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Research article

A novel approach to hesitant multi-fuzzy soft set based decision-making

  • Received: 03 November 2019 Accepted: 13 February 2020 Published: 21 February 2020
  • MSC : 06D72, 90B50, 94D05

  • In this article, we present the idea of a hesitant multi-fuzzy set. We join the characteristics of a hesitant multi-fuzzy set with the parametrization of the soft set and constructs the hesitant multi-fuzzy soft set. We center around the fundamental operations for the instance of the hesitant multi-fuzzy soft subsets. Also, we look at the root mean square sum level soft set or RMSS-level soft set to deal with uncertainties. We also provide the utilization of hesitant multi-fuzzy soft set into the decision-making issues. Finally, we deliver a standard algorithm to resolve decision-making issues and test the effectiveness of it by a socialistic decision-making problem.

    Citation: Asit Dey, Tapan Senapati, Madhumangal Pal, Guiyun Chen. A novel approach to hesitant multi-fuzzy soft set based decision-making[J]. AIMS Mathematics, 2020, 5(3): 1985-2008. doi: 10.3934/math.2020132

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  • In this article, we present the idea of a hesitant multi-fuzzy set. We join the characteristics of a hesitant multi-fuzzy set with the parametrization of the soft set and constructs the hesitant multi-fuzzy soft set. We center around the fundamental operations for the instance of the hesitant multi-fuzzy soft subsets. Also, we look at the root mean square sum level soft set or RMSS-level soft set to deal with uncertainties. We also provide the utilization of hesitant multi-fuzzy soft set into the decision-making issues. Finally, we deliver a standard algorithm to resolve decision-making issues and test the effectiveness of it by a socialistic decision-making problem.


    Notation: For two sequences of positive constants {an,n1} and {bn,n1}, symbols anbn, an=O(bn) and an=o(bn) stand for liman/bn=1, liman/bn(0,) and liman/bn=0, respectively. For simplicity, we shall write P, a.s. and Lp to express the convergence in probability, the almost certain convergence and p-mean convergence, respectively.

    The following concept of superadditive function was introduced in [1].

    Definition 1.1. A function ϕ:RnR is called superadditive if ϕ(xy)+ϕ(xy)ϕ(x)+ϕ(y) for all x,yRn, where is for componentwise maximum and is for componentwise minimum.

    Hu [2] introduced the concept of negatively superadditive-dependent (NSD) based on the above concept of superadditive function.

    Definition 1.2. A random vector X=(X1,X2,,Xn) is said to be NSD if

    Eϕ(X1,X2,,Xn)Eϕ(X1,X2,,Xn),

    where X1,X2,,Xn are independent such that Xi and Xi have the same distribution for each i and ϕ is a superadditive function such that the expectations in the above equation exists. A sequence {Xn,n1} of random variables is said to be NSD if for each n1, (X1,X2,,Xn) is NSD.

    Hu [2] established some basic properties and three structural theorems of NSD random variables. An interesting example was also presented in [2], which illustrated that NSD is not necessarily negatively associated (NA, [3]). Christofides and Vaggelatou [4] showed that NA is NSD. Eghbal et al. [5] derived two maximal inequalities and strong law of large numbers of quadratic forms of NSD random variables. Shen et al. [6] studied almost sure convergence and strong stability for weighted sums of NSD random variables. Wang et al. [7] studied complete convergence for arrays of rowwise NSD random variables, with applications to nonparametric regression. For more research of the limit theory for NSD random variables, the author can refer the reader to [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

    NA random variable has been studied many times and attracted extensive attention, so it is very significant to investigate the limit theorems of this wider NSD class, which is highly desirable and of considerable significance in theory and application.

    A random variable X is called to be a two-tailed Pareto distribution whose density is

    f(x)={qx2ifx1,0if1<x<1,px2ifx1, (1.1)

    where p+q=1.

    Let {Xn,n1} be independent Pareto-Zipf random variables satisfying P(Xn=0)=11/n,

    P(Xnx)=11x+nforallx>0, (1.2)

    and fXn(x)=1(x+n)2I(x>0).

