study on the optimal crew scheduling for the green line of metropolitano … · 2012-08-04 · of...

Nikhil Menon | Joao Fialho

Transportation Systems Analysis – Course Project

STUDY ON THE OPTIMAL CREW SCHEDULING FOR THE GREEN LINE OF

METROPOLITANO DE LISBOA.

A PROJECT REPORT

PREPARED BY

NIKHIL MENON

JOAO FIALHO

MSc (COMPLEX TRANSPORT INFRASTRUCTURE SYSTEMS)

In partial fulfillment of the credit requirements for the course

TRANSPORTATION SYSTEMS ANALYSIS (TSA)

INSTITUTO SUPERIOR TECNICO

1149-001, LISBOA

DECEMBER 2011



CONTENTS

1. INTRODUCTION……………………………………………………….....5

2. CASE STUDY………………………………………………………………5

3. TERMINOLOGY………………………………………………………......7

4. PROBLEM DESCRIPTION……………………………………………....8

5. MATHEMATICAL MODEL…………………………………………......11

5.1. DECISION VARIABLES………………………………………….....11

5.2. OBJECTIVE FUNCTION…………………………………………....11

5.3. CONSTRAINTS……………………………………………………….12

6. RESULTS……………………………………………………………….......14

6.1. INFERENCE…………………………………………………………..15

6.2. SENSITIVITY ANALYSIS…………………………………………...17

6.2.1. SENSITIVITY ANALYSIS ON N……………………………..17

6.2.2. SENSITIVITY ANALYSIS ON w3……………………………18

7. CONCLUSION……………………………………………………………..21

8. REFERENCES……………………………………………………………..21

APPENDIX…………………………………………………………22



LIST OF FIGURES

2.1 NETWORK MAP, LISBOA METRO…………………………...6

4.1 ROSTERING DIAGRAM…………………………..……………8

4.2 DEMAND HISTOGRAM………………………………………...9

5.1 DEMAND CONSTRAINT………………………………………13

6.1 TRAIN DRIVER SCHEDULING………………………………16

6.2 REQUIREMTN OF DRIVERS Vs. DRIVERS SCHEDULED.16

6.3 SENSITIVITY ANALYSIS ON N……………………………….18

6.4 SENSITIVITY ANALYSIS ON N – TYPE OF DRIVERS…….18

6.5 SENSITIVITY ANALYSIS ON w3………………………………19

6.6 SENSITIVITY ANALYSIS ON w3 – TYPE OF DRIVERS…....20



LIST OF TABLES

2.1…………………………………………………………………………………7

4.1…………………………………………………………………………………9

4.2…………………………………………………………………………………10

6.1………………………………………………………………………………....14

6.2…………………………………………………………………………………15

6.3…………………………………………………………………………………17

6.4…………………………………………………………………………………19



1. INTRODUCTION

Crew scheduling is the process of assigning crews to conduct a particular activity. Crew

scheduling has a wide application in transport, where it is used to assign crews to operate

transportation systems. For example, Crew scheduling is a common technique adopted in

railways to assign drivers to operate trains on various routes. The goal of the problem is to assign

a subset of trips to each crew in such a way that no trip is left unassigned. As usual, not every

possible assignment is allowed since a number of constraints must be observed. Additionally, a

cost function has to be minimized. (Yunes, Moura and Souza et al, 2000)

Crew scheduling problems have their great practical importance based on the fact that, in most

companies, employee related expenses may rise to a very significant portion of the total

expenditures. Therefore, these notoriously difficult complex combinatorial problems deserve a

great deal of attention. (Yunes, Moura and Souza et al, 2000) Gomes, M., Cavique, L. and

Themido, I., in their paper, “The crew timetabling problem: An extension to the crew scheduling

problem” (2006) have dealt in depth with the aspect of preparation of performance measures

developed for the crew timetabling problem, which are combined in an objective function.

Through this study, it is intended to determine the optimal requirement of crew for the Green line

of the Lisbon Underground, Metropolitano de Lisboa which is also abbreviated as ML in this

case study.

2. CASE STUDY

Metropolitano de Lisboa (ML) is the metro (subway) system of Lisbon, Portugal. It was opened

on December 29, 1959 thus becoming the first subway system to be opened for the public in

Portugal. As of today, it consists of four lines with 52 stations, spread over 39.6 kilometers. As

of 2009, Metropolitano de Lisboa (ML) carries an estimated 177 million passengers per year.



The figure above shows the network map of Metropolitano de Lisboa (ML) spread over the city

of Lisbon. The Green Line (Linha Verde) is being analyzed as part of the current study. The

green line of the Metropolitano de Lisboa, runs from Cais do Sodre to Telheiras, a distance of 9

kms served by 13 stations. The construction started in 1972, with the link from Restauradores to

Alvalade. In 1993, this was further extended from Alvalade to Campo Grande. The Blue and the

Green lines were split in 1998, with the construction of the section from Baixa Chiado to Cais do

Sodre and it reached the present level of completeness in 2002, with the extension from Campo

Grande to Telheiras.

The services in the Green line commence at 0630 from both the terminals (Cais do Sodre and

Telheiras) and they run upto 0130 with varying frequencies throughout the day. The frequencies

are as described below:

Figure 2.1 – Network Map, Lisboa Metro



Time [hh:mm] Frequency [mm:ss]

06:30 - 07:30 07:30

07:30 - 09:30 03:50

09:30 - 17:00 04:40

17:00 - 20:00 03:50

20:00 - 22:30 05:45

22:30 - 01:30 09:15

As can be seen from the table, the frequencies of the trains on the Green line are classified into 6

periods dictated by the demand required to be met. The morning rush hours between 07:30 and

09:30 and the evening rush hours between 17:00 and 20:00 demand a frequency of 03:50

between each train plying along the Green line. The frequency is lesser during the other periods

of the day, especially during the last period (22:30 – 01:30) when it is 09:15 between two trains.

