boarding or alighting; (ii) feeder buses on any specific line maintain regular headways; (iii) patrons
arrive at the feeder stops randomly without consulting the service timetable; and (iv) all patrons waiting
at the terminal or a feeder bus stop will be picked up by the next arriving bus (i.e., no patron will be left
behind). These assumptions are commonly employed in the literature (e.g., Daganzo, 2010; Chen et al.,
2015; Fan et al., 2018).
The problem aims to minimize the generalized cost of the feeder system, consisting of the patrons’
travel cost and the agency’s operating cost. All cost components are expressed in units of hours; in other
words, the agency cost items originally expressed in monetary units are converted to hours to ensure
they can be added together with the patron costs. The following two sections develop these cost terms
incurred during one hour of bus operations.
2.2 Patrons’ travel time cost
The patrons’ travel time cost comprises three components: (i) the time spent accessing (for the patron-
collection direction) and egressing (for the distribution direction) feeder stops; (ii) the waiting time at
feeder stops and the transfer time at the terminal; and (iii) the in-vehicle travel time on feeder buses.
They are formulated in Sections 2.2.1 to 2.2.3, respectively.4 Note that while bus fare is another
component of patron cost, it is excluded from our generalized cost model. This exclusion is due to the
fare constituting a monetary transfer from patrons to the transit agency, thereby representing a cost for
patrons and a revenue source for the agency.
2.2.1 Access and egress cost
The total access and egress time per hour of bus operations, denoted by 𝐶 , is formulated as follows:
𝐴
𝐶 =
∫𝐿 ∫𝑊 𝑆𝐼(𝑥)+𝐵(𝑥,𝑦)
[𝜆 (𝑥,𝑦)+𝜆 (𝑥,𝑦)]𝑑𝑦𝑑𝑥 (1)
𝐴 𝑥=0 𝑦=0 4𝑣𝑊 𝑝 𝑑
where (𝑆 (𝑥)+𝐵(𝑥,𝑦))⁄4 represents the average access or egress distance per passenger originating
𝐼
from or destined for location (𝑥,𝑦); and 𝑣 denotes the walking speed.5 Note that the double integral
𝑊
part related to 𝐵(𝑥,𝑦) would not appear (i.e., the RHS of (1) would reduce to a single integral of
decision function 𝑆 (𝑥)) if stop spacings are not jointly optimized.
𝐼
2.2.2 Waiting and transfer cost
The total waiting and transfer time per hour of bus operations, denoted by 𝐶 , is formulated as:
𝑊
𝐶 = 𝐶 +𝐶 (2)
𝑊 𝑊𝑝 𝑊𝑑
4 In this paper, the total patron cost is formulated by summing the three patron travel time components. Alternatively, different
weighting coefficients could be applied to these components to account for the varying values of time during different stages
of travel.
5 Eq. (1) is based upon the assumption that a patron always chooses the nearest feeder line and the nearest feeder stop on that
line for boarding or alighting (see assumption (i) in Section 2.1). In a heterogeneous feeder network, a patron’s nearest stop
might not be situated on the closest line. Nevertheless, incorporating the choice of the nearest stop would considerably
complicate the modeling process, while only moderately improving model accuracy. Thus, we opt for this simple (and
conservative) approach for modeling patrons’ stop selection.
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