Agenda

1. Agenda • Introduction • The process from timetable to crew plan • The problem: Shortening the process • Details on TURNI, a crew planning optimization tool • Objective function • A solution: Design of Experiments • Chosen parameters • The experiments • The analysis • The closed form • Validity of the closed form • Future work

2. Goals of this paper • Goal • Reduce the time to agree on a cost efficient crew plan accepted by the union and wanted by the drivers • Tool • Parameter analysis and preparation

3. Dimensions at S-togs • 2*170 km tracks • 80 stations • ~ 1100 departures on a daily basis • 80 trains running • 530 drivers hereof 150 in reserve • 300.000 passengers a day • A target regularity of >95 % • A target reliability of >97 %

4. The S-train network • 3 crew depots (KH, KJ, HI) • 2 break facilities (KH, HL)

5. Line A 11 • North 92 • South 68 • Round 160 min • Frequency 20 min • Rolling stock 8 HI 40 41 KH 22 22 UND 24

6. Lines

7. Efficiency of driver duties 107 106 105 Index (year 2002 = 100) 104 103 102 101 100 2002 2003 2004 2005 2006 Efficiency of driver duties

8. The planning process

9. The decision cycle

10. TURNI • Crew optimization system www.turni.it • Used by bus companies in Italy • Used by railway company NSR in Holland • 2001: Kroon and Fischetti, Crew Scheduling for Netherlands Railways "Destination: Customer” • 2000: Kroon and Fischetti. Scheduling train drivers and guards: the dutch noord-oost case.

11. Maximum duty length Duty examples Minimum transfer time Pre- and post times Meal break rule

12. Maximum percentage late duties Rostering Rules Maximum average duty length Maximum percentage long duties (>8 hrs)

13. The different objective functions

14. Convergence of a TURNI run, OBJ2

15. Parameters of the rules

16. Parameter analysis, methods • Method 1: One parameter at a time • Only one parameter is changed each time • Each parameter can be tried at many levels • Method 2: Lagrange multipliers • TURNI use them. • One multiplier for each restriction in the math model • Measure the improvement of OBJ2 when changing the parameter one unit Method 3: Design of experiments • Used here

17. Design of experiments Full factorial design, 23=8 runs Fractional factorial design 23-1=4 See for instance Design and Analysis of Experiments by Douglas C. Montgomery (2004)

18. Parameter analysis

19. The general linear model • OBJ =  const+A+B+C….+D+E+F • +AB+AC+AD+AE+AF • +BC+BD+BE+BF • +CD+CE+CF • +DE+DF • +EF+error. • No 3. order effects. Model validity 98,5%

20. Results and analysis

21. Interesting features • Synergetic effects: • When changing both C and D, the effect is larger than the sum of the effect of C and D alone: -591-172-113=-876 • Counterintuitive signs: • Difference between OBJ1 and OBJ2 • Significance level 5% • From 1st order effects: leave F out. F is not significant • From 2nd order effects: keep F, since DF is significant • Rule of variation was redefined after this analysis.

22. The closed form • Let fA denote the level of parameter A. • With only two factors you would have • OBJ = const + fAA+fBB+fAfBAB + error • fA = {0,1}

23. The closed form A+B+AB B 1 fAA+fBB+fAfBAB ? fB 0 A 0 fA 1 0

24. The closed form • Let fA denote the level of parameter A. • With only two factors you would have • OBJ = const + fAA+fBB+fAfBAB + error • fA = {0,1} • fA arbitrary

25. Justifying the closed form The last parameter setting is ”best” possible

26. Future work • Introduce center points. Requires a non-linear model • Larger experiments, • Screening (remove insignificant) • Priorities • Use DoE in rolling stock rostering or other planning problems from the railway industry

27. Questions?