Combining Heuristic and Utility Function for Fair Train Crew
Rostering
Ademir Aparecido Constantino
1
, Candido Ferreira Xavier de Mendonça
2
,
Antonio Galvão Novaes
3
and Allainclair Flausino dos Santos
1
1
Department of Computer Science, State University of Maringá, Maringá, PR, Brazil
2
School of Arts, Science and Humanities, State University of São Paulo, São Paulo, SP, Brazil
3
Department of Production Engineering, Federal University of Santa Catarina, Florianópolis, SC, Brazil
Keywords: Crew Rostering Problem, Stated Preferences, Fair Rostering, Bottleneck Assignment Problem.
Abstract: In this paper we address the problem of defining a work assignment for train drivers within a monthly
planning horizon with even distribution of satisfaction based on a real-would problem. We propose an
utility function, in order to measure the individual satisfaction, and a heuristic approach to construct and
assign the rosters. In the first phase we apply stated preference methods to devise a utility function. The
second phase we apply a heuristic algorithm which constructs and assigns the rosters based on the previous
utility function. The heuristic algorithm constructs a cyclic roster in order to find out a minimum number of
train drivers required for the job. The cyclic roster generated is divided into different truncated rosters and
assigned to each driver in such way the satisfactions should be evenly distributed among all drivers as much
as possible. Computational tests are carried out using real data instance of a Brazilian railway company. Our
experiments indicated that the proposed method is feasible to reusing the discrepancies between the
individual rosters.
1 INTRODUCTION
Manpower scheduling is a well-studied NP-hard
problem in computer science and operation research
studied (Lau, 1996; Tien and Kamiyam, 1982).
Several variants of manpower scheduling have been
studied. Bodin et al., (1983) describe the state of the
art about scheduling of crews in transportation
companies. Crew rostering problem (CRP) and crew
scheduling problem (CSP) are related problem we
find in crew management in any large transportation
company (Vera Valdes, 2010; Ernst et al., 2004).
Although they are related, usually CRP and CSP are
solved sequentially, but recently we find propose of
integrated optimization model (Ernst et al. 2001;
Valdes, 2010).
Crew scheduling is related to construct of shits
for a short period of time like a day (or few days e.g.
when there are long trip). In this phase the shifts are
not yet assigned to individual crews.
Crew rostering is a second phase in crew
management in which the shifts generated during the
crew scheduling phase are sequenced together in
order to form a roster for each crew for a larger
planning horizon (typically a week or a month).
Crew scheduling is the main topic studied in the
literature, where the main focus is the cost reduction,
whilst crew rostering is more related to aspects like
quality of life than to costs (Valdes, 2010).
In this paper we are interested in a CRP which
arises in a Brazilian railway company where the
crew scheduling is not considered because usually
the trips are of long duration. A solution in a CRP
must satisfy all the related constraints over the crew
working shifts which stems from the union rules and
regulation covering all aspects over the train drivers
work.
Usually, two approaches to find solutions to the
CRP are reported in the literature (Xie, L. and Suhl,
2015; Tien and Kamiyama, 1982; Lau, 1996;
Khoong et al., 1994), one is cyclical or rotating
(Caprara et al., 1998) and the other is personal or
non-cyclical (Bianco et al., 1992). In cyclic rostering
the work patterns are generated and rotated them
among workers, therefore, theoretically all workers
do the same roster. While the no-cyclical approach
where cyclic and personal rostering are combined
for constructing schedules. Usually this last one use
personal preference and it is more flexible that
cyclic scheduling but also fairer than cyclic rostering
Aparecido Constantino A., Ferreira Xavier de Mendonça C., Galvão Novaes A. and Flausino dos Santos A..
Combining Heuristic and Utility Function for Fair Train Crew Rostering.
DOI: 10.5220/0005364505930602
In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 593-602
ISBN: 978-989-758-096-3
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

