Evolutionary Techniques for the Nurse Scheduling Problem

Mehdi Sadeghilalimi, Malek Mouhoub

and Aymen Ben Said

Department of Computer Science, University of Regina, Regina, Canada

Keywords:

Combinatorial Optimization, Nature-Inspired Techniques, Metaheuristics, Stochastic Optimization, Resource

Allocation, Nurse Scheduling Problem (NSP).

Abstract:

The Nurse Scheduling Problem (NSP) is a combinatorial optimization problem that creates weekly scheduling

solutions for nurses. These solutions must satisfy constraints for the workload coverage requirements while

optimizing one or more objectives related to hospital costs or nurses’ preferences. Although exact methods

may be used to solve the NSP and return the optimal solution, they usually come with an exponential time

cost. Therefore, approximate methods may be considered as they offer a good trade-off between the quality of

the solution and the running time. In this context, we propose a solving method based on Genetic Algorithms

(GAs) to solve the NSP. To evaluate the efﬁciency of our proposed method, we conducted experiments on

various NSP instances. Further, we compared the quality of the returned solutions against solutions obtained

from exact methods and metaheuristics. The experimental results reveal that our proposed method can fairly

compete with B&B in terms of the quality of the solution while delivering the solutions in much faster running

times.

1 INTRODUCTION

The Nurse Scheduling Problem (NSP) is among

the challenging NP-Hard combinatorial optimization

problems in healthcare. The aim is to create weekly

schedules that satisfy the workload constraints and si-

multaneously optimize some objectives. Various ap-

proaches were proposed to solve the NSP using ex-

act and approximate methods. While exact methods

guarantee the return of the optimal solution, they usu-

ally suffer from expensive running time costs. For

this reason, researchers often look for alternatives to

reduce this time complexity by exploring metaheuris-

tics. The latter typically trade the quality of the solu-

tions for better running times. In this context, we pro-

pose a new method based on the Genetic Algorithm

(GA) to solve the NSP. To assess the performance

of our proposed method, we adopted the NSP for-

mulation from (Sadeghilalimi et al., 2023) and con-

ducted multiple experiments using a set of NSP in-

stances. Furthermore, we compared the obtained re-

sults against exact and metaheuristics. For the ex-

act method, we used the Branch & Bound (B&B)

algorithm from (Ben Said and Mouhoub, 2022) that

relies on constraint propagation techniques (Dechter

and Cohen, 2003) as a pre-processing step to remove

https://orcid.org/0000-0001-7381-1064

locally inconsistent values. The latter step allows the

reduction of the search space size before the execu-

tion of B&B. For the metaheuristics, we used variants

of the Whale Optimization Algorithm (WOA) and

Stochastic Local Search (SLS) (Sadeghilalimi et al.,

2023). Note that the WOA variants correspond to

different types of mutations we have considered for

the exploration phase to ensure more diversiﬁcation

in the search and overcome local minima. We have

adopted the same variants for our GA-based method.

The SLS-based method starts by ﬁnding an initial so-

lution using a depth-ﬁrst search (DFS) technique and

then attempts to tune it further to improve the solu-

tion’s quality while maintaining feasibility. Similar

to B&B, SLS uses constraint propagation to optimize

the backtrack search. The experimental results are

very promising as they reveal the efﬁciency of our

proposed GA-based technique in providing a reason-

able trade-off between the quality of the solution re-

turned and the running time compared to the above-

mentioned methods. Although B&B and DFS use

constraint propagation before the search, the results

show they still suffer from their expensive exponen-

tial time costs.

Sadeghilalimi, M., Mouhoub, M. and Ben Said, A.

Evolutionary Techniques for the Nurse Scheduling Problem.

