Masterpiece Optimization Algorithm: A New Method for Solving Engineering Problems

Journal of Soft Computing in Civil Engineering 9-1 (2025) 1-21
How to cite this article: Ghasemi MR, Aghajanpour NH, Ghohani A. Masterpiece optimization algorithm: a new method for
solving engineering problems. J Soft Comput Civ Eng 2025;9(1): 1–21. https://doi.org/10.22115/scce.2024.404534.1677
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Masterpiece Optimization Algorithm: A New Method for Solving
Engineering Problems
Mohammad Reza Ghasemi 1,*
; Nader Haji Aghajanpour 2
; Hamed Ghohani Arab 3
1. Professor, Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
2. Ph.D. Candidate, Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
3. Associate Professor, Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
* Corresponding author: mrghasemi@eng.usb.ac.ir
https://doi.org/10.22115/SCCE.2024.404534.1677
ARTICLE INFO ABSTRACT
Article history:
Received: 21 July 2023
Revised: 03 March 2024
Accepted: 03 March 2024
The "Masterpiece Optimization Algorithm (MOA)" is
introduced as a new nature-inspired meta-heuristic
optimization method that mimics the masterpiece of
Japanese Male Puffer Fishes (MPFs) that succeed in mating.
Actually, the nest prepared by the MPFs for spawning is
known as the masterpiece. Due to the importance of seabed
fine sand to fish reproduction, the proposed algorithm is
based on movement and accumulation of seabed fine sand
particles. In order to find a place with a sufficient amount of
fine sand for spawning, the MPF starts a search on the
seabed. It then constructs the nest so that fine sand covers the
eggs. The location with the greatest quantity of fine sand is
where the majority of egg protection occurs. In the
algorithm, this point is considered the global optimum. Now,
to handle this content, in the first iteration of the proposed
algorithm, different points in the answer zone are introduced
for the MPF. Then, the MPF selects a point for the nest
location using the Evaluator Function (EF). In continuation
of the first iteration, three steps are executed. The point
acquired in the third step represents a local optimum. For the
next iteration, this point is deemed the nest location, and the
aforementioned three steps are also executed. This process is
done iteratively during each iteration. The best point of the
last iteration is the global optimum of the problem in the
answer zone. The obtained solutions of the MOA were then
compared with some available techniques in the literature.
Keywords:
Masterpiece optimization
algorithm;
Masterpiece of puffer fish;
Meta-heuristic methods;
Nature-inspired method.

2 M.R. Ghasemi et al./ Journal of Soft Computing in Civil Engineering 9-1 (2025) 1-21
1. Introduction
1.1. History of meta-heuristic optimization methods
Optimization is one of the universe’s features. There is no doubt that it plays a key role in
formation and stability of the universe. A little thinking about the phenomena of the universe
reminds us that optimality is a fundamental principle of nature. For example, atoms form in a
state where the energy of electrons is optimal [1]. Then, molecules and material crystals form in
an optimal state of energy. This process, which makes it easy to adapt to the environment, is
continued at higher levels. One of the most obvious behaviors observed in animals is to find
food. There are many researchers who have started extensive studies in the field of optimization.
Meta-heuristic optimization methods that have become famous in engineering in recent years are
inspired by nature. The following four groups can be a suitable division for these methods:
human-based, evolution-based, swarm-based, and physic-based algorithms.
The human-based algorithms mimic human behaviors and activities. The following are examples
of human-based methods: Teaching-Learning Based Optimization (TLBO) [2], Harmony-Search
(HS) [3], Taboo Search (TS) [4,5], Group Search Optimizer (GSO) [6], and War Strategy
Optimization algorithm (WSO) [7].
In The evolution-based algorithms that are inspired by natural evolution rules, a random initial
population is created and evolves in the next generation. The most famous evolution-based
method is the Genetic Algorithm (GA) [8–10]. This method is based on Darwinian evolution
theory. In addition, the following methods may be categorized in this group: Evolution Strategy
(ES) [11], Genetic programming (GP) [12], Biogeography Based Optimizer (BBO) [13], and
Population Based Incremental Learning (PBIL) [14].
In the swarm-based algorithms that are inspired by the social behavior of animal groups, the
number of creatures moves in a search space to find the best solution. The most popular method
of the swarm-based group is the Particle Swarm Optimization (PSO) [15]. In addition, the
following methods can also fall into this group: Ant Colony Optimization method (ACO) [16],
Bird Mating Optimizer (BMO) [17], Fruit-fly Optimization Algorithm (FOA) [18], Krill-Herd
(KH) [19], Gannet Optimization Algorithm (GOA) [20], Artificial Rabbits Optimization (ARO)
[21], Starling Murmuration Optimizer (SMO) [22], gaze cues learning-based grey wolf optimizer
(GGWO) [23], GWOEHO [24], and artificial Jellyfish Search (JS) optimizer[25].
The physic-based algorithms mimic physic rules in the universe. The following methods are
examples of this group: Simulated-Annealing (SA) [26], Gravitational Local Search Algorithm
(GLSA) [27], Big-Bang, Big-Crunch (BB-BC) [28], Central Force Optimizer (CFO) [29], and
Gravitational Search Algorithm (GSA) [30].
Some methods are introduced as a combination of two categories, such as: Sine-Cosine hybrid
optimization algorithm with Modified Whale Optimization Algorithm (SCMWOA) [31],
enhanced Quantum behaved Particle Swarm Optimization (e-QPSO) [32], and hybrid PSO-GRG
[33]. In addition, Lamghari et al. (2022) proposed a new metaheuristic method. In this method,
the components from exact algorithms are hybridized [34]. A hybrid Arithmetic Optimization

