Developing an Enterprise Diagnostic Index System Based on Interval-Valued Hesitant Fuzzy Clustering

Abstract

Global economic integration drives the development of dynamic competition. In a dynamic competitive environment, the ever-changing customer demands and technology directly affect the leadership of the core competence of enterprises. Therefore, assessing the performance of enterprises in a timely manner is necessary to adjust business activities and completely adapt to new changes. Enterprise diagnosis is a scientific tool for judging the development status of enterprises, and building a scientific and rational index system is the key to enterprise diagnosis. Considering the large number of enterprise diagnostic indicators and the high similarity among indicators, this study proposes a selection method for enterprise diagnostic indicators based on interval-valued hesitant fuzzy clustering by comparing the existing indicator systems. First, enterprise organizations are considered as the starting point. Through the key analysis of relevant indicators of domestic and foreign enterprise diagnosis, enterprise diagnosis candidate indicators are constructed from three aspects, namely enterprise performance, employee health, and social benefit. In view of the ambiguity and inconsistency of expert judgment, this study proposes an interval-valued hesitant fuzzy set based on the characteristics of hesitant fuzzy sets and interval-valued evaluation. For improving the interval-valued hesitant fuzzy entropy function, an interval-valued hesitant fuzzy similarity measurement formula considering information features is designed to avoid the problem of data length and improve the degree of identification among indicators. Then, the similarity, equivalence, and truncation matrices are constructed, and the interval-valued hesitant fuzzy clustering method is used to eliminate redundant indicators with repeated information. The availability of the proposed method is illustrated via an example, and the key indicators in the enterprise diagnostic index system are found. Finally, the advantages of the proposed method are discussed using comparative analysis with existing methods. A rational and comprehensive enterprise diagnostic index system was constructed. The system can be used as a scientific basis for diagnosing the development of enterprises and providing an objective and effective reference.

1. Introduction

As the economic system of the market continues to improve, the development environment of enterprises is becoming competitive, making enterprise diagnosis an important means for enterprises to achieve sustainable development. To perform effective enterprise diagnosis, some experts and scholars have constructed a multidimensional quantitative enterprise diagnostic index system based on life science and medical theory using the analogy method based on enterprise bionics and organizational health theory [1,2,3,4]. Enterprise diagnosis is an advanced and scientific management method for consulting experts to thoroughly analyze the enterprise through various advanced methods, identify the problems of the enterprise and its causes, and customize effective strategies to improve the management ability and realize the sustainable development of the enterprise [5,6]. A large number of diagnostic indicators help to truly and comprehensively reflect the actual situation of the enterprise, and simultaneously provide data for decision making. However, striving for comprehensiveness leads to too many indicators, which results in a large amount of repeated information, thereby making analysis and calculation difficult and affecting the accuracy of diagnostic results [7]. Improper enterprise diagnosis methods will cause irreversible shocks and risks, bring significant property losses to the enterprise, and even damage the prospects for sustainable growth and development. Therefore, to evaluate and improve the operational efficiency of enterprises, screening diagnostic indicators is crucial to identify the development status of enterprises and build a rational indicator system using the obtained indicators.

Since the enterprise diagnostic index system involves various factors and content [8], the distinction between enterprise diagnostic indicators is low, indicating a certain degree of ambiguity. Fuzzy clustering is a mathematical method to describe and classify objective things according to certain requirements (characteristics, relationship, and similarity) from the perspective of fuzzy sets. Fuzzy clustering has been successfully applied to different fields in recent years, such as healthcare [9], assessing the quality of academic journals [10], wireless sensor networks [11], the common bicycle relocation problem [12], and image segmentation [13,14,15]. Commonly used fuzzy clustering methods include the C-means fuzzy clustering method, direct clustering method, and transitive closure algorithm [16]. The transitive closure algorithm can be particularly used to mine a large amount of uncertain information [17]. The more redundant indexes the diagnostic index system contains, the more chaotic the information reflected by the diagnostic results will be. Through fuzzy clustering, which is based on transitive closure, the diagnostic indexes with high correlation and the same information can be classified into one category. This means different indicators in different categories can reflect different data characteristics, which ensures that the information reflected by the filtered indicators is not repeated and makes a comprehensive diagnosis of the business activities of the enterprise. Therefore, this study will use the transitive closure algorithm to perform fuzzy clustering for screening important enterprise diagnostic indicators.

A comprehensive and effective enterprise diagnosis is performed by several scholars and experts from related fields based on their own experience and knowledge. However, these experts have different judgments and experiences, and will hold different opinions about a diagnostic indicator. This is referred to as a group decision problem. During the decision-making process, disagreement among individuals in the group is a common problem. To overcome this challenge, Torra [18] defined hesitant fuzzy sets (HFS), which consist of a comprehensive set of feature membership functions with indeterminate and disordered values representing group opinion. However, in the actual decision-making process, decision makers face difficulty in expressing opinions with precise values owing to the fuzzy uncertainty of their cognition or the complexity of objects. Considering that the description of opinions with interval values can better reflect the ambiguity of subjective judgments, Chen et al. [19] proposed the concept of interval-valued hesitant fuzzy sets (IVHFS), which characterized the membership degree of elements as the number of different possible interval values in a given collection. IVHFS consider the hesitancy, ambiguity, and validity of information, which correctly fits the human description of problems and expressions, particularly in group decision making. IVHFS can map the opinions of multiple experts into multiple ranges of evaluation information, thereby improving the practicability and flexibility of the descriptive information, and ensuring the integrity of the expert evaluation information. Currently, IVHFS has been successfully applied in various fields, such as green supplier selection [20], steam turbine fault diagnosis [21], project manager recruitment [22], smart refrigerator service systems [23], and M&A target selection problems [24].

Table 1. Methods for enterprise diagnosis.

Reference	Method	Advantage	Disadvantage
[1,2,3,4]	Analogy method	Comprehensive information and multiple dimensions	Poor discrimination
[9,10,11,12,13,14,15]	Fuzzy clustering	Comprehensive information and the indicators are easy to classify and calculate, and the filtered indicators are not repeated	Difficult to deal with individual differences of opinion in group expert groups
[18]	Hesitant fuzzy sets	Group opinions can be more comprehensively expressed	Cognitive uncertainty and difficulty for decision makers to express opinions with precise values
[19]	Interval-valued hesitant fuzzy set	Comprehensive information and the membership degree of elements can be characterized as several different possible interval values	Calculation is highly complex and time consuming

summarizes the various methods used in the enterprise diagnosis, and the current methods have some shortcomings in the construction of enterprise diagnostic indicators.

In this study, IVHFS is used to express the experts’ opinions on enterprise diagnostic indicators and integrate the opinions of most experts. The present study proposes an interval-valued hesitant fuzzy clustering method based on transitive closure to construct an enterprise diagnostic index system. The contributions of this paper are four-fold:

• Based on the theory of organizational health [25] and through literature review, this paper contracts an enterprise diagnostic index system from the three aspects of “enterprise performance, employee health, and social benefits”, and finds a more appropriate method for scientific and effective diagnosis of enterprise development.
• Based on the interval-valued hesitant fuzzy element (IVHFE) suitable for group decision making, the interval-valued hesitant fuzzy entropy function is improved, and a new interval-valued hesitant fuzzy similarity measurement function is proposed.
• The method proposed not only effectively measures the degree of information uncertainty and retains all the opinions of the group experts but also eliminates the problem of data length.
• This study uses the proposed interval-valued hesitant fuzzy clustering method to eliminate redundant information and construct a scientific and rational enterprise diagnostic index system.
• An example is provided to illustrate the effectiveness of the proposed method. The work description and data processing techniques in the example can help other researchers to carry out similar research.

The remainder of this article is organized as follows. The selection and design of enterprise diagnostic indices are presented in Section 2. Section 3 introduces the related concepts and properties of interval-valued hesitant ambiguity. Section 4 presents a novel interval-valued hesitant fuzzy entropy function and compares it with existing methods, thereby illustrating the advantages of the proposed system. Section 5 describes the selection method and implementation steps of the key indicators of the enterprise diagnosis system. Section 6 provides an application and a comparative analysis with other state-of-the-art methods to verify the feasibility and strengths of the proposed method. Conclusions are summarized in Section 7.

2. Enterprise Diagnostic Index Design

The openness and dynamics of the economic environment require enterprises to constantly exchange materials with the outside world in the process of participating in the social division of labor. The nature and form of these material exchanges determine the different degrees of impact on enterprise health problems, for example, enterprise investment and financing, technical exchange and cooperative research, material purchase and product sales [25]. Therefore, diagnosing an enterprise is a complex process. In this process, diagnostic experts often assume important responsibilities and need to screen and analyze a large amount of information based on basic data, historical data, and actual conditions of the enterprise operations. With a large amount of information as the basis, the experts design the final diagnostic program through scientific diagnostic methods. Therefore, to conduct an effective enterprise diagnosis, a rational diagnostic index system must be established.

