PME

I
J

International Journal of 
Production Management 
and Engineering

https://doi.org/10.4995/ijpme.2022.16736 

Received: 2021-11-25 Accepted: 2022-01-24

Comparative Study of Whale Optimization Algorithm and Flower 
Pollination Algorithm to Solve Workers Assignment Problem

Huthaifa Al-Khazraji 

a Control and System Engineering Department, University of Technology-Iraq, Baghdad, Iraq. 
60141@uotechnology.edu.iq

Abstract:
Many important problems in engineering management can be formulated as Resource Assignment Problem 
(RAP). The Workers Assignment Problem (WAP) is considered as a sub-class of RAP which aims to find an 
optimal assignment of workers to a number of tasks in order to optimize certain objectives. WAP is an NP-hard 
combinatorial optimization problem. Due to its importance, several algorithms have been developed to solve 
it. In this paper, it is considered that a manager is required to provide a training course to his workers in order 
to improve their level of skill or experience to have a sustainable competitive advantage in the industry. The 
training cost of each worker to perform a particular job is different. The WAP is to find the best assignment of 
workers to training courses such that the total training cost is minimized. Two metaheuristic optimizations named 
Whale Optimization Algorithm (WOA) and Flower Pollination Algorithm (FPA) are utilized to final the optimal 
solution that reduces the total cost. MATLAB Software is used to perform the simulation of the two proposed 
methods into WAP. The computational results for a set of randomly generated problems of various sizes show 
that the FPA is able to find good quality solutions.

Key words:
Servitization, Resource Assignment Problem, Workers Assignment Problem, Metaheuristic Optimization, Whale 
Optimization Algorithm, Flower Pollination Algorithm. 

1. Introduction

With the increase in competition in the global 
market, industrial companies are forced to improve 
their manufacturing processes via cutting costs and 
increasing process efficiency (Ostadi et al., 2021). 
In this direction, it’s become necessary for decision-
makers to find the best strategies that utilize their 
resources in order to have the best performance (Lin 
and Chiu, 2018). The problem of finding the best 
utilization of resources in the industrial companies 
is named a Resource Assignment Problem (RAP). 
Among many varieties of resources, human 
resources play a significant role in the success of 
the industrial organization if they are well allocated 
to different services or systems, with an aim to 

maximize or minimize certain objectives related to 
performance and productivity (Bouajaja and Dridi, 
2017). Therefore, Worker Assignment Problem 
(WAP) is defined as a process of assigning workers 
among various tasks for maximization of the profit 
(or efficiency) or minimization of the cost (or time). 
A WAP is the most widely used in the context of 
industrial and engineering management such as in 
production planning and maintenance management 
(Krokhmal and Pardalos, 2009).

For some industrial processes, the load needs to be 
distributed among workers in such a way that the 
time is minimized or/and the efficiency is increased. 
Moreover, it can be noticed that there are particular 
workers who can perform some of the jobs with less 

To cite this article: Al-Khazraji, H. (2022). Comparative Study of Whale Optimization Algorithm and Flower Pollination Algorithm to Solve Workers Assignment Problem. 
International Journal of Production Management and Engineering, 10(1), 91-98. https://doi.org/10.4995/ijpme.2022.16736 

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time or more efficient way than others due to their 
experience or skills (Mahmoud, 2009). Caron et al. 
(1999) considered the case of WAP with additional 
constrain that an unassigned worker cannot be given 
a certain job unless the unassigned worker has the 
qualification to perform that particular job. However, 
if the manager focuses only on the workers which 
have skills or experiences to perform the tasks 
without having the right blanching of distrusting the 
loads among all workers, a worker fatigue problem 
might be happed in the long term (Yadav et al., 
2020; Demiral, 2017). Besides, intangible resources 
represented by the skill and experience of the workers 
are valuable and scarce. Moreover, the experiences 
that the workers gained in the past are not enough 
to have a sustainable competitive advantage in the 
present (Ruiz et al., 2020). Therefore, a manager 
has to provide a training course continuously to 
his workers in order to improve their level of skill 
or experience to have a sustainable competitive 
advantage in the industry.

