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Hesitant Fuzzy Linguistic Agglomerative Hierarchical Clustering Algorithm and Its Application in Judicial Practice

: The fuzzy clustering algorithm has become a research hotspot in many fields because of its better clustering effect and data expression ability. However, little research focuses on the clustering of hesitant fuzzy linguistic term sets (HFLTSs). To fill in the research gaps, we extend the data type of clustering to hesitant fuzzy linguistic information. A kind of hesitant fuzzy linguistic agglomerative hierarchical clustering algorithm is proposed. Furthermore, we propose a hesitant fuzzy linguistic Boole matrix clustering algorithm and compare the two clustering algorithms. The proposed clustering algorithms are applied in the field of judicial execution, which provides decision support for the executive judge to determine the focus of the investigation and the control. A clustering example verifies the clustering algorithm’s effectiveness in the context of hesitant fuzzy linguistic decision information.


Introduction
Clustering refers to dividing a collection of physical or abstract objects into different classes or clusters according to some specific standards, which makes the objects in the class similar to each other but different from the objects outside the class [1]. Unlike traditional clustering with an either-or nature, fuzzy clustering is a mathematical method to classify things with both-and characteristics. The fuzzy clustering method has been widely used in multi-attribute decision-making problems such as image processing [1], risk assessment [2] and energy planning [3].
Ruspini [4] proposed the concept of fuzzy division and introduced fuzzy set theory into cluster analysis. Subsequently, scholars have successively proposed fuzzy clustering analysis methods, such as a transitive closure algorithm based on fuzzy equivalence relation [2,5,6], a fuzzy c-means clustering algorithm [7][8][9] and a hierarchical clustering algorithm [3,10]. The hierarchical clustering algorithm is a representative and essential clustering method. Hierarchical clustering decomposes or gathers the data set layer by layer by a particular method until all the classified data in the last layer meet the requirements. According to the different decomposition principles or aggregation principles, hierarchical clustering can be divided into agglomerative and divisive methods, and the agglomerative hierarchical clustering algorithm is more widely used [11].
It is known from the literature that the existing research on fuzzy data types of clustering mainly involves intuitionistic fuzzy information [12,13], type-2 fuzzy information [14,15] and hesitant fuzzy information [16,17]. There are few studies on the clustering of hesitant fuzzy linguistic information. In addition, uncertainty analysis methods such as intuitionistic fuzzy sets, soft multi-set topology and hesitant fuzzy linguistic term sets (HFLTSs) have been widely used in multi-attribute decision-making problems, such as optimal recommendation [18][19][20], risk assessment [21,22] and pattern recogni-tion [23]. The application of the HFLTSs clustering method in the field of judicial execution is rare.
In practical decision-making, the hesitant fuzzy linguistic term sets (HFLTSs) have significant advantages in expressing experts' qualitative judgment. For example, in judicial practice, many attributes of many persons subjected to execution cannot be described quantitatively. However, they can only be assessed qualitatively. The possibility of the hidden property of the persons subjected to execution is mostly presented in the form of linguistic expressions or sentences. At this time, the HFLTSs can scientifically represent the executive judge's qualitative decision information. Therefore, it is of practical significance to study the clustering analysis method for processing HFLTSs.
Based on the above analysis, the main contributions are as follows. (1) The data type of clustering to hesitant fuzzy linguistic information is extended. (2) A hesitant fuzzy linguistic agglomerative hierarchical (HFLAH) clustering algorithm and hesitant fuzzy linguistic Boole matrix (HFLBM) clustering algorithm are proposed, and the two clustering algorithms are compared and analyzed. (3) The fuzzy cluster analysis method is applied in judicial execution, which helps the executive judge determine the critical points of investigation and control scientifically and objectively and realizes the subject's classification of investigation and control under the hesitant fuzzy linguistic environment.
The structure of this paper is as follows. Section 2 reviews some of the related concepts, distance and similarity measures and integrated operations of the HFLTSs. In Section 3, the agglomerative hierarchical clustering algorithm is introduced in detail. On this basis, a kind of HFLAH clustering algorithm is proposed. Section 4 introduces the identification principle and realization steps of the person's concealed property behavior subjected to execution. An example of a cluster analysis of the person subjected to execution based on hesitant fuzzy linguistic decision information is given. The validity of the HFLAH clustering algorithm is verified. Furthermore, an HFLBM clustering algorithm is proposed, and the two clustering algorithms are compared and analyzed. The conclusion is given in Section 5.

Distance and Similarity Measures of Hesitant Fuzzy Linguistic Decision Information
Definition 6 [29]: For a given linguistic term set ={ =-, , 1, 0,1, , } Since the relationship between the distance and similarity measures satisfies   H is defined as (10) is converted to the weighted Hamming distance: If 2 λ = , then Equation (10) is converted to the weighted Euclidean distance:

Hesitant Fuzzy Linguistic Bonferroni Mean Operator
The image of the function ( ) t g s is Since ( ) g τ is monotone, the inverse is obtained as follows: The image of the inverse function defined in Equation (18) is obtained as follows: Definition 11 [31]:

Agglomerative Hierarchical Clustering Algorithm
Agglomerative hierarchical clustering is a bottom-up strategy that treats each object as a cluster first, then merges these atomic clusters into larger and larger clusters until all objects are in a cluster or some terminal condition is satisfied. Most of the hierarchical clustering methods belong to this class, and they just differ in the definition of the similarity between clusters. The steps of the traditional agglomerative hierarchical clustering algorithm are as follows [32]:

Hesitant Fuzzy Linguistic Agglomerative Hierarchical Clustering Algorithm
This paper proposes a hesitant fuzzy linguistic agglomerative hierarchical (HFLAH) clustering algorithm based on the above analysis. The algorithm flow is shown in Figure 1, and the specific implementation steps are as follows: • Step 1: Supposing that the scheme set of a certain multi-attribute problem is , then the attribute set is as a class, and use Equation (11) or (12) to calculate the distance ( , ), ( , )

Cluster Analysis of the Person Subjected to Execution Based on Hesitant Fuzzy Linguistic Decision Information
With the substantial increase in litigation cases, some useful legal documents cannot be implemented, which is referred to by people as difficult to execute. Unenforceable legal documents are called a mere scrap of paper. A person subjected to execution who can repay the debt but refuses to do so is called a deadbeat or laolai (a derogatory term in Chinese used for diehard debtors). The degree of realizing the legitimate rights and interests of the winning parties needs to be improved urgently. In many cases, the prominent phenomenon of hidden properties and limited human resources have become the prominent characteristics of problematic execution.
In judicial practice, the executive judge's measures to the person subjected to execution mainly rely on the experience of handling cases, which is often restricted by subjective and objective factors such as cognitive bias, information asymmetry and personal bias. Many attributes of the executed person can only be evaluated qualitatively. The evaluation results of the possibility of concealing property by the person subjected to execution are presented in the form of linguistic expressions or statements. Avoiding the drawbacks of empirical judicature and scientifically representing the hesitation and qualitative decision information of executive judges is of great significance to improving decision-making.

Screening the Path of Concealment of Property by the Person Subjected to Execution
The quantitative probability of the concealed property was estimated based on the number of samples of laolai and non-laolai in the category to be identified. The specific screening steps for the person subjected to execution for concealing property are as follows: • Step 1: Suppose that the executive judge needs to judge the possibility of concealing property for n persons subjected to execution to be identified is obtained in turn. The greater the probability value is, the greater the concealed property's possibility, and the greater the corresponding control force is. According to the reasonable allocation of execution resources, complete the classification of the person under execution.

Numerical Example
Assume that there are four samples . According to the linguistic term set S and the context-free grammar H G , the executive judge uses the linguistic expression to give the corresponding evaluation information, as shown in Table 1.
To calculate the distance between two HFLTSs more accurately, the HFLTSs with fewer elements add linguistic terms by Equation ( • Step 2: Take each executed person •  Figure 2. The clustering results show the effectiveness of the concealed property behavior evaluation system and the clustering algorithm, especially when the number of cases was large, as scientific cluster analysis of the person subjected to execution could effectively help the executive judge objectively determine the focus of the investigation and control, which is conducive to improving the execution's quality and efficiency.

Contrastive Analysis
Inspired by the intuitionistic fuzzy clustering algorithm based on the Boole matrix proposed by Xu [33], we further propose the hesitant fuzzy linguistic Boole matrix (HFLBM) clustering algorithm. The HFLBM clustering algorithm was used to analyze the above examples, and the result of the HFLBM clustering algorithm was compared with that of the HFLAH clustering algorithm. Some basic definitions follow: The clustering process is shown in Figure 3. The clustering results of the two clustering algorithms are shown in Table 2.  , the classification results of the two algorithms were different. The reason for this is that the HFLAH clustering algorithm was based on the distance measure of the hesitant fuzzy linguistic term sets and the integrated operation of the hesitant fuzzy linguistic information. In contrast, the HFLBM clustering algorithm was based on the similarity measure of the hesitant fuzzy linguistic term sets and the Boole matrix. The clustering principles of the two algorithms are different. Both of the two clustering algorithms have a small amount of computation and cannot lose the decision information. Therefore, both of them can effectively cluster the hesitant fuzzy linguistic data.

Conclusions
To avoid the drawbacks of empirical judicature and improve the quality and efficiency of execution, this paper applied the clustering idea of birds of a feather flock together to the field of judicial execution. It proposed a method for the evaluation of the concealed property behavior of a person subjected to execution. Given the executive judge's fuzzy uncertainty in screening the property behavior involved in concealment, the executive judge's qualitative decision information is represented by the hesitant fuzzy linguistic term sets. The data type of clustering is extended to the hesitant fuzzy linguistic information, and an HFLAH clustering algorithm and HFLBM clustering algorithm are proposed. The evaluation method's effectiveness and the clustering algorithm were verified by the process of clustering analysis of the executors under the background of hesitant fuzzy linguistic decision information. The hesitant fuzzy linguistic clustering algorithms effectively avoided losing decision information and improved the clustering accuracy and decision quality. The cluster analysis results provided decision support for the executive judge to scientifically and objectively determine the focus of the investigation and control, reduce the judicial cost and have specific practical significance and application value.
Due to the limitation of the objective conditions, the sample size of the subjects used for clustering analysis in this paper was small. Only the running process of the clustering algorithm was demonstrated. The research conclusion may be relatively one-sided, and the research results can be further improved on the premise of a sufficient data sample size in the future.