A Variable Neighbourhood Search for Nurse Scheduling with Balanced

Preference Satisfaction

Ademir Aparecido Constantino

, Everton Tozzo

, Rodrigo Lankaites Pinheiro

, Dario Landa-Silva

and Wesley Romão

Informatics Department, State University of Maringá, Maringá, PR, Brazil

ASAP Research Group, School of Computer Science, University of Nottingham, Nottingham, U.K.

Keywords:

Nurse Scheduling Problem, Bottleneck Assignment Problem, Variable Neighbourhood Search, Combinatorial

Optimisation.

Abstract:

The nurse scheduling problem (NSP) is a combinatorial optimisation problem widely tackled in the litera-

ture. Recently, a new variant of this problem was proposed, called nurse scheduling problem with balanced

preference satisfaction (NSPBPS). This paper further investigates this variant of the NSP as we propose a

new algorithm to solve the problem and obtain a better balance of overall preference satisfaction. Initiall, the

algorithm converts the problem to a bottleneck assignment problem and solves it to generate an initial fea-

sible solution for the NSPBPS. Posteriorly, the algorithm applies the Variable Neighbourhood Search (VNS)

metaheuristic using two sets of search neighbourhoods in order to improve the initial solution. We empirically

assess the performance of the algorithm using the NSPLib benchmark instances and we compare our results

to other results found in the literature. The proposed VNS algorithm exhibits good performance by achieving

solutions that are fairer (in terms of preference satisfaction) for the majority of the scenarios.

1 INTRODUCTION

In this paper we consider a variant of the well-known

Nurse Scheduling Problem (NSP) referred as the

Nurse Scheduling Problem with Balanced Preference

Satisfaction (NSPBPS). The NSP is a NP-Hard com-

binatorial optimisation problem (Osogami and Imai,

2000) that consists of: given a period of time, a set of

staff requirements for the given period and a team of

nurses, the aim is to assign shift patterns for the team

of nurses in order to satisfy the staff requirements. On

the basic version of the NSP, the objective is to max-

imise the number of requirements fulﬁlled. The prob-

lem has several variants where the objectives include

the minimisation of the number of nurses used, the

minimisation of the overall assignment costs (given

that the cost of each nurse assignment can be cal-

culated), the satisfaction of nurses preferences, and

more (Petrovic and Vanden Berghe, 2008).

The NSPBPS was proposed by Constantino et al.

(2011) and it is a variant of the NSP where each nurse

can have preferences regarding her/his shift assign-

ments. In the NSP, maximising the overall number

of preferences attended may lead to some nurses be-

ing largely beneﬁted while other nurses are neglected.

The purpose of fulﬁlling nurses’ preferences is to im-

prove work conditions and raise overall staff satis-

faction. However, by providing individual timeta-

bles that are unbalanced in terms of individual pref-

erences satisfaction, provokes that some nurses feel

disfavoured when the level of satisfaction of their in-

dividual preferences is lower than for other nurses.

Therefore, the objective in the NSPBPS is to fulﬁll

staff requirements but attending individual nurse pref-

erences in a balanced manner instead of just maximis-

ing the overall number of preferences. In the NSPBPS

we aim to obtain a fair assignment by minimising the

difference between the preference satisfaction of the

most favoured nurse and the least favoured nurse.

Surveys of the NSP were provided by Cheang

et al. (2003) and Burke et al. (2004), where prob-

lem variants, features and solving methods were cov-

ered and discussed. Both works observed that at

the time there were no sets of benchmark instances

to cross-evaluate solving methodologies. Maenhout

and Vanhoucke (2005) presented a comprehensive

dataset of varied NSP instances called NSPLib. Later,

the NSPLib was further improved with the addition

462

Aparecido Constantino A., Tozzo E., Lankaites Pinheiro R., Landa-Silva D. and Romão W..

A Variable Neighbourhood Search for Nurse Scheduling with Balanced Preference Satisfaction.

DOI: 10.5220/0005364404620470

In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 462-470

ISBN: 978-989-758-096-3

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

of new features and updated results for comparison

(Maenhout and Vanhoucke, 2006, 2007, 2008).

