Research on hesitant fuzzy clustering method based on fuzzy matroids

Abstract

In this paper, to improve the situation of singleness of selecting results in hesitant fuzzy set decision-making and expand the range of choices for decision makers, we construct a hesitant fuzzy set clustering algorithm combined with fuzzy matroid operation. The algorithm synthesizes the r-cut set, fuzzy shrinking matroids in the fuzzy matroids and the operational properties of the fuzzy derived matroids, the r value also is used to connect the two types of fuzzy matroids to form a clustering algorithm. Finally, we apply the algorithm to the hesitant fuzzy set decision-making of job seekers choosing recruitment websites, each recruitment website as an optional scheme is divided into three categories of excellent to inferior schemes to provide job seekers with ideas and methods for favorably selecting recruitment websites.

Keywords

Hesitant fuzzy set decision-making fuzzy matroid contraction matroid derived matroid clustering algorithm

1 Introduction

Matroid was introduced by Whitney [1] in 1935 and looked as the concept of extended linear algebra and graph theory. This theory provides great help for solving the combinatorial optimization problems. Matroid has been developed in combination with other theories over the years. For example, the expansion of matroid of rough set [2], concept of lattice [3, 4], etc. Among them, The theory of fuzzy matroid, which combines matroid and fuzzy sets, was considered as an important extension of matroid. Fuzzy set theory was first proposed by Zadeh [5] in 1965. The research on the way of expending fuzzy implications by Han et al. [6]. In 1988, the theory of fuzzy matroid was first proposed by Goetschel and Voxman in “Fuzzy Matroids” [7]. In the development of fuzzy matroid, the properties of L-fuzzy matroids and generalized fuzzy matroids was discussed by Shi [8, 9]. Li et al. [10] mainly studied and proved the tree structure of G-V fuzzy matroid. Li et al. [11] studied the related definitions and operators of intuitionistic fuzzy matroids and the relations between intuitionistic fuzzy matroids, G-V fuzzy matroids and H fuzzy matroids. Similarly, fuzzy set theory was also applied to the expression of fuzzy problems. In 2009–2010, Torra and Narukawa [12, 13] proposed the concept of hesitant fuzzy sets for the first time. Since then, scholars at home and abroad have started in-depth research on hesitant fuzzy decision-making problems.

In recent years, with the deepening of informationization of social, the application of cluster analysis in hesitant fuzzy set decision-making has attracted much attention. There already had some clustering algorithms for different fuzzy numbers. For intuitionistic fuzzy numbers, there had intuitionistic fuzzy hierarchical clustering and the netting clustering analysis of Xu [14]; for type-2 fuzzy numbers, there had interval C-means fuzzy clustering and the similarity numbers of Hwang et al. [15] and Yang et al. [16]; for hesitant fuzzy numbers, there had hesitant fuzzy clustering with correlation coefficients of Chen et al. [17], hesitant fuzzy hierarchical clustering method for HFAH based on distance matrix and it used cases and hesitant fuzzy minimum spanning tree clustering algorithm combined with graph theory of Zhang et al. [18], hesitant fuzzy K-means clustering with hierarchical clustering as initial result of Chen et al. [19]. In addition, there were some literatures on the application of fuzzy clustering model, Alghamdi et al. [20] used the fuzzy clustering method of C-means to classify the academic buildings in universities, which were three categories with different contribution to the environment, so as to improve the academic buildings with low contribution. Han et al. [21] through scaling of linguistic variables and minimizing cross-entropy principle classified the tourist destinations in different regions and analyzed the robustness of the clustering results. According to the existing garbage management regulations and the different impacts of different types of garbage on the environment, Cao et al. [22] used the fuzzy clustering analysis method to reclassify the garbage, so as to reduce the damage to the environment. Guajardo et al. [23] introduced Mahalanobis distance into the traditional distance clustering method to generalize the distance in the fuzzy system. This method could also identify the distance that could not be measured by the traditional Euclidean distance. Cai et al. [24] constructed a similarity clustering method combining reconstruction coefficient and pairwise distance, which could effectively reduce the influence of outliers. D’Urso et al. [25] proposed the robust fuzzy c-medoids clustering method for imprecise data, it could improve information gain.

