ORIGINAL ARTICLE

An evolutionary algorithm based on constraint set partitioningfor nurse rostering problems

Han Huang • Weijia Lin • Zhiyong Lin •

Zhifeng Hao • Andrew Lim

Received: 24 September 2013 / Accepted: 14 December 2013 / Published online: 3 January 2014

� Springer-Verlag London 2014

Abstract The nurse rostering problem (NRP) is a repre-

sentative of NP-hard combinatorial optimization problems.

The hardness of NRP is mainly due to its multiple complex

constraints. Several approaches, which are based on an

evolutionary algorithm (EA) framework and integrated

with a penalty-function technique, were proposed in the

literature to handle the constraints found in NRP. However,

these approaches are not very efficient in dealing with

large-scale NPR instances and thus need to be improved

upon. In this paper, we investigate a large-scale NRP in a

real-world setting, i.e., Chinese NRP (CNRP), which

requires us to arrange many nurses (up to 30) across a

1-month scheduling period. The CNRP poses various

constraints that lead to a large solution space with multiple

isolated areas of infeasible solutions. We propose a single-

individual EA for the CNRP. The novelty of the proposed

approach is threefold: (1) using a constraint separation to

partition the constraints into hard and soft constraints;

(2) using a revised integer programming to generate a high-

quality initial individual (solution), which then leads the

subsequent EA search to a promising feasible solution

space; and (3) using an efficient mutation operator to

quickly search for a better solution in the restricted feasible

solution space. The experimental results based on extensive

simulations indicate that our proposed approach signifi-

cantly outperforms several existing representative algo-

rithms, in terms of solution quality within the same

calculation times of the objective function.

Keywords Evolutionary algorithm � Nurse rostering

problem � Constraint set partitioning � Integer

programming

1 Introduction

Nurse rostering problem (NRP) is a class of resource-

allocation problems [1–5] that pose various constraints.

Due to the health care system’s improvement and patients’

various requirements, hospitals have the responsibility of

caring for many patients with limited medical resources.

One of the challenging problems is to assign working shifts

to the nurses more effectively. The final shift assignment

will directly determine how fully the medical human

resources are utilized for patient care. A reasonable and

robust shift assignment solution helps to deal with patients’

requirements, reduces nurses’ workload and improves

medical service. Hence, the goals of nurse rostering prob-

lems are to find a best solution for shift assignments that

satisfies multiple constraints such as minimal nurse

demands, maximum work allowances and individual day-

off requirements.

Assigning nurses is a tough job due to patients’ various

requirements and nurses’ requests for certain shifts. These

requirements are classified into hard constraints and soft

H. Huang (&) � W. Lin

School of Software Engineering, South China University of

Technology, Guangzhou 510006, People’s Republic of China

e-mail: bssthh@163.com; hhan@scut.edu.cn

H. Huang � A. Lim

Department of Management Sciences, College of Business,

City University of Hong Kong, Kowloon, Hong Kong

Z. Lin

Department of Computer Science, Guangdong Polytechnic

Normal University, Guangzhou, People’s Republic of China

Z. Hao

Faculty of Computer Science, Guangdong University of

Technology, Guangzhou 510006, People’s Republic of China

123

Neural Comput & Applic (2014) 25:703–715

DOI 10.1007/s00521-013-1536-2

constraints. Hard constraints are those that should be fully

satisfied (e.g., daily coverage requirement), and thus, any

hard constraint violation would result in an invalid solu-

tion. Soft constraints are those conditions that are desirable

but not necessary (e.g., nurse’s complete weekends

request), which are mainly used to evaluate the quality of

the schedule solution. These constraints make nurse

rostering problem a challenging task for the hospital

administrators to handle.

Researchers have been working on many instances and

have developed a variety of methods to handle typical

nurse rostering problems [1, 2]. Most techniques of NRP

solutions can be classified into two categories: exact

algorithms [6–8] and heuristics [9–12]. Exact algorithms

aim to find an optimal solution for a problem by exhaustive

search, in which every possible shift assignment instance is

searched for and evaluated in the solution space. NRP was

also tackled by several heuristics methods such as simu-

lated annealing [13], tabu search [14], variable neighbor-

hood searches [15] and estimation of distribution

algorithms [16]. Heuristics have performed effectively to

solve some NRP instances [9–12], by producing high-

quality feasible solutions that are not always optimal.

However, exact algorithms and heuristic methods are

not available to handle all NRPs, like the large-scale one

tackled in this paper. More and more modern real-world

NRPs have motivated the researchers to find efficient

algorithms for their solution. As the scale of problems

grows larger, the situations become more complex. For

example, hospitals in China would employ many more

nurses (up to 28 or even more) than others [17–19] due to

the large population and involve more rules (up to 16) [20].

These rules aim to control the workload of nurses and

improve their service quality for the needs of the hospital in

China. Traditional approaches [6–8] fail to deal with these

real problems in large hospitals.

The NRP [20, 21] with higher dimensions (more nurses,

more constraints) is denoted as CNRP which is short for

Chinese Nurse Rostering Problem. CNRP contains many

nurses across a longer duration and various constraint sets,

which requires trade-off results between quality and com-

putational time. Exact algorithms and recent heuristics

approaches [9–12] are usually not able to tackle such large-

scale problems since the solution space of CNRP contains a

large area of infeasible solutions. These considerations

provoke us to find an efficient approach by integrating

different algorithms to solve the CNRP like combining

their advantages together. The major contribution of our

study is to raise a hybrid algorithm including integer pro-

gramming (denoted as IP) and evolutionary algorithm

(denoted as EA), on a basis of the set partition for soft

constraints. Constraints set partition is helpful to reduce the

difficulties of solving CNRP. Based on the partition, IP was

used to generate an initial solution with several rigid con-

straints. Then, a single-individual EA was carried out to

optimize the initial solution and obtain the final results

through evolutionary operator per iteration.

