Delay-Robust Event-Scheduling In Memoriam of A. Caprara

Transcription

1 Delay-Robust Event-Scheduling In Memoriam of A. Caprara A. Caprara 1 L. Galli 2 S. Stiller 3 P. Toth 1 1 University of Bologna 2 University of Pisa 3 Technische Universität Berlin 17th Combinatorial Optimization Workshop January 7-11, 2013 Aussois, France

2 Outline Delay-Robust Event-Scheduling Robustness Framework

3 Outline Delay-Robust Event-Scheduling Robustness Framework Train Platforming Problem In and Out TPP deterministic model

4 Outline Delay-Robust Event-Scheduling Robustness Framework Train Platforming Problem In and Out TPP deterministic model Delay-Robust Train Platforming Delay propagation network Buffers linking constraints

5 Outline Delay-Robust Event-Scheduling Robustness Framework Train Platforming Problem In and Out TPP deterministic model Delay-Robust Train Platforming Delay propagation network Buffers linking constraints Computational results

6 Outline Delay-Robust Event-Scheduling Robustness Framework Train Platforming Problem In and Out TPP deterministic model Delay-Robust Train Platforming Delay propagation network Buffers linking constraints Computational results References

7 Delay-Robust Event-Scheduling Robustness Robust Optimization Robust Optimization finds best solutions, which are feasible for all likely scenarios.

8 Delay-Robust Event-Scheduling Robustness Robust Optimization Robust Optimization finds best solutions, which are feasible for all likely scenarios. Pros no knowledge of the underlying distribuition is required

9 Delay-Robust Event-Scheduling Robustness Robust Optimization Robust Optimization finds best solutions, which are feasible for all likely scenarios. Pros no knowledge of the underlying distribuition is required Cons Strict robustness is generally overconservative, because: solutions must cope with every likely scenarios without any recovery

10 Delay-Robust Event-Scheduling Robustness Recoverable Robustness Informally speaking, a solution to an optimization problem is called recovery robust if it can be adjusted to all likely scenarios by limited recovery action. Thus a recovery-robust solution provides a service guarantee (Liebchen et al [3], Stiller 2008 [4]).

11 Delay-Robust Event-Scheduling Robustness Robust Network Buffering We are interested in the special case in which the recovery problem is a delay (d e ) propagation in some directed graph N =(E,A), which is buffered on the arcs against disturbances on the nodes e E. Details in Liebchen et al [3], Stiller 2008 [4].

12 Delay-Robust Event-Scheduling Framework A framework A set E of events to be scheduled. Denoting by F the set of feasible schedulings of events E, i.e., the feasible region, a nominal problem of the form min{c(x):x F}. A set S of possible scenarios, where each scenario s S is defined by the external disturbance δ s e 0 assigned to each event e E. A delay-propagation network N =(E,A), with (e,e) A if a delay d e on event e may propagate to a delay d e on event e. A delay d e on event e may propagate to a delay d e on event e according to the relation d e d e b((e,e),x), where b((e,e),x) is a buffer time function.

13 Delay-Robust Event-Scheduling Framework Mathematical model The delay-robust problem can be formulated as: minc(x)+d, (1) subject to x F, (2) D de, s s S, (3) e E d s e δ s e, e E, s S, (4) d s e d s e f (e,e), (e,e) A, s S, (5) f (e,e) =b((e,e),x), (e,e) A. (6)

14 Delay-Robust Event-Scheduling Framework Scenario set {δ s R E + : δs 1 } (7) These are uncountably many scenarios, which can however be handled easily thanks to the following observation. Proposition For the delay-robust problem (1)-(6), the compact scenario set defined by (7) is equivalent to the finite scenario set S defined by the vectors which contains only E scenarios. {δ s {0, } E : δ s 1 = },

15 Delay-Robust Event-Scheduling Framework Quadratic buffer time function In the most natural case, the buffer time b((e,e),x) only depends on the way in which events e and e are scheduled in x, and is independent of the scheduling of the other events: f (e,e) = ϕ(e,h,e,h) x e,h x e,h, h H e h H e These quadratic constraints can be linearised in a few ways. (e,e) A.