In the existing results of the research of hesitant fuzzy clustering algorithm, most of the algorithms focus on the calculation of the correlation matrix and the distance between the schemes and the ideal schemes, the elements of these algorithms basically include the calculation and weight of hesitant fuzzy numbers and most of clustering results were only a few scheme sets, but there was no further comparison for the clustering results. This paper combines hesitant fuzzy information with fuzzy matroid computation, then the schemes are clustered by computing and transforming the subsets of fuzzy matroids, finally, according to the clustering results, there kinds of scheme sets meaning advantages to disadvantages can be obtained for decision makers to choose.

2 Methods

In this paper, we establish a hesitant fuzzy clustering model through the operation of fuzzy matroids.

Step 1: Data preprocessing and initial matroid establishment

Firstly, by Definition 2.1, we define the sign of hesitant fuzzy evaluation value.

Definition 2.1 [26], let X be a finite properties set, set A ={ 〈 x, h_A (x) 〉 |x ∈ X }. And h_A (x) is the set of numbers in [0,1], it is expressed as the possible membership degree of x elements about set A, h = h_A (x) is called hesitant fuzzy element.

The hesitant fuzzy numbers in this model are all the normalized numbers that length of k, expressed as h ={ t₁, t₂, ⋯ , t_k }.

According to Zeshui Xu’s [27] data processing method, there are two ways: pessimistic rule and optimistic rule. They apply to prudent decision makers and risky decision makers respectively. We obtain the normalized hesitant fuzzy numbers by pessimistic rule. Let all hesitant fuzzy numbers exchange to equal length hesitant fuzzy numbers and reorder the numbers in hesitant fuzzy numbers by value from small to large, expressed as: $\bar{h} = {t_{σ (1)}, t_{σ (2)}, \dots, t_{σ (k)}} .$

In the hesitant fuzzy decision-making, suppose there are m kinds of alternatives, expressed as: A₁,...,A_i,...,A_m and n kinds of related attributes, expressed as: C₁,...,C_j,...,C_n.

Then the normalized matrix of the hesitant fuzzy evaluation matrix can be expressed as: $M = (\begin{matrix} {\bar{h}}_{11} & \dots & {\bar{h}}_{1 j} & \dots & {\bar{h}}_{1 n} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ {\bar{h}}_{i 1} & \dots & {\bar{h}}_{ij} & \dots & {\bar{h}}_{in} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ {\bar{h}}_{m 1} & \dots & {\bar{h}}_{mj} & \dots & {\bar{h}}_{mn} \end{matrix}) = (\begin{matrix} {\bar{h}}_{1} \\ ⋮ \\ {\bar{h}}_{i} \\ ⋮ \\ {\bar{h}}_{m} \end{matrix})$

Let $h^{-} = min {t \in \bar{h} | \bar{h} = (t_{σ (1)}, t_{σ (2)}, \dots, t_{σ (k)})}$

and $h^{+} = max {t \in \bar{h} | \bar{h} \in (t_{σ (1)}, t_{σ (2)}, \dots, t_{σ (k)})}$

Then we define the fuzzy matroids based on definition 2.2 and 2.3.

Definition 2.2 [28, 29], if X is a set, then a fuzzy set μ on X is a mapping μ : X → [0, 1]. We denote the family of fuzzy sets on X by F (X). Let μ ∈ F (E), then

Support set: $supp μ = {x \in X | μ (x) > 0} .$

r-Cut set: $C_{r} (μ) = {x \in X | μ (x) ⩾ r, r \in [0, 1]} .$

R⁺ (μ) = { μ (x) |μ (x) > 0 } .

m (μ) = min R^- + (μ), $M (μ) = max R^{+} (μ) .$

$ω (X, r) = {\begin{matrix} r & x \in X \\ 0 & x \notin X \end{matrix}$ .

|μ| = ∑_x∈Xμ (x).

