−1 −1
2𝛼(𝑥)((𝐻∗ (𝑥)) +(𝐻∗ (𝑥)) )+2𝛾(𝑥)
𝑆∗(𝑥)= min √ 𝐼𝑝 𝐼𝑑 , 𝐾 , 𝐾 (B4)
𝐼 Λ𝑝𝑥(𝑥 2) 𝑣+ 𝑊Λ𝑑𝑥(𝑥) +𝜏𝑏[Λ𝑑𝑥(𝑥)]2𝐻𝑡+𝛽𝑝(𝑥)𝐻 𝐼∗ 𝑝(𝑥)+𝛽𝑑(𝑥)𝐻 𝐼∗ 𝑑(𝑥) Λ𝑝𝑥(𝑥)𝐻 𝐼∗ 𝑝(𝑥) Λ 𝑑𝑥(𝑥)𝐻 𝐼∗ 𝑑(𝑥)
{ }
𝐵∗(𝑥,𝑦) =
√4𝑣𝑤[ 𝜇𝑆𝐼∗1 (𝑥)(𝜋𝑠+ 𝐻𝜋 𝐼∗𝑚 𝑝(𝜏 𝑥0 )+ 𝐻𝜋 𝐼∗𝑚 𝑑(𝜏 𝑥0 ))+𝜏0(Λ𝑝𝑥𝑦(𝑥,𝑦)+Λ𝑑𝑥𝑦(𝑥,𝑦))]
(B5)
𝜆𝑝(𝑥,𝑦)+𝜆𝑑(𝑥,𝑦)
where function 𝑚𝑖𝑑{𝑧 ,𝑧 ,𝑧 } returns the middle one of the three arguments.
1 2 2
Appendix C. Generating a specific feeder system design
Given 𝑆 ∗(𝑥 ) and 𝐵∗(𝑥 ,𝑦 ) (𝑖 =1,2,…,𝑛 ; 𝑗 = 1,2,…,𝑚 ), we use the following three steps to
𝐼 𝑖 𝑖 𝑗
generate line and stop locations. Step 1 generates line locations that closely match 𝑆 ∗(𝑥 ) . Step 2
𝐼 𝑖
determines the continuous stop spacing function for each line generated in Step 1. Step 3 generates the
stop locations for each line.
Step 1. Apply the spline curve fitting method to fit a cubic spline function 𝑆 (𝑥) to 𝑆 ∗(𝑥 ) (𝑖 =
𝐼 𝐼 𝑖
𝑥 𝑑𝑧
1,2,…,𝑛). Next, place one line at every 𝑥 where ∫ = 𝑘+0.5,𝑘 = 1,2,…. Here, 0.5 is added
𝑧=0𝑆𝐼(𝑥)
to the RHS to ensure the first line’s catchment zone resides on both sides of that line. The resulting line
location set is denoted by Ω = {𝑥 :𝑝 = 1,2,…,𝑁𝑆}, where 𝑁𝑆 is the number of lines, and 𝑥𝑆 the
𝑆 𝑝 𝑝
location of the 𝑝-th line satisfying 0 < 𝑥 < 𝑥 < ⋯ < 𝑥 < 𝐿.
1 2 𝑁𝑆
Step 2. Use the gridded data interpolation method to fit a piecewise cubic function 𝐵(𝑥,𝑦) to
𝐵∗(𝑥 ,𝑦 ) (𝑖 = 1,2,…,𝑛;𝑗 = 1,2,…,𝑚 ). The continuous stop spacing function on the 𝑝-th line is
𝑖 𝑗
𝐵(𝑥 ,𝑦).
𝑝
Step 3. For each line 𝑝 ∈{1,2,…,𝑁𝑆}, place one stop at every 𝑦 where ∫𝑦 𝑑𝑧 = 𝑘+0.5,
𝑧=0𝐵(𝑥𝑝,𝑦)
𝑘 = 1,2,…. The resulting stop location set is denoted by Ω𝐵 = {(𝑥 ,𝑦 ):𝑞 = 1,2,…,𝑁𝐵 }, where 𝑁𝐵
𝑝 𝑝 𝑞 𝑝 𝑝
is the number of stops on the 𝑝-th line. Coordinates 𝑦 satisfy 0 < 𝑦 < 𝑦 < ⋯ < 𝑦 < 𝑊.
𝑞 1 2 𝑁𝑝𝐵
Finally, the headways of each line are obtained from 𝐻∗ (𝑥) and 𝐻∗ (𝑥) at 𝑥 ∈Ω .
𝐼𝑝 𝐼𝑑 𝑝 𝑆
𝐿 1
Note that the CA solution renders ∫ 𝑑𝑥 feeder lines, and this number is not always close
𝑥=0𝑆∗(𝑥)
𝐼
to an integer. The above method guarantees finding the nearest integer number of lines. However, when
−1
𝐿 1
that number is small, the percentage error could be relatively large (capped by 0.5(∫ 𝑑𝑥) ).
𝑥=0𝑆∗(𝑥)
𝐼
To further reduce this error and improve the converted specific design, we can evenly allocate the
fractional residual number of lines (positive or negative) to all lines by proportionally increasing or
decreasing the line spacings. Similarly, stop spacings on each line can also be adjusted to eliminate the
fractional residuals. In addition, one can reoptimize stop spacings and headways using the CA model
after line locations are determined by Step 1 of the above method. This will make the specific design
more efficient. For simplicity, these fine-tuning steps are not included in our numerical case studies.
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