3.2.2. Super-SBM Model

Since the traditional DEA model has the deviation of efficiency value caused by the relaxation of input and output, the undesirable output was incorporated into the evaluation system [52]. The non-radial and non-directional Super-SBM model based on relaxation variables was used to achieve the effective ordering of decision-making units [53]. Suppose there are *n* DMU (decision units), and each DMU has *m* input indicators, *s*<sup>1</sup> desirable output indicators, *s*<sup>2</sup> undesirable output indicators, and *x*, *ye* , and *y<sup>u</sup>* are the elements of the corresponding input matrix, desirable output matrix, and undesirable output matrix, respectively. Input matrix *<sup>X</sup>* = [*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3, ··· , *xn*] ∈ *<sup>R</sup>m*×*n*, and desirable output matrix *Y<sup>e</sup>* = *ye* <sup>1</sup>, *<sup>y</sup><sup>e</sup>* <sup>2</sup>, *<sup>y</sup><sup>e</sup>* <sup>3</sup>, ··· , *<sup>y</sup><sup>e</sup> n* ∈ *<sup>R</sup>s*1×*n*. The undesirable output matrix *Y<sup>u</sup>* = *yu* <sup>1</sup> , *<sup>y</sup><sup>u</sup>* <sup>2</sup> , *<sup>y</sup><sup>u</sup>* <sup>3</sup> , ··· , *<sup>y</sup><sup>u</sup> n* <sup>∈</sup> *<sup>R</sup>s*2×*n*. The Super-SBM model containing the undesired outputs is:

$$\begin{cases} \min \rho = \frac{\frac{1}{m} \sum\_{i=1}^{n} \frac{\overline{y}}{\tau\_{ik}^{d}}}{\frac{1}{s\_{1} + s\_{2}} \left( \frac{\overline{y}\_{i-1}^{d} \overline{y}^{d}}{\overline{y}\_{ik}^{d}} + \frac{\overline{y}\_{i-1}^{d} \overline{y}^{d}}{\overline{y}\_{ik}^{d}} \right)} \\\\ \overline{x} \ge \sum\_{j=1, \neq k}^{n} x\_{ij} \lambda\_{j}; \ \overline{y}^{d} \le \sum\_{j=1, \neq k}^{n} y\_{rj}^{c} \lambda\_{j}; \ \overline{y}^{d} \ge \sum\_{j=1, \neq k}^{n} y\_{tj}^{d} \lambda\_{j}; \\\\ \overline{x} \ge x\_{k}; \ \overline{y}^{c} \le y\_{k}^{c}; \ \overline{y}^{u} \ge y\_{k}^{u} \\\ \lambda\_{j} \ge 0, \ i = 1, 2, \cdots, m; j = 1, 2, \cdots, n, \ j \ne 0; \\\ r = 1, 2, \cdots, s\_{1}; t = 1, 2, \cdots, s\_{2} \end{cases} \tag{1}$$

*x*, *ye*, and *y<sup>u</sup>* represent the input, desirable output, and undesirable output vectors considering the slack variables, respectively, *j* represents the decision unit, *n* is the number of decision-making units, *k* is the production period, and *λ<sup>j</sup>* is the weight vector of decisionmaking units. *ρ* is the efficiency value, *ρ* ≥ 1 is a relatively effective decision unit, and 0 < *ρ* < 1 is a relatively invalid decision unit.

*3.3. Evaluation Index System and Evaluation Method for the Tourism Economy's High-Quality Development (TEHQD)*

3.3.1. Evaluation Index System of the Tourism Economy's High-Quality Development (TEHQD)

The evaluation index system of TEHQD was established from five dimensions of "innovation, coordination, green, openness and sharing", as shown in Table 2. A21 tourism R&D expenditure is represented by "the whole society R&D expenditure" multiplied by "the ratio of tourism production value to the gross national economic product". A22 is represented by "R&D personnel in the whole society" multiplied by "ratio of tourism employees to total employment in the region". A23 is represented by "total social fixed asset investment" multiplied by "ratio of tourism output value to GDP". B24 is represented by the difference between "turnover of local passengers" and "total turnover of national passengers". C22 is represented by the ratio of "garden green space area" to "total urban area". D22 is represented by the ratio of "international tourists per 10,000 people" to "tourism employees". E22 is expressed by the ratio of "park area" to "total population".


**Table 2.** Evaluation index system of the tourism economy's high-quality development (TEHQD).

3.3.2. Entropy Value Method

The method of assigning weight to entropy can avoid subjective judgment and ensure a scientific and effective index score [54]. First of all, standardized treatment should be carried out according to the basic indicators, and the formula is as follows:

$$\mathbf{x}\_{ij} = \begin{cases} \frac{\mathbf{X}\_{ij} - \mathbf{X}\_{j,\min}}{\mathbf{X}\_{j,\max} - \mathbf{X}\_{j,\min}} \text{ positive indicates} \\\\ \frac{\mathbf{X}\_{j,\max} - \mathbf{X}\_{ij}}{\mathbf{X}\_{j,\max} - \mathbf{X}\_{j,\min}} \text{ negative indicates} \end{cases} \tag{2}$$

In the formula, *Xij* is the original value of the *j* index of the *i* sample, *xij* is the normalized value of *Xij*. *Xj*,*max* and *Xj*,*min* are the maximum and minimum values of the

*j* index, respectively, and there are *m* samples and *n* indexes. Since there is a value of 0 after normalization, *xij* is shifted to the right by 1 unit to obtain *x ij* prime for logarithmic operation in the information entropy.

Determine the entropy value of item *j*:

$$H\_{\dot{j}} = -\frac{1}{\text{Inm}} \sum\_{i=1}^{\text{on}} \left( P\_{\dot{i}\dot{j}} \times \text{In} P\_{\dot{i}\dot{j}} \right) \; , \; P\_{\dot{i}\dot{j}} = \mathbf{x}\_{\dot{i}\dot{j}}' / \sum\_{i=1}^{\text{on}} \mathbf{x}\_{\dot{i}\dot{j}}' \tag{3}$$

Determine the weight of item *j*:

$$w\_{\dot{\jmath}} = \left(1 - H\_{\dot{\jmath}}\right) / \sum\_{j=1}^{n} \left(1 - H\_{\dot{\jmath}}\right) \tag{4}$$

The linear weighted model was adopted to measure the comprehensive development level of the tourism economy's high-quality development (TEHQD). The formula is as follows:

$$w\_E = \sum\_{j=1}^{n} w\_{j\epsilon} \mathbf{e}\_j \tag{5}$$

*vE* is the value of TEHQD, *wje* is the weight of each index of TEHQD, and *ej* is the standardized value of each index of TEHQD.

*3.4. Symbiotic Interaction Model between Tourism's Carbon Emission Efficiency (TCEE) and the Tourism Economy's High-Quality Development (TEHQD)*
