Research on economic system based on fuzzy set comprehensive evaluation model

Abstract

The development of the economic system is affected by many factors, and the stability of the traditional economic analysis model is difficult to maintain. In order to explore the efficient and stable economic system evaluation and analysis model, based on machine learning ideas, this study uses rough set algorithm as the basic algorithm, and applies the related methods of rough set and catastrophe model theory to the evaluation of ecological economic development level. Moreover, this study reduces the redundant index of the index system and calculates the importance of the index after reduction. Based on the catastrophe set model, this study uses MATLAB software programming to comprehensively quantify the ecological economy, and finally divides the ecological economic grade. In addition, this study combines rough set theory with fuzzy mathematics, and initially establishes a two-branch fuzzy evaluation model. Finally, this study combines the actual situation to use the established model to evaluate the regional eco-economic system. The research results show that the method proposed in this paper has a certain effect, which can provide a reference for subsequent related research.

Keywords

Fuzzy set machine learning comprehensive evaluation economic system

1 Introduction

How to survive better in the world and how to better develop the economy and society are the eternal themes of mankind. China is the world’s largest developing country. After undergoing reform and opening up and economic policy adjustments, China’s economy has developed rapidly, and China’s overall strength has also been rapidly improved. With the gradual improvement of the level of human knowledge and knowledge, people’s understanding of urban economic and social development will continue to deepen. With the advancement of China’s urbanization process and the deepening of economic and political system reform and the continuous expansion of the modernization process, the social development stage, development problems and development strategies of China have attracted great attention from the academic community and all sectors of society. Although China is large in land and rich in resources, it has a large population and a large population base. On average, each person occupies few resources. At present, China is in the development stage of continuous progress in industrialization, marketization, urbanization, and internationalization, and is facing great pressure on resources and the environment [1].

Rough set theory is a digital tool to deal with some incomplete, incomplete, and inaccurate data. After the theory was successfully applied in pattern recognition, machine learning, decision-making and analysis, etc., it attracted the attention of scholars in related academic circles. At present, the theory has been widely used in scientific research fields such as information retrieval, data mining and artificial intelligence. Among them, attribute reduction is one of the core contents of rough set theory. It mainly refers to eliminating some unnecessary information or irrelevant information while keeping the classification ability unchanged. Many scholars have proposed attribute reduction methods from different perspectives, and these proposed algorithms can be divided into three categories: First of all, Zhang Tengfei, Xiao Jianmei and others proposed two reduction methods that use positive regions to directly find kernels based on rough set theory. One of them is to filter out the core attributes of the matrix based on the reduction of the difference matrix, and then select the most frequent attributes in the gradually decreasing difference matrix until it becomes a reduction [2]. The second type is attribute reduction based on Shannon information entropy; The last category is the extension of LiangJY and others on the basis of Shannon’s information entropy method, so that it has the essence of complement set and can obtain the ambiguity of rough set. Since the last century, the development of the application of set theory in logic and modern mathematics has had a great influence, and it has become part of the theory of mathematics and logic [3]. In order to eliminate the paradox caused by the lack of restrictions on the method of defining sets, Cantor created set theory. After that, with the efforts of many mathematicians, a more accurate and complete axiomatized set theory was established in the 20th century. In 1965, Professor L.A. Zadeh proposed the concept of fuzzy sets in order to solve some problems, and then the concept gradually formed the theory of fuzzy sets and was successfully applied in many fields. In the early 1980 s, Z. Pawla first proposed the rough set theory on the basis of set theory, but it was not widely valued by academics from various countries at the time. It was not until the end of the 1980 s that the theory was gradually concerned by many seniors. Since 1992, an exchange research group on rough sets has been formally established to further discuss and explore the theory.

2 Related work

Rough set theory is a theory applied to data analysis and reasoning proposed by Polish scientist Pawlak in 1982 [4], which specifically solves the problem of uncertainty and inaccuracy. In 1991, Pawlak published the book “Rough Sets-The Theory of Data Reasoning”, which marked the birth of rough set theory. The book systematically elaborates rough set theory and lays a foundation for the development of rough sets, thus setting off a research boom in rough set theory. The research on rough set theory started late, but its development is rapid, and some monographs have been published [5]. At present, China has become the backbone of rough set research, and the Chinese Fuzzy Set and Soft Computing Symposium has become a high-level academic exchange meeting. The development of rough set theory in China is booming.

