Similarly, given a solution of (DMC) with objective value K +P
k∈E1wk(xk−πk), M :={a= (i, j)∈ Acirc:vij = 1} is a perfect matching with objective valueK.
Note that a minimum cost perfect matching can be computed in polynomial time, for example in timeO(n3) wheren=|Eend∪ Estart|by using theHungarian method (also known as Kuhn-Munkres algorithm), see [Kuh55] for its original O(n4) formulation or [BDM09] for an overview of available solution approaches.
5.5 Solution Approaches
As we have shown in Section 5.3, (DMC) in general is NP-hard. One approach to nevertheless solve the problem, even if (5.20) is not fulfilled, is to solve the problem on the reduced event-activity network computed by algorithmFix-And-Reduce; for this approach, we have shown rather good results (for delay management without rolling stock circulations) in our case study in Section 3.5. However, even after a reduction of the input instance, it might still take too long to solve the remaining problem to optimality. Hence, in this section, we suggest a generic solution framework which in general does not compute an optimal, but at least a feasible solution.
First, we present the generic framework; the main idea is to solve the delay management problem with fixed circulation decisions and to look for an improvement of the circulation decisions alternately. After presenting the generic framework, we discuss different possible approaches for each step.
Algorithm 5.2: Local-Improvement Step 1: Fix the circulation activities.
Step 2: Solve the corresponding instance of (DM) with fixed circulation activities.
Step 3: Find a local improvement of the circulation activities (i.e. one that allows to decrease the disposition time of at least one eventi∈ Estart). If none found: end.
Step 4: Go to Step 2.
For each of the three first steps of algorithmLocal-Improvement, we roughly discuss some ideas:
In Step 1, an initial assignment for the circulation decisions can for example be given by fixing them as they are in the original plan or by using the following greedy approach:
Fix all connections and all headways according to the original timetable, then run algorithm Cpm. Whenever the actual event ihas outgoing circulation activities, fix that outgoing circulation activity a for which the scheduled time of its end event j is minimal (or for whichxi+La is minimal), provided that j has no fixed incoming circulation activity yet. Note that depending on the circulation activities, this approach might not find a feasible solution even if one exists. To demonstrate this, we consider the event-activity network depicted in Figure 5.6.
1
2
3
4
5
6
Figure 5.6: Depending on the set of available circulation activities, the greedy approach might not find a feasible assignment of trips to trains, although one exists.
Dotted arrows represent circulation activities.
If the first circulation activity that the greedy approach fixes in this example is circulation activity(1,5)and if(2,6) is the next one, then the trip ending with event 3 and the trip starting with event 4 cannot be connected to any other trip – although a feasible solution exists.
Once the circulation decisions are fixed, the solution of the corresponding instance of (DM) in Step 2 of algorithmLocal-Improvementcan be computed exactly (either on the original event-activity network or on the reduced event-activity network computed by one of the reduction techniques suggested in Chapter 3) or by applying one of the heuristic solution approaches suggested in Chapter 4.
As we have shown in Section 5.3, computing optimal circulation decisions in general is NP-hard (as a circulation decision might depend on other wait/depart, priority, or circulation decisions – if all circulation decisions are independent of other decisions, fixing the circulation activities reduces to a polynomially solveable matching problem, see Section 5.4). Hence, in Step 3 of algorithm Local-Improvement, we are not looking for optimal circulation decisions, but for an improvement of the current situation.
5.5 Solution Approaches
For finding such an improvement, one could pairwise swap the assignment of two trips, search for a local optimum in one single station (i.e. exactly solve the matching problem in that station), or use the approach described in the following lemma:
Lemma 5.6. Let(x, z, g, v) be a feasible solution of (DMC) where xis a time-minimal timetable, and let ˜v be another feasible assignment of the circulation decisions that fulfills(5.9)-(5.12) and (5.16). For fixed k∈ Estart∪ Eterm, let i(k),˜i(k) denote those events for which vi(k)k = ˜v˜i(k)k= 1, i.e. the start events of both circulation activities ending in k that have been fixed by v or by v, respectively. If˜
x˜i(k)+L˜i(k)k≤xi(k)+Li(k)k ∀k∈ Estart∪ Eterm, (5.21) then there exists a solution(˜x, z, g,˜v) of (DMC) with x˜j ≤xj∀j∈ E and
f(˜x, z, g,v)˜ ≤f(x, z, g, v).