DOI: 10.5220/0012402300003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 333-340

ISBN: 978-989-758-681-1; ISSN: 2184-4372

333

2 RELATED WORKS

In (Ben Said et al., 2021), the authors proposed an

implicit approach to solve the NSP relying on Ma-

chine Learning (ML) algorithms to learn the frequent

patterns and associations among past scheduling solu-

tions. Although the proposed ML methods may gen-

erate solutions almost instantly, they come with a de-

gree of uncertainty since the constraints and objec-

tives are implicitly learned and represented through

the learned patterns (e.g. association rules, trained

ML models). In (Ben Said and Mouhoub, 2022),

the authors proposed an explicit approach to solve

the NSP using the Weighted Constraints Satisfaction

Problem (WCSP) formalism (Larrosa, 2002; Bidar

and Mouhoub, 2023; Lee and Leung, 2009) to over-

come the uncertainty challenges. The authors formu-

lated the NSP as a WCSP model and further solved it

using B&B (we use this B&B algorithm as a compara-

tive method for our proposed GA variants in the scope

of this paper). B&B was also used in (Baskaran et al.,

2014) to solve the NSP, and as discussed in (Woeg-

inger, 2003), the challenge with exact algorithms like

B&B is that they suffer from their exponential time

cost, particularly when solving large-size problem in-

stances.

Other related work relied on evolutionary methods

based on metaheuristics to solve the NSP (Jan et al.,

2000; Gutjahr and Rauner, 2007; Wu et al., 2013; Ja-

fari and Salmasi, 2015; Rajeswari et al., 2017). These

methods usually use random search to elicit candi-

date solutions while balancing between exploration

and exploitation to escape local minima/maxima. The

latter is an alternative to exact methods due to the time

complexity challenge. However, returning the opti-

mal solution is not guaranteed.

Hybrid methods that combine multiple algorithms

were also explored to solve the NSP (Burke et al.,

2001). For instance, in (Zhang et al., 2011), the au-

thors proposed a hybrid method combining GAs and

Variable Neighborhood Search (VNS). The GA was

used to solve sub-problems and return initial feasible

solutions that are consequently fed into the VNS to

improve them. Experimental results show that the

proposed method can return feasible solutions and

therefore can be used effectively in solving other re-

source allocation problems.

3 PROBLEM FORMULATION

In the following we provide the Mixed-integer pro-

gramming (MILP) formulation of the NSP. In addi-

tion, we also present the NSP modeling as a WCSP.

The WCSP formulation is used by the B&B algorithm

(Ben Said and Mouhoub, 2022) that we have consid-

ered in the comparative experiments reported in Sec-

tion 6.

3.1 MILP Formulation

Table 1 lists the decision variables and all the required

parameters needed for our formulation of the NSP.

Basically, the main goal is to assign nurses to daily

shifts

such that a set of constraints are met (follow-

ing hospital personnel policies) while an overall cost

is minimized.

Constraints

1. Minimum and Maximum Number of Nurses

per Shift. The following constraint expresses

the minimum and maximum assigned number of

nurses per shift j during day k.

≤

∑

i jk

≤ S

(1)

2. Maximum Number of Shifts for a Given Nurse

During the Schedule. The following constraint

sets the maximum number of shifts w

, for a given

nurse i during the schedule.

∑

i jk

≤ w

(2)

3. Maximum Number of Consecutive Shifts. The

following constraint sets the maximum number of

consecutive shifts L for a given nurse i during the

schedule.

∑

i3k

+ x

i1(k+1 mod d)

) ≤ L (3)

4. Maximum Number of Night Shifts. Each nurse

i should not work more than n

night shifts in the

schedule.

∑

i jk

≤ n

j = 3 (4)

Objective: Hospital Costs to Minimize

min(

∑

i j

· x

i jk

) (5)

We assume that all shifts have the same number of

hours.

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334

Table 1: Parameters of the NSP.

Parameters Description

n Number of nurses

i Nurse index

Morning shift ( j = 1)

j A given shift: Evening shift ( j = 2)

Night shift ( j = 3)

d The number of days in the schedule

i j

The cost of nurse i working in shift j for any day

1, if nurse i works in shift j on day k,

i jk

The decision variable:

0, otherwise

Minimum nurses needed for shift j in day k

Maximum number of nurses required for shift j in day k

Maximum number of shifts during the schedule for nurse i

L Maximum consecutive shifts that are allowed (a consecutive shift cor-

responds to j = 3 for a day k, followed by j = 1 for day k +1, mod d)

