Table 8. Optimal designs with a 𝟏×𝟏.𝟓 km2 service region
Uniform line and stop Uniform line and stop
Variable Heterogeneous design Uniform stop spacing
spacing spacing, and headway
𝐾, pax 6 6 9 12
𝐵∗(𝑥,𝑦), km [0.10, 1.00] (0.20) 0.12 0.11 0.13
𝑆(𝑥), km [0.17, 1.50] (0.35) [0.17, 1.50] (0.36) 0.37 0.29
𝐼
𝐻 (𝑥), min [3.6, 30.0] (10.7) [3.9, 30.0] (11.0) [3.0, 30.0] (12.8) 4.4
𝐼𝑝
𝐻 (𝑥), min [5.0, 30.0] (11.2) [5.0, 30.0] (11.4) [5.0, 30.0] (13.6) 5.8
𝐼𝑑
𝐴𝐶, h 17.78 18.90 15.68 28.18
𝑈𝐶, h 85.23 86.35 93.91 108.09
𝐺𝐶, h 103.01 105.25 109.60 136.27
𝐺𝐶 gap 2.13% 3.97% 19.57% –
4.4.4 Sensitivity to the feeder line layout
In this subsection, we compare two alternative feeder line layouts: (i) feeder buses travel along the 𝑦-
direction for patron pick-up or drop-off and move along the 𝑥 -direction to and from the terminal
without stopping; and (ii) feeder buses load and unload passengers along the 𝑥-direction and perform
𝑊
nonstop travel along the 𝑦-direction. We let the rectangular service region’s aspect ratio, , vary from
𝐿
0.1 to 2, and plot the generalized costs per patron for the two layouts in Fig. 8. The service region’s area
𝐿×𝑊 and other parameter values are kept consistent with those in Section 4.3 (with 𝜃 = 20 $/h).
The figure reveals that the generalized cost is consistently lower when buses make stops along the
shorter side of the service region. In this configuration, more bus lines are deployed along the longer
side, resulting in each bus carrying fewer patrons and reduced boarding delays. The cost gap between
𝑊
the two layouts is substantial when the service region is “slim.” For example, with = 2 or 0.5, the
𝐿
more efficient layout outperforms the alternative by 6.49% in generalized cost. This finding highlights
the importance of selecting the appropriate layout, particularly when the service region is more oblong
than square. Notably, this straightforward result has not been previously documented in the literature;
for example, Sivakumaran et al. (2012) arranged the feeder lines along the longer side of the service
region, which our analysis indicates is a suboptimal design.
Fig. 8 Comparison between two alternative feeder line layouts
4.5 Benefits of schedule coordination
We now explore the benefits of schedule coordination. The feeder network is reoptimized considering
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