#### 1. Introduction

Since Zadeh [1] proposed the fuzzy set theory, a large number of scholars have promoted it to varying degrees. For example, the concept of interval-valued fuzzy sets(IVFS) [2] extends the membership degree to the range of interval numbers; intuitionistic fuzzy sets(IFS) [3], [4] introduced non-membership degree, which strengthened the expression ability of fuzzy sets to uncertain information; Torra extended fuzzy sets to hesitant fuzzy sets(HFS) [5], [6], the membership degree is extended to a set containing multiple possible membership degrees, enables it to retain more original information.

At present, the research on fuzzy sets is mostly around IFS and HFS. In terms of theoretical innovation, Chen [7] proposed interval valued hesitation fuzzy sets in combination with IVFS and HFS, Qian [8] proposed generalized hesitant fuzzy sets(GHFS) based on IFS and HFS. Palanikumar [9] present the idea of a possibility Pythagorean cubic fuzzy soft set and discuss how it may be used to solve practical issues. In terms of information aggregation, interval intuitionistic fuzzy power weighted operators [10] and q-rung triangular hesitation ambiguity BM operators [11] are proposed and applied to multi-attribute group decision making. Ahmad [12] discussed a couple of different hesitant fuzzy aggregation operators and developed linear and hyperbolic membership functions under hesitant fuzziness, which contains the concept of hesitant degrees for different objectives. In terms of preference relationship, Li [13] and Yang [14] respectively researched the application of incomplete hesitation fuzzy preference relation and interval intuitionistic fuzzy preference relation in decision making. To address the fuzziness and uncertainty in constellation satellite communication system research, Wang [15] proposed a method under a probabilistic hesitant intuitionistic fuzzy preference relationship and deduced its consistency index to make the consensus of the preference relationship acceptable. In terms of weight promotion, Zeng [16], [17] proposed the concepts of weighted hesitant fuzzy sets(WHFS) and weighted interval hesitant fuzzy sets, and discussed their practicability in decision-making.

Classification is one of the most basic tasks in the field of machine learning. Therefore, ensemble learning is widely used in classification tasks. By using different methods to combine multiple classifiers, ensemble learning can often achieve higher accuracy and generalization ability than a single classifier. Ngo [18] proposed evolutionary bagged ensemble learning, which utilise evolutionary algorithms to evolve the content of the bags in order to iteratively enhance the ensemble by providing diversity in the bags. Dynamic ensemble learning [19] is also an important innovation, it selects the most competent base classifiers for an unseen instance and combines these classifiers to make the final decision. As for application, Gracia [20] implements the bagging and stacking ensemble learning systems, which will be used to detect malicious applications within the Android Mobile System.

In fact, the results given by the classifier are the probabilities that the samples belong to a certain category, so the fuzzy set method is very suitable for dealing with classification problems. Singh [21], [22] introduced two new classification methods by employing IFS and IVFS approach. The proposed methods can classify the objects available in a system called fuzzy decision table into four distinct fuzzy regions, and based on the classified objects, various decision rules are generated from these distinct fuzzy regions. Dai [23] proposed an ensemble learning algorithm based on IFS. The algorithm constructs an intuitionistic fuzzy preference relation matrix to determine the weights of classifiers, then it uses multi-criteria group decision making method to determine the sample classification result. However, the traditional IFS or HFS theory may lose some original information when dealing with classification problems. For example, it is not reliable to use only the accuracy ratio as the membership degree to evaluate the effectiveness of the classification algorithm. To solve this problem, this paper proposes weighted generalized hesitant fuzzy set (WGHFS), and studies its operation rules, aggregation operators, preference relations and other related concepts. In addition, the paper also proposes an ensemble learning algorithm based on weighted generalized hesitant fuzzy sets (WGHFS-EL). Finally, the experiment proves that this algorithm has better accuracy and generalization ability than the traditional ensemble learning algorithm.

The remainder of this article is structured as follows: In Sect. 2, some concepts of IFS, HFS, WHFS, GHFS are reviewed. In Sect. 3, some concepts related to WGHFS are presented. The algorithm and its verification are illustrated in Sect. 4 and Sect. 5. Conclusion is presented in Sect. 6.

#### 2. Preliminaries

In this section, some basic concepts related to IFS, HFS, WHFS and GHFS are reviewed.

##### 2.1 Intuitionistic Fuzzy Set

**Definition 1:** Let *X* be a non-empty set, an intuitionistic fuzzy set *A* can be defined as [3]:

\[\begin{aligned} A=\left\{ \left< x,\mu _A\left( x \right) ,\upsilon _A\left( x \right) \right> \middle| x\in X \right\} \end{aligned}\] |

where \(\mu_A\left(x\right):X\rightarrow\left[0,1\right]\) and \(\upsilon_A\left(x\right):X\rightarrow\left[0,1\right]\) respectively represents the degree of the membership and the degree of the non-membership of the element *x* in set *A*, and the two values meet the condition: \(0\le\mu_A\left(x\right)+\upsilon_A\left(x\right)\le1\) for all element *x* in set *X*. The ordered pair \(\left(\mu_A\left(x\right),\upsilon_A\left(x\right)\right)\) is referred to as an intuitionistic fuzzy value(IFV).

**Definition 2:** Let \(a=\left(\mu_a,\upsilon_a\right)\) and \(b=\left(\mu_b,\upsilon_b\right)\) be two IFVs, then we have [24], [25]:

- \(a^C=\left(\upsilon_a,\mu_a\right)\)
- \(a\cup b=\left(\max\left(\mu_a,\mu_b\right),\min\left(\upsilon_a,\upsilon_b\right)\right)\)
- \(a\cap b=\left(\min\left(\mu_a,\mu_b\right),\max\left(\upsilon_a,\upsilon_b \right)\right)\)
- \(a\oplus b=\left(\mu_a +\mu_b -\mu_a\mu_b,\upsilon_a\upsilon_b\right)\)
- \(a\otimes b=\left(\mu_a\mu_b,\upsilon_a +\upsilon_b -\upsilon_a\upsilon_b\right)\)
- \(a^{\lambda}=\left({\mu_a}^{\lambda},1-\left(1-\upsilon_a\right)^{\lambda}\right)\)
- \(\lambda a=\left(1-\left(1-\mu_a\right)^{\lambda},{\upsilon_a}^{\lambda}\right)\)

##### 2.2 Hesitant Fuzzy Set

**Definition 3:** Let *X* be a non-empty set, a hesitant fuzzy set *E* can be defined as [5]:

\[\begin{aligned} E=\left\{ \left< x,h_E\left( x \right) \right> \middle| x\in X \right\} \end{aligned}\] |

where \(h_E\left(x\right)\subseteq\left[0,1\right]\) is a set of all possible membership degree of the element *x* to set *E*, and \(h_E\left(x\right)\) is referred to as an hesitant fuzzy element (HFE) [26], abbreviated as \(h\left(x\right)\).