Several heuristic techniques have been applied to

solve the NSP and its variants. Examples include

the recent works of Maenhout and Vanhoucke (2007)

which uses the Electromagnetic Method, Maenhout

and Vanhoucke (2006) using Scatter Search, several

applications of Evolutionary Algorithms (Maenhout

and Vanhoucke, 2008; Ohki et al., 2008; Tsai and Li,

2009; Beddoe et al., 2009), Oughalime et al. (2008)

using Tabu Search, Burke et al. (2008) using Variable

Neighbourhood Search, Goodman et al. (2009) which

uses GRASP and Cheng et al. (2007) using Simmu-

lated Annealing. Additionally, exact methods were

also employed to tackle the NSP. We can highlight the

works of Yilmaz (2012) using Integer Programming

and Bard and Purnomo (2005) that employs Column

Generation. Constantino et al. (2013) used a hybrid

algorithm for the NSP with preference satisfaction.

To the best of our knowledge, only Constantino

et al. (2011) presented results for the NSPBPS. In

that work they proposed a hybrid algorithm that uses

a bottleneck assignment problem exact algorithm to

construct an initial solution and a local search to im-

prove that solution. The present work further extends

that concept by using the same mechanism to generate

the initial solution. However, instead of a simple lo-

cal search, we propose the use ofa VNS metaheuristic

with a new set of neighbourhoods. Experimental re-

sults on the NSPLib showed that we managed to im-

prove the balance of preference satisfaction.

Section 2 describes the NSPBPS. In section 3, the

bottleneck assignment problem used to generate the

initial solution, is described. Section 4 presents the

proposed VNS algorithm with the new set of neigh-

bourhoods. Section 5 presents the experimental re-

sults and a discussion on the algorithm performance.

Finally we conclude our work in Section 6.

2 THE NURSE SCHEDULING

PROBLEM WITH BALANCED

PREFERENCE SATISFACTION

In this work we consider the NSP as presented by

Maenhout and Vanhoucke (2007). Nevertheless, in

the NSPBPS the objective is not to minimise the

number of constraints violations but instead to opti-

mise the balance of attended nurses’ preferences. The

problem is composed by a number of hard and soft

constraints to be considered when generating each in-

dividual schedule. The hard constraints deﬁne strict

work regulations and the soft constraints deﬁne work-

ing policies and desirable aspects.

• Hard Constraints. The NSPBPS considers two

hard constraints, both related to avoid certain shift

patterns. The ﬁrst forbids the assignment of an af-

ternoon shift before a morning shift. The second

forbids the assignment of a night shift before a

morning or an afternoon shift.

• Soft Constraints. There are four soft constraints

that represents nurses’ preferences. The ﬁrst one

deﬁnes a minimum and maximum number of

working days within the scheduling period. The

second one deﬁnes a minimum and maximum

number of consecutive working assignments. The

third one represents a minimum and maximum

number of assignments of each shift type in the

scheduling period. The fourth soft constraint sets

a minimum and maximum number of consecutive

shift assignments of the same type.

• Nurses Preferences. The preferences represent the

main objective of the NSPBPS. They are declared

in advance and then a cost (inversely proportional

to the preference) is associated to each shift.

We can deﬁne the NSPBPS as follows. Given a

set N of nurses, where N = {n

, n

, . . . , n

max

}, a set D

of days where D = {d

, d

, . . . , d

max

} and a set S of

shifts where S = {s

, s

, . . . , s

max

}. For each day d

∈

D, each nurse n

∈ N has to be assigned a shift s

∈ S

such that the hard and soft constraints are considered

and the differences of the overall individual costs for

the nurses preferences are minimised.

In this work we use the tests scenarios provided by

the NSPLib, a library with problem instances for the

NSP. For more information, please refer to the work

of Maenhout and Vanhoucke (2005). The NSPLib

uses shifts that refers to a givenworking period (early,

day or night shift) or a rest period (free shift). A duty

roster, or roster, is a sequence of |D| shifts assigned to

a nurse. We consider S

to be the set of shifts required

for day d.

Additionally,let pf

be a sum of the costs (prefer-

ences non-satisfaction) for nurse n

regarding his/her

individual roster. Thus, if we wish to optimise the

overall nurse satisfaction, we must build and assign

rosters to nurses in such way that the sum of pf

for

all n

∈ N is minimised. However, to aim for rosters

with balanced preference satisfaction, one approach

is to minimise the maximum pf

for all n

∈ N. This

objectiveis used in the well-known bottleneck assign-

ment problem as discussed next.