Definition 2.3 [29, 30], E is a finite and nonempty set and ℓ ⊆ F (E) is a nonempty family of fuzzy sets satisfying:

(1) (Hereditary property)

if μ ∈ ℓ , υ ∈ F (E) , υ ⩽ μ, then υ∈ ℓ.

(2) (Exchange property)

if μ, υ ∈ ℓ , |suppμ| < |suppυ|, then exists ω∈ ℓ, such that

μ < ω ⩽ (μ ∨ υ).

m (ω)⩾ min { m (μ) , m (υ) }.

We define the mapping of the article clustering model according to Definition 2.2. The following defines the mapping of the clustering model,

Define the mapping $μ : C \to \bar{h}, μ_{i} = {\begin{matrix} {\bar{h}}_{i 1} & x_{1} = C_{1} \\ {\bar{h}}_{i 2} & x_{2} = C_{2} \\ ⋮ & ⋮ \\ {\bar{h}}_{in} & x_{n} = C_{n} \end{matrix}$

Then, establish the initial evaluation fuzzy matroid, $\begin{matrix} M_{1} = (C, F), \\ C = (C_{1}, \dots, C_{j}, \dots C_{n}), \\ F = (μ_{1}, \dots, μ_{i}, \dots, μ_{m}), \\ calculate {| μ |}_{M_{1}} = n . \end{matrix},$

Step 2: Building systolic fuzzy matroid and taking schemes of I

Secondly, we establish the systolic fuzzy matroid to calculate schemes of I. Systolic fuzzy matroid is contraction of original matroid according to the relation conditions of r-Cut sets and the comparison results of mappings in subset families. Systolic fuzzy matroid can help us to select a class of schemes with higher fuzzy evaluation values, so we cluster and select Systolic fuzzy matroids for the first time. Then we list the conditions we need.

Let $C_{r} (μ) = {x \in C | h^{-} ⩾ r, \bar{h} \in μ}$ (1)

and the other four conditions as follows,

α, β, γ, μ ∈ F, α ⩾ γ

and define, α ⩾ γ calculation formula is $\frac{1}{n} \sum_{i = 1}^{n} \frac{\sum_{d = 1}^{k} t_{d}}{l_{{\bar{h}}_{α}}} ⩾ \frac{1}{n} \sum_{i = 1}^{n} \frac{\sum_{d = 1}^{k} t_{d}}{l_{{\bar{h}}_{γ}}}, t_{d} \in {\bar{h}}_{i},$

$l_{{\bar{h}}_{α}}, l_{{\bar{h}}_{γ}}$ are lengths of α, γ respectively.

C_r (α) , C_r (γ) ⊆ C.

C_r (α) ∩ C_r (γ) = φ.

C_r (α) ∪ C_r (γ) = C.

Let I = { μ ∈ F|μ ⩽ α, ∃ β ∈ F, $β ⩽ γ & μ \lor β \in F}$ (2) we call M₂ = (C, I) as systolic fuzzy maoid, and X₁ ={ μ|μ ∈ F - I } is the set of schemes of I.

Note: let $r = \frac{1}{mn} \sum_{j = 1}^{n} \sum_{i = 1}^{m} \frac{\sum_{d = 1}^{k} t_{d}}{k}, t_{d} \in {\bar{h}}_{ij}, \bar{h} \in μ, \forall μ \in F$ (3)

Step 3: Establishing the derived matroid and the other schemes will be divided into I and III

Thirdly, we use the induced matroid to do the second clustering. Based on the average value of the induced function, the induced matroid can select the schemes with largest number of attributes. Based on the results of the first clustering, the schemes are divided into kinds of II and III.

Related definitions of induced matroid sequences and induced functions are listed in definitions 2.4 and 2.5.

