# An Optimization Model for a Desert Railway Route Scheme Based on Interval Number and TOPSIS
## 2 Research Methods
### Optimization Model of the Desert Railway Route Scheme
#### 2.4.3 TOPSIS Method  
TOPSIS is a suitable approach to solving mixed multi-attribute decision-making problems with the advantages of simple calculation and a wide application range [40]. The optimization of railway alignment schemes is a multi-level and multi-factor decision-making process involving qualitative and quantitative indicators. Therefore, TOPSIS can be effectively employed for the evaluation of railway schemes. The decision-making problem is commonly solved by sorting the positive and negative ideal solutions of the scheme. The basic steps for analyzing unknown problems are provided as follows:  
Step 1: Construct the evaluation matrix for the railway route scheme.  
If there are \(m\) railway schemes and \(n\) evaluation indexes, the scheme set can be stated by \(A\) = {\(A_{1}\), \(A_{2}\),..., \(A_{m}\)}, and the indicators set can be expressed as \(C\) = {\(C_{1}\), \(C_{2}\),..., \(C_{n}\)}, where the factors \(C_{1}\)\(\sim\)\(C_{k}\) represent the quantitative indicators with an accurate value, and \(C_{k+1}\)\(\sim\)\(C_{n}\) stand for the qualitative indicators associated with the interval numbers. The values of the quantitative and qualitative indexes of the scheme \(A_{i}\) for the index \(C_{i}\) are  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost Indicators** & **Benefit Indicators** & **Interval Range** \\ \hline Very small & Very bad & [0, 1.5] \\ Smaller & Worst & [1.5, 3] \\ Small & Bad & [3, 4.5] \\ Average & Average & [4.5, 5.5] \\ Large & Good & [5.5, 7] \\ Larger & Better & [7, 8.5] \\ Very large & Very good & [8.5, 10] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Conversion rule between linguistic scales and interval numbers.  
The positive ideal solution \(Z^{+}\) and the negative ideal solution \(Z^{-}\) of the decision scheme are determined according to the following relations:  
\[Z^{+}=(z_{1}^{+},z_{2}^{+},\cdots,z_{n}^{+})=(\max_{i}[z_{ij}^{L},z_{ij}^{L}]) \tag{21}\]  
\[Z^{-}=(z_{1}^{-},z_{2}^{-},\cdots,z_{n}^{-})=(\max_{i}[z_{ij}^{L},z_{ij}^{L}]) \tag{22}\]  
in which \(z_{ij}^{+}=\left[\underset{i}{\max}\left\{z_{ij}^{L}\right\},\underset{i}{ \max}\left\{z_{ij}^{L}\right\}\right]\), \(z_{ij}^{-}=\left[\underset{i}{\max}\left\{z_{ij}^{L}\right\},\underset{i}{ \max}\left\{z_{ij}^{L}\right\}\right]\).  
Step 5: Determine the distance between the alternative and the ideal  
The distance \(D^{+}\) from each alternative to the positive ideal solution and the distance \(D^{-}\) from the negative ideal solution are, respectively, calculated using the following relations:  
\[D^{+}=\sqrt{\sum_{j=1}^{n}(z_{ij}^{+}-Z_{j}^{+})}=\left(\sum_{j=1}^{n}\frac{ \sqrt{2}}{2}\sqrt{(z_{ij}^{L}-Z_{j}^{L})^{2}-(z_{ij}^{L}-Z_{j}^{L})^{2}} \right)^{1/2} \tag{23}\]  
\[D^{-}=\sqrt{\sum_{j=1}^{n}(z_{ij}^{-}-Z_{j}^{-})}=\left(\sum_{j=1}^{n}\frac{ \sqrt{2}}{2}\sqrt{(z_{ij}^{L}-Z_{j}^{L})^{2}-(z_{ij}^{L}-Z_{j}^{L})^{2}} \right)^{1/2} \tag{24}\]  
Step 6: Calculate relative closeness  
The design schemes are sorted according to the value of relative closeness. The smaller value of \(D_{i}\) indicates that the scheme is better, which is calculated as follows:  
\[D_{i}=\frac{D_{i}^{+}}{D_{i}^{+}+D_{i}^{-}} \tag{25}\]