The attribute reduction method based on importance is a very important attribute reduction algorithm, which has great theoretical guiding significance for the future heuristic attribute reduction research. This kind of algorithm is based on the positive domain of decision-making class and takes attribute dependency as a standard to measure the importance of attributes. By comparing the size of attribute dependency, attributes are continuously added. As the number of attributes increases, the degree of dependency or positive domain continues to increase, and finally the smallest subset of attributes with the same degree of dependency is obtained. The advantage of this method is that if a reduction of the decision table can be found, then the reduction must be the optimal or suboptimal solution, but it is not complete, that is, it may not be possible to find the reduction. Moreover, the time complexity of the algorithm is exponential, and it is difficult to deal with attribute reduction under big data, which limits its practical application. Many scholars have proposed improvements to this algorithm: The literature [7] proposed an incremental calculation method for the positive domain, which opens up the idea of gradually reducing the theoretical domain; The literature [8] proposed a heuristic attribute reduction algorithm based on knowledge granularity, which reduces the algorithm time complexity; The literature [9] proposed a fast parallel attribute reduction algorithm to distribute attribute reduction tasks to multiple processors for simultaneous processing, which further reduces the complexity of the algorithm.

Attribute reduction algorithm based on information entropy is an attribute reduction algorithm based on information view, which plays an important role in heuristic attribute reduction algorithm. The literature [10] offered the concept of rough set theory under the concept of information from the perspective of information theory and proved that the knowledge reduction of the information view is consistent with the algebraic view. In the same year, he further proposed an attribute reduction algorithm based on mutual information. On this basis, the literature [11] proposed an attribute reduction algorithm based on conditional information entropy. However, some literatures point out that the conditional information entropy is inconsistent with the result of attribute reduction under the algebraic perspective. The literature [12] analyzed the shortcomings of the conditional information entropy attribute reduction algorithm, that is, it cannot be equivalently expressed as reduction and high time complexity, and proposed a new conditional information entropy, and gave an efficient algorithm for calculating the conditional information entropy. The literature [13] introduced the concept of decision strength to propose a new attribute importance measure method, which improves the attribute reduction algorithm based on information entropy, thereby overcoming the incompleteness of the definition of classic rough set theory reduction and the inability to obtain the optimal solution.

Attribute reduction algorithm based on difference matrix and improved difference matrix. The literature [14] proposed the attribute reduction algorithm of Skowron discernibility matrix. This method can intuitively find the core attributes and use the minimal disjunction paradigm of the difference function to find all attribute reductions. Through this method, all reductions of the information system can be obtained, so as to find the minimum reduction. However, in terms of time complexity, the attribute reduction algorithm based on the HU difference matrix is not ideal, and both time and space complexity need to be reduced. Therefore, some scholars have successively proposed various improved discernibility matrix forms. The literature [15] proposed an attribute order attribute reduction method sorted by user preference, and literature [16] further developed the method, and proposed an incremental attribute reduction algorithm. To address the storage problem of the difference matrix, the literature [17] proposed a binary difference matrix with l and 0 as matrix elements. Since the incompatibility of the decision table is not taken into consideration, HU’s algorithm cannot guarantee to obtain the correct kernel attributes. Reference [18] separated consistent objects and inconsistent objects to establish a difference matrix, which properly solves the problem of inconsistent decision tables. However, the time complexity of the algorithm is the same as the HU method, and it still needs to be improved. While revising the difference matrix, literature [19] established a new attribute reduction algorithm using attribute frequency as heuristic information. In order to cope with the dynamic changes of the decision table, literature [20] proposed an incremental Pawlak attribute reduction algorithm under attribute order based on the difference matrix. Compared with the non-incremental type, the efficiency of the algorithm is greatly improved. The literature [24] talks about the construction of directed acyclic graph for video coding algorithms for motion estimation in parallel reconfigurable computing systems. The partitioning algorithm also plays a key role in optimizing the encoding of images. The literature [25] dealt with the exploitation of IoT and BigData Analytics using the Hadoop ecosystem in real-time environments. The implementation of IoT-based Smart City is accomplished through the above-mentioned processes. The article [26] centers around IoT and its noteworthy work in sophisticating the human hones and endeavors. This paper moreover overseen the combination of diverse data from distinctive resources that are related with the web. The article [27] talks approximately the different issues within the vehicular communication field with the proposition of agreeable centralized and distributed spectrum detecting model. Due to the execution of the agreeable cognitive model, obstructions and different hidden issues are minimized. The article [28] discusses the problem, such as the tremendous amount of big data, and introduces the SmartBuddy idea of a smart and intelligent world using individual activities and human resources [29, 30].

