). Further, denote by ¯˜Hmq(α) the subclass of ˜Hmq(α) consisting of harmonic functions f=h+¯g so that h and g are of the form

    h(z)=zn=2anzn, and  g(z)=n=1bnzn, |b1|<1. (1.6)

    where an,bn0.

    In the following theorem we shall determine coefficient bounds for harmonic functions belonging to the classes ˜Hmq(α) and ¯˜Hmq(α).

    Theorem 2.1. For 0α<1 and f=h+¯g, let

    n=2~[n]mq(~[n]qα)|an|+n=1~[n]mq(~[n]q+α)|bn|1α, (2.1)

    where h and g are respectively given by (1.1) and (1.2). Then

    (i) f is harmonic univalent in U and f˜Hmq(α) if the inequality (2.1) holds.

    (ii) f is harmonic univalent in U and f¯˜Hmq(α) if and only if the inequality (2.1) holds.

    The equality in (2.1) occurs for harmonic function

    f(z)=z+n=21α~[n]mq(~[n]qα)xnzn+¯n=11α~[n]mq(~[n]q+α)ynzn,

    where

    n=2|xn|+n=1|yn|=1.

    Proof. For part (i): First we need to show that f=h+¯g is locally univalent and orientation-preserving in U. It suffices to show that the second complex dilatation w of f satisfies |w|=|g/h|<1 in U. This is the case since for z=reiθU. We have

    |˜qh(z)|1n=2~[n]q|an|rn1>1n=2~[n]q|an|1n=2~[n]mq(~[n]qα)1α|an|n=1~[n]mq(~[n]q+α)1α|bn|n=1~[n]q|bn|n=1~[n]q|bn|rn1|˜qg(z)|

    in U which implies as q1 that |h(z)|>|g(z)| in U that is the function f is locally univalent and sense-preserving in U. To show that f=h+¯g is univalent in U we use an argument that is due to author [8]. Suppose z1 and z2 are in U so that z1z2. Since U is simply connected and convex, we have z(t)=(1t)z1+tz2U for 0t1. Then for z1z20, we can write

    f(z2)f(z1)z2z1>10(˜qh(z(t))|˜qg(z(t))|)dt.

    On the other hand, we observe that

    (˜qh(z))|˜qg(z)|˜qh(z)n=1~[n]q|bn|1n=2~[n]q|an|n=1~[n]q|bn|1n=2~[n]mq(~[n]qα)1α|an|n=1~[n]mq(~[n]q+α)1α|bn|0.

    Therefore, f=h+¯g is univalent in U. It remains to show that the inequality (1.5) holds if the coefficients of the univalent harmonic function f=h+¯g satisfy the condition (2.1). In other words, for 0α<1, we need to show that

    (˜Dm+1qf(z)˜Dmqf(z))=(˜Dm+1qh(z)+(1)m+1˜Dm+1q¯g(z)˜Dmqh(z)+(1)m˜Dmq¯g(z))α.

    Using the fact that (w)α if and only if |1α+w||1+αw|, it suffices to show that

    |˜Dm+1qf(z)+(1α)˜Dmqf(z)||˜Dm+1qf(z)(1+α)˜Dmqf(z)|0. (2.2)

    Substituting for

    ˜Dmqf(z)=z+n=2~[n]mqanzn+(1)m¯n=1~[n]mqbnzn

    and

    ˜Dm+1qf(z)=z+n=2~[n]m+1qanzn+(1)m+1¯n=1~[n]m+1qbnzn.

    In the left hand side of the inequality (2.2), we obtain

    |˜Dm+1qf(z)+(1α)˜Dmqf(z)||˜Dm+1qf(z)(1+α)˜Dmqf(z)|2(1α)|z|{1n=2~[n]mq(~[n]qα)1α|an||z|n1n=1~[n]mq(~[n]q+α)1α|bn||z|n1}2(1α){1n=2~[n]mq(~[n]qα)1α|an|n=1~[n]mq(~[n]q+α)1α|bn|}.

    This last expression is non-negative by (2.1), and this completes the proof.

