overcome the distance, 𝐶 ; (ii) the time lost to bus deceleration, acceleration, door opening and closing
𝑇1
at stops, 𝐶 ; and (iii) the time lost to patrons’ boarding and alighting behavior, 𝐶 . Formulations of
𝑇2 𝑇3
these cost items are given in (5)–(8).
𝐶 = 𝐶 +𝐶 +𝐶 (5)
𝑇 𝑇1 𝑇2 𝑇3
𝐿 𝑊 (𝑥+𝑦)
𝐶 = ∫ ∫ [𝜆 (𝑥,𝑦)+𝜆 (𝑥,𝑦)]𝑑𝑦𝑑𝑥 (6)
𝑇1 𝑥=0 𝑦=0 𝑣𝐼 𝑝 𝑑
𝑊
𝐶 = 𝜏
∫𝐿 ∫𝑊 ∫ 𝑧=𝑦(𝜆𝑝(𝑥,𝑧)+𝜆𝑑(𝑥,𝑧))𝑑𝑧
𝑑𝑦𝑑𝑥 (7)
𝑇2 0 𝑥=0 𝑦=0 𝐵(𝑥,𝑦)
𝐿 𝑊 𝑊
𝐶 = ∫ ∫ (𝜏 𝑆 (𝑥)𝐻 (𝑥)𝜆 (𝑥,𝑦)∫ 𝜆 (𝑥,𝑧)𝑑𝑧+
𝑇3 𝑥=0 𝑦=0 𝑏 𝐼 𝐼𝑝 𝑝 𝑧=𝑦 𝑝
𝑊
𝜏 𝑆 (𝑥)𝐻 (𝑥)𝜆 (𝑥,𝑦)∫ 𝜆 (𝑥,𝑧)𝑑𝑧)𝑑𝑦𝑑𝑥 (8)
𝑎 𝐼 𝐼𝑑 𝑑 𝑧=𝑦 𝑑
where 𝑣 denotes the cruise speed of feeder buses; and 𝜏 the fixed time loss per bus at a stop due to
𝐼 0
𝑊
bus deceleration, acceleration, door opening and closing. In (7), ∫ 𝜆 (𝑥,𝑧)𝑑𝑧∙𝑑𝑥 and
𝑧=𝑦 𝑝
𝑊
∫ 𝜆 (𝑥,𝑧)𝑑𝑧∙𝑑𝑥 represent the onboard patron flows passing 𝑦 within a swath of width 𝑑𝑥 for
𝑧=𝑦 𝑑
the collection and distribution directions, respectively.
𝑊
The first term on the RHS of (8) is derived as follows. First, 𝑆 (𝑥)𝐻 (𝑥)∫ 𝜆 (𝑥,𝑧)𝑑𝑧
𝐼 𝐼𝑝 𝑧=𝑦 𝑝
represents the number of patrons onboard a bus on the line located at 𝑥 when the bus passes 𝑦 in the
collection direction. Second, 𝜏 𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥 corresponds to the boarding time loss generated by the
𝑏 𝑝
patrons originating from an area of size 𝑑𝑥×𝑑𝑦 at (𝑥,𝑦) . This time loss is then multiplied by
𝑊
𝑆 (𝑥)𝐻 (𝑥)∫ 𝜆 (𝑥,𝑧)𝑑𝑧 since it affects all onboard patrons of a bus passing (𝑥,𝑦). The second
𝐼 𝐼𝑝 𝑧=𝑦 𝑝
term on the RHS of (8) is developed in a similar manner. Note again that (7) would not involve a double
integral of decision function 𝐵(𝑥,𝑦) if stop spacings were not jointly optimized. Moreover, (8) would
not exist if the dependency of bus dwell times on the boarding and alighting patrons were disregarded.
2.3 Agency cost
During each hour of operation, the feeder service operator incurs three types of costs: (i) the stop
infrastructure cost, denoted by 𝐶 ; (ii) the operating cost associated with bus-km traveled (e.g., fuel
𝑠
cost), 𝐶 ; and (iii) the cost related to the bus fleet, including amortized bus purchase and maintenance
𝑣𝑘
costs, as well as driver wages6, 𝐶 . Note that these costs are converted to hours using the average
𝑣ℎ
value of time for patrons, 𝜃. The formulations for these costs are as follows:
𝐶 =
𝜋𝑠∫𝐿 ∫𝑊 1
𝑑𝑥𝑑𝑦 (9)
𝑠 𝜃 𝑥=0 𝑦=0𝑆𝐼(𝑥)𝐵(𝑥,𝑦)
𝐶 =
𝜋𝑣∫𝐿 [(𝑊+𝑥)
(
1
+
1
)]𝑑𝑥 (10)
𝑣𝑘 𝜃 𝑥=0 𝑆𝐼(𝑥) 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥)
𝐶 =
𝜋𝑚𝑉
(11)
𝑣ℎ ℎ
𝜃
where 𝜋 , 𝜋 , and 𝜋 denote the unit costs per feeder stop ($/stop), bus-km traveled ($/bus-km), and
𝑠 𝑣 𝑚
1
bus hour ($/bus-hour), respectively. Note that 𝑑𝑥𝑑𝑦 is the number of stops in the area of
𝑆𝐼(𝑥)𝐵(𝑥,𝑦)
6 In some studies (e.g., Gu et al., 2016), this cost component is alternatively referred to as the time-based or bus-hour based
operating cost.
9