The problem of WAP that is considered in this paper 
can be stated as follows: if the manager needs to 
assign n of jobs to n of workers where the training 
cost of each worker to perform that particular job is 
different. The problem is to find the best assignment 
of workers to training courses such that the total 
training cost is minimized. The problem is NP-hard 
problems with enormous search spaces (Ammar 
et al., 2013).

Pentico (2007) reviewed the mathematical model for 
most of the variations of the Assignment Problem 
(AP) which is the general form of WAP. Bouajaja 
and Dridi (2017) presented a comprehensive review 
study on the numerous approaches that applied to 
solve varieties classes of WAP in different application 
areas. The General AP (GAP) can be formulated as 
integer linear programming. Different methods such 
as exact, heuristic and metaheuristic are developed 
to solve the problem. Kuhn (1955) developed the 
well-known Hungarian method to solve the general 
AP. In Ross and Soland (1975), a Branch and Bound 
(B&B) technique is proposed to solve the general AP. 
Xuezhi and Xuehua (1996) described how AP can be 
solved using dynamic programming. Exact methods 
such as the Hungarian method, B&B technique and 
dynamic programming are only effective in certain 
problems with a small size of decision variables. 
Therefore, larger-sized problems are often solved 
by using heuristic and metaheuristic to obtain high-
quality solutions with reasonable computational time 
(Bouajaja and Dridi, 2017).

In terms of heuristic methods, Cattrysse et al. 
(1994) proposed a column generation heuristic 
method where the problem was formulated as a set 
partitioning problem. On other hand, among many 
metaheuristic methods, Ant Colony Optimization 
(ACO) was the most approach that is utilized 
to solve the problem. For example, Wang et al. 
(2009) presented a detailed procedure to apply 
ACO to WAP where the objective was to maximize 
efficiency. Demiral (2017) implemented ACO for a 
set of randomly generated WAP. Three objectives 
of WAP (minimization cost, maximization sales and 
maximization profit) were investigated in the study. 
Statistical analysis based on mean, standard deviation 
and variance was performed to help decision-
makers to select the best objective based on their 
perspective. In the same direction, Suliman (2019) 
examined the performance of ACO in comparison 
with the traditional Hungarian method for solving the 
WAP with the size of 3×3 in terms of running time, 
number of iteration and quality of solutions. Besides 
ACO, Chu and Beasley (1997) presented a genetic 
algorithm (GA) for solving the AP. Jia and Gong 
(2008) solved the multi-objective WAP using Multi-
Objective Particle Swarm Optimization (MOPSO). 
In this paper, two metaheuristic optimizations named 
Whale Optimization Algorithm (WOA) and Flower 
Pollination Algorithm (FPA) are utilized final the 
optimal solution that reduces the total cost. MATLAB 
Software is used to perform the simulation of the two 
proposed methods into WAP.

2. Training Course

Training can be defined as a process of developing 
programs to ensure that employees are provided 
the right skills that are needed to achieve better 
positive in the market (Halawi and Haydar, 2018). 
The importance of providing training courses to the 
workforce of the industrial companies appears to be 
a recognition strategy toward gaining a competitive 
advantage in the global market competition 
(Sharma, 2014). Workforces are required continuous 
training courses to have sustainability professional 
qualifications to cope with the recent advanced 
technology within industry 4.0. According to Walsh 
and Volini (2017), 80% of human resource managers 
reported that workforce training is one of the 
biggest problems to improve the effectiveness and 
competitiveness of industrial companies.

In this direction, industrial companies are compelled 
to provide workforce training strategies continuously 

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to increase productivity. The present paper consideres 
that a manager is required to assign n of jobs to n of 
workers where the training cost of each worker to 
perform a particular job is different. The problem 
is to find the best assignment of workers to training 
courses such that the total training cost is minimized.