    For part (ii): Since ¯˜Hmq(α) is subset of ˜Hmq(α), we only need to prove the "only if" part of the theorem. Let f¯˜Hmq(α). Then by the required condition (1.5), we have

    ((1α)zn=2~[n]mq(~[n]qα)anzn(1)2mn=1~[n]mq(~[n]q+α)bnznzn=2~[n]mqanzn+(1)2mn=1~[n]mqbnzn)0.

    This must hold for all values of z in U. So, upon choosing the values of z on the positive real axis where 0z=r<1, we have

    1αn=2~[n]mq(~[n]qα)anrn1n=1~[n]mq(~[n]q+α)bnrn11n=2~[n]mqanrn1+n=1~[n]mqbnrn10. (2.3)

    If the condition (2.1) does not hold, then the numerator in (2.3) is negative for r sufficiently close to 1. Hence there exists z0=r0 in (0,1) for which the left hand side of the inequality (2.3) is negative. This contradicts the required condition that f¯˜Hmq(α) and so the proof is complete.

    Example 2.2. The function f=h+¯g given by

    f(z)=z+n=2Anzn+n=1Bn¯zn,

    where

    An=(2+δ)(1α)ϵn2(n+δ)(n+1+δ)~[n]mq(~[n]qα),Bn=(1+δ)(1α)ϵn2(n+δ)(n+1+δ)~[n]mq(~[n]q+α),

    belonging to the class ˜Hmq(α), for δ>2,0α<1,q(0,1),ϵnC,|ϵn|=1. Because, we know that

    n=2~[n]mq(~[n]qα)|An|+n=1~[n]mq(~[n]q+α)|Bn|n=2(2+δ)(1α)2(n+δ)(n+1+δ)+n=1(1+δ)(1α)2(n+δ)(n+1+δ)=(2+δ)(1α)2n=21(n+δ)(n+1+δ)+(1+δ)(1α)2n=11(n+δ)(n+1+δ)=(2+δ)(1α)2n=2(1(n+δ)1(n+1+δ))+(1+δ)(1α)2n=1(1(n+δ)1(n+1+δ))=1α.

    For q1, then Theorem 2.1 reduces to following known results ([11,Theorems 1 and 2]).

    Corollary 2.3. For 0α<1 and f=h+¯g, let

    n=2nm(nα)|an|+n=1nm(n+α)|bn|1α, (2.4)

    where h and g are, respectively, given by (1.1) and (1.2). Then

    (i) f is harmonic univalent in U and fHm(α) if the inequality (2.4) holds.

    (ii) f is harmonic univalent in U and f¯Hm(α) if and only if the inequality (2.4) holds.

    The equality in (2.4) occurs for harmonic functions

    f(z)=z+n=21αnm(nα)xnzn+¯n=11αnm(n+α)ynzn,

    where

    n=2|xn|+n=1|yn|=1.

    The closed convex hull of ¯˜Hmq(α) denoted by clco¯˜Hmq(α), is the smallest closed set containing ¯˜Hmq(α), it is the intersection of all closed convex sets containg ¯˜Hmq(α), In the next theorem we determine the extreme points of the closed convex hull of ¯˜Hmq(α).

    Theorem 2.4. If the functions f=h+¯gclco¯˜Hmq(α) if and only if

    f(z)=n=1(Xnhn(z)+¯Yngn(z)),h1(z)=z,
    hn(z)=z1α~[n]mq(~[n]qα)zn,(n=2,3,...),gn(z)=z+(1)m1α~[n]mq(~[n]q+α)zn,(n=1,2,3,...),
    n=1(Xn+Yn)=1,Xn0,Yn0. (2.5)

    In particular, the extreme points of ¯˜Hmq(α) are hn and gn.

    Proof. For the functions of the form (2.5), we have

    f(z)=n=1(Xn+Yn)zn=21α~[n]mq(~[n]qα)Xnzn+(1)mn=11α~[n]mq(~[n]q+α)¯Ynzn.

    This yields

    n=2~[n]mq(~[n]qα)1αan+n=1~[n]mq(~[n]q+α)1αbn=n=2Xn+n=1Yn=1X11.

    and so f=h+¯gclco˜Hmq(α). Conversely, let f=h+¯gclco˜Hmq(α). Then by setting

    Xn=~[n]mq(~[n]qα)1αan, (n=2,3,...),Yn=~[n]mq(~[n]q+α)1αbn, (n=1,2,3,...),

    where

    n=1(Xn+Yn)=1.