Before the 1990s, people subconsciously believed that “a profitable enterprise is a healthy enterprise”; that is, enterprise health emphasizes the financial success of the enterprise [26]. In the 1990s, people gradually realized the importance of employees’ health and enterprise health, and believed that a healthy enterprise could not be separated from financial success and employee health [27]. In the 21st century, due to the complexity of the living environment and the intensification of market competition, enterprises are faced with high employee requirements. Moreover, corporate health should consider the interests of stakeholders, such as customers, communities, and the environment, while considering financial performance and employee health [28,29]. Generally, the goal of an enterprise is to make profits, and enterprise performance is the most important aspect to reflect the development of an enterprise. Employees are an important part of the enterprise. Therefore, the health of employees, such as satisfaction and physical health, can reflect the cohesion of employees and is an important indicator in the development of the enterprise. In the process of enterprise development, enterprises will provide products and services to the society, employment opportunities, taxes, etc., which have social benefits. In addition, social benefits reflect the overall development of the enterprise. Since the enterprise diagnostic index system involves various factors and the factors to be considered in the selection of diagnostic indicators are complex, by evaluating the available research results [30,31,32,33,34], the present study divides the enterprise diagnostic index system into three aspects: enterprise performance, employee health, and social benefits. Accordingly, 18 second-level indicators and 94 third-level indicators are listed in

Table 2. Candidate indicators for enterprise diagnosis.

First-Level Indicators	Second-Level Indicators		Third-Level Indicators
Enterprise performance $D_1$	General financial situation	Financial structure analysis $d_{1–1}$	Assets and liabilities ratio $d_{1–1–1}$; equity ratio $d_{1–1–2}$; current assets to fixed assets ratio $d_{1–1–3}$; the percentages of current liabilities and long-term liabilities in total assets $d_{1–1–4}$
		Profitability $d_{1–2}$	Sales margin $d_{1–2–1}$; gross profit margin $d_{1–2–2}$; return on assets (ROA) $d_{1–2–3}$; asset net interest rate $d_{1–2–4}$; Return on owner’s equity $d_{1–2–5}$; cost–profit margin $d_{1–2–6}$; working capital index $d_{1–2–7}$
		Solvency $d_{1–3}$	Current ratio $d_{1–3–1}$; quick ratio $d_{1–3–2}$; cash-to-liability ratio $d_{1–3–3}$; assets and liabilities ratio $d_{1–3–4}$; equity ratio $d_{1–3–5}$; long-term liability ratio $d_{1–3–6}$; interest coverage ratio $d_{1–3–7}$
		Asset management status $d_{1–4}$	Business cycle $d_{1–4–1}$; inventory turnover $d_{1–4–2}$; owner’s equity turnover $d_{1–4–3}$; total asset turnover $d_{1–4–4}$; working capital turnover $d_{1–4–5}$
		Fundraising and investment $d_{1–5}$	Cash-to-debt ratio $d_{1–5–1}$; cash flow adequacy ratio $d_{1–5–2}$; inflow and outflow ratio of financing activities $d_{1–5–3}$; inflow and outflow ratio of investment activities $d_{1–5–4}$
		Cost $d_{1–6}$	Main business cost-to-income ratio $d_{1–6–1}$; operating expense profit ratio $d_{1–6–2}$; management expense profit ratio $d_{1–6–3}$; financial expense profit ratio $d_{1–6–4}$
		Listed company $d_{1–7}$	Price/earnings (P/E) ratio $d_{1–7–1}$; dividend per share $d_{1–7–2}$; dividend coverage multiple $d_{1–7–3}$; net assets per share $d_{1–7–4}$; price-to-book (P/B) ratio $d_{1–7–5}$
	Growth $d_{1–8}$		Asset growth rate $d_{1–8–1}$; sales revenue growth rate $d_{1–8–2}$; growth rate of net fixed assets $d_{1–8–3}$; capital accumulation rate $d_{1–8–4}$
	Safety	Solvency risk $d_{1–9}$	Current ratio $d_{1–9–1}$; quick ratio $d_{1–9–2}$; assets and liabilities ratio $d_{1–9–3}$; earned interest multiple $d_{1–9–4}$
		Business risk $d_{1–10}$	Degree of financial leverage (DFL) $d_{1–10–1}$; degree of operating leverage (DOL) $d_{1–10–2}$; degree of total leverage (DTL) $d_{1–10–3}$
		Asset risk $d_{1–11}$	Non-performing loan (NPL) ratio $d_{1–11–1}$; asset loss rate $d_{1–11–2}$
	Innovation $d_{1–12}$		Research and development (R&D) expenses $d_{1–12–1}$; development expenditure $d_{1–12–2}$; R&D intensity $d_{1–12–3}$; R&D capabilities $d_{1–12–4}$; patent grant rate $d_{1–12–5}$; globalization $d_{1–12–6}$; influence $d_{1–12–7}$
Employee health $D_2$	Subjective satisfaction $d_{2–1}$		Task assignment satisfaction $d_{2–1–1}$; work environment satisfaction $d_{2–1–2}$; ability to improve satisfaction $d_{2–1–3}$; salary and benefit satisfaction $d_{2–1–4}$; employee management satisfaction $d_{2–1–5}$; clarity of career goals $d_{2–1–6}$; smooth career development $d_{2–1–7}$; effectiveness of career development coaching $d_{2–1–8}$; sense of belonging to the organization $d_{2–1–9}$
	Objective health $d_{2–2}$		Sickness absence rate $d_{2–2–1}$; sickness absence duration $d_{2–2–2}$; labor intensity $d_{2–2–3}$; employee work safety protection $d_{2–2–4}$
Social benefit $D_3$	Customer $d_{3–1}$		Product and service cost performance ratio $d_{3–1–1}$; standard of after-sales service $d_{3–1–2}$; product return rate $d_{3–1–3}$; authenticity of product advertisements $d_{3–1–4}$; product satisfaction $d_{3–1–5}$; product safety and quality $d_{3–1–6}$; product loyalty $d_{3–1–7}$
	Social environment $d_{3–2}$		Employment contribution rate $d_{3–2–1}$; proportion of charitable donations $d_{3–2–2}$; government policy participation $d_{3–2–3}$; relief for vulnerable groups $d_{3–2–4}$; tax payment $d_{3–2–5}$; community building spending $d_{3–2–6}$; sponsorship of education $d_{3–2–7}$; soundness of the social legal system $d_{3–2–8}$; social responsibility report qualification rate $d_{3–2–9}$
	Natural environment $d_{3–3}$		Resource consumption per unit product $d_{3–3–1}$; investment in environmental protection equipment $d_{3–3–2}$; environmental governance research and development expenses $d_{3–3–3}$; recycling system for old products $d_{3–3–4}$; environmental management system application effect $d_{3–3–5}$
	Business partner $d_{3–4}$		Brand value $d_{3–4–1}$; credit to seller $d_{3–4–2}$; loyalty to suppliers $d_{3–4–3}$; brand awareness $d_{3–4–4}$

.

3. Relevant Concepts and Characteristics of Interval-Valued Hesitant Fuzzy Sets

With the complexity of decision making problems, the uncertainty of the objective world, and the limitations of human cognition, it is difficult for decision makers to express their opinions precisely in a group decision-making process. Since Zadeh [35] first tried to use fuzzy sets to handle uncertain information, fuzzy set theory has developed rapidly. However, because its membership is defined as a specific and precise value in the interval [0, 1], the traditional fuzzy sets encountered some drawbacks when dealing with decision making problems; that is, they cannot effectively reflect decision making information. Therefore, some scholars have gradually expanded the fuzzy sets, such as the intuitionistic fuzzy sets [36], interval-valued intuitionistic fuzzy sets [37], HFS [18] and IVHFS [19]. The membership value of IVHFS is a set of values in several intervals, which retains the characteristics of HFS and the practicability of interval number expression. It provides a precise expression from the decision maker’s viewpoint and retains all the fuzzy information given by the decision maker. Furthermore, using IVHFS is more rational than conventional methods to deal with uncertain information [19]. The definition of IVHFS is as follows:

Definition 1.

Let any nonempty subset be X, and$\tilde{E}=\left\{\langle x,{\tilde{h}}_{\tilde{E}}\rangle |x\in X\right\}$be IVHFS, where${\tilde{h}}_{\tilde{E}}\left(x\right)$, a of unequal interval numbers in the range [0, 1], is the set of possible memberships of x in$\tilde{E}$for X.$\tilde{h}={\tilde{h}}_{\tilde{E}}$is an IVHFE, and${\tilde{h}}_{\tilde{E}}\left(x_i\right)=\left\{\tilde{\gamma }|\tilde{\gamma }\in {\tilde{h}}_{\tilde{E}}\left(x_i\right)\right\}$is the basic unit of IVHFS, where$\tilde{\gamma }=\left[{\tilde{\gamma }}^L,{\tilde{\gamma }}^U\right]$is an interval number and${\tilde{\gamma }}^L=\inf \tilde{\gamma }$and${\tilde{\gamma }}^U=\sup \tilde{\gamma }$are the lower and upper bounds of$\tilde{\gamma }$.

To facilitate the comparison of the size of different IVHFE, the algorithm of IVHFE is given below:

Definition 2.

