are formulated in Sections 2.2 and 2.3. Section 2.4 furnishes the optimization model without schedule
coordination. Schedule coordination between the trunk line service and the feeders is modeled in Section
2.5.
2.1 Problem setup
Figure 1 depicts a schematic representation of a feeder network. A feeder bus terminal, marked by the
black dot, is located at the left-bottom corner of the figure. This terminal can be a trunk-line (e.g., rail
transit) station in a city’s suburban area or an intercity rail station.2 It is assumed that the terminal’s
catchment area, which represents the region served by the feeder network, is a rectangular zone centered
at the terminal, encompassing a dense street grid. A quarter of this catchment area is defined as Ω ≡
{(𝑥,𝑦)|0 ≤ 𝑥 ≤ 𝐿,0 ≤ 𝑦 ≤ 𝑊}; see Fig. 1.
Owing to the symmetric nature of the area, it is only necessary to identify the optimal design for
feeder lines carrying patrons within Ω, both to and from the terminal. We presume that feeder buses
collect or distribute passengers along the 𝑦-axis, while they undertake nonstop travel along the 𝑥-axis
either towards or away from the terminal, as illustrated by the thin solid lines in the figure. The positions
of feeder bus stops are indicated by black squares.
Fig. 1 The feeder system layout
Travel demands in this service area are expressed as continuous, time-invariant density functions.3
Specifically, the demand density from location (𝑥,𝑦) ∈Ω to the terminal (in the patron-collection
direction) is denoted by 𝜆 (𝑥,𝑦), while the demand density from the terminal to (𝑥,𝑦)∈ Ω (in the
𝑝
patron-distribution direction) is denoted by 𝜆 (𝑥,𝑦). The line spacing, 𝑆 (𝑥), and the headways in the
𝑑 𝐼
patron-collection and distribution directions, 𝐻 (𝑥) and 𝐻 (𝑥) respectively, are functions of 𝑥. The
𝐼𝑝 𝐼𝑑
stop spacing, 𝐵(𝑥,𝑦) , is a function of both 𝑥 and 𝑦 . In the context of the CA framework, these
decision functions are assumed to be integrable on 0 ≤ 𝑥 ≤ 𝐿 and Ω, respectively.
We further assume that: (i) each patron chooses the nearest feeder stop on the closest line for
2 The terminal may also represent a central business district in a medium or small-sized city, a shopping center within an urban
district, or an airport. In such instances, the feeder service effectively functions as a shuttle service. Nevertheless, our models
remain applicable to these cases, requiring only minor modifications.
3 The models can readily be adapted to real-world situations where demand densities fluctuate over time, e.g., between peak
and off-peak periods. To account for these variations, one can model the costs associated with different time periods and
aggregate them accordingly; see Wu et al. (2020) for an example.
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