PLANNING BUS DRIVER ROSTERS

Marta Mesquita

, Margarida Moz

, Ana Paias

and Margarida Pato

CIO and ISA-UTL, Tapada da Ajuda, 1349-017 Lisboa, Portugal

CIO and ISEG-UTL, Rua do Quelhas 6, 1200-781 Lisboa, Portugal

CIO and DEIO-FCUL, Bloco C6, Piso 4, 1749-016 Lisboa, Portugal

Keywords: Bus driver rostering, Mixed integer linear programming, Branch-and-bound techniques.

Abstract: This paper proposes a methodology for planning bus driver rosters with days off patterns in public transit

companies. The problem is modeled as a mixed integer linear programming problem which is solved with

special devised branch-and-bound techniques by a standard MILP solver. The new methodology was tested

on instances of two companies operating in Portugal. Two types of days off rules giving rise to rosters with

specific days off patterns are compared. The computational experiment shows promising results which

suggest that the proposed framework can be used as a tool to evaluate and discuss different days off patterns

within public transit companies.

1 INTRODUCTION

In urban public transit companies, bus driver

rostering is the problem of assigning drivers to

vehicle schedules, while satisfying labor law,

contracts and internal regulations. A vehicle

schedule is the sequence of timetabled trips to be

performed by a vehicle during a day. Scheduling for

both the vehicles and the drivers must be defined for

a given time horizon. The solution quality of these

problems has a great impact on transit companies’

operating costs. For example, according to the

“Annual report & accounts 2009” of CARRIS SA,

the main public transit company in Lisbon,

expenditure on staff is 62.5% of operating costs

which, in turn, is 71% of current costs. Hence, this is

an area where savings are urgent.

Due to the computational complexity of these

problems, they are usually solved separately, on a

sequential basis. First, a vehicle scheduling problem

is solved for each day of the time horizon building

the daily schedules for the vehicles that cover the

demand for urban transport. Then, also for each day,

crew duties are defined to cover vehicle schedules,

satisfying daily labor constraints – the crew

scheduling problem. Afterwards, using this

information, a rostering problem is solved to assign

the anonymous daily crew duties to specific

company drivers, thus defining their sequences of

work days and days off for the whole time horizon.

Different mathematical formulations and solution

approaches have been proposed for the rostering

problem in several transport contexts. An extensive

survey may be found in (Ernst, Jiang,

Krishnamoorthy, Nott and Sier 2004). Multilayer

network models have been proposed in (Carraresi

and Gallo, 2004) and (Moz, Respício and Pato,

2009) for bus driver rostering, (Aringhieri and

Cordone, 2004) for refuse collection staff and

(Cappanera and Gallo, 2004) for air crews. Set

covering/partitioning models have been considered

by Catanas and Paixão (1995) for bus driver

rostering and Freling, Lentink and Wagelmans

(2004) for railway and air crews. Recently, Hartog,

Huisman, Abbink, and Kroon (2009) presented an

assignment model with additional constraints and

developed a decision support system for crew

rostering at NS (Netherlands railways) and Nurmi,

Kyngäs, and Post (2011) proposed a population-

based local search algorithm to schedule drivers in a

Finnish bus transit company.

This paper proposes a mathematical formulation

and a computational framework to solve rostering

problems with days off patterns in public transit

companies. The paper is organized as follows: in the

next section we give some definitions and notation;

in section 3, we present a mathematical formulation

for the problem along with a brief description of the

solution approach; computational results concerning

two different roster patterns are reported and

415

Mesquita M., Moz M., Paias A. and Pato M..

PLANNING BUS DRIVER ROSTERS.

DOI: 10.5220/0003757504150420

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 415-420

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

discussed in section 4; finally, in section 5, some

conclusions are drawn.

2 DEFINITIONS AND NOTATION

The bus driver rostering problem, DRP, consists of

assigning a set of M drivers to daily crew duties that

operate the vehicles during a given planning horizon

H. In this paper, we consider a planning horizon of 7

weeks, 49 days. A crew duty is a daily working

period that respects labor law, union contracts and

internal rules of the company such as

maximum/minimum spread (time elapsed between

the beginning and end of a crew duty), maximum

working time without a break, break duration.

