Computational Intelligence and Neuroscience

Volume 2019, Article ID 5370763, 10 pages

https://doi.org/10.1155/2019/5370763

## Study on Hesitant Fuzzy Information Measures and Their Clustering Application

^{1}Institute of Intelligence Engineering and Mathematics, Liaoning Technical University, Fuxin 123000, China^{2}School of Economics and Management, Huainan Normal University, Huainan 232038, China

Correspondence should be addressed to Jin-hui Lv; moc.qq@633656953

Received 23 September 2018; Revised 7 January 2019; Accepted 20 January 2019; Published 3 March 2019

Academic Editor: Paolo Gastaldo

Copyright © 2019 Jin-hui Lv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

At present, research on hesitant fuzzy operations and measures is based on equal length processing, and an equal length processing method will inevitably destroy the original data structure and change the data information. This is an urgent problem to be solved in the development of hesitant fuzzy sets. Aiming at solving this problem, this paper firstly defines a hesitant fuzzy entropy function as the measure of the degree of uncertainty of hesitant fuzzy information and then proposes the concept of hesitant fuzzy information feature vector. The hesitant fuzzy distance measure and similarity measure are studied based on the information feature vector. Finally, the hesitant fuzzy network clustering method based on similarity measure is given, and the effectiveness of our algorithm through a numerical example is illustrated.

#### 1. Introduction

Torra and Narukawa [1, 2] extended fuzzy sets [3] to hesitant fuzzy sets (HFSs) because they found that, under a group setting, it is difficult to determine the membership of an element to a set due to doubts between a few different values. For example, two DMs discuss the membership degree of into . One wants to assign 0.4 and the other 0.6, and they cannot persuade with each other; thus the membership degrees of into can be represented by {0.4, 0.6}. This is obviously different from fuzzy number 0.4 (or 0.6) and the intuitionistic fuzzy number (0.4, 0.6). Therefore, hesitant fuzzy sets can better simulate the hesitant preferences of decision-makers. Since it was put forward, the hesitant fuzzy set has received extensive attention from scholars at home and abroad. The main research work is concentrated in the following aspects: (1) research on various measures in the hesitant fuzzy environment [4–10]; (2) research on the integration operator of hesitant fuzzy information [11–16]; and (3) the expansion of hesitant fuzzy set theory [17–22].

It should be pointed out that the present researches on the operation, sorting, and various measures of hesitant fuzzy sets require that the hesitant fuzzy elements have the same length. In practical application, the length of hesitant fuzzy element is different. The method proposed in [2] is adding some elements to a shorter hesitant fuzzy element, making it equal to another hesitant fuzzy element, or repeating their elements in order to obtain two series with the same length [23]. Obviously, these methods will destroy the original data structure and change the data information. How to overcome the shortcomings has become an urgent problem to be solved in the development process of hesitant fuzzy sets.

Clustering is a basic technique, which is often utilized in a primary step of analyzing unlabeled data with the goal of summarizing structural information [24]. In practical applications, the clustering data are mostly uncertain or fuzzy. To solve the problem of data clustering in different fuzzy environments, fuzzy clustering algorithms [25], intuitionistic fuzzy clustering algorithms [26], and 2-type fuzzy clustering algorithms [27] have been proposed. However, in the group of decision-making environment, the decision information is more suitable to express hesitant fuzzy sets, and the algorithm mentioned above is not suitable for handling the clustering problem of this type of information. If the fuzzy logic is used to handle it, generally take the average value of preference information that are provided by experts or can take the minimum range containing all of the preference information, that is, convert the hesitant fuzzy information into interval value information for processing. This method of data processing is bound to change the original preference information that provided by experts; as a result, the research of clustering problem under the hesitant fuzzy information has a certain scientific significance. One of the advantages of applying the hesitant fuzzy set is that clustering hesitant and vague information permits us to find patterns among hesitant fuzzy data. At present, the clustering researches under the hesitant fuzzy environment are still at the its initial stage, and Chen et al. [28] used the correlation coefficient of hesitant fuzzy set to construct hesitant fuzzy relationship matrix and then conducted hesitant fuzzy clustering analysis based on the relation of equivalence. In order to obtain an equivalence relation matrix, a fuzzy relation matrix needs to be iterated continuously, which not only loses information but also has a large amount of calculation [29]. Due to the existence of uncertainty for the similarity measure of samples, leading to the clustering, results were not precise enough and the divided categories were inconsistent with the fact. In [4], the hesitant fuzzy similarity measure formula based on distance was proposed. The measurement is inconsistent with the facts sometimes, and the resolution is not high enough; in the literature [29], a hesitant fuzzy clustering method based on agglomerative hierarchical clustering [30] was proposed. This method needs to use a hesitant fuzzy average operator to calculate the clustering center repeatedly, and the calculation amount is large; in the literature [31], a hesitant fuzzy clustering algorithm based on minimal spanning tree was proposed. The distance of hesitant fuzzy set used in this method is put forward based on the literature [4], which also has the shortcoming of low resolution and sometimes inconsistent with the fact; in the literature [32], from the point of view of information theory, hesitant fuzzy relative entropy and symmetric interactive entropy are proposed, a new kind of hesitant fuzzy similarity degree is proposed, which is combined with the idea of TOPSIS, and a hesitant fuzzy clustering method is proposed based on the traditional netting clustering method. The premise of all the above methods in the measurement and operation is that the data are equal in length, which is not satisfied by the hesitant fuzzy set. Therefore, it is necessary to add artificial elements for equal length processing, and the processed data will inevitably change the original data information and affect the clustering results.

