Agenda

ƒ Introductionƒ The process from timetable to crew planƒ The problem: Shortening the processƒ Details on TURNI, a crew planning optimization toolƒ Objective functionƒ A solution: Design of Experiments

ƒ Chosen parametersƒ The experimentsƒ The analysis

ƒ The closed formƒ Validity of the closed formƒ Future work

Goals of this paper

Goalƒ Reduce the time to agree on a cost efficient crew plan accepted by the union and wanted by the drivers

Toolƒ Parameter analysis and preparation

Dimensions at S-togs

2*170 km tracks80 stations~ 1100 departures on a daily basis80 trains running530 drivers hereof 150 in reserve300.000 passengers a day A target regularity of >95 % A target reliability of >97 %

ƒ 3 crew depots (KH, KJ, HI)ƒ 2 break facilities (KH, HL)

The S-train network

Line A

KH

HI

UND

24

22

41

11

40

22

North 92South 68Round 160 min

Frequency 20 minRolling stock 8

Lines

Round Rolling stock Line

120 6 C, A+

140 7 B, B+

160 8 A

180 9 E

200 10 H, H+

Efficiency of driver duties

100101102103

104105106107

2002 2003 2004 2005 2006

Index (year 2002 = 100)

Efficiency of driver duties

Timetable

Rolling stock

Crew

Management

The planning process

The decision cycle

Crew

Management

Before (many repetitions)

After (full picture)

Crew

Management

…

TURNI

Crew optimization system www.turni.itƒ Used by bus companies in Italyƒ Used by railway company NSR in Holland

ƒ 2001: Kroon and Fischetti, Crew Scheduling for Netherlands Railways "Destination: Customer”

ƒ 2000: Kroon and Fischetti. Scheduling train drivers and guards: the dutch noord-oost case.

Minimum transfer

time

Pre- and post times

Meal break rule

Maximum duty lengthDuty examples

Rostering Rules

Maximum percentage late duties

Maximum percentage long duties (>8 hrs)

Maximum average duty

length

The different objective functions

S-tog Penalty TURNIEfficiency / Working time Yes Small YesNo of duties implicitly Large Yes Break size implicitly Small YesDuty length Implicitly Small YesPassenger trips (taxis) implicitly Small YesShort duties implicitly Small Yes

Convergence of a TURNI run, OBJ2Total working time

86000

86500

87000

87500

88000

88500

89000

89500

90000

0 100 200 300 400 500 600

Seed 1

Seed 2

Seed 3

Seed 4

Seed 5

Parameters of the rules

Description Factor 0 1 At most 3:20 of driving on line H without a break A 20 30 Max average working time B 07:20 07:30 Rule of variation of specific type C 2 0 Max no of duties longer than 8:00 hours D 10% 20% Amount of extra minutes in connection with breaks E 4 2 Max working time after 17:00 F 07:15 07:45

Parameter analysis, methods

Method 1: One parameter at a timeƒ Only one parameter is changed each timeƒ Each parameter can be tried at many levels

Method 2: Lagrange multipliersƒ TURNI use them.ƒ One multiplier for each restriction in the math modelƒ Measure the improvement of OBJ2 when changing the parameter one unit

Method 3: Design of experimentsƒ Used here

Design of experimentsA B C0 (-) 0 (-) 0 (-)0 (-) 1 (+) 0 (-)1 (+) 0 (-) 1 (+)1 (+) 1 (+) 1 (+)0 (-) 0 (-) 1 (+)0 (-) 1 (+) 1 (+)1 (+) 0 (-) 0 (-)1 (+) 1 (+) 0 (-)

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

Parameter analysis

Method Speed Levels 2. Order effects

3. Order effects

Different objective functions

Change one parameter

Fast Many No No Ok

Lagrange multipliers

Very fast One No No No

Design of experiments

Slower Two Yes Possible Ok

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%

Results and analysis

Effects Value Value/Const (%) P-valueConst 86776A 44 0.1% 0.035B 1144 1.3% 0.000C -591 -0.7% 0.000D -172 -0.2% 0.000E -493 -0.6% 0.000F -36 0.0% 0.477AB -204 -0.2% 0.045AC -174 -0.2% 0.080AD -3 0.0% 0.978AE 129 0.1% 0.179AF -54 -0.1% 0.563BC -292 -0.3% 0.009BD -281 -0.3% 0.010BE -164 -0.2% 0.097BF -121 -0.1% 0.206CD -113 -0.1% 0.236CE 237 0.3% 0.024CF 65 0.1% 0.482DE 30 0.0% 0.742DF 225 0.3% 0.030EF -108 -0.1% 0.254

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 significantFrom 2nd order effects: keep F, since DF is significant

Rule of variation was redefined after this analysis.

The closed form

Let fA denote the level of parameter A.

With only two factors you would haveOBJ = const + fAA+fBB+fAfBAB + error

fA = {0,1}

The closed form

0 1

1

0

B

A

A+B+AB

fB

fA

0

? fAA+fBB+fAfBAB

The closed form

Let fA denote the level of parameter A.

With only two factors you would haveOBJ = const + fAA+fBB+fAfBAB + error

fA = {0,1}

fA arbitrary

Justifying the closed form

A B C D E F OBJ Model value

%-diff

0 0 0 0 0 0 86742 86776 0.04%

0 0 0 0 1 0 86319 86284 -0.04%

0 0 0 1 0 1 86776 86793 0.02%

0 0 0 1 1 1 86239 86222 -0.02%

0 0 1 1 1 0 85646 85674 0.03%

The last parameter setting is ”best” possible

Future work

ƒ Introduce center points. Requires a non-linear modelƒ Larger experiments,

ƒ Screening (remove insignificant)ƒ Priorities

ƒ Use DoE in rolling stock rostering or other planning problems from the railway industry

Questions?