Nurse Rostering with Soft Constraints
Evidence from Chilean Mid-size Health Care Centers
Ricardo Soto
1,2
, Broderick Crawford
1,3
, Rodrigo Bertrand
1
and Eric Monfroy
4
1
Pontificia Universidad Cat´olica de Valpara´ıso, Valpara´ıso, Chile
2
Universidad Aut´onoma de Chile, Santiago, Chile
3
Universidad Finis Terrae, Santiago, Chile
4
CNRS, LINA, Universit´e de Nantes, Nantes, France
Keywords:
Rostering, Constraint Programming, Heuristic Search.
Abstract:
Nurse rostering deals with the shifts arrangements of nursing staff in the daily operation of health care centers.
The design of suitable rosters for nurses is known to be particularly complex due to the number of interre-
lated requirements that must be considered. The literature reports a wide list of works devoted to solve such
a problem. The techniques used range from classic methods such as linear programming to more modern in-
complete methods such as evolutionary computing. However, most works are centered on the performance of
techniques for solving well-known instances of nurse rostering. In this paper, we focus on a real case study of
nurse rostering. The solution is devoted to a set of mid-size Chilean hospitals that use a very uncommon shift
pattern due to proper country legal regulations. We present a new model involving hard and soft constraints
that can be applied generically to any chilean health care center.
1 INTRODUCTION
Nurse rostering consists in producing a schedule of
shift assignments for a given period of time satisfy-
ing a set of constraints. The problem is known to be
challenging mainly because the number of constraints
that must be satisfied. Some constraint involved are
related to the hospital management policies, govern-
ment regulations, minimum area allocation, different
required nurse skills, as well as the fairness among
nurses. Different commercial and generic solutions
have been proposed to tackle this problem. However
they usually fail to satisfy the given requirements due
to the wide variety of constraints from one scenario to
another. Health centers therefore try to design tailored
solutions in order to satisfy their specific policies and
operation mechanisms.
This paper extends previous work (Soto et al.,
2013) done on the study of nurse rostering for mid-
size chilean health care centers. We take as sample
three clinics and four hospital that collect the main
features of mid-size chilean health care centers:
• Size: 30∼50 nurses, 5∼10 chief nurses, and re-
ceive 1000∼2000 patients per month.
• Organization: Urgency, Medical Center, Operat-
ing Rooms, Intensive-care Unit, and Hospitaliza-
tion.
• Shift pattern: the “fourth shift”is used, which is a
very uncommon working shift system.
• The main administrative procedures are computa-
tionally handled. However, nurse rostering is usu-
ally done manually.
Nurse rostering has been largely studied from
the seventies. Classic and exact methods such as
linear, integer, and goal programming were early
used to solve such a problem (Semet et al., 1998;
Miller et al., 1976; Warner and Prawda, 1972).
From the nineties, metaheuristics have taken an im-
portant place in the rostering research. Different
solutions have been proposed, some examples use
tabu search (Dowsland, 1998; Burke et al., 1999),
simulated annealing (Thomson, 1996), genetic al-
gorithms (Aickelin and Dowsland, 2001; Aicke-
lin and Dowsland, 2004; Bai et al., 2010; Maen-
hout and M.Vanhoucke, 2011), scatter search (Burke
et al., 2010), electromagnetic metaheuristics (Maen-
hout and Vanhoucke, 2007), and variable neighbor-
hood search (Burke et al., 2008; L¨u and Hao, 2012).
430
Soto R., Crawford B., Bertrand R. and Monfroy E..
Nurse Rostering with Soft Constraints - Evidence from Chilean Mid-size Health Care Centers.
DOI: 10.5220/0004417404300435
In Proceedings of the 15th International Conference on Enterprise Information Systems (ICEIS-2013), pages 430-435
ISBN: 978-989-8565-59-4
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

In the context of complete search, constraint program-
ming has been also strongly involved in nurse roster-
ing, some examples can be seen in (Abdennadher and
Schlenker, 1999; Bourdais et al., 2003; Pizarro et al.,
2011) and (M´etivier et al., 2009). Hybrids includ-
ing constraint programming and local search, variable
neighborhood search, and tabu search can be seen re-
spectively in (Qu and He, 2009; Qu and He, 2008),
and (Li et al., 2003).