    Obviously, if the random variable Xn satisfies Eq (1.1) or (1.2), then E|Xn|=, n1. Alder [25] considered independent and identically distributed (i.i.d.) random variables satisfying Eq (1.1) and studied the strong law of large numbers. Alder [26] obtained the weak law of large numbers for Pareto-Zipf random variables. For more research on laws of large numbers for i.i.d. random variables with infinite mean, the author can refer to works of Adler [27,28] and Matsumoto and Nakata [29,30,31].

    Yang et al. [24] investigated the law of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means, and obtained the following theorems which extend and improve the corresponding ones in [25,26]:

    Theorem 1.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability

    P(Xn>x)=1x+cnforallx>0andn1, (1.3)

    where {cn} is a nondecreasing constant sequence with cn1 and

    Cn=nj=11cj. (1.4)

    Then we have

    nj=1c1jXjCnlogCnP1. (1.5)

    Theorem 1.2. Let {Xn,n1} be a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1). Then for all β>0 we have

    1logβnnj=1logβ2jjXja.s.pqβ. (1.6)

    In the current work, the author studies the weak and strong laws of large numbers for NSD random variables. The obtained results in this article extend and improve Theorems 1.1 and 1.2. Meanwhile, the author investigates p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24].

    Throughout this paper, the symbol C denotes a positive constant which may differ from one place to another. The symbol I(A) denotes the indicator function of the event A.

    To prove our main results, we first present some technical lemmas.

    Lemma 2.1. ([2]) If (X1,X2,,Xn) is NSD and f1,f2,,fn are all non decreasing, then (f1(X1),f2(X2),,fn(Xn)) is also NSD.

    As we know, moment inequalities are very important tools in establishing the limit theorems for sequences of random variables. Shen et al. [6] presented the following Marcinkiewicz-Zygmund inequality with exponent 2.

    Lemma 2.2. ([6]) Let {Xn,n1} be a sequence of NSD random variables with EXn=0 and EX2n< for n1. Then

    E(max1kn(ki=1Xi)2)2ni=1EX2i,n1.

    By means of similar methods in Shao [32], Wang et al. [7] established the following Rosenthal-type maximal inequality, which is very useful in establishing the convergence properties for NSD random variables:

    Lemma 2.3. ([7]) Let p>1. Let {Xn,n1} be a sequence of NSD random variables with E|Xi|p< for each i1. Then for all n1,

    E(max1kn|ki=1Xi|p)23pni=1E|Xi|pfor1<p2

    and

    E(max1kn|ki=1Xi|p)2(15plnp)p[ni=1E|Xi|p+(EX2i)p/2]forp>2.

    Lemma 2.4. ([6]) Let {Xn,n1} be a sequence of NSD random variables. If

    n=1Var(Xn)<,

    then n=1(XnEXn) almost certainly converges.

    Now we state our main results and the proofs will be presented in next section.

    Theorem 2.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability

    P(Xn>x)=1x+cnforallx>0andn1, (2.1)

    where {cn,n1} is a nondecreasing constant sequence with cn1 and

    Cn=nj=11cj. (2.2)

    Let {Dn,n1} be a sequence of constants satisfying Dn and Cn=o(Dn). Then we have

    1Dnmax1kn|kj=1c1j(XjEXnj)|P0, (2.3)

    where Xnj=XjI(XjDncj)+DncjI(Xj>Dncj), 1jn.

    Take Dn=CnlogCn, then we can obtain the following corollary which extends Theorem 1.1.

    Corollary 2.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability Eq (2.1), where {cn,n1} is a nondecreasing constant sequence satisfying cn1 and Eq (2.2). Then

    1CnlogCnmax1kn|kj=1c1j(XjEXnj)|P0. (2.4)

    Remark 2.1. Yang et al. [24] proved that

    1CnlogCnnj=1c1jEXjI(XjcjCnlogCn)1

    and

    1CnlogCnnj=1c1jE(cjCnlogCnI(Xj>cjCnlogCn))=nj=1P(Xj>cjCnlogCn)0,

    which yields

    1CnlogCnnj=1c1jEXnj1.