3. TERMINOLOGY

For the current study, the following terminologies are adopted:

Timetable: document showing all the trips undertaken by the entire number of trains

on a given day.

Trip: one way movement of a train between two terminuses. Trips are divided into

increments of work for the purpose of crew scheduling.

Increment of work / Period: the portion of work between two adjacent relief times

(or relief points). This is the smallest period into which the time table can be divided.

Relief Point: point along a line where a crew may leave a train and another crew takes

over. This can be done either at the line terminuses or at any intermediate point along

the line.

Rostering: the process of allocation of trains to trips.

Break: a rest period in a crew duty.

Table 2.1: Frequency of Trains on the Green Line



4. PROBLEM DESCRIPTION

The railway company decides to look into the aspect of optimization for the driver’s timetable.

Prior to describing the problem at hand, the following assumptions are to be stated:

Based on the data obtained from ML, the timetable of the trains plying along the

green line is created. The timetable of the trains created is attached on the appendix for

reference.

For the generation of the timetable, the turnaround times of the train are not accounted

for. In saying so, it is meant that when a train arrives at one of the terminal stations, it

is assumed that the same train can perform the return trip as long as the departure time

is after 2 minutes 30 seconds, which might not be the standard turnaround times

observed.

With the timetable prepared, a possible rostering of trains was planned and implemented for the

course of the study. The rostering implemented for the current study is as shown below:

Based on the inputs obtained from the rostering shown above, the demand for the drivers for

each period of time is obtained. It is as shown below in the histogram:

1

2

3

4

5

6

7

8

9

10

11

12

24h0018h00 19h00 20h00 21h00 22h00 23h0012h00 13h00 14h00 15h00 16h00 17h0011h0000h00 01h00 02h00 03h00 04h00 05h00 06h00 07h00 08h00 09h00 10h00

Figure 4.1 – Rostering Diagram



As can be seen from the above two figures, the increment of work adopted for the current study

is 30 minutes. By saying so, it is meant that the smallest period of time into which the timetable

is divided into, is 30 minutes. Considering the timetable generated in which the first train starts

at 0630 and the last train reaches the terminus at 0152 (adopted as 0200 in this study), there are

39 increments of work on a given day. The demand depicted by the histogram can be

alternatively represented on a table as shown below:

Increment of Work dt

Increment of Work dt

Start End t Start End t

06:30 07:00 1 6 16:30 17:00 21 10

07:00 07:30 2 6 17:00 17:30 22 12

07:30 08:00 3 12 17:30 18:00 23 12

08:00 08:30 4 12 18:00 18:30 24 12

08:30 09:00 5 12 18:30 19:00 25 12

09:00 09:30 6 12 19:00 19:30 26 12

09:30 10:00 7 11 19:30 20:00 27 12

10:00 10:30 8 10 20:00 20:30 28 11

10:30 11:00 9 10 20:30 21:00 29 8

11:00 11:30 10 10 21:00 21:30 30 8

11:30 12:00 11 10 21:30 22:00 31 8

12:00 12:30 12 10 22:00 22:30 32 8

12:30 13:00 13 10 22:30 23:00 33 7

13:00 13:30 14 10 23:00 23:30 34 6

13:30 14:00 15 10 23:30 00:00 35 6

14:00 14:30 16 10 00:00 00:30 36 6

14:30 15:00 17 10 00:30 01:00 37 6

15:00 15:30 18 10 01:00 01:30 38 6

15:30 16:00 19 10 01:30 02:00 39 4

16:00 16:30 20 10

12

11

10

9

8

7

6

5

4

3

2

1

20h00 21h00 22h00 23h00 24h0019h0008h00 09h00 10h00 11h00 12h00 13h00 14h00 15h00 16h00 17h00 18h0002h00 03h00 04h00 05h00 06h00 07h0000h00 01h00

Figure 4.2 – Histogram of the demand

Table 4.1: Representation of the demand for all increments of work



With the increments of work and the corresponding demand defined, the number of drivers

required at each instant of time is determined. As far as the drivers are concerned, three major

categories of drivers are resorted to, in the course of this study.

The first type is the full time driver. By saying so, it is meant the type of driver who works for 8

hours (16 increments of work) during the course of the day, all at one stretch. The full time

driver does not take a break during the course of his working day. The premium wages of the full

time driver is fixed at 64€/day.

The second type of the driver is the split shift driver. The working style of the split shift driver is

different from that of the Full time driver. In case of the split shift driver, each driver works for a

period of 4 hours (8 increments of work) at a stretch, then has a break of 2 hours (4 increments of

work) and resumes the work for another 4 hours (8 increments of work) during a working day.

The premium wages of the split shift driver is fixed at 80€/day.

The third type of drivers are the extra drivers. They differ from both the full time and the split

shift drivers in the sense that, they are pressed into service only during situations which demand

their presence. They are not direct employees of the railway company and thus are hired by the

railway company only when required to cover up. The extra drivers work only for a period of 4

hours (8 increments of work) in a working day. Their wages are fixed at 48€/day.

