DITTO Project Deliverable 2.2 – Milestone 5

Scenario generation for train timetabling and train scheduling under uncertainty

Tolga Bektas, Attila Kovacs, Chris Potts

University of Southampton, UK

November 2015

AbstractIn this project, we apply operational research techniques to insert new train services into existing timetables while taking into account reliability and punctuality. The optimisation process is performed in a stochastic environment in which trains are subject to random delays. Typically, the risk of propagating delays over the network is higher when the railway infrastructure is highly utilized, i.e., we can increase capacity only by deteriorating reliability. Additionally, it is unclear how to combine these goals into a common objective function (e.g., how much loss in reliability are we willing to accept for inserting one additional train?). This issue is resolved by applying a multi-objective framework that exploits a special characteristic of the problem, namely that the number of train services that can be inserted into the timetable is small.

The train timetabling and scheduling problem is modelled as a two-stage stochastic program: in the first stage, new trains are inserted into the timetable, following which the reliability of the new timetable is optimised in the second stage under various scenarios. The objective is to generate a new timetable with a higher use of operational capacity and with minimal expected delay. Realistic measures are incorporated to recover from delays, including changing the order in which trains leave the station and approach a platform, and reassignment of trains to different platforms if the initially assigned platform is blocked.

A case study is proposed by considering the rail network around Peterborough station. Delay scenarios will be based on historical records.

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AcronymsACO: Ant colony optimisation.

DEP: Deterministic equivalent program.

FL: Fuzzy logic.

RO: Robust optimisation.

SA: Simulated annealing.

SAA: Sample average approximation.

SCOP: Stochastic combinatorial optimisation problems.

SP: Stochastic programming.

TS: Tabu search.

TSSIP: Two-stage stochastic integer program.

TTSP: Train timetabling and scheduling.

Technical TermsHeuristics: fast solution algorithms yielding good quality solutions in a short amount of computation time; they are often used for hard optimisation problems.

Metaheursitics: high-level heuristics based on searching multiple solutions; they typically generate better solutions than heuristics.

Exact techniques: provide the best feasible, i.e., optimal, solution; they are rarely used in real-world applications as they might require very long computation times.

Random variable: a variable that can take on multiple values (e.g., the number of spots when rolling a dice) because of an uncertain environment.

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Symbols and notation

Ε, ε Greek epsilon; typically denoting a small positive number∈ “element of” symbol

i ∈ N Variable i is element of set N

N = {1,2,3} Set N contains three elements: 1, 2, and 3

Summation: all elements of N are added up

Minimum in set N

Maximum in set N

Ω = {ω1, ω2, . . . , ωn} Set of scenarios, omega. It contains n scenarios. E.g., the scenarios in coin flipping are heads and tails

P = {p1, p2, . . . , pn} Set of probabilities associated with scenarios in Ω. In coin flipping, the set is {½ ,½}

Q(x) Deterministic objective function Q depending on solution x

Q(x, ω1) Objective function Q depending on solution x given scenario 1

Expected objective value over all scenarios Ω with associated probabilities P

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1.IntroductionThe aim of this project is to suggest an adequate response to the steadily growing railway traffic and to the increasing demand for high-quality transportation of passengers.

Clearly, a full passenger train is more cost effective and environmentally friendly than a half-full train. Yet, the rapid growth in passenger travel often results in overcrowded trains with dissatisfied customers. A common strategy to overcome the capacity problem, which is also considered in this project, is to offer more train services in a given time period (by increasing train frequency). A side benefit of this strategy is a greater flexibility for passengers in planning their journeys.

Reliability is the single most important requirement of passengers (Department for Transport [21]). It is expressed by the deviations from the current train timetable that are caused, for example, by bad weather conditions, prolonged boarding and alighting, and technical malfunctions.

Improving reliability and increasing operational capacity tend to be conflicting objectives (Armstrong and Preston [6], Preston et al. [42]). Typically, we cannot increase the number of services without causing reliability to deteriorate. We apply operational research techniques to insert as many new trains as possible into the system while complying with the safety regulations and maintaining reliability – measured by the average and maximum delay that is possibly caused by the new train services – at an acceptable level. In this work, we consider small delays, which are in the range of a few minutes as opposed to major disruptions. Increasing the speed of a train for example might compensate small delays; however, disruptions are unforeseeable events with a significant impact on the functionality of a railway system. Disruptions from accidents, broken tracks, malfunctioning signalling, etc., render the entire timetable infeasible. Effective countermeasures include cancelling trains, introducing alternative train services, and allocating replacement buses. Recovering from a major disruption in the shortest amount of time is a very hard optimisation problem that would exceed the scope of this project. For surveys on disruption management, we refer to Cacchiani et al. [18] and Jespersen-Groth et al. [34].

We make the following assumptions: First, when delays occur, they are mainly propagated at large stations with many incoming and outgoing trains. The track segments close to busy stations are highly utilised, thus causing many conflicts between trains (Armstrong et al. [7], Dollevoet et al. [22], Paraskevopoulos et al. [41]). This observation is also made in the DITTO Project Deliverable 2.1 (Armstrong and Preston [6]). Therefore, stations offer a great potential for mitigating delays. Second, trains are scheduled independently from each other, i.e., we ignore passengers transferring between trains. Third, travel and dwell times are random variables. Train timetables are typically generated with fixed buffer times inserted to protect against possible delays. Our work is based on the

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premise that it is possible to increase the operational capacity of a timetable with minor effects on the reliability if the uncertain nature of delays is taken into account explicitly. If the uncertainty of delays can be described by using probability distributions, then they can be treated as being stochastic. Stochastic optimisation methods set appropriate buffer times so as to increase the overall efficiency of the system. Figure 1 shows an illustrative example with a single platform and six nodes around it. The platform at the top of the figure is used by two trains a and b. Train a arrives from node 1, stops at the platform between nodes 3 and 4, and leaves the station towards node 5. Train b arrives from node 6, stops at the platform, and leaves in the direction of 2. The scheduled arrival times of a and b are 0 and 1 respectively. The dwell time for both trains is one unit of time, respectively. Headways are ignored for the purpose of this example. We assume that train a arrives on time with a probability of 4/5 and it is delayed by five time units with a probability of 1/5. In contrast, b is always on time. In a deterministic approach in which uncertainty is not taken into account, we estimate a reasonable realisation of the random variable denoting the arrival time of a and solve the problem. The estimated value is the mean of the random variable plus a small buffer ε (in our example ε is a value between 0 and 1). Accordingly, the deterministic arrival time of a is 1 + ε, and the optimal sequence in which trains are scheduled at the platform is first b (at time 1) and then a (at time 2). The evaluation of the deterministic solution in a stochastic context is depicted in the middle of the figure with an expected total delay of 2 4/5 + 5 1/5 = 2.6 time units. In the stochastic solution, however, which uses the probability information, the trains are scheduled in a reverse order: a first, b second, and has an expected total delay equal to (5 + 5) 1/5 = 2 time units. The solution of the stochastic problem is shown at the bottom of the figure. In this example, therefore, the stochastic solution would be favoured over the deterministic solution, at least as far as the expected delay is concerned.

Given the high complexity of the operations performed at large railway stations as well as the stochastic environment in which decisions are made, we cannot expect to generate optimal solutions to the problem. Therefore, we propose a fast solution approach based on the tabu search heuristic algorithm developed by Paraskevopoulos et al. [41] in the course of the preceding OCCASION project. For this purpose, we first transform the train scheduling and timetabling problem into a job-shop problem of the type that is used in production management. Job-shop problems are well studied in the literature with numerous solution algorithms available, which provide a solid basis for solving our railway problem. Second, we embed the tabu search heuristic into a framework for stochastic optimisation. By examining the behaviour of the system under minor delay scenarios, we can select a solution (i.e., a timetable) that is robust with respect to many different realisations of the random variables.

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Figure 1: Example: deterministic versus stochastic solution.

2.Related literature on stochastic optimisation

Our project combines two areas of operational research: stochastic optimisation and metaheuristic algorithms. The following sections give an overview of the state-of-the art.

2.1 Stochastic optimisation A straightforward approach for solving stochastic optimisation problems is to replace random variables with expected values and apply an algorithm for the corresponding deterministic problems. This approach provides good solutions in short time when variations in the data are small. The cost of implementing the solution in reality would be only slightly lower or higher than the cost indicated by the model. When variations are large, however, ignoring uncertainty in the model often results in solutions that are extremely costly or even infeasible. In many applications, even simple stochastic optimisation approaches result in significantly better solutions. We expect this to be the case in railway operations. There is always some uncertainty, e.g., a train conductor might get sick, or boarding and alighting may take longer than planned. The full potential of the rail network can only be exploited if uncertainty is considered explicitly. Yet, integrating uncertainty into railway management is extremely demanding due to the complexity of the problems.

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The challenge in optimisation under uncertainty is in formulating models that are realistic, but at the same time simple enough to be solved by modern solution techniques. The process of balancing these two goals has stimulated research that led to a large number of alternatives for formalising uncertainty. These alternatives are characterised, first, by the way uncertain information is incorporated into the models and, second, by the chronology of making decisions and learning about the realisation of the random variables (Bianchi et al. [11]). With regard to the first point, we distinguish several techniques. Stochastic optimisation is probably the most developed technique for considering uncertain information (Birge and Louveaux [14]). The main assumption is that uncertain data is represented by random variables with known probability distributions. In stochastic optimisation models, we typically minimise the expected value of the objective function or bound the probability of violating certain constraints.

Another common approach for dealing with uncertain parameters is robust optimisation (RO); for details, see Ben-Tal et al. [8]. RO models do not require the probability distributions of the uncertain data. In fact, RO models are deterministic; variations are modelled by providing a set of potential realisations of the uncertain data. RO is a worst-case oriented approach. The generated solution is feasible for each scenario in a given set, but is often very inefficient. Therefore, this approach is too conservative for many application areas.

Fuzzy logic (FL)) is a complementary technique to classical stochastic methods; for details, see Zimmermann [52]. In contrast to randomness, which is caused by chance mechanisms (e.g., flipping coins and rolling dice), fuzziness is the result of a subjective perception of events. Therefore, FL is often inappropriate for modelling uncertain information (Riedewald [45]).

Finally, uncertainty can be modelled as a stream of data that becomes gradually available to the decision maker – predictions about future data are impossible (Albers [1]). The solution algorithm, referred to as an online algorithm, generates solutions periodically based on partial information.

Our train timetabling and scheduling problem belongs to the class of stochastic combinatorial optimisation problems (SCOP) – a subclass of stochastic optimisation. Therefore, we will restrict the remainder of this section to SCOPs. Combinatorial optimisation problems have a finite, but usually large, decision space and solutions to the problem could be represented by permutations or binary variables. For example, a solution to the problem of scheduling trains at a station with a single platform could be the sequence in which trains stop at the platform, i.e., a permutation. Binary variables are often used to model yes-no decisions, e.g., to model whether or not a train is assigned to a particular platform.

SCOPs can be further categorised by the chronological order of receiving information and making decisions (Bianchi et al. [11]). In static SCOPs, decisions are made before the realisation of the random variables is known. The resulting

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solution is applied without any further modification, regardless of the actual outcome. Yet, sticking to the planned solution irrespective of the outcomes might be extremely costly or even infeasible. This issue is taken into account in dynamic SCOPs by making a decision without knowledge of the future, but taking a recourse action as more information becomes available. Dynamic SCOPs provide a convenient framework for making decisions at different planning levels. For example, trains are assigned to platforms before knowing the realisation of the random variables (e.g., arrival times) at the tactical planning level. However, trains may be reassigned if there is some delay at the operational level.

There are several techniques for solving SCOPs, e.g., stochastic programming, Markov decision process, and simulation optimisation. Each technique is associated with a particular way of formulating and solving a model under uncertainty. In the following, we restrict ourselves to stochastic programming techniques (SP). The basic idea for solving SPs is to transform a stochastic problem into a deterministic equivalent program (DEP) that can be solved by well-established solution algorithms. Typically, a DEP is created by formulating a sub-problem for each random outcome, and then combining the sub-problems into an integrated model. In the following, we will present DEPs for both, static and dynamic SCOPs.

2.2 Static SCOPs

The simplest SCOP formulation is a stochastic integer program. Random variables appear only in the objective function; the constraints are deterministic. The DEP is created by replacing the uncertain objective function with its expectation, i.e., a sub-problem is formulated for each random outcome and the sub-problems are consolidated into a problem that minimises the expected value of the single objectives. Given the set of outcomes (or scenarios) of a random experiment Ω with a probability distribution, a finite set of feasible solutions X, and an objective function Q(x, ω) that depends on solution x in set X and the random outcome ω in set Ω, the optimisation problem is to find a solution with minimal expected objective value:

In this expression, EP denotes the mathematical expectation with respect to distribution P. If the number of scenarios is finite, i.e., Ω = {ω1, ω2, . . . , ω|Ω|} with associated probabilities p1, p2, . . . , p|Ω|, respectively, we can calculate the expectation as the finite sum

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In some applications, the objective function is deterministic, i.e., Q is independent of the scenarios, but it is difficult to satisfy certain constraints for each random outcome. In railway optimisation, for example, we could maximise the (deterministic) number of train services in a timetable while bounding the (stochastic) knock-on effect. In chance constrained integer programs (CCIP), the problematic constraints are replaced by so called chance constraints. These constraints have to be satisfied at least with a given probability. With a high probability of satisfying a constraint, this formulation guarantees that violations of a constraint are not too frequent. The probabilistic constraints can be transformed into deterministic equivalent constraints if the distribution function of the random variables is known. A deterministic equivalent constraint enforces, for instance, that a connection between two trains is maintained if the delay of the feeder train is not larger than a given quantile of its delay distribution.

2.3 Dynamic SCOPs

In dynamic SCOPs, decisions are made step-by-step at discrete times with different states of information. Making a decision and observing the random outcome are consecutive events that can be repeated several times. Typically, there is a given recourse strategy for responding to a certain random outcome. This strategy is applied to update the solution as soon as new information becomes available. The most flexible, but computationally most expensive recourse strategy, is to re-optimise the entire problem from scratch. The most conservative updating strategy is to observe the realisation of the random variables and their consequences without performing any action. In this case, the problem is equivalent to a static SCOP. If the only action consists of incurring penalties in the objective function, the problem is said to have a simple recourse. Clearly, we can expect better results when applying a more flexible recourse strategy. However, the options for updating a solution are often limited by the available computational resources and the problem domain. For example, it is impossible to update the rail network when we observe a delay of a train.

The most popular dynamic SCOP formulation is the two-stage stochastic integer program (TSSIP). The first-stage corresponds to decisions that are made before knowing the realisation of the random variables. These decisions are denoted by x1 in the solution space X1. Later, when the first-stage decisions have been finalised and uncertainty is revealed, a recourse action may be taken. The recourse decisions are referred to as second-stage decisions (x2 in set X2). The opportunity to take recourse increases the flexibility at the first-stage. Assume, for example, that we assign trains to platforms that might be blocked with a given probability. The chosen assignment remains unchanged for several months. Without recourse, we would try to assign as many trains to reliable platforms as possible in order to avoid train cancellations. By considering a recourse action, we could fully utilise the available resources and temporarily reassign trains if a platform is unavailable. The objective function of the

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stochastic model is to minimise the cost of the first-stage decisions f1(x1) and the expected value of the second-stage decisions. The objective function of the second-stage is denoted by f2(x1 , x2 , ω1 ); it depends on the decisions made in both stages and the current scenario. So, the possibility to respond to the realised random event with a proper recourse strategy is already taken into account when making the first-stage decisions. The effects of the recourse are measured by the recourse function Q(x1, ω):

A special case, which is often assumed in practical applications, is a two-stage stochastic program with relatively complete recourse. In this case, there exists a feasible second-stage solution for each first-stage solution and each random outcome.

TSSIPs can be extended to multi-stage stochastic integer programs by considering several cycles of decision making and observing the realisation of the random variables. A decision is made in each period t of the planning interval for t = 1, 2, ..., T. The set of feasible solutions in period t, depends on the decisions that have been made before and on the previous realisations of the random variables. Figure 2 illustrates the decision making processes that are modelled by two-stage and multi-stage stochastic programs, respectively.

Multi-stage stochastic integer programs are closely related to stochastic dynamic programming and Markov decision processes; for details see Birge and Louveaux [14].

Figure 2: Two-stage and multi-stage decision making under uncertainty.

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2.4 Solving SCOPs

Evaluating the expected value in the objective function, EP[Q(x, ω)], involves solving a large number of optimisation problems, i.e., one for each scenario. Therefore, decision problems with uncertainty become more difficult with a growing number of random outcomes. In fact, in many real world problems, the number of possible scenarios is either infinity (e.g., if Ω has a continuous probability distribution) or grows exponentially with the dimension of the input data. Consider, for example, a railway network with 100 connections. On each connection, trains are either on time, delayed, or cancelled. In this example, set Ω contains 3100 scenarios. Clearly, solving a combinatorial optimisation problem with so many scenarios, i.e., sub-problems, is beyond reach.

One possibility for solving large problems under uncertainty is to replace the expected value in the objective function by an approximation. In two-stage stochastic programming, for example, we are primarily interested in a robust first-stage solution. Given a first-stage solution, the second-stage decisions can be addressed later when full information is available. So, instead of solving the sub-problem for each scenario, we could approximate the value of EP[Q(x, ω)] by an analytical function. Continuous approximation (CA) is often applied in logistics in order to solve complex problems quickly (Daganzo [20], Langevin et al. [37]). In CA, the mathematical programming formulation is replaced by an analytical model that relies on a concise summary of the data rather than on detailed information. The models are simple, but the objective value can often be expressed in closed form. With CA, the robustness of the first-stage decision can be evaluated in constant time regardless of the number of scenarios. The complexity of the problem is significantly reduced by using approximation functions. Yet, even sophisticated approximations introduce a systematic error into the computation of the objective function (i.e., the approximation consistently differs from the true objective value). The larger the approximation error, the less robust the first-stage solution becomes.

Another possibility is to estimate the objective value by simulation. This approach is often applied if there is no accurate closed-form expression for calculating the expected value. The estimate of the true objective value (EP

[Q(x, ω)]) is calculated by solving the problem only for a small sample of the random outcomes. Several sampling based techniques have been proposed to solve stochastic optimisation problems more efficiently. The basic principle is to find a small set of scenarios (i.e., draw a sample from P) that represents the full range of the uncertain parameters. The fewer scenarios, the easier it is to solve the problem. However, it is difficult to accurately incorporate uncertainty into the optimisation process when the sample is small. Even an optimal solution could be infeasible in reality if it was generated by using a too small sample.

Two sampling techniques can be distinguished: interior sampling and exterior sampling. In exterior sampling, a given number of sample scenarios is generated from Ω with regard to the probability distribution P before making a decision.

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Based on these scenarios, the deterministic equivalent of the stochastic problem is solved. In interior sampling, the samples are modified at each iteration of the optimisation process, e.g., by adding new scenarios, drawing a new sample, and using a subset of previously generated samples.

2.4.1 Exterior sampling

Two prominent examples of exterior sampling are scenario-based optimisation and sample average approximation (SAA). In scenario-based optimisation, the expected value function (EP[Q(x, ω)]) is estimated by using a set of specifically constructed scenarios. Scenarios are generated by taking into account the probability distribution of the random variables and the problem-specific experience of experts (Goetschalckx [26]). The resulting solution is robust with respect to the selected scenarios. However, this approach has several shortcomings. First, it is unclear how to select a small number of representative scenarios from an infinite or very large set of possible outcomes. Second, scenarios are often designed to represent an average, optimistic, and pessimistic case. These scenarios may be unrealistic if the correlation between the variables is ignored. Finally, it is difficult to associate suitable probabilities with the constructed scenarios.

Scenarios are based on random sampling rather than being constructed in sample average approximation (Kleywegt et al. [36], Verweij et al. [51]). This approach requires only the distribution of the random parameters. Most commonly, scenarios are selected by Monte-Carlo simulation. The Monte-Carlo principle requires samples to vary randomly in order to imitate a random experiment. Accordingly, a sample of size N is generated by picking N realisations randomly from Ω. The sample is said to be an independently and identically distributed random sample of the set of all possible random outcomes. The objective value of the true problem (i.e., the problem considering all random outcomes) is estimated by the corresponding sample average approximation:

A shortcoming of this approach is that rare events with small probability tend to be under-represented. Additionally, a large number of scenarios is needed to generate an accurate solution. In fact, the larger the sample, the better the results become. By applying the law of large numbers, we observe that the objective value of the sample-average solution converges to the objective value of the true problem as the sample size N increases (Shapiro and de Mello [47]). Yet, choosing a suitable sample size is challenging because of the trade-off between solution quality and computational effort. Despite these shortcomings, SAA is used widely because the problem can be solved by existing software for deterministic problems (e.g., commercial software packages for solving general

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linear programs such as CPLEX and specialised metaheuristics), it has good convergence properties, and the statistical inference is well developed (for example, we refer to Kleywegt et al. [36] and Verweij et al. [51] for validation of solution quality).

2.4.2 Interior sampling

In interior sampling, scenarios are generated within the solution algorithm, where a different sample is typically generated at each iteration. Examples of interior sampling are stochastic decomposition (Higle and Sen [30]) and the statistical L-shaped method (Slyke and Wets [48]). Both approaches have been applied for solving TSSIPs. The idea is to avoid solving the second-stage for each scenario, which is required for calculating the objective value. Instead, the expected cost of the second-stage objective function is replaced by a piecewise-linear approximation. This approximation bounds the recourse cost from below (in minimisation problems) and the bound is iteratively tightened. In each iteration of the algorithm, we solve one master and one (or several) sub-problems. Given a first-stage solution, a second-stage sub-problem is solved for one or several randomly chosen scenarios. The solutions of the sub-problems are used to generate constraints that strengthen the bound on the recourse function without eliminating the optimal solution (i.e., cutting planes). The master problem solves the relaxed first-stage problem by taking into account the bounds on the recourse function. The process of tightening bounds, i.e., generating cuts, and solving the relaxed problem is repeated until a stopping criterion is met (e.g., no more cuts can be identified).

Another method that applies interior sampling is stochastic branch and bound (Norkin et al. [39, 40]). Just as in deterministic branch and bound, the set of solutions is iteratively partitioned into smaller subsets. Within each step of the search process, the optimal solution is bounded from above and from below in order to identify promising subsets and to eliminate subsets that are not worth exploring. In the stochastic approach, these bounds are obtained by taking the expectation of certain random variables. Stochastic bounds are able to guide the partitioning and pruning process and they are also used for estimating the accuracy of the current stochastic solution. Yet, bounds might be invalid as they are calculated by generating samples in a Monte-Carlo fashion. Therefore, subsets cannot be disregarded in a straightforward manner. Rather, a backtracking mechanism must be incorporated into the algorithm that allows revisiting previously evaluated subsets. The algorithm might proceed infinitely, but it provides a probabilistic estimate of the quality of the solution at each iteration.

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2.5 Metaheuristics for stochastic problems

Metaheuristic algorithms provide good solutions quickly and are, therefore, very popular for solving hard optimisation problems. The main feature of metaheuristics is their ability to escape from solutions that are optimal only with respect to a small part of the solution space; such solutions are referred to as local optimum solutions. Given an initial solution, classical search algorithms move from the present solution to a neighbouring solution in the search space. At each iteration, the current solution is improved until a local optimum is reached. Depending on the initial solution, the local optimum can be a global optimum. Most commonly, however, the local optimum solution is considerably worse than the global optimum. One way to escape from local optima is to reinitialise the search process as soon as a local optimum is reached. By using different initial solutions, we can explore a large part of the solution space. Alternatively, we can move to a randomly selected solution from the neighbourhood or to the solution with the smallest improvement in the objective value (Hoos and Stützle [33]). There is an enormous number of papers proposing tailor-made solution approaches based on well-known metaheuristic frameworks such as simulated annealing (SA) and tabu search (TS). Most solution approaches have been developed in a deterministic context. However, there is a growing number of papers proposing efficient metaheuristics for solving problems with uncertain or stochastic information. In fact, metaheuristics are an important alternative to classical solution approaches that are based on mathematical or dynamic programming when it comes to solving stochastic optimisation problems. The lower computational complexity of metaheuristics allows us to solve problems approximately, but with a more realistic formulation of uncertainty.

As described above, there are two approaches for reducing computational efforts of solving SCOPs: replacing the expected objective value by a closed-form approximation and estimating the objective value by simulation. Metaheuristics with approximation functions have been developed mainly for stochastic routing problems. Probabilistic travelling salesman problems and vehicle routing problems with stochastic demand have been addressed using ant colony optimisation (ACO) by Bianchi et al. [12, 13] and Branke and Guntsch [16, 17]; using SA by Teodorovi and Pavkovi [49]; and by TS by Gendreau et al. [25]. Bianchi et al. [9,10] compare five metaheuristics (including SA, TS, iterated local search, and an evolutionary algorithm) by applying different approximation functions. The authors point out that using a fast approximation of the objective function significantly reduces the computation time in the local search phase. Given a solution x, local search identifies all solutions that can be generated by modifying x according to a given rule. The union of these solutions is called the neighbourhood, S(x), of x. The more complicated the rule, the larger the neighbourhood becomes. In order to decide whether or not to move from solution x to another solution in S(x), we need to calculate the change in the objective value for each potential move. In the deterministic case, this evaluation

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is typically performed in constant time. However, computing the cost of a move in the stochastic case is time consuming. Applying approximation functions has a substantial effect in reducing the running time. However, it is difficult to find approximations that are both accurate and easy to calculate.

The mechanism in sample-based metaheuristics is as follows. A solution is selected from the neighbourhood of the current solution. The two solutions are compared by generating a random sample and calculating the sample average estimates as described above.

There are many papers with different ideas of performing each step. For example, new solutions from the neighbourhood S(x) can be selected randomly or based on approximation functions as described above. Alrefaei and Andradttir [4] compare two different neighbourhood structures and conclude that selecting solutions randomly from the entire solution space produces better results than choosing solutions from a more restricted neighbourhood when the solution space is small.

Samples are usually reset in each iteration by assuming that random variables are independent and identically distributed (e.g., Gelfand and Mitter [24], Homem-de Mello [31], and Gutjahr and Pflug [29]). Resetting prevents the algorithm from being dominated by a single sample. However, samples can also be cumulative, i.e., new scenarios are appended to the sample in each iteration (e.g., Fox and Heine [23], Homem-de Mello [31]). This approach reduces computation time because each solution-scenario pair is evaluated only once. The decision of whether or not to accept a new solution as a current incumbent is mostly based on the comparison of the sample estimates. An alternative approach is presented in Alkhamis et al. [3] who use a statistical significance test in order to determine the difference between the new solution and the current solution. Bianchi et al. [11] highlight that applying statistical tests for comparing sample average values is essential for identifying the best solution.

The sample size used to compare solutions is probably the most critical parameter in stochastic metaheuristics. If the number is too small, the comparison of the solutions will be distorted. Alternatively, if the number is too large, the evaluation of the objective value becomes expensive (Bianchi et al. [11]). Homem-de Mello [32] propose an adaptive scheme for the sample size. For each iteration, the sample size is increased only if the results of a t-test indicate that a higher accuracy of the estimates is required. A similar approach is applied by Gutjahr [28].

Bianchi et al. [11] list several issues that are common to all stochastic metaheuristics; these issues are considered as open problems in the community. As mentioned above, metaheuristics are able to balance between intensification, i.e., finding a local optimum solution, and diversification, i.e., exploring a large part of the solution space on the search for a global optimum (Blum and Roli [15]). However, in SCOPs it is unclear how to balance intensification and

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diversification when the objective value is approximated. If the approximation is distorted, a poor solution might be accepted as the current incumbent, leading the search process into unpromising regions of the solution space. Also, it is hard to select a strategy for selecting the best solution based on samples. Most selection-of-the best strategies are designed to derive theoretical properties of convergence of a specific metaheuristic. So, these strategies might be inefficient when solving real-world problems. Adaptive sampling procedures that increase the sample size based on statistical tests, on the other hand, seem to be applicable for many problem variants. Another issue that has to be taken into account when designing metaheuristics for SCOPs is the level of stochasticity of the problem instance (e.g., measured by the variance of the random variables). State-of-the-art metaheuristics for deterministic problems might out-perform metaheuristics for SCOPs with regard to solution quality and computation time, if the level of stochasticity is low. Whether or not this is the case in train timetabling and scheduling is examined in the DITTO project.

3.Computational designWe develop two models for analysing the trade-off between increasing capacity by inserting additional train services and improving the reliability of the railway system. The first model is based on the train timetabling and scheduling problem investigated in the OCCASION project. This model has been developed in a deterministic context where no delays are taken into account, and is described in Section 3.1. For this reason, the ability of this model to incorporate stochastic information and, more importantly, to consider actions to mitigate any delays is limited. This necessitates the need to introduce a second model that is able to incorporate measures for reducing the effect of delays, which we develop and describe in Section 3.2.

3.1 A static timetabling and scheduling problem within a station

The train timetabling and scheduling problem (TTSP-I) is closely related to the model proposed in the OCCASION project. We consider a single station with an infrastructure that can be defined on a graph. Each edge represents a track segment; segments are connected by railway points that are represented by nodes. Figure 3 gives an example of a railway station. The station consists of three platforms for boarding and alighting, a set of railroad points {1, 2, . . . , 20}, a set of incoming/outgoing nodes {O, N}, and a set of track segments. Track segments can be either one-way or two-way. For each train passing through the station, we are given a scheduled arrival time and a scheduled departure time on

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Figure 3: Illustration of a railway station.

a specific track segment (e.g., the track segment next to a platform). A route through a station is a set of track segments that connect the incoming node with the outgoing node. In our example, a route from O to N could be the path (O, 3, 4, 5, 9, 10, N); passengers board and alight at the platform between nodes 4 and 5. A high-speed train from N to O that passes by the station without stopping could be routed on the path (N, 16, 15, 18, 17, O). The route is given for each train and cannot be altered during the optimisation process. Only one train may occupy track segments within the station at a time. Additionally, a train may enter a track segment only after a pre-defined buffer time, referred to as headway, following a previous train on the same track. Track segments at the entry and exit of the train station represent connections between stations and may hold several trains as long as the headway constraints are satisfied. For example, we may dispatch any number of trains from 16 to N as long as the minimum headway time between trains is respected.

Edges adjacent to O and N can represent any layout of the railway network. Therefore, the proposed notation is without loss of generality. For example, (O, 3) and (O, 7) may be two tracks to two different destinations, i.e., (O, 3) to destination X and (O, 7) to destination Y. Another possibility is that (O, 3) and (O, 7) connect the same stations and may be traversed in the same direction, but (O, 3) is a fast line while (O, 7) is a slow line. Each track segment is associated with a minimum travel time and each platform requires a minimum dwell time. Trains may be subject to random delays; this is incorporated by adding a random variable to the travel times of the edges adjacent to O and N, and to the dwell times at the platforms. Travel times within the station are assumed to be deterministic.

Suppose we are given a set of train services that should be inserted into an existing timetable. The route of each new service is fixed, but arrival times and departure times are set during the optimisation process. The following constraints have to be taken into account when inserting a new service: first, a new train service must be scheduled such that the timings should be reasonable in relation to the timings of existing services operating with the same itinerary

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(e.g., a duplicate of a service should not leave shortly before or after the original); second, original services may be shifted backward and forward in time only within a limited interval.

The problem involves defining the arrival time of each train at each track segment while taking into account safety considerations such as headways.

3.2 A dynamic timetabling and scheduling problem on a small network

The second train timetabling and scheduling problem (TTSP-II) is designed to explicitly take into account stochastic events. Given a timetable and multiple delay scenarios, our aim is to set new arrival and departure times such that the expected deviations from the timetable that are caused by random delays are minimised. Several recourse actions are available to recover from a delay as soon as possible (so as to avoid knock-on effects, for example). We can decide on the order in which trains travel between stations, the train-to-platform assignment, and the order in which trains approach a platform. The objective function is the weighted average of the maximum delay that any train experiences and the average delay per train. TTSP-II belongs to the class of two-stage stochastic programs.

In TTSP-II, we consider a small segment of the rail network, an example of which with three stations is given in Figure 4. Station 1 and Station 2 are connected by two tracks with one in each direction. Station 2 and Station 3 are connected by two tracks in each direction: a low-speed track and a high-speed track, respectively. Trains enter and leave the system through a Source/Sink node.

Figure 4: Segment of rail network with three stations.

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The TTSP-II can be defined on an event-activity graph (Dollevoet et al. [22], Schöbel [46], Veelenturf et al. [50]). Events are grouped into arrival events and departure events. Each train service is modelled as a series of alternating arrival and departure events. The activity connecting an arrival event and a departure event is referred to as waiting activity. The activity between a departure event and an arrival event is a travelling activity. Each activity is associated with a given duration (e.g., dwell and travel time).

Figure 5 shows an event-activity graph with three train services that are offered on our network segment. Events are denoted by squares and activities by arcs. For each train, there is one arrival event A and one departure event D per station. Trains 1 and 2 enter the system from the source, pass through all the stations, and head towards the sink node. Train 3 travels into the opposite direction from the source through Station 3 and Station 2 to the sink node (i.e., in a direction different from Station 1). Dashed arcs represent headway activities. Headway activities are incorporated to avoid conflicts between trains using the same resource. Trains 1 and 2, for example, use the same track segment when travelling from Station 1 to Station 2. Clearly, the trains cannot depart simultaneously. Therefore, we have to determine a precedence relation between these two trains: either Train 1 departs before Train 2 or vice versa. Generating a feasible solution involves selecting exactly one headway activity for each pair of conflicting events. The duration of the headway activities is specified by safety regulations. The events of Train 2 at Station 2 are shown by dashed lines to indicate that the train passes through without stopping, i.e., arrival and departure take place at the same time. These “dummy” events are needed to prevent overtaking on single-track segments, e.g., if Train 1 leaves Station 1 before Train 2, then Train 1 should arrive at Station 2 before Train 2. The first-in-first-out property is modelled by additional headway activities at Station 2. In Figure 4 there is only a single track between Station 3 and the Source/Sink node, i.e., we have to determine a precedence relation between all three trains passing through Station 3. This is achieved by extending the graph by one pair of headway activities for each pair of conflicting trains. The duration of a headway, which is also shown as an activity here, between trains travelling in the opposite direction is at least as long as the time required for traversing the single-track segment.

Figure 5: Event-activity graph with three stations and three trains.

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Each station is represented by a sub-network. Station 2, for example, consists of several track segments and points, and four platforms for boarding and alighting (trains can stop at both sides of each platform, see Figure 6). For each pair of trains that are assigned to the same platform at the same station, we define platform activities as illustrated in Figure 7. Similar to the headway activities, there is a pair of mutually exclusive dwell activities for each pair of conflicting events. Let i and j be the arrival events of two trains assigned to the same platform, respectively. Conflicts are resolved by scheduling either i before j, or j before i. Dwell activities ensure that enough time has elapsed between one train leaving a platform and another one approaching it.

We use binary decision variables (i.e., 0 or 1) to model whether or not an arrival is assigned to a certain platform. Some arrival-platform assignments are infeasible, e.g., if the train is longer than the platform. Some other assignments are pre-defined in a way to avoid confusion among passengers.

Figure 6: Illustration of Station 2 with rail tracks and switches.

Figure 7: Pair of dwelling activities for trains using the same platform (adapted from Dollevoet et al. [22]).

The objective function considers both, the worst delay and the average delay. The worst case is expressed by the maximum deviation from the original timetable. The average case is given by the average delay per event. Stochasticity is incorporated into the model by replacing deterministic travel and dwell times by random variables and minimising the expectation of the objective function.

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3.3 Dealing with conflicting objectivesIn operational research, we can distinguish between three approaches for modelling a decision problem with several goals: using hard constraints, soft constraints, or multiple objective functions. With hard constraints, we can for example bound the minimum number of trains that will be added to the timetable. Soft constraints are expressions that are penalised or rewarded in the objective function in addition to optimising the actual objective, e.g., to minimise the weighted sum of the delay minutes minus the number of additional train services. If we want a solution to be good with respect to several features without being able to prioritise one feature over the other, we typically adopt a multi-objective formulation. As mentioned above, our goal is to insert as many trains into the timetable as possible while keeping the reliability of the system at an acceptable level. Our measurement for reliability is the expected weighted sum of average and maximum delay.

Both terms, the number of inserted trains and expected delay, could be used as an objective function. Yet, putting a hard constraint on either measurement might have undesired consequences: the solution might be infeasible if the bound is too tight, whereas if the bound is too loose, we might not use the full potential of the railway system. Also, it is difficult to aggregate operational capacity and reliability in one objective function: how much loss in reliability are we willing to accept for inserting one additional train?

To deal with conflicting objectives we propose a multi-objective framework that considers each objective function explicitly. Our assumption is the following. The number of additional train services that can be inserted into the timetable is small. Results from the OCCASION project indicate that we cannot expect to insert more than 5 new train services in the considered time horizon (morning peak hour) – this number is probably even lower if reliability is taken into account. Therefore, we can enumerate all possible combinations of additional train services. First, we identify different sets of train services to be inserted into the timetable. Hard constraints guarantee that the new services will be incorporated. Then, we solve the stochastic optimisation with the objective of improving the reliability of the updated timetable. If the capacity of the system can be increased by a maximum of 5 train services, then there are 25 = 32 possibilities of forming sets of services, i.e., we have to run the optimisation process 32 times. The output is a graph that shows the reliability of the 32 timetables, each having a different operational capacity.

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3.4 Data sets

The models and solution approaches proposed in this project are generally applicable regardless of the input data. In the development phase, however, we restrict ourselves to a small part of the rail network around Peterborough station. We consider all train services related to this station. Delay scenarios are generated by analysing the historic delay data provided by Network Rail [38]. The data set contains information about all delays and cancellations of passenger train services from December 2013 to May 2015. The large number of delay records enables us to design realistic delay scenarios.

We will examine delay attribution data, including that produced by the Office of Road and Rail Regulation (ORR), and have done some preliminary analysis for Virgin Trains East Coast services. Previous work by Preston et al. [42] has shown how delays can be attributed to Trains Operating Companies (TOCs), Network Rail or external events, but with this breakdown evolving over time. Some of these events (such as routine train and infrastructure failures) will be random. Others may have more systematic features, related to both time and space. For example, delays relating to train operations may be concentrated in peak periods and adverse external conditions concentrated in particular season (e.g., leaves on the line in autumn, and snow in winter).

We will use statistical tools (e.g., regression analysis and distribution fitting) to assign a distribution function to each random variable based on the data characteristics. Given the distribution, we will generate random samples in a Monte-Carlo fashion.

4.Future workIn the next steps of the project, we plan to finalise the model and formulate a corresponding integer linear program. Once the random variables that influence reliability have been identified, we can analyse the delay data provided by Network Rail [38] and generate random scenarios.

The most work-intensive task will be to design and implement an efficient metaheuristic for the two-stage stochastic integer program. Problems of this type have rarely been addressed in the literature. Therefore, we consider this project as original and innovative. Our work will provide new tools for solving complicated train timetabling and scheduling problems and new insights into the conflict between capacity and reliability.

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