because all works’ preferences are considered
simultaneously. Non-cyclic rostering provides more
freedom to take holidays and special events into
account.
Techniques for solving the CRP are reported in
the literature which includes Mathematical
Programming (Glover and McMillan, 1986),
Constraint Logic Programming (Caprara et al.,
1998), heuristic algorithms (Carraresi and Gallo
1984; Bianco et al., 1992) and fuzzy set (El Moudani
and Mora-Camino, 2000). These techniques are
strongly dependent on the domain for which they
were written, therefore, it is an easier work to
develop a new one from scratch rather than to try to
adapt them. Much of this research concentrates on
the point of view of the enterprises minimizing their
operational costs. From point of view of the driver is
interesting to consider the workload and the
preferences. Although, this last point of view reflects
the quality of life of the drivers it have been received
less attention. Sometimes, an even distribution of
the workload among the crew involved is applied
(see Bianco et al., 1992; Carreresi and Gallo, 1984;
Jachnik, 1981; Caprara et al., 1998), but normally
the workload is only based on the time worked. In
the work of Carraresi and Gallo (1984) the drivers
assign weights to each shift; such weights are based
on hours worked and sometimes they take into
account other factors. In the work of Ernst et al.
(1996, 2000) rosters are built trying satisfying some
quality standards (referred to as quality of life) for all
drivers in a train company attempting to satisfy their
personal preferences. The worker’s resting day
preference is also refereed in the work of Lau
(1996). Recently Mühlenthaler and Wanka (2014)
applied the concept of fair distribution in the context
of course timetables, considering the distribution of
resources, costs and penalty over a set of
stakeholders.
In this paper we take into account both workload
and preference represented by a utility function.
Utility function in the context of economic
represents an individual’s or group’s degree of
satisfaction (Mansfield, 1985). The utility function is
a mathematical expression that attempts to model the
worker’s satisfaction. We apply Stated Preference
(SP) methods (Benjamin et al., 2014; Carson and
Louviere, 2011; Kroes and Sheldon, 1988; Ben-
Akiva and Leman, 1985) to devise a utility function
to address the drivers’ preferences and workload
balance. The major advantage of our approach is that
our utility function takes into account all the
following factors: kind of duty, preferred days off,
night shifts, total number of hours and overnight
hours. For the best of our knowledge this is the first
study applying this methodology in order to try to
construct fair rosters so that no worker is favored
over another.
The remaining of this paper is organized as
follows. In section 2 we give a description of the
railway company and the CRP. In section 3 we
describe an application of the stated preferences
methods to CRP for building Personal Rosters, in
this section we also describe the two modules to
solve the addressed CRP where the first module we
propose an algorithm to construct a cyclic roster and
the second module constructs the personal rosters
applying local search on the previous cyclic roster.
In section 4 we state the time complexity of the
modules used. In section 5 we discuss the
computational results over real data. Finally, we
draw the main conclusion in the Section 6.
2 CREW ROSTERING PROBLEM
DESCRIPTION
This work was developed for a Brazilian cargo
railway company which main activity is mineral
transportation. Eventually, a trip may also carry
people and other cargo. Its set of trips is changed
very often imposing to a manager the task to
construct (manually) a new roster. The manager
involved in the task had to deal with two main
problems: how to quickly build a (cyclical)
optimized roster (satisfying union regulations and
operational constraints) and how to distribute the
satisfaction (workload and preference) evenly
among the crew. In the satisfaction it had to be
considered the following factors: total number of
hours and overnight hours, type of duties and
preferred resting days. Naturally, due to the
complexity of the problem, sometimes the above
mentioned task turns to be a hard work.
We present a railway company where a crew
member always start and finish their working shifts
in the same station that is both source and
destination of a duty, this station is called the home
station (the town where the crew live); the duties
(journeys or roundtrips) are repeated every day
regardless it is a Sunday or a holiday; most of the
duties are trips, but there are two other activities
(duties): readiness (a driver stays in the company
premises awaiting for an eventual call) and shunting
(a short trip within a metropolitan area); the trip
duties are divided in two categories: short duties
(trips that take up to 8 hours) and long duties (long

trips consisting of two consecutive short trips
divided by an external rest). Our roster is a railway
station working schedule that must follow some
hard constraints (the union regulations and the
operational constraints) and should follow some soft
constraints (workload properties that improve
drivers’ safety and life style). A working day is the
period of time from 00:00 to 24:00 hours. There are
two hard constraints:
• The minimum resting time between
consecutive duties which is of 11 hours;
• It is allowed a block of at most 7 (5 is
desirable) consecutive working days
between two resting day, we refer this set of
consecutive working days as a block.
There are also two soft constraints:
• A block should have at most 3 consecutive
overnight duties;
• The resting time between blocks should be
as long as possible – our roster deals with
this property postponing the block starting
time.
Soft constraints are desirable but not necessarily
satisfied by the roster. Note that, these constraints
are referred as quality of life of the crew.
Our objective is construct personal rosters (for
each driver) over a 30 day planning horizon using a
minimum number of drivers. Furthermore, the main
concern is to balance the satisfaction among the
drivers involved. The satisfaction is measured by a
utility function (devised from stated preference
methods) which considers several factors described
in the beginning of this Section.
3 THE CRP RESOLUTION
PHASES
To find a solution to our CRP for the railway
company our project was split into the following two
phases:
1. PHASE 1: Apply the SP methods to estimate
a utility function from the crew’s preferences
in order to measure the fairness of the rosters.
2. PHASE 2: A development of an algorithm
divided into two modules: construction
algorithm and weighted distribution module.
• The construction algorithm constructs a
master roster (a cyclic roster covering
whole set of duties minimizing the size of
the crew);
• The weighted distribution module
constructs personal rosters tries to
breaking the cyclic roster constructed in
the previous module. The even
distribution of duties among the crew is
reached by applying a utility function
developed on the phase 1.
3.1 Applying the Stated Preference
Methods
The Stated Preference (SP) methods were originated
in mathematical psychology (Krantz and Tversky,
1971), developed and often used in marketing
research early in the years 70 – by that time these
methods were called “trade-off” analysis. Attention
to these methods regarding transport started late in
this decade and their popularity had grown in the
years 80. For more detail about SP methods the
reader are referred to Benjamin et al., (2014), Carson
and Louviere (2011), Hensher (1994), Koes and
Sheldon (1988), Louviere (1988) and Ben-Akiva and
Lerman (1985).
In a short, SP methods mean a family of
techniques that uses individual respondent
statements about his/her preferences over a set of
alternatives to estimate a utility function (which
describes the structure of the respondent’s
preference). The alternatives (scenarios or options)
are typically descriptions of fictitious situations or
contexts constructed by a researcher. Those
descriptions are printed in a set of cards prepared
statistically by experimental design. Subsets of these
cards are provided to respondents that are asked to
express their preferences by sorting the alternatives
in decreasing order of preferences, or by giving a
rating value for each card. The next step is to devise
the implicit utility (preference) function related to
the respondent. This function can be inferred by the
use of appropriate statistical techniques (e.g.
maximum likelihood LOGIT, non-metric regression,
Monanova). In SP methods it is usual to assume that
the utility function is a linear weighted function as
follows:
U =