3 Fuzzy set theory

Fuzzy mathematics is a type of mathematics that studies and deals with the phenomenon of ambiguity. Since fuzzy concepts cannot be described by classical sets, fuzzy set theory came into being. Considering some negative factors in fuzzy decision-making and fuzzy control, based on Zadeh’s fuzzy set theory, Chinese scholar KaiquanShi proposed the two-branch fuzzy set theory and applied it to fuzzy decision-making and fuzzy recognition [21].

We assume that the domain U is a set of certain, identifiable factors. For ∀A ⊆ U, a feature function A (x) can be introduced, namely: $A (x) = {\begin{matrix} 1 & x \in A \\ 0, & X \notin A \end{matrix}$ (1)

The characteristic function in U is a mapping from U to [0, 1], and any characteristic function in U can also completely determine a classic subset of U, namely: $A = {x \in U : A (x) = 1}$ (2)

From the perspective of feature functions, the classic set is an explicit set, which corresponds to binary logic. However, from the perspective of set theory, the objects in a discourse domain only belong to or do not belong to the set. However, when dealing with practical problems, binary logic does not reflect the actual situation very well. For example, “Wang Wu is tall and Li Si is short”, which cannot be fully reflected in binary logic. There is no clear boundary between “tall man” and “non-tall man”, “short man” and “non-short man”, so it is a transitional state in a certain sense. For this state that cannot be described clearly, we extend the feature function to a membership function. The so-called membership function on the universe U refers to a mapping of U on [0, 1] [22].

We define a fuzzy set A on the universe U to be represented by a membership function on U, that is, formula (3). $A : U \to [0, 1]$ (3)

Thus, for an element X in the universe U and a fuzzy set A in the universe U, we cannot simply say that x is “must” belong or not belong to A, but only to what extent X belongs A.The degree of membership A (x) is a quantitative index of the degree to which X belongs to A. If A (x) = 0, it is considered that X does not belong to A at all. If A (x) = 1, it is considered that X belongs to A completely. If 0 < A (x) < 1, it is considered that x belongs to A to some extent, which is a transitional state between elements that belong to and do not belong to A at all.

In general, a fuzzy set A can be expressed as: $A = {(x, A (x)) : x \in U}$ (4)

If the universe U is a countable set or a finite set, it can be expressed as: $A = \sum x_{i} / A (x_{i})$ (5)

If the universe U is an infinite and uncountable set, then it can be expressed as: $A = \int x / A (x)$ (6)

The overall fuzzy set on U is F (U).

Definition: We assume A, B ⊆ F (U). If for any x ∈ U, there is A (x) ⩽ B (x), then it is said that B contains A or A is contained in B, denoted A ⊆ B. If A ⊆ B and B ⊆ A are both true at the same time, then A and B are equal, denoted as A = B. The empty set φ means that the membership function of the fuzzy set is always 0, and U means that the membership function of the set membership is always l. A ∪ B is the union of fuzzy sets A and B, and its membership function is defined as [23]: $(A \cup B) (x) = A (x) \lor B (x) = max {A (x), B (x)}$ (7)

A ∩ B is the intersection of fuzzy sets A and B, and its membership function is defined as: $(A \cap B) (x) = A (x) \land B (x) = min {A (x), B (x)}$ (8)

A^C or ∼A is the complement of set A, and its membership function is defined as: $A^{C} = 1 - A (x)$ (9)

A - B is the difference between A and B, and its membership function is defined as; $(A - B) (x) = [A \cap B^{C}] (x) = A (x) \land (1 - B (x))$ (10)

The intersection, complement, union, and inclusion of fuzzy sets represent the extraction, negation, conjunction, and implication of fuzzy concepts in turn, which is of great significance for analyzing and solving practical problems and theoretical research.