Maximum night shifts during the schedule for nurse i

3.2 WCSP Formulation

The CSP is a powerful framework for modeling and

solving combinatorial problems (Dechter and Cohen,

2003). Modeling a given problem using the CSP

consists in formulating it in terms of a set of vari-

ables X = {x

, . . . , x

}, where each variable x

is deﬁned on a non-empty domain of possible val-

ues dom(x

; and a set of constraints C = {c

, ..., c

}

restricting variables’ assignment combinations. The

goal of solving a CSP is to ﬁnd a consistent assign-

ment of values to all the variables from their domains

such that all the constraints are satisﬁed. The WCSP

(Larrosa, 2002; Bidar and Mouhoub, 2023; Lee and

Leung, 2009) is an extension of the CSP which con-

siders violation costs related to soft constraints or

weights associated with variable domain values. In

addition to ﬁnding a solution that satisﬁes all the con-

straints, the target of a WCSP is to optimize the solu-

tion’s total cost.

More formally, a WCSP is deﬁned by the tuple

(X, D,C, K), where X, D, and C are variables,

domains, and constraints respectively. K is the

largest numerical cost value for for a given variable

assignment.

Variables: X = {X

, ..., X

}, the set of nurses.

Domain: D is the set of shift patterns.

Constraints: C = {const

, ..., const

}

To elicit the NSP constraints, we rely on func-

tion A(i, j, k, s) that returns 1 if Nurse i is assigned

shift pattern j, and j covers shift s on day k, or 0 oth-

erwise. Note that constraint const

is a global con-

straint that involves all the variables and constraints

const

, const

are unary constraints. For the

WCSP formulation, we rely on the following param-

eters and indices to elicit the constraints.

Parameters and Indices:











n = Number of nurses

m = Number of possible shift patterns

i j

= Cost of assigning nurse i the shift pattern j

= Minimum nurses needed for shift s

in day k

= Maximum number of nurses required

for shift s in day k

= Maximum number of shifts for nurse i during

the schedule

y = Maximum number of consecutive shifts

(night shift followed by a morning shift)

= Maximum number of night shifts for

nurse i during the schedule

i = {1, .., n}: nurse index

j = {1, ..., m}: index of the weekly shift pattern

k = {1, .., 7}: day index

s = {1, .., 3}: shift index within a given day

z = {1, .., 21}: index of shifts in a shift pattern

Constraints:

1. Minimum and Maximum Number of Nurses

per Shift: const

≤

∑

i=1

A(X

, j

, k, s) ≥ q

∀k, ∀s, a

∈ D (6)

Evolutionary Techniques for the Nurse Scheduling Problem

335

2. Maximum Number of Shifts for a Given Nurse

During the Schedule: const

∑

k=1

∑

s=1

A(X

, j

, k, s) ≤ h

∀X

, a

∈ D (7)

3. Maximum Number of Consecutive Shifts

(Night Shifts Followed by Morning Shifts):

const

∑

k=1

A(X

, j

, k, 3) + A(X

, j

, k + 1, 1) ≤ y ∀X

, a

∈ D

(8)

4. Maximum Number of Night Shifts: const

∑

k=1

A(X

, j

, k, 3) ≤ b

∀X

, a

∈ D (9)

Soft Constraint:

: a

∈ D → c

i j

(10)

Objective: Hospital Costs to Minimize

Minimize(

∑

i=1

i j

) a

∈ D (11)

4 PROPOSED SOLVING

APPROACH

Our proposed method ﬁrst enforces the satisfaction of

all constraints. Then, the obtained consistent outputs

are given to the GA algorithm to search for the opti-

mal solutions.

More precisely, after generating a random popula-

tion of potential solutions (schedules), the ﬁrst part

of the solving method consists in enforcing satisﬁ-

ability by eliminating any detected constraint viola-

tion. After conducting a preliminary work, we came

to the conclusion that constraints should be satisﬁed

using the following order. First, constraint 4, which

is related to the maximum number of night shifts, is

satisﬁed. Then constraint 3, followed by constraint 2

are enforced. Finally, constraint 1, which is the most

challenging constraint, is satisﬁed. To solve each of

the above constraints, the algorithm ﬁrst detects the

variables involved in the constraint violations. Then,

some of these variables are randomly selected, and

their values are ﬂipped in order to satisfy the re-

lated constraint. After enforcing the constraints as

described earlier, the GA operators are applied on the

feasible solutions to ﬁnd the optimal one. In this con-

text, the one-point crossover is ﬁrst applied on some

selected chromosomes. Then, mutation is conducted

to ensure some diversity. Constraint satisﬁability is

enforced on the offsprings after each of these two op-

erations. Figures 1 and 2 illustrate the crossover and

mutation operations, respectively. In our example, we

have 5 nurses working over 7 days. Like for the WOA

(Sadeghilalimi et al., 2023), we consider the follow-

ing variants for the mutation operator.