    Then we can find that Theorem 1.1 is a special case of Corollary 2.1 for k=n. Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem 1.1.

    Theorem 2.2. Let {Xn,n1} be a sequence of identically distributed NSD random variables. Let {dn,n1} be a sequence of positive constants satisfying dn, and {cn,n1} be a sequence of positive constants such that φ(n)cndn satisfies φ(n) as n,

    m=n1φ2(m)=O(nφ2(n)) (2.5)

    and

    n=1P(|X1|>φ(n))<. (2.6)

    Then

    1dnnj=1c1j(XjE~Xj)0a.s., (2.7)

    where ~Xj=φ(j)I(Xj<φ(j))+XjI(|Xj|φ(j))+φ(j)I(Xj>φ(j)), 1jn.

    Remark 2.2. We will show that Theorem 1.2 is a special case of Theorem 2.2. In fact, if we assume that {Xn,n1} is a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1), and take cn=nlog2βn and dn=logβn (β>0), then φ(n)=cndn=nlog2n. We can verify that φ(n)=nlog2n satisfies the conditions stated in Theorem 2.2.

    First, it is clear that φ(n)=nlog2n satisfies φ(n) as n.

    Second, we have by standard calculations that

    m=n1φ2(m)n1x2log4xdx=O(n1log4(n))=O(nφ2(n)),

    which shows that Eq (2.5) is verified.

    Next, we have by Eq (1.1) and φ(n)=nlog2n that

    n=1P(|X1|>φ(n))=n=1P(|X1|>nlog2n)=n=1(nlog2nqx2dx+nlog2npx2dx)=n=1p+qnlog2n=n=11nlog2n<

    and then Eq (2.6) is verified.

    Finally, we also obtain by Eq (1.1) and φ(j)=jlog2j that

    1dnnj=1c1jE~Xj=1dnnj=1(djP(Xj<φ(j))+c1jEXjI(|Xj|<φ(j))+djP(Xj>φ(j)))=pqlogβnnj=1logβ2jj+pqlogβnnj=1logβ1jj=:J1+J2.

    By similar argument as in the proof of H0 in [24], we can obtain J10. By similar argument as in the proof of Eq (3.5) in [24], we can prove J2pqβ. Then we obtain by Eq (2.7) that

    1logβnnj=1logβ2jjXja.s.pqβ.

    To sum up, Theorem 1.2 is a special case of Theorem 2.2 and then Theorem 2.2 extends Theorem 1.2.

    Next, we present a new theorem of p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24,25,26].

    Theorem 2.3. Let {Xn,n1} be a sequence of NSD random variables satisfying

    limxsupj1xαP(|Xj|>x)<,α(1,2). (2.8)

    Let {dn,n1} be a sequence of positive constants satisfying dn, and {cn,n1} be a sequence of positive constants such that cj1 and

    nj=1cαj=o(dαn). (2.9)

    Then for p(1,α),

    1dnmax1kn|kj=1c1j(XjE^Xnj)|Lp0, (2.10)

    where ^Xnj=dncjI(Xj<dncj)+XjI(|Xj|dncj)+dncjI(Xj>dncj), 1jn.

    Proof of Theorem 2.1. We first observe that for every ε>0,

    P(1Dnmax1kn|kj=1c1j(XjEXnj)|>2ε)P(max1kn|kj=1c1j(XjXnj)|>Dnε)+P(max1kn|kj=1c1j(XnjEXnj)|>Dnε)=:H1+H2.

    To prove Eq (2.3), we need only to show that Hi0 as n, i=1,2. For H1, we have by the definition of Xnj, Cn=o(Dn), Eqs (2.1) and (2.2) that

    H1P(nj=1(XjXnj))nj=1P(Xj>Dncj)=nj=11Dncj+cj=1Dn+1nj=1c1j=CnDn+10.