As can be seen, the split shift wage is more than the full time wage. The full and split shift

drivers are drawn from the same pool of persons. A driver of any class can work at most one

shift per day. The data shown above is aggregated in the table below:

So to sum up, the problem aims at determining the number of drivers required from each class so

as to meet the demand. The goal is to minimize the total wage cost. The railway company

currently has total employee strength of 24. The said 24 employees can work either as a full time

Nr of hours [h]Increments of

WorkNr of hours [h]

Increments of

WorkNr of hours [h]

Increments of

Work

Full Time Driver 64 8 16 - - - -

Split Shift Driver 80 4 8 2 4 4 8

Extra Driver 48 4 8 - - - -

1st Work Period Break Period 2nd Work PeriodType of Driver Wage [€/day]

Table 4.2: Extensive Driver data



driver or a split shift driver. The rest of the drivers (if required), can be hired as extra drivers.

There is no restriction on the number of extra drivers hired. The extra drivers are always

available as long as they are required to cover up.

5. MATHEMATICAL MODEL

In this section, the formulation of the model is explained, which works as a basis towards

obtaining the solution to the problem at hand.

5.1 DECISION VARIABLES

As is known, the primary objective of this mathematical model is to identify the decision

variables. The main concern in this study is to determine the number of drivers of each

type to be pressed into service at each instant of time. Therefore, the decision variables

chosen are described in the general form, below:

where,

and

{ }

All the decision variables are integer numbers because they represent the requirement of

train drivers at each instant of time.

5.2 OBJECTIVE FUNCTION

The objective of this problem is to minimize the total wage spent by the railway company

in hiring the train drivers, irrespective of their type to fulfill the requirements of the

timetable. This is achieved by summing the product of all train drivers hired with their

respective wages over all the increments of work. This is as shown below:

∑∑



where, wi is the wage/day for each train driver of type i.

[ ] [ ]

5.3 CONSTRAINTS

- Demand: The demand of train drivers must be fulfilled at all times. This is achieved by

formulating the following demand constraint, as shown below:

∑

∑

∑

∑

For every increment of work t, the number of train drivers must be equal to or higher than

the demand. In order to ensure this, it is necessary to check how many full time drivers

have started to work in the past 8 hours (16 increments of work), or until the beginning of

the service [06:30].

Also in case of split shift drivers, it is necessary to check how many drivers have started

to work in the past 4 hours (8 increments of work), or until the beginning of the service

[06:30]. It is also necessary to check how many drivers started to work in the period

between the last 6 and 10 hours, (since they work on a split shift) or until the beginning of

the service [06:30].

And in case of extra drivers, it is necessary to check how many extra drivers have started

to work in past 4 hours.

This description is illustrated by the figure below. As can be seen from the figure below,

the requirement for a driver is pressed in at the 18th increment of work, corresponding to

the period between 15:00 and 15:30. In case of the full time driver, it is necessary to

check how many drivers have started to work in the past 16 increments (3rd

increment of

work) or until the beginning of service [06:30], which happens earlier in this case. For the

split shift driver, it is necessary to check into the past 8 increments (11th) of work and

follow it up with a check on the past 20 increments from the past 12 increments (6th) or

until the beginning of service [06:30], which in this case is the key. For the extra time

driver, it is necessary to check into the past 8 increments (till the 11th increment of work).



- Number of drivers who are employees of the company:

The railway company has discretion on the number of drivers who are employees of the

company. Extra drivers do not belong the employee class of the company and they are

hired only when there is a need for cover up. And thus, they have a different pay scale

from the employees of the railway company. For the purpose of the model formulation,

the number of drivers who are employees of the company are fixed at 24.

The mathematical formulation of this fact has been described below.

∑∑

- subject to the following condition:

Figure 5.1 – Demand Constraint



6. RESULTS

The mathematical model illustrated above was input on XPRESS 7.2 IVE and used to find the

optimal solution. The results obtained give an idea on the number of drivers that are required of

each type to fulfill demand at each instant of time. The XPRESS model used is attached

alongside, on the Appendix.

The results obtained on XPRESS are summarized as below:

Period x1t x2t x3t

Start End t

06:30 07:00 1 6

07:00 07:30 2

07:30 08:00 3 4 2

08:00 08:30 4

08:30 09:00 5

09:00 09:30 6

09:30 10:00 7

10:00 10:30 8

10:30 11:00 9

11:00 11:30 10

11:30 12:00 11

12:00 12:30 12

12:30 13:00 13 2

13:00 13:30 14

13:30 14:00 15

14:00 14:30 16

14:30 15:00 17 2

15:00 15:30 18

15:30 16:00 19 4

16:00 16:30 20 1

Period x1t x2t x3t

Start End t

16:30 17:00 21

17:00 17:30 22 1

17:30 18:00 23 2

18:00 18:30 24

18:30 19:00 25

19:00 19:30 26

19:30 20:00 27

20:00 20:30 28

20:30 21:00 29

21:00 21:30 30

21:30 22:00 31

22:00 22:30 32

22:30 23:00 33

23:00 23:30 34

23:30 00:00 35 2

00:00 00:30 36

00:30 01:00 37

01:00 01:30 38

01:30 02:00 39

Σ 20 4 2

The objective function obtained for the above result is 1696€/day.

Table 6.1: Result of the Mathematical Model



6.1 INFERENCE

1. The first point of importance, based on the results obtained is shown in the figure

above. The total number of drivers required for the complete working day is

estimated to be 26, by the model.