For A ⊆ F (U) anf λ ∈ [0, 1]: $\begin{matrix} A_{λ} = {x \in U | A (x) ⩾ λ} \\ A_{λ}^{S} = {x \in U | A (x) > λ} \end{matrix}$ (11)

Among them, A_λ is called the cut set of set A, $A_{λ}^{S}$ is called the strong cut set of set A, and λ is called the threshold or confidence level. When λ = 1, A₁ is called the core of set A.The set {x∈ U|A (x) > 0 } is called the support of A, which is denoted SuppA.

When solving practical problems, it is sometimes necessary to make a clear decision and a clear understanding of the fuzzy concept and to determine the specific attribution of an object to the fuzzy set. This requires transformation between fuzzy sets and classic sets according to some established rules. At this time, the cut set of the fuzzy set is a more satisfactory solution to the problem.

If it is assumed that U and V are two domains, and R is a fuzzy set on U × V, then R is said to be a fuzzy relationship from U to V.In particular, when U = V, R is called a fuzzy relation on U.

If both U and V are finite sets, then a correspondence can be established between the fuzzy relationship and the fuzzy matrix, and the fuzzy matrix means that any element in the matrix is on [0, 1].

Definition: If we assume that U, V, and W are three domains, R is a fuzzy relationship from U to V, and S is a fuzzy relationship from V to W, then the synthesis of R to S can be regarded as a fuzzy relationship from U to W, which is written as R ∘ S. Its membership function is defined as: $\begin{matrix} (R \circ S) (x, z) = \underset{y \in V}{\lor} (R (x, y) \land S (y, z)) \\ (x, z) \in U \times W \end{matrix}$ (12)

In the fuzzy relationship R, for each pair of elements (x, y), there is a number R (x, y) in between [0, 1], which indicates the degree of correlation of x to Y with respect to the relationship R.

Definition: We assume that R is a fuzzy relation on the universe U, and its membership function is R (x, y). If there is R (x, y) = 1 for any x ∈ U, then R is reflexive. If for any x, y ∈ U, there is R (x, y) = R (y, x), then R is symmetric. If R ∘ R ⊆ R, then R is transitive. If R is reflexive, symmetric and transitive fuzzy relation, then R is equivalent, and if R is reflexive, symmetric fuzzy relation, then R is similar.

Definition: We assume that R is a fuzzy relation on U.Then, only when R_λ is a reflexive ordinary relationship for any λ ∈ [0, 1], R is reflexive; only when R_λ is a symmetric ordinary relationship for any λ ∈ [0, 1], R is symmetric; only when R_λ is a transitive ordinary relationship for any λ ∈ [0, 1], R is transitive; only when R_λ is a equivalent ordinary relationship for any λ ∈ [0, 1], R isequivalent; only when R_λ is a similar ordinary relationship for any λ ∈ [0, 1], R is similar.

4 Establishment of a two-branch fuzzy evaluation model

Coal floor water inrush is a non-linear, multi-level and fuzzy complex problem affected by various factors including confined water and mining pressure. Different factors that affect water inrush from coal floor have different effects on promoting and inhibiting water inrush. Therefore, it is necessary to start from the two factors of promoting and inhibiting the water inrush from the coal seam floor to conduct a complete and comprehensive evaluation of the safety of the water inrush from the floor. Therefore, this paper introduces two-branch fuzzy mathematics, and from the actual situation of each influencing factor, establish two different factor sets. One set is a set that promotes floor water inrush and the other set is a set that suppresses floor water inrush. Based on the above ideas, this study combines with the two-branch fuzzy decision theory to establish an evaluation model to more objectively evaluate the safety of floor water inrush, so as to obtain a reasonable and accurate evaluation results.

In the promotion factor set X⁺, the positive target eigenvalue vector of the factor for the evaluation state is: $X^{+} = {(x_{1}, x_{2}, \dots, x_{α})}^{T}$ (13)

If a target evaluates the state of α sample, the target eigenvalue matrix is obtained, and the eigenvalue matrix is represented by X⁺: $X^{+} = {(\begin{matrix} x_{11} & x_{12} & \dots & x_{1 n} \\ x_{21} & x_{22} & \dots & x_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{α 1} & x_{α 2} & \dots & x_{α n} \end{matrix})}^{T} = (x_{ij}^{+})$ (14)

For the larger the better, the relative priority formula of the promotion direction is: $r_{ij}^{+} = (x_{ij} - \underset{j}{\land} x_{ij}) / (\underset{j}{\lor} x_{ij} - \underset{j}{\land} x_{ij})$ (15)

Among them, ∨ is a max operation, and ∧ is a minoperation.

