(𝑊+𝑥)
size 𝑑𝑥×𝑑𝑦 at (𝑥,𝑦), and 𝑑𝑥 is the bus line length contributed by a “vertical” swath of width
𝑆𝐼(𝑥)
𝑑𝑥 at 𝑥. The 𝑉 in (11) denotes the required bus fleet size, which comprises (i) the number of buses
ℎ
that are actively travelling between stops, 𝑉 ; and (ii) the number of buses that are dwelling at stops.
ℎ1
The latter component can be further divided into the number of buses needed due to deceleration,
acceleration, door opening and closing at stops, 𝑉 , as well as the number of buses required for
ℎ2
passenger boarding and alighting, 𝑉 . The three components 𝑉 , 𝑉 , and 𝑉 are formulated in
ℎ3 ℎ1 ℎ2 ℎ3
(12)–(14), respectively:
𝐿 (𝑊+𝑥) 1 1
𝑉 = ∫ [ ( + )]𝑑𝑥 (12)
ℎ1 𝑥=0 𝑆𝐼(𝑥)𝑣𝐼 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥)
𝑉 =
∫𝐿 ∫𝑊 𝜏0
(
1
+
1
)𝑑𝑥𝑑𝑦 (13)
ℎ2 𝑥=0 𝑦=0𝑆𝐼(𝑥)𝐵(𝑥,𝑦) 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥)
𝐿 𝑊
𝑉 = (𝜏 +𝜏 )∫ ∫ [𝜆 (𝑥,𝑦)+𝜆 (𝑥,𝑦)]𝑑𝑦𝑑𝑥 (14)
ℎ3 𝑎 𝑏 𝑥=0 𝑦=0 𝑝 𝑑
Derivation of (12)–(14) is similar to those presented earlier in this section.
2.4 Optimization model
The generalized cost can be expressed as the sum of all the above cost components:
𝐺𝐶 = 𝐶 +𝐶 +𝐶 +𝐶 +𝐶 +𝐶 (15a)
𝑆 𝑣𝑘 𝑣ℎ 𝐴 𝑊 𝑇
The optimization problem is formulated as follows:
min𝐺𝐶 (15b)
subject to:
𝑊
∫ 𝜆 (𝑥,𝑦)𝑑𝑦∙𝑆 (𝑥)𝐻 (𝑥)≤ 𝐾 (15c)
𝑦=0 𝑝 𝐼 𝐼𝑝
𝑊
∫ 𝜆 (𝑥,𝑦)𝑑𝑦∙𝑆 (𝑥)𝐻 (𝑥)≤ 𝐾 (15d)
𝑦=0 𝑑 𝐼 𝐼𝑑
𝐻 ≤ 𝐻 (𝑥)≤ 𝐻 (15e)
𝑚𝑖𝑛 𝐼𝑝 𝑚𝑎𝑥
max{𝐻 ,𝐻 }≤ 𝐻 (𝑥) ≤ 𝐻 (15f)
𝑚𝑖𝑛 𝑡 𝐼𝑑 𝑚𝑎𝑥
where 𝐾 denotes a feeder bus’s patron-carrying capacity; and 𝐻 , 𝐻 the minimum and
𝑚𝑖𝑛 𝑚𝑎𝑥
maximum headways, respectively. Constraints (15c) and (15d) ensure that the feeder bus capacity is
sufficient for carrying the patrons. Constraints (15e)–(15f) are boundary constraints. Note that the feeder
headway in the distribution direction, 𝐻 (𝑥), must be greater than or equal to the trunk-line headway,
𝐼𝑑
𝐻 . Otherwise, some feeder buses would not have any passengers to collect at the terminal.
𝑡
2.5 Modeling schedule coordination
Feeder and trunk-line schedules can be coordinated to minimize waiting times during transfers
(Sivakumaran et al., 2012). Coordination in both patron-collection and distribution directions can
effectively eliminate the waiting time components,
𝐻𝑡
in Eq. (3) and
𝐻𝐼𝑑(𝑥)
in Eq. (4). However, the
2 2
coordination schemes in the two directions are different. In the patron-collection direction, coordination
necessitates that a trunk-line vehicle is dwelling at the terminal when patrons alighting from a feeder
bus arrive at the trunk platform. This is only feasible if trunk-line vehicles bound for all destinations
arrive at the terminal simultaneously. This may occur when a single trunk line serves the terminal and
most transferring patrons travel in the same direction (e.g., during rush hour commuting). Meanwhile,
coordination in the distribution direction merely requires a feeder bus to be ready for departure when
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