2. The total number of drivers of type 1 and 2 number 24, which is the number of

employees of the railway company. The remaining 2 employees are belonging to

the type of extra drivers.

3. The next point of inference is given by the figure below. It gives the total

requirement of drivers at all increments of work, the number of drivers available for

the same and also the number of excess drivers present.

Numerically stating, there is a requirement of 367 increments of work which

require drivers in a day. The optimal solution of the problem obtained indicates an

availability of 400, which are 33 more than that required. This means that there are

a proportion of drivers who are non - utilized during the course of the working day.

It is given by the following relation:

Start End t

06:30 07:00 1 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 12 2

11:00 11:30 10 10 12 2

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 14 4

14:00 14:30 16 10 14 4

14:30 15:00 17 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 10 10 0

16:00 16:30 20 10 11 1

16:30 17:00 21 10 11 1

17:00 17:30 22 12 12 0

17:30 18:00 23 12 12 0

Excess of

Drivers

Perioddt

Total

Drivers Start End t

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 12 12 0

20:00 20:30 28 11 11 0

20:30 21:00 29 8 9 1

21:00 21:30 30 8 8 0

21:30 22:00 31 8 8 0

22:00 22:30 32 8 9 1

22:30 23:00 33 7 7 0

23:00 23:30 34 6 8 2

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 - 0 3 3

02:30 03:00 - 0 3 3

03:00 03:30 - 0 2 2

03:30 04:00 - 0 0 0

04:00 04:30 - 0 0 0

04:30 05:00 - 0 0 0

∑ 367 400 33

Excess of

Driversdt

Total

Drivers

Period

Table 6.2: Comprehensive Analysis – Driver Scheduling



4. The results are also used to cross analyze with the rostering planned, done in

chapter 3, to generate a possible schedule for the train drivers. The representation is

as shown below:

1

2

3

4

5

6

7

8

9

10

11

12

R

R

R

R

Legend

Full Time Driver number x

Split Shift Driver number y

Extra Driver number z

11h0000h00 01h00 02h00 03h00 04h00 05h00 06h00 07h00 08h00 09h00 10h00 23h0012h00 13h00 14h00 15h00 16h00 17h00 24h0018h00 19h00 20h00 21h00 22h00

1

2

3

4

5

6

7

8

9

10

11

21

13

14

15

16

17

18

2221

23

24

252

6

12

11

12

19

20

11

11

12

12

13

14

22

21

22

22

18

15

19

20

20

2121

2222

23

24

25

26

22

25

26

y

x

z

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

14h0000h00 01h00 02h00 03h00 04h00 05h00 06h00 07h00 08h00 09h00 10h00 11h00 12h00 13h00 21h00 22h00 23h00 24h0015h00 16h00 17h00 18h00 19h00 20h00

Figure 6.1 – Train Driver Scheduling

Figure 6.2 – Requirement of drivers Vs Train drivers scheduled



From the figures above, the rate of non - utilization of drivers is also evident, as is

the split of the drivers according to the type.

6.2 SENSITIVITY ANALYSIS

Sensitivity Analysis is the study of how the variation (uncertainty) in the output of a

mathematical model can be attributed to different variations in the inputs of the

model. Put another way, it is a technique for systematically changing the variables in

a model to determine the effects of such changes.

In the current study undertaken, sensitivity analysis is done on two parameters that

are under the influence of the railway company:

Sensitivity Analysis on the number of drivers who are employees of the company

(N).

Sensitivity Analysis on the wages of the type – “extra drivers” (w3).

6.2.1 Sensitivity Analysis on N

Through this process, it is desired to determine the influence of the employee

dimensions of the company on the output, which in this case is the total wage spent

on the drivers. It also is an indicator on the rate of non-utilized drivers. The

Sensitivity Analysis on N would thus indicate the optimal dimension of the work

force of the company and also indicate the corresponding levels of efficiency with

each value of N. It is to be noted that the constraint on the number of employees

belonging to the company is imposed to be equal to N rather than it being lesser than

or equal to N, as is the case during the model formulation. The observations of the

sensitivity analysis obtained, are as shown below:

N

22 23 24 25 26

W 1728 1712 1696 1696 1728

Rate of non - utilized Drivers 8,3% 8,3% 8,3% 8,3% 11,8%

x1t 20 20 20 19 22

x2t 2 3 4 6 4

x3t 6 4 2 0 0

N 22 23 24 25 26

From the table above, it can be seen that the output, which in this case is the total

wage is the highest (1728 €) when N=22 and N=26. It is the lowest (1696 €) when

N=24 and N=25. In case of the rate of non-utilized workers, it is the highest (11.8%)

Table 6.3: Sensitivity Analysis on N



at N=26 and the lowest (8.3%) for all other values of N. The split of drivers based on

the various types are also shown. It is of importance to note here that N=x1t + x2t.

As a conclusion, it can be seen that the company will minimize the total wage paid to

the drivers, by employing N=24 or N=25 employees. Thus, the company currently

works under the ideal size, in accordance with the problem description stated earlier.