The target eigenvalue matrix formula (14) is converted into a target relative priority matrix. $R^{+} = {(\begin{matrix} r_{11} & r_{12} & \dots & r_{1 n} \\ r_{21} & r_{22} & \dots & r_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ r_{α 1} & r_{α 2} & \dots & r_{α n} \end{matrix})}^{T} = (r_{ij}^{+})$ (16)

Among them, $r_{ij}^{+}$ is the target goodness.

If g⁺ is the maximum goodness of promotion (the target goodness of the excellent state), then the relative superiority vector of the excellent state corresponding to each index is: $\begin{matrix} g^{+} = (\lor_{1, j = 1}^{n} r_{1, j} \land_{2, j = 1}^{n} r_{2, j}, \dots, \lor_{α, j = 1}^{n} r_{α, j}) = \\ {(g_{1}^{+}, g_{2}^{+}, \dots, g_{α}^{+})}^{T} \end{matrix}$ (17)

If b⁺ is the promotion to the minimum goodness (the target goodness of the inferior state), then the relative superiority vector of the inferior state corresponding to each index is: $\begin{matrix} b^{+} = (\lor_{1, j = 1}^{n} r_{1, j} \land_{2, j = 1}^{n} r_{2, j}, \dots, \lor_{α, j = 1}^{n} r_{α, j}) = \\ {(b_{1}^{+}, b_{2}^{+}, \dots, b_{α}^{+})}^{T} \end{matrix}$ (18)

If $u_{j}^{+}$ is the decision degree of sample j for superior decision, ${\bar{u}}_{j}^{+}$ is the decision degree of sample j for inferior decision, then ${\bar{u}}_{j}^{+} = 1 - u_{j}^{+}$ (19)

We assume that the α indicators for promotion have different weights w⁺, and $\begin{matrix} w^{+} = (w_{1}^{+}, w_{2}^{+}, w_{3}^{+}, \dots, w_{α}^{+}) \\ \sum_{i = 1}^{α} w_{i}^{+} = 1 \end{matrix}$ (20)

The object of evaluation is regarded as a water inrush system. In the process of this system being triggered, a more obvious characteristic is non-linearity. Therefore, the established evaluation model should be a nonlinear model. When expressing the difference between the superiority vector R⁺ of the promotion and the superior state of the promotion, it is expressed by the Euclidean distance $d_{g}^{+}$ , which is written as: $d_{g}^{+} = \sqrt{\sum_{i = 1}^{α} {(w_{i}^{+} (g_{i}^{+} - r_{ij}^{+}))}^{2}}$ (21)

When expressing the difference between the superiority vector R⁺ of the promotion and the inferior state of the promotion, it is expressed by the Euclidean distance $d_{b}^{+}$ , which is written as: $d_{b}^{+} = \sqrt{\sum_{i = 1}^{α} {(w_{ij}^{+} (r_{ij}^{+} - b_{i}^{+}))}^{2}}$ (22)

From Equation (16), Equation (21), and Equation (22), The optimal distance $D_{j}^{+}$ of weighted distance and the inferior distance $D_{b}^{+}$ of weighted distance are respectively obtained. $D_{g}^{+} = = u_{j}^{+} d_{g}^{+} = u_{j}^{+} \sqrt{\sum_{i = 1}^{α} {(w_{i}^{+} (g_{i}^{+} - r_{ij}^{+}))}^{2}}$ (23)

$\begin{matrix} D_{b}^{+} = = & (1 - u_{j}^{+}) d_{b}^{+} \\ = (1 - u_{j}^{+}) \sqrt{\sum_{i = 1}^{α} {(w_{i}^{+} (r_{ij}^{+} - b_{i}^{+}))}^{2}} \end{matrix}$ (24)

In order to obtain the optimal solution for the relative membership degree $u_{j}^{+}$ , the following optimization criterion is implemented: the sum of squares of the optimal distance of the weighted distance and the inferior distance of the weighted distance is the smallest.