• Random Resetting Mutation (RRM). A num-

ber of entries of the NSP potential solution are

randomly selected and their values are randomly

changed. This process is depicted in Figure 2.

• Swap Mutation (SwM). After selecting a pair of

variables, their values are swapped.

• Scramble Mutation (ScM). A subset of contigu-

ous entries are selected from the potential solu-

tion. Then these values are randomly scrambled.

• Inverse Mutation (IM). A subset of contiguous

entries are selected and inverted.

The ﬁtness function corresponds to the NSP objective

deﬁned in Equation 5.

5 B&B, SLS, AND WOA FOR THE

NSP

5.1 B&B

We adapted the B&B algorithm from (Ben Said and

Mouhoub, 2022) to reﬂect the minimization variant

of the NSP. B&B uses the Depth First Search (DFS)

strategy to explore all candidate solutions and relies

on the Upper Bound (UB) and Lower Bound (LB) pa-

rameters for pruning the non-optimal and infeasible

solutions. In this context, the UB is used to record

the cost of the best solution found during the search,

and the LB is used to overestimate the quality of the

solutions that may be obtained at any given node.

The latter may be seen as a forward-checking tech-

nique; if the estimated LB is greater than the cur-

rent UB, then there will be no need to continue ex-

ploring the current decision because it would deﬁ-

nitely not lead to a better solution. Note that the

LB is estimated by computing the real costs of al-

ready explored nodes in a given sub-branch plus the

minimum costs of shift patterns that could possibly

be assigned to the remaining variables/nurses. Al-

though B&B guarantees the solution’s optimality, it

may come with an exponential time cost given the

number of nurses and the domain size to be consid-

ered (O(d

), where d is the domain size and n the

number of nurses). To overcome this limitation, con-

straint propagation (Dechter and Cohen, 2003) is used

to reduce the search space and consequently minimize

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

336

Figure 1: An example of the crossover operation.

Figure 2: An example of the RRM Mutation.

the B&B execution time. Constraint propagation is

also conducted as a pre-processing step before the ex-

ecution of B&B to reduce the variable domain sizes

by removing locally inconsistent values that violate

unary, binary, and k-ary constraints. Enforcing con-

straint propagation may result in two different sce-

narios; the ﬁrst one consists in proving the inconsis-

tency of the problem in the case of removing all the

values from a given variable domain (no solution ex-

ists). The second scenario corresponds to removing

some of the values and obtaining a reduced-size do-

main which will requires less effort from the B&B al-

gorithm to search for the optimal solution. We apply

Node consistency (NC) (Larrosa, 2002) through the

unary constraints 2, 3, and 4 (in Section 3), and we

call B&B + NC, the search method using NC as a pre-

processing phase before running B&B. Furthermore,

we apply the Generalized Arc Consistency (GAC) al-

gorithm (Lecoutre and Szymanek, 2006; Cheng and

Yap, 2010) through global constraint 1 (in Section 3)

to eliminate inconsistent domain values that are not

part of any feasible solution, and we call, B&B + NC

+ GAC, the method using both NC and GAC as a pre-

processing step before running B& B.

5.2 SLS

We consider SLS variants: SLS, DFS+SLS, and DFS

+ NC + GAC + SLS. These variants work by obtain-

ing an initial solution and then trying to enhance it by

sequentially looking for a better value substitution for

each variable while maintaining the solution’s feasi-

bility. The difference between the three variants is the

method used to get the initial solution. For SLS, the

initial solution is obtained using a random search. In

DFS+SLS, the initial solution is found after running

a Depth-First-Search (DFS) and ﬁnding the ﬁrst fea-

sible solution. Finally, for DFS + NC + GAC + SLS,

the initial solution is found following a DFS search

after enforcing constraint propagation (NC and GAC)

as a preprocessing step to eliminate inconsistent do-

main values and tackle the exponential time cost that

may come from DFS.

