1. Erasmus Center for Optimization in Public Transport Bus Scheduling & Delays Dennis Huisman Email: huisman@few.eur.nl Joint work with: Richard Freling and Albert P.M. Wagelmans May 23, 2002

2. Erasmus Center for Optimization in Public Transport 2 Contents Introduction Static versus Dynamic Scheduling Dynamic Vehicle Scheduling: –single-depot –multiple-depot Computational Experience Conclusions and Future Research

3. Erasmus Center for Optimization in Public Transport 3 Introduction static dynamic

4. Erasmus Center for Optimization in Public Transport 4 Vehicle Scheduling Problem Minimise total vehicle costs Constraints: –every trip has to be assigned to exactly one vehicle; –every vehicle is associated with a single depot; –some trips have to be assigned to vehicles from a certain set of depots; –…

5. Erasmus Center for Optimization in Public Transport 5 Static versus Dynamic Scheduling (1) Traditional: static vehicle scheduling Disadvantage: a lot of delays Solution? --> Fixed buffer times??? No!!! Idea: dynamic vehicle scheduling

6. Erasmus Center for Optimization in Public Transport 6 Static versus Dynamic Scheduling (2) Example: –2 trips (1 & 2) end at location A at time 10:00 –1 trip (3) starts at A at time 10:05 –1 trip (4) starts at A at time 10:15 Static optimal solution: 1  3 and 2  4 Suppose trip 1 has a delay of 10 minutes Dynamic scheduling: change schedule to 1  4 and 2  3

7. Erasmus Center for Optimization in Public Transport 7 Static versus Dynamic Scheduling (3) Dynamic vehicle scheduling: –reschedule a few times per day –take into account delays in the past --> scenarios

8. Erasmus Center for Optimization in Public Transport 8 Dynamic vehicle scheduling At time point T, we make decisions for the period [T,T+l). Assumption: travel times are known for this period. For the period after T+l, we consider different scenarios for the travel times based on historical data, or one average scenario. Consequence: the smaller l, the more realistic, but the quality of the solution decreases and the cpu time increases.

9. Erasmus Center for Optimization in Public Transport 9 TT+l end of the day start of the day scenario 1 scenario 2 scenario 5 scenario 3 scenario 4 Iteration i end of the day T scenario 1 scenario 2 scenario 5 scenario 3 scenario 4 start of the day T+l Iteration i+1 Dynamic Vehicle Scheduling Example with 5 scenarios

10. Erasmus Center for Optimization in Public Transport 10 Vehicle Scheduling Network (single-depot) G=(V,A) with V nodes and A arcs –Nodes for every trip, source r and sink t –Arcs between source r and every trip; two trips i and j, if trips i and j are compatible; every trip and sink t. 2 rt 1 3 4

11. Erasmus Center for Optimization in Public Transport 11 Dynamic Vehicle Scheduling (single-depot) Notation –N: set of trips –S: set of scenarios –A 1 : set of arcs in period [T,T+l) –A 2 : set of arcs in period after T+l –c: fixed vehicle cost –c’ ij (c s ij ): variable vehicle & delay cost of arc i->j (in scenario s) –p s : probability of scenario s –Decision variables:

12. Erasmus Center for Optimization in Public Transport 12 Assumption (1) Special cost structure: –fixed costs for every vehicle; –variable costs per time unit that a vehicle is without passengers outside the depot.

13. Erasmus Center for Optimization in Public Transport 13 Assumption (2) Consequences: –if it is possible, a vehicle returns to the depot –delete the arcs, where c’ ij  c’ it + c’ rj and c s ij  c s it + c s rj –add a restriction for the number of vehicles B s –Extra notation: H is the set of all relevant time points (all possible moments that a bus can leave just before a possible arrival) b sh is the number of trips at time point h

14. Erasmus Center for Optimization in Public Transport 14 Dynamic Vehicle Scheduling (single-depot) Mathematical Model

15. Erasmus Center for Optimization in Public Transport 15 Dynamic Vehicle Scheduling (multiple-depot) Size of the problem is very large Cluster-Reschedule Heuristic: –cluster the trips via the static MDVSP –reschedule per depot via the dynamic SDVSP Lagrangean Relaxation for computing lower bounds

16. Erasmus Center for Optimization in Public Transport 16 Data (1) Data from Connexxion 1104 trips and 4 depots Rotterdam, Utrecht and Dordrecht Average depot group size: 1.71

17. Erasmus Center for Optimization in Public Transport 17 Data (2)

18. Erasmus Center for Optimization in Public Transport 18 Computational Experience (1) Results static scheduling: –109 vehicles –average number of trips starting too late: 17.2% –average delay costs: 107,830 (10x 2 )

19. Erasmus Center for Optimization in Public Transport 19 Computational Experience (2) Results static scheduling with fixed buffer times: –Buffer times have only a small impact on large delays, but reduce the number of delays significantly, because the small ones disappear. –The number of vehicles used is the same for all days, which need not be necessary.

20. Erasmus Center for Optimization in Public Transport 20 Computational Experience (3) Dynamic scheduling: –fixed cost per delay; –cost for a delay is equal to the fixed cost per bus; –9 scenarios (I) or 1 average scenario (II); –different values of l: 1, 5, 10, 15, 30, 60 and 120 minutes.

21. Erasmus Center for Optimization in Public Transport 21 Computational Experience (4) Results dynamic scheduling (average over all days): –Cpu time: max. 55 seconds for one iteration and one depot (Pentium III, 450 MHz)

22. Erasmus Center for Optimization in Public Transport 22 Computational Experience (5) Lower bound: –gap between the cluster-reschedule heuristic and the lower bound is in the first iteration about 3.5% (I) and 5.7% (II) Perfect information: –optimal: 110.6 vehicles –heuristic: 114.5 vehicles Sensitivity analysis: –small mistakes in the estimated travel times have a small influence on the quality of the solution

23. Erasmus Center for Optimization in Public Transport 23 Conclusions and Future Research An optimal solution for the static vehicle scheduling may lead to a lot of delays. Dynamic vehicle scheduling performs better in both the number of vehicles & the number of trips starting late than static vehicle scheduling with fixed buffer times. Future: –integration with crew scheduling.

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