*3.1. Model Summary*

The entropy-TOPSIS-coupling coordination degree model is constructed to horizontally and vertically measure the regional green economy development, the specific model for which is shown in Figure 2. First, Index weight reflects the different importance of indicators in the evaluation process, and it is a comprehensive measure of subjective and objective responses to the relative importance of indicators in decision-making (or evaluation) issues. Decancq, K. and Lugo, A. [31] summarized eight methods for setting indicator weights and highlight their strengths and weaknesses. The entropy method employs the inherent information in the evaluation indicators to discriminate the utility value of the indicators, which avoids any subjective factors, and therefore has higher credibility than subjective weighting methods such as Delphi and AHP [32,33]. TOPSIS (technique for order preference by similarity to an ideal solution), is simple to calculate and produces reasonable results as it is able to obtain the relative proximity between each evaluation object and the optimal solution by calculating the distance between each evaluation object and the optimal solution and the worst solution, after which the evaluation objects are ranked based on relative proximity [34]. The combination of these two methods (entropy weight-TOPIS) is able to more objectively and accurately reflect the evolutionary regional green economy development trends using a simple and practical calculation method.

Second, coupling is a physics concept that refers to a phenomenon whereby two or more systems or forms of motion interact [35]. The coordination degree is the degree to which the internal system factors are in harmony during the development process and reflects the system trends as it moves from disorder to order [36,37]. A coupling coordination degree is introduced to quantitatively analyze the degree of internal system coupling in regional green economy development, determine whether the coordination status of each subsystem is good or bad, clarify the role of each subsystem in the green economy development, and determine a lateral regional green economy development measurement.

**Figure 2.** Research framework for the sustainable development of the green economy.

#### *3.2. Model Solution*

**Step 1.** Calculate the normalized measurement matrix.

With *n* indicators to measure the regional green economy development over m years, the initial measurement matrix can be expressed as

$$A = \begin{bmatrix} a\_{11} & a\_{12} & \cdots & a\_{1n} \\ a\_{21} & a\_{22} & \cdots & a\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a\_{m1} & a\_{m2} & \cdots & a\_{mn} \end{bmatrix}$$

where *aij* represents the measurement value of the *j*-th measurement indicator in the *i*-th year.

To eliminate the different dimensions and orders of magnitude in the original variable sequences and guarantee the reliability of the measurement analysis results, a linear dimensionless processing method is applied to the initial measurement matrix. Where *aij* is the benefit criteria set,

$$a\_{ij}^{'} = \frac{a\_{ij} - \min\_{1 \le i \le m} a\_{ij}}{\max\_{1 \le i \le m} a\_{ij} - \min\_{1 \le i \le m} a\_{ij}} \tag{1}$$

and where *aij* is the cost criteria set,

$$a\_{ij}^{-\prime} = \frac{\max\_{1 \le i \le m} a\_{ij} - a\_{ij}}{\max\_{1 \le i \le m} a\_{ij} - \min\_{1 \le i \le m} a\_{ij}} \tag{2}$$

To eliminate the impact of the index value normalized logarithmic calculations, it is necessary to coordinate the *aij* translation, which is expressed as

$$f\_{\text{ij}} = a\_{\text{ij}}^{\prime} + \theta \tag{3}$$

where *θ* is the translational amplitude for the relevant reference, *θ* > min *aij* ; when the value for *θ* is closer to min *aij* , the measurement result is more significant [38]; therefore, this study uses *θ* = 0.001; and the measured matrix F = [ *fij*] *<sup>m</sup>*×*<sup>n</sup>* is then obtained.

**Step 2.** Determine the entropy weight for each indicator.

From basic information theory principles, information is a measure of the degree of order in a system and entropy is a measure of the degree of disorder in a system; therefore, the smaller the indicator information entropy, the greater the information provided by the indicator, the greater the effect in the comprehensive evaluation, and the higher the weight. The entropy weight rule is an objective weighting method that makes weight judgments based on the size of the data information load. Here, the objective weight for each index is determined by the degree of dispersion in the measurement index, as this reduces the influence of human subjectivity on the evaluation result and makes the evaluation results more realistic.

From the information entropy definition, the entropy value and entropy weight for the *j*-th measure index are calculated as

$$E\_j = -\frac{1}{\ln m} \sum\_{i=1}^{m} \frac{\mathbf{f}\_{ij}}{f\_j} \ln \frac{\mathbf{f}\_{ij}}{\mathbf{f}\_j} \text{ ( $j = 1, 2, \dots, n$ )}\tag{4}$$

$$w\_{\vec{j}} = \frac{1 - E\_{\vec{j}}}{n - \sum\_{j=1}^{n} E\_{\vec{j}}} (\vec{j} = 1, \, 2, \, \dots, n) \tag{5}$$

$$\text{where } f\_{\bar{j}} = \sum\_{i=1}^{m} f\_{\bar{i}j} \tag{6}$$

**Step 3.** Determine the entropy weight for each subsystem.

For the multi-layered structural measurement system, based on the addition of the information entropy weight, the entropy weight *Wk* for each subsystem in the corresponding criterion layer is calculated using the entropy weight *wj* of each index layer. The entropy weight *Pj* in the subsystems corresponding to each index is then obtained, after which the development scores in each subsystem *Zik* are calculated. It is assumed that there are s measure indicators under the *k*-th subsystem.