Assuming X is a nonempty subset, three IVHFEs defined on X, namely$\tilde{h}=\left\{\tilde{\gamma }=\left[{\tilde{\gamma }}_{}^L,{\tilde{\gamma }}_{}^U\right]|{\tilde{\gamma }}_{}\in {\tilde{h}}_{}\right\}$,${\tilde{h}}_1=\left\{{\tilde{\gamma }}_1=\left[{\tilde{\gamma }}_1^L,{\tilde{\gamma }}_1^U\right]|{\tilde{\gamma }}_1\in {\tilde{h}}_1\right\}$, and${\tilde{h}}_2=\left\{{\tilde{\gamma }}_2=\left[{\tilde{\gamma }}_2^L,{\tilde{\gamma }}_2^U\right]|{\tilde{\gamma }}_2\in {\tilde{h}}_2\right\}$, satisfy the following rules of calculation:

(a) 

${\tilde{h}}^c=\left\{\left[1-{\tilde{\gamma }}^L,1-{\tilde{\gamma }}^U\right]|\tilde{\gamma }\in \tilde{h}\right\}$

(b) 

${\tilde{h}}_1\cup {\tilde{h}}_2=\left\{\max \left({\tilde{\gamma }}_1^L,{\tilde{\gamma }}_2^L\right),\max \left({\tilde{\gamma }}_1^U,{\tilde{\gamma }}_2^U\right)|{\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2\right\}$

(c) 

$\lambda \tilde{h}=\left\{\left[1-{\left(1-{\tilde{\gamma }}^L\right)}^{\lambda },1-{\left(1-{\tilde{\gamma }}^U\right)}^{\lambda }\right]|\tilde{\gamma }\in \tilde{h}\right\},\lambda >0$

(d) 

${\tilde{h}}_1\oplus {\tilde{h}}_2=\left\{{\tilde{\gamma }}_1^L+{\tilde{\gamma }}_2^L-{\tilde{\gamma }}_1^L\times {\tilde{\gamma }}_2^L,{\tilde{\gamma }}_1^U+{\tilde{\gamma }}_2^U-{\tilde{\gamma }}_1^U\times {\tilde{\gamma }}_2^U|{\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2\right\}$

Definitions 1 and 2 illustrate that it is difficult to compare any two IVHFS. For this, Quirós et al. [38] presented a comparison method for IVHFE as follows:

Definition 3.

Let any IVHFE be$\tilde{h}=\left\{\tilde{\gamma }=\left[{\tilde{\gamma }}_{}^L,{\tilde{\gamma }}_{}^U\right]|{\tilde{\gamma }}_{}\in {\tilde{h}}_{}\right\}$; then, its score function$M\left(\tilde{h}\right)$and the exact function$H\left(\tilde{h}\right)$are defined as

$$M\left(\tilde{h}\right)=\frac{1}{\#\tilde{h}}{\displaystyle {\sum }_{\tilde{\gamma }\in \tilde{h}}\left[\frac{{\tilde{\gamma }}^L+{\tilde{\gamma }}^U}{2}\right]}$$

(1)

$$H\left(\tilde{h}\right)=\frac{1}{\#\tilde{h}}{\displaystyle {\sum }_{\tilde{\gamma }\in \tilde{h}}\left({\tilde{\gamma }}^U-{\tilde{\gamma }}^L\right)}$$

(2)

where$\#\tilde{h}$is the number of interval values in$\tilde{h}$.

Let there be two IVHFEs ${\tilde{h}}_1$ and ${\tilde{h}}_2$, and $\#{\tilde{h}}_1$ and $\#{\tilde{h}}_2$ are the number of interval values in ${\tilde{h}}_1$ and ${\tilde{h}}_2$, respectively, then:

(a)

If $M\left({\tilde{h}}_1\right)<M\left({\tilde{h}}_2\right)$, then ${\tilde{h}}_1\le {\tilde{h}}_2$;

(b)

If $M\left({\tilde{h}}_1\right)=M\left({\tilde{h}}_2\right)$ and $H\left({\tilde{h}}_1\right)<H\left({\tilde{h}}_2\right)$, then ${\tilde{h}}_1\ge {\tilde{h}}_2$;

(c)

If $M\left({\tilde{h}}_1\right)=M\left({\tilde{h}}_2\right)$, $H\left({\tilde{h}}_1\right)=H\left({\tilde{h}}_2\right)$, and $\#{\tilde{h}}_1<\#{\tilde{h}}_2$, then ${\tilde{h}}_1\le {\tilde{h}}_2$.

4. A New Interval-Valued Hesitant Fuzzy Entropy Function

Entropy is an effective tool for measuring the degree of information uncertainty, and it is also a popular research direction for many scholars in the field of uncertain decision making. Although some experts have conducted corresponding discussions and research on the construction of interval-valued hesitant fuzzy entropy, they assumed that the number of elements in the IVHFE is the same [39]. Usually, the number of elements in the IVHFE is not equal, and if the elements are added subjectively, it will easily affect the accuracy of the calculation results and increase the complexity of the calculation [40]. Additionally, the decision maker’s hesitation and ambiguity are important factors reflecting the uncertainty of fuzzy numbers [41]. To solve the shortcomings of the current interval-valued hesitant fuzzy entropy function, this study defines a new interval-valued hesitant fuzzy measurement method as follows.

Definition 4.

Let an IVHFE be$\tilde{h}=\left\{\tilde{\gamma }=\left[{\tilde{\gamma }}_{}^L,{\tilde{\gamma }}_{}^U\right]|{\tilde{\gamma }}_{}\in {\tilde{h}}_{}\right\}$and$\#\tilde{h}$is the number of elements in the IVHFE. The entropy function$E:IVHF\to \left[0,1\right]$used on the IVHFE$\tilde{h}$can be expressed as a binary function $E\left(x,y\right)=x^2+y^2$:

$$x=\frac{1}{2\#\tilde{h}}{\displaystyle \sum _{i=1}^{\#\tilde{h}}\left(1-2\left|{\gamma }_i^L-0.5\right|+1-2\left|{\gamma }_i^U-0.5\right|\right)}$$

(3)

$$y=\{\begin{array}{ll}\frac{1}{\#\tilde{h}\left(\#\tilde{h}-1\right)}{\displaystyle \sum _{i,j=1,i<j}^{\#\tilde{h}}\left(\left|{\gamma }_i^L-{\gamma }_j^L\right|+\left|{\gamma }_i^U-{\gamma }_j^U\right|\right)},& \#\tilde{h}>1\\ y=0,& \#\tilde{h}=1\end{array}$$

(4)

In the equation, x is the ambiguity of $\tilde{h}$ and y is the hesitation degree of $\tilde{h}$.

According to the axiomatic definition of hesitant fuzzy entropy [35], an axiomatic definition of interval-valued hesitant fuzzy entropy is presented as follows.

If $E\left(x,y\right)=x^2+y^2$ satisfies the following four conditions, $E\left(x,y\right)$ is an interval-valued hesitant fuzzy entropy function:

(a)

$E\left(x,y\right)=0$, if and only if $x=0$ and $y=0$;

(b)

$\frac{\partial E}{\partial x}>0$ and $\frac{\partial E}{\partial y}>0$, $\frac{\partial E}{\partial x^2}>0$ and $\frac{\partial E}{\partial y^2}>0$;

(c)

$E\left(x,y\right)=1$, if and only if $\left(x,y\right)=\left(1,0\right)$ or $\left(x,y\right)=\left(0,1\right)$;

(d)

$E\left(x,y\right)=E\left(y,x\right)$;

To prove that $E\left(x,y\right)=x^2+y^2$ is the entropy of IVHFE $\tilde{h}$, it must be proven that $E\left(x,y\right)=x^2+y^2$ satisfies the four conditions above.

(a)

$y=0\leftrightarrow \#\tilde{h}=1,x=0\leftrightarrow \tilde{h}=\left\{\left[0,0\right]\right\}\vee \tilde{h}=\left\{\left[1,1\right]\right\}$, then $x=0\wedge y=0\leftrightarrow \tilde{h}=\left\{\left[0,0\right]\right\}\vee \tilde{h}=\left\{\left[1,1\right]\right\}$. $\tilde{h}=\left\{\left[0,0\right]\right\}$ or $\tilde{h}=\left\{\left[1,1\right]\right\}$ means that $\tilde{h}$ is a clear set and the entropy $E\left(\tilde{h}\right)=E\left(x,y\right)=0$.

(b)

$\frac{\partial E}{\partial x}=2x,\frac{\partial E}{\partial y}=2y,\frac{\partial E}{\partial x^2}=2,\frac{\partial E}{\partial y^2}=2$. Since $x\ge 0,y\ge 0$, then $\frac{\partial E}{\partial x}>0$ and $\frac{\partial E}{\partial y}>0$, $\frac{\partial E}{\partial x^2}>0$ and $\frac{\partial E}{\partial y^2}>0$ are satisfied.