6.2.2 Sensitivity Analysis on w3

Through this process, it is desired to understand the influence of the variation in the

wages of the extra time drivers to the output of the railway company, which in this

case is the total wage paid to the drivers. The variation on w3 is more likely to occur

since it involves outsourcing of drivers and thus has a better scope for obtaining a

1728 1712

1696 1696

1728 8,3% 8,3%

8,3% 8,3%

11,8%

0,0%

3,0%

6,0%

9,0%

12,0%

15,0%

1600

1640

1680

1720

1760

1800

22 23 24 25 26

€/d

ay

N

Sensitivity Analysis on N

W

Rate ofnon -utilizedDrivers

0

5

10

15

20

25

30

22 23 24 25 26

Tra

in D

rive

rs

N

Sensitivity Analysis on N - Type of train drivers

x1t

x2t

x3t

N

Figure 6.4: Sensitivity Analysis on N - type of train drivers

Figure 6.3: Sensitivity Analysis on N



good bargain. It also gives an idea of the work force dimensions of the company and

the corresponding split of drives based on their types. The observations are as shown

under:

w3 [€/day]

80% 85% 90% 95% 100%

W 1664 1681,6 1686,4 1691,2 1696

Rate of non - utilized Drivers 8,3% 8,3% 8,3% 8,3% 8,3%

x1t 20 20 20 20 20

x2t 0 4 4 4 4

x3t 10 2 2 2 2

N 20 24 24 24 24

From the table above, it can be seen that the output, which in this case is the total

wage is the highest (1696 €) when the wage is as present now. It is the lowest

(1664 €) when the revised wage is 80% the existing wage. The split of drivers based

on the various types are also shown. It is of importance to note here that N=x1t + x2t.

1664 1681,6 1686,4 1691,2 1696

8,3% 8,3% 8,3% 8,3% 8,3%

0,0%

3,0%

6,0%

9,0%

12,0%

15,0%

1600

1640

1680

1720

1760

1800

80% 85% 90% 95% 100%

€/d

ay

% change in w3

Sensitivity Analysis on w3

W

Rate ofnon -utilizedDrivers

Figure 6.5: Sensitivity Analysis on w3

Table 6.4: Sensitivity Analysis on w3



As expected, if the wage of the extra driver is lower, it is possible to obtain a better

outcome in terms of the total wage paid to the drivers. If the wages of the extra driver

are reduced to 80% of that is prevailing at the present, the size of the railway

company can be reduced from the existing 24 to the revised strength of 20

employees.

0

5

10

15

20

25

30

80% 85% 90% 95% 100%

Nr.

Tra

in D

rive

rs

% change in w3

Sensitivity Analysis on w3 - Type of train drivers

x1t

x2t

x3t

N

Figure 6.6: Sensitivity Analysis on w3 - type of train drivers



7. CONCLUSION

The objectives proposed during the commencement of the study have been achieved. The

total wage paid by the railway company towards the drivers is minimized.

Some of the simplifications employed during the current study may not be relevant to

what is the standard observed on a global scale. For instance, the relief points are assumed

to be either of the line terminuses or any intermediate point along the line where the

drivers change their duty. Another aspect is the non-accounting for the break/rest periods

for the full time drivers, which are not bound to be in line with the common practices

adopted.

So, as a suggestion for improving the model, some of the above given aspects could be

taken into account which will work in the betterment of the accuracy of this study. Even

with the simplifications adopted, the mathematical model used in this study could be

applied to any real world situation of mass transit planning, even though this study has

been restricted to Linha Verde of Metropolitano de Lisboa.

8. REFERENCES

1. 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.

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.



APPENDIX

1. Timetable and Rostering Generation



2. XPRESS – IVE Model

model "Crew_Schedule"

uses "mmxprs","mmive"

declarations

Period = 1..39 !Number of Time Periods

TypeDriver = 1..3 !Type of Train Driver

D: array(Period) of real !Demand

w: array(TypeDriver) of real !Wage for type of driver

x: array(TypeDriver, Period) of mpvar !Decisions Variables

end-declarations

D:: [ 6, 6, 12, 12, 12, 12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12,

11, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 4]

w:: [ 64, 80, 45.6 ]

N:=24 !Number of drivers who are employed of the company

!Constraints

!Demand

D(1)<=x(1,1)+x(2,1)+x(3,1) !t=1

D(2)<=x(1,2)+x(1,1)+x(2,2)+x(2,1)+x(3,2)+x(3,1) !t=2 D(3)<=x(1,3)+x(1,2)+x(1,1)+x(2,3)+x(2,2)+x(2,1)+x(3,3)+x(3,2)+x(3,1) !t=3 D(4)<=x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,4)+x(3,3)+x(3,2)+x(3,1) !t=4 D(5)<=x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,5)+x(3,4)+x(3,3)+x(3,2)+x(3,1) !t=5 D(6)<=x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,6)+x(3,5)+x(3,4)+x(3,3)+x(3,2)+x(3,1) !t=6 D(7)<=x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,7)+x(3,6)+x(3,5)+x(3,4)+x(3,3)+x(3,2)+x(3,1) !t=7 D(8)<=x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,8)+x(3,7)+x(3,6)+x(3,5)+x(3,4)+x(3,3)+x(3,2)+x(3,1) !t=8 D(9)<=x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(3,9)+x(3,8)+x(3,7)+x(3,6)+x(3,5)+x(3,4)+x(3,3)+x(3,2) !t=9 D(10)<=x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(3,10)+x(3,9)+x(3,8)+x(3,7)+x(3,6)+x(3,5)+x(3,4)+x(3,3) !t=10