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463

3 THE BOTTLENECK

ASSIGNMENT PROBLEM

The NSP considered here involves the construction of

the duty roster for a number of nurses in a scheduling

period of D days. For each nurse, a shift must be as-

signed for each day in the scheduling period. In this

paper, we model the NSP as an acyclic multipartite

graph with |D|+1 partitions, where the vertices in the

ﬁrst partition corresponds to nurses and the vertices in

the remaining |D| partitions correspond to shifts, that

is, one partition per day d

. An edge represents the

possibility of assigning a shift to a nurse or shift (de-

pends on the algorithm phase).

Formally, lets have a graph G = (V, E), where V

is the set of vertices and E is the set of edges. The set

V is composed by the subsets (partitions) of vertices

, ...,V

max

, where V

is the set of vertices rep-

resenting the nurses and V

(d from d

to d

max

) is the

set of vertices representing day d of the scheduling

period.

Figure 1 presents a diagram of the problem repre-

sentation for two nurses in a seven day period. The

two leftmost (rectangular) vertices, ﬁrst on each par-

tition, are the vertices representing the nurses, respec-

tively nurses one and two. The vertices to the right

correspond to a day of the schedule and can contain

one of the following shifts: (E)arly, (D)ay, (N)ight,

and (F)ree shift. Additionally, the * symbol deﬁnes a

spare shift.

The optimisation problem is to ﬁnd n

max

paths go-

ing from the ﬁrst to the last partition while minimis-

ing the total cost (each path corresponds to an indi-

vidual roster). For this, we propose a heuristic al-

gorithm that solves successive bottleneck assignment

problems each corresponding to two consecutive par-

titions. The goal in the bottleneck assignment prob-

lem (BAP) is to make a one-to-one assignment be-

tween two sets of the same size in such way that

the maximum assignment costs is minimised. Then,

solving each successive BAP in our approach corre-

sponds to minimising the maximum cost due to the

non-satisfaction of individual preferences. The bot-

tleneck assignment problem is formulated as follows:

minimise max

1≤i, j≤n

subject to

∑

i=1

= 1, j = 1, . . . , n

∑

j=1

= 1, i = 1, . . . , n

∈ {0, 1}, i, j = 1, . . . , n

In this case, index i sometimes corresponds to a

nurse and other times to a shift while index j can be

a shift or a roster (more details are given below). The

term c

corresponds to the cost of assigning i to j.

The main advantage of tackling the NSP in this way is

that the bottleneck assignment problem can be solved

in polynomial time using the algorithm proposed by

Carpaneto and Toth (1981).

4 THE PROPOSED VNS

ALGORITHM

This work presents a two-phases algorithm to tackle

the NSPBPS. First, an initial solution is obtained by

solving a bottleneck assignment problem for each day

of the rostering period using the algorithm proposed

by Carpaneto and Toth (1981). Then, a general VNS

metaheuristic (Mladenovi

c and Hansen, 1997) is ap-

plied in order to improve this initial solution by ex-

ploring two domain-speciﬁc neighbourhoods, Cutting

and Recombination procedure and k-Swap.

4.1 Initial Solution

We now brieﬂy present the construction algorithm for

the initial solution. Note that we use the same tech-

nique as proposed by Constantino et al. (2011). For

further information, please refer to that work.

The construction phase starts by generating a mul-

tipartite graph as deﬁned in Section 3. An initial solu-

tion is obtained through the resolution of |D| succes-

sive BAPs from the ﬁrst to the last day. Each BAP is

deﬁned by the square cost matrix [c

] of order n

max

where c

is the cost of assigning shift j to nurse i on

Figure 1: An initial solution for a problem containing two nurses. The letters E, D, N, F mean Early, Day, Night and Free

shift, respectively, and * means spare shift.

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day d. This cost is deﬁned by the following function:

f(i, j, d) = pc(i, j, d) + P

× nHCV + P

× nSVC

where pc(i, j, d) is the preference cost associated to

assigning a nurse i to shift j on day d; nHCV is the

number of hard constraint violations; P

is the penalty

due to the violation of a hard constraint; nSCV is

the number of soft constraint violations and P

is the

penalty cost due to the violation of a soft constraint.