Based on the above analysis, this paper firstly proposes the concept of hesitant fuzzy entropy function and hesitant fuzzy information feature vector, aiming at solving the problem of processing data of hesitant fuzzy set, sorting, and various measures in the study of different lengths. Furthermore, the hesitant fuzzy uncertainty measure, distance measure, and similarity measure are studied. Finally, based on the similarity measure and the traditional network clustering method, the network clustering method for hesitant fuzzy information is given. And then we illustrate its effectiveness via numerical examples.

#### 2. Preliminary

*Definition 2.1 [1, 2]. *Let be a fixed set; a hesitant fuzzy set (HFS) on is represented by a function that when applied to , it returns a subset of [0, 1], which can be expressed by a mathematical symbol:where is a set of some values in , denoting the possible membership degrees of the element to the set . is called the hesitant fuzzy number or hesitant fuzzy element. If it does not cause confusion, it can be abbreviated as . The hesitant fuzzy number can be expressed in more detail as . Among which, denotes the number of elements in a hesitant fuzzy number . Obviously, when , the hesitant fuzzy set degenerates into the traditional fuzzy set.

*Definition 2.2 [1, 2]. *Set as a given nonempty set, then is the complement of the hesitant fuzzy set , among whichDistance measure and similarity measure are important research contents in fuzzy set theory and have a wide application background. In the literature [4], the axiomatic definitions of distance and similarity measure of hesitant fuzzy sets are given.

*Definition 2.3 [4]. *Sets *A*, *B* be the two hesitant fuzzy sets defined on , and then the distance measure between *A* and *B* satisfies the following conditions:(1)(2), if and only if (3)

*Definition 2.4 [4]. *Sets *A*, *B* be the two hesitant fuzzy sets defined on , and then the similarity measure between *A* and *B* satisfies the following conditions:(1)(2), if and only if (3)Definition 2.3 is proposed to facilitate the use of distance measures to define similarity measures. In practice, the distance can only be satisfied with .

#### 3. A New Kind of Hesitant Fuzzy Entropy

Entropy is the measurement of the degree of uncertainty of information, and it has always been an important research object in uncertainty decision analysis. A new hesitant fuzzy entropy measure function is proposed by analyzing the shortcomings in the current research results on hesitant fuzzy entropy.

*Definition 3.1. *Assign the hesitant fuzzy element , where is the number of element in the hesitant fuzzy element, and recordwhere represents the fuzzy degree of the hesitant fuzzy element and represents the hesitant degree of the hesitant fuzzy element . Then the real valued function on the hesitant fuzzy element can be expressed by a binary function , if the meets the following conditions:(1) if and only if and (2) if and only if or (3) and , and (4)Then, can be called as a hesitant fuzzy entropy function.

##### 3.1. Interpretation and Analysis

(1), . Then indicates that is a clear set, then the entropy is 0.(2)When , since , , then it can get , and it is concluded that the domain of the entropy function is because the entropy function is concave increase with respect to and , and the maximum value of is 1 when or is obtained; that is, when or , the uncertainty reaches the maximum. is completely contradictory information, and is completely fuzzy information; in both cases, the uncertainty is maximized and in line with intuitive judgment.(3)It ensures that the entropy function is concavely increased with respect to fuzziness and hesitation degree, conforms to human cognitive characteristics, and improves discrimination.(4)Fuzziness and hesitancy have the same effect on entropy.

Based on the above analysis, function obviously satisfies the above conditions in Definition 3.1, so it can be regarded as an entropy function. For example, if , then ; if , then ; and if , , then , , where . The above judgment results are consistent with the intuition.

*Property 3.1. *Set hesitant fuzzy element , when ; the hesitant fuzzy element degenerates into a fuzzy number, and the entropy of fuzzy value is .

*Proof. *(1), that is, or , where is a clear set.(2)According to condition (2) or because when , , so , that is, .(3)According to condition (3), it is known that increases monotonously with respect to , so when is closer to 0.5, the larger the is, the larger the entropy of the fuzzy value is.The property 3.1 indicates that the fuzzy entropy is a special case of the hesitant fuzzy entropy function, and the hesitant fuzzy entropy function can also be applied to the fuzzy set.

In order to illustrate the advantage of the entropy function proposed in this paper in measuring uncertainty, the following is compared with the existing entropy formula: at present, the common formulas of hesitating fuzzy entropy include the entropy formula proposed by Xu and Xia and the entropy formula proposed by Farhadinia, in whichwhere indicates the number of elements in a hesitant fuzzy number and indicates the element of the largest *i*th in the hesitant fuzzy number where is strictly monotonically decreasing function, which may get , , , ; (, where indicates the number of elements contained in the fuzzy number ).

Set hesitant fuzzy number , , , , , , , and . The entropy formula proposed by Xu and Xia and the entropy formula proposed by Farhadinia are compared with the entropy function proposed in this paper. The results are shown in Table 1.

Because the entropy formula proposed by Farhadinia only considers fuzziness and neglects the influence of hesitancy, the result is quite different from that of the method proposed in this paper and the method proposed by Xu. It is not difficult to find from the above table that the method proposed in this paper is obviously higher in the discrimination than that proposed by Xu, and the comparison result is close to it, and the individual results are inconsistent. For example, ; however, according to the method presented in this paper, the result is ; this is because the starting point is inconsistent and the hesitant fuzzy entropy proposed by Xu requires that the number of elements contained in the two pairs be equal and that the elements should be artificially added when the number of elements is different. Therefore, the proposed method is bound to deviate from intuitive judgment for the comparison of entropy of two hesitant fuzzy numbers with a different number of elements contained. The entropy measure function proposed in this paper not only considers the influence of fuzziness on the entropy value but also considers the effect of hesitation degree on the entropy value, which can more reasonably depict the uncertainty degree of the hesitation fuzzy number, so the result is more consistent with our intuition.