=
f
n
i
ii
x
1
α
(1)
were
U = total weighted utility function;
n
f
= number of factor (variables or attributes)
considered;
x
i
= factor i;
α
i
= utility weight of factor i.

Table 1: The set of 96 scenarios broken down into 12 groups with 8 alternatives (cards) each.
Card Group
1
Group
2
Group
3
Group
4
Group
5
Group
6
Group
7
Group
8
Group
9
Group
10
Group
11
Group
12
1
0000 0021 0012 0010 0022 0001 5010 5022 5001 5020 5011 5022
2
4112 4100 4121 4101 4110 4122 6001 6010 6022 6022 6020 6011
3
1021 1012 1000 1021 1001 1010 7022 7001 7010 7011 7022 7020
4
2000 2021 2012 2010 2022 2001 0110 0122 0101 0120 0111 0102
5
5111 5100 5121 5101 5110 5122 4001 4010 4022 4002 4020 4011
6
3021 3012 3000 3022 3001 3010 1122 1101 1110 1111 1102 1120
7
6100 6121 6111 6110 6122 6101 2110 2122 2101 2120 2111 2102
8
7112 7100 7121 7101 7110 7122 3101 3110 3122 3102 3120 3111
Note that other mathematical forms of utility
functions expressing different hypothesis about the
way respondents combine their overall preferences
can be tested (Lerman and Louviere, 1978).
The first step in the design of a SP exercise is the
definitions of the factors of interest and their
corresponding levels (discrete values). Those factors
are needed to be evaluated by the respondents. In
our application we identified (chosen) four
important factors in a typical roster based on the
company requirements described in the Section 2.
The four factors are the following:
x
1
: is a mix of different duties (long trip, short
trip, readiness shunting) in a roster.
x
2
: is related to weekday with resting.
x
3
: is the perceptual of overnight hours worked in
the block.
x
4
: is the number of progressiveness violation.
We define progressiveness when a duty starts
earlier than the consecutive duty. The
objective is to avoid the non-occurrence of
the progressiveness between two consecutive
duties.
For each factor was considered different levels as
follows:
x
1
: We considered eight levels (0 to 7), each one
is a possible combination of different duties
in a block.
x
2
: two levels:
0: resting in a Sunday or holiday;
1: resting in a weekday or Saturday.
x
3
: three levels:
0: 0 to 33%;
1: 33% to 66%;
2: 66% to 100%.
x
4
: thee levels:
0: zero or one occurrence;
1: two occurrences;
2: three or four occurrences;
In our SP exercise each alternative (that will be
printed in a card) is a descriptions of a possible
block of duties over a period of consecutive working
days (each alternative is a combination of the factors
levels). Of course, the total number of all possible
combinations of factor levels is enormous.
Therefore, respondents can only evaluate a fairly
limited number of alternatives at a time which
typically falls between 9 and 16 (Kroes and Sheldon,
1988). The full factorial design (all possible
combinations) can only be used if there are very few
factors with very few levels each. In our case, a full
factorial design consists of 144 (2
3
×2
1
×3
1
×3
1
=
2
4
×3
2
) possible alternatives from which we selected
a set of 96 in the same fashion of the “fractional
factorial design (2/3)” by Mc Lean and Anderson
(1994).
To conserve space we do not reproduce here the
exercise that generated the 144 alternatives (all
possible combination of factors and levels) nether
the selection of the 96 alternatives. The set of 96
alternatives was broken down into 12 groups with 8
alternatives each (Table 1) and were printed on cards
(illustrated in Figure 1). The groups of these cards
were provided to a single respondent to sort the
alternatives in decreasing order of preferences. Due
to operational difficulty we selected the railway
company’s manager as the respondent to represent
ROSTER (Block)
START Week-Day SYMBOL TASK
00:00 MO SHT SHUNTING
22:00 TU MIN MINERAL
WE CNT continuation
10:00 TH CRG CARGO
FR CNT continuation
SA RST REST
Figure 1: An example of a card with fictitious scenario
(roster).