In this problem the number of nurses is always

greater than or equal to the number of demanded

tasks, i.e. n

max

≥ |S

|, then it might be necessary

to complete the cost matrix with spare shifts in or-

der to form a square matrix. Hence, matrix [c

] may

be divided into two blocks. Block I contains the shifts

that satisfy the required coverage of that day. Block

II contains the added spare shifts (in the case that in

day d the number of nurses is greater than the num-

ber of demanded tasks). Since the minimum coverage

requirement is guaranteed by block I, then any shift

assignment in block II (spare shifts) is permitted, in-

cluding the assignment of a free shift (with no given

coverage requirement).

One BAP is constructed and solved for each day

d of the scheduling period. Note that in the ﬁrst as-

signment there is no shift previously assigned to each

nurse. But from the second partition, previously as-

signed shifts are considered in order to calculate the

cost matrix. At the end, after solving the |D| BAPs,

we have constructed an initial solution (duty roster)

for the set N of nurses. The construction phase works

as follows:

Procedure Construction

begin

for d ←

to d

max

construct the cost matrix for day

;

solve the BAP for the cost matrix;

assign shifts to nurses according to

the BAP solution;

end-for

end Construction

4.2 Solution Improvement: Variable

Neighbourhood Search

The proposed algorithm implements a general Vari-

able Neighbourhood Search (VNS) as proposed by

Mladenovi

c and Hansen (1997). The VNS starts from

an initial solution and randomly picks a neighbour in

order to disturb the current solution and avoid local

optima. Then, it systematically inspects the neigh-

bourhoods around that solution in pursuance of im-

proving the current solution using a Variable Neigh-

bourhood Descent (VND) algorithm. The VND is

a simpler version of the VNS that does not possess

the stochastic factor, hence it cycles through a set of

neighbourhoods, accepting improvements, until the

solution cannot be improved.

Below we present the pseudocode of the general

VNS:

Procedure VNS

begin

normalize the input data.

s ←

initial solution;

while stop criteria not satisfied do

for i ←

to k do

′

←

choose random neighbour of

;

′′

←

VND(

′

);

if f(s

′′

) < f(s) then

s ← s

′′

;

i ← 0

;

else

i ← i+ 1

;

end-if

end-for

end-while

return s

;

end VNS

The chosen neighbourhoods plays a major role in

the performance of the algorithm. Hence, we tailored

two types of neighbourhoods in order to make bet-

ter use of the data structures selected and improve the

convergence of the algorithm to good solutions. We

now describe the proposed neighbourhoods.

4.2.1 Neighbourhood 1: Cutting and

Recombination Procedure (CRP)

The ﬁrst neighbourhood, Cutting and Recombination

Procedure, is the same neighbourhood employed in

the local search technique proposed by Constantino

et al. (2011). The procedure takes a current solution

and performs successive ’cuts’ between each pair of

consecutive days. For each cut, it recombines all ros-

ters and uses the BAP to search for the best feasible

solution.

For every d

cut

∈ (d

, d

, . . . , d

max

− 1), the proce-

dure virtually splits the duty roster in two sides, the

left, containing the roster for the days (d

, d

, . . . , d

cut

)

and the right, containing the roster for the remaining

days (d

cut

+ 1, . . . , d

max

). Now, each duty roster on

the left side can be recombined with |N| rosters on

the right side (including the original combination), re-

sulting in a total of |N|

possible recombinations. The

procedure then creates a new cost matrix [c

cut

], with

size |N| × |N|. Each value c

cut

of the cost matrix is

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465

(a) Possible recombinations for the CRP on the second cut point.

(b) New timetable after choosing a better combination.

Figure 2: Example of CRP application for two nurses.

calculated as the cost of assigning roster j to the nurse

i from the day d

cut

onwards. After the entire matrix

is calculated, a BAP is solved using [c

cut

] and the re-

sults are used to recombine the current solution.

The pseudocode for the CRP procedure is shown

below:

Procedure CRP

begin

for d

cut

←

to d

max

− 1 do

construct the cost matrix for day

cut

;

solve the BAP for the cost matrix;

recombine the rosters according to

the BAP solution;

end-for

end CRP

Figure 2 presents two diagrams with an example

of the application of the CRP procedure. The exam-

ple considers the cut between days one and two for

a two nurses scenario. In Figure 2(a) we have the

two possible recombinations – the current assignment

with solid lines and the other possibility with dashed

lines. Then, in Figure 2(b), assuming that the previ-

ous dashed lines assignment poses a better solution,

we have the new solution after effecting the changes.