Definition 2.4 [7], let M = (E, ℓ) is a fuzzy matroid, ∀r ∈ (0, 1], define $I_{r} = {C_{r} (μ) | \forall μ \in ℓ}$ (4)

then M_r = (E, I_r) is the matroid about E. ∃0 = r₀ < r₁ < ⋯ < r_n ⩽ 1 is fundamental sequence on M, the induced matroid sequence is expressed as: $\begin{matrix} M_{{\bar{r}}_{1}} = (E, I_{{\bar{r}}_{1}}) \supset M_{{\bar{r}}_{2}} = (E, I_{{\bar{r}}_{2}}) \\ \supset \dots \dots \supset M_{{\bar{r}}_{n}} = (E, I_{{\bar{r}}_{n}}) \end{matrix}$ (5) ${\bar{r}}_{i} = \frac{(r_{i - 1} + r_{i})}{2}, 1 ⩽ i ⩽ n .$

Definition 2.5 [7], define $Σ_{M} : P (E) \to [0, 1]$ is the function of power set P (E) of set E, such that $\forall A \in P (E),$

if $A \in I_{{\bar{r}}_{1}}$ , then $Σ_{M} (A) = sup {r \in (0, 1) | μ \in ℓ, C_{r} (μ) \supseteq A}$ (6)

if $A \notin I_{{\bar{r}}_{1}}$ , then Σ_M (A) = 0. We definite function Σ_M as the set function induced from matroid M. Let $λ_{Σ} = {Σ_{M} (A) | \forall A \in P (E), Σ_{M} (A) > 0}$ (7)

According to these definitions, we list the formulas for calculating the induced functions.

Define number sequence r₁,...,r_q, let $r_{q} = \frac{r}{2} + 0.1 \times (q - 1), r_{q} ⩽ r, q = 1, 2, \dots$ (8)

We establish the induced matroid sequences: $\begin{matrix} M_{2 {\bar{r}}_{1}} = (C, I_{{\bar{r}}_{1}}) \supset M_{2 {\bar{r}}_{2}} = (C, I_{{\bar{r}}_{2}}) \\ \supset \dots \dots \supset M_{2 {\bar{r}}_{q}} = (E, I_{{\bar{r}}_{q}}) \end{matrix}$ (9) and the function Σ_M about set C, then let $Σ_{M} (A) = sup {\hat{r} \in (0, r] | \exists μ \in I, C_{\hat{r}} (μ) \supseteq A}$ (10)

Let $r^{'} = \frac{\sum Σ_{M}}{| Σ_{M} |}$ (11) use ∑Σ_M to represent the sum of all Σ_M and |Σ_M| represent the total numbers of Σ_M, we can calculate induced matroid, $M_{3} = (C, I_{r^{'}}),$ $I_{r^{'}} = {C_{r^{'}} (μ) | \forall μ \in I}$

In that, and $I_{1} = {μ | | C_{r^{'}} (μ) | ⩾ 3} .$

X₂ ={ μ|μ ∈ I₁ } is the set of schemes of I, X₃ ={ μ| ∀ μ ∈ I - I₁ } is the set of schemes of III.

Step 4: Getting comprehensive results

The last step is to get the evaluation clustering results, we list the results of the second and third steps, and the clustering results are shown as follows.

The elements in set X₁ are the first kind, them in set X₂ are the second kind, them in set X₃ are the third kind.

Now, we express the whole algorithm process through Fig. 1.

Fig. 1

Algorithm flow chat.

3 Experimental study/ numerical study

In the second part of this paper, we have constructed a complete clustering model. In this part, we choose the problem of selection of the recruitment website for job seekers to illustrate the use of this method. Let’s talk about the importance of solving this problem and preliminary establishment of the model.

