Study on the optimal crew scheduling for the green line of Metropolitano de Lisboa

1/22

- Joao Fialho & Nikhil Menon

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Outline

2/22

1. Introduction/Case Study

2. Problem Description

3. Mathematical Model

4. Results

5. Conclusion

6. Reference(s)

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1.Introduction/Case Study

3/22

• Case studyGreen line of ML

• ObjectiveOptimal crew scheduling that minimizes total wage paid to the train drivers

• MethodologySolving an Integer Linear Programming problem by a mathematical model

• ToolXPRESS-IVE 7.2

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2.Problem Description

4/22

• Train drivers requirementsFrequency→Timetable →Rostering →Request Drivers

• AssumptionsType of driversCompany work force dimensionIncrement of work

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2.Problem Description

5/22

• Train drivers requirementsFrequency→Timetable

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2.Problem Description

6/22

• Train drivers requirementsFrequency→Timetable →Rostering

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2.Problem Description

7/22

• Train drivers requirementsFrequency→Timetable →Rostering

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2.Problem Description

8/22

• Train drivers requirementsFrequency→Timetable →Rostering →Request Drivers

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2.Problem Description

9/22

• Assumptions

Type of drivers3 type of train drivers: Full time driver, split shift driver and extra driver (outsourced)

Company work force dimensionThe railway company has currently 24 employees that can work either as a full time driver or a split shift driverThere is no restriction on the number of extra drivers that can be hired

Increment of work30 minutes (basic time unit of work)

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3. Mathematical Model

10/22

• Decision Variables

xit → Number of train drivers of type i that started to work in increment of work t

where,

i=1 → train drivers that work on full timei=2 → train drivers that work on split shifti=3 → extra train drivers that work 4 hours shifts

and

t = ,1..39- → increments of work

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3. Mathematical Model

11/22

• Objective Function

Minimize total wage paid to train drivers

where,

wi = * w1 w2 w3 + = * 64 80 48 + → wage/day for each train driver of type i

𝑀𝑖𝑛 𝑊 = 𝑤𝑖

39

𝑡=1

3

𝑖=1

∙ 𝑥𝑖𝑡

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3. Mathematical Model

12/22

• Constraints

1. Demand – the demand of train drivers must be fulfilled at all times

𝑥1(𝑡−𝑖)

15

𝑖=0

+ 𝑥2(𝑡−𝑖)

7

𝑖=0

+ 𝑥2(𝑡−𝑖)

19

𝑖=12

+ 𝑥3(𝑡−𝑖)

7

𝑖=0

≥ 𝑑𝑡 ∀ 𝑡 > 𝑖 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑

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3. Mathematical Model

13/22

• Constraints

2. Number of drivers who are employees of the company - work force dimension of the company

𝑥𝑖𝑡

39

𝑡=1

2

𝑖=1

≤ 24

3. Subject to

𝑥𝑖𝑡 ≥ 0,∀ 𝑡 ∈ 𝑃𝑒𝑟𝑖𝑜𝑑

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4. Results

14/22

• Output of the model

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4. Results

15/22

• Possible Crew Scheduling Solution

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4. Results

16/22

• Schedule of the train drivers according to the various types over the demand histogram.

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4. Results

17/22

• Sensitivity Analysis on N

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4. Results

18/22

• Sensitivity Analysis on N

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4. Results

19/22

• Sensitivity Analysis on w3

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4. Results

20/22

• Sensitivity Analysis on w3

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5. Conclusions

21/22

• The objectives proposed have been achieved

• The developed model has some limitations

o Relief points

o Break/rest periods

• This model can be applied in real world situations of mass transit planning

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• Yunes, T., Moura, A. and Souza, C., “Solving Very Large Crew Scheduling Problems to Optimality” ACM 0-89791-88-6/97/05., pp 1-2.

• Gomes, M., Cavique, L. and Themido, I., (2006). “The crew timetabling problem: An extension of the crew scheduling problem” Springer Science + Business Media, LLC 2006, pp 2-2.

22/22

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6. Reference(s)