    For fixed n1, Xnj is the nondecreasing function of Xj. Hence, it follows by Lemma 2.1 that {Xnj,1jn} is a sequence of NSD random variables. Hence we have by Markov's inequality and Lemma 2.3 with 1<p2,

    H2CDpnE(max1kn|kj=1c1j(XnjEXnj)|)pCDpnnj=1cpjE|Xnj|p=CDpnnj=1cpjE|Xj|pI(XjDncj)+Cnj=1P(Xj>Dncj)=CDpnnj=1cpj(Dncj)p0P(|Xj|pI(XjDncj)t)dt+Cnj=11Dncj+cj=CDpnnj=1cpj(Dncj)p0P(|Xj|pt)dt+CCnDn+1=CDpnnj=1cpj(Dncj)p01t1/p+cjdt+CCnDn+1(by(2.1))CDpnnj=1cpj(Dncj)p01t1/pdt+CCnDn+1=CCnDn+CCnDn+10.

    The proof is completed.

    Proof of Theorem 2.2. Obviously, to prove Eq (2.7), we need only to show

    1dnnj=1c1j(Xj~Xj)0a.s. (3.1)

    and

    1dnnj=1c1j(~XjE~Xj)0a.s.. (3.2)

    By Eq (2.6), dn and the Borel-Cantelli lemma, we obtain

    1dnnj=1c1j|Xj|I(|Xj|>φ(j))0a.s..

    Noting that

    |Xj+φ(j)|I(Xj<φ(j))+|Xjφ(j)|I(Xj>φ(j))|Xj|I(|Xj|>φ(j)).

    Then

    |1dnnj=1c1j(Xj~Xj)|=|1dnnj=1c1jXjI(|Xj|>φ(j))+(Xj+φ(j))I(Xj<φ(j))+(Xjφ(j))I(Xj>φ(j))|2dnnj=1c1j|Xj|I(|Xj|>φ(j))0a.s.,

    which yields Eq (3.1).

    It follows by the definition of ~Xj that

    j=11φ2(j)E(~XjE~Xj)2Cj=11φ2(j)EX2jI(|Xj|φ(j))+Cj=1P(|Xj|>φ(j))=:I1+I2.

    We obtain directly by Eq (2.6) that I2<. Let F(x) be the distribution of X1, then

    I1=Cj=11φ2(j)EX21I(|X1|φ(j))=Cj=11φ2(j)x2I(|X1|φ(j))dF(x)
    =Cx2j:φ(j)|x|1φ2(j)dF(x). (3.3)

    Define N(|x|)={j:φ(j)<|x|} and j=inf{j:φ(j)|x|}. Hence we can obtain N(|x|)j1 and

    j:φ(j)|x|1φ2(j)j=j1φ2(j)Cjφ2(j)(by Eq (2.5))Cjx2
    CN(|x|)+1x2. (3.4)

    It follows by Eqs (2.6), (3.3) and (3.4) that

    I1C(N(|x|)+1)dF(x)=CEN(|X1|)+C=CE[j=1I(|X1|>φ(j))]+C=Cj=1P(|X1|>φ(j))+C<.

    Now we obtain by I1< and I2< that

    j=11φ2(j)E(~XjE~Xj)2<. (3.5)

    Consequently, by Lemma 2.4 and Eq (3.5), we get

    j=11φ(j)(~XjE~Xj)convergesa.s.,

    which implies Eq (3.2) by Kronecker's lemma, together with the condition dn.

    The proof is completed.

    Proof of Theorem 2.3. Noting that

    E{1dnmax1kn|kj=1c1j(XjE^Xnj)|}p1dpnE{max1kn|kj=1c1j(^XnjE^Xnj)|}p+1dpnE{max1kn|kj=1c1j(Xj^Xnj)|}p1dpn{E(max1kn|kj=1c1j(^XnjE^Xnj)|)2}p/2+1dpnE{max1kn|kj=1c1j(Xj^Xnj)|}p=:J1+J2.

    To prove Eq (2.10), it is sufficient to prove J10 and J20. By Lemma 2.1 and the fact that ^Xnj is the nondecreasing function of Xj, {^Xnj,1jn} is also a sequence of NSD random variables.

    We have by Lemma 2.2 that

    J2/p1=1d2nE{max1kn|kj=1c1j(^XnjE^Xnj)|}2Cd2nnj=1c2jE(^XnjE^Xnj)2Cd2nnj=1c2jEX2jI(|Xj|dncj)+Cnj=1P(|Xj|>dncj)=:J3+J4.