Calculate the entropy weight of each subsystem in the corresponding criterion layer *Wk*

$$\mathcal{W}\_k = \sum\_{j=s(k-1)+1}^s w\_j \begin{pmatrix} k=1 \ 2 \ 3 \ 4 \end{pmatrix} \tag{7}$$

Calculate the entropy weight in the subsystem corresponding to each index *Pj*

$$p\_j = \frac{w\_j}{W\_k} \text{ ( $j = 1, 2, \dots, n$ )}\tag{8}$$

Calculate the development scores for each subsystem in the *i*-th year *Zik*

$$Z\_{ik} = \sum\_{j=s(k-1)+1}^{s} f\_{i\bar{j}} p\_{\bar{j}}\ (k=1,2,3,4) \tag{9}$$

**Step 4.** Calculate the comprehensive evaluation coefficient *Cj* for the sustainable development of the green economy system based on TOPSIS.

A larger *Cj* value indicates that the green economy is more sustainable; therefore, *Cj* indicates the overall development of the regional green economy at the macro level. The specific steps are

(1) Determine the positive ideal solution *Z*<sup>+</sup> *ik* and the negative ideal solution *Zik* for each subsystem

$$Z\_{ik}^{+} = \{ \max Z\_{ik} | i = 1, 2, 3, \dots, m \} = \{ Z\_{i1}^{+}, Z\_{i2}^{+}, Z\_{i3}^{+}, Z\_{i4}^{+} \} \tag{10}$$

*Sustainability* **2019**, *11*, 280

$$Z\_{i\bar{k}} = \{ \min Z\_{i\bar{k}} | i = 1, 2, 3, \dots, m \} = \{ Z\_{i1}^-, Z\_{i2}^-, Z\_{i3}^-, Z\_{i4}^- \} \tag{11}$$

(2) The distance from the *i*-th year weighted value to the positive *Z*<sup>+</sup> *ik* and the negative ideal solution *Zik* can be calculated as

$$D\_i^+ = \sqrt{\sum\_{k=1}^4 \left(Z\_{ik} - Z\_{ik}^+\right)^2} \tag{12}$$

$$D\_i^- = \sqrt{\sum\_{k=1}^4 \left(Z\_{ik} - Z\_{ik}\right)^2} \tag{13}$$

(3) Calculate the relative closeness to the ideal solution and rank the performance order. The comprehensive evaluation coefficient *Ci* for the sustainable development of the regional green economy development system in the *i*-th year is expressed as

$$\mathcal{C}\_{i} = \frac{\mathcal{D}\_{i}^{-}}{\mathcal{D}\_{i}^{+} + \mathcal{D}\_{i}^{-}}, \ (0 \le \mathcal{C}\_{i} \le 1) \tag{14}$$

**Step 5.** Calculate the coupling coordination degree in the regional green economy.

Calculate the coupling coordination degree *K<sup>i</sup>* of the regional sustainable green economy in the *i*-th year to account for the coupling degree and the coordination degree correlation characteristics, which can then be used to assess the intensity and orderly development of the annual coupling between the internal subsystems in the regional sustainable green economy. The specific steps are as follows:

(1) *Mi* is the sustainable green economy multi-factor coupling degree in the *i*-th year; the larger the value, the better the state of the sustainable regional green economy, the calculation formula for which is

$$\mathcal{M}\_{l} = \left| \prod\_{k=1}^{k} \mathcal{Z}\_{lk} / \left( \frac{\sum\_{k=1}^{k} \mathcal{Z}\_{lk}}{k} \right)^{k} \right|^{\frac{1}{k}} \tag{15}$$

where *k* is the number of coupled subsystems needed to calculate the sustainable green economy development; that is, 2 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> 4, *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>+.

(2) *Qi* is the sustainable green economy comprehensive coordination index in the *i*-th year, which reflects the orderly and/or disorderly development; the more orderly the system, the better the development of the sustainable green economy; the calculation formula for which is

$$\mathbf{Q}\_{i} = \sum\_{k=1}^{4} Z\_{ik} \times \mathbf{W}\_{k} \tag{16}$$

(3) The coupling degree and comprehensive coordination index are combined to determine the coupling coordination degree, which is expressed as

$$\mathbf{K}\_{\dot{l}} = \left| \sqrt{\mathbf{M}\_{\dot{l}} \times \mathbf{Q}\_{\dot{l}}} \right| \tag{17}$$

The larger the *Ki*, the more cooperative the subsystems in that year, and the higher the coordination. The coordination degree is divided into 8 levels [39], as shown in Table 2.


**Table 2.** Coordination levels.
