# An Optimization Model for a Desert Railway Route Scheme Based on Interval Number and TOPSIS
## 3 Case Study
### Index Weight Calculation
#### 3.2.2 Weight of the Maximum Dispersion Method  
According to the standardized interval number decision matrix and the maximum deviation method, the objective weight of the indicators is obtained by using Equations (1)-(6). The results are shown in Table 5.  
The ratio of bridges and tunnels, impact on nature reserves, and impact on the vegetation cover have the largest weight among the secondary indicators evaluated by the dispersion approach. The ratio of bridges and tunnels in Scheme A is 28.59%, which is 1.41 times that of Scheme C (i.e., the smallest one). Compared with the subgrade, the excavation volume and land area of the bridge are commonly smaller, which reduces the damage of the railway construction with respect to the surface. The wind-blown sand flow can pass under the bridge to avoid the problem of railway sand damage [41]. The Hotan-Ruoqiang railway is a single-track railway that employs diesel locomotives to provide power. During operation, the locomotive directly discharges CO, SO\({}_{2}\), and other exhaust gases into the areas along the line. These pollutants can damage the biological diversity and water safety in the wetland park.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{C_{1}}\) & **Sub-Criteria** & \(\mathbf{C_{11}}\) & \(\mathbf{C_{12}}\) & \(\mathbf{C_{13}}\) & \(\mathbf{C_{14}}\) & \(\mathbf{C_{15}}\) \\ \hline \(C_{11}\) & Length of the railway line & 1 & 4 & 2 & 5 & 3 \\ \(C_{12}\) & Land requisition amount & 1/4 & 1 & 1/3 & 2 & 1/2 \\ \(C_{13}\) & Project investment & 1/2 & 3 & 1 & 4 & 2 \\ \(C_{14}\) & Minimum curve radius & 1/5 & 1/2 & 1/2 & 1 & 1/3 \\ \(C_{15}\) & The ratio of the bridges and tunnels & 1/3 & 2 & 1/2 & 3 & 1 \\  & & & \multicolumn{4}{c}{\(CR=0.0152\)} \\ \hline \(\mathbf{C_{2}}\) & **Sub-Criteria** & \(\mathbf{C_{21}}\) & \(\mathbf{C_{22}}\) & \(\mathbf{C_{23}}\) & \(\mathbf{C_{24}}\) & \(\mathbf{C_{25}}\) \\ \hline \(C_{21}\) & Railway length in the moveable dune area & 1 & 4 & 3 & 2 & 5 \\ \(C_{22}\) & The angle between the railway and the wind direction & 1/4 & 1 & 1/2 & 1/3 & 2 \\ \(C_{23}\) & Temporary land occupation & 1/3 & 2 & 1 & 1/2 & 3 \\ \(C_{24}\) & Feasibility of the plant sand control & 1/2 & 3 & 2 & 1 & 4 \\ \(C_{25}\) & The scale of the sand control work & 1/5 & 1/2 & 1/3 & 1/4 & 1 \\  & & & & \multicolumn{4}{c}{\(CR=0.0152\)} \\ \hline \(\mathbf{C_{3}}\) & **Sub-Criteria** & \(\mathbf{C_{31}}\) & \(\mathbf{C_{32}}\) & \(\mathbf{C_{33}}\) & \(\mathbf{C_{34}}\) & \(\mathbf{C_{35}}\) \\ \hline \(C_{31}\) & Impact on nature reserves & 1 & 2 & 4 & 3 & 9/2 \\ \(C_{32}\) & Impact on the water environment & 1/2 & 1 & 3 & 2 & 5/2 \\ \(C_{33}\) & Impact on the wildlife & 1/4 & 1/3 & 1 & 1/2 & 1 \\ \(C_{34}\) & Impact on the vegetation cover & 1/3 & 1/2 & 2 & 1 & 3/2 \\ \(C_{35}\) & Impact on the basic farmland & 2/9 & 2/5 & 1 & 2/3 & 1 \\  & & & & \multicolumn{4}{c}{\(CR=0.0060\)} \\ \hline  & **Sub-Criteria** & \(\mathbf{C_{41}}\) & \(\mathbf{C_{42}}\) & \(\mathbf{C_{43}}\) & \\ \hline \(C_{41}\) & Speed limit & 1 & 2/3 & 2/5 & \\ \(C_{42}\) & Track maintenance & 3/2 & 1 & 2/3 & \\ \(C_{43}\) & Operation safety & 5/2 & 3/2 & 1 & \\  & & & & \multicolumn{4}{c}{\(CR=0.0010\)} \\ \hline \end{tabular}
\end{table}
Table 4: The judgment matrix of the second-level index.