**Definition 4:** Given two *HFEs* \(h_1\) and \(h_2\), for \(\lambda>0\), we have [5]:

- \({h_1}^C=\bigcup_{\gamma_1\in h_1}{\left\{1-\gamma_1\right\}}\)
- \(h_1\cup h_2=\bigcup_{\gamma_1\in h_1,\gamma_2\in h_2}{\max\left\{\gamma_1,\gamma_2\right\}}\)
- \(h_1\cap h_2=\bigcup_{\gamma_1\in h_1,\gamma_2\in h_2}{\min\left\{\gamma_1,\gamma_2\right\}}\)
- \(h_1\oplus h_2=\bigcup_{\gamma_1\in h_1,\gamma_2\in h_2}{\left\{\gamma_1 +\gamma_2 -\gamma_1\gamma_2 \right\}}\)
- \(h_1\otimes h_2=\bigcup_{\gamma_1\in h_1,\gamma_2\in h_2}{\left\{\gamma_1\gamma_2\right\}}\)
- \({h_1}^{\lambda}=\bigcup_{\gamma_1\in h_1}{\left\{{\gamma_1}^{\lambda}\right\}}\)
- \(\lambda h_1=\bigcup_{\gamma_1\in h_1}{\left\{1-\left(1-\gamma_1\right)^{\lambda}\right\}}\)

##### 2.3 Weight Hesitant Fuzzy Set

**Definition 5:** Let *X* be a non-empty set, a weight hesitant fuzzy set \(E^{\omega}\) can be defined as [16]:

\[E^{\omega}=\left\{\left<\left.x,h_{E^{\omega}}\left(x\right)\right.\right>\left|x\in X\right.\right\}\] |

where \(h_{E^{\omega}}\left(x\right)\subseteq\left[0,1\right]\) is a set of all possible membership degree of the element *x* to set \(E^{\omega}\), and \(h_{E^{\omega}}\left(x\right)\) is referred to as a weight hesitant fuzzy element (WHFE) [13], abbreviated as \(h^{\omega}\left( x \right)\).

**Definition 6:** Given two *WHFEs* \({h_1}^{\omega}\) and \({h_2}^{\omega}\), for \(\lambda>0\), we have:

- \(\left( {h_1}^{\omega} \right) ^C=\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega}}{\left( 1-\gamma _i,\omega _i \right)}\)
- \({h_1}^{\omega}\cup {h_2}^{\omega} \\ \!=\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega},\left( \gamma _j,\omega _j \right) \in {h_2}^{\omega}}{\left( \max \left\{ \gamma _i,\gamma _j \right\} ,{\omega _{\max \left\{ \gamma _i,\gamma _j \right\}}}^{\prime} \right)}\)

where set \({\omega _{\max \left\{ \gamma _i,\gamma _j \right\}}}^{\prime}\) is the normalization of weights set \({\omega _{\max \left\{ \gamma _i,\gamma _j \right\}}}\). - \({h_1}^{\omega}\cap {h_2}^{\omega} \\ =\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega},\left( \gamma _j,\omega _j \right) \in {h_2}^{\omega}}{\left( \min \left\{ \gamma _i,\gamma _j \right\} ,{\omega _{\min \left\{ \gamma _i,\gamma _j \right\}}}^{\prime} \right)}\)

where set \({\omega _{\min \left\{ \gamma _i,\gamma _j \right\}}}^{\prime}\) is the normalization of weights set \({\omega _{\min \left\{ \gamma _i,\gamma _j \right\}}}\). - \({h_1}^{\omega}\oplus {h_2}^{\omega} \\ =\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega},\left( \gamma _j,\omega _j \right) \in {h_2}^{\omega}}{\left( \gamma _i+\gamma _j-\gamma _i\gamma _j,\omega _i\omega _j \right)}\)
- \({h_1}^{\omega}\otimes {h_2}^{\omega} \\ =\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega},\left( \gamma _j,\omega _j \right) \in {h_2}^{\omega}}{\left( \gamma _i\gamma _j,\omega _i\omega _j \right)}\)
- \(\left( {h_1}^{\omega} \right) ^{\lambda}=\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega}}{\left( {\gamma _i}^{\lambda},\omega _i \right)}\)
- \(\lambda \left( {h_1}^{\omega} \right) =\bigcup_{\left( \gamma _i,\omega _i \right) \in {h_1}^{\omega}}{\left( 1-\left( 1-\gamma _i \right) ^{\lambda},\omega _i \right)}\)

##### 2.4 Generalized Hesitant Fuzzy Set

**Definition 7:** Let *X* be a non-empty set, a generalized hesitant fuzzy set *M* can be defined as [8]:

\[\begin{aligned} M=\left\{\left<x,h_M\left(x\right)\right>\middle|x\in X\right\} \end{aligned}\] |

where \(h_M\left(x\right)=\left\{\alpha_i=\left(\mu_i,\upsilon_i\right)\left|i=1,2,\ldots ,N\right.\right\}\) is a set of all possible membership degree of the element *x* to set *M*, expressed in the form of multiple IFVs, \(h_M\left(x\right)\) is referred to as a generalized hesitant fuzzy element (GHFE), abbreviated as \(Gh\left(x\right)\).

**Definition 8:** Given two *GHFEs* \(Gh_1\) and \(Gh_2\), for \(\lambda>0\), we have [8]:

- \({Gh_1}^C=\bigcup_{\alpha_1\in Gh_1}{{\alpha_1}^C}\)
- \(Gh_1\cup Gh_2=\bigcup_{\alpha_1\in Gh_1,\alpha_2\in Gh_2}{\left(\alpha_1\cup\alpha_2\right)}\)
- \(Gh_1\cap Gh_2=\bigcup_{\alpha_1\in Gh_1,\alpha_2\in Gh_2}{\left(\alpha_1\cap\alpha_2\right)}\)
- \(Gh_1\oplus Gh_2=\bigcup_{\alpha_1\in Gh_1,\alpha_2\in Gh_2}{\left(\alpha_1\oplus\alpha_2\right)}\)
- \(Gh_1\otimes Gh_2=\bigcup_{\alpha_1\in Gh_1,\alpha_2\in Gh_2}{\left(\alpha_1\otimes\alpha_2\right)}\)
- \({Gh_1}^{\lambda}=\bigcup_{\alpha_1\in Gh_1}{{\alpha_1}^{\lambda}}\)
- \(\lambda Gh_1=\bigcup_{\alpha_1\in Gh_1}{\lambda\alpha_1}\)

#### 3. Weighted Generalized Hesitant Fuzzy Set

##### 3.1 Definition and Operation of WGHFS

**Definition 9:** Let *X* be a non-empty set, a weighted generalized hesitant fuzzy set \(M^{\omega}\) can be defined as:

\[\begin{aligned} M^{\omega}=\left\{\left<\left.x,h_{M^{\omega}}\left(x\right)\right.\right>\left|x\in X\right.\right\} \end{aligned}\] |

where \(h_{M^{\omega}}\left(x\right)=\left\{\left(\alpha_i,\omega_i\right)\left|i=1,2,\ldots ,N\right.\right\}\) is a set of all possible membership degree of the element *x* to set \(M^{\omega}\), expressed in the form of multiple weighted IFVs, \({\omega}_i\) is the weight of \({\alpha}_i\), and \(\sum_{i=1}^N{\omega _i}=1\). \(h_{M^{\omega}}(x)\) is referred to as a weighted generalized hesitant fuzzy element (WGHFE), abbreviated as \(Gh^{\omega}\left(x\right)\).

