A Branch and Bound Algorithm and Iterative Reordering Strategies for Inserting Additional Trains in Real Time: A Case Study in Germany

Table 1

Characteristics of adding paths problem and solution approaches.

Publication	Background	Model	Objective	Solution	Real time	Connection	Problem size evaluated for
[17]	(F)	ILP	Maximizes the number of additional trains and minimize the violations to the ideal insertion	(HA)	No	No	679/202/24-48-96, 520/202/24-48-96, 0/202/188-338-554-64
[18]	(P)	MIP	Minimizes the total weighted time window violations and the makespan	(CA), (HA)	No	No	6/3/1-5, 6/5/1-5, 10/10/1-5, 24/10/1-5, 15/5/1-5, 54/30/1-5, 20/20/1-5, 20/12/1-5, 20/24/1-5
[19]	(F/P)	CSM	Minimizes the average traversal time of new train	(DP), (PR)	No	No	81/65/20
[20]	(F/P)	LPM, CSM	Support railway planners by computing a set of Pareto optimal solutions with respect to travel time and expected delay to additional trains	(SP)	No	No	—/—/1
Our paper	(F/P)	MIP	Minimizes consecutive delay to existing timetable	(BB), (AG),; (SP), (PR)	Yes	Yes	36/60/1-15

(i) Background: passenger trains insertion (P); freight trains insertion (F).
(ii) Model: mixed integer programming (MIP); computer simulation model (CSM); integer linear programming (ILP); linear regression model (LPM).
(iii) Solution: constructive algorithm (CA); alternative graphs (AG); shortest path algorithm (SP); branch-and-bound (BB); heuristics algorithm (HA); dynamic programming (DP); local search (LS); practical rules (PR).
(iv) Problem size evaluated for number of initial trains/number of stations or block sections/number of additional trains. —: between double / means missing the information.