Branching Strategies to Improve Regularity of Crew Schedules in Ex-Urban Public Transit

1. International Graduate School of Dynamic Intelligent Systems Branching Strategies to Improve Regularity of Crew Schedules in Ex-Urban Public Transit Leena Suhl University of Paderborn, Germany joint work with Ingmar Steinzen and Natalia Kliewer

2. Outline • Introduction • Ex-urban vehicle and crew scheduling problem • Problem definition • Irregular timetables • Solution Approach • Column Generation with Lagrangian relaxation • Distance measure • modified Ryan/Foster branching rule • Local Branching • Computational results

3. Introduction lines / service network timetable of one line service trip: 21:45 -- 22:00 from Westerntor to Liethstaudamm

4. line+frequency planning timetabling timetable/service trips vehicle scheduling vehicle blocks/tasks crew scheduling crew duties crew rostering crew rosters Introduction relief points labour regulations

5. D1 D1 block 1 D1 D1 block 2 D2 D2 block 3 Multi-Depot Vehicle Scheduling Problem (MDVSP) • Given: set of service trips of a timetable • Task: find an assignment of trips to vehicles such that • Each trip is covered exactly once • Each vehicle performs a feasible sequence of trips (vehicle block) • Each sequence of trips starts and ends at the same depot • (vehicle capital and operational) costs are minimized

6. D1 D1 block 1 break D1 D1 block 2 Crew Scheduling Problem (CSP) • Given: set of tasks • From vehicle blocks and relief points (sequential CSP) • From timetable and relief points (integrated CSP) • Task: assign tasks to crew duties at minimum cost such that • Each task is covered (exactly) once • Each duty starts/ends at the same depot • Each duty satifies (complex) governmental and in-house regulations

7. trip piece of work-related duty-related deadhead constraints relief point Crew Scheduling Problem (CSP) duty piece of work 2 piece of work 1 break task 1 task 4

8. Crew Scheduling Problem (CSP) • Minimize total crew costs • Constraints • Cover all tasks of vehicle schedule (sequential) • Cover all tasks of timetable (independent) I set of all tasks K set of all feasible duties K(i) set of all duties covering task i set partitioning orset coveringformulation possible

9. Ex-urban Vehicle and Crew Scheduling Problem (VCSP) • Given: set of service trips of a timetable and set of relief points • Task: find a set of vehicle blocks and crew duties such that • Vehicle and crew schedule are feasible • Vehicle and crew schedule are mutually compatible • Sum of vehicle and crew costs is minimized • Only few relief points in ex-urban settings • Assumption: All relief points in depot (typical for ex-urban settings)

10. Irregular Timetables • Timetable consists of • regular (daily) trips • irregular trips (e.g. to school or plants): about 1-5% of all trips • similar situation: timetable modifications • similar and regular crew schedules • easier to manage in crew rostering phase • less error-prone for drivers regular trips trips day A trips day B

11. instance: Monheim (423 trips) 2% of trips different timetable Monday timetable Tuesday 66% of vehicle blocks different vehicle schedule vehicle schedule 100% of crew duties different crew schedule crew schedule 93% of crew duties different crew schedule crew schedule Irregular Timetables • Naive approach: plan all periods sequentially, but • Modifications of timetable have a strong impact on regularity of vehicle and crew scheduling solutions

12.  fix (regular) duties  C: set of remaining (unfixed) tasks trade-off large problems low regularity small problems many deadheads, high costs Irregular Timetables • No literature on irregular timetables in public transport • Simple heuristics from practice • Solve problem with all trips of periods • Solve problem with regular and irregular trips of periods separately

13. Outline • Introduction • Ex-urban vehicle and crew scheduling problem • Problem definition • Irregular timetables • Solution Approach • Column Generation with Lagrangian relaxation • Distance measure • modified Ryan/Foster branching rule • Local Branching • Computational results