Following the operation of intuitionistic fuzzy sets and hesitant fuzzy sets, the operation of weighted generalized hesitant fuzzy sets is given.

**Definition 10:** Given two *WGHFEs* \({Gh_1}^{\omega}\) and \({Gh_2}^{\omega}\), for \(\lambda>0\), we have:

- \(\begin{aligned} &\left( {Gh_1}^{\omega} \right) ^C=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega}}{\left( {\alpha _i}^C,\omega _i \right)} \end{aligned}\)
- \({Gh_1}^{\omega}\cap {Gh_2}^{\omega}\\ \begin{aligned} &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \alpha _i\cup \alpha _j,{\omega _{\max}}^{\prime} \right)}\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \begin{array}{c} \left( \max \left( \mu _i,\mu _j \right) ,\min \left( \upsilon _i,\upsilon _j \right) \right) ,\\ \left( \frac{\omega _{\max \left( \mu _i,\mu _j \right)}+\omega _{\min \left( \upsilon _i,\upsilon _j \right)}}{2} \right) ^{\prime}\\ \end{array} \right)\\ \end{aligned}\)

where set \({\omega_{\max}}^{\prime}\) is the normalization of weights set \(\omega_{\max}\), \(\omega_{\max\left(\mu_i,\mu_j\right)}\) represents the weight of the WGHFE corresponding to the maximum value of \(\left(\mu_i,\mu_j\right)\), \(\omega_{\min\left(\upsilon_i,\upsilon_j\right)}\) represents the weight of the WGHFE corresponding to the minimum value of \(\left(\upsilon_i,\upsilon_j\right)\). - \({Gh_1}^{\omega}\cap {Gh_2}^{\omega}\\ \begin{aligned} &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \alpha _i\cap \alpha _j,{\omega _{\min}}^{\prime} \right)}\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \begin{array}{c} \left( \min \left( \mu _i,\mu _j \right) ,\max \left( \upsilon _i,\upsilon _j \right) \right) ,\\ \left( \frac{\omega _{\min \left( \mu _i,\mu _j \right)}+\omega _{\max \left( \upsilon _i,\upsilon _j \right)}}{2} \right) ^{\prime}\\ \end{array} \right)\\ \end{aligned}\)

where set \({\omega_{\min}}^{\prime}\) is the normalization of weights set \(\omega_{\min}\), \(\omega_{\min\left(\mu_i,\mu_j\right)}\) represents the weight of the WGHFE corresponding to the minimum value of \(\left(\mu_i,\mu_j\right)\), \(\omega_{\max\left(\upsilon_i,\upsilon_j\right)}\) represents the weight of the WGHFE corresponding to the maximum value of \(\left(\upsilon_i,\upsilon_j\right)\). - \({Gh_1}^{\omega}\oplus {Gh_2}^{\omega}\\ \begin{aligned} &=\bigcup_{\left(\alpha_i,\omega_i\right)\in{Gh_1}^{\omega},\left(\alpha_j,\omega_j\right)\in {Gh_2}^{\omega}}{\left(\alpha_i\oplus\alpha_j,\omega_i\omega_j\right)}\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \left( \mu _i+\mu _j-\mu _i\mu _j,\upsilon _i\upsilon _j \right) ,\omega _i\omega _j \right)\\ \end{aligned}\)
- \({Gh_1}^{\omega}\otimes {Gh_2}^{\omega}\\ \begin{aligned} &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \alpha _i\otimes \alpha _j,\omega _i\omega _j \right)}\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \left( \mu _i\mu _j,\upsilon _i+\upsilon _j-\upsilon _i\upsilon _j \right) ,\omega _i\omega _j \right)\\ \end{aligned}\)
- \(\begin{aligned} &\left( {Gh_1}^{\omega} \right) ^{\lambda}=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega}}{\left( {\alpha _i}^{\lambda},\omega _i \right)} \end{aligned} \\ \begin{aligned} &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega}}{\left( \left( {\mu _i}^{\lambda},1-\left( 1-\upsilon _i \right) ^{\lambda} \right) ,\omega _i \right)}\\ \end{aligned}\)
- \(\begin{aligned} &\lambda \left( {Gh_1}^{\omega} \right) =\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega}}{\left( \lambda \alpha _i,\omega _i \right)}\\ \end{aligned} \\ \begin{aligned} &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega}}{\left( \left( 1-\left( 1-\mu _i \right) ^{\lambda},{\upsilon _i}^{\lambda} \right) ,\omega _i \right)}\\ \end{aligned}\)

**Theorem 1:** If \({Gh_1}^{\omega}\) and \({Gh_2}^{\omega}\) are two *WGHFEs* and \(\lambda>0\), then \({Gh_1}^{\omega}\cup{Gh_2}^{\omega}\), \({Gh_1}^{\omega}\cap{Gh_2}^{\omega}\), \({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\), \({Gh_1}^{\omega}\otimes {Gh_2}^{\omega}\), \(\left({Gh_1}^{\omega}\right)^{\lambda}\) and \(\lambda\left({Gh_1}^{\omega}\right)\) are also *WGHFEs*.

**Proof:1** Let \({Gh_1}^{\omega}=\left\{\left(\left(\mu_i,\upsilon_i\right),\omega_i\right)\left|i=1,2,\ldots ,m\right.\right\}\), \({Gh_2}^{\omega}=\left\{\left(\left(\mu_j,\upsilon_j\right),\omega_j\right)\left|j=1,2,\ldots ,n\right.\right\}\) be two *WGHFEs*. So that \(0\leqslant\mu_i,\upsilon_i\leqslant 1\); \(0\leqslant\mu_j,\upsilon_j\leqslant 1\); \(0\leqslant\mu_i+\upsilon_i\leqslant 1\); \(0\leqslant\mu_j+\upsilon_j\leqslant 1\); \(\sum_{i=1}^m{\omega_i}=1\); \(\sum_{j=1}^n{\omega_j}=1\). Take \({Gh_3}^{\omega}={Gh_1}^{\omega}\oplus{Gh_2}^{\omega}=\left\{\left(\left(\mu_k,\upsilon_k\right),\omega_k\right)\middle|k=1,2,\ldots ,l\right\}\), where \(\mu_k=\mu_i+\mu_j-\mu_i\mu_j\); \(\upsilon_k=\upsilon_i\upsilon_j\); \(\omega_k=\omega_i\omega_j\); \(l=mn\).

