𝐶 =
∫𝐿 ∫𝑊 𝐻𝐼𝑝(𝑥)
𝜆
(𝑥,𝑦)𝑑𝑦𝑑𝑥+∫𝐿 ∫𝑊 (𝜏𝑎𝑆
(𝑥)𝐻
(𝑥)∙∫𝑊
𝜆 (𝑥,𝑦)𝑑𝑦+𝑡 +
𝑊𝑝 𝑥=0 𝑦=0 2 𝑝 𝑥=0 𝑦=0 2 𝐼 𝐼𝑝 𝑦=0 𝑝 𝑓−𝑡
𝐻𝑡)𝜆
(𝑥,𝑦)𝑑𝑦𝑑𝑥 (3)
𝑝
2
𝐶 =
∫𝐿 ∫𝑊
(𝑡
+𝐻𝐼𝑑(𝑥) +𝜏𝑏𝑆
(𝑥)𝐻
∙∫𝑊
𝜆 (𝑥,𝑦)𝑑𝑦)𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥 (4)
𝑊𝑑 𝑥=0 𝑦=0 𝑡−𝑓 2 2 𝐼 𝑡 𝑦=0 𝑑 𝑑
where 𝐶 and 𝐶 denote the total waiting and transfer times for patrons traveling to and from the
𝑊𝑝 𝑊𝑑
terminal, respectively. The first term on the RHS of (3) corresponds to the total waiting time at the
feeder stops, with 𝐻 (𝑥)⁄2 being the average waiting time per patron for the bus line located at 𝑥.
𝐼𝑝
The second term on the RHS of (3) accounts for the total patron delay at the terminal, which comprises
three components: the time lost due to patrons’ alighting process from feeder buses, the transfer delay
from feeder to trunk-line transit, and the waiting time for a trunk-line vehicle. Here, 𝜏 denotes the
𝑎
alighting time per patron, 𝑡 the transfer delay per patron (including walking time), and 𝐻 the
𝑓−𝑡 𝑡
trunk-line service headway at the terminal. These components are illustrated in Fig. 2. Note in particular
𝑊
that 𝑆 (𝑥)𝐻 (𝑥)∫ 𝜆 (𝑥,𝑦)𝑑𝑦 represents the number of onboard patrons per bus on a line located
𝐼 𝐼𝑝 𝑦=0 𝑝
at 𝑥, while
𝜏𝑎𝑆
(𝑥)𝐻
(𝑥)∫𝑊
𝜆 (𝑥,𝑦)𝑑𝑦 corresponds to the average time loss per patron during the
2 𝐼 𝐼𝑝 𝑦=0 𝑝
alighting process at the terminal. Similarly, the three components on the RHS of (4) are: the transfer
delay from trunk-line to feeder, the waiting time for the feeder service, and the boarding time loss,
where 𝑡 denotes the transfer delay per patron and 𝜏 the boarding time per patron.
𝑡−𝑓 𝑏
The boarding and alighting time losses in (3) and (4) are significant, as all patrons will board or
alight at the terminal, resulting in the longest bus dwell time at this location. Regrettably, these factors
were overlooked in previous feeder design models (e.g., Chang et al., 1991; Sivakumaran et al., 2014;
Su et al., 2019; Badia et al., 2020; Yang et al., 2020).
Additionally, note that the formulas for 𝐶 and 𝐶 are asymmetrical. Thus, modeling a single
𝑊𝑝 𝑊𝑑
direction of travel or two symmetrical directions (e.g., Chang and Schonfeld, 1991; Kim and Schonfeld,
2012; Sivakumaran et al., 2012) is insufficient for optimal feeder system design.
Fig. 2 Patron delays at the terminal (in the patron-collection direction)
2.2.3 In-vehicle travel cost
The total in-vehicle travel time per operation hour, 𝐶 , is composed of three parts: (i) the time to
𝑇
8