As illustrated, a considerable number of relevant
studies can be found in the topic. However most work
is focused on the efficient solving of well-known in-
stances, and the work devoted to real cases of nurse
rostering is quite limited. This paper presents the
study of a real case of nurse rostering in mid-size
chilean health care centers. The problem is modeled
as a Constraint Satisfaction Problem (CSP) involving
soft and hard constraints and solved with state-of-the-
art Constraint Programming (CP) techniques via the
JaCoP solver. We believe that this work will be use-
ful to increase the experience in real cases of nurse
rostering as well as to future NRP researchers.
This paper is structured as follows: Section 2
gives an overview of CP and related solving tech-
niques. A real case of nurse rostering based on the
study of mid-size health care centers is described and
modeled in Section 3. The experiments are presented
in Section 4. Finally, we conclude and give some di-
rections for future work.
2 CONSTRAINT
PROGRAMMING
Constraint Programming (CP) is a programming
paradigm devoted to the efficient solving of constraint
satisfaction and optimization problems. The main
features of this paradigm are inherited from differ-
ent and relevant domains of computer science such as
operational research, artificial intelligence, and pro-
gramming languages. In CP, a problem is formally
stated as a Constraint Satisfaction Problem (CSP),
which is defined by a triple N = hX, D,Ci:
• X is a finite sequence of integer variables X =
(x
1
,... ,x
n
).
• D is the corresponding set of domains for X, that
is, D = D(x
1
) × . .. × D(x
n
), where D(x
i
) ⊂ Z is
the finite set of values that variable x
i
can take.
• C is a set of constraints C = {c
1
,... ,c
e
}, where
variables in X(c
j
) are in X.
A solution to a CSP is an assignment
{x
1
→ a
1
,... ,x
n
→ a
n
} such that a
i
∈ d
i
for
i = 1,... ,n and the set C is satisfied.
2.1 CSP Solving
In CP, a CSP is commonly solved by using a forward
checking algorithm that can be seen as a combina-
tion of a backtracking procedure with a filtering tech-
nique. The backtracking algorithm is responsible for
exploring the combinatorial space while the filtering
technique attempt to reduce it by deleting those val-
ues that do not lead to any solution. This is carried
out in a process called constraint propagation where
a local consistency is enforced. A local consistency
is indeed a property that the problem must satisfy (a
detailed explanation about constraint propagation can
be found in (C. Bessi`ere, 2006) ).
Algorithm 1.
Input: C , D
1 While ¬success or failure do
2 Variable
Selection(D)
3 Value Selection(D)
4 Propagate
C
(D)
5 If empty
domain in future var
6 Shallow
Backtrack()
7 If empty domain in current var
8 Backtrack()
9 End While
Algorithm 1 illustrates a classic procedure for
solving CSPs. It receives as input the set of con-
straints C and the sequence of domains D of the prob-
lem. Then, a while loop encloses a set of actions to
be performed until a solution is reached or no solu-
tion is encountered. The first action to be performed
is to select a variable and a value from its domain.
This selection allows to build the potential solution to
be verified whether is feasible. At line 4, the propa-
gation procedure is triggered, which tries to filter the
domains. Finally, two procedures are responsible for
bactracking. The backtracking come back to the most
recent viable state, and the shallow backtrack tries the
following remaing value from the domain of the vari-
able currently verified.
3 THE NRP MODEL
Mid-size health care centers in Chile employs a par-
ticular type of shift assignment called the “fourth
shift” system. This system is quite uncommon do not
NurseRosteringwithSoftConstraints-EvidencefromChileanMid-sizeHealthCareCenters
431

fitting with the common 8 hours-shift system used in
most hospital and research papers. The “fourth shift”
system considers two shifts of 12 hours per day as de-
noted in the following:
• Day Shift (D): starts at 8:00 AM and ends at 8:00
PM.
• Night Shift (N): starts at 8:00 PM and ends at 8:00
AM.
The “fourth shift” is used for nurses and
paramedics, for chief nurses a different system applies
to be presented in section 3.3. The complete set of
health care centers uses almost the same constraints,
obviously the preferences may vary from one center
to another but this can be adapted with a slight mod-
ification on the preference constraints. In the follow-
ing, we illustrate the constraints for nurses and chief
nurses. The constraints for paramedics are omitted
since they are analogous to the ones of nurses.