In this paper, we aim to: (1) analyze the characteristics

of constraints in CNRP and divide the constraints into two

sets; (2) propose an IP ? EA algorithm to produce solu-

tions satisfying the constraints respectively; (3) build a

basic and simplified NRP problem model at the IP stage,

conquer the first constraint set and obtain an initial solution

of high quality (low penalty-function value); (4) design an

EA stage that does not violate the first constraint set and

satisfies the second constraint set, to improve the initialized

solution in finite evolutionary iterations; EA will eventu-

ally result the final solution that obeys all the constraints;

and (5) compare the proposed IP ? EA algorithm with

other approaches for NRPs to find out their advantages and

disadvantages in different CNRP instances.

The remainder of this paper is structured as follows.

Section 2 describes a brief literature review. Section 3

introduces the basic mathematical model of CNRP.

Section 4 presents constraint analysis, constrain set parti-

tion and the procedure of IP ? EA algorithm for CNRP.

Section 5 shows the experimental results and analysis by

comparing the proposed algorithm with other approaches

[17, 22]. Finally, Sect. 6 presents the conclusion of this

paper.

2 Literature review

This section will present a brief overview of the existing

research on solving nurse rostering problems.

Studies [23–25] of nurse rostering problems date back to

the early 1960s. Most of the researchers [6–8, 26–28]

adopted convention optimization approaches to generate

solutions with minimum cost. They are always able to

obtain the optimal solution if there is no time limit.

However, the methods were only effective with small-scale

NRPs with simple constraints since they strongly replied on

the particularity of the problems.

Because exact algorithms were impractical for real-

world NRPs, many heuristic methods [5, 9–12, 29] were

proposed to improve the feasibility. Different from exact

algorithms, the heuristics cannot always guarantee the

optimal solution for each run, but they always result in a

solution approximate to the optimal in the limited runtime.

Thus, they generate a feasible solution in quality and

efficiency, which is practical for real NRPs. After early

attempts, other metaheuristic algorithms were used for

solving NRPs, such as simulated annealing [13], tabu

search method [14], variable neighborhood search [15] and

estimation of distribution method [16]. More heuristic

704 Neural Comput & Applic (2014) 25:703–715

123

methods are necessary to the high-quality solutions of real

NRPs that involve many constraints from hospital, patients

and nurses, due to their difficulty in handling highly con-

strained problems.

Given the advantages and disadvantages of exact algo-

rithms and heuristics, some researchers applied hybrid

methods to solve NRP, by combining the assets of different

methods. Substantial results have been presented in recent

years. Burke et al. [17] proposed a two-stage hybrid

method of IP and variable neighborhood search (VNS).

This approach gained better solutions when compared with

a genetic algorithm (GA) method from ORTEC’s Harmony

[30] and a hybrid VNS approach based on heuristic ranking

method [31]. Moreover, the results showed that it outper-

formed pure IP method or pure VNS method. Tsai and Li

[18] developed a two-stage mathematical modeling and

applied GA in the two-stage processing. Their results

showed that GA is an efficient tool for the NRP and this

model could be easily modified to suit different cases. Bai

et al. [22] formed a hybrid EA combining stochastic

ranking and simulated annealing method for a classical

NRP problem [32], in which there was one hard constraint

and three soft constraints. They compared the hybrid EA

approach with other four approaches (TSHH [33], IGA

[34], EDA [35] and SAHH [36]) and found that the hybrid

EA approach obtained better performance for the NRP

instances.

As the scale of a nurse rostering problem increases

(more nurses and longer scheduling period) and the con-

straints become more complex, research on large-size nurse

rostering problems is greatly needed. As a result, several

studies on large-size NRPs such as CNRP [21] have been

presented. Yet, we wish to find an approach to improve

upon this research. Our research would handle a complex

nurse rostering problem [21] proposed from hospitals in

China. This paper will present an EA for solving CNRP, a

large-scale NRP with many complex constraints. The

CNRP possesses two main features: comparatively large

number of nurses and an additional constraint of balancing

the nurses’ workload. Therefore, the proposed results will

help further the research on solutions for large complex

rostering problems.

3 The nurse rostering problem

CNRP, a large-size nurse rostering problem with complex

constraints, will be introduced in this section. In the nurse

shifting system of hospitals [20, 21], the day shifts can be

classified into three types, as follows:

A-shift: 8:00–15:00

P shift: 15:00–22:00

N-shift: 22:00–8:00

A nurse works at most one shift per day. The goal of the

problem is to come up with a shifts-assignment solution.

CNRP contains several hard and soft constraints. The hard

constraints should be satisfied, including:

HC1 The number of nurses should satisfy daily coverage

requirements for each shift type.

HC2 The number of total working days for each nurse

should range between the maximum boundary and the

minimum boundary.

HC3 An N-shift followed by an A-shift is not allowed.

The soft constraints are those to be satisfied as much as

possible, which also serve as the criteria for evaluating the

quality of the solution. The soft constraints are described as

follows:

SC1 (Fair workload) The difference between the number

of different working shifts for each nurse and the

corresponding average value should be no more than 1.

SC2 The number of consecutive working days of each

nurse should range between three and seven.

SC3 There should be at most five consecutive working

night shifts for each nurse.

SC4 (Complete weekends) There should be either no

shift or two shifts on a weekend.

SC5 There should be at most four working days on

weekends in the scheduling period.

SC6 There should be at least 2 days of rest after a series

of working days.

According to the requirement [21], the first four soft

constraints have the same priority and the last two soft

constraints have lower priority. All of the soft constraints’

priorities correspond to the soft constraints’ weights in

quality evaluation. According to the complete statement of

the problem above, we present a mathematical model of

CNRP, which is similar to the one raised by Burke [17].

The NRP contains a task of shift assignment of M days’

scheduling period which involves N nurses. Let I be the set

of nurses, J be the set of days in the period and K be the set

of nurses’ shift types. The decision variable xijk denotes

whether nurse i works on the shift k in day j, in which:

k ¼1;works on A shift

2;works on P shift

3;works on N shift

8<

:

Hence,

xijk ¼1; nurse i works on the shift k in day j

0; otherwise

�

;

where i 2 I and j 2 J.