(c)

If $\#\tilde{h}>1$,

$$\begin{array}{cc}\hfill x& =\frac{1}{2\#\tilde{h}}{\displaystyle \sum _{i=1}^{\#\tilde{h}}\left(1-2\left|{\gamma }_i^L-0.5\right|+1-2\left|{\gamma }_i^U-0.5\right|\right)}\hfill \\ & \le \frac{1}{2\#\tilde{h}\left(\#\tilde{h}-1\right)}{\displaystyle \sum _{i,j=1,i<j}^{\#\tilde{h}}\left[\left(1-2\left|{\gamma }_i^L-0.5\right|+1-2\left|{\gamma }_i^U-0.5\right|\right)+\left(1-2\left|{\gamma }_j^L-0.5\right|+1-2\left|{\gamma }_j^U-0.5\right|\right)\right]}\hfill \\ & \le \frac{2}{\#\tilde{h}\left(\#\tilde{h}-1\right)}-\frac{1}{\#\tilde{h}\left(\#\tilde{h}-1\right)}{\displaystyle \sum _{i,j=1,i<j}^{\#\tilde{h}}\left[\left(\left|{\gamma }_i^L-0.5\right|+\left|{\gamma }_i^U-0.5\right|\right)+\left(\left|{\gamma }_j^L-0.5\right|+\left|{\gamma }_j^U-0.5\right|\right)\right]}\hfill \end{array}$$

$$\begin{array}{cc}y& =\frac{1}{\#\tilde{h}\left(\#\tilde{h}-1\right)}{\displaystyle \sum _{i,j=1,i<j}^{\#\tilde{h}}\left(\left|{\gamma }_i^L-{\gamma }_j^L\right|+\left|{\gamma }_i^U-{\gamma }_j^U\right|\right)}\hfill \\ & =\frac{1}{\#\tilde{h}\left(\#\tilde{h}-1\right)}{\displaystyle \sum _{i,j=1,i<j}^{\#\tilde{h}}\left(\left|\left({\gamma }_i^L-0.5\right)+\left(0.5-{\gamma }_j^L\right)\right|+\left|\left({\gamma }_i^U-0.5\right)+\left(0.5-{\gamma }_j^U\right)\right|\right)}\hfill \end{array}$$

Then, $x+y\le 1$, and the domain of entropy function $E\left(x,y\right)$ is $\left\{\left(x,y\right)|x\ge 0,y\ge 0,x+y\le 1\right\}$; since x and y belong to concave increase at $E\left(x,y\right)$,$\left(x,y\right)=\left(1,0\right)\vee \left(x,y\right)=\left(0,1\right)\to E\left(x,y\right)=1$.

(d)

Since $E\left(x,y\right)=x^2+y^2$ and $E\left(y,x\right)=y^2+x^2$, it is obvious that $E\left(x,y\right)=E\left(y,x\right)$ is true.

Based on the abovementioned analysis, $E\left(x,y\right)=x^2+y^2$ can be used as an interval-valued hesitant fuzzy entropy function. To illustrate the superiority of the proposed method, a comparative analysis of the proposed entropy function is conducted to illustrate its advantages in measuring uncertainty.

Let IVHFE be $\tilde{h}=\left\{\tilde{\gamma }=\left[{\tilde{\gamma }}_{}^L,{\tilde{\gamma }}_{}^U\right]|{\tilde{\gamma }}_{}\in {\tilde{h}}_{}\right\}$ and $\#\tilde{h}$ is the number of elements in IVHFE. The entropy mapping is $E_i:IVHF\to \left[0,1\right]$ and the entropy $E\left(\tilde{h}\right)$ of IVHFE is defined as follows.

(a)

The entropy formula defined in [39].

$$E_1\left(\tilde{h}\right)=\frac{1}{\left(\sqrt{2}-1\right)\#\tilde{h}}{\displaystyle \sum _{i=1}^{\#\tilde{h}}\left[\sin \frac{\left({\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U\right)\pi }{4}+\cos \frac{\left({\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U\right)\pi }{4}-1\right]}$$

$$E_2\left(\tilde{h}\right)=\frac{-1}{\#\tilde{h}\ln 2}{\displaystyle \sum _{i=1}^{\#\tilde{h}}\left[\frac{{\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U}{2}\ln \frac{{\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U}{2}+\left(1-\frac{{\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U}{2}\right)\ln \left(1-\frac{{\tilde{\gamma }}_i^L+{\tilde{\gamma }}_{\#h-i+1}^U}{2}\right)\right]}$$

(b)

The entropy formula defined in [42].

$$E_3\left(\tilde{h}\right)=1-\frac{1}{\#\tilde{h}}{\displaystyle \sum _{i=1}^{\#\tilde{h}}{\left(\left|{\tilde{\gamma }}_i^L-0.5\right|+\left|{\tilde{\gamma }}_{\#h-i+1}^U-0.5\right|\right)}^2}$$

Set the interval-valued hesitant fuzzy numbers ${\tilde{h}}_1=\left\{\left[0.1,0.2\right],\left[0.2,0.3\right]\right\}$,${\tilde{h}}_2=\left\{\left[0.2,0.4\right],\left[0.3,0.4\right],\left[0.5,0.6\right]\right\}$, ${\tilde{h}}_3=\left\{\left[0.3,0.5\right],\left[0.5,0.6\right],\left[0.8,0.9\right]\right\}$,${\tilde{h}}_4=\left\{\left[0.2,0.3\right],\left[0.7,0.9\right]\right\}$, ${\tilde{h}}_5=\left\{\left[0.4,0.6\right],\left[0.5,0.7\right]\right\}$, ${\tilde{h}}_6=\left\{\left[0.2,0.5\right],\left[0.6,0.8\right],\left[0.7,0.8\right]\right\}$. The entropy formulas defined in [39,42] are compared with that of the present study, and the results are listed in

Table 3. Interval-valued hesitant fuzzy entropy comparison.

	${\tilde{\mathit{h}}}_1$	${\tilde{\mathit{h}}}_2$	${\tilde{\mathit{h}}}_3$	${\tilde{\mathit{h}}}_4$	${\tilde{\mathit{h}}}_5$	${\tilde{\mathit{h}}}_6$
$E_1\left(\tilde{h}\right)$	0.605	0.734	0.720	0.947	0.963	0.760
$E_2\left(\tilde{h}\right)$	0.722	0.936	0.868	0.996	0.993	0.951
$E_3\left(\tilde{h}\right)$	0.630	0.913	0.820	0.695	0.980	0.830
$E\left(\tilde{h}\right)$	0.170	0.566	0.534	0.505	0.650	0.431

.

The entropy formulas defined in [39,42] only consider ambiguity and ignore hesitation; therefore, the results are quite different from those in the present study (Table 3). The degree of discrimination of the proposed method is obviously higher than that of the methods proposed in [39,42]. This is because the hesitant fuzzy entropy function of the interval value has a concave increasing relationship with the degree of hesitation and ambiguity, and adapting to the cognitive characteristics of decision makers can improve discrimination. In addition, ref. [39] is only applicable to the comparison of two IVHFEs with the same number of elements. If the method is used to compare two IVHFEs with different number of elements, the result will deviate from intuitive judgment. The proposed entropy measure function considers ambiguity and hesitation, which can more reasonably reflect the uncertainty of IVHFS; thus, the result is more in line with the characteristics of human intuition.

5. The Selection Method of the Key Indicators of the Enterprise Diagnosis System

Table 2 illustrates that the preselected diagnostic indicators at all levels have different degrees of influence on the diagnosis of enterprise status and the boundaries between them are unclear, displaying ambiguity. Fuzzy clustering is a method of classifying objective things by establishing a fuzzy clustering similarity relationship based on the attributes, relationship, similarity, and distance among things [43]. However, irrational expression of expert group opinions is a disadvantage in the selection of enterprise diagnostic indicators via traditional fuzzy clustering, and describing the fuzzy nature of the objective world in a more detailed manner is difficult [44]. Therefore, the study paper proposes an interval-valued hesitant fuzzy clustering method, which first uses IVHFS to characterize individual opinions in the expert group, and then uses the transitive closure algorithm to perform fuzzy clustering to select key indicators for enterprise diagnosis.

5.1. Fuzzy Similarity Matrix

IVHFE similarity measurement is an essential component of IVHFS and plays a key role in decision making research in the context of IVHFS. IVHFE has been widely used in cluster analysis and multi-attribute decision making [39,45]. Although some scholars have discussed the measurement of interval-valued hesitant fuzzy similarity, one should assume that the number of IVHFEs is the same and the IVHFEs need to be arranged sequentially [39,45], which easily affects the accuracy of the calculation results. Furthermore, for an IVHFE $\tilde{h}=\left\{\tilde{\gamma }=\left[{\tilde{\gamma }}_{}^L,{\tilde{\gamma }}_{}^U\right]|{\tilde{\gamma }}_{}\in {\tilde{h}}_{}\right\}$, the most important information is the value of $\tilde{\gamma }$ and the amount of uncertainty. Therefore, the present study proposes an interval-valued hesitant fuzzy similarity measurement method based on the $\tilde{\gamma }$ value and uncertainty of interval-valued hesitant information. This method does not require the subjective addition of elements in IVHFE, nor does it need the sequential arrangement of IVHFE.

