EC techniques for solving Railway Rescheduling Problems

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The application of EC techniques to the train rescheduling problem is, at the present time, a relatively unexplored research area compared to the application of non-EC techniques. One of the most popular EC techniques to use for railway rescheduling is a genetic algorithm (GA). Research approaches differ in the way the chromosome is encoded. One approach is to use the chromosome to encode a sequence of trains to pass through a junction [47, 48, 2], whereas in other approaches the chromosome encodes a vector of train times [49]. Table 3.1 summarises the different approaches to railway rescheduling using EC techniques.

Table 3.1: Summary of EC approaches for single objective rescheduling problems

Approach Authors

GA Ho and Yeung (2000) [47], Ho and Yeung (2001) [48], Semet and Schoenauer (2005) [41], Fan et al. (2012) [2], D¨undar et al. (2013) [50],

DE Chen et al. (2010) [49], Chen et al. (2015) [51] Other EAs Semet and Schoenauer (2005) [41]

ACO Fan et al. (2012) [2] ACO Hybrid Sam`a et al. (2015) [23]

Railway Rescheduling using a GA

Ho and Yeung [47] considered a rescheduling problem on a model of a junction on a metro line. The junction consisted of two train lines converging into one resulting in a decision needing to be made about the order that trains should enter the junction. They used an event-based traffic flow model based on fixed block signalling involving eight trains, four on each line into the junction. They tackled the problem using a GA. Each chromosome represented a feasible sequence of trains to pass through the junction and the starting population was created from the set of all possible feasible solutions. To prevent infeasible solutions from being produced by crossover or muta- tion, they replaced these two operations with the selection of a nearest neighbour to the best solution. The nearest neighbour replaced the worst solution out of the two population members. They found that their algorithm could find a solution within less than 5% of the optimal but with a much shorter computation time than using dynamic programming (DP). In 2001, Ho and Yeung [48] extended the work by also comparing their algorithm to the performance of simulated annealing (SA) and a tabu search (TS). They found that all methods provided a similar balance between computation time and optimality and all performed better than DP. These studies show the value of using a GA for a single junction scheduling problem, however, the fact that they only consider a simple two track junction with a maximum of four trains on each side of the junction makes it a very limited problem. In addition, the static nature of the problem assumes that no other trains will arrive while the trains are being fed through the junction, which may be an unrealistic situation. Further, using a starting selection pool of all possible feasible solutions, means that the computation complexity and therefore execution time will grow as more trains are added to the problem.

considered a rescheduling problem on the British railway network on the Derby to Birmingham line taking in the North Stafford and Stenson junctions. Their aim was to compare and contrast eight different rescheduling approaches to determine which approach performed best. They compared Brute Force (BF), First Come First Served (FCFS), TS, SA, GA, ACO, DP and decision-tree based elimination (DTBE). A description of these algorithms can be found in the glossary at the beginning of this thesis on page xviii. FCFS is a simple heuristic often used by railway dispatchers [32]. In FCFS, trains are allowed to pass through the junction in the order that they arrive at the junction. In the GA the chromosomes encoded the sequence of trains to pass through the junction. New solutions were produced by using a two-point crossover operator and infeasible solutions were repaired by pushing the train that violated the constraint further back in the train sequence, until the point where the violation no longer occurred. The problem involved 12 trains of 3 types, Class 150, Class 200 and F2-mixed freight trains. Trains had different masses, acceleration, lengths and braking rates. The scenarios differed in the size of the delay and the number of trains delayed. They found that the GA was slightly slower than TS and DTBE, taking an average of 88 s over all the delay scenarios, but gave 28% improvement in total delay penalty compared to FCFS.

D¨undar et al. [50] also used a GA but this time encoded the chromosomes to represent which out of a pair of trains should access a passing place first on a stretch of single track on the Turkish State Railway. The track was 150km long with 18 passing points and they considered from 6 to 17 trains. Their objective was to min- imise total weighted delay, with the weights representing the train priorities. They compared the GA with an artificial neural network (ANN), trained with dispatchers’ responses, and an exact model executed on the LINDO software package. Compared to the ANN the GA was able to find the optimal solution for small sized problems in shorter time and to reduce the total delay by around half. The exact method was run on the small problems with up to six trains and the GA was found to produce the same optimal solution. However, in terms of computation time the GA could produce a solution in under 85 s while the exact solution took 320 s. This is an interesting solution to a static railway rescheduling problem, but modelling more than a single track could result in a very complex chromosome.

Railway Rescheduling using Differential Evolution (DE)

Some researchers have applied a DE algorithm to the railway rescheduling problem. A DE is a population based EC technique where individuals in the population consist of a vector of real values. New individuals are generated by adding a weighted

difference between two individuals to a third individual. If the new individual is superior to the comparison individual it replaces it.

Chen et al. [49] used a DE algorithm to tackle the problem of rescheduling at a single junction. The junction under consideration, the St. Pancras Midland Road Junction, has two routes into the junction and one conflict point. The aim was to reschedule 24 trains, 10 on one route and 14 on the other route in a 1 hour window. They used a modified DE strategy (DE JRM), where the modification consists of an additional operation after mutation and crossover that ensures the solutions produced are feasible. The single objective was that of minimising the weighted average delay for passengers and was calculated by considering the number of passengers that alight from the train at the stop after the delay. The test scenarios were generated using a simulator based on the Monte-Carlo method and they tested the performance of the algorithm by comparing it to FCFS.

They found that the DE JRM performed significantly better than FCFS on both the short and long delay test cases. In the case of short delay scenarios most of the trains could be rescheduled close to the nominal timetable, in the case of long train delay scenarios, the algorithm could significantly reduce the delay by re-sequencing the trains before they arrived at the junction point.

Chen et al. [51] also applied the modified DE algorithm (DE JRM) to a bottle- neck junction on the Thameslink. The section of the network under consideration has trains from four different origins converging into the bottlenecks and up to 24 trains per hour in each direction in peak time. The objective was to minimise the weighted average delay, where the weights reflected the priorities of the trains. They compared the results of the DE JRM with FCFS and with a conventional automatic route setting (ARS) strategy which decides the order of trains to be run based on each train’s estimated delay and its weighting. They found that DE JRM decreased the average weighted delay significantly compared to FCFS and ARS. It could also produce a solution in around 2-3 s.

A limitation of this work is that it assumes all delays are known about in advance and that no new incidents occur during the rescheduling process. In a real life situation, while trains are queued to pass through the junction, later trains may be building up behind them and experiencing knock on delays.

Railway Rescheduling using other Evolutionary Algorithms (EAs)

Semet and Schoenauer [41] tackled the problem of rescheduling at multiple junctions. They modelled the problem as a graph with nodes representing the stations and edges representing tracks between stations. The model used was based on real data

provided by SNCF, the French national railroad company. To solve the rescheduling problem they created a hybrid system using an EA combined with CPLEX. The purpose of the EA is to generate an optimal permutation of trains which, when passed through a scheduler module, produce an optimal schedule. The scheduler module handles the constraints and uses a semi-greedy method to allocate the trains to the schedule, one at a time, in the order given in the EA’s chromosome. The scheduler is considered semi-greedy because it only considers optimality at one node (station) level rather than at the global level. The single objective was to minimise the total accumulated delay of the trains. To generate disruption they delayed a train for 10 minutes at a large connecting node around the middle of the simulation period. To test the efficiency of their algorithm, they compared it to a solution obtained using CPLEX alone.

They found that CPLEX took several hours to reach an acceptable solution for an average size problem and that increasing the size or complexity of the problem greatly affected the computational efficiency. In contrast, the EA needed only 15 minutes to reach a good but sub-optimal fitness level. However, they found that the EA was unable to improve the solution after around 15 minutes. They suggest that this is because the scheduler was semi-greedy rather than greedy meaning that train insertions into the schedule were only optimal at the station level rather than globally. They speculate that the system could be improved by allowing it to look ahead to future stations. A limitation of the research is that the system was only applied to one problem scenario, they did not investigate problems with different patterns of delay. In addition, they assumed that all trains had the same priority level, which may not always be the case.

Railway Rescheduling using ACO

Despite the fact that many railway rescheduling problems can be considered to be combinatorial optimisation problems, there seems to have been little work using ACO for railway rescheduling. In their research based on Stenson junction, Fan et al. [2] also applied an ACO algorithm to determine the order that the trains should pass through the junction to minimise the delay penalty. They found that the ACO performed slightly better than the GA taking an average of 77 s to find a solution and producing solutions 30% better than those found by FCFS. They concluded that overall ACO gave the closest results to the optimum within a practical computation time.

Railway Rescheduling using Hybrid ACO

Sam`a et al. [23] used a hybrid system to find a solution to the railway rescheduling problem they named the real-time Railway Traffic Management Problem (rtRTMP). They used the ACO algorithm MMAS combined with a local search to discover an effective set of train routes for input to a MILP solver. The rtRTMP involves the detection and resolution of conflicting resource requests after perturbations in the railway system with the aim, in this case, of minimising the sum of secondary delays at the end station.

The objective function used for MMAS was to minimise the sum of overlap be- tween trains requiring the same track resource. Using MMAS to find good candidate solutions for the routes simplified the work performed by the MILP solver. Exper- iments were carried out in the laboratory using real-world data from the French railway network around the city of Rouen. Each delay scenario had an average of 13 trains and each train had a maximum of 192 routings. The use of MMAS to create the train routes was compared to randomly created train routes on twenty randomly generated delay scenarios with delays of between 5 and 15 minutes. MMAS almost always gave better results than choosing random train routes and it decreased the average objective function value by more than 50 seconds. In 9 out of 20 of the investigated scenarios the improvement was significant.

This work illustrates how selecting good candidate solutions for the trains’ routes before resolving the conflicts is a feasible and efficient approach to the problem. However, this is a static problem and considers a one-time delay scenario, it does not take into account other unforeseen delays that may occur during the time period of the problem.

From the above research it is apparent that EC techniques have the potential to make a contribution to the difficult problem of railway rescheduling. However, none of the rescheduling problems investigated makes an allowance for the dynamic nature of the problem. They do not consider the possibility that more trains may arrive or more train delays may occur as trains are running to their rescheduled timetable.