Neural Comput & Applic (2014) 25:703–715 705

123

For the hard constraints shown by Exp. (2)–(4), let rjk be

the minimum coverage requirement of shift k in day j,

workmin be the minimum working days of a nurse and

workmax be the maximum working days of a nurse. For the

soft constraints shown in Exp. (6)–(15), let p1–p6 be the

corresponding penalty weights and s1–s6 be the number of

different violations of the current solution. Therefore, the

complete mathematical model of CNRP is described as

follows:

min f ¼X6

t¼1

ptst ð1Þ

s:t:X

k2K

xijk � 1 8i 2 I; j 2 J ð2Þ

HC1:X

i2I

xijk � rjk 8j 2 J; k 2 K ð3Þ

HC2: workmin�X

j2J;k2K

xijk �workmax 8i 2 I ð4Þ

HC3: xij3 þ xiðjþ1Þ1� 1 j 2 f1; 2; . . .;M � 1g ð5Þ

Inequality constraint (2) ensures the basic feature of

CNRP, that is, a nurse works at most one shift in a day.

Inequality constraints (3), (4) and (5) are corresponding to

HC1, HC2 and HC3, respectively.

The details of s1–s6 are shown as follows, in which

workave is defined as the average working shifts of the

nurses. s1–s6 are corresponding to all the soft constraints

from SC1–SC6.

workave ¼1

N

X

i2I

X

j2J

X

k2K

xijk ð6Þ

SC1: s1 ¼X

i2I

maxX

j2J

X

k2K

xijk � workave

�����

�����; 1

!

� 1

" #

ð7Þ

s21 ¼X

i2I

maxXrþ7

j¼r

X

k2K

xijk � 7; 0

!

r 2 1; 2; . . .;M � 5f gð8Þ

s22 ¼X

i2I

X

j2J0max

X

k2K

xijk �X

k2K

xiðj�1Þk �X

k2K

xiðjþ1Þk; 0

!

J0 ¼ 2; 3; . . .;M � 1f gð9Þ

s23 ¼X

i2I

X

j2J00max

max

X

k2K

xijk þX

k2K

xiðj�1Þk �X

k2K

xiðjþ1Þk

�X

k2K

xiðjþ2Þk; 0

!

� 1; 0

!

J00 ¼ 2; 3; . . .;M � 2f g

ð10Þ

SC2: s2 ¼ s21 þ s22 þ s23 ð11Þ

SC3: s3 ¼P

i2I

maxPrþ5

j¼r

xij3 � 5; 0

!

r 2 1; 2; . . .;M � 5f gð12Þ

SC4: s4 ¼P

i2I

P

j2S

minP

k2K

xiðj�1Þk � xijk

��

��; 1

� �

S ¼ 7; 14; 21; 28f gð13Þ

SC5: s5 ¼P

i2I

maxP

j2S

P

k2K

ðxiðj�1Þk þ xijkÞ � 4; 0

!

S ¼ 7; 14; 21; 28f gð14Þ

SC6: s6 ¼P

i2I

P

j2J 0max

P

k2K

xiðj�1Þk�P

k2K

xijkþP

k2K

xiðjþ1Þk

����

�����1;0

� �

J0 ¼ 2;3; . . .;M�1f gð15Þ

4 Hybrid approach based on constraint set partitioning

The main innovation of our IP ? EA method is to reduce

the complexity of solving CNRP and to satisfy the con-

straints step-by-step. Within the proposed algorithm, the

constraints are fully analyzed and handled. They are also

divided into two different constraint sets (Set A and Set B)

according to their features by meeting these constraints

separately. By taking the advantage of the precision of IP

for Set A, we narrow the solution space into the one of high

quality. In the next stage, with the improvement of EA

process, the new solutions are in the direction of satisfying

Set B with little violation of Set A. Finally, we obtain a

solution that satisfies as many soft constraints as possible.

4.1 Constraint set partitioning

The problem includes three hard constraints and six soft

constraints. It is useful for the solution to find that some of

the constraints are highly relative or have similar forms by

analyzing their properties, such as:

1. SC1 is related to the total workload of a nurse, and it is

easy to satisfy by controlling the total workload.

Workload control is a global issue of the rostering

problem, and it has a great impact on the probability of

obtaining feasible solutions. A heavy workload will

make it harder to achieve an acceptable solution,

because it splits the solution space into many regions,

most of which are infeasible. A small workload is good

for finding the best solution, but this kind of situation

seldom happens in real life and possesses no practical

value.

706 Neural Comput & Applic (2014) 25:703–715

123

2. SC4 and SC5 are related to the shift assignment on

weekends. We can do a special arrangement for these

work assignments.

3. SC2, SC3 and SC6 have similar forms, which are

concerned with the number of consecutive day-on or

day-off. Therefore, we can describe them using a

similar model.

According to the analysis above, the hard constraints

HC1–HC3 and the soft constraints SC2, SC4, SC5 and SC6

need to be considered first. The hard constraints should be

satisfied through designing an algorithm strategy. The

calculation of SC2, SC4, SC5 and SC6 is easier than other

soft constraints, so they are considered first in order to

reduce the computational complexity of the solution. The

constraints will be met in steps for special purposes in the

proposed algorithm, which helps to obtain better solutions

by lowering the complexity of the rostering problem.

In most studies [6–12, 26–28], mathematical models

were built for the tackled nurse rostering problem, and the

final results were obtained by using evaluation function and

various searching strategies. NRP problems involve many

complex constraints, and the evaluation function is related

to all of the constraints, which will cost the solution

approaches much computational time. Therefore, in the

proposed algorithm, the constraint set is divided into sub-

sets effectively. The most infeasible and poor solutions are

abandoned and improved step-by-step. As a result, it takes

a shorter time to evaluate the value of the remaining

solutions that seldom violate the constraint rules. There-

fore, we raise the idea of combining IP and EA to improve

the solution of NRP.