D(11)<=x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(3,11)+x(3,10)+x(3,9)+x(3,8)+x(3,7)+x(3,6)+x(3,5)+x(3,4) !t=11 D(12)<=x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(3,12)+x(3,11)+x(3,10)+x(3,9)+x(3,8)+x(3,7)+x(3,6)+x(3,5) !t=12 D(13)<=x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,1)+x(3,13)+x(3,12)+x(3,11)+x(3,10)+x(3,9)+x(3,8)+x(3,7)+x(3,6) !t=13 D(14)<=x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,2)+x(2,1)+x(3,14)+x(3,13)+x(3,12)+x(3,11)+x(3,10)+x(3,9)+x(3,8)+x(3,7) !t=14 D(15)<=x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,3)+x(2,2)+x(2,1)+x(3,15)+x(3,14)+x(3,13)+x(3,12)+x(3,11)+x(3,10)+x(3,9)+x(3,8) !t=15 D(16)<=x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(1,1)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,16)+x(3,15)+x(3,14)+x(3,13)+x(3,12)+x(3,11)+x(3,10)+x(3,9) !t=16 D(17)<=x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(1,2)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,17)+x(3,16)+x(3,15)+x(3,14)+x(3,13)+x(3,12)+x(3,11)+x(3,10) !t=17 D(18)<=x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(1,3)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,18)+x(3,17)+x(3,16)+x(3,15)+x(3,14)+x(3,13)+x(3,12)+x(3,11) !t=18 D(19)<=x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(1,4)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,19)+x(3,18)+x(3,17)+x(3,16)+x(3,15)+x(3,14)+x(3,13)+x(3,12) !t=19 D(20)<=x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(1,5)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(2,1)+x(3,20)+x(3,19)+x(3,18)+x(3,17)+x(3,16)+x(3,15)+x(3,14)+x(3,13) !t=20 D(21)<=x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(1,6)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(2,2)+x(3,21)+x(3,20)+x(3,19)+x(3,18)+x(3,17)+x(3,16)+x(3,15)+x(3,14) !t=21



D(22)<=x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(1,7)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(2,3)+x(3,22)+x(3,21)+x(3,20)+x(3,19)+x(3,18)+x(3,17)+x(3,16)+x(3,15) !t=22 D(23)<=x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(1,8)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(2,4)+x(3,23)+x(3,22)+x(3,21)+x(3,20)+x(3,19)+x(3,18)+x(3,17)+x(3,16) !t=23 D(24)<=x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(1,9)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(2,5)+x(3,24)+x(3,23)+x(3,22)+x(3,21)+x(3,20)+x(3,19)+x(3,18)+x(3,17) !t=24 D(25)<=x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(1,10)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(2,6)+x(3,25)+x(3,24)+x(3,23)+x(3,22)+x(3,21)+x(3,20)+x(3,19)+x(3,18) !t=25 D(26)<=x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(1,11)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(2,7)+x(3,26)+x(3,25)+x(3,24)+x(3,23)+x(3,22)+x(3,21)+x(3,20)+x(3,19) !t=26 D(27)<=x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(1,12)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(2,8)+x(3,27)+x(3,26)+x(3,25)+x(3,24)+x(3,23)+x(3,22)+x(3,21)+x(3,20) !t=27 D(28)<=x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(1,13)+x(2,28)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(2,9)+x(3,28)+x(3,27)+x(3,26)+x(3,25)+x(3,24)+x(3,23)+x(3,22)+x(3,21) !t=28 D(29)<=x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(1,14)+x(2,29)+x(2,28)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(2,10)+x(3,29)+x(3,28)+x(3,27)+x(3,26)+x(3,25)+x(3,24)+x(3,23)+x(3,22) !t=29 D(30)<=x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(1,15)+x(2,30)+x(2,29)+x(2,28)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(2,11)+x(3,30)+x(3,29)+x(3,28)+x(3,27)+x(3,26)+x(3,25)+x(3,24)+x(3,23) !t=30



D(31)<=x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(1,16)+x(2,31)+x(2,30)+x(2,29)+x(2,28)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(2,12)+x(3,31)+x(3,30)+x(3,29)+x(3,28)+x(3,27)+x(3,26)+x(3,25)+x(3,24) !t=31 D(32)<=x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(1,17)+x(2,32)+x(2,31)+x(2,30)+x(2,29)+x(2,28)+x(2,27)+x(2,26)+x(2,25)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(2,13)+x(3,32)+x(3,31)+x(3,30)+x(3,29)+x(3,28)+x(3,27)+x(3,26)+x(3,25) !t=32 D(33)<=x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(1,18)+x(2,33)+x(2,32)+x(2,31)+x(2,30)+x(2,29)+x(2,28)+x(2,27)+x(2,26)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(2,14)+x(3,33)+x(3,32)+x(3,31)+x(3,30)+x(3,29)+x(3,28)+x(3,27)+x(3,26) !t=33 D(34)<=x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(1,19)+x(2,34)+x(2,33)+x(2,32)+x(2,31)+x(2,30)+x(2,29)+x(2,28)+x(2,27)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(2,15)+x(3,34)+x(3,33)+x(3,32)+x(3,31)+x(3,30)+x(3,29)+x(3,28)+x(3,27) !t=34 D(35)<=x(1,35)+x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(1,20)+x(2,35)+x(2,34)+x(2,33)+x(2,32)+x(2,31)+x(2,30)+x(2,29)+x(2,28)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(2,16)+x(3,35)+x(3,34)+x(3,33)+x(3,32)+x(3,31)+x(3,30)+x(3,29)+x(3,28) !t=35 D(36)<=x(1,36)+x(1,35)+x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(1,21)+x(2,36)+x(2,35)+x(2,34)+x(2,33)+x(2,32)+x(2,31)+x(2,30)+x(2,29)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(2,17)+x(3,36)+x(3,35)+x(3,34)+x(3,33)+x(3,32)+x(3,31)+x(3,30)+x(3,29) !t=36 D(37)<=x(1,37)+x(1,36)+x(1,35)+x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(1,22)+x(2,37)+x(2,36)+x(2,35)+x(2,34)+x(2,33)+x(2,32)+x(2,31)+x(2,30)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(2,18)+x(3,37)+x(3,36)+x(3,35)+x(3,34)+x(3,33)+x(3,32)+x(3,31)+x(3,30) !t=37 D(38)<=x(1,38)+x(1,37)+x(1,36)+x(1,35)+x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(1,23)+x(2,38)+x(2,37)+x(2,36)+x(2,35)+x(2,34)+x(2,33)+x(2,32)+x(2,31)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(2,19)+x(3,38)+x(3,37)+x(3,36)+x(3,35)+x(3,34)+x(3,33)+x(3,32)+x(3,31) !t=38 D(39)<=x(1,39)+x(1,38)+x(1,37)+x(1,36)+x(1,35)+x(1,34)+x(1,33)+x(1,32)+x(1,31)+x(1,30)+x(1,29)+x(1,28)+x(1,27)+x(1,26)+x(1,25)+x(1,24)+x(2,39)+x(2,38)+x(2,37)+x(2,36)+x(2,35)+x(2,34)+x(2,33)+x(2,32)+x(2,27)+x(2,26)+x(2,25)+x(2,24)+x(2,23)+x(2,22)+x(2,21)+x(2,20)+x(3,39)+x(3,38)+x(3,37)+x(3,36)+x(3,35)+x(3,34)+x(3,33)+x(3,32) !t=39