The sequence of crew duties and days off, one

per day, assigned to a particular driver during the

planning horizon is called a line of work. The set of

lines of work, covering all crew duties, assigned to

the drivers of the company is the roster. A roster

must satisfy a set of constraints related with labor

union contracts as well as internal rules of the

company. These constraints concern the minimum

number of days off per week, specific days off per

week, minimum number of Sundays off in the

planning horizon, minimum number of consecutive

days off, maximum number of consecutive

workdays and minimum number of rest hours

between consecutive crew duties.

Different policies may be followed in a

company, or in different companies, to build the

roster. Some groups of drivers are scheduled in a

cyclic basis so that all drivers in a group are

assigned to the same type of work and rest periods.

In order to be able to perform all crew duties within

cyclic rostering, drivers in the same group usually

share the same characteristics, namely seniority,

same bus and route knowledge. In this paper we deal

with a group of drivers whose contracts allow more

flexibility on the rosters. These drivers work

according to a pre-defined days off pattern where

they get the same type of rest periods but not

necessarily the same type of crew duties.

Each days off pattern is a priori defined and

includes one, or more than one, days off schedules.

Each days off schedule is a template for a line of

work that fixes the days off and the working days to

be filled with crew duties. Each days off schedule

satisfies, a priori, a subset of the above mentioned

constraints: the minimum number of days off per

week is 1; the minimum number of Sundays off in

the planning horizon is 2; the minimum number of

consecutive days off is 2; the maximum number of

consecutive workdays is 6. The remaining

constraints will be explicitly considered in the

mathematical model presented in the next section.

The 0-1 matrix in Table 1 gives an example of a

days off pattern where each day off is denoted by 0

and each workday by 1. This pattern covers a

planning horizon of 7 weeks and includes 7 days off

schedules. Schematically, each days off schedule

starts in row 1 of any column, a Monday, and

consists of 7 consecutive columns being the last day

(Sunday) of column i followed by the first day

(Monday) of column i+1. Note that, column 1

follows column 7. For example, a driver assigned to

the days off schedule that starts with column 6

works and rests according to columns 6, 7, 1, 2, 3, 4

and 5 during weeks 1, 2, 3, 4, 5, 6 and 7,

respectively.

Table 1: Example of days off pattern.

1 2 3 4 5 6 7

Mon 0 1 1 1 1 1 0

Tue 0 0 1 1 1 1 1

Wed 1 0 0 1 1 1 1

Thu 1 1 0 0 1 1 1

Fri 1 1 1 0 0 1 1

Sat 1 1 1 1 0 0 1

Sun 1 1 1 1 0 0 1

The days off pattern described in Table 1 is

followed by a group of drivers from a public transit

company in the city of Lisbon. According to it, all

drivers have 4 consecutive rest periods of 2 days off

and 2 consecutive rest periods of 3 days off which

include Saturday and Sunday. Moreover, during the

planning horizon all drivers rest two Mondays, two

Tuesdays,.., and two Sundays. Consequently, all

drivers share the same type of rest periods and days

off, and a roster built with these schedules has a

cyclic nature in what concerns the days off.

In the next sections we present a methodology to

solve the rostering bus driver problem with pre-

defined days off pattern. As an additional tool, the

underlying computational framework may be used to

compare rosters built under different days off

patterns regarding the rostering problem objectives:

minimizing the number of drivers assigned to work

and evenly distribute the workload among the

drivers during the planning horizon.

3 MATHEMATICAL MODEL

Each daily crew duty has to be assigned to a driver

that works according to one of the schedules

included in the days off pattern in use. Let S be the

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

416

set of days off schedules and let

L be the set of

crew duties to be performed on day

Hh∈ .

According to the crew duty starting time,

L is

partitioned into

, set of early crew duties starting

before 3:30 p.m. and

, set of late crew duties

starting after 3:30 p.m. According to the crew duty

spread

L is partitioned into

, set of short duties

with a maximum spread of 5 hours (without lunch);

, set of normal duties with spread

[]

9,5∈

hours

and

, set of long duties with spread

]]

75.10,9∈

hours (with overtime).

The DRP can be formulated as an

assignment/covering problem with additional

constraints, as stated in (Mesquita, Moz, Paias and

Pato, 2011) for the integrated vehicle-crew-roster

problem.

The mathematical model includes three types of

decision variables. Let

, if driver m performs

crew duty

on day h, or 0 otherwise. Let

, if

driver m is assigned to schedule s, or 0 otherwise. A

cost

, related with driver m salary, is associated

with variables

. The objective function is devised

to minimize the number of drivers assigned to work

as well as to evenly distribute the workload among

the drivers. That is, the undesirable types of daily

crew duties - short and long - must be equitably

partitioned among the lines of work assigned to

drivers. To balance the workload we define a third

type of decision variables

and

, which

represent, respectively, the maximum number of

short and long crew duties assigned to a driver

during H. Penalties

and

are associated with

and

, respectively.