The main process of our proposed algorithm is briefly

introduced in Fig. 1. First, the soft constraint set is divided

into two subsets: Set A and Set B. Thus, IP ? EA hybrid

approach takes advantage of the characteristics of IP and

EA, respectively, and satisfies these two constraint sets

step-by-step. Second, the hybrid approach solves the sim-

plified problem (Set B is not included) by using IP algo-

rithm and produces an initial solution. Obviously, the

initial solution satisfies the hard constraints, as well as the

constraints in Set A. Later, temporary solutions will be

improved by an EA (shown by Fig. 1), aiming to satisfy the

constraints in Set B and obtain the final optimized result.

According to the analysis of the properties of all the soft

constraints above, the division strategy of soft constraints

set is indicated as follows:

Set A contains SC2, SC4, SC5 and SC6. These con-

straints have strong restraining force, which leads to gen-

erating many areas of infeasible solutions in the solution

space. If non-precision algorithms are used to deal with

them, the running process will converge to a local optimal

solution in a short time, which deviates from the original

goal of optimization. Therefore, these soft constraints are

also considered to design the IP stage, aiming to discard a

large number of infeasible solutions and generate an initial

solution for the evolutionary stage. The initialization is

required to satisfy the strongly-restrained constraints of Set

A and ensure the good quality of the consecutive

population.

Set B contains SC1 and SC3. Satisfying SC1 will

involve a large amount of computation; therefore, it should

be met late in Set B. Also, since SC2 involves the

arrangement of consecutive working days, the process of

satisfying SC2 also helps to satisfy SC3 (consecutive

working nights). Hence, SC3 is supposed to be included in

Set B, which reduces the computation of the IP stage. Thus,

because they possess weaker restraining force than the

constraints in Set A, another feasible solution can be

generated from the original one easily by using a chain-

move operation. This feature is helpful to generate new

populations in the EA. The proposed algorithm will use

evolutionary operators in selection and mutation operations

to ensure that all the individuals do not violate the hard

constraints and rarely violate the constraints in Set A.

Figure 1 shows the analysis of the constraint sets.

4.2 Procedure of IP ? EA algorithm

With the partition of soft constraint set, the first stage of the

IP ? EA algorithm handles a nurse rostering problem

which mainly deals with the hard constraints and the

Fig. 1 The partition of constraints and the corresponding algorithm

stages

Neural Comput & Applic (2014) 25:703–715 707

123

constraints in Set A. The goal is not to obtain the final

solution directly, but to obtain an excellent initial solution

by taking advantage of precision of IP approach to handle

those strict constraints of Set A. The second stage of

IP ? EA algorithm deals with the complete problem.

Figure 2 presents the flow chart of IP ? EA algorithm.

In the IP stage, a simplified CNRP problem is solved by

applying IP to generate initial solution for the EA stage. All

the constraints in Set B will be excluded (i.e., only the hard

constraints and the constraints in Set A are included) to

generate a simplified problem. As a result, the simplified

problem is made from the original NRP presented in Sec-

tion III, by removing the constraints shown in Expressions

(7) and (15) and replacing the hard constraint shown in

Expression (4) with the following expression:

workmin�X

j2J;k2K

xijkworkmin þ 1 8i 2 I ð16Þ

The new hard constraint does not violate the original

hard constraint (HC2), and it also satisfies SC1 by

controlling the total workload of nurses. It is trivial to

prove that all the solutions of the simplified problem are

also the solutions of the original NRP. The simplified

problem has a downsized solution space since a strict hard

constraint (Expression 16) replaces a relatively relaxed

hard constraint (Expression 4). So we adopt IP approach to

produce a solution in limited iteration steps. The best one

generated from IP stage is selected as the initial solution of

the next steps, which highly satisfies the constraints in Set

A.

In the EA stage, the original problem is tackled by

evolutionary operators shown in Figs. 3 and 4. On the basis

of the initial solution produced by the IP stage, the EA

stage is to satisfy HC1–HC3 strictly, minimize the sum of

penalty-function value by Exp. (1) and gain the final

solution of NRP. The population size of EA is one, i.e.,

only an offspring will be generated from an individual

parent. The offspring will replace its parent (best-so-far

solution) unless it is superior to the best-so-far individual.

The evaluation function of the individual is calculated by

the weighted sum of the penalty value. Thus, a lower

penalty value is better for the individual.

The pseudocode of EA stage is presented in Fig. 3.

4.3 Mutation operator

The results are given in Table 4, which once again show

the average of the best solutions ± standard deviation (SD)

for each algorithm; the problems are approximately

ordered from simple to complex. Although ELPSO could

not obtain the optimal solution for the Sphere and Rosen-

brock problems, its overall performance among all the

examined algorithms is the best. In particular, ELPSO is

clearly superior when solving multimodal function

problems.

Step 3 is the core operator of EA stage. Item delta is a

parameter for the subalgorithm Mutation(nurse_x, delta),

which generates an offspring offs_x from nurse_x.

Obj(offs_x) is an evaluation function calculated by Exp.

(1). As a result, evaluation function is implemented only

once per iteration. There is only one individual in the

population, and so current individual nurse_x will be

replaced with the offspring offs_x if the evaluation value

offs_s is less than nurse_s. Figure 4 describes the

pseudocode of the evolutionary operation Muta-

tion(nurse_x, delta).

In the mutation process, variable pmuta is a threshold

value. The value of pmuta determines the probability of the

mutation for weekend and workday shifts. If the value of

random probability is smaller than threshold pmuta, evo-

lutionary operation will be implemented. As the iteration

number increases, the value of pmuta decreases according

to Step 3 of Fig. 3 and Step 1 of Fig. 4. For example, at the

beginning of the EA stage, the mutation runs by high

probability since the value of pmuta is high. However,

when closing in on the end of the evolutionary process, the

solution gradually converges because the value of pmuta is

reduced to a lower mutation probability.