Because \(0\leqslant\mu_i,\upsilon_i\leqslant 1\) and \(0\leqslant\mu_j,\upsilon_j\leqslant 1\), it is obvious that \(0\leqslant\mu_i+\mu_j-\mu_i\mu_j\) and \(0\leqslant\upsilon_i\upsilon_j\leqslant 1\), further, \(\mu_i+\mu_j-\mu_i\mu_j=\mu_i+\left(1-\mu_i\right)\mu_j\leqslant\mu_i+1-\mu_i=1\), hence \(0\leqslant\mu_k\leqslant 1\) and \(0\leqslant\upsilon_k\leqslant 1\). Also, because \(0\leqslant\mu_i+\upsilon_i\leqslant 1\), \(0\leqslant\mu_j+\upsilon_j\leqslant 1\), which implies that \(\mu_i\leqslant 1-\upsilon_i\) and \(\mu_j\leqslant 1-\upsilon_j\), therefore, \(\mu_i+\mu_j-\mu_i\mu_j+\upsilon _i\upsilon_j\leqslant 1-\upsilon_i+1-\upsilon_j-\left(1-\upsilon_i\right)\left(1-\upsilon_j\right)+\upsilon_i\upsilon_j=1\), thus \(0\leqslant\mu_k+\upsilon_k\leqslant 1\), which means \(\left(\mu_k,\upsilon_k\right)\) is also an IFV. On the other hand, since \(\sum_{i=1}^m{\omega_i}=1\); \(\sum_{j=1}^n{\omega_j}=1\); \(l=mn\), \(\sum_{k=1}^l{\omega_k}=\sum_{i=1}^m{\sum_{j=1}^n{\omega_i\omega_j}}=1\). Hence, \({Gh_3}^{\omega}=\left\{\left(\left(\mu_k,\upsilon_k\right),\omega_k\right)\middle|k=1,2,\ldots ,l\right\}\) satisfy the definition of *WGHFE*. Thus, we get \({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\) is a *WGHFE*. Similarly, it can be proved that \({Gh_1}^{\omega}\cup{Gh_2}^{\omega}\), \({Gh_1}^{\omega}\cap{Gh_2}^{\omega}\), \({Gh_1}^{\omega}\otimes{Gh_2}^{\omega}\), \(\left({Gh_1}^{\omega}\right)^{\lambda}\) and \(\lambda\left({Gh_1}^{\omega}\right)\) are also *WGHFEs*.

**Theorem 2:** Given three *WGHFEs* \({Gh_1}^{\omega}\), \({Gh_2}^{\omega}\), \({Gh_3}^{\omega}\), we have:

- \({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}={Gh_2}^{\omega}\oplus{Gh_1}^{\omega}\)
- \({Gh_1}^{\omega}\otimes{Gh_2}^{\omega}={Gh_2}^{\omega}\otimes{Gh_1}^{\omega}\)
- \(\left({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\right)\oplus{Gh_3}^{\omega}={Gh_1}^{\omega}\oplus\left( {Gh_2}^{\omega}\oplus{Gh_3}^{\omega}\right)\)
- \(\left({Gh_1}^{\omega}\otimes{Gh_2}^{\omega}\right)\otimes{Gh_3}^{\omega}={Gh_1}^{\omega}\otimes\left( {Gh_2}^{\omega}\otimes{Gh_3}^{\omega}\right)\)
- \(\lambda\left({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\right)={\lambda Gh_1}^{\omega}\oplus{\lambda Gh_2}^{\omega}\)
- \(\left({Gh_1}^{\omega}\otimes{Gh_2}^{\omega}\right)^{\lambda}=\left({Gh_1}^{\omega}\right) ^{\lambda}\otimes\left({Gh_2}^{\omega}\right)^{\lambda}\)

**Proof:2** To be brief, here is the proof of (1), (3) and (5), for others are similar:

- (1) Because \({Gh_1}^{\omega}\) and \({Gh_2}^{\omega}\) are
*WGHFEs*,

\(\begin{aligned} &{Gh_1}^{\omega}\oplus {Gh_2}^{\omega}\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \alpha _i\oplus \alpha _j,\omega _i\omega _j \right)}\\ &=\bigcup_{\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega},\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega}}{\left( \alpha _j\oplus \alpha _i,\omega _j\omega _i \right)}\\ &={Gh_2}^{\omega}\oplus {Gh_1}^{\omega}\\ \end{aligned}\)

Hence, \({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}={Gh_2}^{\omega}\oplus{Gh_1}^{\omega}\). - (3) Because \({Gh_1}^{\omega}\), \({Gh_2}^{\omega}\) and \({Gh_3}^{\omega}\) are
*WGHFEs*,

\(\begin{aligned} &\left( {Gh_1}^{\omega}\oplus {Gh_2}^{\omega} \right) \oplus {Gh_3}^{\omega}\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega},\left( \alpha _k,\omega _k \right) \in {Gh_3}^{\omega}}{}\\ &\quad \left( \left( \alpha _i\oplus \alpha _j \right) \oplus \alpha _k,\left( \omega _i\omega _j \right) \omega _k \right)\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega},\left( \left( \mu _k,\upsilon _k \right) ,\omega _k \right) \in {Gh_3}^{\omega} }{}\\ &\quad \left( \left( \begin{array}{c} \mu _i+\mu _j+\mu _k-\mu _i\mu _j-\mu _j\mu _k-\\ \mu _i\mu _k+\mu _i\mu _j\mu _k,\upsilon _i\upsilon _j\upsilon _k\\ \end{array} \right) ,\omega _i\omega _j\omega _k \right)\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega},\left( \alpha _k,\omega _k \right) \in {Gh_3}^{\omega}}{}\\ &\quad \left( \alpha _i\oplus \left( \alpha _j\oplus \alpha _k \right) ,\omega _i\left( \omega _j\omega _k \right) \right)\\ &={Gh_1}^{\omega}\oplus \left( {Gh_2}^{\omega}\oplus {Gh_3}^{\omega} \right)\\ \end{aligned}\)

Hence, \(\left({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\right)\oplus{Gh_3}^{\omega}={Gh_1}^{\omega}\oplus\left( {Gh_2}^{\omega}\oplus{Gh_3}^{\omega}\right)\) - (5) Because \({Gh_1}^{\omega}\), \({Gh_2}^{\omega}\) are
*WGHFEs*, and \(\lambda>0\),

\(\begin{aligned} &\lambda \left( {Gh_1}^{\omega}\oplus {Gh_2}^{\omega} \right)\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \lambda \left( \alpha _i\oplus \alpha _j \right) ,\omega _i\omega _j \right)}\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \lambda \left( \mu _i+\mu _j-\mu _i\mu _j,\upsilon _i\upsilon _j \right) ,\omega _i\omega _j \right)\\ &=\bigcup_{\left( \left( \mu _i,\upsilon _i \right) ,\omega _i \right) \in {Gh_1}^{\omega},\left( \left( \mu _j,\upsilon _j \right) ,\omega _j \right) \in {Gh_2}^{\omega}}{}\\ &\quad \left( \left( 1-\left( 1-\mu _i \right) ^{\lambda}\left( 1-\mu _j \right) ^{\lambda},{\upsilon _i}^{\lambda}{\upsilon _j}^{\lambda} \right) ,\omega _i\omega _j \right)\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega},\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \lambda \alpha _i\oplus \lambda \alpha _j,\omega _i\omega _j \right)}\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_1}^{\omega}}{\left( \lambda \alpha _i,\omega _i \right)}\oplus \bigcup_{\left( \alpha _j,\omega _j \right) \in {Gh_2}^{\omega}}{\left( \lambda \alpha _j,\omega _j \right)}\\ &={\lambda Gh_1}^{\omega}\oplus {\lambda Gh_2}^{\omega}\\ \end{aligned}\)

Hence, \(\lambda\left({Gh_1}^{\omega}\oplus{Gh_2}^{\omega}\right)={\lambda Gh_1}^{\omega}\oplus{\lambda Gh_2}^{\omega}\)

##### 3.2 The Comparison of WGHFEs

**Definition 11:** If \(\alpha =(\mu,\upsilon)\) is an IFV, then its expectation can be defined as:

\[\begin{equation*} E(\alpha)=\frac{1}{2}(\mu +1-\upsilon) \tag{1} \end{equation*}\] |

According to (1), here is the definition of score function and discrete function of *WGHFE*.