$Min {\begin{matrix} F (u_{j}^{+}) + D_{g}^{+^{2}} + D_{b}^{+^{2}} = u_{j}^{+^{2}} \sum_{i = 1}^{α} {[w_{i}^{+} (g_{i}^{+} - r_{ij}^{+})]}^{2} \\ + {(1 - u_{j}^{+})}^{2} \sum_{i = 1}^{α} {[w_{i}^{+} (r_{ij}^{+} - b_{i}^{+})]}^{2} \end{matrix}}$ (25)

Formula (25) is derived to satisfy the following conditions: $\frac{dF (u_{j}^{+})}{{du}_{j}^{+}} = 2 u_{j}^{+} d_{g}^{+^{2}} - 2 (1 - u_{j}^{+}) d_{b}^{+^{2}} = 0$ (26)

By solving formula (26), the following results can be obtained: $u_{j}^{+} = \frac{1}{1 + \frac{\sum_{i = 1}^{α} {[w_{i}^{+} (g_{i}^{+} - r_{ij}^{+})]}^{2}}{\sum_{i = 1}^{α} {[w_{i}^{+} (r_{ij}^{+} - b_{i}^{+})]}^{2}}}$ (27)

When formula (17) and formula (18) satisfy formula (28), $\begin{matrix} g^{+} = {(1, 1, \dots, 1)}^{T} \\ b^{+} = {(0, 0, \dots, 0)}^{T} \end{matrix}$ (28)

formula (16) can be simplified as: $u_{j}^{+} = \frac{1}{1 + \frac{\sum_{i = 1}^{α} {[w_{i}^{+} (1 - r_{ij}^{+})]}^{2}}{\sum_{i = 1}^{α} {(w_{i}^{+} r_{ij}^{+})}^{2}}}$ (29)

$u_{j}^{+}$ is the relative membership degree of the influencing factors in the promotion factor set X⁺.

5 Experimental research

This study takes area A as an example to conduct research and analysis. The evaluation and research of the socioeconomic development level of the A region is composed of many different disciplines, including social sciences, social sciences and politics, economy, resources, society, environment and ecology. Moreover, the scope of various fields is very wide. Because many indexes are involved, the index system is divided into four criteria layers, which are: economic layer, resource layer, environmental layer, and social layer. Under these four criteria levels, this study selects specific evaluation indicators that are representative, prominent, and reliable to establish an appropriate, scientific, and targeted indicator system.

According to the principles and methods of constructing the index system, starting from its four system contents, through consulting data and consulting experts, this study proposes an evaluation index system to evaluate the level of eco-economic development in the A region, as shown in Fig. 1.

Fig. 1

Framework diagram of the evaluation system of ecological economic development level.

According to the previous discussion and various types of basic principles, this article takes “ resource, environment, economic and social evaluation of area A” as the target layer and “economic system, social system, resource system, environmental system” as the standard layer. In addition, this study takes “regional GDP (100 million yuan), tertiary industry’s share of GDP (%), fixed asset investment (100 million yuan), added value of industrial enterprises above designated size (100 million yuan),total retail sales of social consumer goods (100 million yuan), urbanization rate (%), old-age insurance (ten thousand people), number of people covered by medical insurance (ten thousand people), number of health technicians per thousand population (person),the proportion of socially employed persons in the total population (%), newly added labor force (10,000 people), per capita electricity consumption (kWh), industrial energy consumption above designated size (10,000 tons of standard coal),water consumption per 10,000 yuan of GDP (cubic meters), water consumption per 10,000 yuan of industrial added value (cubic meters), proportion of nature reserves in the area under jurisdiction (%), forest coverage (%), general industrial solid waste comprehensive utilization rate (%), industrial water reuse rate (%), waste gas treatment facility processing capacity (10,000 cubic meters / hour) “as the indicator layer. Among them, the indicators of the indicator layer can be divided into positive indicators and negative indicators by their nature. Since the basic units of the collected data are mostly different, in order to run the subsequent work accurately, it is necessary to eliminate the dimensional influence of different data and use the range method to process the original data without dimensions.