Figure 5 explains the reason why we set a descending

value of pmuta. By comparing different values of pmu-

ta(descending value, ascending value and fixed value) inFig. 2 IP ? EA algorithm flow chart

708 Neural Comput & Applic (2014) 25:703–715

123

the evolutionary process (11 sample points of the average

penalty value in five experiments of 20,000 generations are

captured), the descent policy helps to generate the final

shift solution with less penalty value. By contrast, the other

two policies stop optimizing the results before the termi-

nation condition is reached. According to the result of

Fig. 5, since the mutation operation involves no policy that

guarantees better solutions, high mutation rate may easily

cause much disturbance on the best-so-far solution, so it is

difficult to ensure that a better solution can be generated.

The details of mutation are given by Figs. 6 and 7.

1. Figure 6 shows an example of the mutation for the

weekends, in which we swap the shifts of two working

nurses chosen randomly. In Fig. 5, the swapped objects

are weekend shifts (Sat and Sun) of Nurse 1 and Nurse

2. The swapped shifts still satisfy SC4 and SC5.

2. Figure 7 shows an example of shift swapping on

workdays. For the working days, two nurses (rows) in

the shifts are randomly chosen, and then their shifts on

two days are swapped. For example, the swapped

objects are workdays of Nurse 1 and Nurse 3 in Fig. 6.

The total workload of each nurse remains unchanged.

The EA stage will take effect to a certain degree,

making up for the loss of several excellent solutions. The

operation in EA stage will cause only minor violations in

Set A. For the weekends, in order to strictly satisfy SC4

and SC5, both the shifts for the two days in weekend are

swapped. For the working days, the total workload of the

two nurses involved in shift swapping remains unchanged

on working days. Though the workload of different shift

types for each nurse may change, we tolerate this distur-

bance by penalty function [Exp. (1)] in the EA stage.

Fig. 3 Pseudocode of EA stage

Fig. 4 Pseudocode of

Mutation(nurse_x, delta)

function

Neural Comput & Applic (2014) 25:703–715 709

123

In fact, the IP strategy in our approach may cause the

loss of several excellent solutions. These solutions do not

satisfy the constraints in Set A, but with the satisfaction

of Set B, they possess overall lower penalty values.

Therefore, the solution obtained by EA stage is not

strictly restrained by Set A, but the mutation operator will

expand the searching area of EA slightly, in order to

involve the solutions that have been abandoned in the IP

stage.

The implementation and investigation of the proposed

algorithm will be presented in the Sect. 5.

5 Experiment and results

This section will present the experimental setting and

results. The experiments aim to adopt the proposed algo-

rithm to solve the CNRP and compare the results with

those produced by other recent approaches [17, 22].

5.1 Experimental settings

There were 20 CNRP instances generated in the experi-

ment. The two main differences between these instances

are in daily coverage requirement of each shift type and

minimum number of working days for each nurse. For the

soft constraints, we assigned them different penalty

weights referring the studies [17, 21, 22]. The value of

p1–p4 is 10, and the value of p5–p6 is 1.

CNRP is a large-scale nurse roster problem that contains

a huge solution space including many infeasible areas. The

size of CNRP instances was larger than those tested in

other references [17–19, 21, 22]. The CNRP instances

studied in the experiments involve 30 nurses across a

30-day time horizon and include three hard constraints and

six soft constraints. Compared with the problems in the

literature (Index of Table 1 including the numbers of nur-

ses and days determine the size of the problem), the

problem of size presented here is relatively the largest one.

Fig. 6 Mutation for weekend

shift

Fig. 5 The comparison of

different values of parameter

pmuta

710 Neural Comput & Applic (2014) 25:703–715

123

The experiments will compare the performance of

relaxed IP method, IP ? VNS algorithm [17], simple EA

algorithm, hybrid evolutionary approach (HEA) [22],

modified harmony search algorithm and the proposed

IP ? EA algorithm. The implementation of the algorithms

in the experiment is described as follows.

The relaxed IP is the algorithm only containing an IP

stage of Fig. 2. Because there is no available result obtained

by the IP strategy [17] after 1,000,000 iterations, IP ? VNS

algorithm was implemented by modifying its IP to be a

relaxed IP. VNS was programmed strictly according to Ref.

[17]. Simple EA algorithm is the algorithm of EA stage in

Fig. 2. Hybrid evolutionary approach (HEA) [22] was

carried out by adding a relaxed IP for initialization since it is

difficult to obtain a feasible solution for CNRP instances

without any modification. In brief, the proposed IP ? EA

algorithm includes a relaxed IP (IP stage in Fig. 2) and a

simple EA (EA stage in Fig. 2). MHSA [37] is carried out in

20,000 iterations, which is the same with EA stage.

The purposes of different comparisons for the IP ? EA

algorithm and other methods are to:

1. compare IP ? VNS with IP ? EA and determine

whether the proposed EA is better than VNS strategy

[17] with the same initial solution;

2. compare the relaxed IP and IP ? EA and determine

the effectiveness of IP ? EA is better than single IP

for solving CNRP;

3. compare IP ? EA and simple EA to determine

whether IP stage for initialization is necessary.

4. compare IP ? EA and HEA [22] and indicate which

evolutionary approach is better in solving CNRP.

5. compare IP ? EA and MHSA [37] and indicate which

approach is better in solving CNRP.

Table 2 presents the termination condition of each

approach. The iteration number is considered as a major

index of computational time and experiment termination

since the evaluation function of Exp. (1) is calculated once

per iteration in most of the compared algorithms. To make

sure that the approaches involving the EA method can be

compared under the same condition, the maximum number

of iteration of all EA stages is assigned with the same

value. Also, to make sure the VNS operation and the IP

method could be fully executed, the maximum numbers of

iterations are assigned with very high values. Specially, for

the proposed approach, the simple EA and HEA are run 30

times on each experiment, and the best result is recorded in

Table 3. Furthermore, the means in average and SDs are

shown in Table 4.

