Time Analysis

The experiments were run on a 2.9GHz Intel Xeon E5-2666 v3 (Haswell) proces- sor. Table 6.14 shows the average execution for all of the algorithms on the most disrupted scenario (m = 30, f = 10), while table 6.15 shows the average execution time on the least disrupted delay scenario (m = 10, f = 30). It is apparent that all algorithms can find a solution in a very short computation time even for the most complex high magnitude, high frequency scenarios. For example, MMAS-R took an average of 19 s for change 1 and 34 s for change 6 for the highly disrupted delay scenario (m = 30, f = 10) and 8 s for change 1 and 11 s for change 2 for the smallest disrupted delay scenario (m = 10, f = 30). The execution time increases with each

Table 6.15: Execution times for each algorithm in seconds, for delay scenario m=10, f=30. Change 0 1 PACO-R 0.07 0.09 PACO-S 0.20 0.17 MMAS-R 0.08 0.11 MMAS-S 0.19 0.21 ACS-R 0.08 0.11 ACS-S 0.16 0.19 TABU-R 0.03 0.04 TABU-S 0.02 0.03 VNS-R 0.15 0.22 VNS-S 0.15 0.22 MC1 0.27 0.34 MC2 0.28 0.30

successive change as with each additional disruption more trains will be affected increasing the time taken to detect and resolve the conflict.

These values are within the maximum rescheduling time of 180 s suggested by the French operating company SNCF [58] and within the 5 min maximum rescheduling interval suggested by Meng and Zhou [67]. The two-colony algorithms, MC1 and MC2 have an unsurprisingly longer execution time. For example, MC1 has an computation time of 58s for change 1 and 99s for change 6 on scenario m = 30, f = 10. However, this is still within the acceptable range suggested by [58] and [67].

6.8

Summary

Rescheduling trains at stations after a delay is a complex task, made more com- plicated by the fact that it can be a dynamic problem that changes over time as more delayed trains arrive at the station. The DSRP can be considered to be two separate sub-problems. The first problem is to decide the platform to allocate to the delayed train; the second is to decide the order that the trains should leave the station. Solving these sub-problems can allow trains to overtake other trains at the station. In this work two colonies of ants have been used to address each of these

sub-problems. Investigations have been carried out to examine the effectiveness of using each colony separately and of combining them into a multi-colony algorithm. In order to investigate the DSRP, a model was created from real-world train schedule data for Leicester station, a busy UK railway station. The model allows the impact of the local decisions made at the station to be projected into the future by accessing the impact the decisions have on a train’s ongoing journey.

To evaluate the effectiveness of using ACO algorithms the results were compared with those obtained using TS, VNS and running with no platform reallocation or resequencing (NO-ALG). Results showed that the best performing ACO algorithms were MMAS and PACO and that they outperformed VNS and TS on the majority of the delay scenarios. It addition, it was observed that simply changing the order of the trains as they leave the station is not sufficient to obtain a good outcome. In contrast, just reallocating the trains to new platforms performs well in almost all cases. This is most likely because the reallocation process also performs some reordering of train departures, as explained in Section 6.2.2. Combining the reallo- cation and resequencing colonies, to form a multi-colony algorithm, did not change the performance on the high and medium magnitude delay scenarios but gave an improved performance on the low magnitude delay scenarios. However, this was only the case if reallocation was performed before resequencing.

The use of a platform displacement heuristic combined with a novel best-so-far ant replacement scheme worked well to give the ants the intelligence to minimise unnecessary platform changes. However, from the results it seems that a decrease in delay often results in an increase in platform displacement, especially for MC2 (see Section. 6.7.2). This suggest that minimising platform displacement and minimising delay may be conflicting objectives and future work will consider applying a dynamic, multi-objective ACO algorithm to this problem.

The algorithm executed in an acceptable time frame. For the single colony algorithm where the delay scenario had a small amount of disruption a solution could be found in an average of 19 sec for change 1 and 34 sec for change 6. The two- colony algorithm MC1 took longer to find a solution, which suggests that repeatedly iterating between the two colonies may take too long to be used in a real-world rescheduling scenario, however, this is also an area for future investigation.

Another possible avenue for future work is that of investigating the outcome of allowing the reallocation process to change the dwell time of the train. In this current investigation, dwell time can be changed by resequencing but not by reallocation. In some cases, dwell time may be large enough to absorb some of the train delays and reducing it may provide further improvement to the reallocation algorithm.

The fact that VNS performs better than MC1 on the delay scenario m = 10, f = 20 suggests that there are some delay scenarios that ACO does not perform as well on. An investigation of how to improve the performance of ACO on these scenarios warrants future work. As VNS is based on a local search, this suggests that the addition of a local search to the MC1 algorithm may improve it even further.

It is recognised that there are some limitations with the model, mainly as a result of the paucity of information available in the schedule feed. However, the model works as a ‘proof-of-principle’ and any additional information that could be obtained in the future could be used to refine it. An advantage of the model is that it does not rely on knowing the exact position of each train at every moment in time, via GPS sensors or similar, to work out the impact of the local station decisions on the trains’ ongoing journeys. Any new information about the trains’ positions could be used to update the train details before running for the next dynamic change period. This would allow new solutions to be made based on the current train positions.

The motivation for this work is to contribute towards the development of al- gorithms that could be implemented within a computer-based dispatching system to support the railway controller in solving schedule conflicts after a perturbation. The next step is to model a more complex station, such as Nottingham station, in order to apply the same algorithms. It is possible that in a busier station the two-colony algorithm may provide even more improvements. However, the creation of a model for Nottingham station would require more information than is avail- able in the schedule feed. It is hoped that the results shown in the current work will encourage the provision of such information and will allow the work to be expanded.

Conclusion

Delays are a common occurrence on the British railway network. Minimising the impact of a delay on the railway system is an important concern of railway opera- tors and the effectiveness of a rescheduling solution has an impact both on railway passengers and train operating companies. However, railway rescheduling is a dif- ficult problem, made more complicated by the fact it can be both dynamic and multi-objective. It is dynamic because the railway system is in a constant state of movement, while trains are in the process of being rescheduled, more delays may occur or trains with higher priority may arrive changing the nature of the problem over time. The problem is multi-objective because a train dispatcher may need to simultaneously minimise several conflicting consequences of the perturbation, such as delay, timetable deviation, energy consumption and missed connections. The con- flicting nature of these objectives means that increasing the quality of one objective may have a detrimental effect on the quality of another.

Dynamic railway rescheduling problems are rarely addressed in the literature. Most train rescheduling problems are regarded as static in that all delays are known about in advance and it is assumed that no further unforeseen incidents occur while the original delay is being resolved. In this case, if a further incident occurs the algorithm would have to be restarted again from scratch with the loss of potentially useful information from before the change that could have been used in the new environment. There are a growing number of researchers that recognise a need for such work [5, 3, 7, 6] and one of the aims of this thesis was to take a step towards the investigation and resolution of such dynamic railway rescheduling problems.

Multi-objective railway rescheduling problems are more commonly considered but in most cases the objectives are weighted and combined to create a single ob- jective and the problem is solved as a single objective problem. The disadvantage of this method is that the weights have to be determined in advance using domain

knowledge and it assumes that the relative importance of each objective does not change over time. A more flexible approach is to produce a set of trade-off solutions to provide the decision maker with a choice of solutions. This will allow them to make a decision as to which solution best matches their requirements at a particular moment in time.

The goal of this thesis was to address a gap in railway rescheduling research, that of rescheduling trains in dynamic environments for both single-objective and multi- objective problems. This is seen as an important step towards the development of algorithms for computerised dispatching systems as many real-world scheduling problems may be both multi-objective and dynamic.

To achieve this goal three different dynamic multi-objective problems were cre- ated. The first was a single objective dynamic junction rescheduling problem based on the Stenson junction on the British railway network. The second was an exten- sion of the Stenson junction problem, to make it a dynamic multi-objective problem, by the addition of another objective, minimising energy consumption. The third was based on a model created from Network Rail’s schedule data feed for Leicester sta- tion. This model considered the effect that the local rescheduling decisions made at the station had on the wider network.

A summary of the findings for each of those problems is given below:

7.1

Railway Junction Rescheduling in Dynamic

Environments

This problem is referred to as the dynamic railway junction rescheduling problem (DRJRP) in this work. In the DRJRP, the environmental change is a result of the arrival of new timetabled trains while the original trains are waiting to be rescheduled at the junction. In the extended DRJRP multiple unrelated delays occur over the time period of the investigation. The extra disruptions are caused by trains being delayed at the stations that feed into the railway junction.

The results showed that:

• P-ACO outperforms FCFS on the high to medium magnitude and high to medium frequency changes.

• When considering multiple delays, with a change magnitude of eight trains, FCFS was outperformed by all the P-ACO algorithms.

cope with dynamic changes better than an ACO algorithm that uses only pheromone evaporation (MMAS) to remove redundant pheromone trails.

• In P-ACO, when the ants in memory cannot be modified to make them feasible in the new environment, replacing the memory with elite immigrants after a change works effectively.

• Random immigrants were found to be unsuitable to replace the ants in memory when changes were of a high magnitude and a high frequency. The larger search space appears to demand the knowledge carried over from previous environments.

• Adding the ability to sequence trains at the stations was not beneficial to this set of dynamic problems. This may be because the extra decisions that the ants have to make increases the size of the search space and the ants struggle to explore it adequately. This suggestion is supported by the fact that when the search space is relatively small in the low frequency scenarios, algorithms that sequence the trains at the stations (EI-PACO, HI-PACO and MMAS) perform slightly better than the algorithm without station sequencing (NSS-PACO).