where again M is a constant which is “large enough” (see Theorem 5.1). Note that the objective does not change and that constraints (5.3)-(5.7) and (5.13)-(5.15) are the same as constraints (2.2)-(2.9) in the original IP formulation of (DM) – we have included them here for the sake of clarity only.
We shortly explain the meaning of the additional constraints:
• Constraints (5.8) ensure that if activityais chosen, i.e. ifvij = 1, then its lower bound has to be respected. Otherwise, due to M being “large enough”, we do not further restrict x.
• Each trip needs exactly one predecessor (constraints (5.9)-(5.10)) and exactly one successor (constraints (5.11)-(5.12)), and no train is allowed to directly drive from its start depot to its end depot.
If the circulation activities that are used when operating the original timetable π are all included inAcirc, then (DMC) is feasible; one feasible solution for example can be computed by fixing wait/depart decisions, priority decisions, and circulation decisions as they are in the original timetable and solving the resulting instance of (PP(Afix)).
Since (DM) is NP-hard as already mentioned in the beginning of Chapter 4, (DMC) as a generalization of (DM) is NP-hard, too. Note that the additional constraints (5.9)-(5.12) and (5.16) form a matching problem. Although “pure” matching problems can be solved in polynomial time (see for example [BDM09]), we show in Theorem 5.3 and Theorem 5.4 that even if Achange=Ahead=∅, the matching problem in the context of delay management is NP-hard even in very simple special cases. However, we first show that most of the results from Chapter 3 can be extended to the delay management problem with integrated rolling stock circulations.
5.2 Analyzing the Model
Due to the fact that we only take into account circulation activities satisfying (5.1), Theorem 3.1 and Corollary 3.2 still hold for (DMC):
Theorem 5.1. Given an instance of (DMC), M = max
i∈E di+ X
a∈Atrain
da+ X
(i,j)∈Abackhead
πi−πj +Lij
is “large enough”.
Proof. The proof of Theorem 3.1 only needs some minor modifications: For a solution (x, z, g, v)of (DMC), we denote the set of fixed circulation activities in this solution by
Afixcirc:={(i, j)∈ Acirc:vij = 1}.
To take into account those fixed circulation activities when applying the Critical Path Method (Algorithm 2.1) for computing the disposition timetablex, we define˜
Afix:=Afixchange∪ Afixhead∪ Afixcirc.
Parts 1), 2) a), and 2) b) of the proof keep the same, and for the case that in part 2), the activitya= (i, k)that carries over the delay is a circulation activity, we add part 2) c) as follows:
c) Ifa= (i, k)∈ Afixcirc, then
˜
xk= ˜xi+La
≤πi+Ui+La
≤πk−La+Ui+La
=πk+Ui
≤πk+Uk
where the second line is a direct consequence of the induction hypothesis for event i≺kand the third line is due to the fact that we only allow circulation activities satisfying (5.1).
This completes the proof of Theorem 3.1. To extend the proof of Corollary 3.2 to the delay management problem with integrated rolling stock circulations, it remains to show that in an optimal solution, for any (i, j) ∈ Acirc with vij = 0, (5.8) is satisfied “automatically”. From the extension of Theorem 3.1 to delay management with integrated rolling stock circulations that we just have shown, we know thatxi ≤πi+M, and asx satisfies constraints (5.3), we havexj ≥πj. As we require (5.1) to be satisfied by each circulation activity, we haveπj−πi≥La, hence
M(1−vij) +xj−xi=M+xj−xi
≥M+πj −(πi+M)
=πj−πi
≥La.
If we do not limit the set of possible circulation activities by (5.1), but allow circulation activities between all trips served by compatible trains, Theorem 5.1 still holds if we
5.2 Analyzing the Model
take into account the additional delay caused by fixing a circulation activitya= (i, j) withπj < πi+La, i.e. if we choose
As the proof is rather technical and very similar to the proofs of Theorem 3.1, Corol-lary 3.2, and Theorem 5.1, we omit it here.
If we bound the delay of each event analogously to problem (BDM) presented in Section 3.2, we obtain the delay management problem with integrated rolling stock circulations and bounded delay (BDMC):
and such that (5.3)-(5.16) are satisfied.
As in Section 3.2, we can use this restriction to give a tight bound on the constantMand to yield a reduction approach for reducing the size of the input instance. Theorem 3.5 still holds for (BDMC):
Theorem 5.2. Given an instance of (BDMC), M =Y + max
(i,j)∈Ahead(πj−πi+Lji) is “large enough”.