5.3 WOA

The WOA that we have adopted (Sadeghilalimi et al.,

2023) is an adaptation of the original (Mirjalili and

Lewis, 2016) for the NSP. WOA is inspired by the

Evolutionary Techniques for the Nurse Scheduling Problem

337

behavior of humpback whales and can be seen as a

combination of both the moth ﬂame and the grey wolf

techniques (Camacho-Villal

on et al., 2022). In WOA,

each whale corresponds to a chromosome as illus-

trated in Figures 1 and 2. Exploration is performed

with random whales movements through RRM, ScM,

SwM, and IM mutations. Exploitation is Achieved by

having each whale X move toward the best whale X

∗

through shrinking encircling and spiral motions. In

the case of the NSP, these operators are deﬁned as fol-

lows.

Shrinking Encircling

D = |C ·X

∗

(t) − X(t)| (12)

X(t + 1) = X

∗

(t) − A · D (13)

A = 2a · r − a (14)

C = 2 · r (15)

a and r are random parameters in [0,2] and [0,1] re-

spectively. C is set to 1. Equation 13 will then allow

whale X to move closer to X

∗

by reducing (according

to A) the number of entries that are different in both

whales (using the Hamming distance).

Spiral Attack

Each whale (represented by X(t)) approaches its prey

(best whale, X

∗

) by following a spiral curve. b is a

constant and l is a random variable between [-1, 1].

X(t + 1) = D · e

· cos(2πl) + X

∗

(t) (16)

X(t + 1) = X

∗

(t) − A · D

′

(17)

A = e

· cos(2πl) (18)

To balance shrinking and spiral attacks, a random

parameter, p, is generated between [0, 1] to choose

between the two attacks as follows.

A =

(

2a · r −a p < 0.5

· cos(2πl) p ≥ 0.5

(19)

6 EXPERIMENTATION

To evaluate the performance of our proposed GA-

based method, we conducted comparative exper-

iments against variants of B&B (Ben Said and

Mouhoub, 2022) and metaheuristics (SLS and WOA)

(Sadeghilalimi et al., 2023). All the algorithms are

implemented in MATLAB software on a computer

with a Intel Core i5-6200U processor at 2.3 GHz and

Table 2: The NSP parameters used in the experiments.

n d Q

L n

5 7 1 4 5 2 3

10 7 1 7 5 2 3

15 7 1 12 5 2 3

20 7 1 15 5 2 3

30 7 1 25 5 2 3

50 7 1 35 5 2 3

60 7 1 45 5 2 3

80 7 1 65 5 2 3

Table 3: The cost table.

Nurse no.

c(i, j)

Shift 1 Shift 2 Shift 3

1 0.81 0.16 0.64

2 0.90 0.79 0.37

3 0.12 0.31 0.81

4 0.91 0.52 0.53

5 0.63 0.16 0.35

6 0.09 0.60 0.93

7 0.27 0.26 0.87

8 0.54 0.65 0.55

9 0.95 0.68 0.62

10 0.96 0.74 0.58

11 0.15 0.45 0.20

12 0.97 0.08 0.30

13 0.95 0.22 0.47

14 0.48 0.91 0.23

15 0.80 0.22 0.84

8 GB of RAM. The NSP instances parameters and

cost functions are depicted in Tables 2 and 3. The pa-

rameters guiding the techniques we used in the exper-

iments are tuned, to their best using Chess Rating Sys-

tem (CRS-Tuning) (Ve

cek et al., 2016). While CRS-

Tuning is comparable to other known tuning methods

such as Revac (Nannen and Eiben, 2007) and F-Race

(Birattari et al., 2002), it has several advantages as re-

ported in (Ve

cek et al., 2016). For WOA, a and r are

randomly selected from [0, 1] while l is randomly se-

lected from [−1, +1]. In the case of our proposed GA

method, a one-point crossover is chosen with proba-

bility 0.8, and the roulette wheel method is adopted.