!Number of drivers who are employed of the company

sum(t in Period) x(1, t)+ sum(t in Period) x(2, t)<= N



!Objective function: Minimize total wages cost

Wage:= sum(i in TypeDriver, t in Period) w(i)*x(i,t)

minimize(Wage)

end-model



3. Sensitivity Analysis on N – Results

Start End t

06:30 07:00 1 4 2 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 6 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 10 0

11:00 11:30 10 10 10 0

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 10 10 0

13:00 13:30 14 10 10 0

13:30 14:00 15 10 10 0

14:00 14:30 16 10 10 0

14:30 15:00 17 4 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 6 10 10 0

16:00 16:30 20 10 10 0

16:30 17:00 21 10 10 0

17:00 17:30 22 2 12 12 0

17:30 18:00 23 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 12 12 0

20:00 20:30 28 11 12 1

20:30 21:00 29 8 12 4

21:00 21:30 30 8 10 2

21:30 22:00 31 8 10 2

22:00 22:30 32 8 10 2

22:30 23:00 33 1 7 7 0

23:00 23:30 34 6 9 3

23:30 00:00 35 3 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 6 2

02:00 02:30 40 0 6 6

02:30 03:00 41 0 5 5

03:00 03:30 42 0 3 3

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1728

Excess of

Drivers

N=22

Periodx1t x2t x3t dt

Total

Drivers



Start End t

06:30 07:00 1 5 1 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 5 1 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 11 1

11:00 11:30 10 10 11 1

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 10 10 0

13:00 13:30 14 10 10 0

13:30 14:00 15 10 11 1

14:00 14:30 16 10 11 1

14:30 15:00 17 4 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 5 10 10 0

16:00 16:30 20 10 10 0

16:30 17:00 21 1 10 11 1

17:00 17:30 22 1 12 12 0

17:30 18:00 23 1 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 12 12 0

20:00 20:30 28 11 12 1

20:30 21:00 29 8 11 3

21:00 21:30 30 8 10 2

21:30 22:00 31 8 10 2

22:00 22:30 32 8 10 2

22:30 23:00 33 7 7 0

23:00 23:30 34 6 8 2

23:30 00:00 35 3 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 5 1

02:00 02:30 40 0 5 5

02:30 03:00 41 0 4 4

03:00 03:30 42 0 3 3

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1712

N=23


Total

Drivers

Excess of

Drivers



Start End t

06:30 07:00 1 6 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 4 2 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 12 2

11:00 11:30 10 10 12 2

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 10 10 0

13:00 13:30 14 10 10 0

13:30 14:00 15 10 12 2

14:00 14:30 16 10 12 2

14:30 15:00 17 4 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 3 1 10 10 0

16:00 16:30 20 10 10 0

16:30 17:00 21 1 10 11 1

17:00 17:30 22 2 12 13 1

17:30 18:00 23 1 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 1 12 12 0

20:00 20:30 28 11 12 1

20:30 21:00 29 8 11 3

21:00 21:30 30 8 9 1

21:30 22:00 31 8 10 2

22:00 22:30 32 8 10 2

22:30 23:00 33 7 7 0

23:00 23:30 34 6 9 3

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 4 4

02:30 03:00 41 0 3 3

03:00 03:30 42 0 1 1

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1696

Excess of

Drivers

N=25


Total

Drivers



Start End t

06:30 07:00 1 6 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 6 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 12 2