Definition 5.

Suppose${\tilde{h}}_{\alpha }=\left\{{\tilde{\gamma }}_{\alpha }=\left[{\tilde{\gamma }}_{\alpha }^L,{\tilde{\gamma }}_{\alpha }^U\right]|{\tilde{\gamma }}_{\alpha }\in {\tilde{h}}_{\alpha }\right\}$and${\tilde{h}}_{\beta }=\left\{{\tilde{\gamma }}_{\beta }=\left[{\tilde{\gamma }}_{\beta }^L,{\tilde{\gamma }}_{\beta }^U\right]|{\tilde{\gamma }}_{\beta }\in {\tilde{h}}_{\beta }\right\}$are two IVHFEs,$\#{\tilde{h}}_{\alpha }$and$\#{\tilde{h}}_{\beta }$are the number of elements in IVHFE${\tilde{h}}_{\alpha }$and${\tilde{h}}_{\beta }$, respectively; then, the interval value hesitant fuzzy similarity$S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)$is

$$S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)=1-\frac{\left(\left|M\left({\tilde{h}}_{\alpha }\right)-M\left({\tilde{h}}_{\beta }\right)\right|+\left|E\left({\tilde{h}}_{\alpha }\right)-E\left({\tilde{h}}_{\beta }\right)\right|\right)}{2}$$

(5)

where$M\left({\tilde{h}}_{\alpha }\right)$and$M\left({\tilde{h}}_{\beta }\right)$are the scoring functions of${\tilde{h}}_{\alpha }$and${\tilde{h}}_{\beta }$, respectively, indicating the magnitude of the numerical values;$E\left({\tilde{h}}_{\alpha }\right)$and$E\left({\tilde{h}}_{\beta }\right)$are the entropies of${\tilde{h}}_{\alpha }$and${\tilde{h}}_{\beta }$, respectively, indicating the amount of uncertainty. By rearranging the terms of Equation (5), we obtain

$$\begin{array}{l}S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)\\ =1-\frac{1}{2}(|\frac{1}{\#{\tilde{h}}_{\alpha }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\alpha }}\left(\frac{{\tilde{\gamma }}_{\alpha i}^L+{\tilde{\gamma }}_{\alpha i}^U}{2}\right)}-\frac{1}{\#{\tilde{h}}_{\beta }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\beta }}\left(\frac{{\tilde{\gamma }}_{\beta i}^L+{\tilde{\gamma }}_{\beta i}^U}{2}\right)}|+|{\left(\frac{1}{2\#{\tilde{h}}_{\alpha }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\alpha }}\left(1-2\left|{\gamma }_{\alpha i}^L-0.5\right|+1-2\left|{\gamma }_{\alpha i}^U-0.5\right|\right)}\right)}^2\\ +{\left(\frac{1}{\#{\tilde{h}}_{\alpha }\left(\#{\tilde{h}}_{\alpha }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\alpha }}\left(\left|{\gamma }_{\alpha i}^L-{\gamma }_{\alpha l}^L\right|+\left|{\gamma }_{\alpha i}^U-{\gamma }_{\alpha l}^U\right|\right)}\right)}^2-{\left(\frac{1}{2\#{\tilde{h}}_{\beta }}{\displaystyle \sum _{i=1}^{{\tilde{h}}_{\beta }}\left(1-2\left|{\gamma }_{\beta i}^L-0.5\right|+1-2\left|{\gamma }_{\beta j}^U-0.5\right|\right)}\right)}^2\\ -{\left(\frac{1}{\#{\tilde{h}}_{\beta }\left(\#{\tilde{h}}_{\beta }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\beta }}\left(|{\gamma }_{\beta i}^L-{\gamma }_{\beta l}^L|+\left|{\gamma }_{\beta i}^U-{\gamma }_{\beta l}^U\right|\right)}\right)}^2|)\end{array}$$

(6)

The following proves whether Equation (6) satisfies the axiom of IVHFS similarity [46].

(a)

If ${\tilde{h}}_{\alpha }={\tilde{h}}_{\beta }$,

$${\left(\frac{1}{2\#{\tilde{h}}_{\alpha }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\alpha }}\left(1-2\left|{\gamma }_{\alpha i}^L-0.5\right|+1-2\left|{\gamma }_{\alpha i}^U-0.5\right|\right)}\right)}^2={\left(\frac{1}{2\#{\tilde{h}}_{\beta }}{\displaystyle \sum _{i=1}^{{\tilde{h}}_{\beta }}\left(1-2\left|{\gamma }_{\beta i}^L-0.5\right|+1-2\left|{\gamma }_{\beta j}^U-0.5\right|\right)}\right)}^2$$

$${\left(\frac{1}{\#{\tilde{h}}_{\alpha }\left(\#{\tilde{h}}_{\alpha }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\alpha }}\left(\left|{\gamma }_{\alpha i}^L-{\gamma }_{\alpha l}^L\right|+\left|{\gamma }_{\alpha i}^U-{\gamma }_{\alpha l}^U\right|\right)}\right)}^2={\left(\frac{1}{\#{\tilde{h}}_{\beta }\left(\#{\tilde{h}}_{\beta }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\beta }}\left(\left|{\gamma }_{\beta i}^L-{\gamma }_{\beta l}^L\right|+\left|{\gamma }_{\beta i}^U-{\gamma }_{\beta l}^U\right|\right)}\right)}^2$$

Therefore, $S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)=1$;

If $S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)=1$, $\frac{1}{\#{\tilde{h}}_{\alpha }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\alpha }}\left(\frac{{\tilde{\gamma }}_{\alpha i}^L+{\tilde{\gamma }}_{\alpha i}^U}{2}\right)}-\frac{1}{\#{\tilde{h}}_{\beta }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\beta }}\left(\frac{{\tilde{\gamma }}_{\beta i}^L+{\tilde{\gamma }}_{\beta i}^U}{2}\right)}=0$,

$${\left(\frac{1}{2\#{\tilde{h}}_{\alpha }}{\displaystyle \sum _{i=1}^{\#{\tilde{h}}_{\alpha }}\left(1-2\left|{\gamma }_{\alpha i}^L-0.5\right|+1-2\left|{\gamma }_{\alpha i}^U-0.5\right|\right)}\right)}^2-{\left(\frac{1}{2\#{\tilde{h}}_{\beta }}{\displaystyle \sum _{i=1}^{{\tilde{h}}_{\beta }}\left(1-2\left|{\gamma }_{\beta i}^L-0.5\right|+1-2\left|{\gamma }_{\beta j}^U-0.5\right|\right)}\right)}^2=0$$

$${\left(\frac{1}{\#{\tilde{h}}_{\alpha }\left(\#{\tilde{h}}_{\alpha }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\alpha }}\left(\left|{\gamma }_{\alpha i}^L-{\gamma }_{\alpha l}^L\right|+\left|{\gamma }_{\alpha i}^U-{\gamma }_{\alpha l}^U\right|\right)}\right)}^2-{\left(\frac{1}{\#{\tilde{h}}_{\beta }\left(\#{\tilde{h}}_{\beta }-1\right)}{\displaystyle \sum _{i,l=1,i<l}^{\#{\tilde{h}}_{\beta }}\left(\left|{\gamma }_{\beta i}^L-{\gamma }_{\beta l}^L\right|+\left|{\gamma }_{\beta i}^U-{\gamma }_{\beta l}^U\right|\right)}\right)}^2=0$$

Therefore, ${\tilde{h}}_{\alpha }={\tilde{h}}_{\beta }$.

(b)

It is obvious that $S\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)=S\left({\tilde{h}}_{\beta },{\tilde{h}}_{\alpha }\right)$.

(c)

Since $0\le M\left({\tilde{h}}_{\alpha }\right)\le 1$, $0\le M\left({\tilde{h}}_{\beta }\right)\le 1$, so $0\le \left|M\left({\tilde{h}}_{\alpha }\right)-M\left({\tilde{h}}_{\beta }\right)\right|\le 1$; since $0\le E\left({\tilde{h}}_{\alpha }\right)\le 1$,$0\le E\left({\tilde{h}}_{\beta }\right)\le 1$, so $0\le \left|E\left({\tilde{h}}_{\alpha }\right)-E\left({\tilde{h}}_{\beta }\right)\right|\le 1$. Therefore $0\le 1-\frac{\left(\left|M\left({\tilde{h}}_{\alpha }\right)-M\left({\tilde{h}}_{\beta }\right)\right|+\left|E\left({\tilde{h}}_{\alpha }\right)-E\left({\tilde{h}}_{\beta }\right)\right|\right)}{2}\le 1$ and $0\le \mathrm{S}\left({\tilde{h}}_{\alpha },{\tilde{h}}_{\beta }\right)\le 1$.

Theorem 1.