    By dn, Eqs (2.8) and (2.9), we have

    J4C1dαnnj=1cαj0asn. (3.6)

    Now we will show J30. Observing

    J3=Cd2nnj=1c2j(dncj)20P(X2jI(|Xj|dncj)t)dtCd2nnj=1c2j(dncj)20P(X2jt)dt.

    Let t=u2, then

    J3Cd2nnj=1c2jdncj0uP(|Xj|u)du.

    From Eq (2.8), we know that, there exists M>0 and N0N such that

    P(|Xj|u)Muαforu>N0. (3.7)

    Since dn and cj1, while n is sufficiently large, we can obtain dncj>N0. Hence

    J3Cd2nnj=1c2jN00uP(|Xj|u)du+CMd2nnj=1c2jdncjN0u1αdu=:J3+J3

    By \alpha < 2 , c_j\geq1 and Eq (2.9), we have

    \begin{eqnarray*} J_3'&\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{N_0}u{\mathrm{d}}u\;\leq\;\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\\ &\leq&\frac{C}{d_n^{2-\alpha}}\frac{1}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty \end{eqnarray*}

    and

    \begin{eqnarray*} J_3''&\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\bigl[(d_nc_j)^{2-\alpha}-N_0^{2-\alpha}\bigr]\\ &\leq&\frac{C}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{eqnarray*}

    Finally, we need only to show J_2\rightarrow0 as n\rightarrow \infty . Let

    Z_{nj} = X_{j}-\widehat{X_{nj}} = (X_{j}+d_nc_j)\mathbb{I}(X_j < - d_nc_j)+(X_{j}-d_nc_j)\mathbb{I}(X_j > d_nc_j).

    We first prove that

    \begin{align} \mathbb{E}Z_{nj}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{align} (3.8)

    Observing

    \begin{eqnarray*} |\mathbb{E}Z_{nj}|&\leq&\mathbb{E}|Z_{nj}|\, \leq\, \mathbb{E}|X_{j}|\mathbb{I}(|X_{j}| > d_nc_j)\\ & = &\Biggl(\int_0^{d_nc_j}+\int_{d_nc_j}^{\infty}\Biggr)\mathbb{P}(|X_{j}|\mathbb{I}(|X_{j}| > d_nc_j)\geq t) {\mathrm{d}}t\\ & = &\int_0^{d_nc_j}\mathbb{P}(|X_{j}| > d_nc_j) {\mathrm{d}}t+\int_{d_nc_j}^{\infty}\mathbb{P}(|X_{j}|\geq t) {\mathrm{d}}t\\ & = &d_nc_j\mathbb{P}(|X_{j}| > d_nc_j)+\int_{d_nc_j}^{\infty}\mathbb{P}(|X_{j}|\geq t) {\mathrm{d}}t\\ & = \;:&J_5+J_6. \end{eqnarray*}

    By Eq (3.7) and \alpha > 1 , we have

    J_5\leq \frac{M}{(d_nc_j)^{\alpha-1}}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty

    and

    J_6\leq M\int_{d_nc_j}^{\infty}t^{-\alpha} {\mathrm{d}}t\leq\frac{CM}{(d_nc_j)^{\alpha-1}}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty,

    which yields Eq (3.8). Therefore, we obtain by Lemma 2.3 that

    \begin{eqnarray*} J_2&\leq&\frac{1}{d_n^p}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(Z_{nj}-\mathbb{E}Z_{nj}\bigr)\Biggr|\Biggr\}^p\\ &\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|Z_{nj}|^p\\ &\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|X_{j}|^p\mathbb{I}(|X_{j}| > d_nc_j).\quad\quad(\mbox{by}\;\mbox{the}\, \mbox{definition}\, \mbox{of}\, Z_{nj}) \end{eqnarray*}

    By similar arguments as in the proof of Eq (3.8), we can obtain

    \mathbb{E}|X_{j}|^p\mathbb{I}(|X_{j}| > d_nc_j) = (d_nc_j)^p\mathbb{P}(|X_{j}| > d_nc_j)+\int_{(d_nc_j)^p}^{\infty}\mathbb{P}(|X_{j}|^p\geq t) {\mathrm{d}}t.