3. Mathematical Model

Consider there are (n) of workers (w) are required to 
be assigned to (n) of training courses (s) as shown in 
Figure 1. Each worker wi (i=1,2,…,n) can be assign 
to any course sj (j=1,2,…,n) with different cost 
(ci,j). The problem is to find the best assignment of 
workers to courses such that the total cost of training 
is minimized. The number of the workers was 
assumed equal to the number of courses in this study. 
The WAP is formulated as (Krokhmal and Pardalos, 
2009):

∑∑
= =

n

i

n

j
ijijxcMin

1 1

 (1)

s.t

∑
=

=
n

i
ijx

1
1 (j=1,2,…,n) (2)

∑
=

=
n

j
ijx

1
1 (i=1,2,…,n) (3)

where xij has two values, either 1 if the worker i is 
assigned to job j, otherwise is zero. The constraint 
in Equation (2) satisfies that each training course is 
assign to a worker and the constraint in Equation (3) 
satisfies that each worker is assign to a training 
course.

Figure 1. Workers Assignment Problem.

4. Solution Approach

Bio-inspired algorithms are considered powerful 
in solving NP-hard combinatorial optimization 
problems (Yang, 2009). Therefore, two algorithms 
which are inspired from the biological systems in 
nature are proposed to final the optimal solution 
that reduces the total cost. These two algorithms are 
Whale Optimization Algorithm (WOA) and Flower 
Pollination Algorithm (FPA). The next subsections 
explain these two algorithms.

4.1. Whale Optimization Algorithm
Whale Optimization Algorithm (WOA) is a 
population-based swarm optimization algorithm. It 
was developed by Mirjalili and Lewis in 2016. WOA 
mimics the bubble-net hunting behavior of humpback 
whales. The mathematical model of this algorithm 
consists of two processes named exploitation and 
exploration.

In the exploitation process, the position of the 
humpback whale is updated based on the location of 
the prey using a bubble-net attacking strategy. In this 
strategy, the movement towards the prey is performed 
by two mechanisms (Satapathy et al., 2018). The 
first one is the shrinking encircling mechanism. This 
behavior is represented by the following equations 
(Mirjalili and Lewis, 2016):

Q=2r1 (4)

A=2ar2–a (5)

D=Q·p*(t)–p(t) (6)

p(t+1)=p*(t)–A·D (7)

where

 - a  Coefficient value linearly decreased from 2 to 
0 for each iteration

 - Q  Coefficient value calculated as given in 
Equation (4)

 - A  Coefficient value calculated as given in 
Equation (5)

 - D  Coefficient value calculated as given in 
Equation (6)

 - r1,r2 Random value between [0,1]

 - t Current iteration

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 - p* Position of the prey

 - p Position of the whale

The second mechanism is named spiral updating 
position where in this strategy the whale moves 
towards the prey in a helix-shaped movement. This 
behavior is represented by the following equations 
(Mirjalili and Lewis, 2016):

D'=│ p*(t)–p(t) │ (8)

p(t+1)=D;ebl cos(2πl)+p*(t) (9)

where

 - D'  Coefficient value calculated as given in 
Equation (8)

 - b  Constant used to define the shape of the 
logarithmic spiral

 - l Random value between [0,1]

To model the changes between these two strategies, 
it was assumed that there is a probability of 50% 
to select between the shrinking encircling strategy 
and the spiral one to update the position of the 
current whale in the simulation of the algorithm. 
The mathematical model of this scenario can be 
formulated by selecting a random value (Rand), then 
if the value of the random value >50 the movement 
of the individual will be performed based on 
Equation (7), otherwise wiii be performed based on 
Equation (9) (Mirjalili and Lewis, 2016).

In the exploration process, the position of the 
humpback whale is updated randomly. The basic 
idea of this strategy is to ensure the search space 
explored globally. This strategy is represented by the 
following equations (Mirjalili and Lewis, 2016):

D=│Q·prand(t)–p(t) │ (10)

p(t+1)=prand (p)+A·D (11)

where

 - prand  Random position chosen from the current 
population

The pseudo code of FPA is illustrated in Figure 2.

