schedule coordination in the collection direction only, in the distribution direction only, and in both
directions simultaneously. The percentage of generalized cost savings compared to the uncoordinated
case is plotted against 𝐻 ∈ {5,7.5,10} minutes in Fig. 9. Other parameters remain the same as those
𝑡
in Section 4.3 (with 𝜃 = 20$/h).
Fig. 9 demonstrates the sizeable benefits of schedule coordination. Coordination in the collection
direction brings cost savings ranging from 7.42% to 10.95%. Coordination in the distribution direction
yields a smaller but still noteworthy benefit (-0.04% to 7.35%), particularly when 𝐻 is large.
𝑡
Moreover, coordinating in both directions achieves savings of up to 20.10%. Intriguingly, this value
slightly exceeds the combined percentage cost savings of coordinating in individual directions. This can
be attributed to two possible reasons: (i) waiting time constitutes a large portion of the generalized cost;
and (ii) simultaneously coordinating the schedules in both directions may facilitate more efficient
optimization of line spacings, stop spacings, and the vehicle size (factors that concurrently impact the
generalized costs in both directions).
Additionally, the figure indicates that coordination becomes increasingly advantageous as trunk-
line headway grows. This is owing to two reasons: (i) the waiting time saved by coordination rises with
trunk-line headway (see Section 2.5); and (ii) the agency cost savings also grow as larger feeder bus
headways (in multiples of 𝐻 ) are enforced.
𝑡
Fig. 9 Generalized cost saving of schedule coordination
5. Conclusions
We propose continuous approximation models for designing an optimal heterogeneous feeder bus
network catering to spatially heterogeneous demand to and from a rail terminal. These models identify
the optimal feeder service headways, line spacings, stop spacings, and vehicle size that minimize the
generalized system cost. An iterative method was developed to find the optimal solution, leveraging the
analytical properties of the optimal solution. Extensive numerical analyses were conducted to validate
the accuracy of the CA models, showcase the effectiveness of our solution method, and unveil new
insights that have practical implications.
Our models stand out by considering the effects of passenger boarding and alighting on bus dwell
times and transfer times, while optimizing heterogeneous stop spacings alongside line spacings and
headways. Numerical results show that more accurately modeling dwell and transfer times yields a 2–
5% difference in generalized cost compared to prior models with constant dwell times. Furthermore,
optimizing heterogeneous stop spacings results in additional cost savings up to 4% compared to designs
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