**Definition 12:** If \(Gh^{\omega}\) is a *WGHFE*, then its score function can be defined as:

\[\begin{equation*} s\left(Gh^{\omega}\right)=\sum_{\left(\alpha,\omega\right)\in Gh^{\omega}}{\omega E\left(\alpha\right)} \tag{2} \end{equation*}\] |

its discrete function can be defined as:

\[\begin{equation*} d\left( Gh^{\omega} \right) =\sqrt{\sum_{\left( \alpha ,\omega \right) \in Gh^{\omega}}{\omega \left( E\left( \alpha \right) -s\left( Gh^{\omega} \right) \right) ^2}} \tag{3} \end{equation*}\] |

**Definition 13:** Given two *WGHFEs* \({Gh_1}^{\omega}\) and \({Gh_2}^{\omega}\), according to (2) and (3), here is the comparison rules:

- (1) If \(s\left({Gh_1}^{\omega}\right)>s\left({Gh_2}^{\omega}\right)\), then \({Gh_1}^{\omega}>{Gh_2}^{\omega}\)
- (2) If \(s\left({Gh_1}^{\omega}\right)=s\left({Gh_2}^{\omega}\right)\), then:
- (2.1) if \(d\left({Gh_1}^{\omega}\right)>d\left({Gh_2}^{\omega}\right)\), then \({Gh_1}^{\omega}<{Gh_2}^{\omega}\)
- (2.2) if \(d\left({Gh_1}^{\omega}\right)=d\left({Gh_2}^{\omega}\right)\), then \({Gh_1}^{\omega}={Gh_2}^{\omega}\)

##### 3.3 The Aggregation Operators of WGHFS

**Definition 14:** If \({Gh_1}^{\omega},{Gh_2}^{\omega},\ldots ,{Gh_n}^{\omega}\) are *WGHFEs*, then:

\[\begin{aligned} \begin{aligned} &WGHFWA\left( {Gh_1}^{\omega},{Gh_2}^{\omega},\ldots ,{Gh_n}^{\omega} \right)\\ &=\overset{n}{\mathop {\oplus}_{i=1}}\left( w_i{Gh_i}^{\omega} \right) =\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_i}^{\omega}}{\left( \overset{n}{\mathop {\oplus}_{i=1}}\,w_i\alpha _i,\prod_{i=1}^n{\omega _i} \right)}\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_i}^{\omega}}{\left( \left( 1-\prod_{i=1}^n{\left( 1-\mu _i \right) ^{w_i},\prod_{i=1}^n{{\upsilon _i}^{w_i}}} \right) ,\prod_{i=1}^n{\omega _i} \right)}\\ \end{aligned} \end{aligned}\] |

is referred to as the weighted generalized hesitant fuzzy weighted averaging operator(WGHFWA), where \(w_i\) is the weight of \({Gh_i}^{\omega}\), and \(\sum_{i=1}^n{w_i}=1\).

**Definition 15:** If \({Gh_1}^{\omega},{Gh_2}^{\omega},\ldots ,{Gh_n}^{\omega}\) are *WGHFEs*, then:

\[\begin{aligned} \begin{aligned} &WGHFWG\left( {Gh_1}^{\omega},{Gh_2}^{\omega},\ldots ,{Gh_n}^{\omega} \right)\\ &=\overset{n}{\mathop {\otimes}_{i=1}}\left( w_i{Gh_i}^{\omega} \right) =\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_i}^{\omega}}{\left( \overset{n}{\mathop {\otimes}_{i=1}}\,w_i\alpha _i,\prod_{i=1}^n{\omega _i} \right)}\\ &=\bigcup_{\left( \alpha _i,\omega _i \right) \in {Gh_i}^{\omega}}{\left( \left( \prod_{i=1}^n{{\mu _i}^{w_i}},1-\prod_{i=1}^n{\left( 1-\upsilon _i \right) ^{w_i}} \right) ,\prod_{i=1}^n{\omega _i} \right)}\\ \end{aligned} \end{aligned}\] |

is referred to as the weighted generalized hesitant fuzzy weighted geometric operator(WGHFWG), where \(w_i\) is the weight of \({Gh_i}^{\omega}\), and \(\sum_{i=1}^n{w_i}=1\).

##### 3.4 Weighted Generalized Hesitant Fuzzy Preference Relation

In practical problems, decision-makers sometimes need to choose the optimal scheme by judging the preference of different schemes or describing the selection tendency, while hesitant fuzzy preference relation can describe decision-makers’ preference evaluation of different schemes in more detail when facing a large number of uncertain and complex data [16]. Based on hesitant fuzzy preference relation, here is the definition of weighted generalized hesitant fuzzy preference relations(WGHFPR) [27].

**Definition 16:** The weighted generalized hesitant fuzzy preference relation on scheme set \(C=\left\{c_1,c_2,\ldots ,c_m \right\}\) is a matrix \(\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times m}\) and \(\tilde{r}_{ij}=\Big\{\Big({\alpha_{ij}}^{\left(s\right)},{\omega_{ij}}^{\left(s\right)}\Big)\big| s=1,2, \ldots ,l\Big\}\), where \({\alpha_{ij}}^{\left(s\right)}\) is an IFV, \({\mu_{ij}}^{\left(s\right)}\) represents the degree to which scheme \(c_i\) is superior to scheme \(c_j\), \({\upsilon_{ij}}^{\left(s\right)}\) represents the degree to which scheme \(c_i\) is inferior to scheme \(c_j\), \(l\) represents the numbers of elements in each WGHFE \(\tilde{r}_{ij}\), \({\omega_{ij}}^{\left(s\right)}\) represents the weight of \({\alpha_{ij}}^{\left(s\right)}\) in WGHFE \(\tilde{r}_{ij}\), and \(\sum_{s=1}^l{{\omega_{ij}}^{\left(s\right)}}=1\); when \(i<j\), the elements in \(\tilde{r}_{ij}\) sort in ascending order and the elements in \(\tilde{r}_{ji}\) sort in descending order, \({\alpha_{ij}}^{\left(s\right)}\) is the \(s\) smallest element in \(\tilde{r}_{ij}\), and also the \(s\) biggest element in \(\tilde{r}_{ji}\).