16 Delay-Robust Event-Scheduling Framework Projecting out delay variables One may directly optimise over the feasible region obtained by projecting out the d s e variables (and removing the constraints (3)-(5)). Let σ f (s,ē) be the value of the shortest path from s to ē E on the delay propagation network N =(E,A) with arc lengths f (e,e) for (e,e) A. Moreover, let P f (s,ē) A denote such a shortest path. Lemma Given buffer time values f (e,e), an external disturbance equal to on event s E (and null for the other events) propagates to a delay of value max{0, σ f (s,ē)} on event ē E.

17 Delay-Robust Event-Scheduling Framework Projecting out delay variables (cont.) A direct consequence is: Lemma Given buffer time values f (e,e) and maximum cumulative delay D, for every scenario s S there exist values of the d s e variables satisfying (3)-(5) if and only if e E max{0, σ f (s,e)} D. Otherwise, for the considered scenario s, letting E f (s) E be the set of events such that σ f (s,e)>0, a valid inequality that is violated by f (e,e) and D is f a E f (s) D. e E f (s) a P f (s,e)

18 Train Platforming Problem Problem definition The objective of train platforming is assigning trains to platforms in a railway station.

19 Train Platforming Problem Problem definition The objective of train platforming is assigning trains to platforms in a railway station. Platforming is carried out:

20 Train Platforming Problem Problem definition The objective of train platforming is assigning trains to platforms in a railway station. Platforming is carried out: for a specific railway station

21 Train Platforming Problem Problem definition The objective of train platforming is assigning trains to platforms in a railway station. Platforming is carried out: for a specific railway station after the timetable has been defined

22 Train Platforming Problem In and Out In and Out

23 Train Platforming Problem In and Out In and Out Input Train schedule : arrival, departure times, directions and allowed shifts Railway station topology: platforms, paths and directions

24 Train Platforming Problem In and Out In and Out Input Train schedule : arrival, departure times, directions and allowed shifts Railway station topology: platforms, paths and directions Output Assign each train a platform and two paths for arrival and departure s.t. no operational constraint is violated

25 Train Platforming Problem In and Out Train schedule PALERMO C.LE FN DIR L arrival and departure times directions and preference list shifts Figure: Train Schedule The train schedule of a railway station contains info on arrival and departure times, directions and allowed shifts of each train passing through it.

26 Train Platforming Problem In and Out Railway station topology MILANO (1) 1 PADOVA (2) 2 FIRENZE (3) 3 4 DEPOT (4) Figure: Topology The topology of a railway station includes platforms, paths and directions.

27 Train Platforming Problem TPP deterministic model Resources and operational constraints A B C D Platform conflicts are forbidden, path conflicts are allowed to some extent. E F G :00 23:59 MILANO (1) PADOVA (2) FIRENZE (3) A,D,G B,E C,F 4 DEPOT (4) Figure: Platform and path. A pattern P for a train t is a 5-tuple defining: platform, arrival/departure paths and shifts. Operational constraints can be expressed using an incompatibility graph among patterns. Figure: Incompatibility graph.

28 Train Platforming Problem TPP deterministic model TPP deterministic model s.t. min c t,p x t,p t T P P t P P t x t,p =1, x t,p 1, (t,p) K x t,p {0,1}, Details in Caprara et al [1]. t T K K t T, P P t

29 Delay-Robust Train Platforming Delay propagation network Delay propagation network The platforming gives rise to a network in which the delay caused by disturbances propagates.

30 Delay-Robust Train Platforming Delay propagation network Delay propagation network The platforming gives rise to a network in which the delay caused by disturbances propagates. We consider two type of disturbances: (i) the train arrives late at the station, and (ii) the platform operations take longer than required.

31 Delay-Robust Train Platforming Delay propagation network Delay propagation network The platforming gives rise to a network in which the delay caused by disturbances propagates. We consider two type of disturbances: (i) the train arrives late at the station, and (ii) the platform operations take longer than required. Hence, the following two events are associated with each train t T and define E: arrival a t : the train occupies a given arrival path at a given instant departure p t : the train frees its platform at a given instant

32 Delay-Robust Train Platforming Delay propagation network Example Assume that in the nominal solution trains t and t stop at the same platform, event p t is scheduled at 10:00, and event a t at 10:04 with 3 minutes to travel along the arrival path, so that t occupies the platform at 10:07. Moreover, assume that the headway time that must elapse between t freeing the platform and t occupying it is 2 minutes and that the 3-minute travel time for t along its arrival path is fixed. If the departure path of t and the arrival path of t are compatible, the buffer time f (pt,a t ) is equal to 5 minutes, as a delay of up to 5 minutes on p t does not affect a t, whereas a larger one does.

