## 2 TBM Selection Decision-Making Method
### TBM Selection Decision-Making Method  
The basic principle of the TOPSIS method is to rank the evaluation objects by the distance between the positive ideal solution and the negative ideal solution in the multi-objective decision-making problem. Each index of the positive ideal solution is optimal, which can be understood as a virtual optimal solution, while the negative ideal solution is the complete opposite. The TOPSIS method sorts the evaluation objects according to the proximity of the evaluation objects to the idealized target and evaluates the relative advantages and disadvantages of the existing objects. If the evaluation object is closest to the positive ideal solution, it is the optimal value, otherwise, it is the worst value. TOPSIS is a commonly used and effective method in multi-objective decision analysis. The specific steps of the model developed in this paper are as follows:  
Step 1. Build the initial evaluation matrix. Let the scheme set \(P=\{P_{1},\,P_{2},\,\cdots,\,P_{m}\}\) and the evaluation index set of each scheme \(r=\{r_{1},\,r_{2},\,\cdots,\,r_{n}\}\). The evaluation index \(r_{ij}\) refers to the \(j\)th evaluation index of the \(i\)th scheme, where \(i\in[1,m]\) and \(j\in[1,n]\), and the initial evaluation matrix can be expressed as Equation (8). \[P=\left\{r_{ij}\right\}_{n\times m}= \begin{pmatrix}r_{11}&r_{12}&\cdots&r_{1j} &\cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2j}&\cdots&r_{2n}\\ \vdots&\vdots&\ddots&\vdots&&\vdots\\ r_{i1}&r_{i2}&\cdots&r_{ij}&\cdots&r_{in}\\ \vdots&\vdots&&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\cdots&r_{nj}&\cdots&r_{mn}\\ \end{pmatrix}  \tag{8}\]  
Step 2. Weighted standardized decision matrix. In the TOPSIS method, the evaluation indexes can be divided into consumption indexes and profit indexes. For consumption indexes, the smaller the value, the better. For profit indexes, the larger the value, the better. Since each evaluation index has different dimensions and units of dimensions, it does not have comparability. In order to eliminate the incommensurability of the indexes, it is necessary to carry out dimensional unification of the evaluation indexes. For the standardized decision matrix, \(D=(\mathrm{d}_{ij})m\times n\). The calculation formula is shown as Equation (9):  
\[\left\{ \begin{array}{l}\mathrm{d}_{ij}=\dfrac{r_{ij}-\min(r_{ij})}{\max(r_ {ij})-\min(r_{ij})},\text{ when }\mathrm{d}_{ij}\text{ is profitability index}\\ \mathrm{d}_{ij}=\dfrac{\max(r_{ij})-r_{ij}}{\max(r_{ij})-\min(r_{ij})},\text{ when }\mathrm{d}_{ij}\text{ is consumptive index}\\ \end{array} \right. \tag{9}\]  
Multiply the column vector of matrix \(D\) by the total ranking weight \(X\) of the index level determined by the AHP method to obtain the weighted standardized decision matrix \(R\), shown as Equation (10). \[R=\left\{r_{ij}\right\}_{n\times n}= \begin{pmatrix}x_{1}d_{11}&x_{2}d_{12}& \cdots&x_{j}d_{1j}&\cdots&x_{n}d_{1n}\\ x_{1}d_{21}&x_{2}d_{22}&\cdots&x_{j}d_{2j}&\cdots&x_{n}d_{2n}\\ \vdots&\vdots&\ddots&\vdots&&\vdots\\ x_{1}d_{11}&x_{2}d_{22}&\cdots&x_{j}d_{ij}&\cdots&x_{n}d_{in}\\ \vdots&\vdots&&\vdots&\ddots&\vdots\\ x_{1}d_{n1}&x_{2}d_{n2}&\cdots&x_{j}d_{nj}&\cdots&x_{n}d_{nm}\\ \end{pmatrix}  \tag{10}\]  
Step 3. Calculate the distance between positive and negative ideal solutions. The positive ideal solution of the revenue index set \(J_{1}\) is the maximum value of the row vector, and the negative ideal solution is the minimum value of the row vector. The value of the consumption index set \(J_{2}\) is opposite to that and can be expressed as Equation (11). \[R^{+} =\{\left(\max_{n}d_{mn}|m\in J_{1}\right),\left(\min_{x_{n}}d_{mn}|m \in J_{2}\right)\} \tag{11}\] \[R^{-} =\{\left(\min_{x_{n}}d_{mn}|m\in J_{1}\right),\left(\min_{x_{n}}d_{ mn}|m\in J_{2}\right)\}\]  
where \(R^{+}\) and \(R^{-}\) are the positive ideal solution and negative ideal solution, respectively. The distance between the evaluation object and the ideal solution is shown in Equation (12).