Proof. The proof is similar to the proof of Theorem 3.5. It remains to show that for eacha= (i, j) ∈ Acirc with vij = 0, constraint (5.8) is fulfilled “automatically” in an optimal solution:
where the last inequality is due to the fact that we allow circulation activities only between two events i and j if (5.1) is satisfied – with this restriction, a circulation activity can only carry over an existing delay, but not increase it (in a time-minimal solution, we have xj −πj = xi+La−πj ≤xi −πi if (5.1) is fulfilled). However, if we drop assumption (5.1) and allow circulation activities between all trips served by compatible trains, then Theorem 5.2 still holds if
M =Y + max
(i,j)∈Ahead
(πj−πi+Lji) + max
a=(i,j)∈Acirc
(πi−πj+La).
Analogously to the proof above, in this case we have M(1−vij) +xj −xi ≥πj−πi+ max
a=(i,j)∈Acirc
(πi−πj +La)
≥La.
Furthermore, Theorem 3.6 and the consequences which yield algorithmFix-Headways still hold, so algorithmFix-Headwaysalso can be applied to an instance of (BDMC).
In addition, the reduction techniques suggested in Section 3.3 are valid for problem (DMC), too, if we apply the following two slight modifications to algorithm Reduce (see page 38).
First, we have to consider the circulation activities inAπ, i.e. we have to modify (3.4) in the following way:
Aπ :=Atrain∪ Achange∪ Aforwhead∪ Acirc. (5.17) This reflects the fact that each circulation activity could carry over a delay. The modified set Aπ has to be used whenEmark is computed according to equation (3.5).
Secondly, we have to apply a rather technical change in the definition of Ereduced in Step 2 of algorithmReduce which we explain using a small example, see Figure 5.2.
In that example, event 1 is source delayed, hence Emark = {1,2,3,5,6}. Event 4 is not contained inEmark, hence it is not contained inEreduced, so in Step 3 of algorithm Reduce, the circulation activities(4,2)and(4,5)are not contained inAreduced, yielding an infeasible problem. Thus we have to modifyEreduced in Step 2 of algorithm Reduce as follows:
Ereduced :=Emark∪ {i∈ E :∃a= (i, j)∈ Atrain withda>0}
∪ {i∈ E :∃(i, j)∈ Acircwith j∈ Emark}. (5.18) With these modifications, algorithmReducelooks as follows:
5.2 Analyzing the Model
1 2 3
4 5 6
Figure 5.2: If Ereduced in algorithm Reduce is not adapted, too many activities are deleted. Solid arrows represent driving activities, dotted arrows are circulation activities. In grey: events contained in Emark if event 1 is source delayed.
Algorithm 5.1: AlgorithmReducefor problem (DMC)
Input: The event-activity networkN = (E,A), a timetableπ, and source delays d∈N|E|+|Atrain|.
Step 1: CalculateEmark according to (3.5), using the definition of Aπ from (5.17).
Step 2: ComputeEreduced according to (5.18).
Step 3: ComputeAreduced :={(i, j)∈ A:i, j∈ Ereduced}.
Output: The reduced event-activity networkNreduced = (Ereduced,Areduced).
This modification of algorithmReduce(and hence also the corresponding modification of algorithm Fix-And-Reduce) can also be applied to an instance of (BDMC) as Theorem 3.8, Theorem 3.9, and Theorem 3.10 are still valid as long as (5.1) is satisfied (as we have shown in the proof of Theorem 5.2, in this case, a circulation activity can only carry over an existing delay, but not cause an additional delay). For the proof of Theorem 3.8, the circulation activities can be treated like the other activities in Aπ. After solving the problem on the reduced network computed by algorithmReduce, for all(i, j)∈ Acirc\ Areduced, we have to set vij = 1if (i, j) is chosen in the original plan, 0 otherwise.
Finally, as long as (5.1) is satisfied, by definingAπ as in (5.17), we can also transfer the results from Section 3.4 to the delay management problem with integrated rolling stock
circulations, especially the definition of the never-meet property: Definition 3.12 does not change (except for the fact that the underlying setAπ has changed), Lemma 3.13 still holds as it depends only on Theorem 3.10 and Theorem 3.8 (which, as already mentioned above, also are valid for the extended model), Corollary 3.14 is a direct consequence of Lemma 3.13, and Theorem 3.15 which depends on Lemma 3.13 and Corollary 3.14 is still valid, too.