# A Heuristic Approach to Solving the Train Traffic Re-Scheduling Problem in Real Time
## 6 Conclusions and Future Work  
This paper addresses the real-time train traffic re-scheduling problem. A mixed graph is used for modeling the problem as a blocking job-shop scheduling problem. A heuristic algorithm is proposed that benefits from re-timing, re-ordering, and local re-routing. The algorithm benefits also from a dynamic update of data, which accelerates the computations.  
The response time for such a real-time computational scheduling tool is a vital factor. In the proposed solution approach, the problem for a 1 h time window is solved in less than 10 s, and for a 1.5 h time window, the computational time is less than 20 s. It is also unknown what time horizon is necessary to consider in different situations and what role this uncertainty would play. Interviews with dispatchers suggest that it differs a lot depending on the situation, context and associated working load.  
The \(TFD_{j}^{+3}\) objective function is a relevant criterion to control the lateness of trains. However, based on the situation and type of disturbance scenario, the dispatchers also have other concerns and objectives. The investigation of other objective functions and useful solution quality metrics is necessary to investigate in future research. The graph \(G\) is represented by an adjacency matrix in the current implementation. Using alternative data structures such as adjacency lists, can be an option to investigate the possibility to reduce computational time further.  
**Acknowledgments:** The presented research work was conducted within the research projects FLOAT (FLexibel Omplanering Av Taglagen i drift) and BLXTEN (Beslutstod for trafikLedare: approxlmatva och eXakta opTimEraNde metoder), funded by the Swedish Transport Administration (Trafikverket) with the KAJT (KApacitet i jarnvagsTrafiken) research programme. We are thankful for the financial support, data, and significant guidance provided by Trafikverket.  
**Author Contributions:** [PERSON] has developed the graph model and heuristic algorithms. The algorithmic approach has been implemented in Java by [PERSON]. [PERSON] has provided all input data required to build and perform the experiments, and she is the main developer of the visualization tool used for visual inspection of re-scheduling solutions. [PERSON] has also managed the associated research project, handled all contacts with the industry partner, Trafikverket, and acquired the associated research funding. [PERSON] has generated the disturbance scenarios, performed the experiments with the algorithmic approach and the associated performance assessment including  
Figure 5: The computational time increases quadratically, based on the number of events.  
the analysis of the resulting data and conclusions. [PERSON] has formulated and implemented the corresponding MIP formulation in java, which was applied to solve the same disturbance scenarios in order to evaluate the level of optimality and efficiency of the algorithmic approach. Both [PERSON] and [PERSON] contributed significantly to writing the manuscript. Both authors have read and approved the final manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.