1. Input
✓  Objective function (fitness function), population size (N), coefficient value 

a, number of iteration T
2. Initialization
✓  Initialize population
✓  Evaluate Objective function
✓  Find p*

3. Loop:
✓  For t = 1:T
✓  For i = 1: N
✓  Update Q as in Eq. (4), A as in Eq. (5) and select random value θ
✓  If θ > 0.5
➢ If |A| < 1:Update the position of the current whale based on Eq. (7)
➢ If |A) > 1:Update the position of the current whale based on Eq. (9)

✓  Else
➢ Update the position of the current whale based on Eq. (11)

✓  Perform greedy selection and update p"
✓  If there is no convergence of the current solution & if t >T go to Loop

4. Print the optimal solution

Figure 2. The pseudo code of WOA.

4.2. Flower Pollination Algorithm

Flower Pollination Algorithm (FPA) is a swarm-
based meta-heuristic optimization developed by Yang 
in 2012. The FPA mimics the pollination phenomena 
in the flower. The main idea of the pollination in 
flower is to transfer the pollen from the male into the 
female. This process can be classified based on the 
way that pollen is transferred into biotic and abiotic. 
In the biotic, the pollinator can be animal or insect, 
whereas in the abiotic, the pollinator is the wind and 
diffusion in water (Abdel-Basset and Shawky, 2019). 
The optimization procedure of the FPA starts with 
randomly initialized a population of N of flowers 
within the search space as given in Equation (12):

yi = yl+Rand*(yu–yl)  (12)

where

 - i  Counter (i= 1,2,3,…)

 - Rand  Initial solution

 - yi  Random value between [0,1]

 - yl  Lower bound of the search space

 - yu  Upper bound of the search space

In the FPA, there are two ways to search of the 
optimum value. These are the global pollination 
and local pollination (Yang, 2012). In the global 
pollination stage, the movement of each individual 
in the population is directed by the one that has the 

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best cost function found yet. This can be represented 
mathematically as:

yinew = yiold + σ*(yiold –yg) (13)

where

 - yinew  New solution

 - yiold  Old solution

 - yg  The current best solution

 - σ  The step size

The step size σ can be set fixed or follow a random 
steps such as Lèvy flight. In terms of local pollination, 
the algorithm selects two solutions randomly, and 
then the new solution is generated based on the 
following (Yang, 2012):

yinew = yiold + ε*(yj –yk) (14)

where

 - yj  A solution chosen randomly

 - yk  A solution chosen randomly

 - ε  Random value between [0,1]

The pseudo code of FPA is illustrated in Figure 3

1. Input
✓  Objective function (fitness function), population size (N), switch 

probability (p), number of iteration (T)
2. Initialization

✓  Initialize population N flowers based on Eq. (12)
✓  Evaluate objective function and assign yz

3. Loop:
✓  For t = 1: T
✓  For i = 1: N
✓  If rand < p

➢  Generate a step size (σ)
➢  Generate a new solution based on Eq. (13) (global pollination)

✓  Else
➢  Choose two solutions randomly among all solutions
➢  Generate a new solution based on Eq. (14) (local pollination)

✓  Perform greedy selection and update yg
✓  If there is no convergence of the current solution & if t > T go 

to Loop
4. Print the optimal solution

Figure 3. The pseudo code of FPA

5. Simulation Study

For evaluating the performance of the two algorithms 
(WOA and FPA) to solve WAP, a set of different size 

randomly generated problems have been used. Three 
sizes (n=5,10,15) of WAP are considered to perform 
the evaluation as given in the Appendix I. MATLAB 
Software is used to perform the simulation. MATLAB 
becomes a powerful tool in wide applications in 
engineering, economics and management. It can 
handle different computational algorithms with a 
reasonable time. On other words, different algorithms 
could be tested and evaluated with less time. As a 
consequence of using simulation, more knowledge 
and insight can be gained to enhance the solution of 
WAP. All simulations were conducted on a computer 
with Intel(R) Core(TM) CPU i7-4500 and 8 GB 
RAM. The WOA and FPA parameters are presented 
in Table 1. For justify the comparison between the 
WOA and FPA, the size of the population and the 
number of iteration are set equally. The coefficient 
value (a) is set to 2 as recommended by the (Mirjalili 
and Lewis, 2016). In the same way, the value of the 
switch probability (p) is set to 0.5 as recommended 
by the Yang (2012).

