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Estimation of crew demand in S-tog
Agenda
ƒ The planning processƒ Motivation for the modelƒ What is the purpose of the modelƒ Input to the modelƒ Objectivesƒ Experiments and conclusionƒ Future work
Michael FolkmannOperational Researcher Production Planning, DSB S-tog
Process of the (sequential) planning
Number of Passengers
Time Table
Draft of Product
Economystructure
Evaluation
Rolling Stock plan
Crew plan
New product and changes in the product.
Basis plan
Planning period ≈ one year
Track
Work
Track
Work Track
Work
Track
Work
Track
Work
…….. … Track
Work
Motivation for the model
Good early estimation of crew costs in reasonable time for a plan
Crew plans were time consuming
Crew targetƒ Train drivers
ƒ First generationƒ Needs Rolling Stock Planƒ Duty planning with Turni integrated ƒ Adjustment for track works
ƒ Second generationƒ Change the input
ƒ Other groups of personnelƒ No Rolling Stock Plan
What is the purpose of the model
Breifly:Cover a workload with duty template in relation to a number of constraints
DayTypesƒ Weekdaysƒ Saturdayƒ Sunday
Size of time intervals; 15 minutes
Inputsƒ Workload
ƒ Coming from Rolling Stock Plan which needs a driverƒ Duty templatesƒ Parameters for constraints
Use of the resultsƒ Not actual planƒ Crew demand
Workload
Fictitious Workload
0
10
20
30
40
50
60
03:0
0
3:45
4:30
5:15
6:00
6:45
7:30
8:15
9:00
9:45
10:3
0
11:1
5
12:0
0
12:4
5
13:3
0
14:1
5
15:0
0
15:4
5
16:3
0
17:1
5
18:0
0
18:4
5
19:3
0
20:1
5
21:0
0
21:4
5
22:3
0
23:1
5
24:0
0
24:4
5
25:3
0
26:1
5
Time
Rollin
g S
tock
s
Duty Templates
ƒWorking dayƒAmount of drivingƒCheck In/out – work – Breaks
ƒSix different template variationsƒBy hand or with model
ƒFractionel
Template Type 1
0
0,2
0,4
0,6
0,8
1
1,2
00:0
0
00:3
0
01:0
0
01:3
0
02:0
0
02:3
0
03:0
0
03:3
0
04:0
0
04:3
0
05:0
0
05:3
0
06:0
0
06:3
0
07:0
0
Time
Am
ount
Duty Templates variants
ƒ For all six templates, i.e. three sets with six templates
Template Type 2
0
0,2
0,4
0,6
0,8
1
1,2
00:0
0
00:3
0
01:0
0
01:3
0
02:0
0
02:3
0
03:0
0
03:3
0
04:0
0
04:3
0
05:0
0
05:3
0
06:0
0
06:3
0
07:0
0Time
Am
ount
Template Type 3
0
0,2
0,4
0,6
0,8
1
1,2
00:0
0
00:3
0
01:0
0
01:3
0
02:0
0
02:3
0
03:0
0
03:3
0
04:0
0
04:3
0
05:0
0
05:3
0
06:0
0
06:3
0
07:0
0
Time
Am
ount
Difference in Template Types
Template Type T1
Template Type T2
Template Type T3
Working TimeNo break Utilization % Utilization % Utilization %
Template 1 07:15 2 71% 74% 79%
Template 2 06:30 2 69% 73% 77%
Template 3 06:15 2 64% 66% 72%
Template 4 07:15 1 73% 83% 83%
Template 5 06:45 1 77% 79% 81%
Template 6 07:15 2 67% 69% 71%
Working Rules to be modelled
Each Duty - Inputƒ Length of break in dutiesƒ Duty lengthƒ Length of driving blockƒ …
All Duties - Constraintsƒ Average working time per dutyƒ Average breaks per duty
Variables
Check-in cIn
Duty Template Dt
Number of train drivers check-in in interval cIn following template Dt
x(cIn,Dt) - Integer
Average Working Time per Duty
Total amount of working time divided by the number of duties
Reformulated:
AverageWTDtcInx
DtlengthDtcInx
Dt cIn
Dt cIn
),(
))(()),((
0)))(()),(( Dt cIn
AverageWTDtlengthDtcInx
Average Breaks per Duty
Reference solution ƒ241 dutiesƒBreak Average 1.722 (=AB)
ƒTotal number of breaks in duties divided by the number of duties.
ƒReformulated constraints with variation
0))1()(()),(( Dt cIn
VariationABDtbreaksDtcInx
One break Two breaks
(Total 241 duties)
Variation At most At least
10.00% 109 133
5.00% 88 153
4.00% 84 157
3.00% 79 162
2.00% 75 166
1.00% 71 170
0.10% 67 174
0.01% 67 174
0.00% 67 174
Model Objective
ƒMinimise the total amount of working time in the solution (Objective1)
ƒMinimise the total number of templates used (Objective 2)
cIn Tj
cIn,Dt)x(min
Dt cIn
DtlengthDtcInx ))(()),((min
Adjustment and Experiments
Decisionƒ Size of time interval
Adjustmentsƒ Importance of the average breaksƒ Importance of the templates types
Experimentsƒ Two objectivesƒ Gap - solution quality
Runs
One daytype for one plan – one reference solution
Two fictitious plan – relativ evaluation
ƒ Running time 6:00 hoursƒ Zero gap tolerance
ƒ Cplex 8.1.0 (using Gams 2.0.23.10)ƒ Windows 2000ƒ Pentium M 1700 MHzƒ 2GB ram
Average Breaks and Templates
Three Templates Types - Objective 1
1500
1550
1600
1650
1700
0,01% 0,10% 1,00% 2,00% 3,00% 4,00% 5,00% 10,00% No limit
Average Break variation
No h
ours T1
T2
T3
Average Breaks and Templates
Three Templates Types - Objective 2
205
210
215
220
225
230
235
240
245
0,01% 0,10% 1,00% 2,00% 3,00% 4,00% 5,00% 10,00% No limit
Average Break variation
No D
uties
T1
T2
T3
Quality of solutions
Gap in solution - Objective 1
0,00%
0,50%
1,00%
1,50%
2,00%
2,50%
3,00%
3,50%
0,01% 0,10% 1,00% 2,00% 3,00% 4,00% 5,00% 10,00% No limit
Average Break variation
Gap
T1
T2
T3
Quality of solutions
Gap in solution - Objective 2
0,00%
0,50%
1,00%
1,50%
2,00%
2,50%
3,00%
3,50%
4,00%
4,50%
5,00%
0,01% 0,10% 1,00% 2,00% 3,00% 4,00% 5,00% 10,00% No limit
Average Break variation
Gap
T1
T2
T3
Gap progress
Running time for (almost) no progress in gap compared to 360 minutesƒTemplate Type T1ƒBreak Average 0.1% and 1.0%
Break Average variation
0.10%Break Average variation
1.0%
Objective 1 325 minutes 45 minutes
Objective 2 81 minutes 1.25 minutes
Results and conclusion
ƒ Decreasing in objective – as expected
ƒ Two Objectives
ƒ Average Breaks – important but not strict
ƒ Running time
Have initial parameter setting:ƒ Minimising number of duties with T1 duty templates with 15 minutes intervals and Average Breaks variation 1.0% for a short running time
Future work
More reference solution from Turni
Different templatesƒ effect of small changesƒ fewer/more templates
Develop of the template model
Other constraintsƒ difference during the dayƒ limited number of specific templates
Change the model into second generation – i.e. Public Time Table as input
Questions?
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