satellite city connected to a trunk transit system (e.g., the city of Langfang connected to the intercity
high-speed rail network, and cities in the San Francisco Bay Area connected to the BART system). The
relationships between 𝜋 , 𝜋 and 𝐾, 𝜃 are obtained using empirical data; see Appendix D for
𝑣 𝑚
details.
Regarding the hyperparameters of the solution algorithm, we find that 𝑚 = 30 and 𝑛 = 20
yield sufficiently accurate numerical solutions. Further increasing 𝑚 or 𝑛 renders a relative
percentage change in the generalized cost of less than 0.5%. Thus, the values of 𝑚 and 𝑛 are set to
30 and 20, respectively. The error tolerance 𝜀 is set to 0.0001. Each numerical instance is solved using
various combinations of initial values, consistently converging to the same optimal solution. A typical
instance’s solution is found within 30 seconds.
4.2 Validating the accuracy of the CA cost functions
To examine the accuracy of the CA cost functions, we first convert the optimal CA solution, which
comprises continuous functions for headways, line and stop spacings, into a specific feeder system
design. This is done using the method described in Appendix C. Following this, we recalculate the
generalized cost and each cost component for comparison. We apply a demand pattern with Λ = Λ =
𝑝 𝑑
1200 patrons/h, 𝜇 = 𝜇 = 0 , 𝜇 = 𝜇 = 0 , 𝜎 = 𝜎 = 𝐿⁄4 , and 𝜎 = 𝜎 = 𝑊⁄4 .
𝑥𝑝 𝑥𝑑 𝑦𝑝 𝑦𝑑 𝑥𝑝 𝑥𝑑 𝑦𝑝 𝑦𝑑
These parameter values suggest that the demand is concentrated near the terminal, a common scenario
where the trunk transit station is located in an area with high neighboring demand. The spatial
distribution of the demand is illustrated in Fig. 4. We set the bus capacity at 𝐾 = 10 (e.g., a 10-
passenger Ford Transit Van). The converted feeder network, comprising the layout of feeder lines and
the locations of stops (denoted by square markers), is illustrated in Fig. 5. The figure reveals that the
bus lines and stops are more densely clustered near the terminal, reflecting the demand distribution
shown in Fig. 4. It is worth noting that the line and stop locations can be fine-tuned to align with the
local street network. For a case study on adapting a conceptual design to a local city street network,
please refer to Estrada et al. (2011).
Fig. 4 The demand densities (patron/km2/h) in the collection direction for a terminal-centered demand
Percentage errors are computed between the recalculated cost terms and those of the CA cost
functions. The results are displayed in Table 3. Note that the error in generalized cost is only 0.73%,
and the error in each cost component remains consistently below 2%. This demonstrates the accuracy
of our CA cost functions.8
8 The error can be further reduced by an improved method for generating the specific network design; see the details in
16