The MATLAB software is used to realize the rough set attribute reduction and information entropy importance calculation of the information decision table. In the attribute reduction, 10 redundant indicators are screened out, which are regional GDP (100 million yuan), total retail sales of consumer goods (100 million yuan), urbanization rate (%), pension insurance (ten thousand people),number of health technicians per 1,000 population (person), newly added labor force (10,000 people), water consumption per 10,000 yuan of GDP (cubic meters),the proportion of nature reserves in areas under jurisdiction (%), forest coverage (%), and comprehensive utilization rate of general industrial solid waste (%). The specific attribute reduction is shown in Table 1 and Fig. 2.

Table 1

The result table of eco-economic regionalization index at-tribute reduction and importance calculation

Specific indicators	Importance
C12 Tertiary industry’s share of GDP (%)	4.697
C13 Investment in fixed assets (100 million yuan)	4.679
C14 Value-added of industrial enterprises above designated size (100 million yuan)	4.671
C23 Number of medical insurance participants (ten thousand people)	2.140
C25 Number of social employees in the total population (%)	2.193
C32 electricity consumption per capita (kWh)	4.393
C33 Industrial energy consumption above designated size (10,000 tons of standard coal)	4.375
C35 million of industrial value-added water consumption (cubic meters)	4.428
C44 Industrial water reuse rate (%)	2.116
C45 waste gas treatment facility treatment capacity (10,000 cubic meters / hour)	2.177

Fig. 2

The result diagram of eco-economic regionalization index attribute reduction and importance calculation.

According to the calculation results of attribute index reduction and importance of eco-economic development evaluation in A region, the catastrophe model for eco-economic development evaluation in A region is determined (Fig. 3).

Fig. 3

Catastrophe model of ecological economic development evaluation.

Through MATLAB software programming to achieve the calculated comprehensive quantitative value, Moreover, this study takes the total membership function value A as sample data, uses SPSS17.0 software to perform K-mean cluster analysis, and divides the 11 division units in the A area into 4 categories. It can be seen from the clustering results that it is generally consistent with the evaluation results of the mutation series model, as shown in Table 2 and Fig. 4.

Table 2

The result table calculated by the mutation series method

	B1	B2	B3	B4	A	A	classification
1	1	0.8954	0.9209	0.7973	0.9273	0.93	I
2	0.3146	0.7258	0.8994	0.4934	0.7902	0.79	III
3	0.6478	0.7212	0.818	0.8525	0.9044	0.9	I
4	0.7947	0.9173	0.6322	0.8802	0.7951	0.8	II
5	0.6998	0.5022	1.2042	0.9979	0.8418	0.84	II
6	0.3131	0.5	0.864	0.4773	0.7815	0.78	III
7	0.8067	0.8031	0.8465	0.222	0.6055	0.61	IV
8	0.5724	0.5568	0.5766	0.554	0.7593	0.76	III
9	0.477	0.7353	0.3562	0.7449	0.5968	0.6	IV
10	0.4289	0.3429	0.8149	0.2685	0.6451	0.65	IV
11	0.6708	0.8603	0.8117	0.9116	0.9009	0.9	I

Fig. 4

The result diagram calculated by the mutation series method.

It can be seen from Fig. 4 that city a, as the capital city of the province, has the highest economic score, with a score of 1.0, and each indicator ranks first in the province. The cities that follow are cities g and d. Their overall economic score is around 0.8, and their fixed asset investments are 1041.25 and 117.846 billion yuan respectively. In contrast, the economic scores of cities f, b, and j are all below 0.4, ranking first, second, and third from the bottom, which shows that there are certain problems in the economic development of these cities. From the perspective of selected economic factors, the average fixed asset investment of cities f, b, and j in 2010–2014 is 31.32, 42.855, and 62.891 billion yuan, respectively, which is a low level in area A. The overall score of the remaining cities is above 0.55, which belongs to the middle level in the province.

It can be seen from Fig. 5 that city a has the highest economic score as the capital city of the province, with a score of 1.0, and each index ranks first in the province, followed by cities with better economy in city g and city d, and the overall economic score is around 0.8 The investment in fixed assets was 1041.25 and 117.846 billion yuan respectively, which was second to Nanchang in the province. In contrast, the economic scores of f city, Jingdezhen and Fuzhou are all lower than 0.4, ranking first, second, and third indicate that there are certain problems in economic development. From the perspective of selected economic factors, f city, b city and j city In 2010–2014, the average fixed asset investment was 31.32, 428.55 and 62.891 billion yuan respectively, which was at a low level in Area A, and the overall scores of the remaining cities were all above 0.55, which was within the provincial medium level.