4.2.2 Neighbourhoods 2. . . k: k-Swap

The following neighbourhoods are obtained through

a new procedure proposed in this work: k-Swap. The

k-Swap is inspired in the original CRP as it is also

based on cutting points and recombination. However,

the k-swap seeks to cut entire blocks of size k and

recombine these blocks, thus, each neighbourhood is

composed by the allowed recombinations of blocks of

size k.

The procedure works as follows: for every d

cut

∈

, d

, . . . , d

max

−k−1), the procedurevirtually splits

the duty roster in three parts, namely:

• left side: the roster for the days (d

, d

, . . . , d

cut

);

• k-block: the roster for the days (d

cut

1, . . . , d

cut

+ k); and

• right side the roster for the days (d

cut

+ k +

1, . . . , d

max

Now, instead of recalculating all possible recom-

binations between the block and the sides, which

would generate |N|

possible recombinations (and ar-

guably be a slow procedure), the k-Swap recombines

the block such that each recombination on the left side

is symmetric to the right side, hence for every left side

recombination, there is only one possible outcome on

the right side, generating a total of |N|

possible re-

combinations. The procedure also creates a new cost

matrix [c

cut

], with size |N| × |N|, where each value

cut

is the cost of assigning the nurse i (along with

the left roster i) to the k-block roster j and the k-block

roster j to the right roster i. After [c

cut

] is obtained,

the BAP is solved and the results are used to recom-

bine the current solution.

The pseudocode for the k-Swap procedure is

shown below:

Procedure k-Swap

begin

for d

cut

←

to d

max

− k − 1 do

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(a) Possible recombinations for the k-Swap on the second block.

(b) Improved timetable after applying the k-Swap on the second block.

Figure 3: Example of k-Swap application for two nurses using k = 3.

construct the cost matrix for the

block

cut

... d

cut

+ k

;

solve the BAP for the cost matrix;

recombine the rosters according to

the BAP solution;

end-for

end k-Swap

Figure 3 presents a sample application of the k-

Swap procedure for k = 3 to a scenario with two

nurses and seven days. In Figure 3(a) we see the de-

limitation of each of the possible four blocks. The

3-Block 2, presented in solid lines, is the one chosen

for the current step of the procedure. Still, in Fig-

ure 3(a) we can observe all possible recombinations

of rosters considering the current block, in this case,

two recombinations. The ﬁrst one, presented in solid

lines is the current assignment while the second, pre-

sented in dashed lines is the alternative. Figure 3(b)

shows the resulting roster assuming that the dashed-

lined alternative from the previous ﬁgure gives better

results.

5 EXPERIMENTAL RESULTS

AND DISCUSSION

In order to evaluate the proposed algorithm, we used

the same instances evaluated by Constantino et al.

(2011). These instances were obtained from the

NSPLib (Maenhout and Vanhoucke, 2005), a digital

repository for instances of the NSP.

There are two types of problem instances avail-

able in the NSPLib. The ﬁrst is the set of instances

that contains weekly schedules. The number of nurses

varies from 25 to 100 with an increment of 25 nurses.

For each set of nurses there are 7290 ﬁles with differ-

ent daily requirements and constraint values, hence

totalling 29160 distinct scenarios. Additionally there

are 8 separated ﬁles containing coverage constraints

and preferences information that can be combined

to any scenario, hence totalling 233280 problem in-

stances that are availablein the library. The second set

of instances considers monthly schedules (28 days)

but restricts the number of available nurses to 30 and

60. There are 960 ﬁles for each problem size and

an additional 8 ﬁles containing preferences and cov-

erage constraints, totalling 15360 possible scenarios.

Therefore, when considering all the possible scenar-

ios, we have a total of 248640 instances.