Both algorithms were run 10 times and the statistical 
data such as the average (Avg.), the maximum (Max), 
the minimum (Min) and the standard deviation (Std.) 
were recorded for each algorithm. Table 2 presents 
the statistical data of the experiments for the three 
problem size of WAP using WOA and FPA.

Table 1. WOA and FPA algorithms parameters.

Parameters WOA FPA
Number of population (N) 50 50
Number of iteration (T) 100 100
Coefficient value (a) - 2
Switch probability (p) 0.5 -

Table 2. Statistics for solving WAP using WOA and FPA

Size Method Avg. Min Max Std.

5
WOA 30 30 30 0
FPA 30 30 30 0

10
WOA 64.6 63 67 1.175
FPA 62.2 61 63 0.78

15
WOA 104.1 103 105 0.738
FPA 101.8 101 102 0.422

It can be noticed form Table 2 that in general for 
small size problem, both algorithms are obtained a 
good solution results. However, Table 2 show that 
if the size of the problem increased (i.e. n=10 and 
n=15), the FPA is recommended to solve the WAP. 
FPA shows better performance in terms of obtaining 
a less average value, less minimum value, less 
maximum value and less standard deviation.

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6. Conclusions

The Worker Assignment Problem (WAP) is 
an important problem faced by manufacturing 
companies. WAP is an NP-hard combinatorial 
optimization problem. Human resources play 
a significant role in the overall performance of 
manufacturing companies. Workforce training is one 
of the biggest problems in the industrial companies 
to improve effectiveness and competitiveness. The 
paper considers the problem of assigning workers to 
training courses in order to improve the level of skill 
or experience of the worker to have a sustainable 
competitive advantage in the industry. The training 

cost to perform a particular job of each worker is 
different. The WAP is to find the best assignment of 
workers to training courses such that the total training 
cost is minimized. Two metaheuristic optimizations 
named Whale Optimization Algorithm (WOA) and 
Flower Pollination Algorithm (FPA) are utilized 
to final the optimal solution that reduces the total 
cost. The simulations results reveal that for a small 
size problem, both algorithms are obtained a good 
solution result. However, for a large size problem (i.e. 
n=10 and n=15), the FPA is recommended to solve 
the WAP. FPA shows better performance in terms of 
obtaining a less average value, less minimum value, 
less maximum value and less standard deviation.

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http://creativecommons.org/licenses/by-nc-nd/4.0/


Appendix I

This appendix presents the three randomly generated WAP that were used in this study.

Size Example (ci,j) Optimal Assignment (xi,j)
Minimum 

Cost

5

⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

67978
108776
710968
89688
768710

⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

10000
00001
00010
00100
01000

30

10

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

101096789788
1167988108710
79787710976
101069789788
119798988610
811107987678
910796108779
77787610978
61098789788
119798868710

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

0001000000
0100000000
0000000001
0010000000
0000000010
0000000100
0000100000
0000010000
1000000000
0000001000

60

15

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

1161099711109799788
10117971167988108710
877111099781098769
77679810877878910
8781197778786979
788108910796108778
6118981187988108710
118771079787710976
89768119798988810
7119871010910769788
11897861098789788
791079810877879610
878116798797871011
78978101096789788
97789101069789788

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

010900000000000
000000100000000
000000000000010
001000000000000
000000000001000
000000000100000
100000000000000
000000000000001
000100000000000
000000000010000
000001000000000
000000000000010
000010000000000
000000001000000
000000010000000

90

Int. J. Prod. Manag. Eng. (2022) 10(1), 91-98 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Al-Khazraji

98

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