**Definition 17:** If \(\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times m}\) is the weighted generalized hesitant fuzzy preference relation on scheme set \(C=\left\{c_1,c_2,\ldots ,c_m \right\}\), \(\tilde{R}\) is completely consistent if for any \(s\in\left\{ 1,2,\ldots ,l\right\}\) there is always [28]:

\[\begin{equation*} \begin{aligned} {\mu _{ij}}^{\left( s \right)}{\mu _{jk}}^{\left( s \right)}{\mu _{ki}}^{\left( s \right)}={\upsilon _{ij}}^{\left( s \right)}{\upsilon _{jk}}^{\left( s \right)}{\upsilon _{ki}}^{\left( s \right)} \\ \forall i,j,k\in \left\{ 1,2,\ldots ,m \right\} \end{aligned} \tag{4} \end{equation*}\] |

**Definition 18:** If \(\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times m}\) is the weighted generalized hesitant fuzzy preference relation on scheme set \(C=\left\{c_1,c_2,\ldots ,c_m \right\}\), its consistency index \(CI\left(\tilde{R}\right)\) can be defined as:

\[\begin{equation*} \begin{aligned} &CI\left( \tilde{R} \right) =\frac{1}{C_{m}^{3}}\\ &\sum_{1\le i\le j\le k\leqslant m}{\sum_{s=1}^l{\left| \begin{array}{c} w_{ij}\left( \ln {\mu _{ij}}^{\left( s \right)}-\ln {\upsilon _{ij}}^{\left( s \right)} \right) +\\ w_{jk}\left( \ln {\mu _{jk}}^{\left( s \right)}-\ln {\upsilon _{jk}}^{\left( s \right)} \right) +\\ w_{ki}\left( \ln {\mu _{ki}}^{\left( s \right)}-\ln {\upsilon _{ki}}^{\left( s \right)} \right)\\ \end{array} \right|}}\\ \end{aligned} \tag{5} \end{equation*}\] |

where \(w_{ij}\), \(w_{jk}\), \(w_{ki}\) respectively represent the weight of \(\tilde{r}_{ij}\), \(\tilde{r}_{jk}\), \(\tilde{r}_{ki}\), The smaller the value of \(CI\left(\tilde{R}\right)\), the more consistent the WGHFPR \(\tilde{R}\). The acceptable threshold is set as 0.1, if \(CI\left(\tilde{R}\right) \le 0.1\), then the WGHFPR \(\tilde{R}\) is acceptably consistent.

Sometimes the original WGHFPR \(\tilde{R}\) is not acceptably consistent and needs to be adjusted. The method of goal programming is used here to find such a WGHFPR \(\bar{R}\), which on the one hand should be as close as possible to the original WGHFPR \(\tilde{R}\), and on the other hand should meet the definition of acceptable consistency. Therefore, the absolute deviation between the two WGHFPRs can be used as the optimization goal, and the definition of acceptable consistency can be used as constraints to construct the following goal programming model.

\[\begin{aligned} &\min \sum_{i=1}^{m-1}{\sum_{j=i+1}^m{\sum_{s=1}^l{\left( \left| {u_{ij}}^{\left( s \right)}-\bar{u}_{ij}^{\left( s \right)} \right|+\left| v_{ij}^{\left( s \right)}-\bar{v}_{ij}^{\left( s \right)} \right| \right)}}}\\ &s.t.\begin{cases} \begin{aligned} &\frac{1}{C_{m}^{3}}\sum_{1\le i\le j\le k\leqslant m}{\sum_{s=1}^l{\left| \begin{array}{c} w_{ij}\left( \ln {\mu _{ij}}^{\left( s \right)}-\ln {\upsilon _{ij}}^{\left( s \right)} \right) +\\ w_{jk}\left( \ln {\mu _{jk}}^{\left( s \right)}-\ln {\upsilon _{jk}}^{\left( s \right)} \right) +\\ w_{ki}\left( \ln {\mu _{ki}}^{\left( s \right)}-\ln {\upsilon _{ki}}^{\left( s \right)} \right)\\ \end{array} \right|\leqslant 0.1}}\\ &\bar{u}_{ii}^{\left( s \right)}=\bar{v}_{ii}^{\left( s \right)}=0.5,\bar{u}_{ij}^{\left( s \right)}=\bar{v}_{ji}^{\left( s \right)}\\ & \bar{u}_{ij}^{\left( s \right)}, \bar{v}_{ij}^{\left( s \right)}\in \left[ 0,1 \right] , \bar{u}_{ij}^{\left( s \right)}+\bar{v}_{ij}^{\left( s \right)}\leqslant 1\\ &i,j=1,2,\ldots ,m;s=1,2,\ldots ,l\\ \end{aligned} \end{cases}\\ \end{aligned} \!\!\!\!\!\!\!\!\!\!\] |

By solving this model, the adjusted WGHFPR \(\bar{R}\) can be obtained, which has the smallest deviation from the original matrix and satisfies acceptable consistency.

#### 4. The Ensemble Learning Application of WGHFS

##### 4.1 Algorithm Process

The WGHFS-EL proposed in this paper is mainly divided into 7 steps, and the process is shown in Fig. 1.

- Preprocess the dataset and use the hold-out method and stratified sampling to divide the original datasets into training sets and testing sets in a 7:3 ratio.
- Use different classifiers to train and predict the training sets, and get the classification indicators.
- Use entropy weight method to get the weights of each classification indicator.
- Build WGHFPR according to the classification indicators.
- Check the consistency of WGHFPR, if the consistency index can reach the acceptable threshold, then go to step 6, otherwise return to step 4 to adjust the WGHFPR.
- Use the WGHFWA to aggregate the WGHFPR, and get the weights of each classifier.
- According to the prediction results of each classifier on the test set, combining with the weight aggregation voting, get the final classification results.

##### 4.2 Base Classifiers

To achieve the contrast effect, the paper use Support Vector Machine (SVM), Logistic Regression (LR) and Naive Bayes (NB) as the base classifiers of ensemble learning.

SVM is a classification method that predict classes based on patterns from the results of the training process. Classification is carried out with a dividing line (hyperplane) that separates the positive and negative classes. Intuitively, a good delimiter is the one with the greatest distance to the closest training data point of each class, because generally the larger the margin, the lower the generalization error of the classifier.

LR is a classification algorithm to find the relationship between dichotomous (scaled with two categories) or polychotomous (scaled with more than two categories) dependent variables with continuous or categorical independent variables. It uses the coefficient weighted linear combination of input variables to classify, and can give the corresponding class distribution estimation on any given class.

Naive Bayes is a classification method using probability and statistical methods. This algorithm works by calculating the probability by adding up the combinations and frequencies of the test data. By using Bayes theorem method, assuming the dependence of one variable on another with classification [29].

To sum up, though SVM, LR and NB have different principles, they all have the characteristics of simple structure and fast training speed, so they are suitable as base classifiers for ensemble learning.

##### 4.3 Entropy Weight Method

Entropy weight method is an objective weighting method, it calculates the information entropy of each indicator by using the information entropy formula, weights them, and finally gets the objective weight of each indicator. The main principles are as follows:

If there are \(m\) schemes, each scheme has \(n\) evaluation indicators, then there is an evaluation matrix \(R=\left(r_{ij}\right)_{m\times n}\). For any indicator \(j\), the information entropy of \(j\) can be expressed as: \(E_j=-\sum_{i=1}^m{f_{ij}\ln f_{ij}}\), while \(f_{ij}=r_{ij}/\sum_{i=1}^m{r_{ij}}\).

It can be known from the formula that the greater the dispersion of an indicator, the smaller its information entropy, the more information it contains, and the higher its weight. On the contrary, the greater the information entropy, the less information it contains and the lower its weight. Compared with traditional methods such as analytic hierarchy process, entropy weight method can avoid the subjective influence of decision-makers and get more accurate weights.