3.1 Constraints for Nurses
The “fourth shift” considers 12 hours shifts ordered in
the following sequence:
Day 1: D, Day 2: N, Day 3: Off, Day 4: Off
where D corresponds to the day shift and N to the night
shift. We consider a complete planning of 28 days
(see Figure 1) to cover the lowest common multiple
of 4 (the number of shifts within a cycle) and 7 (the
number of week days). Then, taking into account a
set of nurses {1,..., nurses}, a set of days {1,...,28},
and a set of variables V
i, j
∈ {0,1, 2}, where 0 denotes
day off, 1 denotes day shift, and 2 denotes night shift,
the constraints are modeled as follows:
• A day shift must be followed by a night shift
which in turn is followed by an off day, for i ∈
{1,... ,nurses} ∧ j ∈ {1,. ..,27}
(V
i, j
= 1 ⇒ V
i, j+1
= 2)∧
(V
i, j
= 2 ⇒ V
i, j+1
= 0)
• A night shift must be followed by two off days,
which in turn must be followed by a day shift for
i ∈ {1,. ..,nurses} ∧ j ∈ {1,. ..,26}
(V
i, j
= 2) ⇒ (V
i, j+1
= 0∧V
i, j+2
= 0)∧
(V
i, j
= 0∧V
i, j+1
= 0) ⇒ (V
i, j+2
= 1)
Now, in order to guarantee the required nurse al-
location and off nurses per day the
occurrences
global constraint is included. The global constraint
occurrences (Value,Vars,N) ensures that Value oc-
curs N times in the Vars list. Let V
j
denote the set
of variables {V
1, j
,... ,V
nurses, j
} for j ∈ {1,. ..,28}.
Then, the required constraints are as follows:
• nurses/2 nurses must be off per day
occurrences (0,V
j
,nurses/2)
• nurses/4 day shifts per day
occurrences (1,V
j
,nurses/4)
• nurses/4 night shifts per day
occurrences (2,V
j
,nurses/4)
3.2 Soft Constraints
Soft constraints allow to model the general prefer-
ences of staff nurses. For instance, in the studied
health-care centers, there exist nurses holding a se-
nior position able to suggest some preferences on
the schedule. As common CP solvers do note pro-
vide primitives to directly handle soft constraints, we
model the soft constraints by using reified constraints.
A reified constraint
reified (c
k
,B
k
) assigns 1 ( true )
to the variable B
k
if the constraint c
k
is satisfied; and
it assigns
0 ( false ) otherwise. Then, the idea is to
maximize the sum of B
k
variables as follows, assum-
ing k ∈ {1, ..., p}, where p is the number of prefer-
ences.
maximize
∑
p
k=1
B
k
Then, the constraint satisfaction problem is natu-
rally transformed into an optimization problem. As
an example, let us consider three preferences:
• c
1
: Nurse 5 prefers not to be off on day 1
• c
2
: 2 and 4 prefer to start with a day shift
• c
3
: 11 prefers to be off the third weekend of the
cycle
Those constraints can be modeled as follows:
• c
1
:V
5,1
6= 0
• c
2
:V
2,1
= 1 ∧ V
4,1
= 1
• c
3
:V
11,20
= 0 ∧ V
11,21
= 0
Then, we impose three reified constraints and the
corresponding objective function.
reified (c
1
,B
1
)∧
reified (c
2
,B
2
)∧
reified (c
3
,B
3
)
maximize
∑
3
k=1
B
k
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432

1 2 3 ... 26 27 28
1 0 0 1 2 0 0 ... 1 2 0 0 1 2
2 0 0 1 2 0 0 ... 1 2 0 0 1 2
1 2 0 0 1 2 ... 0 0 1 2 0 0
.
.
.
.
.
.
0 1 2 0 0 1 ... 2 0 0 1 2 0
nurses 2 0 0 1 2 0 ... 0 0 1 2 0 0
1 2 0 0 1 2 0 ... 0 0 1 2 0 0
2 2 0 0 1 2 0 ... 0 0 1 2 0 0
1 2 0 0 1 2 ... 1 2 0 0 1 2
.
.
.
.
.
.
1 2 0 0 1 2 ... 1 2 0 0 1 2
chief nurses 1 1 0 2 0 2 ... 1 1 0 2 0 2
Figure 1: Example of roster generation.
