𝜏 Boarding time per patron h/patron
𝑏
𝑡 Transfer delay from feeder stop to the trunk-line platform per patron h/patron
𝑓−𝑡
𝑡 Transfer delay from trunk-line platform to feeder stop per patron h/patron
𝑓−𝑡
𝐻 Headway of the trunk line h
𝑡
𝐻 Minimum headway of a feeder line h
𝑚𝑖𝑛
𝐻 Maximum headway of a feeder line h
𝑚𝑎𝑥
Appendix B. Derivation of the optimality properties
Combining all the cost metrics, we can rewrite the generalized cost (15a) as:
𝐺𝐶 =
{𝜋𝑚(𝜏
+𝜏
)Λ+∫𝐿 (𝑥
(Λ (𝑥)+Λ (𝑥))+(𝑡
+𝐻𝑡)Λ
(𝑥)+𝑡 Λ (𝑥))𝑑𝑥+
𝜇 𝑎 𝑏 𝑥=0 𝑣𝐼 𝑝𝑥 𝑑𝑥 𝑓−𝑡 2 𝑝𝑥 𝑡−𝑓 𝑑𝑥
∫𝑊 𝑦
(Λ (𝑦)+Λ
(𝑦))𝑑𝑦}+{∫𝐿 [𝜋𝑣(𝑊+𝑥)
(
1
+
1 )+𝜋𝑚(𝑊+𝑥)
(
1
+
1
)+
𝑦=0𝑣𝐼 𝑝𝑦 𝑑𝑦 𝑥=0 𝜃𝑆𝐼(𝑥) 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥) 𝜃𝑆𝐼(𝑥)𝑣𝐼 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥)
𝑆𝐼(𝑥)
(Λ (𝑥)+Λ
(𝑥))+𝐻𝐼𝑝(𝑥)
Λ
(𝑥)+𝐻𝐼𝑑(𝑥)
Λ
(𝑥)+𝜏𝑎𝑆
(𝑥)𝐻 (𝑥)[Λ
(𝑥)]2
+
𝑝𝑥 𝑑𝑥 𝑝𝑥 𝑑𝑥 𝐼 𝐼𝑝 𝑝𝑥
4𝑣𝑊 2 2 2
𝜏𝑏𝑆 (𝑥)𝐻 [Λ (𝑥)]2+𝜏 𝑆 (𝑥)𝐻 (𝑥)𝛭 (𝑥)+𝜏 𝑆 (𝑥)𝐻 (𝑥)𝛭 (𝑥)]𝑑𝑥}+
𝐼 𝑡 𝑑𝑥 𝑏 𝐼 𝐼𝑝 𝑝𝑥 𝑎 𝐼 𝐼𝑑 𝑑𝑥
2
{∫𝐿 ∫𝑊 [𝜋𝑠 1 +𝜋𝑚 𝜏0
(
1
+
1 )+𝐵(𝑥,𝑦)
[𝜆 (𝑥,𝑦)+𝜆 (𝑥,𝑦)]+
𝑥=0 𝑦=0 𝜃 𝑆𝐼(𝑥)𝐵(𝑥,𝑦) 𝜃 𝑆𝐼(𝑥)𝐵(𝑥,𝑦) 𝐻𝐼𝑝(𝑥) 𝐻𝐼𝑑(𝑥) 4𝑣𝑊 𝑝 𝑑
𝜏0Λ𝑝𝑥𝑦(𝑥,𝑦) +𝜏0Λ𝑑𝑥𝑦(𝑥,𝑦)
]𝑑𝑦𝑑𝑥} (B1)
𝐵(𝑥,𝑦) 𝐵(𝑥,𝑦)
The part of the objective function in the first pair of braces is constant. The part in the second pair
of braces contains single integrals relating to either 𝑥 or 𝑦 (involving decision functions 𝑆 (𝑥) ,
𝐼
𝐻 (𝑥), 𝐻 (𝑥)). Meanwhile, the part in the third pair of braces contains double integrals related to
𝐼𝑝 𝐼𝑑
both 𝑥 and 𝑦 (involving the decision function 𝐵(𝑥,𝑦)).
Note that (B1) is a convex function when only one decision variable (𝑆 (𝑥), 𝐻 (𝑥), 𝐻 (𝑥), or
𝐼 𝐼𝑝 𝐼𝑑
𝐵(𝑥,𝑦) ) is considered, and the other variables are fixed. (This type of functions is known as the
coordinate-wise convex functions, which are not necessarily convex themselves.) Further note that
constraints (15c)–(15f) act as boundary constraints when only one decision variable is considered. As a
result, the optimal solution for each decision variable can be developed analytically from first-order
derivatives, assuming the other variables are fixed. These optimal solutions are:
−1
𝐻∗ (𝑥)= 𝑚𝑖𝑑{𝐻 ,min{ 𝐾 ,𝐻 } ,√ 2𝛼(𝑥)(𝑆 𝐼∗(𝑥)) } (B2)
𝐼𝑝 𝑚𝑖𝑛 Λ𝑝𝑥(𝑥)𝑆 𝐼∗(𝑥) 𝑚𝑎𝑥 Λ𝑝𝑥(𝑥)+𝛽𝑝(𝑥)𝑆 𝐼∗(𝑥)
−1
𝐻∗ (𝑥)= 𝑚𝑖𝑑{max{𝐻 ,𝐻 },min{ 𝐾 ,𝐻 },√ 2𝛼(𝑥)(𝑆 𝐼∗(𝑥)) } (B3)
𝐼𝑑 𝑚𝑖𝑛 𝑡 Λ𝑑𝑥(𝑥)𝑆 𝐼∗(𝑥) 𝑚𝑎𝑥 Λ𝑑𝑥(𝑥)+𝛽𝑑(𝑥)𝑆 𝐼∗(𝑥)
26