33 Delay-Robust Train Platforming Delay propagation network Delay-robust mathematical model subject to min c t,p x t,p +D t T P P t P P t x t,p =1, t T, x t,p 1, K K, (t,p) K x t,p {0,1}, t T, P P t, f (e,e) = c P1,P 2,a x t1,p 1 x t2,p 2, P 1 P t1 P 2 P t2 (e,e) A, D de, s s S, e E d s e δ s e, e E, s S, d s e d s e f (e,e), (e,e) A, s S.

34 Delay-Robust Train Platforming Buffers linking constraints Buffers linking constraints A straightforward link between the buffer value of a given arc a A(N) associated with train pair (t 1,t 2 ) T 2 and the choice of patterns for the given pair of trains is the following: c P1,P 2,a x t1,p 1 x t2,p 2 P 1 P t1 P 2 P t2 where c a,p1,p 2 is a constant associated to arc a and to the corresponding choice of patterns (P 1,P 2 ) for trains (t 1,t 2 ).

35 Delay-Robust Train Platforming Buffers linking constraints Buffers linking constraints f a P 1 P t1 α a P 1 x t1,p 1 + P 2 P t2 β a P 2 x t2,p 2 γ a, a A(N), (α,β,γ) F a (8) Following Caprara et al [1], the separation of Constraints (8) is done by a sort of polyhedral brute force, given that, for each pair of trains t 1,t 2, and for each arc a A(N) the number of vertices in Q t1,t 2,a is small. Specifically, Q t1,t 2,a has P t1 P t2 vertices and lies in R P t 1 + P t2 +1, so we can separate over it by solving an LP with P t1 P t2 variables and P t1 + P t2 +1 constraints.

36 Computational results Computational results: Palermo C.Le. [1] Solution Delay-Robust LP Delay-Robust Solution instance D time D time D %gap %reduction time PA I % 29% 9788 PA II % 16% 3892 PA III % 22% 5411 PA IV % 13% 1417 Table: Results for the real-world instances of Rete Ferroviaria Italiana.

37 Computational results Computational results: Genova P.Principe [1] Solution Delay-Robust LP Delay-Robust Solution instance D time D time D %gap %reduction time GE I % 19% 147 GE II % 35% GE III % 19% 1086 GE IV % 34% 274 Table: Results for the real-world instances of Rete Ferroviaria Italiana.

38 Computational results Computational results: Bari C.Le. [1] Solution Delay-Robust LP Delay-Robust Solution instance D time D time D %gap %reduction time BA I % 23% 1095 BA II % 0% 5 BA III % 8% 525 BA IV % 6% 2431 BA V % 7% 3204 BA VI % 0% 5 BA VII % 27% 4126 BA VIII % 19% 535 Table: Results for the real-world instances of Rete Ferroviaria Italiana.

39 Computational results Computational results: Milano C.Le. [1] Solution Delay-Robust LP Delay-Robust Solution instance D time D time D %gap %reduction time MI I % 1% 201 MI II % 13% 2415 MI III % 25% 938 MI IV % 11% 2547 MI V % 44% 2503 MI VI % 9% 618 Table: Results for the real-world instances of Rete Ferroviaria Italiana.

40 Questions Thank you!

41 References References A. Caprara, L. Galli and P. Toth (2011). Solution of the Train Platforming Problem. Tran. Sci., 45(2), A. Caprara, L. Galli and P. Toth (2012). Delay-Robust Event-Scheduling Submitted. C. Liebchen, M. Lübbecke, R.H. Möhring, and S.Stiller (2009). The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications. In R.K. Ahuja, R.H. Möhring and C.D. Zaroliagis (eds.), Robust and On-Line Large Scale Optimization. Lecture Notes in Computer Science 5868, Springer-Verlag, S. Stiller (2008). Extending Concepts of Reliability PhD thesis, Institut für Mathematik, Technische Universität Berlin, Berlin, Germany.