the preference of the group of the crews. This phase
of study consumed some weeks of work.
After applying this survey and collect the data,
we applied a maximum likelihood (LOGIT)
technique to analyses the respondent selection (Ben-
Akiva and Lerman, 1985).
After computational adjustment we detected that
the factor x
4
was not statistically significant, and
then it was removed. For the other hand, we
decomposed the factor x
1
into five factors in order to
enter explicitly some information as follows:
x
11
: number of trips with load of minerals;
x
12
: number of trips with heterogeneous load
(named cargo);
x
13
: number of shunting;
x
14
: number of readiness;
x
15
: total sum of hours worked in the block;
As the most applications of SP, in this work we
adopted the multinomial logit (MNL) model. The
results of maximum likelihood estimation of MNL
model are summarized in Table 2. All parameters
are statistically significant at 99% confidence level,
which indicates a good model fit.
With this result we have a decomposition of the
overall preferences into the following utility
weighted function:
U =α
11
x
11
+α
12
x
12
+α
13
x
13
+α
14
x
14
+α
15
x
15
+
α
2
x
2
+α
3
x
3
(2)
This function is used in section 3.3 for estimating
the level of the satisfaction of the crew and for
assigning personal rosters to drivers.
Table 2: Estimation Results of Multinomial Logit.
Weight Value t-statistic
α
11
-5.4235 -19.16
α
12
-5.9182 -15.74
α
13
-2.1471 -15.10
α
14
-3.0419 -13.46
α
15
-12.2412 -4.40
α
2
7.7781 13.12
α
3
-1.7855 -3.09
N
umber of observations 672
Lo
g
-likelihood at zero -127.255
Lo
g
-likelihood at conver
g
ence -56.947
Lo
g
-likelihood ratio (ad
j
usted ρ2 ) 0.551
3.2 The Construction Algorithm
This module builds a cyclic master roster that
determines the minimum size of the crew needed for
the job. This roster is used by the weighted
distribution module to build the personal rosters. Let
D = {a
1
, a
2
, ..., a
n
} be a set of n duties of a given day
where each duty has a start time, s
i
and end time, f
i
.
Since all duties are repeated seven days a week it is
possible to construct cyclic roster as illustrated in
Figure 2. This (cyclic) master roster consists of a
sequence of 27 duties that spans into 42 days. The
27 duties are divided into 7 blocks where each block
spans into 5 days with resting times between blocks
with at least one entire weekday. According with
this roster a crew of 42 drivers are needed to
perform the daily occurrences of a
1
, ..., a
27
as
follows. Driver number 1 performs: duty a
1
on day
x, duty a
2
on day x+1, ..., no duty on day x +41, duty
a
1
on day x+42 again, and so forth. Driver number 2
performs: duty a
2
on day x, duty a
3
on day x+1, ...,
duty a
1
on day x+ 41, duty a
2
on day x+42 again,
and so forth. Finally, driver number 42 stays off on
day x, performs duty a
1
on day x+1, ..., duty a
27
on
day x+41, stays off on day x+42 again, and so forth.
workday 1 workday 2 workday 3 workday 4 workday 5 workday 6 workday 7
a
1
a
2
a
3
a
4
a
5
a
6
a
7
a
8
a
9
a
10
a
11
a
12
a
13
a
14
a
15
a
16
a
17
a
18
a
19
a
20
a
21
a
22
a
23
a
24
a
25
a
26
a
27
X
X
X
X
X
X X
Figure 2: An example of a roster with 42 days where X means a resting day.

Before running the construction algorithm we
must set up the following parameters:
n
blocks
: the number of blocks;
start
min
: the minimum start time of the blocks;
rest
min
: the minimum resting time between two
consecutive duties;
block
max
: the maximum number of consecutive
working days;
overnight
max
: the maximum number of
consecutive overnight duties.
When the module stops mislaying a duty these
parameters must modified to run the module again
as we explain below.
Let us define w as the length of a master roster
(in days) which is equal to the size of the crew
needed to cover the daily duties. The maximum
length of a block is defined by q.
Our construction algorithm was inspired on ideas
proposed by Caprara et al., (1998) which uses
information provides by an assignment problem in
order to construct only one sequence (roster) adding
a duty at time until have no duty remaining. Our
algorithm constructs several sequences (blocks)
simultaneously adding a duty to each block in each
iteration until assign all duties. An assignment
problem (AP) is associated to a cost matrix C
n
×
n
where each c
ij
is the minimum time, in minutes,
between the start of the duty i and the start of the
duty j if they can be sequenced directly, otherwise
c
ij
= ∞ (naturally c
ii
= ∞). The AP is defined as
follows.