    We obtain the functions of the form (2.5) as required.

    Remark 2.5. For q1, then Theorem 2.4 reduces to the known results proved in [11].

    Finally, we give the following distortion bounds which yields a covering result for ¯˜Hmq(α).

    Theorem 2.6. If f=h+¯g¯˜Hmq(α), then for |z|=r<1, we have the distortion bounds

    |f(z)|(1|b1|)r1~[2]mq(1α~[2]qα1+α~[2]qα|b1|)r2, (2.6)
    |f(z)|(1+|b1|)r+1~[2]mq(1α~[2]qα1+α~[2]qα|b1|)r2. (2.7)

    The bounds given in (2.6) and (2.7) for the function f(z)=h+¯g of the form (1.6) also hold for functions of the form (1.1), (1.2), if the coefficient condition (2.1) is satisfied. Let the

    f(z)=(1+|b1|)¯z+1~[2]mq(1α~[2]qα1+α~[2]qα|b1|)¯z2

    and

    f(z)=(1|b1|)z1~[2]mq(1α~[2]qα1+α~[2]qα|b1|)z2,

    for

    |b1|1α1+α,

    show that the bounds given in Theorem 2.6 are sharp.

    Proof. We will only prove the inequality (2.7) of the Theorem 2.6. The arguments for the inequality (2.6) is similar and so we omit it. Let f¯˜Hmq(α). Taking the absolute value of f(z), we obtain

    |f(z)|(1+|b1|)r+n=2(|an|+|bn|)rn(1+|b1|)r+n=2(|an|+|bn|)r2(1+|b1|)r+1α~[2]mq(~[2]qα)×n=2(~[2]mq(~[2]qα)1α|an|+~[2]mq(~[2]q+α)1α|bn|)r2(1+|b1|)r+1α~[2]mq(~[2]qα)×n=2(~[2]mq(~[n]qα)1α|an|+~[2]mq(~[n]q+α)1α|bn|)r2(1+|b1|)r+1α~[2]mq(~[2]qα)n=2(11+α1α|b1|)r2(1+|b1|)r+1~[2]mqn=2(1α~[2]qα1+α~[2]qα|b1|)r2.

    The following covering result follows the inequality (2.7) in Theorem (2.6).

    Theorem 2.7. If f¯˜Hmq(α), then for |z|=r<1, we have

    {w:|w|<~[2]mq(~[2]qα)(1α)~[2]mq(~[2]qα){~[2]mq(~[2]qα)(1+α)}|b1|~[2]mq(~[2]qα)}f(U).

    Proof. Using the inequality (2.6) of Theorem 2.6 and letting r1, it follows that

    (1|b1|)1~[2]mq(1α~[2]qα1+α~[2]qα|b1|)=(1|b1|)1~[2]mq(~[2]qα){1α(1+α)|b1|}=(1|b1|)~[2]mq(~[2]qα)(1α)+(1+α)|b1|~[2]mq(~[2]qα)=~[2]mq(~[2]qα)(1α)~[2]mq(~[2]qα){~[2]mq(~[2]qα)(1+α)}|b1|~[2]mq(~[2]qα)f(U).

    Research on symmetric q-calculus in connection with Geometric Function Theory and especially harmonic univalent functions is fairly new and not much is published on this topic. In this paper we have made use of the symmetric quantum (or q-) calculus to defined and investigated new classes of harmonic univalent functions by using newly defined symmetric Salagean q-differential operator for complex harmonic functions and obtained sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlighted some known consequence of our main results.

    Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [28,p.351–352] and [29,p.328]). Moreover, in this recently-published survey-cum-expository review article by Srivastava [29], the so-called (p,q)-calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant (see, for details, [29,p.340], see also [15,16,17,18,19]). This observation by Srivastava [29] will indeed apply also to any attempt to produce the rather straightforward (p,q)-variations of the results which we have presented in this paper.

    The authors declare no conflict of interest.