    Then

    \begin{eqnarray*} J_2&\leq&C\sum\limits_{j = 1}^n\mathbb{P}(|X_{j}| > d_nc_j)+\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{(d_nc_j)^p}^{\infty}\mathbb{P}(|X_{j}|^p\geq t) {\mathrm{d}}t\\ & = \;:&J_2'+J_2''. \end{eqnarray*}

    By similar arguments as the proof of J_4\rightarrow0 , we obtain J_2'\rightarrow0 . We also have by Eq (3.7), p < \alpha and Eq (2.9) that

    \begin{eqnarray*} J_2''&\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{(d_nc_j)^p}^{\infty}t^{-\alpha/p} {\mathrm{d}}t\\ &\leq&\frac{C}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{eqnarray*}

    The proof is completed.

    In this work the author investigated the limit theorems for negatively superadditive-dependent random variables, and obtained some new results on the law of large numbers and mean convergence under some appropriate conditions. As a future work, we propose to consider some other strong convergence for sequence of negatively superadditive-dependent random variables.

    The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Social Sciences Planning Project of Anhui Province (AHSKY2018D98).

    The author declares that he has no conflict of interest.



    [1] M. Akram, A. Adeel, J. C. R. Alcantud, Group decision-making methods based on hesitant N-soft sets, Expert Syst. Appl., 115 (2019), 95-105. doi: 10.1016/j.eswa.2018.07.060
    [2] J. C. R. Alcantud, A. Giarlotta, Necessary and possible hesitant fuzzy sets: A novel model for group decision making, Inform. Fusion, 46 (2019), 63-76. doi: 10.1016/j.inffus.2018.05.005
    [3] M. Ali, F. Feng, X. Liu, et al. On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547-1553. doi: 10.1016/j.camwa.2008.11.009
    [4] B. Bedregal, R. Reiser, H. Bustince, et al. Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms, Inform. Sci., 255 (2014), 82-99. doi: 10.1016/j.ins.2013.08.024
    [5] N. Cagman, S. Enginoglu, Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207 (2010), 848-855. doi: 10.1016/j.ejor.2010.05.004
    [6] N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010), 3308-3314. doi: 10.1016/j.camwa.2010.03.015
    [7] D. Chen, E. C. C. Tsang, D. S. Yeung, et al. The parameterization reduction of soft sets and its applications, Comput. Math. Appl., 49 (2005), 757-763. doi: 10.1016/j.camwa.2004.10.036
    [8] J. Chen, X. Huang, J. Tang, Distance measures for higher order dual hesitant fuzzy sets, Comput. Appl. Math., 37 (2018), 1784-1806. doi: 10.1007/s40314-017-0423-3
    [9] N. Chen, Z. S. Xu, M. M. Xia, Interval-valued hesitant preference relations and their applications to group decision making, Knowl. Based Syst., 37 (2013), 528-540. doi: 10.1016/j.knosys.2012.09.009
    [10] A. Dey, M. Pal, Genelalised multi-fuzzy soft set and its application in decision making, Pac. Sci. Rev. A, 17 (2015), 23-28.
    [11] A. Dey, M. Pal, On hesitant multi-fuzzy soft topology, Pac. Sci. Rev. B, 1 (2015), 124-130.
    [12] F. Feng, Y. B. Jun, X. Liu, et al. An adjustable approach to fuzzy soft set based decision making, J. Comput. Appl. Math., 234 (2010), 10-20. doi: 10.1016/j.cam.2009.11.055
    [13] F. Feng, Y. Li, N. Cagman, Generalized uni-int decision making schemes based on choice value soft sets, Eur. J. Oper. Res., 220 (2012), 162-170. doi: 10.1016/j.ejor.2012.01.015
    [14] X. Guan, Y. Li, F. Feng, A new order relation on fuzzy soft sets and its application, Soft Comput., 17 (2013), 63-70. doi: 10.1007/s00500-012-0903-8