3.3 Chief Nurses Constraints
Most of health care centers need also chief nurses,
which are responsible for management and coordina-
tion of specific nurse tasks. However, chief nurses
usually do not follow the same pattern that regu-
lar nurses. They must take either 4 D per week or
3 N per week. Then in order to balance the to-
tal amount of nurses per day, for a chief nurse i ∈
{1,... ,chief nurses} and a day j ∈ {1,. ..,28} the
following set of constraints are required.
• Let X
j
denote the set of variables
{X
1, j
,..., X
chief
nurses, j
} for j ∈ {1, ..., 28}.
If the occurrence of N is less than D for a
determined day the chief nurse take a night shift.
Then, for j ∈ {1, ...,28}:
(
occurrences (1,V
j
,N)∧
occurrences (2,V
j
,D) ∧ (N < D)
∧
occurrences (2,X
j
,S)) ⇒ (S = D− N)
• If the occurrence of D is less than N for a deter-
mined day, the chief nurse take a day shift.
( occurrences (1,V
j
,N) ∧
occurrences (2,V
j
,D) ∧ (N > D)
∧
occurrences (1,X
j
,S)) ⇒ (S = N − D)
• Four day shifts or three night shifts per week, and
two days off, for j ∈ {1,... ,days}
(
occurrences (1,X
j
,4) ∨
occurrences (2,X
j
,3)) ∧
occurrences (0,X
j
,2)
4 EXPERIMENTS
In this section we describe the experimental evalua-
tion of the presented approach. We have tested 10 in-
stances of the problem by considering different num-
ber of nurses (see Table 1). Experiments have been
performed on a 2.0 Ghz Intel Core2Duo T7250 with
2GB RAM computer running Windows 7 and the pre-
sented model has been implemented in the JaCoP
solver. For each instance we provide the following
information from left to right.
• The number of nurses of the instance.
• The number of constraints of the instance.
• The number of visited nodes of the search tree
during the solving process.
• The number of decision taken during the search-
ing.
• The maximum depth of the tree reached.
• The maximum depth of the tree reached.
• The solving time.
The results illustrate an explosion of indicators
when the number of nurses increases (see Fig. 2, 3, 4,
and 5). The visited nodes, decisions, and solving time
exhibit an obvious exponential growth as the problem
belongs to the NP-Hard class. Despite of this expo-
nential growth, the proposed solution satisfies the re-
quirement of mid-size health care centers. Rosters
are automatically generated in a reasonable amount
of time (about 22 minutes for 40 nurses) instead of
a manual generation as usually done in this kind of
health centers.
NurseRosteringwithSoftConstraints-EvidencefromChileanMid-sizeHealthCareCenters
433

Table 1: Statistics of solving process.
Nurses Constraints Nodes Decisions Depth Solving time (ms)
4 433 12 11 7 31
8 729 34 29 14 32
12 1065 86 61 22 47
16 1381 306 177 31 94
20 1697 1422 741 40 343
24 2013 7378 3725 49 1623
28 2329 39734 19909 58 8299
32 2645 217186 108641 67 47471
36 2961 1196942 598525 76 243361
40 3277 6635010 3317565 85 1365273
Figure 2: Visited nodes during the solving process of differ-
ent instances of the problem.
Figure 3: Decisions taken during the solving process of dif-
ferent instances of the problem.
5 CONCLUSIONS
AND FUTURE WORK
In this paper, we have addressed a study of nurse ros-
tering for a set of chilean mid-size health care cen-
ters. We have presented a model that handles a very
uncommon shift pattern and the nurse preferences via
soft constraints. Soft constraints have been modeled
by using reified constraints in conjunction with an
objetive function in order to maximize the number
of preferences satisfied. In this way, the model be-
comes an optimization problem that has been solved
Figure 4: Solving times for different instances of the prob-
lem.
Figure 5: Depth of the tree reached during the solving pro-
cess of different instances of the problem.
in the JaCoP solver. We lllustrated experimental re-
sults where the optimizer is able to solve the problem
in a reasonable amount of time (about 22 minutes con-
sidering 40 nurses).
The solution introduced here is ongoing work,
and it can clearly be extended by considering big-
ger health care centers and more complex soft con-
straints. Another interesting research direction to pur-
sue is about the integration of autonomous search in
the solving process, which in many cases has demon-
strated excellent results (Crawford et al., 2013; Mon-
froy et al., 2013; Crawford et al., 2012; Soto et al.,
2012; Crawford et al., 2011b; Crawford et al., 2011c;
Crawford et al., 2011b; Crawford et al., 2011a).
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