#### 5. Algorithm Verification and Analysis

##### 5.1 Datasets

To verify the practicability of WGHFS-EL, 6 datasets in UCI machine learning database are used for the classification experiment. For convenience, the datasets used in this paper are all binary datasets and preprocessed. The specific attributes of each dataset are shown in Table 1.

\[\begin{align} \left[ \begin{matrix} \left[ \begin{array}{c} \left[ \left( 0.500,0.500 \right) ,0.172 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.066 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.532,0.468 \right) ,0.172 \right] ,\\ \left[ \left( 0.471,0.529 \right) ,0.066 \right] ,\\ \left[ \left( 0.542,0.458 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.500,0.500 \right) ,0.172 \right] ,\\ \left[ \left( 0.470,0.530 \right) ,0.066 \right] ,\\ \left[ \left( 0.530,0.470 \right) ,0.762 \right]\\ \end{array} \right]\\ \left[ \begin{array}{c} \left[ \left( 0.468,0.532 \right) ,0.172 \right] ,\\ \left[ \left( 0.529,0.471 \right) ,0.066 \right] ,\\ \left[ \left( 0.458,0.542 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.500,0.500 \right) ,0.172 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.066 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.468,0.532 \right) ,0.172 \right] ,\\ \left[ \left( 0.474,0.526 \right) ,0.066 \right] ,\\ \left[ \left( 0.463,0.537 \right) ,0.762 \right]\\ \end{array} \right]\\ \left[ \begin{array}{c} \left[ \left( 0.500,0.500 \right) ,0.172 \right] ,\\ \left[ \left( 0.530,0.470 \right) ,0.066 \right] ,\\ \left[ \left( 0.470,0.530 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.532,0.468 \right) ,0.172 \right] ,\\ \left[ \left( 0.525,0.474 \right) ,0.066 \right] ,\\ \left[ \left( 0.537,0.463 \right) ,0.762 \right]\\ \end{array} \right]&\left[ \begin{array}{c} \left[ \left( 0.500,0.500 \right) ,0.172 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.066 \right] ,\\ \left[ \left( 0.500,0.500 \right) ,0.762 \right]\\ \end{array} \right]\\ \end{matrix} \right] \tag{6} \end{align}\] |

##### 5.2 Evaluation Indicators

Research shows that for unbalanced datasets, it is sometimes unreliable to use accuracy only to evaluate the classification performance of the classifiers, therefore, this paper introduces accuracy, precision and recall into the algorithm to obtain more accurate classification results.

For any binary classification problem, its results can be expressed as the confusion matrix shown in Table 2 which contain four indicators, respectively, true positive (TP), false positive (FP), true negative (TN) and false negative (FN).

According to these four indicators, the definitions of accuracy, precision and recall are as follows:

- \(Accuracy=\frac{TP+TN}{TP+FP+TN+FN}\)
- \(Precision=\frac{TP}{TP+FP}\)
- \(Recall=\frac{TP}{TP+FN}\)

In addition, in the final comparison of the effects of various classification algorithms, the paper also use Receiver Operating Characteristic (ROC) and Area Under ROC Curve (AUC) to evaluate the algorithm. In general, the closer the ROC curve is to the upper left corner, the better the classifier performance. AUC is the area enclosed by ROC curve and x axis. In general, the closer the AUC value is to 1, the better the classifier performance.

##### 5.3 Experiment Process

Take Heart dataset as an example to illustrate the process of this algorithm:

- Preprocess the dataset and use the hold-out method and stratified sampling to divide the original datasets into training sets and testing sets in a 7:3 ratio.
- Use SVM, LR and NB to train and predict the training set, the evaluation matrix which contains three indicators is shown as Table 3.
- Use entropy weight method to ensure the weights of three classification indicators as [0.172 0.066 0.762]
- Build WGHFPR matrix \(\tilde{R}\) (Eq. (6))
- Calculate \(CI\left(\tilde{R}\right)=0.083<0.1\), so the WGHFPR is considered as acceptably consistent.
- Use WGHFWA to aggregate each line of \(\tilde{R}\), and calculate its score function to get the weights of classifiers as [0.346 0.318 0.336].
- According to the weights of classifiers, the final classification results are shown in Fig. 2.

##### 5.4 Experiment Results and Comparative Analysis

To verify the practicability of WGHFS-EL, in addition to three base classifiers SVM, LR, NB, two classical ensemble learning algorithms Bootstrap aggregating (Bagging) and Adaptive Boosting (Adaboost) are also selected for comparison.

Bagging algorithm firstly conducts bootstrap sampling of the original training set for many times and puts it back to form multiple different training sets, then builds multiple homogeneous weak classifiers in parallel on training sets, and combines the prediction results of multiple classifiers with equal voting to get the final classification result.

Adaboost algorithm will firstly assigns the same weights to all training samples. In each turn of training, the samples with wrong classification results will get bigger weights, while the samples with correct classification results will get smaller weights. Then the algorithm uses all the samples with updated weights for the training of the next classifier, serially builds multiple heterogeneous weak classifiers, improves the weight of classifiers with high accuracy and reduces the weight of classifiers with low accuracy, finally gets the classification results by weighted voting.

The performance indicators of the proposed algorithm and other five classification algorithms on six datasets are shown in Table 4 and Fig. 3.

As can be seen from Table 4, among the 24 indicators in 6 datasets, the WGHFS-EL algorithm has the highest value in 13 of them. Among them, 4 indicators are the highest in the Diabetes dataset, and 9 of the remaining 11 indicators are the next highest value.

It can be known from Fig. 3 that the average values of all four indicators of the WGHFS-EL algorithm are the highest. Among them, the accuracy and recall rates are 2.5\(\%\) and 0.6\(\%\) higher than SVM, which is the highest among the base classifiers. The precision rate is 1.3\(\%\) higher than NB, which is also the highest among the base classifiers. In addition, the accuracy, precision and recall rates of WGHFS-EL are 3.0\(\%\), 2.1\(\%\) and 1.8\(\%\) higher than Adaboost, which performs best among traditional ensemble learning algorithms. In terms of AUC indicators, WGHFS-EL is also 1.2\(\%\) and 2.7\(\%\) higher than SVM and Adaboost respectively.

In the field of combining fuzzy set theory with ensemble learning, Dai proposed an ensemble learning algorithm based on intuitionistic fuzzy sets (IFS-EL). The paper selects four datasets, and conducts comparative experiments with the same base classifiers. The experiment results are shown in Fig. 4.

It can be known from Fig. 4 that except for the same accuracy of two algorithms on Sonar dataset, the accuracy of WGHFS-EL is slightly higher than IFS-EL on the other three datasets. And the average accuracy is improved by 1.3\(\%\), which means that WGHFS-EL proposed in this paper has certain practicability.

#### 6. Conclusion

Traditional fuzzy set theory is easy to lose some original information when dealing with complex uncertain information, and sometimes it may lead to wrong decisions. By using WGHFS and retaining the weighted membership degrees in the form of multiple IFVs, the loss of original information can be minimized and the correct decision can be ensured as far as possible. However, even with the advantage of expressing information more comprehensively, WGHFS has a more complex structure compared to other forms of fuzzy sets, which leads to longer running time for algorithms based on it.