    [15] J. Ignatius, S. Motlagh, M. Sepehri, et al. Hybrid models in decision making under uncertainty: The case of training provider evaluation, J. Intell. Fuzzy Syst., 21 (2010), 147-162. doi: 10.3233/IFS-2010-0443
    [16] P. Kakati, S. Borkotokey, S. Rahman, et al. Interval neutrosophic hesitant fuzzy Einstein Choquet integral operator for multicriteria decision making, Artif. Intell. Rev., (2019), 1-36.
    [17] Z. Kong, L. Gao, L. Wang, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math., 223 (2009), 540-542. doi: 10.1016/j.cam.2008.01.011
    [18] Z. Kong, L. Wang, Z. Wu, Application of fuzzy soft set in decision making problems based on grey theory, J. Comput. Appl. Math., 236 (2011), 1521-1530. doi: 10.1016/j.cam.2011.09.016
    [19] D. C. Liang, D. Liu, A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy information, IEEE Trans. Fuzzy Syst., 23 (2015), 237-247. doi: 10.1109/TFUZZ.2014.2310495
    [20] D. C. Liang, Z. S. Xu, D. Liu, Three-way decisions based on decision-theoretic rough sets with dual hesitant fuzzy information, Inform. Sci., 396 (2017), 127-143. doi: 10.1016/j.ins.2017.02.038
    [21] D. C. Liang, M. W. Wang, Z. S. Xu, et al. Risk appetite dual hesitant fuzzy three-way decisions with todim, Inform. Sci., 507 (2020), 585-605. doi: 10.1016/j.ins.2018.12.017
    [22] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9 (2001), 589-602.
    [23] P. K. Maji, R. Biswas, A. R. Roy, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077-1083. doi: 10.1016/S0898-1221(02)00216-X
    [24] P. Majumdar, S. K. Samanta, Generalised fuzzy soft sets, Comput. Math. Appl., 59 (2010), 1425-1432. doi: 10.1016/j.camwa.2009.12.006
    [25] D. A. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19-31.
    [26] S. Naz, M. Akram, Novel decision-making approach based on hesitant fuzzy sets and graph theory, Comput. Appl. Math., 38 (2019), 7.
    [27] B. Ozkan, E. Ozceylan, M. Kabak, et al. Evaluating the websites of academic departments through SEO criteria: A hesitant fuzzy linguistic MCDM approach, Artif. Intell. Rev., 53 (2020), 875-905. doi: 10.1007/s10462-019-09681-z
    [28] X. Qi, D. Liang, J. Zhang, Multiple attribute group decision making based on generalized power aggregation operators under interval-valued dual hesitant fuzzy linguistic environment, Int. J. Mach. Learn. Cybern., 7 (2016), 1147-1193. doi: 10.1007/s13042-015-0445-3
    [29] Z. Ren, C. Wei, A multi-attribute decision-making method with prioritization relationship and dual hesitant fuzzy decision information, Int. J. Mach. Learn. Cybern., 8 (2017), 755-763. doi: 10.1007/s13042-015-0356-3
    [30] R. M. Rodryguez, L. Martynez, V. Torra, et al. Hesitant fuzzy sets: state of the art and future directions, Int. J. Intell. Syst., 29 (2014), 495-524. doi: 10.1002/int.21654
    [31] A. R. Roy, P. K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math., 203 (2007), 412-418. doi: 10.1016/j.cam.2006.04.008
    [32] S. Sebastian, T. V. Ramakrishnan, Multi-fuzzy sets: An extension of fuzzy sets, Fuzzy Inf. Eng., 1 (2011), 35-43.
    [33] C. Song, Y. Zhang, Z. Xu, An improved structure learning algorithm of Bayesian Network based on the hesitant fuzzy information flow, Appl. Soft Comput., 82 (2019), 105549.
    [34] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst., 25 (2010), 529-539.
    [35] F. Wang, X. Li, X. Chen, Hesitant fuzzy soft set and its applications in multicriteria decision making, J. Appl. Math., 2014 (2014), 643785.