Table 1 Size comparison of our problem and other problems in previous literature

Approach The number of Published in

Nurses Days Hard constraints Soft constraints

The proposed EA 30 30 3 6

IP ? VNS algorithm [17] 16 30 10 7 2010

Hybrid evolutionary approach [22] 30 7 2 1 2010

Utopic Pareto genetic heuristic [19] 5 7 5 3 2008

Two-stage modeling with genetic algorithm [18] 15 30 7 5 2009

Modified harmony search algorithm [37] 10 28 2 10 2011

Fig. 7 Mutation for workday

shift

Neural Comput & Applic (2014) 25:703–715 711

123

The experiments are all carried out on an Intel Core Duo

(2.40 GHz and 2.40 GHz) PC with 2G RAM under Win-

dow 7. The IP stage is implemented by software lingo 11.0

(http://www.lindo.com/). The EA stage is programmed and

running on MATLAB R2010b (http://www.mathworks.cn/).

The data of CNRP instances and the source code of the

proposed IP ? EA algorithm can be downloaded on the

web address http://www.rayfile.com/zh-cn/files/cb5aaf8f-

4c28-11e2-bf3b-0015c55db73d/.

5.2 Experimental results

The important index of experiment is the penalty-function

value [by Exp. (1)] of the final solution for each compared

approach. Lower penalty-function values imply better

solutions if the solution is feasible for all of the hard

constraints. Tables 3, 4 and 5 present the overall experi-

mental results of 20 CNRP instances.

Table 3 presents the comparison of the solution results.

From the overall view on the penalty value of all the

approach, the proposed hybrid approach (IP ? EA) gen-

erates solutions with the least penalty values in 17 instan-

ces. That is, our approach is regarded as an effective

method for the CNRP, and it outperforms other approaches

in terms of resulting low-penalty results.

Table 4 presents the percentage deviation of the average

penalty value from the lowest penalty value in each

instance. For the simple EA, the hybrid EA [22] and

MSHA [37], they can produce stable results (little differ-

ence among each run) since the average deviation is no

more than 1 %. Specially, the proposed IP ? EA algorithm

is the most stable of all since it produced the same final

solution in each run. Moreover, since the IP ? VNS

method and IP method are deterministic algorithms, all the

Table 3 Best penalty value of the proposed EA and other NRP approaches

Exp no. Object value

IP ? EA (best) Simple EA (best) IP ? VNS IP Hybrid EA (best) MHSA (best)

1 110 307 253 642 594 609

2 300 321 653 523 570 568

3 360 372 770 683 925 603

4 378 344 365 745 666 707

5 199 219 239 593 604 585

6 237 346 494 657 649 631

7 248 286 271 612 593 632

8 337 311 589 649 601 571

9 320 398 456 617 549 521

10 381 442 396 853 803 817

11 258 258 695 548 511 560

12 269 205 257 506 487 475

13 191 290 255 570 600 627

14 167 229 326 941 911 929

15 191 219 239 593 604 647

16 196 200 542 667 576 575

17 183 211 653 600 601 647

18 270 220 646 543 502 481

19 389 709 418 937 849 819

20 277 339 317 740 694 675

Avg. 263.05 311.3 441.7 660.95 644.45 640.2

The items colored in red are the best of all

Table 2 Termination condition (maximum iterations)

Method Maximum number of iteration

IP ? EA IP: 150,000 Simple EA: 20,000

IP ? VNS IP: 150,000 VNS: 500,000

Relaxed IP 300,000

Simple EA 20,000

IP ? HEA IP: 150,000 HEA: 20,000

MHSA 20,000

712 Neural Comput & Applic (2014) 25:703–715

123

approaches involved in the experiments are relatively

stable.

Table 5 presents the runtime cost of each approach. As

the maximum iteration number was fixed in Table 2,

Table 5 only indicates the runtime per iteration of each

algorithm on average instead of the solution speed. For

different problem instances, the runtime may vary from

300 to 900 s. Also, the average runtime cost varies from

540 s (9 min) to 630 s (about 11 min). For a CNRP, it is

trivial to compare the time cost of each approach as more

time cost does not ensure better results.

Based on the same initial solution from IP stage, the

results of IP ? EA are obviously better than the IP ? VNS

approach in 18 instances, except for the fourth and 12th

ones. Table 2 shows that the iteration number of

IP ? VNS is much more than that of IP ? EA. Therefore,

IP ? VNS has more computational time than IP ? EA if

calculating the evaluation function of Exp. (1) is consid-

ered as a basic operator of the compared algorithms. The

results of fourth instance and 12th instance only mean that

IP ? EA needs more iterations to obtain the better solution

than IP ? VNS.

IP ? EA performs better than simple EA in 15 instan-

ces, equally in the 11th instance and slightly worse in the

4th, 8th, 12th and 18th instances. It uses a promising initial

solution from IP stage. On the contrary, the initial solution

of simple EA approach is generated randomly. Therefore,

high-quality initial solution is useful in the final optimi-

zation, and IP stage is necessary. The three instances in

which the simple EA method performs slightly better may

have traps of local optimization. IP ? EA cannot jump out

of local optimal when the initial solution from IP stage falls

in the trap of local optimization. Random initialization can

produce various solutions per run, so it has higher proba-

bility to avoid falling in local optimization. Thus, the

simple EA is better than IP ? EA in solving the three

instances.

The results from the relaxed IP method have 2–5 times

function value as the hybrid approach: That is, the relaxed

IP method performs notably worse. Though the maximum

number of iteration of IP approach is very large, it is not

enough to obtain a low-penalty solution. For IP approach,

the slight improvement on solutions needs a large amount

of runtime, which is not a practical way to solve the CNRP

instance. The results in Fig. 7 also indicate that efficiency

does not mainly depend upon the function of IP stage. The

EA stage plays a role in the solution of CNRP.

Apparently, HEA achieves poor solutions with a much

higher penalty value. HEA is able to adopt valid

Table 4 Percentage deviation of the average penalty value

Exp

no.

Mean and SD of penalty value

IP ? EA Simple EA Hybrid EA MHSA

Mean Std. Mean Std. Mean Std. Mean Std.