=
n
i
n
j
ijij
xcMin z
1=1
.=
(3)
nx
n
i
ij
1,...,=j 1 :s.t.
1
∀=

=
nx
n
j
ij
1,...,=i 1
1
∀=

=
njix
ij
1,...,=, }1,0{ ∀∈
After solving AP we have the optimal value of Z. It
is easy to note that the value (z/1440) gives a lower
bound for the length of the master roster. This value
must be an integer number of days (1 day = 1440
minutes). The minimum number of blocks in the
master roster can be easily estimated by:








=
q
z
n
blocks
1440
1
,
(4)
where q = 5 (desirable value of hard constraint).
The block construction procedure is outlined as
follows.
Initialisation: i) define a list of n
blocks
empty blocks;
ii) select an initial duty a
0
to the block
k, k=1,..., n
p
.
STEP 0: start a list with n
blocks
(empty) blocks and
select an initial duty a
0
for a block k,
k=1,..., n
blocks
.
STEP 1: For k=1 to n
blocks
do
Select the best duty that can
sequenced after the last duty of block
k;
STEP 2: If all duties have been sequenced or no
duty was selected in the STEP 1 or all
block lengths reached its maximum value
q STOP, otherwise return to STEP 1.
The follow we give a more detailed description
of the steps of the algorithm.
Choice the initial duty
The initial duty a
0
for each block is heuristically
chosen with base in the best value of a score. The
score is calculated with base in the number of
reduced costs equal to zero found in the rows and
columns relative to activity a. As larger the number
of zero reduced costs found in the row of the activity
a, larger the number of activities that can be
sequenced after the activity a without causing great
increments in the function objective from AP. A
similar interpretation can be made for the reduced
costs in the column. Since the activity will be first
duty performed at the block, we give preference to
the duty with the smallest number of costs reduced
zero in the column and a large number of zero
reduced costs in row from the duty a. Moreover,
since the initial duty should be followed by other
duties in the same block, we penalise the duties that
have no zero reduced cost in the column.
Choice de next duty
The choice of the duty j that can be sequenced
soon after the last duty i, in the block k, also follows
some similar criteria. For each candidate duty j
which can be sequenced after duty i in the block k
receive a score P(j). This score takes into an account
the increasing of the objective function z when the
sequencing of j after i is impose in the AP’s solution.
Additional weight can be impose in the score in
order to sequence the duties considered “critical”,
i.e., duties having few number of zero reduced in the
matrix’s row and column, long journey, or specific
attribute. The duty j having better score P(j) is
sequenced after duty i in the block k.
Update the AP
After each pair of duties has been consecutively

