1. Solving the dispatching problem by MILP MIMIC (Mixed Integer Maurizio Igor Carlo) Carlo Mannino SINTEF ICT (Oslo), University of Rome Maurizio Boccia University of Sannio, Benevento Igor Vasiliev (Russian Academy of Sciences, Irkutsk) INFORMS RAILWAY COMPETITION 2012 Phoenix, 2012, October 14

2. Outline 1 The model: representing train movements and conflicts resolution 2 Reduce complexity: simplifying the network and conflicts representation 3 Casting into Mixed Integer Linear Program. 4 Computations

3. Modelling train movements 15 16 17 18 5 13 14 D WANT: find a schedule t which minimize c(t) (complicated) D E Train movements are represented by oriented path (routes) time train D enters the segment starting at u

4. Conflicts For every possible conflict decide which train goes first t D 14 D E No two trains can occupy simultaneously the same segment (conflict) Conflict point: first or last point of a common segment 15 16 17 18 5 13 14 D E 13 19 18 17 16 15 Routes Graph

5. Modelling train movements D E For every possible conflict point decide which train goes first 15 16 1718 13 14 D E 15 16 17 18 5 13 14 D E 13 19 18 17 16 15 disjunctive precendence constraint

6. The problem Given: set of networks N = {( T 1, R 1 ), …, ( T n, R n )} fading matrix [ A tj ] t  T, j  R frequency domain F t, t  R revenue function u ( C ( p, f, s )) = u ( p, f, s ) of the coverage Sdsdsdfasdasa Given a rail network a set of trains T and their current position a cost function c Find: a routing for T a route for each train in T and a schedule t a starting time t i u for each train i and each segment uv in the route of i So that: no conflicts occur and c(t) is minimized Real-time train dispatching problem No conflicts: decide disjunction for each common segment Typically we have too many routes, too many disjunctions.

7. Approaches to train dispatching Lines with multiple tracks, sidings, cross-overs, stations: very complex. Literature focused on sub-problems: Singletrack (no station): Afonso and Bispo 2011, Mladenovic and Cangalovic 2007, Romeiro de Jesus 2010, Sahin et al 2008, Sindaravalli et al 2010, Lamorgese and Mannino 2012, Single (complcated) station (no connecting tracks) Caimi 2009, Luethi et al 2006. Limited regions, junctions Mazzarello and Ottaviani 2005, Rodriguez 2005, Luethi et al. 2006, Mannino and Mascis 2009, Acuna-Agost et al 2010, Corman et al. 2010, Harrod 2011, Lusby et al 2011

8. Reducing complexity: routes We neglect sidings 1 route substitutes 2 k routes We need to take into account that trains can use sidings to meet or pass each other 1 route 4 routes

9. Handling meet/pass events S is represented in train routes by two nodes (inS and outS) 15 18inS 22 26 A outS 21 B 15 18outS 12inS 21 22 A and B meet (pass) in the siding section S if and only if: AND A enters S before B leaves S AND B enters S before A leaves S This can be expressed by two (standard) precedence constraints siding section S

10. Back to conflicts DCBA Even if some fixing may help, it is crucial to reduce the number of disjunctions explicitly represented in the model. t A1 t Aa t A2 t Ad t A5 t Ag t A7 t Am t A8 t B1 t Ba t B2 t Bd t B5 t Bg t B7 t Bm t B8 t C5 t Ca t C2 t Cd t C1 t D8 t Dm t D7 t Dg t D5 t Dd t D2 t Da t D1 Each potential conflict is represented by a disjunction

11. Limiting explicit disjunctions 15 16 1718 D E 19 20 2122 23 2425 26 1 2 3 4 5 6 78 9 1011 12 OR E enters 15 before D OR D enters 22 before E 15 18S1 22 26 D E S2 21 15 18S2 12 S1 21 22 OR E D It suffices to consider only the extreme points of the common tracks main 2 main 1

12. Modelling meet-pass events 15 16 1718 D E 19 20 2122 23 2425 26 27 28 1 2 3 4 5 6 78 9 1011 12 OR OR E enters 15 before D OR D and E meet at siding OR D enters 22 before E 15 18inS 22 26 D E outS 21 15 18outS 12inS 21 22 OR OR E D With a siding then we have a trisjunctrion: main 2 main 1

13. Mixed Integer Linear Programming Our initial experiments ruled out time-indexed formulations (unsuitable for our real-time application) We represent and solve the problem by using Mixed Integer Linear Programming Sub-problems: 1. scheduling, 2. routing, 3. meet/pass decisions Modelling scheduling Two competing approaches (in the literature) Time indexed: - better bounds, many variables and constraints Continuous : - worse bounds, but “few” variables and constraints If train i enters segment s at time t the time train i enters segment s

14. Modelling Routing Dantzig-Wolfe Exactly one route assigned to each train Reformulation multi-commodity Modelling Routing: two classical approaches The feasible routes are then the solutions to a (multi-commodity flow) system of linear equations:

15. Meet-pass decision variables The candidate meet-pass points P ij are either extremes of common tracks or siding sections 15 18S1 22 26 D E S2 21 15 18S2 12 S1 21 22 OR OR E D If i, j meet/pass at p, a set A p of precedence constraints holds i and j must meet

16. Back to the problem There are many more though minor constraints to take into account Let alone the objective function, which is very complicated and needs additional mixed integer variables and constraints The MILPs arising from the instances were too large to be solved to optimality (using CPLEX) We resort to use the MILPs as building blocks in heuristics

17. The algorithm Two heuristic strategies: 1. Limit “number of trains” (Fixed Trains Heuristic) Solve a sequence of instances with growing sets of trains T 0  T 1  …  T Each obtained by adding k trains to the previous set Routing and meeting variables for trains in T p are fixed when solving the instance relative to T p+1 2. Limit "number of routings“ (Fixed Routes Heuristic) Choose in advance some most promising routes for each train We have route generation mechanisms

18. 12 tr. Fixed RoutesFixed Trains 1-1Fixed Trains 2-2Fixed Trains 6-2 3 min15 minPath form.Arc form.Path form.Arc form.Path form.Arc form. ObjVal 842,2-919,7 919,9 Time 31,0-23,0504,145,41086,8217,72182,7 18 tr. Fixed RoutesFixed Trains 1-1Fixed Trains 2-2Fixed Trains 6-2 3 min15 minPath form.Arc form.Path form.Arc form.Path form.Arc form. ObjVal 8522,14249,82883,6 2787,42883,62787,4 - Time 18090071,8890,3323,82827,61290,2- 20 tr. Fixed RoutesFixed Trains 1-1Fixed Trains 2-2Fixed Trains 6-2 3 min15 minPath form.Arc form.Path form.Arc form.Path form.Arc form. ObjVal 9516,67845,78826,9 8827,78948.13-- Time 18090068,61063,7393,03165,5-- Results The fixed routes needs more time Path and arc formulations. Have similar results in terms of cost Path formulation 10-20 times faster than arc Path formulation finds satisfactory solutions within time limit.