1 110 0.0 309 4.47 600 5.48 566 0

2 300 0.0 323 4.47 572 4.47 596 0

3 360 0.0 372 0.00 929 5.48 657 0

4 378 0.0 344 0.00 666 0.00 674 4.47

5 199 0.0 223 5.48 606 4.47 653 0

6 237 0.0 348 4.47 653 5.48 691 0

7 248 0.0 286 0.00 593 0.00 601 0

8 337 0.0 313 4.47 607 5.48 613 5.48

9 320 0.0 398 0.00 553 5.48 531 0

10 381 0.0 442 0.00 803 0.00 849 4.47

11 258 0.0 258 0.00 511 0.00 504 0

12 269 0.0 211 8.94 491 5.48 457 0

13 191 0.0 292 4.47 602 4.47 645 4.47

14 167 0.0 233 8.94 913 4.47 933 0

15 191 0.0 219 0.00 606 4.47 612 4.47

16 196 0.0 200 0.00 576 0.00 572 0

17 183 0.0 211 0.00 601 0.00 581 4.47

18 270 0.0 222 4.47 506 8.94 520 0

19 389 0.0 711 4.47 851 4.47 877 0

20 277 0.0 339 0.00 696 4.47 672 0

Avg. 263.05 0.0 312.7 2.73 646.75 3.66 640.2 1.3915

Table 5 Comparison of running time cost

Exp

no.

Running time cost (s)

IP ? EA IP ? VNS EA

(avg)

IP Hybrid

EA (avg)

MHSA

(avg)

1 444 408 446 672 533 527

2 361 343 367 603 409 409

3 488 484 499 611 506 507

4 421 422 491 677 451 464

5 576 522 614 671 475 492

6 711 661 719 599 526 540

7 602 551 848 701 646 638

8 558 509 562 768 598 599

9 490 456 490 659 551 545

10 382 356 386 577 629 631

11 403 365 406 774 586 590

12 648 352 395 646 590 602

13 752 691 711 722 545 558

14 592 548 567 587 611 629

15 659 593 693 710 602 611

16 788 677 786 658 623 632

17 654 657 608 658 533 537

18 789 737 689 656 523 517

19 880 898 838 617 617 617

20 704 691 686 676 474 485

Avg. 595.1 546.05 590.05 662.1 551.4 555.1

Neural Comput & Applic (2014) 25:703–715 713

123

evolutionary operation for the small problems with rela-

tively few hard constraints. Moreover, it heavily relies on

the simplicity of the hard constraints, so that HEA can

adopt the means of penalty function (some infeasible

solutions are accepted) and rectify the solutions in heuristic

methods (only the feasible solutions survive). For the

CNRP, it is impossible to include the infeasible solutions,

because the solution space is huge and unpredictable,

which is easy to trap in the infeasible solution space so that

no valid result can be produced. Therefore, the hybrid EA

method is problem-specific for its own problem instance,

making it hard to solve multiple-hard-constraint NRPs like

CNRP.

Finally, MHSA [37] performs much worse than our

proposed method. Though MHSA has been proved to solve

some specific nurse rostering problems, it is hard to handle

the problems that include more hard constraints as well as

the problem in this paper. That is because MHSA heavily

relies on the stable optional patterns of problems, which

means less feasible solution can be obtained. Since pro-

cessing the problem in this paper will cause many infea-

sible solutions, MHSA is not able to correct and handle

these solutions, and so it produces many bad solutions to

the results.

All the results presented above show that the IP stage is

able to generate an initial solution under the hard con-

straints and part of the soft constraints, which provides the

basis of EA improvement. With an initial solution for the

CNRP (the problem in this paper involves 30 nurses in

30 days), the EA stage is well designed for the specific

problem and obtains a good result in the accepted runtime.

IP ? EA combines the precision of IP and rapid evolve-

ment of EA and outperforms the other four approaches

[17, 21, 22] in 20 instances on average.

6 Conclusion

This paper has considered a large-scale nurse rostering

problem (named as CNRP) with many complex constraints.

By analyzing the features of the constraints and problem

modeling, we partitioned the constraints into two sets

according to their features and tried to satisfy them

respectively. On the basis of this, we proposed the hybrid

approach of IP and EA stages to achieve global optimiza-

tion. The IP stage is used as the initialization of EA stage

by solving a simplified problem with constraints partition.

The EA stage adopted an evolutionary operation by slightly

violating the previous constraints and tended to satisfy as

many of the soft constraints as possible. The proposed

algorithm took advantage of a deterministic approach and

evolutionary operator to solve the CNRP. In 20 instances,

the proposed algorithm outperformed the other four

approaches on average, which indicates that the IP stage

can produce a high-quality initial solution for evolution and

the EA stage has capacity for global optimization. This

paper sparks the idea of flexible partitioning constraints

and overcoming them respectively, which helps to reduce

the computational complexity of solving NRP problems.

According to the experimental results, it is worth further

study to improve the proposed algorithm for other complex

nurse rostering problems.

One possible direction of future study is to use the

proposed IP ? EA for solving other complex nurse

rostering problems and scheduling optimization problems

in which there are multiple constraints. Moreover, the

combination of IP and EA needs to be studied in detail to

adjust the hybrid algorithm for better optimization.

Acknowledgments This work was supported by National Natural

Science Foundation of China (61370102, 61170193, 61202453,

61203310), Guangdong Natural Science Foundation

(S2011040002890, S2012010010613), the Fundamental Research

Funds for the Central Universities, SCUT (2012ZZ0087,

2014ZG0043) and The Pearl River Science&Technology Star Project

(2012J2200007). The authors thank Dr. Kyle McIntosh for his

proofreading.