The DRP can be stated as the following MILP:

Min

OOTT

MmSs

ηληλ

∑∑

∈∈

(1)

HhLy

∈∈=

∑

∈

,,1 l

(2)

Mmx

∈≤

∑

∈

(3)

HhMmxay

∈∈≤

∑

−

∑

∈

,,0

(4)

}1{,,1

)1(

−∈∈≤

∑

−

∈

−

∈

HhMmyy

(5)

}1{,,1

)1(

−∈∈≤

∑

−

∈

−

∈

HhMmyy

(6)

},{,,0 OTtMmy

∈∈≤−

∑∑

∈

(7)

HhMmLy

hmh

∈∈∈∈ ,,},1,0{ l

(8)

MmSsx

∈∈∈ ,},1,0{

(9)

0, ≥

(10)

Constraints (2) guarantee that each crew duty is

assigned to one and only one driver. Constraints (3)

ensure that a driver is assigned to one schedule of set

S or is available for other services in the company.

Inequalities (4), where parameter

if h is a

workday on schedule s or

if h is a day off on

schedule s, link variables

and

. These

inequalities establish that a driver assigned to a duty

ℓ on day h works according to a schedule s where h

is a workday. Constraints (5) prevent undesirable

sequences of crew duties in which, on consecutive

days, a driver performs an early duty followed by a

late duty. Constraints (6) ensure the implementation

of labor laws with regard to minimum rest periods

between consecutive working days. That is, (6)

forbids the assignment of a driver to an early duty in

the day after he performed a late duty. Both (5) and

(6) impose a day off period between different crew

duty types. Inequalities (7) define the variables

and

which determine the maximum number of

short/long duties assigned to a driver.

Branch-and-bound techniques are used to obtain

optimal/near optimal solutions for DRP. Due to the

combinatorial nature of the problem only small real

instances can be solved directly with a software

package. To reduce the size of the instances under

resolution different branching strategies combined

with variable fixing have been tested and compared.

The mathematical model (1) to (10) includes two

sets of integer decision variables. Variables x define

the assignment of drivers to the days off schedules

thus establishing, for each driver, the sequence of

rest periods. Variables y define the assignment of

drivers to the crew duties thus determining the

sequence of crew duties that each driver has to

perform along the planning horizon. These two sets

of integer variables suggest different branching

strategies according to the subset of variables

branching dichotomy is based on. The first

branching rule (R1) looks to the linear programming

relaxation solution and fixes to 1 decision variables

MmSsx

∈∈∀> ,,75.0

and decision variables

MmHhLy

hmh

∈∈∈∀> ,,,999.0 l

. The second

PLANNING BUS DRIVER ROSTERS

417

rule (R2) performs branching over the subset of

variables

MmHhLy

hmh

∈∈∈∀ ,,, l

, fixing to 1

variables

85.0>

at the end of the root node.

The third rule (R3) performs branching over the two

subsets of variables

MmHhLy

hmh

∈∈∈∀ ,,, l

and

Mmx

∈∀

∑

∈

, fixing to 1 variables

95.0>

at the end of the root node.

Computational results are shown in the next

section.

4 COMPUTATIONAL RESULTS

We have compared two days off patterns denoted by

PI and PII. Pattern PI was described in section 2,

Table 1. Pattern PII, described in Table 2, includes

two different sets of days off schedules. One is the

set of days off schedules defined by PI. The other set

includes a single schedule, denoted by s

, containing

7 rest periods, which always occur on Saturday and

Sunday. That is, during the 7 weeks of the planning

horizon, a driver assigned to s

will have his rest

periods always on Saturday and Sunday. PII arises to

counterbalance the lower demand, in what concerns

the number of crew duties to cover during weekends.

Within pattern PII, the group of drivers is partitioned

into two sub-groups: one sub-group will work

according to the days off schedules, s

,…,s

, defined

by PI and the other sub-group according to s

Pattern PII is not a cyclic pattern since drivers

assigned to different subgroups no longer have the

same type of rest periods.

Table 2: Days off pattern PII.

