High-Speed Rail Train Timetabling Problem: A Time-Space Network Based Method with an Improved Branch-and-Price Algorithm

Table 1

The parameters for IBAP.

𝑉∶ the set of nodes generated in branch-and-price tree, indexed by 𝑣	𝑋(𝑣): the set of un-split 𝑥𝑝 a node 𝑣
𝑉: the set of nodes unsearched in 𝑉, indexed by 𝑣	Ω: the set of time-space path, and Ω⊆𝑃
Co(𝑣): the set of all the constraints in a node 𝑣	𝐺^𝑇𝑟,dual: the time-space network with the value of dual variables
UB(𝑣): upper bound of a node 𝑣	𝜂𝑝: reduce cost of time-space path 𝑝
LB(𝑣): lower bound of a node 𝑣	LB=min𝑣∈𝑉LB(𝑣): is the lower bound of branch-and-price tree
	UB=min𝑣∈𝑉UB(𝑣): is the upper bound of branch-and-price tree