feeder line layout, is presented. Lastly, the benefits of schedule coordination between the trunk line and
feeder lines are discussed in Section 4.5.
4.1 Experimental setup
Our models are applicable to any continuous demand functions. For illustrative purposes, we use the
following demand functions in our numerical case studies:
𝜆 (𝑥,𝑦)= 𝑇𝑟𝑁(𝑥|𝜇 ,𝜎 ,0,𝐿)𝑇𝑟𝑁(𝑦|𝜇 ,𝜎 ,0,𝑊)Λ (23a)
𝑝 𝑥𝑝 𝑥𝑝 𝑦𝑝 𝑦𝑝 𝑝
𝜆 (𝑥,𝑦)= 𝑇𝑟𝑁(𝑥|𝜇 ,𝜎 ,0,𝐿)𝑇𝑟𝑁(𝑦|𝜇 ,𝜎 ,0,𝑊)Λ (23b)
𝑑 𝑥𝑑 𝑥𝑑 𝑦𝑑 𝑦𝑑 𝑑
where 𝑇𝑟𝑁(𝑥|𝜇,𝜎,𝑎,𝑏) indicates the probability density function of a truncated normal distribution
with mean 𝜇 , standard deviation 𝜎 , and support [𝑎,𝑏] . Note that 𝜎 , 𝜎 , 𝜎 , and 𝜎 can be
𝑥𝑝 𝑦𝑝 𝑥𝑑 𝑥𝑑
viewed as proxies for demand heterogeneity. A larger value of 𝜎 , 𝜎 , 𝜎 , or 𝜎 signifies that
𝑥𝑝 𝑦𝑝 𝑥𝑑 𝑥𝑑
the spatial distribution of trip origins or destinations is “flatter.” Particularly, if 𝜎 = 𝜎 = 𝜎 =
𝑥𝑝 𝑦𝑝 𝑥𝑑
𝜎 = ∞, the demand is uniformly distributed over the service region.
𝑥𝑑
Additionally, note that patrons in close proximity to the terminal may prefer to walk rather than
take feeder buses. To account for this preference, we assume that patrons with an origin or destination
within 300 meters from the terminal, using the Manhattan distance measure, will not take the feeder
buses.7
Table 2. Parameter values
Notation Description Value Unit
𝐿 Length of the service region 3 km
𝑊 Width of the service region 2 km
𝜃 Value of time 20 $/h
Amortized unit cost per feeder stop infrastructure per
𝜋 0 $/stop/h
𝑠 operation hour
𝜋 Unit cost per bus-km traveled 0.0314+0.0039𝐾 $/vehicle⋅km
𝑣
𝜋 Unit cost per bus hour 2.068+0.108𝐾+2𝜃 $/vehicle⋅h
𝑚
𝜏 Bus dwell time per stop 12/3600 h/stop
0
𝜏 Alighting time per patron 2/3600 h/patron
𝑎
𝜏 Boarding time per patron 4/3600 h/patron
𝑏
𝑣 Walking speed 2 km/h
𝑊
𝑣 Bus cruise speed 25 km/h
𝐼
𝑡 Transfer delay from feeder to trunk transit per patron 3/60 h/patron
𝑓−𝑡
𝑡 Transfer delay from trunk transit to feeder per patron 3/60 h/patron
𝑡−𝑓
𝐻 Minimum feeder headway 3/60 h
𝑚𝑖𝑛
𝐻 Maximum feeder headway 30/60 h
𝑚𝑎𝑥
𝐻 Headway of the trunk line 5/60 h
𝑡
𝜀 error tolerance 0.0001 –
Numerical instances in this section use the parameter values displayed in Table 2 (Jaiswal, 2010;
Gu et al., 2016; Chen et al., 2017; Mei et al., 2021; Sangveraphunsiri et al., 2022) unless otherwise
specified. The service region size (2×3 km2) is suitable for representing a quarter of a small town or
7 Sensitivity analysis shows that our main findings remain valid when this distance threshold varies within a reasonable range.
For instance, when the walking zone size increases from 300 m to 500 m, the change in generalized cost savings for our
heterogeneous design is less than 0.1%.
15