    [36] G. Wei, X. Zhao, R. Lin, Some hesitant intervalvalued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowl. Based Syst, 46 (2013), 43-53. doi: 10.1016/j.knosys.2013.03.004
    [37] G. Wei, X. Zhao, Induced hesitant interval-valued fuzzy Einstein aggregation operators and their application to multiple attribute decision making, J. Intell. Fuzzy Syst., 24 (2013), 789-803. doi: 10.3233/IFS-2012-0598
    [38] G. Wei, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, Int. J. Mach. Learn. Cybern., 7 (2016), 1093-1114. doi: 10.1007/s13042-015-0433-7
    [39] M. Xia, Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason., 52 (2011), 395-407. doi: 10.1016/j.ijar.2010.09.002
    [40] M. Xia, Z. S. Xu, Some studies on properties of hesitant fuzzy sets, Int. J. Mach. Learn. Cybern., 8 (2017), 489-495. doi: 10.1007/s13042-015-0340-y
    [41] Z. S. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inform. Sci., 181 (2011), 2128-2138. doi: 10.1016/j.ins.2011.01.028
    [42] Z. Xu, W. Zhou, Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment, Fuzzy Optim. Decis. Ma., 16 (2017), 481-503. doi: 10.1007/s10700-016-9257-5
    [43] W. Xue, Z. Xu, H. Wang, et al. Hazard assessment of landslide dams using the evidential reasoning algorithm with multi-scale hesitant fuzzy linguistic information, Appl. Soft Comput., 79 (2019), 74-86. doi: 10.1016/j.asoc.2019.03.032
    [44] Y. Yang, X. Tan, C. Meng, The multi-fuzzy soft set and its application in decision making, Appl. Math. Model., 37 (2013), 4915-4923. doi: 10.1016/j.apm.2012.10.015
    [45] Y. Yang, X. Peng, H. Chen, A decision making approach based on bipolar multi-fuzzy soft set theory, J. Intell. Fuzzy Syst., 27 (2014), 1861-1872. doi: 10.3233/IFS-141152
    [46] D. Yu, D. F. Li, J. M. Merigo, Dual hesitant fuzzy group decision making method and its application to supplier selection, Int. J. Mach. Learn. Cybern., 7 (2016), 819-831. doi: 10.1007/s13042-015-0400-3
    [47] C. Zhang, D. Li, J. Liang, Hesitant fuzzy linguistic rough set over two universes model and its applications, Int. J. Mach. Learn. Cybern., 9 (2018), 577-588. doi: 10.1007/s13042-016-0541-z
    [48] C. Zhang, D. Li, J. Liang, Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes, Inform. Sci., 507 (2020), 665-683. doi: 10.1016/j.ins.2019.01.033
    [49] C. Zhang, D. Li, J. Liang, Interval-valued hesitant fuzzy multi-granularity three-way decisions in consensus processes with applications to multi-attribute group decision making, Inform. Sci., 511 (2020), 192-211. doi: 10.1016/j.ins.2019.09.037
    [50] Z. Zhang, Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making, Inform. Sci., 234 (2013), 150-181. doi: 10.1016/j.ins.2013.01.002
    [51] Z. Zhang, C. Wang, D. Tian, et al. Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making, Comput. Ind. Eng., 67 (2014), 116-138. doi: 10.1016/j.cie.2013.10.011
    [52] Z. Zhang, H. Dong, S. Lan, Possibility multi-fuzzy soft set and its application in decision making, J. Intell. Fuzzy Syst., 27 (2013), 2115-2125.
    [53] H. Zhang, L. Xiong, W. Ma, On interval-valued hesitant fuzzy soft sets, Math. Probl. Eng., 2015 (2015), 254764.
    [54] F. Zhang, J. Ignatius, C. P. Lim, et al. A new method for ranking fuzzy numbers and its application to group decision making, Appl. Math. Model., 38 (2014), 1563-1582. doi: 10.1016/j.apm.2013.09.002
    [55] W. Zhou, Z. Xu, Hesitant fuzzy linguistic portfolio model with variable risk appetite and its application in the investment ratio calculation, Appl. Soft Comput., 84 (2019), 105719.
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