Table 6. Optimal designs when demand is centered at the remote end
Uniform line and stop Uniform line and stop
Variable Heterogeneous design Uniform stop spacing
spacing spacing, and headway
𝐾, pax 11 11 12 22
𝐵∗(𝑥,𝑦), km [0.08, 2.00] (0.28) 0.24 0.24 0.25
𝑆(𝑥), km [0.10, 2.29] (0.19) [0.10, 2.41] (0.19) 0.18 0.20
𝐼
𝐻 (𝑥), min [4.4, 30.0] (10.1) [4.5, 30.0] (10.2) [3.1, 30.0] (14.1) 5.2
𝐼𝑝
𝐻 (𝑥), min [5.0, 30.0] (10.3) [5.0, 30.0] (10.5) [5.0, 30.0] (14.6) 5.3
𝐼𝑑
𝐴𝐶, h 122.23 122.72 117.34 145.66
𝑈𝐶, h 813.40 853.58 862.96 877.85
𝐺𝐶, h 935.62 976.30 980.30 1023.50
𝐺𝐶 gap 4.17% 0.41% 4.22% –
4.4.2 Sensitivity to the demand rate
In this subsection, we examine the terminal-centered demand pattern with Λ = Λ taking a smaller
𝑝 𝑑
value (400 patrons/h). The results are provided in Table 7. Comparing the cost savings with those in
Table 4 reveals that the percentage cost savings of the optimal heterogeneous design increase
moderately as demand diminishes. In addition, lower demand entails fewer lines served by smaller
vehicles operating at lower frequencies, which is a logical outcome. However, the stop spacings remain
largely unaffected by the demand rate.
4.4.3 Sensitivity to the service region’s size
In this analysis, we continue to use the terminal-centered demand pattern but examine a smaller service
region with 𝐿 = 1.5 km and 𝑊 = 1 km. To ensure a fair comparison, we keep the average demand
density (i.e., Λ /𝐿𝑊 and Λ /𝐿𝑊) consistent with the instance in Table 4. The results are summarized
𝑝 𝑑
in Table 8. The table indicates that the percentage cost savings of the heterogeneous design increase as
the service region shrinks, especially the cost saving resulting from spatially-heterogenizing headways.
This time, the headways and stop spacings demonstrate relative insensitivity to changes in service
region size, while the average line spacing increases significantly as the service region contracts. The
optimal 𝐾 also decreases, since the demand carried by each line is lower.
Table 7. Optimal designs under terminal-centered demand with lower rates
Uniform line and stop Uniform line and stop
Variable Heterogeneous design Uniform stop spacing
spacing spacing, and headway
𝐾, pax 7 7 8 18
𝐵∗(𝑥,𝑦), km [0.13, 2.00] (0.25) 0.17 0.16 0.17
𝑆(𝑥), km [0.22, 3.00] (0.49) [0.22, 3.00] (0.49) 0.40 0.42
𝐼
𝐻 (𝑥), min [4.4, 30.0] (13.3) [4.6, 30.0] (13.5) [3.2, 30.0] (15.4) 6.7
𝐼𝑝
𝐻 (𝑥), min [5.0, 30.0] (13.5) [5.0, 30.0] (13.7) [5.0, 30.0] (15.8) 6.7
𝐼𝑑
𝐴𝐶, h 43.77 40.62 40.10 54.89
𝑈𝐶, h 173.70 175.65 182.58 195.01
𝐺𝐶, h 212.74 216.27 222.68 249.90
𝐺𝐶 gap 1.63% 2.88% 10.89% –
The results in Tables 4–8 reveal that spatially-heterogenizing headways always yields the largest
benefit. Nevertheless, heterogenizing stop spacings also contributes to moderate yet consistent
improvements.
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