3. Solution method
For brevity, this section only focuses on the solution approach for (15a)–(15f). The model involving
schedule coordination (i.e., Eq. (18) and (19)) can be solved in a highly similar fashion. Section 3.1
derives the properties characterizing a globally optimal solution. Section 3.2 furnishes a heuristic
solution algorithm built upon these optimality properties.
3.1 Optimality properties
To simplify the presentation, we begin by defining the following aggregate demand quantities and
functions:
Λ = Λ +Λ ; (20a)
𝑝 𝑑
𝐿 𝑊
Λ = ∫ ∫ 𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥; (20b)
𝑝 𝑥=0 𝑦=0 𝑝
𝐿 𝑊
Λ = ∫ ∫ 𝜆 (𝑥,𝑦)𝑑𝑦𝑑𝑥; (20c)
𝑑 𝑥=0 𝑦=0 𝑑
𝑊
Λ (𝑥)= ∫ 𝜆 (𝑥,𝑦)𝑑𝑦, 0 ≤ 𝑥 ≤ 𝐿; (20d)
𝑝𝑥 𝑦=0 𝑝
𝑊
Λ (𝑥)= ∫ 𝜆 (𝑥,𝑦)𝑑𝑦, 0 ≤ 𝑥 ≤ 𝐿; (20e)
𝑑𝑥 𝑦=0 𝑑
𝑊
Λ (𝑥,𝑦)= ∫ 𝜆 (𝑥,𝑧)𝑑𝑧,0 ≤ 𝑥 ≤ 𝐿,0 ≤ 𝑦 ≤ 𝑊; (20f)
𝑝𝑥𝑦 𝑧=𝑦 𝑝
𝑊
Λ (𝑥,𝑦)= ∫ 𝜆 (𝑥,𝑧)𝑑𝑧,0 ≤ 𝑥 ≤ 𝐿,0 ≤ 𝑦 ≤ 𝑊; (20g)
𝑑𝑥𝑦 𝑧=𝑦 𝑑
𝑊
𝛭 (𝑥)= ∫ 𝜆 (𝑥,𝑦)Λ (𝑥,𝑦)𝑑𝑦,0 ≤ 𝑥 ≤ 𝐿; (20h)
𝑝𝑥 𝑦=0 𝑝 𝑝𝑥𝑦
𝑊
𝛭 (𝑥)= ∫ 𝜆 (𝑥,𝑦)Λ (𝑥,𝑦)𝑑𝑦,0 ≤ 𝑥 ≤ 𝐿. (20i)
𝑑𝑥 𝑦=0 𝑑 𝑑𝑥𝑦
where Λ, Λ , and Λ represent the total hourly demand for both directions combined, the collection
𝑝 𝑑
direction, and the distribution direction, respectively; Λ (𝑥) and Λ (𝑥) the demand densities
𝑝𝑥 𝑑𝑥
integrated over the 𝑦-direction at location 𝑥 for collection and distribution travels, respectively; and
Λ (𝑥,𝑦) and Λ (𝑥,𝑦) the demand densities at location 𝑥, integrated from 𝑦 to the top edge of
𝑝𝑥𝑦 𝑑𝑥𝑦
Ω , for collection and distribution travels, respectively (note that Λ (𝑥,0) = Λ (𝑥) and
𝑝𝑥𝑦 𝑝𝑥
Λ (𝑥,0)= Λ (𝑥)). The Λ (𝑥,𝑦) and Λ (𝑥,𝑦) are utilized to calculate the onboard patron
𝑑𝑥𝑦 𝑑𝑥 𝑝𝑥𝑦 𝑑𝑥𝑦
numbers at 𝑦 for a bus on a line located at 𝑥. Lastly, 𝛭 (𝑥) and 𝛭 (𝑥) are employed to calculate
𝑝𝑥 𝑑𝑥
the total dwell time loss for all patrons combined.
The following functions are defined for further simplification of the optimality properties:
𝛼(𝑥)=
1
(𝜋
(𝑊+𝑥)+𝜋𝑚(𝑊+𝑥)
+𝜋 𝜏
∫𝑊 1
𝑑𝑦),0 ≤ 𝑥 ≤ 𝐿; (21a)
𝜃 𝑣 𝑣𝐼 𝑚 0 𝑦=0𝐵(𝑥,𝑦)
2
𝛽 (𝑥)= 𝜏 [Λ (𝑥)] +2𝜏 𝛭 (𝑥),0 ≤ 𝑥 ≤ 𝐿; (21b)
𝑝 𝑎 𝑝𝑥 𝑏 𝑝𝑥
𝛽 (𝑥)= 2𝜏 𝛭 (𝑥),0 ≤ 𝑥 ≤ 𝐿; (21c)
𝑑 𝑎 𝑑𝑥
1 𝑊 1
𝛾(𝑥)= 𝜋 ∫ 𝑑𝑦,0 ≤ 𝑥 ≤ 𝐿. (21d)
𝜃 𝑠 𝑦=0𝐵(𝑥,𝑦)
Here 𝛼(𝑥) denotes the agency cost rate per bus on a line located at 𝑥 , excluding the agency cost
associated with boarding and alighting patrons; 𝛽 (𝑥) and 𝛽 (𝑥) are related to the boarding and
𝑝 𝑑
alighting time loss in the collection and distribution travel directions, respectively; and 𝛾(𝑥) represents
the stop infrastructure cost rate for a line located at 𝑥.
Note that Λ (𝑥,𝑦), Λ (𝑥,𝑦), 𝛭 (𝑥), 𝛭 (𝑥), 𝛽 (𝑥), and 𝛽 (𝑥) are needed because the
𝑝𝑥𝑦 𝑑𝑥𝑦 𝑝𝑥 𝑑𝑥 𝑝 𝑑
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