11:00 11:30 10 10 12 2

11:30 12:00 11 10 12 2

12:00 12:30 12 10 12 2

12:30 13:00 13 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 12 2

14:00 14:30 16 10 12 2

14:30 15:00 17 5 10 11 1

15:00 15:30 18 10 11 1

15:30 16:00 19 3 2 10 10 0

16:00 16:30 20 10 10 0

16:30 17:00 21 10 10 0

17:00 17:30 22 2 12 12 0

17:30 18:00 23 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 2 12 12 0

20:00 20:30 28 11 12 1

20:30 21:00 29 8 12 4

21:00 21:30 30 8 10 2

21:30 22:00 31 8 12 4

22:00 22:30 32 8 12 4

22:30 23:00 33 7 7 0

23:00 23:30 34 6 9 3

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 4 4

02:30 03:00 41 0 4 4

03:00 03:30 42 0 2 2

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 416 49

W 1728

N=26


Total

Drivers

Excess of

Drivers



4. Sensitivity Analysis on w3 – Results

Start End t

06:30 07:00 1 6 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 4 2 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 4 10 10 0

11:00 11:30 10 10 10 0

11:30 12:00 11 2 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 2 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 12 2

14:00 14:30 16 10 12 2

14:30 15:00 17 10 12 2

15:00 15:30 18 10 12 2

15:30 16:00 19 2 10 10 0

16:00 16:30 20 1 10 11 1

16:30 17:00 21 1 10 12 2

17:00 17:30 22 12 12 0

17:30 18:00 23 2 12 14 2

18:00 18:30 24 12 14 2

18:30 19:00 25 2 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 2 12 12 0

20:00 20:30 28 11 11 0

20:30 21:00 29 8 8 0

21:00 21:30 30 8 8 0

21:30 22:00 31 8 8 0

22:00 22:30 32 8 8 0

22:30 23:00 33 7 8 1

23:00 23:30 34 6 8 2

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 4 4

02:30 03:00 41 0 2 2

03:00 03:30 42 0 2 2

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1664

80% w3


Total

Drivers

Excess of

Drivers



Start End t

06:30 07:00 1 4 2 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 4 2 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 10 0

11:00 11:30 10 2 10 12 2

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 2 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 14 4

14:00 14:30 16 10 14 4

14:30 15:00 17 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 4 10 10 0

16:00 16:30 20 1 10 11 1

16:30 17:00 21 10 11 1

17:00 17:30 22 1 12 12 0

17:30 18:00 23 2 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 2 12 12 0

19:30 20:00 27 12 12 0

20:00 20:30 28 11 11 0

20:30 21:00 29 8 9 1

21:00 21:30 30 8 8 0

21:30 22:00 31 8 8 0

22:00 22:30 32 8 9 1

22:30 23:00 33 7 9 2

23:00 23:30 34 6 10 4

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 3 3

02:30 03:00 41 0 3 3

03:00 03:30 42 0 0 0

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1681,6

85% w3


Total

Drivers

Excess of

Drivers



Start End t

06:30 07:00 1 4 2 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 6 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 10 0

11:00 11:30 10 10 10 0

11:30 12:00 11 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 2 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 12 2

14:00 14:30 16 10 12 2

14:30 15:00 17 2 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 4 2 10 10 0

16:00 16:30 20 1 10 11 1

16:30 17:00 21 10 11 1

17:00 17:30 22 1 12 12 0

17:30 18:00 23 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 2 12 12 0

20:00 20:30 28 11 11 0

20:30 21:00 29 8 9 1

21:00 21:30 30 8 8 0

21:30 22:00 31 8 10 2

22:00 22:30 32 8 11 3

22:30 23:00 33 7 9 2

23:00 23:30 34 6 10 4

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 3 3

02:30 03:00 41 0 3 3

03:00 03:30 42 0 2 2

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1686,4

90% w3


Total

Drivers

Excess of

Drivers



Start End t

06:30 07:00 1 4 2 6 6 0

07:00 07:30 2 6 6 0

07:30 08:00 3 4 2 12 12 0

08:00 08:30 4 12 12 0

08:30 09:00 5 12 12 0

09:00 09:30 6 12 12 0

09:30 10:00 7 11 12 1

10:00 10:30 8 10 12 2

10:30 11:00 9 10 10 0

11:00 11:30 10 10 10 0

11:30 12:00 11 2 10 10 0

12:00 12:30 12 10 10 0

12:30 13:00 13 2 10 12 2

13:00 13:30 14 10 12 2

13:30 14:00 15 10 14 4

14:00 14:30 16 10 14 4

14:30 15:00 17 10 10 0

15:00 15:30 18 10 10 0

15:30 16:00 19 4 10 10 0

16:00 16:30 20 1 10 11 1

16:30 17:00 21 10 11 1

17:00 17:30 22 1 12 12 0

17:30 18:00 23 2 12 12 0

18:00 18:30 24 12 12 0

18:30 19:00 25 12 12 0

19:00 19:30 26 12 12 0

19:30 20:00 27 2 12 12 0

20:00 20:30 28 11 11 0

20:30 21:00 29 8 9 1

21:00 21:30 30 8 8 0

21:30 22:00 31 8 8 0

22:00 22:30 32 8 9 1

22:30 23:00 33 7 9 2

23:00 23:30 34 6 10 4

23:30 00:00 35 6 6 0

00:00 00:30 36 6 6 0

00:30 01:00 37 6 6 0

01:00 01:30 38 6 6 0

01:30 02:00 39 4 4 0

02:00 02:30 40 0 3 3

02:30 03:00 41 0 3 3

03:00 03:30 42 0 2 2

03:30 04:00 43 0 0 0

04:00 04:30 44 0 0 0

04:30 05:00 45 0 0 0

SUM 367 400 33

W 1691,2

95% w3


Total

Drivers

Excess of

Drivers

study on the optimal crew scheduling for the green line of metropolitano … · 2012-08-04 · of...

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