# Multicriteria Route Planning for In-Operation Mass Transit under Urban Data
## 4 Multicriteria Route Planning for Mass Transit
### The Algorithm
The overall performance is evaluated in mean and standard deviation of solutions derived by different methods under various constrained settings for Table 4, and in hypervolume (HV) of Pareto set obtained by different methods under various constrained settings for Table 5. Additionally, Table 6 shows the probability for BiasSpan to retrieve a better solution compared to other methods. Figure 4: Pareto-efficient frontier and solutions under different area ranges in Chicago (**left**) and Tainan (**right**) dataset. All solutions for BiasSpan are located at (1, 1) and achieve the Pareto optimal. The illustrated Pareto-efficient frontier indicates that the solutions that are located on which retains no neighboring solutions that could obtain an improved satisfaction in certain criteria without losing in some other criteria (e.g., either length ratio or PF ratio for our evaluation criteria); that is, if there is no scope for further Pareto improvement, the solutions achieve the Pareto optimal among all candidates and constitute the Pareto-efficient frontier. Figure 5: PF ratio (**left**) and length ratio (**right**) for methods under area range in Chicago dataset. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multicolumn{2}{c}{**Instance\(\backslash\)Method**} & \multicolumn{1}{c}{**BiasSpan**} & **Dijlstra’s** & **BPS** & **IDDFS** & **Best-First** & **Distance-A\({}^{\star}\)** & **PF-A\({}^{\star}\)** \\ \hline
**City** & **Area** & **Indicator** & & & & & & & \\ \hline \multirow{11}{*}{Chicago} & \multirow{4}{*}{Overall} & Unit PF & **0.25 \(\pm\)



\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multicolumn{2}{c}{**Instance**} & \multicolumn{1}{c}{**Method**} & \multicolumn{1}{c}{**BiasSpan**} & \multicolumn{1}{c}{**Dijkstra/s**} & \multicolumn{1}{c}{**BFS**} & \multicolumn{1}{c}{**IDDFS**} & \multicolumn{1}{c}{**Best-First**} & \multicolumn{1}{c}{**Distance-A\({}^{\ast}\)**} & \multicolumn{1}{c}{**PF-A\({}^{\ast}\)**} \\ \hline
**City** & **Area** & **Indicator** & & & & & & & & \\ \hline \multirow{6}{*}{Middle} & Unit FF & **0.02 \(\pm\) 0.01** & 0.01 \(\pm\) 0.01 & \(5.2\times 10^{-3}\pm 3.7\) & \(3.6\times 10^{-3}\pm 2.6\times\) & \(3.9\times 10^{-3}\pm 3.2\



Figure 8: PF ratio (**left**) and length ratio (**right**) for methods under number of must-visit stations in Tainan dataset. Figure 6: PF ratio (**left**) and length ratio (**right**) for methods under area range in Tainan dataset. Figure 7: PF ratio (**left**) and length ratio (**right**) for methods under number of must-visit stations in Chicago dataset. Figure 3:0 Unit PF ratio (**left**) and runtime (**right**) for different methods under area range in Chicago dataset. For methods with more than half failure cases, their performance would not be displayed. Figure 3:1 Unit PF ratio (**left**) and runtime (**right**) for different methods under number of must-visit stations in Chicago dataset. Figure 3:2 Unit PF ratio (**left**) and runtime (**right**) for different methods under number of must-visit stations in Tainan dataset. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Instance\textbackslash{Method}} & \multicolumn{1}{c}{**BiasSpan**} & \multicolumn{1}{c}{**Dijkstra’s**} & \multicolumn{1}{c}{**BFS**} & \multicolumn{1}{c}{**IDDFS**} & \multicolumn{1}{c}{**Best-First**} & \multicolumn{1}{c}{**Distance-A\textasciiti



#### 5.3.1 Pareto-Optimal between Profit and Cost

Overall experimental results indicate that BiasSpan maintains high PF per unit length and success rate in the middle-scale and large-scale region, where its length ratio referring to the cost of route outperforms that of other methods. First, we compare the length and corresponding PF of routes generated by different methods in Figures 5-12.