Let${\tilde{h}}_i$and${\tilde{h}}_j$be any two IVHFSs, and$i,j=1,2,\dots ,n$. The matrix$C={\left(c_{ij}\right)}_{n\times n}$is constructed based on interval hesitant fuzzy similarity, in which

$$c_{ij}=S\left({\tilde{h}}_i,{\tilde{h}}_j\right)$$

(7)

Then matrix$C$is a fuzzy similarity matrix.

Proof of Theorem 1.

(a) Reflexivity. $\forall i=1,2,\dots ,n$, from Equation (7), we get $S\left({\tilde{h}}_i,{\tilde{h}}_i\right)=1$, so $c_{ii}=1$. (b) Symmetry. Since $S\left({\tilde{h}}_i,{\tilde{h}}_j\right)=S\left({\tilde{h}}_j,{\tilde{h}}_i\right)$, $c_{ij}=c_{ji}$. □

5.2. Operational Steps of the Proposed Method

In this section, we develop an enterprise diagnostic index system approach in the following four steps using interval-valued hesitant fuzzy clustering, and further illustrate it in Figure 1 intuitively.

Step 1: Construct a scoring matrix.

Due to factors such as time pressure and the degree of understanding of the enterprise’s diagnosis problems, each group of experts may display hesitation between some interval evaluation values during a diagnosis. The group of experts gives an evaluation value, ${\tilde{a}}_{ij}$, preselected indicators, $d_j$, in the attribute $c_i$. ${\tilde{a}}_{ij}=\left\{{\tilde{\gamma }}_{ij}=\left[{\tilde{\gamma }}_{ij}^L,{\tilde{\gamma }}_{ij}^U\right]|{\tilde{\gamma }}_{ij}\in {\tilde{a}}_{ij}\right\}$, in which ${\tilde{\gamma }}_{ij}=\left[{\tilde{\gamma }}_{ij}^L,{\tilde{\gamma }}_{ij}^U\right]$ is an interval value. The kth expert group ${\rm E}=\left\{e_1,\dots ,e_q,\dots ,e_k\right\}$ presents m indexes $D=\left\{d_1,\dots ,d_j,\dots ,d_m\right\}$, scoring information for n indicator properties $C=\left\{c_1,\dots ,c_i,\dots ,c_n\right\}$, which is used to construct the scoring matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$ as follows:

$$\tilde{A}=\left[\begin{array}{cccc}{\displaystyle \underset{\left[{\tilde{\gamma }}_{11}^L,{\tilde{\gamma }}_{11}^U\right]\in {\tilde{a}}_{11}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{11}^L,{\tilde{\gamma }}_{11}^U\right]\right\}}& {\displaystyle \underset{\left[{\tilde{\gamma }}_{12}^L,{\tilde{\gamma }}_{12}^U\right]\in {\tilde{a}}_{12}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{12}^L,{\tilde{\gamma }}_{12}^U\right]\right\}}& \cdots & {\displaystyle \underset{\left[{\tilde{\gamma }}_{1n}^L,{\tilde{\gamma }}_{1n}^U\right]\in {\tilde{a}}_{1n}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{1n}^L,{\tilde{\gamma }}_{1n}^U\right]\right\}}\\ {\displaystyle \underset{\left[{\tilde{\gamma }}_{21}^L,{\tilde{\gamma }}_{21}^U\right]\in {\tilde{a}}_{21}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{21}^L,{\tilde{\gamma }}_{21}^U\right]\right\}}& {\displaystyle \underset{\left[{\tilde{\gamma }}_{22}^L,{\tilde{\gamma }}_{22}^U\right]\in {\tilde{a}}_{22}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{22}^L,{\tilde{\gamma }}_{22}^U\right]\right\}}& \cdots & {\displaystyle \underset{\left[{\tilde{\gamma }}_{2n}^L,{\tilde{\gamma }}_{2n}^U\right]\in {\tilde{a}}_{2n}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{2n}^L,{\tilde{\gamma }}_{2n}^U\right]\right\}}\\ \cdots & \cdots & \cdots & \cdots \\ {\displaystyle \underset{\left[{\tilde{\gamma }}_{m1}^L,{\tilde{\gamma }}_{m1}^U\right]\in {\tilde{a}}_{m1}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{m1}^L,{\tilde{\gamma }}_{m1}^U\right]\right\}}& {\displaystyle \underset{\left[{\tilde{\gamma }}_{m2}^L,{\tilde{\gamma }}_{m2}^U\right]\in {\tilde{a}}_{m2}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{m2}^L,{\tilde{\gamma }}_{m2}^U\right]\right\}}& \cdots & {\displaystyle \underset{\left[{\tilde{\gamma }}_{mn}^L,{\tilde{\gamma }}_{mn}^U\right]\in {\tilde{a}}_{mn}}{{\displaystyle \cup }}\left\{\left[{\tilde{\gamma }}_{mn}^L,{\tilde{\gamma }}_{mn}^U\right]\right\}}\end{array}\right]$$

(8)

Step 2: Construct the fuzzy similarity matrix.

A standardized scoring matrix is obtained according to the expert scores, as shown by Equation (8). To obtain the correlation between the indicators, a fuzzy similarity matrix should be constructed. According to Definition 5, the formula for obtaining the fuzzy similarity matrix $\tilde{R}$ is:

$$\tilde{R}=1-\frac{\left(\left|M\left({\tilde{h}}_i\right)-M\left({\tilde{h}}_j\right)\right|+\left|E\left({\tilde{h}}_i\right)-E\left({\tilde{h}}_j\right)\right|\right)}{2}$$

(9)

where $i,j=1,2,\dots ,m$.

Step 3: Construct the fuzzy equivalent matrix.

The fuzzy similarity matrix $\tilde{R}$ is symmetric and reflexive, but not transitive [9]. To solve this problem, the present study uses the squared method to find the transitive closure, namely: $\tilde{R}\to {\tilde{R}}^2\to {\tilde{R}}^4\to \cdots \to {\tilde{R}}^{2t}\to \cdots$. When the calculation reaches the $t+1$ step, there exists ${\tilde{R}}^{\prime }={\tilde{R}}^{2t}={\tilde{R}}^{2t-1}$; then, the fuzzy similarity matrix $\tilde{R}$ is transformed into a fuzzy equivalent matrix ${\tilde{R}}^{\prime }$ [16] such that it is transitive.

Step 4: Determine the optimal threshold $\lambda$.

Since ${\tilde{R}}^{\prime }$ is a fuzzy equivalent matrix, when the $\lambda \in \left[0,1\right]$ values are different, there are different classifications. This shows that the value of the threshold is important, which will directly affect the clustering of indicators. The formula for the relationship between threshold and classification is:

$$c_{ij}=\{\begin{array}{l}1,c_{ij}\ge \lambda \\ 0,c_{ij}<\lambda \end{array}$$

(10)

where 1 and 0 indicates that the two indicators are related and unrelated, respectively.

6. Results

In this section, an example of a hardware manufacturer wanting to improve operation and management performance is applied to demonstrate the validity of the proposed approach. In the process of enterprise diagnosis, due to the resource limitations, experts usually prefer to express their views in an interval format. However, the different experiences and subjective preferences of enterprise diagnostic experts have a certain impact on the accuracy and credibility of the index screening results. In response to this problem and to minimize the impact of objective factors and the ambiguity on the index screening results, this study used IVHFS as the form of expert opinion expression and the recruited 6 experts with more than 10 years of enterprise diagnosis experience to form 3 evaluation groups, denoted as $e_1$, $e_2$, and $e_3$. Table 2 is used by the experts of each group make an empirical judgment on the attribute of each index, give the interval value within [0, 1] as their own evaluation value, and then give their own reasons. Through the multiple rounds of discussion, each group of experts eventually reduce their views to three or less until no compromise can be made, and then summarize them into the form of IVHFS. Appendix A contains the raw data which is used in the analysis of this section.

Specific, measurable, achievable, relevant, time-bound, and strategic (SMARTS) framework is a useful approach for creating goals to evaluate and improve enterprise performance [47]. The SMARTS goal-setting framework facilitates the consulting experts in thoroughly analyzing the enterprise and ensuring that the diagnostic indicators can effectively reflect the actual situation of the enterprise, providing a basis for decision making [48]. Therefore, as the cornerstone of enterprise diagnosis, the enterprise diagnostic index system can only ensure credible and reliable diagnostic results if it uses SMARTS. As a result, we use the SMARTS, denoted as c₁, c₂, c₃, c₄, c₅ and c₆, as the evaluation criteria of enterprise diagnostic indicators. In the following, we use social benefits $D_3$ as an example to illustrate detailed operation of the proposed method and how to get the enterprise diagnostic index system for the hardware manufacturer.

Step 1: Construct a scoring matrix.