Due to time constraints we performed the tests on

weekly schedules only, however we remind the reader

that they constitute the majority of the scenarios in the

NSPLib. We used all options of available nurses (25,

50, 75 and 100). In the objective evaluation function

of our proposed algorithm, we considered the penalty

for violating a hard constraint to be P

= 1000 and the

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Table 1: Summary of the results.

n |D| Case # of Instances

Constantino’s Algorithm Proposed Algorithm

Best Worst Range Best Worst Range

25 7 1 7290 12864 19099 6235 16051 19669 3619

25 7 2 7290 11774 18657 6884 16041 19637 3596

25 7 3 7290 13984 19415 5431 15406 19715 4309

25 7 4 7290 12344 19031 6688 16054 19682 3627

25 7 5 7290 13585 19492 5907 15765 19789 4024

25 7 6 7290 11812 18690 6877 16040 19650 3610

25 7 7 7290 14560 20102 5541 14219 20066 5847

25 7 8 7290 13030 19180 6149 15129 19834 4705

50 7 1 7290 13459 19842 6383 15719 19648 3929

50 7 2 7290 12480 19489 7009 15727 19584 3857

50 7 3 7290 14412 20147 5735 15143 19798 4655

50 7 4 7290 13034 19788 6754 15732 19644 3912

50 7 5 7290 14122 20213 6091 15364 19858 4493

50 7 6 7290 12541 19515 6974 15713 19603 3890

50 7 7 7290 14923 20787 5864 14083 20225 6142

50 7 8 7290 13539 19943 6404 14868 20004 5135

75 7 1 7290 13833 19530 5697 15464 19522 4058

75 7 2 7290 13122 19164 6042 15455 19493 4038

75 7 3 7290 14650 19927 5276 14901 19618 4716

75 7 4 7290 13398 19359 5961 15457 19526 4069

75 7 5 7290 14415 19944 5529 15156 19688 4532

75 7 6 7290 13148 19179 6031 15452 19502 4050

75 7 7 7290 15163 20642 5479 13952 19955 6002

75 7 8 7290 13955 19735 5780 14669 19643 4974

100 7 1 7290 11464 19425 7961 15026 20509 5483

100 7 2 7290 10545 19067 8522 15001 20468 5467

100 7 3 7290 12749 19862 7113 14660 20855 6195

100 7 4 7290 10966 19287 8321 15032 20524 5492

100 7 5 7290 12179 19831 7653 14770 20771 6001

100 7 6 7290 10589 19081 8491 15015 20478 5463

100 7 7 7290 13229 20529 7301 13386 21501 8115

100 7 8 7290 11699 19587 7888 14188 21051 6863

penalty for violating a soft constraint to be P

= 100.

Table 1 summarises the results. The column n

indicates the number of nurses considered. The col-

umn |D| represents the number of days. The column

Case indicate the preferences ﬁle used. The follow-

ing column indicates the number of instances evalu-

ated. The column Best indicates the average cost of

the route of the most favoured nurse in the duty roster,

i.e. the nurse whose evaluation cost is lower. Analo-

gously,the column Worst represent the average cost of

the route for the least favoured nurse, thus, the nurse

whose evaluation cost is higher. The column Range

presents the difference between the best and the worst

columns values. A high value in the column Range

means that there exists a higher discrepancy between

the preferences considered for the nurses, hence some

have many preferences considered in detriment of at-

tending other nurses. Therefore, a low value in that

column means a fairer rostering regarding the nurses

preferences.

The results show that for almost all test scenar-

ios, the proposed VNS algorithm provided fairer so-

lutions when comparing the route costs of the most

favoured nurse against the least favoured nurse. An

interesting ﬁnding is that the reduction in the gap be-

tween the routes with highest and lowest costs, causes

an increase in the average costs for the most favoured

nurses, and sometimes also it increases the cost of the

roster for the least favoured nurses. The proposed al-

gorithm minimises the gap of nurses preferences by

reducing the overall satisfaction of the nurses, hence

raising the overall assignment costs. The reason is

that the normalisation is applied over the preferences

of all nurses. This process, aside from maintaining

a ﬁxed priority for all work shifts, alters the costs

of the original preferences, approximating the value

to 3. Therefore, when the normalisation is applied,

the algorithm bias towards a solution that is close to

a central preference value, and does not necessarily

minimises the costs.

