patrons disembarking a trunk-line vehicle arrive at the feeder stop. In this case, coordination can be
accomplished even when the terminal serves multiple trunk lines operating on distinct schedules.
For simplicity, we assume that the terminal serves only one trunk line, with patrons traveling to
and from a city center using this line. This assumption facilitates coordination between feeder and trunk-
line services. Specifically, to achieve coordination in the collection direction, a feeder line’s headway
must be an integer multiple of 𝐻 , i.e.,
𝑡
𝐻 (𝑥)= 𝑘𝐻 (𝑘 = 1,2,3,…); (16)
𝐼𝑝 𝑡
while in the distribution direction, all feeder-line headways must equal 𝐻 , i.e.,
𝑡
𝐻 (𝑥)= 𝐻 , ∀𝑥 ∈[0,𝐿]. (17)
𝐼𝑑 𝑡
This highlights the asymmetry in the two travel directions when coordination is implemented. We
further assume that feeder buses always arrive on schedule and that feeder bus layover time costs are
disregarded (e.g., Kim and Schonfeld, 2014; Mei et al., 2021).
Under schedule coordination, the total waiting and transfer times in both travel directions,
originally represented by Eqs. (3) and (4), are replaced by the following equations:
𝐶 ′ = ∫𝐿 ∫𝑊 𝐻𝐼𝑝(𝑥) 𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥+∫𝐿 ∫𝑊 (𝜏 𝑆 (𝑥)𝐻 (𝑥)∙∫𝑊 𝜆 (𝑥,𝑦)𝑑𝑦+
𝑊𝑝 𝑥=0 𝑦=0 2 𝑝 𝑥=0 𝑦=0 𝑎 𝐼 𝐼𝑝 𝑦=0 𝑝
𝑡 )𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥 (18)
𝑓−𝑡 𝑝
𝐶 ′ = ∫𝐿 ∫𝑊 (𝑡 +𝜏 𝑆 (𝑥)𝐻 ∙∫𝑊 𝜆 (𝑥,𝑦)𝑑𝑦)𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥 (19)
𝑊𝑑 𝑥=0 𝑦=0 𝑡−𝑓 𝑏 𝐼 𝑡 𝑦=0 𝑑 𝑑
In Eq. (18), the second term consists of the total patron alighting time loss and the total transfer
delay. Note that the waiting time for the trunk-line vehicle,
𝐻𝑡,
is eliminated due to the coordination,
2
and the alighting time loss term in Eq. (3) is doubled. The derivation is illustrated in Fig. 3 for a case of
𝑘 = 1. Eq. (19) is developed in a similar manner.
The remaining part of the optimization model involving schedule coordination is identical to (15a)–
(15f).
Fig. 3 Patron delays at the terminal under schedule coordination with 𝒌=𝟏 (the collection direction)
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