sequenced, this condition imposes a new AP’s
solution. The new AP’s solution required can be
easily update solving the AP parametrically in the
O(n
2
) time. This procedure gives us a new cost
reduced matrix.
Refining Procedure
After the solution is found, we can try to improve
it modifying the parameters and heuristic algorithm
is re-applied. This procedure can do manually,
providing to company manager a way to test
alternatives in order to take decision.
When the procedure stops leaving a duty not
sequenced, so we manually modify some of the
initial parameters (e.g. increase the number n
blocks
of
blocks, the decrease the start time of the blocks -
start
min
, etc.) and run the procedure again. In this
procedure the heuristic used for choosing the first
duty for a block (STEP 0) and the candidate duty to
follow the duties already sequenced in a block
(STEP 1) is the same outlined in Caprara et al.,
(1998). However, our approach differs from theirs
where they build the blocks sequentially
(individually) and we start all blocks together and
select one duty to each block on each iteration. This
is particularly better approach since critical duties
can be assigned earlier, because we have n
blocks
blocks instead of only one sequence as proposed by
Caprara et al. (1998).
3.3 The Weighted Distribution Module
If a cyclical roster runs for a long period of time
where all drivers follow the “same” roster, this
ensures that a balanced workload among the crew
and gives to all drivers a similar track experience.
However, sometimes an accident or cancellation of
some programmed trip generates an unbalance. In
this case, a manager tries to correct such distortion
assigning new rosters to each driver. However, the
balance of the workload cannot be guaranteed for
the reason that sometimes the measure of the
workload is based on subjective aspects over
intrinsic attributes of the roster. Furthermore,
driver’s satisfaction is subjective information related
to driver’s preference. We propose the following as
an alternative solution to correct such balancing (fair
distribution). The general idea is to break a cyclical
roster into w (number of drivers) smaller personal
rosters and make a balanced workload distribution of
these new rosters among the crew. The workload is
measured by a utility function (devised from stated
preference methods) which considers both the
workload history of each driver and the driver’s
preferences as described in Section 3.1.
Normally, the planning horizon is over a month.
In general, the length of a master roster is greater
than a month. However, we intend to create personal
rosters of a month each or less. Therefore, a master
roster is broken down into a set of w smaller
separate rosters as follows. Note that w is both the
number of days needed to perform all duties of the
master roster and the size of the crew needed to
cover the master roster.
Let M = (m
0
, m
1
, ..., m
1440w-1
) be the master roster
consisting of a sequence of integers (cyclic roster
with l440w minutes). Each m
i
, i = 0, 1, ..., 1440w-1,
is an integer field corresponding to a duty number.
When a field m
i
is set to a value between 1 and n,
say j, it means that the minute i of master roster M is
spent on duty j, otherwise it is set to 0 meaning it is
free (resting time). Let length
j
, j = 1, 2, ..., n, be an
integer value corresponding to the length of duty j in
minutes. For example, if duty j starts at the first k-th
minute of the l-th day of the master roster M, in this
case,
jm
wkl
=
−+ 1440 mod)11440(
, ...,
jm
wlenghtkl
j
=
++ 1440 mod)1440(
(duty j starts at minute
(1440l+k-1) mod 1440w and ends at minute
(1440l+k-1+length
j
) on the master roster M). Let h
be the number of days (generally a month) of the
desired planning horizon. Let S
l
, l = 1, 2, ..., w, be a
copy of the subsequence
wl
m
1440 1))mod-(1440(
,
wl
m
1440 1)mod1)-(1440( +
, ... ,
wlenghtl
j
m
1440 1)mod1)-(1440( +
and
'
1400
'
2
'
1
,...,,(
hl
mmmS =
. Let the subsequence S
l
be the l-th truncated roster
of size 1440h minutes (corresponding to h days)
starting in the first minute of day l and ending in the
last minute of day ((l+h-2) mod w)+1. When the first
minute of a truncated roster l is not free (
0
1
≠= jm )
and its length
j
-th minute is not spent on the last
minute of duty j (
jm
j
length
≠ ) we say that the
truncated roster l starts with a “broken” duty. In this
case, starting from
1
m we follow all consecutive
values equal to j in the truncated roster l, let
λ
m be
the last of these values, we unset duty j from the
truncated roster l by setting
0=
σ
m , for all
consecutive
σ =1, 2, ..., λ .
Let U(b
ϕ
), ϕ = 1, ..., n
blocks
, be the measure of the
block satisfaction value (values of the utility
function) of the block b
ϕ
belonging the master roster.
Let the truncated roster satisfaction value (for roster
S
ϕ
) be the sum of the measures U(b
ϕ
) over all blocks
b
ϕ
belonging S
l
. Clearly, truncate rosters S
1
, ..., S
w
are different, and expectedly, having different
truncate roster satisfaction values, therefore, they

must be balanced. An important aspect we consider
is the work performed in the past, called historical
roster. Let H
i
be the historical roster of the driver i,
S
j
be the truncate roster j, for i = 1, 2, ..., w and j = 1,
2, ..., w.
Now we describe the mathematical model of the
Bottleneck Assignment Problem and outline the
weighted distribution module.
The mathematical model of the Bottleneck
Assignment Problem (BAP) is formulated as
follows.
; ,...,1=, }1,0{
; ,...,1=
; ,...,1= ; 1
; ,...,1= ; 1 :s.t.
1
1
wlkx
wlycx
wkx
wlx
Min y
kl
kl
kl
w
l
kl
w
k
kl
∈
≤
=
=