    [1] J. W. Wang, C. H. Cheng, K. C. Huang, Fuzzy hierarchical TOPSIS for supplier selection, Appl. Soft Comput., 9 (2009), 377–386. https://doi.org/10.1016/j.asoc.2008.04.014 doi: 10.1016/j.asoc.2008.04.014
    [2] S. Q. Weng, G. H. Huang, Y. P. Li, An integrated scenario-based multi-criteria decision support system for water resources management and planning: a case study in the Haihe River Basin, Expert Syst. Appl., 37 (2010), 8242–8254. https://doi.org/10.1016/j.eswa.2010.05.061 doi: 10.1016/j.eswa.2010.05.061
    [3] A. Noori, H. Bonakdari, K. Morovati, B. Gharabaghi, The optimal dam site selection using a group decision-making method through fuzzy TOPSIS model, Environ. Syst. Decisions, 38 (2018), 471–488. https://doi.org/10.1007/s10669-018-9673-x doi: 10.1007/s10669-018-9673-x
    [4] Z. L. Yang, M. W. Lin, Y. C. Li, W. Zhou, B. Xu, Assessment and selection of smart agriculture solutions using an information error-based Pythagorean fuzzy cloud algorithm, Int. J. Intell. Syst., 36 (2021), 6387–6418. https://doi.org/10.1002/int.22554 doi: 10.1002/int.22554
    [5] M. W. Lin, Z. Y. Chen, R. Q. Chen, H. Fujita, Evaluation of startup companies using multicriteria decision making based on hesitant fuzzy linguistic information envelopment analysis models, Int. J. Intell. Syst., 36 (2021), 2292–2322. https://doi.org/10.1002/int.22379 doi: 10.1002/int.22379
    [6] J. Deng, J. M. Zhan, Z. Sh. Xu, E. Herrera-Viedma, Regret-theoretic multiattribute decision-making model using three-way framework in multiscale information systems, IEEE Trans. Cybernetics, 2022, 1–14. https://doi.org/10.1109/TCYB.2022.3173374 doi: 10.1109/TCYB.2022.3173374
    [7] H. B. Wang, F. Smarandache, Y. Q. Zhang, R. Sunderraman, Single valued neutrosophic sets, Tech. Sci. Appl. Math., 2010.
    [8] 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
    [9] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
    [10] B. C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybernetics, 30 (2014), 409–420. http://doi.org/10.15625/1813-9663/30/4/5032 doi: 10.15625/1813-9663/30/4/5032
    [11] F. Smarandache, Neutrosophy neutrosophic probability, set and logic: analytic synthesis synthetic analysis, American Research Press, 1998.
    [12] R. Şahin, A. Küçük, Subsethood measure for single valued neutrosophic sets, J. Intell. Fuzzy Syst., 29 (2015), 525–530. http://doi.org/10.3233/IFS-141304 doi: 10.3233/IFS-141304
    [13] H. L. Huang, New distance measure of single-valued neutrosophic sets and its application, Int. J. Intell. Syst., 31 (2016), 1021–1032. https://doi.org/10.1002/int.21815 doi: 10.1002/int.21815
    [14] S. Pramanik, S. Dalapati, S. Alam, F. Smarandache, T. K. Roy, NS-cross entropy-based MAGDM under single-valued neutrosophic set environment. Information, 9 (2018), 37. https://doi.org/10.3390/info9020037 doi: 10.3390/info9020037
    [15] A. Aydoğdu, On similarity and entropy of single valued neutrosophic sets, Gen. math. notes, 29 (2015), 67–74.
    [16] P. Biswas, S. Pramanik, B. C. Giri, TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment, Neural comput. Appl., 27 (2016), 727–737. https://doi.org/10.1007/s00521-015-1891-2 doi: 10.1007/s00521-015-1891-2
    [17] H. M. A. Farid, M. Riaz, Single-valued neutrosophic Einstein interactive aggregation operators with applications for material selection in engineering design: case study of cryogenic storage tank, Complex Intell. Syst., 8 (2022), 2131–2149.
    [18] J. Ling, M. W. Lin, L. L. Zhang, Medical waste treatment scheme selection based on single-valued neutrosophic numbers, AIMS Mathematics, 6 (2021), 10540–10564. https://doi.org/10.3934/math.2021612 doi: 10.3934/math.2021612