Fig. 5

Analysis diagram of economic factor scores.

As can be seen from Fig. 6, the social construction of city d is the first in the province, with a score of 0.9173, and cities a, k, and g are second only to city d, ranking second, third, and fourth in the province. The scores of social constructions in cities a and k are also nearly 0.9, and the score in city g is also above 0.8.Among the 4 cities with the highest social construction scores in the province, the overall number of people participating in the medical insurance is high in the province, which are 59.356, 86.36, 60.42 and 622,500 respectively. The number of social employees also accounts for more than 60% of the total population. The city’s social construction score is the worst in the province. The score is the only city below 0.5. The number of medical insurance insurers is 370,000, and the number of social employees in the total population is 55.15. These are all low levels in the province. The social construction of cityb, city c, and city i belongto the middle level in the province, with equal scores above 0.7. The social constructions in remaining cities e, f, and h all belong to poor levels in the province, and their social construction scores are all between 0.5 and 0.6.

Fig. 6

Analysis diagram of social factor score.

It can be seen from Fig. 7 that the resource utilization score of ecity is the first in the province, with a score of 1.2, which exceeds city a. Moreover, its water consumption per 10,000 yuan of GDP and water consumption per 10,000 yuan of industrial added value are 106 and 90 cubic meters, which are not high in the province. It shows that the city is relatively reasonable in resource utilization. City a, city b, city c, city f, city g, city j, and city k all have resource utilization scores above 0.8, which are within the upper middle level of the province. In contrast, the utilization of resources in cities d and h is general, with a score of around 0.6.The use of resources in city i is the worst in the province. The city’s per capita electricity consumption, water consumption per 10,000 yuan of GDP, and water consumption per 10,000 yuan of industrial value added are 274 kWh, 344 cubic meters, and 218.6 cubic meters.

Fig. 7

Analysis diagram of resource element score.

As can be seen from Fig. 8, the eco-environmental protection score of e city ranks first in the province, with a score of nearly 1.0, its industrial and industrial water reuse rate reaches 93.68%, and the treatment capacity of waste gas treatment facilities reaches 42.564 million cubic meters. City k, city d and city c rank second, third and fourth in the province respectively in terms of ecological environment protection, with scores of 0.9116, 0.8802 and 0.8525 respectively. Among them, the environmental protection scores of city a is nearly 0.8, and the environmental protection score of city i is 0.7449, both of which are in the middle level of the province. The two cities with the worst environmental protection in the province are city g and city j, respectively. Their environmental scores is 0.222, 0.2685 respectively, and their industrial water reuse rate is below 50%, and their exhaust gas treatment facility treatment capacity are 15.99886 million cubic meters and 10.79158 million cubic meters, respectively. Among them, the environmental protection scores of city a is nearly 0.8, and the environmental protection score of city i is 0.7449, both of which are medium levels in the province.

Fig. 8

Analysis diagram of environmental factor score.

6 Conclusion

With the goal of coordination, unification, and sustainable development of economy, society, resources, and environment, this paper is dedicated to exploring and exploring a suitable development model. According to the actual situation of the municipalities in the studied area and the current status of socio-economic development, the status of resource systems, the availability of natural environment and indicators, and the requirements of reliability, scientificity, operability, integrity and stability of various indicators, this study establishes an indicator system for coordinated development of resources, environment, society and economy. In this paper, the related methods of rough set and catastrophe model theory are applied to the evaluation of the ecological economic development level of a certain area. Based on the indistinguishable principle of rough set and reduction algorithm, the redundant index reduction of the index system is realized, and the importance of the index after reduction is calculated. Moreover, based on the catastrophe set model, this study uses MATLAB software programming to comprehensively quantify the ecological economy, and finally divides the ecological economic grade. This not only expands the applied theory of rough set and catastrophe series method, but also makes the final data more practical.

Footnotes

Acknowledgments

The fourth economic census of Jiangxi Province: Evaluation of the impact of optimizing business environment on the development of private enterprises (Serial number: 16).

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