In traditional classification tasks, the performance of classification algorithms is often evaluated by a single evaluation indicator. The WGHFS-EL proposed in this paper can take multiple weighting indicators into consideration and allow them to be uncertain in the form of IFVs. The comparison tests on 6 UCI datasets and 5 different classification algorithms show that the WGHFS-EL proposed in this paper not only improves the performance of the base classifiers, but also outperforms the traditional Bagging and Adaboost algorithms. It is obvious that the WGHFS-EL proposed in this paper has scientific and practical value.

Due to the mathematical form of WGHFSs, using them to store data can slow down the algorithm’s running speed. Therefore, further optimization of the algorithm structure should be carried out to accelerate its running speed. In addition, the performance of the base classifiers used in this ensemble learning algorithm is not high, and in the future, higher performance base classifiers can be considered. As future work, we will further explore the operational laws related to WGHFS and explore the possibility of improving its mathematical form. Furthermore, we will also consider applying WGHFS to convolutional neural networks or graph neural networks, combining WGHFS theory with attention mechanisms in neural networks, in order to achieve better results in areas such as object detection or recommendation algorithms.

#### References

[1] L.A. Zadeh, “Fuzzy sets,” Information and control, vol.8, no.3, pp.338-353, 1965.

CrossRef

[2] L.A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning―i,” Information sciences, vol.8, no.3, pp.199-249, 1975.

CrossRef

[3] K.T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol.20, no.1, pp.87-96, 1986.

CrossRef

[4] K.T. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol.31, no.3, pp.343-349, 1989.

CrossRef

[5] V. Torra, “Hesitant fuzzy sets,” International journal of intelligent systems, vol.25, no.6, pp.529-539, 2010.

CrossRef

[6] V. Torra and Y. Narukawa, “On hesitant fuzzy sets and decision,” 2009 IEEE international conference on fuzzy systems, pp.1378-1382, IEEE, 2009.

CrossRef

[7] S. Chen and L. Cai, “Interval-valued hesitant fuzzy sets,” Fuzzy Systems and Mathematics, vol.27, no.6, pp.38-44, 2013.

[8] G. Qian, H. Wang, and X. Feng, “Generalized hesitant fuzzy sets and their application in decision support system,” Knowledge-Based Systems, vol.37, pp.357-365, 2013.

CrossRef

[9] M. Palanikumar, K. Arulmozhi, A. Iampan, and L.J. Manavalan, “Novel possibility pythagorean cubic fuzzy soft sets and their applications,” International Journal of Innovative Computing Information and Control, vol.19, no.02, p.325-337, 2023.

[10] B. Chen, Y. Guo, X. Gao, Y. Wang, and X. Du, “Dynamic multi-attribute decision-making method with interval-valued intuitionistic fuzzy power weighted operators,” Syst. Eng. Electron, vol.41, no.4, pp.850-855, 2019.

[11] Y. Ren, X. Yuan, and L. Huang, “q-rung hesitant triangular fuzzy bm operator and its application in multiple criteria decision making,” Syst. Eng. Electron, vol.44, no.1, pp.181-191, 2022.

[12] F. Ahmad, A.Y. Adhami, B. John, and A. Reza, “A novel approach for the solution of multiobjective optimization problem using hesitant fuzzy aggregation operator,” RAIRO-Operations Research, vol.56, no.1, pp.275-292, 2022.

CrossRef

[13] J. Li, J. Ye, L.-L. Niu, Q. Chen, and Z.-X. Wang, “Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation,” Soft Computing, vol.26, no.7, pp.3129-3145, 2022.

CrossRef

[14] Z. Yang, L. Zhang, and C. Liang, “A method for group decision making with multiplicative consistent interval-valued intuitionistic fuzzy preference relation.,” IAENG International Journal of Applied Mathematics, vol.51, no.1, 2021.

[15] Y. Wang, X. Hu, and L. Wang, “Effectiveness evaluation method of constellation satellite communication system with acceptable consistency and consensus under probability hesitant intuitionistic fuzzy preference relationship,” Soft Computing, vol.26, no.22, pp.12559-12581, 2022.

CrossRef

[16] W. Zeng, D. Li, and Q. Yin, “Group decision making approach of weighted hesitant fuzzy sets,” Control and Decision, vol.34, no.3, pp.527-534, 2019.

[17] W. Zeng, D. Li, and Q. Yin, “Weighted interval-valued hesitant fuzzy sets and its application in group decision making,” International Journal of Fuzzy Systems, vol.21, pp.421-432, 2019.

CrossRef

[18] G. Ngo, R. Beard, and R. Chandra, “Evolutionary bagging for ensemble learning,” Neurocomputing, vol.510, pp.1-14, 2022.

CrossRef

[19] X. Zhu, J. Li, J. Ren, J. Wang, and G. Wang, “Dynamic ensemble learning for multi-label classification,” Information Sciences, vol.623, pp.94-111, 2023.

CrossRef

[20] N.G. Angelica and G. Dennis, “Ensemble learning for permission-based mobile malware detection in android,” ICIC Express Letters, Part B: Applications, vol.14, no.07, p.727-734, 2023.

[21] P. Singh, Y.-P. Huang, and S.-I. Wu, “An intuitionistic fuzzy set approach for multi-attribute information classification and decision-making,” International Journal of Fuzzy Systems, vol.22, pp.1506-1520, 2020.

CrossRef

[22] P. Singh and Y.-P. Huang, “A four-way decision-making approach using interval-valued fuzzy sets, rough set and granular computing: a new approach in data classification and decision-making,” Granular Computing, vol.5, pp.397-409, 2020.

CrossRef

[23] Z. Dai, K. Hu, and J. Xie, “Ensemble learning algorithm based on intuitionistic fuzzy sets,” Computer Science, vol.48, no.S1, pp.270-274, 2021.

[24] Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Trans. Fuzzy Syst., vol.15, no.6, pp.1179-1187, 2007.

CrossRef

[25] Z. Xu and R.R. Yager, “Some geometric aggregation operators based on intuitionistic fuzzy sets,” International journal of general systems, vol.35, no.4, pp.417-433, 2006.

CrossRef

[26] M. Xia and Z. Xu, “Hesitant fuzzy information aggregation in decision making,” International journal of approximate reasoning, vol.52, no.3, pp.395-407, 2011.

CrossRef

[27] Z. Xu and P. Ren, “A survey of decision making with hesitant fuzzy preference relations: Progress and prospect,” System Engineering Theory and Practice, vol.40, no.8, pp.2193-2202, 2020.

[28] J. Yu and J. Wang, “Multiplicative consistency of hesitant intuitionistic fuzzy preference relations and its applications in group decision making,” Fuzzy Systems and Mathematics, vol.34, pp.114-121, 2020.

[29] B. Satya, M.H. SJ, M. Rahardi, and F.F. Abdulloh, “Sentiment analysis of review sestyc using support vector machine, naive bayes, and logistic regression algorithm,” 2022 5th International Conference on Information and Communications Technology (ICOIACT), pp.188-193, IEEE, 2022.

CrossRef