    [19] M. Ali, F. Smarandache, Complex neutrosophic set, Neural comput. Appl., 28 (2017), 1817–1834.
    [20] D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst., 10 (2022), 171–186. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
    [21] A. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, In International conference on fundamental and applied sciences, 1482 (2012), 464–470. https://doi.org/10.1063/1.4757515
    [22] 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
    [23] J. M. Zhan, J. C. R. Alcantud, A novel type of soft rough covering and its application to multicriteria group decision making, Artif. Intell. Rev., 52 (2019), 2381–2410. https://doi.org/10.1007/s10462-018-9617-3 doi: 10.1007/s10462-018-9617-3
    [24] C. Jana, M. Pal, A robust single-valued neutrosophic soft aggregation operators in multi-criteria decision making, Symmetry, 11 (2019), 110. https://doi.org/10.3390/sym11010110 doi: 10.3390/sym11010110
    [25] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9 (2001), 589–602.
    [26] N. Çaǧman, S. Karataş, Intuitionistic fuzzy soft set theory and its decision making, J. Intell. Fuzzy Syst., 24 (2013), 829–836. https://doi.org/10.3233/IFS-2012-0601 doi: 10.3233/IFS-2012-0601
    [27] P. K. Maji, Neutrosophic soft set, Ann. Fuzzy Math. Inform., 5 (2013), 157–168.
    [28] P. Thirunavukarasu, R. Suresh, V. Ashokkumar, Theory of complex fuzzy soft set and its applications, Int. J. Innov. Res. Sci. Technol., 3 (2017), 13–18.
    [29] T. Kumar, R. K. Bajaj, On complex intuitionistic fuzzy soft sets with distance measures and entropies, J. Math., 2014 (2014), 1–12. http://doi.org/10.1155/2014/972198 doi: 10.1155/2014/972198
    [30] S. Broumi, A. Bakali, F. Smarandache, M. Talea, M. Ali, G. Selvachandran, Complex neutrosophic soft set, In: 2017 FUZZ-IEEE Conference on Fuzzy Systems, 2017.
    [31] K. Y. Zhu, J. M. Zhan, Fuzzy parameterized fuzzy soft sets and decision making, Int. J. Machine Learn. Cyb., 7 (2016), 1207–1212.
    [32] E. Sulukan, N. Çağman, T. Aydin, Fuzzy parameterized intuitionistic fuzzy soft sets and their application to a performance-based value assignment problem, J. New Theory, 29 (2019), 79–88.
    [33] F. Smarandache, Extension of soft set of hypersoft set, and then to plithogenic hypersoft set, Neutrosophic Sets Syst., 22 (2018), 168–170.
    [34] A. U. Rahman, M. Saeed, F. Smarandache, M. R. Ahmad, Development of hybrids of hypersoft set with complex fuzzy set, complex intuitionistic fuzzy set and complex neutrosophic set, Neutrosophic Sets Syst., 38 (2020), 335–355.
    [35] A. U. Rahman, M. Saeed, S. S. Alodhaibi, H. A. E. W. Khalifa, Decision making algorithmic approaches based on parameterization of neutrosophic set under hypersoft set environment with fuzzy, intuitionistic fuzzy and neutrosophic settings, CMES Comput. Model. Eng. Sci., 128 (2021), 743–777. https://doi.org/10.32604/cmes.2021.016736 doi: 10.32604/cmes.2021.016736
    [36] A. U. Rahman, M. Saeed, A. Alburaikan, H. A. E. W. Khalifa, An intelligent multiattribute decision-support framework based on parameterization of neutrosophic hypersoft set, Comput. Intel. Neurosc., 2022, 6229947. https://doi.org/10.1155/2022/6229947 doi: 10.1155/2022/6229947
    [37] P. Rezaei, K. Rezaie, S. Nazari-Shirkouhi, M. R. J. Tajabadi, Application of fuzzy multi-criteria decision making analysis for evaluating and selecting the best location for construction of underground dam, Acta Polytech. Hung., 10 (2013), 187–205.
    [38] J. Chezgi, H. R. Pourghasemi, S. A. Naghibi, H. R. Moradi, M. K. Zarkesh, Assessment of a spatial multi-criteria evaluation to site selection underground dams in the Alborz Province, Iran, Geocarto Int., 31 (2016), 628–646. https://doi.org/10.1080/10106049.2015.1073366 doi: 10.1080/10106049.2015.1073366