For the mutation, we consider Swap Mutation, Se-

quence Swap Mutation, and Inversion Mutation, re-

spectively with probability 0.1, 0.02, and 0.08.

We have adopted the same population size for both

GAs and WOA. This is justiﬁed by the fact that the

function evaluation (ﬁtness computation) cost is the

most expensive part in a nature-inspired technique,

and its related running time is directly affected by the

number of individuals in a population. Table 4 reports

on the experiment results comparing variants of the

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338

Table 4: Comparative running time and quality of the solution for all the methods.

Method

Number of Nurses

5 10 15 20 30 50 60 80

BS RT (s) BS RT (s) BS RT (s) BS RT (s) BS RT (s) BS RT (s) BS RT (s) BS RT (s)

GA + RRM 8.87 3.85 16.95 6.77 22.92 8.32 34.74 11.15 49.95 21.95 88.84 150.66 114.38 285.57 156.06 225.70

GA + SwM 9.01 6.31 17.06 6.82 24.58 8.45 35.19 11.57 51.2 22.01 89.12 189.72 114.99 298.75 156.89 331.64

GA + ScM 9.45 6.68 17.20 7.11 25.23 9.22 35.20 11.89 54.70 22.48 89.65 214.43 115.58 312.71 157.69 350.14

GA + IM 9.51 7.03 17.25 8.32 25.64 10.18 35.43 12.89 54.97 22.84 90.15 205.98 116.11 322.44 158.67 358.15

WOA 10.22 1.01 21.29 1.62 33.39 3.03 40.83 2.90 69.93 37.86 106.25 244.34 124.75 2221.78 177.66 3393.23

WOA + RRM 10.82 1.50 21.72 1.46 30.56 1.50 43.09 9.98 65.04 44.33 105.04 412.05 129.19 653.03 170.18 4196.58

WOA + SwM 10.29 0.95 22.81 1.53 32.48 0.97 41.85 14.96 65.28 24.40 103.26 340.61 127.09 1073.41 175.55 959.06

WOA + ScM 9.78 1.73 19.56 0.11 29.51 3.49 40.68 6.01 63.38 42.05 102.01 332.04 123.40 240.24 169.48 1301.61

WOA + IM 10.45 1.85 22.19 0.21 33.01 1.60 41.21 15.41 63.49 4.14 103.78 277.68 127.66 599.18 173.49 2005.55

SLS 11.57 0.69 24.10 0.50 32.99 1.23 47.28 1.12 72.32 1.17 109.49 1.49 142.92 1.97 189.89 2.79

DFS + SLS 14.86 18.90 28.68 164.11 38.91 345.57 43.62 404.75 76.08 1060.31 119.21 3873.79 148.34 4351.57 189.96 6901.90

DFS + NC + GAC + SLS 12.34 5.49 25.81 89.24 32.28 100.16 49.33 246.71 67.98 1394.28 109.96 3830.05 140.86 4781.31 185.54 5813.59

B&B + NC 9.68 1894 18.86 14415 25.55 22689 38.58 28137 52.34 34259 89.35 41459 - - - -

B&B + NC + GAC 9.68 534 18.86 11400 25.55 16211 38.58 23418 52.34 29768 89.35 37108 - - - -

Table 5: Comparative quality of the solution in Best, Average, and Standard deviation.

Method

Number of Nurses

5 10 15 20 30 50 60 80

Best Ave. Dev. Best Ave. Dev. Best Ave. Dev. Best Ave. Dev. Best Ave. Dev. Best Ave. Dev. Best Ave. Dev. Best Ave. Dev.