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Table 2: Number of Violations.

n |D| Case # of Instances

Constantino’s Algorithm Proposed Algorithm

Range Avg. # of Range Avg. # of Avg. # of Avg. # of

Violations Hard Viol. Soft Viol. Violations

50 7 1 7290 6383 1031 3929 987 102 1090

50 7 2 7290 7009 937 3857 983 9 992

50 7 3 7290 5735 2364 4655 949 2216 3165

50 7 4 7290 6754 1053 3912 978 135 1112

50 7 5 7290 6091 2744 4493 1406 1188 2594

50 7 6 7290 6974 955 3890 985 60 1045

50 7 7 7290 5864 6091 6142 1510 6824 8333

50 7 8 7290 6404 2890 5135 1555 3106 4661

Table 3: Comparison – average increase of violations × average decrease of range.

n |D| Case # of Instances Increase on the number of Evaluations Decrease on the Range

50 7 1 7290 5.72% 38.45%

50 7 2 7290 5.87% 44.97%

50 7 3 7290 33.88% 18.83%

50 7 4 7290 5.60% 42.08%

50 7 5 7290 -5.47% 26.24%

50 7 6 7290 9.42% 44.22%

50 7 7 7290 36.81% -4.74%

50 7 8 7290 61.28% 19.82%

Table 2 presents the average number of constraints

violations for 50 nurses over 7 days. In order to sim-

plify the visualisation, we chose only one set of in-

stances, however it is important to emphasise that all

other instances presented similar results. We compare

our results with the results obtained by Constantino

et al. (2011). As it can be seen, the proposed algo-

rithm violated more constraints than the compared al-

gorithm, except for preferences case 5. Nonetheless,

when comparing the increase of the number of viola-

tions against the decrease of the range (Table 3), we

can verify that, for the majority of the test cases, the

decrease of the range highly surpasses the increase of

violations. Notably, in case 7 the proposed algorithm

was not able to improve the range, while on case 5

it was able to improve the range and the number of

violations.

It is interesting to observe that the proposed algo-

rithm could provide better results regarding the fair-

ness of the routes but was unable to attend a same

ammount of overall preferences. This gives some in-

dication that there might be a trade-off in the sense

that increasing fairness may necessarily decrease the

overall preferences satisfaction.

Therefore, for the majority of the test instances,

the proposed algorithm was able to provide fairer re-

sults regarding nurses preferences, hence providing

better results. In general, there was an increase in

the number of constraint violations, but for most cases

the fairness of the comparison surpassed this increase.

Nonetheless, the ultimate goal of the NSPBPS is to

obtain a duty roster that evenly distributes the pref-

erence satisfaction among all nurses and for that the

proposed algorithm achieved better results.

6 CONCLUSION

This work proposed an hybrid algorithm that uses

an exact method (for solving bottleneck assignment

problems) and a VNS metaheuristic, to solve the

nurse scheduling problem with balanced preferences

satisfaction (NSPBPS), a variant of the nurse schedul-

ing problem (NSP). In the NSPBPS the aim is to sat-

isfy nurses’ preferences in such a way that the indi-

vidual satisfaction of all nurses is evenly attended.

Therefore, the ﬁnal timetable should be fair regarding

the satisfaction of individual preferences.

The proposed algorithm employs an exact method

to solve a series of bottleneck assignment prob-

lems in order to generate an initial feasible solu-

tion. Then, a VNS metahueristic was employed to

improve the initial solution using two types of neigh-

bourhoods, the CRP (cutting and recombination pro-

cedure) suggested by Constantino et al. (2011) and a

new neighbourhood structure k-Swap. Experimental

results showed that the proposed algorithm managed

to achieve a better balance of the individual prefer-

ences satisfaction. However, the overall number of

constraint violations raised. Nonetheless, as the main

goal in solving the NSPBPS is to present a duty roster

that is more balanced in terms of individual prefer-

ences satisfaction, the proposed algorithm achieved a

AVariableNeighbourhoodSearchforNurseSchedulingwithBalancedPreferenceSatisfaction

469

signiﬁcant improvement over the algorithm by Con-

stantino et al. (2011).

After evaluating the experimental results, it be-

came evident that providing a more balanced roster

means that the algorithm looses ability to minimise

the overall number of constraint violations. Hence,

we can conjecture that a trade-off exists between bal-

ancing the solution against minimising constraint vi-

olations. Hence, future work could investigate the

multi-objective aspect of the NSPBPS and evaluate

the feasibility of applying a multi-objective technique

to solve the problem. Additionally, we aim to pro-

vide results for the monthly scenarios available in the

NSPLib.

ACKNOWLEDGEMENTS

We thank CAPES (Coordenação de Aperfeiçoa-

mento de Pessoal de Nível Superior - Brazilian Min-

istry of Education), Araucária Foundation (Fundação

Araucária do Paraná) and CNPQ (National Council

for Scientiﬁc and Technological Development) for

their ﬁnancial support of this research.

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