Proof. Given a feasible solution(x, z, g, v)of (DMC) and another feasible assignment ˜v of the circulation decisions, we define
Afix:={a∈ Achange:za= 0} ∪ {(i, j)∈ Ahead :gij = 0} ∪ {(i, j)∈ Ahead : ˜vij = 0}
and compute a disposition timetablex˜ by applying algorithmCpm. AsCpmcomputes a time-minimal timetable and since (5.21) is satisfied, x˜j ≤xj for all j ∈ E. Hence, f(˜x, z, g,v)˜ ≤f(x, z, g, v).
The statement of this lemma is the following: If we can change the circulation decisions in such a way that we improve the “right” side of the matching problem, i.e. if we can reduce the time of some events in Estart without increasing it for others, then this local change improves the whole solution.
In general, algorithmLocal-Improvementdoes not compute an optimal solution of (DMC) – however, by fixing the circulation decisions in Step 1 as they are in the original plan, Local-Improvementnever computes a solution that is worse than in the case of (DM) with fixed circulations where no changes to the rolling stock circulations are allowed at all (if the same delay management strategy is used), but it might improve the solution significantly in some cases.
Another approach for solving the problem is to use the known branch and bound approaches for the delay management problem (see for example [Sch06] and [Job08]) where lower bounds can be derived by solving relaxations of (DMC), while algorithm Local-Improvementyields upper bounds. Within the branch and bound, we can additionally use the circulation variables for branching.
Chapter 6
Robustness Aspects
In the previous chapters, we investigated different strategies for delay management and focused on the operational phase (when the trains are on the track) and on how toreact when delays occur. However, it makes sense to already take into account delays during the strategicplanning phase to reduce the probability and the propagation of delays.
One possibility is to already take into account robustness when planning the lines. To make a line plan robust against delays, one approach is to distribute the traffic evenly across the network, see for example [SS06]. However, research on line planning goes beyond the scope of this work.
Other approaches consider robustness aspects when computing the timetable and try to add slack times to the timetable in a “smart” way to make it robust against small delays.
In Section 6.1, we shortly resume results from a joint research project on computing delay resistant railway timetables; it has been published in [LSS+10]. In Sections 6.2-6.5, we present the concept of recoverable robustness, its extension to multi-stage recoverable robustness and the application of both concepts to robust timetabling. We conclude our investigation of recoverable robustness by summarizing the results in Section 6.6.
Sections 6.2-6.6 are based on a joint research project and partly have been published in [CDSS08] and [CDD+09b].
6.1 Computing Delay Resistant Railway Timetables
As mentioned in the overview of related work, many different approaches on robust timetabling exist in the literature. In short, we subsume some results from a joint research project, a case study on robust timetabling, published in [LSS+10]. There, the focus is on taking into account delay management already in the objective of the
timetabling step: Instead of minimizing the weighted sum of the durations of the activities, we added a second term to the objective to count the expected delay. Of course the expected delay depends not only on the distribution of the source delays, but also on the delay management strategy that is applied during the operational phase. As periodic timetabling and delay management are both already NP-hard, it is too ambitious to take into account an optimal delay management strategy in the objective of timetabling. Hence, we assumed a simplified delay management strategy, namely a strict no-wait policy. Then, for different distributions of the source delays, we computed robust timetables, according to the modified objective function, evaluated their robustness under optimal delay management and compared them to an optimal, non-robust timetable. It turned out that the simplified delay management in the objective of timetabling is a good estimate of the optimal delay management strategy.
Furthermore, the results imply that we can significantly reduce the passengers’ delays at a rather small price of robustness, i.e. the increase of nominal travel times is rather small, compared to the level of robustness which we achieved. For further details on this approach, we refer to [LSS+10].
We now introduce the concepts of recoverable robustness and multi-stage recoverable robustness and show how to apply them to the timetabling problem.