Next, we compare our heterogeneous design against three benchmark designs, including: (ⅰ) an
optimal design with heterogeneous line spacings and headways but uniform stop spacing; (ⅱ) an optimal
design featuring spatially-heterogeneous headways with uniform line and stop spacings; and (ⅲ) an
optimal design with a uniform line spacing, stop spacing, and headway. All four optimal designs use
the same demand pattern and a value of time set at 20 $/h. Results are presented in Table 4, where the
range of optimal values for each heterogeneous decision variable is shown in brackets, followed by the
average value in parentheses.
Table 4. Optimal heterogeneous, partially-homogeneous, and homogeneous designs under terminal-centered demand
Uniform line and Uniform line and stop
Variable Heterogeneous design Uniform stop spacing
stop spacings spacings, and headway
𝐾∗ 9 9 12 22
𝐵∗(𝑥,𝑦), km [0.12, 2.00] (0.23) 0.16 0.15 0.16
𝑆∗(𝑥), km [0.08, 3.00] (0.24) [0.09, 3.00] (0.25) 0.20 0.22
𝐼
𝐻∗ (𝑥), min [3.9, 30.0] (10.7) [4.0, 30.0] (11.0) [3.0, 30.0] (14.2) 5.2
𝐼𝑝
𝐻∗ (𝑥), min [5.0, 30.0] (11.1) [5.0, 30.0] (11.3) [5.0, 30.0] (14.8) 5.2
𝐼𝑑
𝐴𝐶, h 98.51 102.02 94.23 139.89
𝑈𝐶, h 465.89 470.19 487.80 507.19
𝐺𝐶, h 564.39 572.20 582.04 647.09
1.36% (compared to 1.68% (compared to 10.05% (compared
𝐺𝐶 saving design with uniform design with uniform line to the homogeneous –
stop spacing) and stop spacings) design)
The last row of Table 4 demonstrates that, compared to the homogeneous design, heterogenizing
the headway results in a cost saving of 10.05%. Additionally, further heterogenizing the line spacing
yields an extra saving of 1.68%, while heterogenizing the stop spacing contributes an added saving of
1.36%. Note that each number is computed by comparing the 𝐺𝐶 against a “more homogeneous”
design, which is the one in the column immediately to the right of the current design in Table 4. The
last number is a conservative estimate, as it is derived from a design with optimized (though uniform)
stop spacing. Should stop spacing be disregarded or assumed constant in the model, as in previous
studies such as Chang and Schonfeld (1991) and Yang et al. (2020), even greater savings would be
realized. For example, compared to using a fixed stop spacing of 500 m, the cost saving stemmed from
our heterogeneous design amounts to 11.37%. Furthermore, larger savings are also observed under
alternative demand patterns, as will be discussed shortly.
Lastly, we compare the generalized cost of the heterogeneous design to the cost of the optimal
design without accounting for the impacts of boarding and alighting passengers on dwell times and
transfer times (details are omitted in the interest of brevity). In the latter design, the bus dwell time is
set at 30 seconds, as in Gu et al. (2016). Numerical results across various demand levels reveal that the
relative percentage error lies between 2–5%. This underscores the importance of incorporating
passenger boarding and alighting processes in the model.
4.4 Sensitivity analyses
In this section, we examine the sensitivity of the optimal heterogeneous design and cost savings in
comparison to the benchmark designs. Sensitivity analyses concerning the demand patterns, demand
rates, service region sizes, and feeder line layouts are furnished in subsections 4.4.1–4.4.4, respectively.
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