=
=
(5)
where c
kl
= – (U(H
k
) + U(S
l
)) if the truncated roster
S
l
can be assigned to driver k (satisfying the union
regulations and operations constrains), otherwise c
kl
= ∞. We consider U(H
k
) and U(S
l
) to be the utility
(satisfaction) of H
k
and S
l
, respectively. The
objective this model is finding w assignments that
minimizes the maximum workload, i.e., that
maximizes the minimum satisfaction, using the
algorithm proposed by Carraresi and Galo (1984).
Generating New Truncated Rosters
Let G be a complete graph such that each vertex is a
block of the master roster. Now, to generate new
truncated rosters we solve the associated Travelling
Salesman Problem (TSP). In this problem the cost of
the Hamiltonian cycle is given by objective function
of the BAP where edges of G have no cost. Each
new Hamiltonian cycle is associated (generated) to
new truncated rosters. In this way, the first
Hamiltonian cycle is given by the order of mater
roster. A new Hamiltonian cycle is found applying
classical improvement heuristic procedures such as
2-opt and 3-opt (Lin, 1965).
So, the algorithm that distributes the truncated
rosters is presented as follows.
Distribution Algorithm:
STEP1: Solve the Bottleneck Assignment
Problem associated with S
l
, for l = 1, 2,
..., w;
STEP2: New truncated rosters are generated by
permutations of the blocks belonging to
the original master roster. Return to
STEP 1.
The two steps are repeated to all cyclical
permutations.
4 COMPLEXITY OF THE
ALGORITMS
The master roster construction in the construction
algorithm is found by solving the AP which can be
solved in time complexity O(n
3
) by any classical
Primal-Dual algorithms: Hungarian Method or
Shortest Augmenting Path method (Sorevik, 1993;
Carpaneto and Toth, 1987), we used the latter. In our
case the AP is solved parametrically every time that
a duty is sequenced where this step takes time
complexity O(n
2
) (in the worst case), this is done in
sequentially (for each duty). Therefore, the overall
time complexity of the whole step remains O(n
3
).
The personal rosters construction in the weighted
distribution module are found by solving BAP which
can be solved in time complexity O(w
2
) by an
algorithm by Carraresi and Galo (1884). A BAP is
run each time a 2-opt procedure improves the
Hamiltonian cycles where this 2-opt procedure (Lin ,
1965) takes time complexity O(n
blocks
2
), this time the
BAP is solved on each iteration increasing the time
complexity of this step to O((w n
blocks
)
2
). Therefore,
the overall time complexity is O((w n
blocks
)
2
). Note
that w > n > n
blocks
.
5 COMPUTATIONAL RESULTS
The master roster construction and personal rosters
construction defined in previous sections have been
implemented in PASCAL. The resulting code was
tested on the real-world instance provided by a
public Brazilian railway company. The master roster
solution constructed by the algorithm is always
fairly better in “quality” and in optimality than the
roster constructed manually by the company
personnel.
We compared the result with four previous cyclic
roster provided by the company. Our rosters
decreased an average of about 2% on the number of
drivers needed. The Table 3 illustrates the some
results obtained. In addition, our rosters always
respected all union regulations and all operational
constraints. However, this was not observed in
several rosters prepared by company. The
computational results were obtained in an Intel Core
i3CPU running Windows in less than a minute for a
roster.