14. Column generation in combination with Lagrangean relaxation duties= initial column set while duties≠  and no termination criteria satisfied Add duties to master Compute dual multipliers by solving Lagrangean dual problem with current set of columns Volume Algorithm Delete duties with high positive reduced costs Partial Pricing with Dynamic Programming Algorithm duties = Generate new negative reduced cost columns Find integer solution Solution approach crew scheduling Construct feasible vehicle schedule (pieces of work correspond to service trips) vehicle scheduling

15. Space Time Piece generation network Network Models for a Decomposed Pricing Problem pieces of work pieces of work connection-based duty generation network (Freling et al. 1997, 2003) aggregated time-space duty generation network (Steinzen et al. 2006) network size: O(#tasks2) network size: O(#tasks4)

16. Guided IP Branch-and-Bound search • Average number of different optima for ICSP • Idea: guide IP solution method to „favorable“ solutions (concerning distance to reference solution) • Follow-on branching • Adaptive local branching • Adaptive local branching with follow-on branching test set from Huisman, abort search after 2500 optima set partitioning, independent crew scheduling, variable costs

17. service trips service trips ti si timetable A timetable B 2 6 9 14 21 56 duties Gi duties Hi 1 2 3 4 5 … 1 2 3 4 5 … 2 6 84 9 24 56 irregular trip Distance measure for crew duties crew schedule G crew schedule H trip chain T1={2,6,9} Reference solution

18. Follow-on Branching • Ryan/Foster branching rule for fractional solution of a set partitioning problem and two rows r and s • Create two subproblems • Choose r and s with max f(r,s) • Follow-on branching: allow only consecutive tasks (rows)

19. Initialize set Sk of trip chains Ti with Sk={Ti: 0<f(Ti)<1} Branch on trip chain (r,s) with 0<f(r,s)<1 and max(f(r,s)) Yes No Initialize Skmax={Ti:max(|Ti|)} and branch on Ti Skmax with max(f(Ti)) FOR2 Sk=? Follow-on branching to create regular crew schedules • Follow-on branching strategies • DEF: Original • FOR1: Sequences from reference schedule • FOR2: Piece of work from reference schedule • FOR3: Maximum length sequence from reference schedule

20. Local Branching • Strategic local search heuristic controls „tactical“ MIP solver • Local branching cuts equal Hamming distance with L0={kK: xk’=1} • Exact solution approach

21. Local Branching to create regular crew schedules • Use local branching to search subspaces that contain „regular“ solutions first • Initial solution • modify cost function ck’ = ck+dkwith dk distance of duty to reference crew schedule  weight of distance • Adapt neighbourhood size if necessary (time limit exceeded) • Optional: use follow-on branching in subproblem

22. Outline • Introduction • Ex-urban vehicle and crew scheduling problem • Problem definition • Irregular timetables • Solution Approach • Column Generation with Lagrangian relaxation • Distance measure • modified Ryan/Foster branching rule • Local Branching • Computational results

23. Computational Results • Tests with both real-world and artificial data • Artificial data generated like Huisman (2004) with 320/400/640/800 trips (two instances each), relief points only in depots • Real-world data with ~430 trips (German town with ~45.000 inh.) • Irregular trips: 5% (artificial), 2-3% (real-world) • Reference crew schedule is known for all instances • All tests on Intel Pentium IV 2.2GHz/2 GB RAM with CPLEX 9.1.3 • Limited branch-and-bound time to 2 hours

24. Computational Results(Column Generation) irr% - percentage of irregular trips cpu_ma – cpu time (sec) for the master problem cpu_pr – cpu time (sec) for the pricing subproblem

25. Computational Results(Regularity of Crew Schedules) prd% - percentage of duties (completely) preserved from reference crew schedule prp% - percentage of trip sequences preserved from reference avcl% - percentage of average trip sequence length preserved from reference

26. International Graduate School of Dynamic Intelligent Systems Thank you very muchfor your attention