4.4.1 Sensitivity to the demand pattern
We investigate the benefit of heterogenous design under two distinct demand patterns for a high-wage
city (𝜃 = 20 $/h). The first pattern remains terminal-centered but exhibits a more concentrated spatial
distribution with 𝜎 = 𝜎 = 𝐿⁄8 and 𝜎 = 𝜎 = 𝑊⁄8 . The other parameter values are
𝑥𝑝 𝑥𝑑 𝑦𝑝 𝑦𝑑
consistent with those in Section 4.3. The outcomes of the optimal heterogeneous design and the three
benchmark designs are presented in Table 5. The last row of this table is derived in a similar fashion to
Table 4.
Table 5. Optimal designs when demand is more concentrated near the terminal
Uniform line and stop Uniform line and stop
Variable Heterogeneous design Uniform stop spacing
spacing spacing, and headway
𝐾, pax 8 13 13 33
𝐵∗(𝑥,𝑦), km [0.09, 2.00] (0.34) 0.13 0.13 0.14
𝑆(𝑥), km [0.03, 3.00] (0.23) [0.04, 3.00] (0.26) 0.13 0.20
𝐼
𝐻 (𝑥), min [3.9, 30.0] (19.4) [4.1, 30.0] (19.7) [3.0, 30.0] (21.6) 5.9
𝐼𝑝
𝐻 (𝑥), min [5.0, 30.0] (19.6) [5.0, 30.0] (19.8) [5.0, 30.0] (21.9) 5.9
𝐼𝑑
𝐴𝐶, h 83.42 88.21 96.09 142.72
𝑈𝐶, h 313.50 322.88 330.84 384.98
𝐺𝐶, h 396.92 411.09 426.93 527.70
𝐺𝐶 gap 3.45% 3.71% 19.1% –
Table 5 shows significantly larger percentage cost savings under the more concentrated demand
pattern compared to Table 4. For example, heterogenizing the stop spacings now produces a 3.45% cost
saving. This suggests that the heterogeneous design is more advantageous when demand is more
spatially heterogeneous. Also, designs with higher degrees of heterogeneity tend to favor smaller feeder
buses, as the optimal spatially-heterogeneous line and stop spacings align more closely with the
heterogeneous demand (see Fig. 7), rendering the patrons more evenly distributed across the lines and
vehicles.
The second demand pattern employs the same standard deviation parameters as in Section 4.3, but
with demand centered at the far end of the service region, i.e., 𝜇 = 𝜇 = 𝐿 and 𝜇 = 𝜇 = 𝑊.
𝑥𝑝 𝑥𝑑 𝑦𝑝 𝑦𝑑
This remotely-centered pattern also has real-world examples.10 Other parameters remain the same as
before. The results under this demand pattern are provided in Table 6.
Table 6 demonstrates that merely heterogenizing stop spacings results in an even greater cost
saving of 4.17%. This is likely due to the better alignment of optimal stop spacings with this demand
pattern. Specifically, 𝐵∗(𝑥,𝑦) is small near the service region’s top edge due to high demand density
and low onboard patron flow, while it is large near the bottom owing to low demand density and high
onboard patron flow; see Eq. (20g). Recall that the advantages of our design would be even more
pronounced when compared to previous feeder optimization models that utilize a constant, non-
optimized stop spacing.
10 For instance, the Palm Springs and Fairview Park in Hong Kong are large-scale residential estates housing most of the
population in the surrounding region. However, these estates are situated several kilometers away from the nearest metro
station.
20