1 2 3 4 5 6 7 s

Mon 0 1 1 1 1 1 0 1

Tue 0 0 1 1 1 1 1 1

Wed 1 0 0 1 1 1 1 1

Thu 1 1 0 0 1 1 1 1

Fri 1 1 1 0 0 1 1 1

Sat 1 1 1 1 0 0 1 0

Sun 1 1 1 1 0 0 1 0

The algorithms were coded in C++ and the

programs ran on a PC Pentium IV 3.2 GHz. Branch-

and-bound schemes were tackled with CPLEX 11.0.

We have considered 8 instances: 5 instances,

denoted by L1,…,L5, derived from a bus company

operating in the city of Lisbon and 3 instances,

denoted by P1, P2, P3, derived from a bus company

operating in the city of OPorto. Characteristics of the

test instances are described in Table 3. Columns 2

and 3 show the number of short/normal/long daily

crew duties to be covered, respectively, from

Monday to Friday and on weekend days. Column 4

refers to the total number of (short/normal/long)

crew duties to be assigned during the planning

horizon.

Table 3: Data set description.

Daily crew duties Total crew duties

Mon-Fri Sat-Sun Mon-Sun

T / N / O T / N / O T / N / O

L1 1 / 8 / 8 4 / 3 / 2 91 / 322 / 308

L2 4 / 14 / 20 7 / 6 / 6 238 / 574 / 784

L3 0 / 9 / 30 1 / 4 / 11 14 / 371 / 1204

L4 0 / 8 / 26 0 / 6 / 9 0 / 364 / 1036

L5 5 / 13 / 36 11 / 8 / 8 329 / 567 / 1372

P1 3 / 9 / 16 6 / 6 / 4 189 / 399 / 616

P2 3 / 28 / 12 10 / 8 / 8 245 / 1092 / 532

P3 0 / 9 / 30 7 / 4 / 12 98 / 371 / 1218

Parameters

and

in (1) were set to

5.0=

and

, as long crew duties are more

undesirable than short crew duties since contain

overtime.

A time limit of 10800 seconds has been imposed

as stopping criterion for solving MILP problems.

Computational experiments on the different

branching strategies have shown that rule (R3)

yielded the best branch-and-bound results

concerning either CPU time and solution quality. On

the one hand, (R1) proved to be ineffective for

solving medium/large size instances due to

excessive CPU times. On the other hand, both (R2)

and (R3) led to feasible solutions within the time

limit but (R3) gave a better solution in 11 out of 16

instances. As for the branch-and-bound CPU time,

Table 4 shows, for each instance, the corresponding

time, in seconds, for both (R2)/(R3) and PI/PII,

excluding root node CPU. The last two rows present,

respectively, the average time and the number of

times branch-and-bound stopped due to the time

limit.

Results reported in Table 4 strengthen the choice

of (R3) to compare the quality of PI and PII

solutions.

Table 4: Comparing (R2) and (R3) – CPU time.

PI PII

(R2) (R3) (R2) (R3)

L1 30.4 46.9 10800 10800

L2 15.3 70.8 10800 528.6

L3 0.5 15.0 10800 571.6

L4 0.1 3.1 10800 82.5

L5 107.4 786.9 10800 10800

P1 343.1 254.0 10800 1095.3

P2 8225.9 10800 10800 10800

P3 2352.2 355.1 10800 575.0

av 1384.4 1541.5 10800 4406.6

#tl 0 1 8 3

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

418

Table 5 presents computational results for PI and

PII. Column 3 shows the number of drivers assigned

to the best roster attained, while in brackets the

number of drivers assigned to s

is reported. Column

4 presents the maximum number of short/long crew

duties assigned to a driver during the planning

horizon. The last column displays total CPU times,

in seconds, spent by the enhanced branch-and-bound

algorithm (LP relaxation CPU + MILP CPU).

Table 5: Computational results for PI vs PII.

# driv #T / #O CPU (sec)

PI 25 4 / 14 106

PII 24 (8) 4 / 15 10805

PI 54 6 / 16 85

PII 47 (18) 9 / 20 575

PI 55 1 / 22 22

PII 47 (22) 1 / 26 599

PI 49 0 / 22 18

PII 40 (17) 0 / 35 93

PI 77 5 / 18 828

PII 68 (26) 8 / 23 10924

PI 40 14 / 17 288

PII 36 (12) 7 / 18 1112

PI 61 5 / 9 10838

PII 57 (17) 13 / 13 10853

PI 59 4 / 25 394

PII 50 (15) 4 / 28 628

From Table 5, we can see that with PII one can

save on the number of drivers assigned to work. In

fact, it is important to know the minimum workforce

needed to operate the fleet of vehicles in order to

have a pool of drivers available to replace those

absent or to be assigned to other services in the

company. Column 4 gives an idea of the roster

quality concerning the worst scenario of short/long

duties assigned to a driver in the planning horizon.