Table 3: Computational results on real instances.
Compan
y
Al
g
orith
m
N
ame n w w
Inst1 21 43 42
Inst2 33 55 54
Inst3 33 52 51
Inst4 31 62 60
6 CONCLUSIONS
In the methodological point of view we presented a
solid new approach which combines the cyclical
approach with the individualized approach. Our
approach by the mean of a utility function
establishes a fair balance of the personal rosters
taking into account several factors regarding the
crew quality of life not considered in the literature.
Furthermore, the ideas presented here were applied
with sensible success on real data.
Our weighted distribution module based on
utility function could not be fully evaluated and
compared with real data from company because it
requires closer investigation during a long period of
time. Nevertheless, we believe that the utility
function used in this paper provides a fair way to
distribute and consider crew preferences on the
workload. Note that in this work a single utility
function was used to evaluate the workload of the
whole crew. The development of particular utility
function for each crew member seems to be a quite
interesting improvement to be investigated.
ACKNOWLEDGEMENTS
We thank CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior -
Brazilian Ministry of Education), Araucária
Foundation (Fundação Araucária do Paraná) and
CNPq (National Council for Scientific and
Technological Developmen) for the financial
support.
REFERENCES
Ben-Akiva, M. and Lerman, S. R., 1985. Discrete Choice
Analysis: Theory and Application to Travel Demand.
The Mit Press, Cambridge, Massachusetts.
Benjamin, D. J., Heffetz, O. and Kimball, M. S. 2014.
Beyond Happiness and Satisfaction: Toward Well-
Being Indices Based on Stated Preference. The
American Economic Review, 104(9), pp. 2698-2735.
Bianco, L., Bielli, M., Mingozzi, A., Ricciardelli, S. and
Spadoni, M., 1992. A Heuristic Procedure for the
Crew Rostering Problem. European Journal of
Operational Research, 58(2), 272-283.
Bodin, L., Golden, B., Assad, A. and Ball, M., 1983.
Routing and Scheduling of Vehicles and Crews.
Computers and Operations Research, 10(2), 127-147.
Caprara, A., Fischetti, M., Toth, P. and Vigo, D., 1998.
Modeling and Solving the Crew Rostering Problem.
Operatons Research, 46(6), pp. 820-830.
Caprara, A., Focacci, F., Lamma, E., Mello, P., Milano,
M., Toth, P. and Vigo, D., 1998. Integrating
Constraint Logic Programming and Operations
Research Techniques for the Crew Rostering Problem.
Software Practice and Experience, 28, 49-76.
Carpaneto G. and Toth P., 1987. ‘Primal-Dual Algorithms
for the Assignment Problem. Discrete Applied
Mathematics, 18, 137-153.
Carraresi, P. and Gallo, G., 1984. A Multi-level
Bottleneck Assignment Approach to the Bus Drivers’
Rostering Problem. European Journal of Operational
Research, 16, pp. 163-173.
Carson, R. T. and Louviere, J. J., 2011. A Common
Nomenclature for Stated Preference Elicitation
Approaches, Environmental and Resource Economics,
49 (4), pp. 539-559.
EL Moudani, W., Mora-Camino, F., 2000. A fuzzy
solution approach for the roster planning problem.
Proceeding of 9th IEEE International Conference on
Fuzzy Systems (FUZZ-IEEE 2000), San Antonio
(USA), 7-10 Mai 2000, pp. 346-351.
Ernst, A. H. Jiang, T., Krishnamoorthy, M., Nott, H. and
Sier, D., 2000. Rail crew scheduling and rostering:
Optimisation Algorithms. Proceeding of 8th
International Conference: Computer Aided Scheduling
of Public Transport, (Berlin, Germany).
Ernst, A. H. Jiang, T., Krishnamoorthy, M., Nott, H.,
Owens, B. and Sier, D., 2004. An Anotated
Bibliography of Personnel Scheduling and Rostering.
Annals of OR, 127, pp. 21-144.
Ernst, A. T., Jiang, H., Krishnamoorthy, M., Nott, H., and
Sier, D., 2001. Na Integrated Optimizations Model for
Train Crew Management. Annals of OR, 108, pp. 211-
224.
Ernst, A. T., Krishnamoorthy, M. and Dowling, D., 1996.
Train Driver Rostering using Simulated Annealing.
Technical Report. Division of Mathematics and
Statistics, CSIRO, Australia. pp. 1-14.
Glover, F. and McMilla, C., 1986. The General Employee
Scheduling Problem: An Integration of MSand AI.
Computers and Operations Research, 13(5), 563–573.
Hensher, D. A., 1994. Stated Preference Analysis of
Travel Choices: the State of Practice. Transportation,
21(2), pp. 107-133.
Jachnik, J. K., 1981. Attendance and Rostering System. In
A. Wren, editor, Computer Scheduling of Public
Transport, pp. 337-343. North-Holland Publishing Co.
Khoong, C. M., Lay, H. C. and Chew, L. W., 1994.
Automated Manpower Rostering: Techniques and

Experience. International Transaction in Operational
Research, Vol. 1, pp 353-361.
Krantz, D.H., Tversky, A.,1971. Conjoint measurement
analysis of composition rules in psychology.
Psychological Review 78(2), pp. 151–169.
Kroes, E. P. and Sheldon, R., 1988. Stated Preference
Methods: An Introduction Journal of Transport
Economics and Policy, 22(1), pp. 11-25.
Lau, H. C., 1996. Preference-Based Scheduling Via
Constraint Satisfaction. Technical Report. Information
Technology Institute. Singapore.
Lerman, S. R., and Louviere, J. J., 1978. On the Use of
Functional Measurement to Identify the Functional
Form of the Utility Expression in Travel Demand
Models. Transportation Research Record, 673, pp. 78-
86.
Lin, S., 1965. Computer Solutions of the Traveling
Salesman Problem. Bell System Technical Jornal, 44,
pp. 2245-2269.
Louviere, J. J., 1988. Conjoint Analysis Modelling of
Stated Preferences: a Review of Theory, Methods,
Recent Developments and External Validity. Journal
of Transport Economics and Policy, 22(1), pp. 93-119.
Mansfield, E., 1985. Microeconomics Theory/
Applications, 5th edn. New York: W.W. Norton.
Mc Lean, R. and Anderson, V. L., 1984. Applied
Factorial and Fractional Designs. Marcel Dekker Inc,
New York.
Mühlenthaler, M. and Wanka, R., 2014. Fairness in
Academic Course Timetabling. Annals of Operations
Research, February 14, 2014. doi:10.1007/s10479-
014-1553-2.
Sorevik, T., 1993. Average Case Complexity Analysis of
Algorithms for the Linear Assignment Problem. In
Proceedings of NOAS '93, Trondheim.
Tien, J. and Kamiyama, A., 1982. On Manpower
Scheduling Algorithms. SIAM Review, 24(3), pp. 275-
287.
Vera Valdes, V. A., 2010. Integrating Crew Scheduling
and Rostering Problems. Phd Thesis.University of
Bologna.
Xie, L. and Suhl L., 2015. Cyclic and non-cyclic crew
rostering problems in public bus transit.
OR Spectrum, 37, pp. 99-136.