𝐿
(0.1) Discretize the service region Ω into a 𝑚×𝑛 lattice of points, setting 𝑥 = (𝑖−0.5) , 𝑖 =
𝑖
𝑛
𝑊
1,2,…,𝑛, and 𝑦 = (𝑗−0.5) , 𝑖 = 1,2,…,𝑚.
𝑗
𝑚
(0) (0) (0)
(0.2) Initialize 𝐻 (𝑥 ) , 𝐻 (𝑥 ) , and 𝑆 (𝑥 ) to satisfy the boundary constraints for 𝑖 =
𝐼𝑝 𝑖 𝐼𝑑 𝑖 𝐼 𝑖
1,2,…,𝑛. Set the outer-loop iteration count 𝑘 = 1 and proceed to Stage 1.
Stage 1: Compute 𝑩(𝒙,𝒚).
Calculate 𝐵(𝑘)(𝑥 ,𝑦 ) using Eq. (22d) for 𝑖 = 1,2,…,𝑛 , 𝑗 = 1,2,…,𝑚 , taking 𝐻(𝑘−1) (𝑥 ) ,
𝑖 𝑗 𝐼𝑝 𝑖
(𝑘−1) (𝑘−1)
𝐻 (𝑥 ) , and 𝑆 (𝑥 ) as inputs. Note that the integrals are approximated by summations.
𝐼𝑑 𝑖 𝐼 𝑖
Proceed to Stage 2.
Stage 2: Compute 𝑯 (𝒙), 𝑯 (𝒙) and 𝑺 (𝒙).
𝑰𝒑 𝑰𝒅 𝑰
(2.1) Set 𝑆̃(𝑘,0) (𝑥 )= 𝑆(𝑘−1) (𝑥 ), 𝑖 = 1,2,…,𝑛. Initialize the inner-loop iteration count 𝑘′ = 1.
𝐼 𝑖 𝐼 𝑖
(2.2) Taking 𝐵(𝑘)(𝑥 ,𝑦 ) and
𝑆̃(𝑘,𝑘′−1)
(𝑥 ) (𝑖 = 1,2,…,𝑛 , 𝑗 = 1,2,…,𝑚 ) as inputs, calculate
𝑖 𝑗 𝐼 𝑖
𝐻̃
(𝑘,𝑘′)
(𝑥 ) and 𝐻̃
(𝑘,𝑘′)
(𝑥 ) using Eqs. (22a)–(22b).
𝐼𝑝 𝑖 𝐼𝑑 𝑖
(2.3) Taking 𝐵(𝑘)(𝑥 ,𝑦 ) , 𝐻̃
(𝑘,𝑘′)
(𝑥 ) , and 𝐻̃
(𝑘,𝑘′)
(𝑥 ) (𝑖 = 1,2,…,𝑛 , 𝑗 = 1,2,…,𝑚 ) as
𝑖 𝑗 𝐼𝑝 𝑖 𝐼𝑑 𝑖
inputs, calculate
𝑆̃(𝑘,𝑘′)
(𝑥 ) by (22c).
𝐼 𝑖
(2.4) Set 𝑆(𝑘) (𝑥 )= 𝑆̃(𝑘,𝑘′) (𝑥 ) , 𝐻 (𝑘)(𝑥 ) = 𝐻̃ (𝑘,𝑘′) (𝑥 ) , and 𝐻 (𝑘)(𝑥 )= 𝐻̃ (𝑘,𝑘′) (𝑥 ) .
𝐼 𝑖 𝐼 𝑖 𝐼𝑝 𝑖 𝐼𝑝 𝑖 𝐼𝑑 𝑖 𝐼𝑑 𝑖
Check for convergence using the predefined error tolerance 𝜀. If the following convergency
criteria are satisfied: ∑𝑛
|𝑆̃(𝑘,𝑘′)
(𝑥
)−𝑆̃(𝑘,𝑘′−1)
(𝑥 )| ≤ 𝑛𝜀 and ∑𝑛 (|𝐻̃
(𝑘,𝑘′)
(𝑥 )−
𝑖=1 𝐼 𝑖 𝐼 𝑖 𝑖=1 𝐼𝑝 𝑖
𝐻̃
(𝑘,𝑘′−1)
(𝑥 )|+|𝐻̃
(𝑘,𝑘′)
(𝑥 )−𝐻̃
(𝑘,𝑘′−1)
(𝑥 )|)≤ 2𝑛𝜀 , then proceed to the next step.
𝐼𝑝 𝑖 𝐼𝑑 𝑖 𝐼𝑑 𝑖
Otherwise, update 𝑘′ ←𝑘′+1 and repeat steps (2.2) and (2.3).
(2.5) Check convergency. If ∑𝑛 |𝑆(𝑘) (𝑥 )−𝑆(𝑘−1) (𝑥 )|≤ 𝑛𝜀 , ∑𝑛 (|𝐻 (𝑘)(𝑥 )−
𝑖=1 𝐼 𝑖 𝐼 𝑖 𝑖=1 𝐼𝑝 𝑖
𝐻 (𝑘−1)(𝑥 )|+|𝐻 (𝑘)(𝑥 )−𝐻 (𝑘−1)(𝑥 )|) ≤ 2𝑛𝜀 , and ∑𝑛 ∑𝑚 |𝐵(𝑘)(𝑥 ,𝑦 )−
𝐼𝑝 𝑖 𝐼𝑑 𝑖 𝐼𝑑 𝑖 𝑖=1 𝑗=1 𝑖 𝑗
𝐵(𝑘−1)(𝑥 ,𝑦 )| ≤ 𝑚𝑛𝜀 are all satisfied, set 𝑆∗(𝑥 )= 𝑆(𝑘) (𝑥 ) , 𝐻 ∗(𝑥 )= 𝐻 (𝑘)(𝑥 ) ,
𝑖 𝑗 𝐼 𝑖 𝐼 𝑖 𝐼𝑝 𝑖 𝐼𝑝 𝑖
𝐻 ∗(𝑥 )= 𝐻 (𝑘)(𝑥 ), and 𝐵∗(𝑥 ,𝑦 ) = 𝐵(𝑘)(𝑥 ,𝑦 ) for 𝑖 = 1,2,…,𝑛, 𝑗 = 1,2,…,𝑚, and
𝐼𝑑 𝑖 𝐼𝑑 𝑖 𝑖 𝑗 𝑖 𝑗
end the search. Otherwise, update 𝑘 ← 𝑘+1 and return to Stage 1.
The optimal CA solution, obtained on the point lattice, can be transformed into a specific layout of
feeder lines. The methodology for this conversion is detailed in Appendix C.
4. Numerical case studies
Section 4.1 outlines the setup of numerical case studies. The accuracy of our CA models is examined
in Section 4.2. Optimal feeder system designs for a typical case in both high- and low-wage cities are
presented and discussed in Section 4.3. In Section 4.4, the sensitivity of the optimal feeder design to
various key operating factors, such as demand pattern, demand rate, size of the service region, and
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