Taking the scoring results of the third-level indicators of customer $d_{3–1}$ subordinates as an example, the constructed scoring matrix is shown in

Table 4. Scoring matrix of customer $d_{3–1}$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$d_{3–1–1}$	$\left\{\left[0.3,0.4\right]\right\}$	$\left\{\left[0.5,0.6\right],\left[0.6,0.7\right]\right\}$	$\left\{\left[0.1,0.2\right]\right\}$	$\left\{\left[0.3,0.5\right]\right\}$	$\left\{\left[0.5,0.6\right]\right\}$	$\left\{\begin{array}{l}\left[0.2,0.5\right],\left[0.6,0.8\right]\\ \left[0.7,0.8\right]\end{array}\right\}$
$d_{3–1–2}$	$\left\{\left[0.6,0.7\right]\right\}$	$\left\{\left[0.1,0.2\right],\left[0.2,0.3\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$	$\left\{\left[0.4,0.6\right],\left[0.5,0.7\right]\right\}$	$\left\{\left[0.3,0.5\right]\right\}$	$\left\{\left[0.4,0.6\right]\right\}$
$d_{3–1–3}$	$\left\{\left[0.7,0.8\right],\left[0.8,0.9\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$	$\left\{\begin{array}{l}\left[0.4,0.5\right],\left[0.6,0.7\right]\\ \left[0.8,0.9\right]\end{array}\right\}$	$\left\{\left[0.4,0.6\right]\right\}$	$\left\{\left[0.7,0.8\right],\left[0.8,0.9\right]\right\}$	$\left\{\left[0.6,0.8\right],\left[0.7,0.8\right]\right\}$
$d_{3–1–4}$	$\left\{\begin{array}{l}\left[0.5,0.6\right],\left[0.7,0.8\right]\\ \left[0.6,0.7\right]\end{array}\right\}$	$\left\{\left[0.6,0.8\right]\right\}$	$\left\{\left[0.5,0.6\right],\left[0.7,0.9\right]\right\}$	$\left\{\left[0.6,0.8\right]\right\}$	$\left\{\left[0.2,0.3\right],\left[0.7,0.9\right]\right\}$	$\left\{\left[0.6,0.7\right],\left[0.7,0.8\right]\right\}$
$d_{3–1–5}$	$\left\{\left[0.3,0.4\right],\left[0.6,0.7\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$	$\left\{\left[0.2,0.4\right],\left[0.7,0.8\right]\right\}$	$\left\{\begin{array}{l}\left[0.3,0.5\right],\left[0.4,0.6\right]\\ \left[0.5,0.7\right]\end{array}\right\}$	$\left\{\left[0.4,0.6\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$
$d_{3–1–6}$	$\left\{\left[0.8,0.9\right]\right\}$	$\left\{\left[0.6,0.7\right],\left[0.7,0.8\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$	$\left\{\left[0.8,0.9\right]\right\}$	$\left\{\begin{array}{l}\left[0.3,0.5\right],\left[0.5,0.6\right]\\ \left[0.8,0.9\right]\end{array}\right\}$	$\left\{\left[0.6,0.7\right]\right\}$
$d_{3–1–7}$	$\left\{\left[0.6,0.7\right]\right\}$	$\left\{\left[0.1,0.3\right]\right\}$	$\left\{\left[0.5,0.6\right]\right\}$	$\left\{\left[0.6,0.7\right],\left[0.7,0.8\right]\right\}$	$\left\{\left[0.7,0.8\right]\right\}$	$\left\{\left[0.6,0.7\right],\left[0.7,0.8\right]\right\}$

.

Step 2: Construct the fuzzy similarity matrix.

First, Equations (3)–(6) are used to obtain the comprehensive score function value and comprehensive entropy value of the customer $d_{3–1}$. The results are shown in

Table 5. Comprehensive score function value of customer $d_{3–1}$.

${\mathit{d}}_{3–1–1}$	${\mathit{d}}_{3–1–2}$	${\mathit{d}}_{3–1–3}$	${\mathit{d}}_{3–1–4}$	${\mathit{d}}_{3–1–5}$	${\mathit{d}}_{3–1–6}$	${\mathit{d}}_{3–1–7}$
0.75	0.92	0.76	0.94	0.76	1.04	0.82

and

Table 6. Comprehensive entropy value of customer $d_{3–1}$.

${\mathit{d}}_{3–1–1}$	${\mathit{d}}_{3–1–2}$	${\mathit{d}}_{3–1–3}$	${\mathit{d}}_{3–1–4}$	${\mathit{d}}_{3–1–5}$	${\mathit{d}}_{3–1–6}$	${\mathit{d}}_{3–1–7}$
0.92	0.91	0.88	0.94	1.15	0.58	0.62

.

Then, the similarity between the indicators is calculated by using the Equation (8), and the fuzzy similarity matrix $\tilde{R}$ is obtained:

$$\tilde{R}=\left[\begin{array}{ccccccc}1& 0.86& 0.78& 0.79& 0.80& 0.72& 0.88\\ 0.86& 1& 0.74& 0.86& 0.94& 0.68& 0.87\\ 0.78& 0.74& 1& 0.88& 0.80& 0.94& 0.87\\ 0.79& 0.86& 0.88& 1& 0.92& 0.82& 0.91\\ 0.80& 0.94& 0.80& 0.92& 1& 0.74& 0.93\\ 0.72& 0.68& 0.94& 0.82& 0.74& 1& 0.81\\ 0.88& 0.87& 0.87& 0.91& 0.93& 0.81& 1\end{array}\right]$$

Step 3: Construct the fuzzy equivalent matrix.

The transitive closure algorithm is used on the fuzzy similarity matrix $\tilde{R}$ to obtain the fuzzy equivalent matrix ${\tilde{R}}^{\prime }$.

$${\tilde{R}}^{\prime }={\tilde{R}}^8={\tilde{R}}^4\times {\tilde{R}}^4=\left[\begin{array}{ccccccc}1& 0.88& 0.88& 0.88& 0.88& 0.88& 0.88\\ 0.88& 1& 0.88& 0.92& 0.94& 0.88& 0.93\\ 0.88& 0.88& 1& 0.88& 0.88& 0.94& 0.88\\ 0.88& 0.92& 0.88& 1& 0.92& 0.88& 0.92\\ 0.88& 0.94& 0.88& 0.92& 1& 0.88& 0.93\\ 0.88& 0.88& 0.94& 0.88& 0.88& 1& 0.88\\ 0.88& 0.93& 0.88& 0.92& 0.93& 0.88& 1\end{array}\right]={\tilde{R}}^4$$

Step 4: Determine the optimal threshold $\lambda$.

Categorize fuzzy equivalent matrices based on the different values of $\lambda$.

(a)

If λ = 0.88, candidate diagnostic indicators are grouped into a single category:

$$\left\{d_{3–1–1},d_{3–1–2},d_{3–1–3},d_{3–1–4},d_{3–1–5},d_{3–1–6},d_{3–1–7}\right\};$$

(b)

If λ = 0.92, candidate diagnostic indicators are grouped into three categories:

$$\left\{d_{3–1–1}\right\},\left\{d_{3–1–2},d_{3–1–4},d_{3–1–5},d_{3–1–7}\right\},\left\{d_{3–1–3},d_{3–1–6}\right\};$$

(c)

If λ = 0.93, candidate diagnostic indicators are grouped into four categories:

$$\left\{d_{3–1–1}\right\},\left\{d_{3–1–2},d_{3–1–5},d_{3–1–7}\right\},\left\{d_{3–1–3},d_{3–1–6}\right\},\left\{d_{3–1–4}\right\};$$

(d)

If λ = 0.94, candidate diagnostic indicators are grouped into five categories:

$$\left\{d_{3–1–1}\right\},\left\{d_{3–1–2},d_{3–1–5}\right\},\left\{d_{3–1–3},d_{3–1–6}\right\},\left\{d_{3–1–4}\right\},\left\{d_{3–1–7}\right\};$$

(e)

If λ = 1, candidate diagnostic indicators are grouped into seven categories:

$$\left\{d_{3–1–1}\right\},\left\{d_{3–1–2}\right\},\left\{d_{3–1–3}\right\},\left\{d_{3–1–4}\right\},\left\{d_{3–1–5}\right\},\left\{d_{3–1–6}\right\},\left\{d_{3–1–7}\right\}.$$

According to the abovementioned clustering results, when the values of λ fall in different intervals, the classification results of the candidate indicators are different.

Table 7. λ-cut matrix of ${\tilde{R}}^{\prime }$ of customer $d_{3–1}$ with λ = 0.93.