In the digital age, the operation of all walks of life is closely related to the Internet, and online recruitment has gradually become an extremely important recruitment method, how to choose the recruitment website has become the primary problem for online job seekers. Many scholars have put forward some evaluation methods for this problem. From the two-way perspective of enterprises and job seekers, Dineen et al. [31] pointed out that the feedback of online recruitment was effective for job seekers; Coyle et al. [32] believed that the recruitment website could attract the attention of job seekers in form and expand the scope of website users; Cober et al. [33] pointed out that the website which focused on the recruitment process and screening process was more attractive to job seekers, etc. Most of the evaluation focused on the judgment of the influencing factors or the analysis of them, but the overall evaluation of each recruitment website is slightly weak.

We establish one clustering model combined with fuzzy matroid to study the methods to solve this kind of problems, considering the overall advantages and disadvantages of recruitment websites and the fuzziness of website evaluation. The influencing factors used in the example were the five attributes which affect the satisfaction of recruitment website constructed by Noh [34].

They are:

C₁: information quality perception,

C₂: website resource richness,

C₃: website usability,

C₄: website responsiveness,

C₅: safety and reliability.

The following ten representative recruitment websites are selected for the example:

A₁: 51job.com,

A₂: zhaopin.com,

A₃: chinahr.com,

A₄: 58.com,

A₅: chinajob.com,

A₆: liepin.com,

A₇: jobcn.com,

A₈:job5156.com,

A₉: wutongguo.com,

A₁₀: job168.com.

And three experts made fuzzy evaluation on these websites, the hesitant fuzzy evaluation values are shown in Table 1.

Table 1
The Hesitant Fuzzy Evaluation Values

C ₁ C ₂ C ₃ C ₄ C ₅

A ₁ {0.5, 0.6, 0.7} {0.7, 0.8, 0.6} {0.8, 0.6} {0.5, 0.6} {0.6, 0.7}

A ₂ {0.6, 0.7, 0.8} {0.7, 0.8} {0.6, 0.7} {0.6} {0.6, 0.7, 0.8}

A ₃ {0.4, 0.5} {0.5, 0.6, 0.7} {0.5, 0.6} {0.4, 0.5, 0.6} {0.5, 0.6}

A ₄ {0.5, 0.7} {0.7, 0.8} {0.6, 0.7, 0.8} {0.5, 0.6} {0.5, 0.6, 0.8}

A ₅ {0.5, 0.7, 0.8} {0.6, 0.7} {0.4, 0.5, 0.7} {0.5} {0.6, 0.7, 0.8}

A ₆ {0.6, 0.7} {0.5, 0.7, 0.8} {0.6} {0.6, 0.7} {0.5, 0.6, 0.7}

A ₇ {0.7, 0.8} {0.5, 0.6, 0.7} {0.6, 0.7} {0.5} {0.5, 0.6}

A ₈ {0.5, 0.6, 0.7} {0.6, 0.7} {0.5, 0.6} {0.5, 0.6} {0.6}

A ₉ {0.6, 0.7, 0.8} {0.5, 0.7} {0.6} {0.5, 0.6} {0.5, 0.7, 0.8}

A ₁₀ {0.6, 0.7} {0.6} {0.7, 0.5} {0.6} {0.5, 0.6}

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	{0.5, 0.6, 0.7}	{0.7, 0.8, 0.6}	{0.8, 0.6}	{0.5, 0.6}	{0.6, 0.7}
A ₂	{0.6, 0.7, 0.8}	{0.7, 0.8}	{0.6, 0.7}	{0.6}	{0.6, 0.7, 0.8}
A ₃	{0.4, 0.5}	{0.5, 0.6, 0.7}	{0.5, 0.6}	{0.4, 0.5, 0.6}	{0.5, 0.6}
A ₄	{0.5, 0.7}	{0.7, 0.8}	{0.6, 0.7, 0.8}	{0.5, 0.6}	{0.5, 0.6, 0.8}
A ₅	{0.5, 0.7, 0.8}	{0.6, 0.7}	{0.4, 0.5, 0.7}	{0.5}	{0.6, 0.7, 0.8}
A ₆	{0.6, 0.7}	{0.5, 0.7, 0.8}	{0.6}	{0.6, 0.7}	{0.5, 0.6, 0.7}
A ₇	{0.7, 0.8}	{0.5, 0.6, 0.7}	{0.6, 0.7}	{0.5}	{0.5, 0.6}
A ₈	{0.5, 0.6, 0.7}	{0.6, 0.7}	{0.5, 0.6}	{0.5, 0.6}	{0.6}
A ₉	{0.6, 0.7, 0.8}	{0.5, 0.7}	{0.6}	{0.5, 0.6}	{0.5, 0.7, 0.8}
A ₁₀	{0.6, 0.7}	{0.6}	{0.7, 0.5}	{0.6}	{0.5, 0.6}

We combine the example to apply the model in this paper, including four steps.

Step 1: It can be seen from Table 1 that the length of hesitant fuzzy evaluation value of this problem is different. Thus, we need to process the data so that all values have the same length. For example, in project A₁, the maximum length of the hesitant fuzzy evaluation value of all attributes is 3, so we need to extend the value whose length is less than 3, as {0.8, 0.6}, {0.5, 0.6}, {0.6, 0.7}. To expand the hesitant fuzzy values, selecting the smallest number in each corresponding value and arranging them in order. Finally, expanding {0.8, 0.6}, {0.5, 0.6}, {0.6,0.7} to {0.6, 0.6, 0.8}, {0.5, 0.5, 0.6}, {0.6,0.6, 0.7}.

The normalized evaluation matrix M is expressed as Table 2.

Table 2

The Normalized Evaluation Matrix

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	{0.5, 0.6, 0.7}	{0.6, 0.7, 0.8}	{0.6, 0.6, 0.8}	{0.5, 0.5, 0.6}	{0.6, 0.6, 0.7}
A ₂	{0.6, 0.7, 0.8}	{0.7, 0.7, 0.8}	{0.6, 0.6, 0.7}	{0.6, 0.6, 0.6}	{0.6, 0.7, 0.8}
A ₃	{0.4, 0.4, 0.5}	{0.5, 0.6, 0.7}	{0.5, 0.5, 0.6}	{0.4, 0.5, 0.6}	{0.5, 0.5, 0.6}
A ₄	{0.5, 0.5, 0.7}	{0.7, 0.7, 0.8}	{0.6, 0.7, 0.8}	{0.5, 0.5, 0.6}	{0.5, 0.6, 0.8}
A ₅	{0.5, 0.7, 0.8}	{0.6, 0.6, 0.7}	{0.4, 0.5, 0.7}	{0.5, 0.5, 0.5}	{0.6, 0.7, 0.8}
A ₆	{0.6, 0.6, 0.7}	{0.5, 0.7, 0.8}	{0.6, 0.6, 0.6}	{0.6, 0.6, 0.7}	{0.5, 0.6, 0.7}
A ₇	{0.7, 0.7, 0.8}	{0.5, 0.6, 0.7}	{0.6, 0.6, 0.7}	{0.5, 0.5, 0.5}	{0.5, 0.5, 0.6}
A ₈	{0.5, 0.6, 0.7}	{0.6, 0.6, 0.7}	{0.5, 0.5, 0.6}	{0.5, 0.5, 0.6}	{0.6, 0.6, 0.6}
A ₉	{0.6, 0.7, 0.8}	{0.5, 0.5, 0.7}	{0.6, 0.6, 0.6}	{0.5, 0.5, 0.6}	{0.5, 0.7, 0.8}
A ₁₀	{0.6, 0.6, 0.7}	{0.6, 0.6, 0.6}	{0.5, 0.5, 0.7}	{0.6, 0.6, 0.6}	{0.5, 0.5, 0.6}

$\begin{matrix} μ_{1} = {\begin{matrix} {0.5, 0.6, 0.7} & x_{1} = C_{1} \\ {0.6, 0.7, 0.8} & x_{2} = C_{2} \\ ⋮ & ⋮ \\ {0.6, 0.6, 0.7} & x_{5} = C_{5} \end{matrix} \\ ⋮ \\ μ_{10} = {\begin{matrix} {0.6, 0.6, 0.7} & x_{1} = C_{1} \\ {0.6, 0.6, 0.6} & x_{2} = C_{2} \\ ⋮ & ⋮ \\ {0.5, 0.5, 0.6} & x_{5} = C_{5} \end{matrix} \end{matrix}$

Those are the ten mappings in fuzzy set, the set of mappings is expressed as: $F = (μ_{1}, \dots, μ_{10})$

then establish evaluation matroid, $M_{1} = (C, F) .$

Step 2: Firstly, we get the value of r through formula (3). In our experimental example, according to the hesitant fuzzy evaluation values, k value is 3, m value is 10 and n value is 5.

Take the r-value, $r = \frac{1}{50} \times \begin{matrix} (0.6 + 0.7 + 0.67 + 0.53 + \dots \\ + 0.6 + 0.57 + 0.6 + 0.53) \end{matrix} = 0.60$

Then we can calculate the r-Cut set of each mapping according to formula (1) and select several r-Cut sets satisfying the four conditions 1), 2), 3), 4). And we can get the projects set I through formula (2) to be filtered in the Step 3.

Calculate C_r (μ_i) , i = 1, ⋯ , 10,

we can get $C_{r} (μ_{5}) = {x_{2}, x_{5}}, C_{r} (μ_{6}) = {x_{1}, x_{3}, x_{5}}$

when the four conditions are satisfied, $α = C_{6}, γ = C_{5},$

get the set I, get $I = {μ_{1}, μ_{3}, μ_{5}, μ_{6}, μ_{7}, μ_{8}, μ_{9}, μ_{10}},$

establishing one contractive fuzzy matroid, $M_{2} = (C, I),$

the schemes set of I is obtained as follows, $X_{1} = {μ_{2}, μ_{4}},$

including the two schemes A2 and A4.

Step 3: We can get the r sequence from formula (8).

The result of calculation is $r_{1} = 0.3, I_{{\bar{r}}_{1}} = 0.15 .$

According to the value of r1, family of sets A can be obtained, and the power numbers of A would be got by formula (10). Then the mean value of the power numbers can be used as r threshold value to select the projects with no less than 3 excellent attributes.

The available set is a subset family of all attributes and Σ_M = (0.6, 0.6, 0.6, ⋯ , 0.5, 0.5, 0.5, 0.5, 0.5) $r^{'} = \frac{\sum Σ_{M}}{| Σ_{M} |} = 0.57$ , get the r’-Cut set and C_r′ (μ) , μ ∈ I, the result set is I₁ ={ μ₁, μ₆, μ₁₀ }.

The schemes of II are A1, A6, A10. The schemes of III are A3, A5, A7, A8, A9.

Step 4: Summing up the projects sets from the two clustering processes in Step 2 and Step 3.

The model results show that, when choosing the recruitment website, the choices should be sorted according to the following preferences: $\begin{matrix} {A_{2}, A_{4}} ≻ {A_{1}, A_{6}, A_{10}} ≻ \\ {A_{3}, A_{5}, A_{7}, A_{8}, A_{9}} . \end{matrix}$

The selection method is more profitable method for job seekers.

4 Discussion

In this paper, from the results calculated by the clustering model, we can see that the preferred recruitment websites for job seeker are 58.com and zhaopin.com, the second type of recruitment websites are 51job.com, liepin.com and job168.com, the third type of recruitment websites which are slightly worse choices for job seekers are chinahr.com, chinajob.com, jobcn.com, job5156.com and wutongguo.com.

In this part, we will compare the results of the algorithm case in this paper with existing ranking websites results.

Now, we compare the results with the ranking websites. According to the ranking of most recruitment websites on November 20, 2020, as shown in Table 3.

Table 3
Comprehensive ranking of the most recruitment websites

Ranking Website Ranking Website

1 51job.com 6 job168.com

2 zhaopin.com 7 jobcn.com

3 58.com 8 chinahr.com

4 liepin.com 9 wutongguo.com

5 job5156.com 10 chinajob.com

Ranking	Website	Ranking	Website
1	51job.com	6	job168.com
2	zhaopin.com	7	jobcn.com
3	58.com	8	chinahr.com
4	liepin.com	9	wutongguo.com
5	job5156.com	10	chinajob.com

Data Sources: top.chinaz.com.

The ranking from top.chinaz.com in Table 3 is based on the numbers of Alexa weekly ranking, Baidu weight, PR and anti-chains, it can reflect the popularity and importance of the websites from the two aspects of website traffic and website quality up to November 20, 2020.

In Table 3, except that the position of 51job.com and 58.com, job5156.com and job168.com are contrary to the results of clustering model in this paper, the difference between the ranking and calculation results of these four websites may be due to different calculation properties and evaluation methods, the ranking of other websites basically corresponds to the calculation results of the model. This shows that the algorithm model in this paper is applicability.

This clustering model is different from the traditional ranking in that it classifies and sorts the alternatives by fuzzy matroid. As present, many scholars have studied the hesitant fuzzy decision-making problems with multiple alternatives, either single ranking results or clustering results of similar schemes, but there are few sorted clustering results. The algorithm in this paper can be applied to the selection of multiple schemes, it is easy to calculate and the clustering results are concise. But the clustering threshold r value of this algorithm is too dependent on the evaluation value, which would lead to high subjectivity of the calculation results, and it does not distinguish the weight of evaluation attributes, the work of increasing attribute weight and objectivity of evaluation can be considered in the next stage.

5 Conclusions

Based on the calculation of contractive matroid and induced matroid in fuzzy matroid theory, this paper establishes a clustering model. The model can divide the schemes into three categories, and provide decision-makers with choice ideas from the optimal arrangement. At the end of the paper, taking how job seekers choose recruitment websites as an example, we use this clustering model and make the final results. The calculation process of the model shows that the model focuses on the integrity of the attributes in each scheme, in the process of scheme screening, the r value used by the first cluster is related to the evaluation number, the second clustering is related to the number of qualified attributes in the scheme. the clustering method is more comprehensive in numerical aspect.

However, the data processing in the model, as well as the correlation between the attributes and consideration of the attribute weight, these two aspects are slightly deficient and still need to be improved, there is also a lack of data analysis in the model, we will strengthen the study of a lot kinds of algorithm analysis and the applications for different purposes. Especially in data analysis methods, now many methods of data analysis have been researched. For example, several better methods for this paper, Haghbin et al. [35] researched the asymptotic distribution of sample proportion applied to confidence interval and hypothesis. Mahmoudi et al. [36, 37] studied the comparison method of regression models of in dependent data sets and the goodness of fit test based on the multiple testing. In the future, we will improve the model by learning analysis methods suitable for hesitant fuzzy data and focus on the ranking function of hesitant fuzzy matroid and the reasonable attribute weight in the clustering method of hesitant fuzzy matroid, and improve the theoretical system of hesitant fuzzy matroid.

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Research on hesitant fuzzy clustering method based on fuzzy matroids

Abstract

Keywords

1 Introduction

2 Methods

Table 3 Comprehensive ranking of the most recruitment websites Ranking Website Ranking Website 1 51job.com 6 job168.com 2 zhaopin.com 7 jobcn.com 3 58.com 8 chinahr.com 4 liepin.com 9 wutongguo.com 5 job5156.com 10 chinajob.com

References

Table 3
Comprehensive ranking of the most recruitment websites

Ranking Website Ranking Website

1 51job.com 6 job168.com

2 zhaopin.com 7 jobcn.com

3 58.com 8 chinahr.com

4 liepin.com 9 wutongguo.com

5 job5156.com 10 chinajob.com