Columns 3 and 4 show that the reduction on the

number of drivers with PII comes with an increase in

the maximum number of long crew duties assigned

to a driver. This was expectable since the number of

long crew duties is the same in PI and PII but in PII

is divided by fewer drivers.

Although there is no guarantee of obtaining an

optimal solution to DRP, one can see that the

number of long crew duties is fairly distributed

among de drivers. For example, concerning instance

P1, a total of 616 long duties (table 3) must be

assigned to 40 drivers under pattern PI and to 36

drivers under PII (table 5) which yields on average

15.4 and 17.1 for PI and PII, respectively. The

solution provided by our methodology leads to a

maximum of 17 and 18 long duties assigned to a

driver, respectively, for PI and PII.

The last column shows that CPU times spent to

solve the tested instances are quite reasonable.

5 CONCLUSIONS

Expenditures on staff have a great impact on

operating costs of public transit companies. One of

the main objectives of the rostering problem is to

minimize such costs. However, it is important that

the rosters comply with driver preferences. Some

preferences concerning rest periods can be drawn a

priori through the days off pattern. This paper

proposes a computational framework that, given a

pre-defined days off pattern, builds the bus driver

roster for a planning horizon. The methodology has

been tested over two sets of real instances derived

from bus companies and proved to be an effective

tool for building and comparing rosters under

different days off patterns.

ACKNOWLEDGEMENTS

This work is supported by National Funding from

FCT - Fundação para a Ciência e a Tecnologia,

under the project: PEst-OE/MAT/UI0152.

The authors thank to CARRIS, SA for the helpful

discussions.

REFERENCES

Aringhieri, R., Cordone, R., 2004. The multicommodity

multilevel bottleneck assignment problem. Electronic

Notes in Discrete Mathematics, 17, pp 35-40.

Catanas, F., Paixão, J. M. P., 1995. A new approach for

the crew rostering problem. In Daduna, J., Branco, I.

M., Paixão J. M. P. (eds.), Computer-Aided

Scheduling of Public Transport. Lecture Notes in

Economics and Mathematical Systems, 430. Springer,

-Verlag, Berlin, pp 267-277.

CPLEX Manual (version 11.0), 2007. Using the CPLEX

Callable Library and CPLEX Mixed Integer Library.

ILOG INC., Incline Village, Nevada, USA.

Cappanera, P., Gallo, G., 2004. A multicommodity flow

approach to the crew rostering problem. Operations

Research, 52(4), pp 583-596.

Carraresi, P., Gallo, G., 1984. A multi-level bottleneck

assignment approach to the bus drivers rostering

problem. European Journal of Operational Research,

16, pp 163-173.

Ernst, A., Jiang, H., Krishnamoorthy, M., Nott, H., Sier,

D., 2001. An Integrated Optimization Model for Train

Crew Management. Annals of Operations Research,

108, pp 211-224

Freling, R., Lentink, R., Wagelmans, A., 2004. A decision

support system for crew planning in passenger

transportation using a flexible branch-and-price

algorithm. Annals of Operations Research, 127(1-4),

pp 203-222.

PLANNING BUS DRIVER ROSTERS

419

Hartog, A., Huisman, D., Abbink, E. J. W., Kroon, L. G.,

2006. Decision support for crew rostering at NS.

Public Transport 1(2), pp 121-133.

Mesquita, M., Moz, M., Paias, A., Pato, M., 2011. An

integrated vehicle-crew-roster problem with days off

pattern. In Hu, B., Morasch, K., Pickl, S., Siegle, M.,

(eds.), Operations Research Proceedings 2010,

Selected papers of the Annual International

Conference of the German Operations Research

Society. Springer-Verlag, Berlin, pp 245-250.

Moz, M., Respício, A., Pato, M. V., 2009. Bi-objective

evolutionary heuristics for bus driver rostering. Public

Transport, 1(3), pp 189-210.

Nurmi, K., Kyngäs, J., Post, G., 2011. Driver Rostering

for Bus Transit Companies. Engineering Letters,

IAENG, Hong Kong, 19(2), pp 125-132,

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

420