GA + RRM 8.87 10.12 0.95 16.95 18.12 1.12 22.92 24.01 1.42 34.74 36.21 1.39 49.95 51.54 1.51 88.84 90.27 1.61 114.38 116.30 1.67 156.06 158.39 1.49

GA + SwM 9.01 10.28 1.09 17.06 18.53 1.53 24.58 25.93 1.42 35.19 36.33 1.54 51.2 52.56 0.61 89.12 91.29 1.13 114.99 116.81 1.08 156.89 158.84 1.01

GA + ScM 9.45 10.37 0.73 17.20 18.79 0.64 25.23 26.39 0.59 35.20 36.77 0.46 54.70 56.22 1.23 89.65 93.02 1.75 115.58 117.11 1.32 157.69 159.55 1.18

GA + IM 9.51 10.85 0.69 17.25 18.74 0.64 25.64 27.2 1.08 35.43 37.14 1.82 54.97 56.89 1.49 90.15 93.41 1.81 116.11 118.21 1.82 158.67 161.01 1.48

WOA 10.22 11.79 0.76 21.29 22.33 0.88 33.39 35.21 1.45 40.83 42.25 1.77 69.93 71.09 1.31 106.25 109.2 1.75 124.75 126.5 1.4 177.66 179.29 0.91

WOA + RRM 10.82 11.28 0.69 21.72 23.01 1.93 30.56 32.15 1.2 43.09 44.98 1.38 65.04 67.25 1.39 105.04 107.34 2.18 129.19 131.35 1.67 170.18 172.71 1.48

WOA + SwM 10.29 11.89 1.54 22.81 24.51 1.45 32.48 34.33 1.69 41.85 43.39 1.67 65.28 68.01 2.18 103.26 105.21 1.88 127.09 129.92 2.22 175.55 178.18 1.49

WOA + ScM 9.78 11.15 1.4 19.56 21.57 1.27 29.51 31.35 2.13 40.68 42.59 1.69 63.38 65.83 1.86 102.01 105.2 1.12 123.40 125.35 2.58 169.48 171.96 1.46

WOA + IM 10.45 11.87 1.58 22.19 23.94 1.88 33.01 35.13 1.42 41.21 43.84 1.38 63.49 65.42 1.04 103.78 105.86 2.27 127.66 129.63 1.84 173.49 175.83 1.14

SLS 11.57 14.44 1.33 24.10 26.81 1.48 32.99 38.09 2.28 47.28 51.45 2.03 72.32 77.75 2.49 109.49 122.74 4.67 142.92 147.69 3.67 189.89 202.84 6.91

GA and WOA algorithms with B&B and SLS, as de-

scribed previously. The quality of the best solution re-

turned (BS) and the corresponding running time (RT)

were used as comparison criteria. All the results are

averaged over 20 runs. The experiments were con-

ducted on several NSP instances, with the number of

nurses varying from 5 to 80. Given that it is an exact

method, B& B returns the optimal solution for each

problem instance, up to 50 nurses. When the number

of nurses exceeds 50, B & B fails to return the optimal

solution due to its exponential time cost. For these

large instances, GA with RRM is the best method re-

garding the returned solution’s quality. The same re-

mark regarding the exponential time cost can be said

when adding DFS as an initial step for the SLS algo-

rithm. In both B&B and SLS, constraint propagation

does help lower the running time (as a consequence

of reducing the running time). However, this effort

is insufﬁcient to compete with WOA and GA vari-

ants. SLS is the best method in terms of running time.

However, the quality of the solution returned by the

latter is inferior to those returned by variants of GA

and WOA. Given that our experiments are conducted

over 20 runs, we conducted a statistical analysis on

the solution returned by each metaheuristic. We re-

ported the results in Table 5 regarding the quality of

the returned solution.

7 CONCLUSION AND FUTURE

WORK

We have proposed a GA-based method to solve the

NSP. To evaluate the efﬁciency of our method, we

conducted experiments against exact and approximate

techniques. The obtained results are promising com-

pared to B&B, which suffers from its exponential

time cost and fails in returning solutions for large-size

problem instances.

In the near future, we plan to explore other nature-

inspired techniques (Hmer and Mouhoub, 2016; Ko-

rani and Mouhoub, 2021) including Particle Swarm

Optimization (PSO) (Kennedy and Eberhart, 1995)

and further plan to tackle the dynamic variant of the

NSP problem, which is characterized by the unpre-

dictable occurrence of events such as nurses call-

ing in sick or sudden changes in hospital demand

in particular scenarios (such as emergencies) (Bidar

and Mouhoub, 2022; Mouhoub, 2003). We will also

use variable ordering heuristics (Yong and Mouhoub,

2018) to improve the time efﬁciency of the B& B al-

gorithm.

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