	${\mathit{d}}_{3–1–1}$	${\mathit{d}}_{3–1–2}$	${\mathit{d}}_{3–1–3}$	${\mathit{d}}_{3–1–4}$	${\mathit{d}}_{3–1–5}$	${\mathit{d}}_{3–1–6}$	${\mathit{d}}_{3–1–7}$
$d_{3–1–1}$	1	0	0	0	0	0	0
$d_{3–1–2}$	0	1	0	0	1	0	1
$d_{3–1–3}$	0	0	1	0	0	1	0
$d_{3–1–4}$	0	0	0	1	0	0	0
$d_{3–1–5}$	0	1	0	0	1	0	1
$d_{3–1–6}$	0	0	1	0	0	1	0
$d_{3–1–7}$	0	1	0	0	1	0	1

shows the λ-cut matrix of the fuzzy equivalent matrix ${\tilde{R}}^{\prime }$ with λ = 0.93. As shown in Table 4, the third-level indicators of customer $d_{3–1}$ subordinates are divided into four categories. Due to the relatively high similarity between the two indicators $d_{3–1–3}$ (product return rate) and $d_{3–1–6}$ (product safety and quality), the two indicators are placed into a single category. However, Table 4 demonstrated that the diagnosis experts scored $d_{3–1–6}$ (product safety and quality) the highest. Therefore, this indicator is retained, and another indicator of the same category, namely $d_{3–1–3}$ (product return rate), is deleted. In addition, the similarity between the three indicators $d_{3–1–2}$ (standard of after-sales service), $d_{3–1–5}$ (product satisfaction), and $d_{3–1–7}$ (product loyalty) is also relatively high. According to the above principles, $d_{3–1–2}$ (standard of after-sales service) is retained in this category, whereas $d_{3–1–5}$ (product satisfaction) and $d_{3–1–7}$ (product loyalty) are deleted. Therefore, the final selected indicators are $d_{3–1–1}$ (product and service cost performance ratio), $d_{3–1–2}$ (standard of after-sales service), $d_{3–1–4}$ (product advertisement authenticity), and $d_{3–1–6}$ (product safety and quality). The classification results are shown in

Table 8. Third-level indicators of social benefit $D_3$ and customers $d_{3–1}$ after screening.

First-Level Indicators	Second-Level Indicators	Third-Level Indicators
Social Benefit $D_3$	Customers $d_{3–1}$	Product and service cost performance ratio $d_{3–1–1}$
		Standard of after-sales service $d_{3–1–2}$
		Product advertisement authenticity $d_{3–1–4}$
		Product safety and quality $d_{3–1–6}$

.

Similarly, according to the principles and steps of enterprise diagnosis index determination, the remaining index selected via the proposed interval-valued hesitant fuzzy clustering method is used as the final enterprise diagnosis index.

Table 9. Values of λ-cut matrix of each second-level indicator.

$d_{1–1}$	$d_{1–2}$	$d_{1–3}$	$d_{1–4}$	$d_{1–5}$	$d_{1–6}$
0.74	0.91	0.94	0.85	0.87	0.86
$d_{1–7}$	$d_{1–8}$	$d_{1–9}$	$d_{1–10}$	$d_{1–11}$	$d_{1–12}$
0.85	0.91	0.82	0.87	0.62	0.90
$d_{2–1}$	$d_{2–2}$	$d_{3–2}$	$d_{3–3}$	$d_{3–4}$
0.88	0.70	0.92	0.88	0.89

gives the values of λ-cut matrix of all the second-level indicators. Finally, by sorting out the screening results of all indicators, an enterprise diagnosis index system can be constructed, as shown in

Table 10. Indicators for enterprise diagnosis.

First-Level Indicators	Second-Level Indicators		Third-Level Indicators
Enterprise performance $D_1$	General financial situation	Financial structure analysis $d_{1–1}$	Current assets to fixed assets ratio $d_{1–1–3}$; the percentages of current liabilities and long-term liabilities in total assets $d_{1–1–4}$
		Profitability $d_{1–2}$	Asset net interest rate $d_{1–2–4}$; return on owner’s equity $d_{1–2–5}$; cost–profit margin $d_{1–2–6}$; working capital index $d_{1–2–7}$
		Solvency $d_{1–3}$	Current ratio $d_{1–3–1}$; quick ratio $d_{1–3–2}$; cash-to-liability ratio $d_{1–3–3}$; long-term liability ratio $d_{1–3–6}$
		Asset management status $d_{1–4}$	Owner’s equity turnover $d_{1–4–3}$; working capital turnover $d_{1–4–5}$
		Fundraising and investment $d_{1–5}$	Inflow and outflow ratio of financing activities $d_{1–5–3}$; inflow and outflow ratio of investment activities $d_{1–5–4}$
		Cost $d_{1–6}$	Operating expense profit ratio $d_{1–6–2}$; management expense profit ratio $d_{1–6–3}$
		Listed company $d_{1–7}$	Price/earnings (P/E) ratio $d_{1–7–1}$; dividend coverage multiple $d_{1–7–3}$; net assets per share $d_{1–7–4}$
	Growth $d_{1–8}$		Sales revenue growth rate $d_{1–8–2}$; growth rate of net fixed assets $d_{1–8–3}$
	Safety	Solvency risk $d_{1–9}$	Current ratio $d_{1–9–1}$; quick ratio $d_{1–9–2}$
		Business risk $d_{1–10}$	Degree of financial leverage (DFL) $d_{1–10–1}$; degree of operating leverage (DOL) $d_{1–10–2}$
		Asset risk $d_{1–11}$	Non-performing loan (NPL) ratio $d_{1–11–1}$; asset loss rate $d_{1–11–2}$
	Innovation $d_{1–12}$		Research and development (R&D) expenses $d_{1–12–1}$; development expenditure $d_{1–12–2}$; influence $d_{1–12–7}$
Employee health $D_2$	Subjective satisfaction $d_{2–1}$		Task assignment satisfaction $d_{2–1–1}$; ability to improve satisfaction $d_{2–1–3}$; salary and benefit satisfaction $d_{2–1–4}$
	Objective health $d_{2–2}$		Sickness absence rate $d_{2–2–1}$; employee work safety protection $d_{2–2–4}$
Social benefit $D_3$	Customer $d_{3–1}$		Product and service cost performance ratio $d_{3–1–1}$; standard of after-sales service $d_{3–1–2}$; authenticity of product advertisements $d_{3–1–4}$; product safety and quality $d_{3–1–6}$
	Social environment $d_{3–2}$		Employment contribution rate $d_{3–2–1}$; sponsorship of education $d_{3–2–7}$; social responsibility report qualification rate $d_{3–2–9}$
	Natural environment $d_{3–3}$		Investment in environmental protection equipment $d_{3–3–2}$; recycling system for old products $d_{3–3–4}$; environmental management system application effect $d_{3–3–5}$
	Business partner $d_{3–4}$		Credit to seller $d_{3–4–2}$; brand awareness $d_{3–4–4}$

.

To verify the effectiveness and superiority of the proposed method, a comparative analysis with other fuzzy clustering methods is provided. As shown in

Table 11. Comparative analysis with other approaches.

Proposed by	Fuzzy Number	Interval Value	Hesitation	Characteristics of Rapid Convergence
Alptekin [9]	√
Wang et al. [14]	√	√
Huang et al. [17]	√			√
This paper	√	√	√	√

, the approach developed by Alptekin [9] and Huang et al. [17] ignored the cognitive uncertainty of decision makers and the complexity of research objects in the decision-making process, as well as they directly used precise values to express decision information, thereby reducing the objectivity of screening results. Wang et al. [14] did not consider the expert group’s hesitation of individual opinions, causing the inability to accurately reflect the original information. In addition, the fuzzy clustering method proposed by Alptekin [9] and Wang et al. [14] did not have the characteristics of rapid convergence, making it impossible to quickly obtain a variety of results of index classification. The abovementioned limitations are overcome via the method proposed in this study. The proposed method not only applies IVHFS to deal with the scoring of experts but also presents an interval-valued hesitant fuzzy similarity measurement formula, considering fuzziness and hesitation for ensuring result accuracy. In addition, it combines the characteristics of rapid convergence in the transitive closure algorithm to quickly obtain various results of index classification. In conclusion, the selected enterprise diagnosis index system can be reasonable and scientific using the proposed method.

7. Conclusions

This study uses interval-valued hesitant fuzzy clustering based on transitive closure to construct the enterprise diagnostic index system. The proposed method conducts a more comprehensive analysis of enterprise diagnostic indicators from three aspects: enterprise performance, employee health, and social benefits. Furthermore, the proposed method eliminates redundant indicators. Finally, 3 first-level indicators, 18 second-level indicators, and 47 third-level indicators are chosen. This indicator system is based on the criteria of specific (c₁), measurable (c₂), achievable (c₃), relevant (c₄), time-bound (c₅), and strategic (c₆).

Through the analysis of case results, the proposed interval-valued hesitant fuzzy clustering method based on transitive closure has the following advantages. (a) The interval-valued hesitant fuzzy entropy function constructed based on hesitation and ambiguity not only effectively measures the degree of information uncertainty, but also eliminates the problem of data length. (b) The value of IVHFE and the similarity measurement of the degree of uncertainty are considered, thereby improving the resolution of the indicators. (c) IVHFS reflects the opinions of the expert group, which not only improves the credibility of evaluation information but also helps to avoid the problem of information loss during the evaluation process, thereby improving the credibility of the evaluation. (d) Since the squaring function in the transitive closure algorithm has the characteristics of rapid convergence, the interval-valued hesitant fuzzy clustering can quickly obtain various results of index classification, thus improving the efficiency of evaluation. (e) Indexes are divided and classified according to different thresholds, thereby improving the rationality of the indexes.

Although IVHFS is used here to deal with expert scoring, actual situations will inevitably be affected by factors such as personal experience and preferences [49,50,51]. Therefore, data collection method and scoring rules should be strengthened in the future to improve the index system and realize the objective quantification of evaluation.