    [39] A. Dortaj, S. Maghsoudy, F. D. Ardejani, Z. Eskandari, A hybrid multi-criteria decision making method for site selection of subsurface dams in semi-arid region of Iran, Groundwater Sustain. Dev., 10 (2020), 100284. https://doi.org/10.1016/j.gsd.2019.100284 doi: 10.1016/j.gsd.2019.100284
    [40] C. B. Karakuş, S. Yıldız, Gis-multi criteria decision analysis-based land suitability assessment for dam site selection, Int. J. Environ. Sci. Technol., 19 (2022), 12561–12580.
    [41] F. S. Chien, C. N. Wang, V. T Nguyen, V. T. Nguyen, K. Y. Chau, An evaluation model of quantitative and qualitative fuzzy multi-criteria decision-making approach for hydroelectric plant location selection, Energies, 13 (2020), 2783. https://doi.org/10.3390/en13112783 doi: 10.3390/en13112783
    [42] V. Esavi, J. Karami, A. Alimohammadi, S. A. Niknezhad, Comparison the AHP and fuzzy-AHP decision making methods in underground dam site selection in Taleghan basin, Sci. Quart. J. Geosci., 22 (2012), 27–34.
    [43] S. Narayanamoorthy, V. Annapoorani, S. Kalaiselvan, D. Kang, Hybrid hesitant fuzzy multi-criteria decision making method: A symmetric analysis of the selection of the best water distribution system, Symmetry, 12 (2020), 2096. https://doi.org/10.3390/sym12122096 doi: 10.3390/sym12122096
    [44] S. Janjua, I. Hassan, Fuzzy AHP-TOPSIS multi-criteria decision analysis applied to the Indus Reservoir system in Pakistan, Water Supply, 20 (2020), 1933–1949. https://doi.org/10.2166/ws.2020.103 doi: 10.2166/ws.2020.103
    [45] O. I. Adeyanju, A. A. Adedeji, Application of hybrid fuzzy-topsis for decision making in dam site selection, Int. J. Civ. Eng., 4 (2017), 40–52. https://doi.org/10.14445/23488352/IJCE-V4I7P106 doi: 10.14445/23488352/IJCE-V4I7P106
    [46] M. Deveci, D. Pamucar, E. Oguz, Floating photovoltaic site selection using fuzzy rough numbers based LAAW and RAFSI model, Appl. Energy, 324 (2022), 119597. https://doi.org/10.1016/j.apenergy.2022.119597 doi: 10.1016/j.apenergy.2022.119597
    [47] S. H. R. Ahmadi, Y. Noorollahi, S. Ghanbari, M. Ebrahimi, H. Hosseini, A. Foroozani, et al., Hybrid fuzzy decision making approach for wind-powered pumped storage power plant site selection: a case study, Sustain. Energy Techn., 42 (2020), 100838. https://doi.org/10.1016/j.seta.2020.100838 doi: 10.1016/j.seta.2020.100838
    [48] M. Heidarimozaffar, M. Shahavand, Spatial zoning of Kabodarahang plain using fuzzy logic in geospatial information system for the construction of an underground dam, Geogr. Data., 30 (2021), 95–110.
    [49] S. S. Haghshenas, M. A. L. Neshaei, P. Pourkazem, S. S. Haghshenas, The risk assessment of dam construction projects using fuzzy TOPSIS (case study: Alavian Earth Dam), Civ. Eng. J., 2 (2016), 158–167. http://doi.org/10.28991/cej-2016-00000022 doi: 10.28991/cej-2016-00000022
    [50] A. Jozaghi, B. Alizadeh, M. Hatami, I. Flood, M. Khorrami, N. Khodaei, et al., A comparative study of the AHP and TOPSIS techniques for dam site selection using GIS: a case study of Sistan and Baluchestan Province, Iran, Geosciences, 8 (2018), 494. http://doi.org/10.20944/preprints201810.0773.v1 doi: 10.20944/preprints201810.0773.v1
    [51] F. Tufail, M. Shabir, E. S. A. Abo-Tabl, A comparison of Promethee and TOPSIS techniques based on bipolar soft covering-based rough sets, IEEE Access, 10 (2022), 37586–37602. https://doi.org/10.1109/ACCESS.2022.3161470 doi: 10.1109/ACCESS.2022.3161470
    [52] M. Akram, G. Ali, M. A. Butt, J. C. R. Alcantud, Novel MCGDM analysis under m-polar fuzzy soft expert sets, Neural Comput. Appl., 33 (2021), 12051–12071.
  • This article has been cited by:

    1. Mohammad Faisal Khan, Teodor Bulboaca, Certain New Class of Harmonic Functions Involving Quantum Calculus, 2022, 2022, 2314-8888, 1, 10.1155/2022/6996639
    2. S. Santhiya, K. Thilagavathi, Ming Sheng Li, Geometric Properties of Analytic Functions Defined by the (p, q) Derivative Operator Involving the Poisson Distribution, 2023, 2023, 2314-4785, 1, 10.1155/2023/2097976
    3. Mohammad Faisal Khan, Isra Al-Shbeil, Najla Aloraini, Nazar Khan, Shahid Khan, Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions, 2022, 14, 2073-8994, 2188, 10.3390/sym14102188
    4. Mohammad Faisal Khan, Isra Al-shbeil, Shahid Khan, Nazar Khan, Wasim Ul Haq, Jianhua Gong, Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag–Leffler Functions, 2022, 14, 2073-8994, 1905, 10.3390/sym14091905
    5. Shahid Khan, Nazar Khan, Aftab Hussain, Serkan Araci, Bilal Khan, Hamed H. Al-Sulami, Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions, 2022, 14, 2073-8994, 803, 10.3390/sym14040803
    6. Afis Saliu, Isra Al-Shbeil, Jianhua Gong, Sarfraz Nawaz Malik, Najla Aloraini, Properties of q-Symmetric Starlike Functions of Janowski Type, 2022, 14, 2073-8994, 1907, 10.3390/sym14091907
    7. Chetan Swarup, Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus, 2023, 15, 2073-8994, 1407, 10.3390/sym15071407
    8. Munirah Rossdy, Rashidah Omar, Shaharuddin Cik Soh, 2024, 3109, 0094-243X, 050005, 10.1063/5.0204775
    9. Mustafa I. Hameed, Shaheed Jameel Al-Dulaimi, Hussaini Joshua, Ismael I. Hameed, Israa A. Ibrahim, Kayode Oshinubi, Teodor Bulboaca, Applications of Subordination for Holomorphic Functions Stated by Generalized By‐Product Operator, 2024, 2024, 0161-1712, 10.1155/2024/8279226
    10. Suha B. Al-Shaikh, New Applications of the Sălăgean Quantum Differential Operator for New Subclasses of q-Starlike and q-Convex Functions Associated with the Cardioid Domain, 2023, 15, 2073-8994, 1185, 10.3390/sym15061185
    11. Isra Al-Shbeil, Shahid Khan, Fairouz Tchier, Ferdous M. O. Tawfiq, Amani Shatarah, Adriana Cătaş, Sharp Estimates Involving a Generalized Symmetric Sălăgean q-Differential Operator for Harmonic Functions via Quantum Calculus, 2023, 15, 2073-8994, 2156, 10.3390/sym15122156
    12. Suha B. Al-Shaikh, Ahmad A. Abubaker, Khaled Matarneh, Mohammad Faisal Khan, Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions, 2023, 7, 2504-3110, 411, 10.3390/fractalfract7050411
    13. Munirah Rossdy, Rashidah Omar, Shaharuddin Cik Soh, 2024, 3023, 0094-243X, 070001, 10.1063/5.0171850
    14. Khadeejah Rasheed Alhindi, Application of the q-derivative operator to a specialized class of harmonic functions exhibiting positive real part, 2025, 10, 2473-6988, 1935, 10.3934/math.2025090
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1948) PDF downloads(129) Cited by(7)

Figures and Tables

Figures(5)  /  Tables(11)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog