Two-phase heuristic for SNCF rolling stock problem

This paper considers the problem of rolling stock unit management at railway sites, as defined by French Railways (SNCF) and proposed at the ROADEF/EURO Challenge 2014. The problem involves managing trains between their arrivals and departures at terminal stations. The purpose of this paper is to describe the method developed by our team upon taking part in this challenge. The main goal was to achieve the best possible results in relation to the other competitors. This problem is currently being jointly addressed by several SNCF Departments, by breaking it down into several sub-problems (train assignment, scheduling constraints, track group conflicts, platform assignment, maintenance planning, etc.) to be solved sequentially. Consequently, the integrated problem formulation dealt with here (in the challenge) actually reflects a forward-looking approach. Between terminal station arrivals and departures, the trains in fact are always present. This aspect, unfortunately, often gets neglected in railway optimization methods (Ramond and Marcos 2014). In the past however, rail networks possessed sufficient capacity to accommodate all trains without introducing excessive constraints—but this is no longer the case. Traffic has indeed increased considerably during recent years, and a number of stations are now experiencing serious congestion (Ramond and Marcos 2014; Corman et al. 2010; Huisman et al. 2005). Current traffic trends will make this phenomenon even more challenging over the next several years. The proposed model focuses on the multiple dimensions inherent in this problem, by taking into account many different aspects. The model’s scope remains within geographically limited boundaries, typically over just a few km in urban environments: train stations and their surrounding railway infrastructures are targeted by this model. The solutions to such problems involve temporary parking and shunting on station locations, which typically consists of platforms, maintenance facilities, rail yards adjacent to stations and the set of tracks linking these various resources (this infrastructure constitutes what is referred to as the “system”).

This paper will be organized as follows. A description of the problem is provided in Sect. 2. The description given herein was borrowed from the official competition subject (Ramond and Marcos 2014). Section 3 addresses a related work. Our two-phase approach is described in Sect. 4: solving the problem of matching (assigning) trains to departures is considered first, followed by the problem of scheduling trains inside the station. An iterative improvement procedure, based on a local search, is presented at the end of this section. The computational results obtained from the available set of instances, provided by SNCF, are detailed in Sect. 5. The last section presents the authors’ final remarks and conclusion.

A comprehensive description of the problem can be found in the competition subject (Ramond and Marcos 2014). Herein, we give a simplified, more concise formulation which is sufficient to describe our approach and make our presentation clearer and more readable.

2.4 Infrastructure resources

Between arrivals and departures, trains are either moving or parked on tracks that we consider to be resources. Let \(\mathcal{R}\) be the set of all resources. Resources can be either platforms, maintenance facilities, yards or track groups. Platforms and maintenance facilities represent portions of tracks considered individually, while track groups and yards are aggregated types of resources which usually contain more than one track as well as switches to physically link the different tracks together. An example of resources configuration is given in Fig. 1. A resource \(r \in \mathcal{R}\) has a set of neighbouring resources, defining the possible transitions for trains, in a symmetrical way. Resources represent railways infrastructure elements which are in general linear and can be accessed from both sides, and in some cases from only one side. Given this aspect, the neighbours of a resource can then be divided into two subsets, one being physically associated with each side of the linear element. The transitions between a resource, r, and one of its neighbours, are performed through one of the entry/exit points, which we call gates, located on either side of the resource. These gates are the physical tracks linking the various resources. Platforms and facilities, representing individual tracks, have at most one gate on each side. Only track groups and yards may have more than one gate on each side. For these types of resources, two adjacent resources might be linked by more than one gate: the gate used for the transition between two resources has to be specified. For example, track group 1 in Fig. 1 has 8 gates and 8 neighbouring resources on the right side, and 14 gates and 2 neighbouring resources to the left.

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Platforms represent tracks within the train station where passengers can board and disembark. Maintenance facility resources are special tracks inside maintenance workshops. They are used to periodically reset the train remDBM. Platforms and maintenance facilities are both characterized by a length which may not be exceeded by the total length of trains using it (length of platforms in the example shown in Fig. 1 varies from 270 to 400, while all facilities have a length of 350 or 450).

Yards are sets of tracks used to park trains. Trains can stay in yards with no time restriction, but a yard has a limited physical capacity in terms of the number of trains it can handle simultaneously (25 for the yard in Fig. 1 example). Capacity is the only constraint when parking the trains in the yards, i.e. there are no constraints on movement of trains in the yards as is the case on platforms or in maintenance facilities.

Track groups are sets of tracks used by trains to move throughout the system. Its real physical configuration in terms of tracks and switches linking them can be very complex. Its complexity is not taken into account herein; we have considered it as a black box with some indications on how to identify conflicts. The duration of use of track group k by any train (i.e. travel time) is a constant denoted by \(trTime_k\). It is the time required by a train to enter the track group k on one side and exit from the opposite side. Indeed, a train entering on one side of a track group must exit from the other side. All gates on one side are reachable from all gates of the opposite side. In addition, \(hwTime_k\) represents the headway of the track group: this is a safety time which must be adhered to at any place between two trains. In Fig. 1, trTime and hwTime are shown for each track group in the top left and bottom right corners respectively; for example, for track group 2 we have \(trTime = 30\,\hbox {s}\) and \(hwTime = 3\,\hbox {min}\). When several trains use a track group over the same time period, there may be conflicts between them. A conflict occurs if the paths of two trains on the track group intersect and headway time hwTime between them is not complied with. Conflicts are considered to be unfeasible. Each arrival and each departure has a fixed arrival/departure sequence. These sequences represent the routing of trains on the tracks during the last few km before reaching the platforms. Sequence is defined by an ordered set of track groups that a train has to use when arriving/departing. In Fig. 1, possible arrival sequences are TrackGroup3-TrackGroup1 and TrackGroup4-TrackGroup2-TrackGroup1.

A large body of literature on train routing problems is available. However, any exact or even similar matches from previous research with the current problem could not be identified. Only variations to some of the sub-problems occurring here can be found in publications such as Lentink et al. (2003) and Freling et al. (2005). Furthermore, there is a broad range of optimization models for specific problem variants. We will not therefore be emphasizing any of the papers or related problem variants herein.

Recently, both during and after the competition, a few papers (or technical reports) were published on this topic. Cambazard and Catusse (2014) propose a methodology heavily based on modelling with both integer programming (IP) and Constraint Programming technologies for problem resolution. These authors mainly concentrate on solving the problem of assigning trains to departures and using an integer programming approach similar to that explained in this paper. Haahr and Bull (2014) propose two exact methods, IP and Column Generation, for solving the same sub-problem (called “Train Departure Matching Problem” in their paper). They report that solving the problem of assigning trains to departures exactly is very difficult, if not impossible, for a given set of instances. Most of the teams competing in the ROADEF/EURO Challenge 2014 proposed algorithms that rely on greedy procedures or integer programming or a combination of both. Modelling an entire problem, or a significant part of one, using IP is theoretically possible and has been achieved by a number of competitors, yet the outcome proved incapable of producing satisfactory results on the given set of instances. IP techniques therefore are mainly used to solve only specific sub-problems. The breakdown of a problem into two dependent sub-problems, i.e. assignment and scheduling problems, is quite a natural step given the complexity of the initial problem and was implemented in most of the approaches presented. To the best of our knowledge, none of competitors conducted a local search (at least as a significant component of their research).

In our method, the problem has been broken down into two sub-problems, which are then solved sequentially. During the first phase, a train assignment problem is solved by combining a greedy heuristic procedure with integer programming (IP). The main objective here is to maximize the number of assigned departures while meeting technical constraints. Other objectives are taken into account as well, with the aim of obtaining “better” input for the subsequent phase. During the second phase, the train scheduling problem, which consists of scheduling the trains inside the station, is solved using a constructive heuristic model. The goal is to schedule as many assignments as possible, by using station resources and complying with all the constraints. An iterative improvement procedure is implemented in order to improve the resulting schedule.

4.1 Assignment problem

This section will describe the method adopted to solve the problem of matching (assigning) trains to departures. Assigning train \(t \in \mathcal{T}\) to departure \(d \in \mathcal{D}\) must meet the following technical constraints:

• The remaining distance before the maintenance of train t must be sufficient for departure d: \(remDBM_t \ge reqD_d\).

• The time difference between arrival and departure must be large enough (for maintenance operations, parking,...).

We call an assignment (t, d) of a train \(t\in \mathcal{T}\) to a departure \(d\in \mathcal{D}\) feasible if the following holds:

1. 1.

   \(arrTime_t + maintTime(t,d) + addTime(t,d) \le depTime_d\),

    
2. 2.

   \(remDBM_t \ge reqD_d\),

    

where:

• maintTime(t, d) is the total maintenance duration required for scheduling t to d (0 if maintenance not required),

• addTime(t, d) is an additional time necessary for parking and handling the train, i.e. in the case where the train is required to leave the arrivals platform before departure (non-immediate departure). The train may be parked either at the maintenance facility to undergo servicing or at any authorized resource before being scheduled for departure.

Additional time, addTime(t, d), is a variable value to be determined; it is used to increase the chance of finding a feasible schedule, yet an excessive value of this variable may also decrease the number of assigned departures. This value has been experimentally set to lie within the range of 5–60 min. Only feasible assignments (t, d) are to be considered. Furthermore, the solution to the assignment problem must abide by the maintenance limit constraint i.e. the total number of maintenance operations during any day must not exceed a given number maxMaint.

The following objectives are considered in the assignment phase:

1. 1.

   maximize the number of assigned departures,

    
2. 2.

   maximize the number of immediate departures,¹

    
3. 3.

   minimize the number of maintenance operations,

    
4. 4.

   minimize the number of assignments with a large difference between departure time and arrival time (greater than 10 h for example).

    

These objectives are mixed and exact importance (weight) of each objective part will be given while describing the methods used for solving the problem. The reason for introducing objectives (2) and (3) is to minimize the use of track groups since minimizing track group use will obviously decrease the chance of conflict. Another goal of inserting (3) is to minimize the use of maintenance facilities, which are considered critical resources. The aim in avoiding long waiting time between arrival and departure is to minimize the use of parking resources. The following definitions will be used herein:

• nmbM(t, d): the number of maintenances required to schedule train t to departure d (equals 0 or 1);

• imm(t, d): equals 1 if d is immediate, 0 otherwise;

• long(t, d): equals 1 if \(depTime_d - arrTime_t > L\), where L is a parameter, 0 otherwise.

The assignment problem is made significantly more complex as the remaining distance for some trains is not known before scheduling the linked departures. The maximum number of maintenance operations per day constraint complicates the problem even more. A combination of greedy and integer programming (IP) algorithms has been implemented to solve this assignment problem.

4.1.1 Greedy procedure

The greedy procedure tries to match departures one by one. For each departure d, the best train is chosen in consideration of the defined objectives. The procedure can be formalised as follows:

• sort departures \(d\in \mathcal{D}\) in ascending order with respect to departure time;

• for each departure \(d\in \mathcal{D}\), find the “best” available train.

Departure assigned to train t will be denoted by d(t) (\(d(t)=-1\) in case no departure is assigned to t). After assigning train t to linked departure d, the value remDBM of the corresponding linked train is updated according to the constraint. The exact choice of train for each departure is accurately described in the pseudo-code of the greedy assignment (Alg. 1). The following rules are informally applied when choosing the train for each departure d:

• consider only currently unassigned trains;

• whenever possible, choose an immediate assignment;

• assignments without required maintenance and trains with a small remDBM are preferable for non-linked departures;

• in contrast to the previous rule, assignments with required maintenance and trains with a large remDBM are preferable for linked departures;

• long waiting times between arrival and departure are not desirable.

The ‘maximum number of maintenance operations per day’ constraint is taken into account in the following manner. For each interval of days \([day_1,day_2]\), we define a current number of maintenance operations performed between days \(day_1\) and \(day_2\) by \(m(day_1,day_2)\). Obviously, \(m(day_1,day_2)\) must not exceed \((day_2 - day_1 + 1) \times maxMaint\). Each time a maintenance operation needs to be performed for assignment (t, d), the value \(m(day_1,day_2)\) is updated for each interval of days \([day_1,day_2]\) containing \([arrDay_t, depDay_d]\), where \(arrDay_t\) and \(depDay_d\) represent arrival day of train t and the day of departure d respectively. Adhering to the given bound for each interval of days guarantees a feasible choice of days for each required maintenance operation; a simple procedure for choosing the exact day of maintenance, along with the proof of correctness, is given in Sect. 4.1.3. This same concept is found in the IP model, which enables a daily maintenance limit constraint to be represented linearly.

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4.1.2 IP model

To enhance solutions to the assignment problem, the greedy procedure explained above is combined with a integer programming (IP) approach. The main difficulty in applying IP directly (independently) to solve the assignment problem involves linked departures. More specifically, the remaining distance before maintenance (remDBM) of some trains is not known before assigning the linked departures. The greedy procedure described in the previous section and IP have thus been combined as follows:

• The assignment problem is solved by a greedy procedure;

• Linked departure assignments are fixed (in updating the data on linked trains);

• The resulting assignment problem is solved once again with IP.

Let \(\mathcal{F} = \{(ft_1,fd_1), (ft_2,fd_2),\ldots , (ft_k,fd_k)\} \subseteq \mathcal{T} \times \mathcal{D}\) be the set of fixed assignments. Let the set of possible assignments be defined as:

$$\begin{aligned} \mathcal{S} = \{(t_1,d_1), (t_2,d_2),\ldots , (t_n,d_n)\} \subseteq \mathcal{T} \times \mathcal{D}, \end{aligned}$$

where:

• each assignment \((t_i,d_i)\) is feasible;

• none of the departures \(d_1,\ldots ,d_n\) is linked;

• none of the trains \(t_1,\ldots ,t_n\) has been assigned to linked departure i.e.

  $$\begin{aligned} \forall i \in \{1,\ldots ,n\}, j \in \{1,\ldots ,k\} t_i \ne ft_j. \end{aligned}$$

For each pair \((t_j,d_j)\) in \(\mathcal S\), a decision variable \(x_j\) is defined: \(x_j=1\) if \(t_j\) is assigned to \(d_j\), \(x_j=0\) otherwise. All the objective parts are merged into a single objective function by applying a non-negative weight for each of them. The objective function to be maximized is formalized as follows:

$$\begin{aligned} \begin{array}{lll} \displaystyle \sum _{j=1}^{n} x_j \times &{}\left( assignmentWeight + durationWeight \times \left( 1 - long(t_j,d_j)\right) \right. &{} \\ &{}+\left. immWeight \times imm(t_j,d_j) +\, maintWeight \times \left( 1 - nmbM(t_j,d_j)\right) \right) \end{array} \end{aligned}$$

The weights are chosen experimentally, and all results in the paper have been obtained with the following choices: \(assignmentWeight = 1000\), \(durationWeight = 100\), \(immWeight = 10\) and \(maintWeight = 1\). Next, we define the constraints included in this model. Let \(nmbFixed(day_1,day_2)\) be the number of maintenance operations between days \(day_1\) and \(day_2\) required by fixed assignments i.e.

$$\begin{aligned} \displaystyle nmbFixed(day_1,day_2) = \sum _{i=1}^{k} nmbM(ft_i,fd_i) \times \mathbbm {1}_{[arrDay_{ft_i},depDay_{fd_i}] \subseteq [day_1,day_2]} \end{aligned}$$

The constraint on maintenance is formulated by the following:

$$\begin{aligned} \begin{array}{lll} &{}\forall [day_1,day_2] \displaystyle \sum _{j=1}^{n} nmbM(t_j,d_j) \times x_j \times \mathbbm {1}_{[arrDay_{t_j},depDay_{d_j}] \subseteq [day_1,day_2]} \\ &{} \quad \le (day_2-day_1 + 1) \times maxMaint - nmbFixed(day_1,day_2) &{}\nonumber \\ \end{array} \end{aligned}$$

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4.1.3 Choosing maintenance days

As described earlier, the maximum number of maintenance operations per day constraint is met by complying with the limit for each interval of days [day1, day2], while updating the number of maintenance operations for each interval that contains \([arrDay_t,depDay_d]\) when servicing has to be performed for the assignment (t, d). The exact day for each maintenance operation still needs to be determined. A simple procedure, along with its proof of correctness, is presented in this section. This procedure functions as follows:

• sorting assignments (requiring maintenance) in ascending order by departure day and then by arrival day (example is given in Fig. 2);

• for each assignment (t, d) in a sorted list:

  • choose the first available day for maintenance, i.e. the first day in \(\{arrDay_t,\ldots , depDay_d\}\) for which the maintenance limit has not been reached.

We will now prove the correctness of this procedure. Let \(M=\{m_1,m_2,\ldots ,m_k\}\) be the set of all assignments requiring maintenance. Next, let \(A_i\) and \(D_i\) be the arrival and departure days of assignment \(m_i\) respectively. We can now write \(m_i = (A_i, D_i)\), \(A_i, D_i \in \{1,2,\ldots ,nDays\}\).

Claim

If the following inequality holds for each interval of days \([day_1,day_2]\):

$$\begin{aligned} \sum _{i=1}^{k} \mathbbm {1}_{[A_i,D_i] \subseteq [day_1,day_2]} \le (day_2 - day_1 + 1) \times maxMaint \end{aligned}$$

(1)

then the assignment of maintenance days using the procedure described above meets the ‘maximum number of maintenance operations per day’ constraint.

Proof

This claim will be proven by way of contradiction. Let \(m_j = (A_j, D_j)\) be the first assignment for which a servicing day cannot be chosen by following the given procedure and \(M_{j-1} = \{m_1, m_2,\ldots , m_{j-1}\}\) be the set of assignments with servicing days already set. This set-up means that the maintenance limit is reached for each day in \([A_j, D_j]\). (*) Let \(day_0\) be the first day such that the maintenance limit is reached for each day in interval \([day_0,D_j]\) (the existence of \(day_0\) is obvious and \(day_0 \le A_j\)). Clearly,

• \(day_0 = 1\) or

• the limit is not reached for \(day_0 - 1\). (**)

Let \(F = \{f_1, f_2,\ldots ,f_l\} \subset M_{j-1}\) be the set of assignments that have already been assigned to one of the days in \([day_0,D_j]\) (\(l = (D_j - day_0 + 1) \times maxMaint \), since the limit has been reached for the entire interval). For each \(f_i = (A_{fi}, D_{fi}) \in F\), we have:

• \(A_{fi} \ge day_0\): otherwise, the chosen servicing day would not be greater than \(day_0 - 1\) because of the assignment rule and (**),

• \(D_{fi} \le D_j\): as a result of the sorting order.

The same holds for \(m_j\): \(A_{j} \ge day_0\) and \(D_j \le D_j\). Let us now consider the set \(U = F \cup m_j\). As previously shown, the maintenance interval for each element of U lies in \([day_0,D_j]\), thus:

$$\begin{aligned} \sum _{i=1}^{k} \mathbbm {1}_{[A_i,D_i] \subseteq [day_0,D_j]} \ge \sum _{i \in U} \mathbbm {1}_{[A_i,D_i] \subseteq [day_0,D_j]} = |U| = (D_j - day_0 + 1) \times maxMaint + 1 \end{aligned}$$

which is contradictory to the main assumption (1).

4.3 Iterative improvement procedure

This section recommends an iterative procedure for improving the schedule. This procedure operates as follows:

• (1) schedule more trains by allowing conflicts in the track groups,

• (2) resolve conflicts by means of a local search,

• repeat steps (1)–(2) until the stopping criteria are met.

The entire solution procedure is illustrated in Fig. 3.

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4.3.1 Feasible to infeasible solution with more trains

The first step of this improvement procedure consists of adding more trains (and departures) to the feasible schedule by allowing conflicts. For each train added, the allowed track group conflicts are limited to a given number (e.g. a maximum of 3 conflicts per train). This scheduling procedure is the same as the one previously explained, but without adhering to the constraint on track group conflicts. All other constraints are to be met. An infeasible solution generated in this manner will serve as input for the local search procedure described below.

4.3.2 Local search to resolve track group conflicts

An infeasible solution is repaired by means of a local search procedure. The aim of this procedure is to change the choice of gates in order to reduce the number of conflicts to zero. The entry and exit times are to remain unchanged for each visit, as is the list of resources allocated for each train. Accordingly, a complete schedule for the train will either remain the same or be deleted (in the case of cancelling the train). The initial configuration (solution) is an infeasible set of visits on track groups. A visit is represented by a pair of gates (g1, g2). The configuration is denoted by \(\mathcal{V}=\{(g1_1,g2_1),\ldots ,(g1_n,g2_n)\}\) and the initial one by \(\mathcal{V}^0=\{(g1^0_1,g2^0_1),\ldots ,(g1^0_n,g2^0_n)\}\). The domain of each variable \(g1_i\) (\(g2_i\)) includes all gates connecting the same pair of resources as \(g1^0_i\) (\(g2^0_i\)) and \(\textsc {null}\) value. A visit corresponding to \((g1, \textsc {null})\), \((\textsc {null},g2)\) or \((\textsc {null},\textsc {null})\) is called a partial visit. It is assumed that no conflict occurs with a partial visit. A configuration \(\mathcal{V}\) is called partial if it contains a partial visit.

Remark

Two successive visits of the same train \((g1_i,g2_i)\) and \((g1_j,g2_j)\) share a common gate, i.e. \(g2_i=g1_j\). For these cases, the local search procedure will perform the same modification at both gates. The objective of the local search procedure is to minimize the number of cancelled trains. A train is cancelled if one of its visits is partial.

Greedy procedure to resolve track group conflicts

The first part of a local search is the greedy procedure to clear conflicts by deleting gates, i.e. setting the gate values to \(\textsc {null}\). The objective here is to compute a partial, but feasible, configuration (set of visits). The heuristic is simple: delete the gate that will decrease conflicts by the greatest number until conflicts no longer exist.

Tabu search on the partial feasible configuration

The tabu search procedure starts from a partial, but feasible, configuration of gates given by the previous procedure. The goal is to assign gate values to partial visits while keeping the configuration feasible. The following two elementary moves are carried out:

• ADD gate means one of 2 possible moves:

  • \(({\textsc {null}},g2_i) \Rightarrow (g1_i,g2_i)\) (\(g1_i \ne {\textsc {null}}\) and \(g1_i\) leads to the same neighbour resource as \(g1^0_i\))

  • \((g1_i,{\textsc {null}}) \Rightarrow (g1_i,g2_i)\) (\(g2_i \ne {\textsc {null}}\) and \(g2_i\) leads to the same neighbour resource as \(g2^0_i\))

• DROP gate means one of 2 possible moves:

  • \((g1_i,g2_i) \Rightarrow ({\textsc {null}},g2_i)\) (delete g1)

  • \((g1_i,g2_i) \Rightarrow (g1_i,{\textsc {null}})\) (delete g2)

These modifications are also applied to the next/previous visit of the given train (see remark in Sect. 4.3.2). The local search move consists of a single ADD move and, should conflicts occur, is to be followed by a few DROP moves in order to clear the conflicts. Deleting a gate \(g_i\) (DROP) is allowed only if \(g_i\) has not been added for a given number of iterations (i.e. if setting \(g_i\) to \(\textsc {null}\) is not tabu). The number of iterations for which deleting a gate is tabu equals the frequency of adding this gate. The current configuration is evaluated by the following hierarchical function:

1. 1.

   number of trains cancelled,

    
2. 2.

   number of deleted gates, i.e. the number of gates with a null value.

    

At each iteration, a non-tabu move that minimizes this function is performed. If there are two or more moves with the same objective, then a random choice is made. This process is repeated until no non-tabu move exists or until a maximum number of moves without improvement has been reached. In all reported experiments this limit has been set to 300.

An illustration of the objective function change during this improvement procedure is given in Fig. 4. It can be noted that most of the improvement occurs during the early stage and not many new trains are added at the end. The improved results due to the local search are illustrated in Fig. 5.

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MIP instead of Local Search The problem of repairing the conflicts explained above, and solved by local search procedure, may also be solved in different ways. One possible approach that we tested is to define a problem as a mixed integer program (which is not too complicated), with the objective of determining the maximum subset (in terms of number of scheduled trains) of train schedules without conflicts on track groups. The overall improvement procedure remains the same, we only try to “repair conflicts” by using mixed integer programming instead of local search. If we consider a single improvement phase iteration (add trains + repair) i.e. solving the problem—given a set of train schedules with conflicts, choose the gate values so that a minimum number of departures/arrivals is cancelled—it is clear that MIP will produce better (or at least the same) solutions than local search. However, using MIP instead of local search has not made any (or negligible) improvements to the final results using presented improvement framework, particularly for a 10-min running time when using MIP usually produces poorer results. The main reason for this is the fact that after many improvement phase iterations (e.g. 100) solution values obtained by local search come very close to the values of infeasible solutions, i.e. only a few trains can be added to the feasible solutions. In addition, many more iterations can be done in the same running time if local search is used, which is crucial for some instances. Similar conclusions can be drawn when MIP is used with a given running time limit (thus, transforming an exact solution approach into an heuristic one).

4.4 Final algorithm

The pseudo-code of the final algorithm is given in Algorithm 2.

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There is clearly a significant difference between the average and best solutions, sometimes more than 5%. This is due to the randomness in the heuristics proposed. However, running the algorithm for more seeds (e.g. 100 instead of 20) or with larger time limit will not yield a significant further improvement for a given set of instances.

The percentage of arrivals and departures cancelled varies from 11.90% on the B3 instance to 44.21% on B12, the average value being 26.17%. Train cancellation is mainly due to the fact that track group conflicts are prohibited. These conflicts are not always the actual conflicts due to a “black-box” approach introduced to model track groups in the current problem. The approach does not appear to lead to a solution that can be used in practice, due to a large number of cancelled arrivals and departures. As previously stated, we have been trying to solve the problem ‘as posted’ in the competition, adhering to all the given constraints to obtain the best possible result. However, the problem needs to be formulated differently in order to recommend solutions that can be used in practice. It is important to mention that the original problem formulated at the start of the competition allowed track group conflicts and there was a penalty in the objective function for each conflict. But the organizers changed this in a later phase.

This rolling stock unit management problem on a railway site is extremely difficult to solve for several reasons. Most induced sub-problems, such as the train assignment, scheduling, track group conflicts and platform assignment, are indeed complicated. In order to solve this problem, we implemented a two-phase approach that combines exact and heuristic methods. A natural way of approaching the problem consists of dividing it into two sub-problems, the first consisting of matching (assigning) trains to departures and the second consisting of planning train movements (scheduling) inside the station, and then solving both sequentially. It would also be quite natural to consider a break down of the problem by days (and subsequently combining the days), as there are hardly any arrivals and departures at night. However, the experiments we performed have shown that the quality of the solutions decreases, while not greatly increasing algorithm simplicity or computational time. This was the case, in particular, for the first version of the problem (i.e. with conflicts on track group as soft constraints), where many more trains have not been assigned to any departure. It should also be noted that the assignment procedure presented in this paper tends to schedule the arrival and departure of a train on the same day.

The presence of linked departures and a constraint on the daily maintenance limit further complicate the assignment problem. Otherwise, the problem could be solved in a polynomial time by means, for instance, of the maximum weighted matching algorithm. A common strategy for overcoming these constraints calls for modelling the assignment problem as a mixed integer program. The number of linked departures in the given set of instances however makes the IP model unworkable due to the tremendous number of variables to be generated in order to cope with the linked departures. A greedy procedure has therefore been used to solve the assignment problem. The departures are sorted by departure time and matched to the trains one by one, in a greedy manner. A simple function was used to evaluate the quality of assignment (t, d). Solutions to the assignment problem are improved by combining a greedy procedure with IP, whereby all train assignments with linked departures obtained by the greedy procedure are fixed and the remaining assignment is solved by IP. We recommended the simple idea (and proved its correctness) of modelling the constraint on the daily maintenance limit as a linear one.

The second step in solving the problem is to plan train movements inside the station while complying with all resource constraints and cancelling as few trains as possible. This problem is very complex and contains a huge number of decision variables (resources, gates and entry/exit times must be specified for each train); hence, modelling the problem exactly and solving it efficiently are likely to be impossible. We therefore opted for a constructive procedure to schedule the trains. They are scheduled sequentially, one by one, after being ordered by a given set of criteria. The possible train movements are restricted to just a few of critical importance in order to reduce problem complexity. For this purpose, a set of potential paths that trains are allowed to use was compiled at the beginning of the procedure. A concept of virtual visits was offered to expand the choice of starting times for each travel in order to address track group conflicts. An iterative procedure was adopted to improve scheduling phase solutions, by allowing infeasible solutions as regards track group conflicts and then resolving these solutions by means of a local search. The introduction of both virtual visits and an iterative improvement procedure helped greatly improve these solutions. The algorithm described in this paper was ranked first at the ROADEF/EURO Challenge 2014.