References

1. Burke EK, De Causmaecker P, Vanden Berghe G, Landeghem H

(2004) The state of the art of nurse rostering. J Sched

7(6):441–499

2. Sitompul D, Randhawa S (1990) Nurse scheduling models: a

state-of-the-art review. J Soc Health Syst 2:62–72

3. Hung R (1995) Hospital nurse scheduling. J Nurs Adm 25(7–8):

21–23

4. Cheang B, Li H, Lim A, Rodrigues B (2003) Nurse rostering

problems—a bibliographic survey. Eur J Oper Res 151:447–460

5. Aickelin U, Burke EK, Li J (2009) An evolutionary squeaky

wheel optimization approach to personnel scheduling. IEEE

Trans Evol Comput 13(2):433–443

6. Beaumont N (1997) Scheduling staff using mixed integer pro-

gramming. Eur J Oper Res 98:473–484

7. Warner M, Prawda J (1972) A mathematical programming model

for scheduling nursing personnel in a hospital. Manag Sci

19:411–422

8. Warner DW (1976) Scheduling nurse personnel according to

nursing preference: a mathematical programming approach. Oper

Res 24(5):842–856

9. Aickelin U, Dowsland K (2000) Exploiting problem structure in a

genetic algorithm approach to a nurse rostering problem. J Sched

3:139–153

10. Burke EK, Cowling P, De Causmaecker P, Vanden Berghe G

(2001) A memetic approach to the nurse rostering problem. Appl

Intell 15:199–214

11. Easton FF, Mansour N (1999) A distributed genetic algorithm for

deterministic and stochastic labor scheduling problems. Eur J

Oper Res 118:505–523

12. Kawanaka H, Yamamoto K, Yoshikawa T, Shinigi T, Tsuruoka S

(2001) Genetic algorithm with the constraints for nurse sched-

uling problem. In: Proceedings of congress on evolutionary

computation (CEC), pp 1123–1130

714 Neural Comput & Applic (2014) 25:703–715

123

13. Burke EK, De Causmaecker P, Vanden Berghe G (1999) A

hybrid tabu search algorithm for the nurse rostering problem. In:

Proceedings of the simulated evolution learning: selected papers

2nd Asia-Pacific conference simulated evolution learning.

(SEAL), Lecture Notes in Computer Science Series 1585.

Springer, Berlin, pp 187–194

14. Burke EK, De Causmaecker P, Petrovic S, Vanden Berghe G

(2001) A multicriteria meta-heuristic approach to nurse rostering.

In: Proceedings of the practice theory automated timetabling:

selected revised papers 3rd practice theory automated timetabling

international conference, Lecture Notes in Computer Science

Series 2079. Springer, Berlin, pp 118–131

15. Burke EK, De Causmaecker P, Petrovic S, Vanden Berghe G

(2006) Metaheuristics for handling time interval coverage con-

straints in nurse scheduling. Appl Artif Intell 20(9):743–766

16. Beddoe G, Petrovic S (2006) Selecting and weighting features

using a genetic algorithm in a case-based reasoning approach to

personnel rostering. Eur J Oper Res 175(2):649–671

17. Burke EK, Li J, Qu R (2010) A hybrid model of integer pro-

gramming and variable neighbourhood search for highly-con-

strained nurse rostering problems. Eur J Oper Res 203:484–493

18. Tsai C-C, Li SHA (2009) A two-stage modeling with genetic

algorithms for the nurse scheduling problem. Expert Syst Appl

36:9506–9512

19. Pato MV, Moz M (2008) Solving a bi-objective nurse rerostering

problem by using a utopic Pareto genetic heuristic. J Heuristics

14:359–374

20. Cheng BMW, Lee JHM, Wu JCK (1997) A constraint-based

nurse rostering system using a redundant modeling approach.

IEEE Trans Inf Technol Biomed 1:44–54

21. Sheer BB, Wong FKY (2008) The development of advanced

nursing practice globally. J Nurs Scholarsh 40(3):204–211

22. Bai R, Burke EK, Kendall G, Li J, McCollum B (2010) A hybrid

evolutionary approach to the nurse rostering problem. IEEE

Trans Evol Comput 14(4):580–590

23. Wolfe H, Young JP (1965) Staffing the nursing unit: part I. Nurs

Res 14(3):236–243

24. Wolfe H, Young JP (1965) Staffing the nursing unit: part II. Nurs

Res 14(4):199–303

25. Howell JP (1966) Cyclical scheduling of nursing personnel.

Hospitals 40(2):77–85

26. Bard J, Purnomo HW (2005) Preference scheduling for nurses

using column generation. Eur J Oper Res 164:510–534

27. Berrada I, Ferland JA, Michelon P (1996) A multi-objective

approach to nurse scheduling with both hard and soft constraints.

Socio Econ Plan Sci 30:183–193

28. Ikegami A, Niwa A (2003) A subproblem-centric model and

approach to the nurse scheduling problem. Math Program

97:517–541

29. Hadwan M, Ayob M, Sabar NR, Qu R (2013) A harmony search

algorithm for nurse rostering problems. Inf Sci 233:126–140

30. Post G, Veltman B (2004) Harmonious personnel scheduling. In:

Proceedings of the fifth international conference on practice and

automated timetabling (PATAT), pp 557–559

31. Burke EK, Curtis T, Post G, Qu R, Veltman B (2008) A hybrid

heuristic ordering and variable neighbourhood search for the

nurse rostering problem. Eur J Oper Res 188:330–341

32. Dowsland KA (1998) Nurse scheduling with tabu search and

strategic oscillation. Eur J Oper Res 106(2–3):393–407

33. Burke EK, Kendall G, Soubeiga E (2003) A tabu-search hype-

rheuristic for timetabling and rostering. J Heuristics 9(6):451–470

34. Aickelin U, Dowsland KA (2003) An indirect genetic algorithm

for a nurse scheduling problem. Comput Oper Res 31(5):761–778

35. Aickelin U, Burke EK, Li J (2007) An estimation of distribution

algorithm with intelligent local search for rule-based nurse ro-

stering. Eur J Oper Res 58(12):1574–1585

36. Bai R, Blazewicz J, Burke EK, Kendall G, McCollum B (2007) A

simulated annealing hyper-heuristic methodology for flexible

decision support. School of CSiT, Univ. Nottingham, Notting-

ham, UK, Tech. Rep. NOTTCS-TR-2007-8

37. Awadallah MA, Khader AT, Al-Betar MA et al (2011) Nurse

rostering using modified harmony search algorithm/Swarm,

evolutionary, and memetic computing. Springer, Berlin, pp 27–37

Neural Comput & Applic (2014) 25:703–715 715

123