comprehensive CA models to explore the optimal fixed-route feeder design problem under spatially
heterogeneous demand. The models aim to minimize the sum of the feeder system’s operating cost and
the feeder patrons’ travel cost, as well as their transfer cost at the rail terminal. They account for the
impacts of boarding and alighting passengers on dwell times and transfer times, and optimize the
spatially varied stop spacings for each feeder line1, as well as the line spacings, headways, and vehicle
size. Additionally, our models can effectively harness the greater potential of schedule coordination by
incorporating coordination efforts in both patron collection and distribution directions.
The models are solved by combining the first-order condition results with an iterative search
algorithm. Extensive numerical analyses are performed to examine: (i) the accuracy of the CA models;
(ii) the properties of the optimal design; (iii) the sensitivity of cost savings to the value of time, demand
pattern, demand rate, service region size, and alternative feeder line layouts; and (iv) the effects of
schedule coordination for each service direction individually, as well as for both directions
simultaneously. The results unveil new insights with practical implications.
The main contributions of this paper are summarized as follows:
(i) To our best knowledge, we are the first to model the optimal fixed-route feeder network design
with heterogeneous stop spacings under heterogeneous demand by incorporating more realistic
dwell time and transfer delay models and the vehicle capacity constraint. Our results indicate
that the system cost reduction by considering heterogeneous stop spacings is moderate but
consistent, and is larger when the demand is more heterogeneous.
(ii) We explore the fuller benefit of schedule coordination between trunk and feeder services in
both service directions, acknowledging the asymmetry in schedule coordination between the
two service directions.
(iii) Through numerical analyses, we demonstrate how various operating factors (e.g., the demand’s
spatial pattern, demand rate, and service region size) affect the optimal feeder network design
and the cost savings compared to homogeneous designs and designs not considering
heterogeneous stop spacings.
In addition, we compare two alternative feeder line layouts and find that it consistently results in
lower costs for feeder buses to travel along the shorter side of the service region to make stops and
along the longer side to the rail station without stopping.
The rest of the paper is structured as follows. Section 2 presents the problem setup and the CA
model formulation. Analytical properties of the optimal solution and the solution algorithm are furnished
in Section 3. Section 4 discusses the numerical case study results and findings. Finally, conclusions and
future work are outlined in Section 5.
2. Methodology
Assumptions and settings of our feeder design problem are presented in Section 2.1. The cost models
1 Incorporating the optimization of heterogeneous stop spacing increases the modeling complexity, as our CA models involve
a bivariate decision function (i.e., stop spacing is a function of two spatial coordinates, 𝑥 and 𝑦) and double integrals of that
decision function; see Section 2.
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