M.R. Ghasemi et al./ Journal of Soft Computing in Civil Engineering 9-1 (2025) 1-21 3
Algorithm (AOA) with Gold-SA was presented by Liu et al. (2022). This method, called
HAGSA, improves the performance of AOA [35].
Some researchers improved the performance of the existing methods. A Balanced Teaching-
Learning Based Optimization was presented by Taheri et al. (2021). This method, called
BTLBO, improves the performance of TLBO [36]. The memory-based Grey Wolf Optimizer was
proposed by Gupta and Deep (2020). This method, called mGWO, is a modification of Grey
Wolf Optimizer (GWO) [37]. In another study, Dhawale et al. (2023) enhanced the Harris
Hawk’s Optimizer (HHO) in both exploration and exploitation phases [38].
The MOA that is presented in this article is placed in the swarm-based group.
1.2. Masterpiece of japanese MPF
Every creature struggling to survive may have its most important task in life. The latest research
about the Japanese puffer fish shows optimal performance in reproductive behavior [39]. The
MPF plays an essential role in reproduction, including determining the spawning location,
building the nest, and providing coverage for the eggs until they hatch. Due to the seven to nine
days of continuous nest construction and the nest's unique shape, it has been dubbed the
masterpiece or mysterious circle. In the nest, the prepared spawning bed is composed of fine
sand. On the other hand, the cover protecting the eggs is composed of fine sand, too. After the
eggs are hatched, due to the reduction of fine sand, the MPF doesn’t reuse a nest [39]. It seems
the fine sand is necessary for the reproduction of Japanese puffer fishes, and the optimal
performance is for the MPF to find a place with a sufficient amount of fine sand to construct the
nest in. The female fish then enters the nest and spawns. The eggs that are well protected will
then turn into fish. Thus, after spawning, there is a point with the maximum volume of fine sand
to cover eggs. In the algorithm, this point is assumed the global optimum. The second section of
the paper describes how to reach this point in an optimization problem by drawing inspiration
from the masterpiece. An illustration of the seabed search and the masterpiece constructed by
MPF is shown in Fig. 1.
(a) Fine sand exploration (b) Masterpiece constructed by MPF (spawning nest)
Fig. 1. The role of the MPF in reproduction.

2. Masterpiece optimization algorithm
As stated earlier, the fine sand accumulated on the seabed is treasured for the fish because it
ensures their regeneration and survival over time. The MOA described in this section tries to find
the points on the seabed (response area) that have the maximum volume of fine sand with the
help of spawning nest geometry as well as water flow. It is important to note that many MPFs
attempt to mate, but only one is chosen by the female after examining their prepared nest.
Therefore, the MPF may not be chosen and mated [39]. The proposed algorithm is written for an
MPF that succeeds in reproduction.
Since many of the problems are constrained, use has been made of an Evaluator Function (EF) to
evaluate the volume of the fine sand in the searching zone [40]. Thus, it gives:
 
1
2
10
1 .
1.1
i
i i
p
i
p p
EF GF r CV
r r 
 
 
 
 
 
  
 
 
(1)
Where, GF is the Goal Function, CV is the Constraint Violation, and i
p
r is the penalty
coefficient for constraint violation in iteration i. It is based on statistical studies that cause a
strong convergence. [] shows the integer part, and the plus and minus signs are used for the
minimization and maximization, respectively.
In the beginning of the algorithm, some random search points are created in the response area.
Then the volume of fine sand is checked in them using Eq. (1). The best point of these created
points is the point with the minimum EF in minimization problems or the maximum EF in
maximization problems. In the first iteration, the location of this best point is considered the
center of the nest. In continuation of the first iteration and also in each iteration of the algorithm,
the following three steps are executed:
2.1. Step 1: Choosing the best point inside the nest
Due to the circular shape of the puffer fish’s masterpiece, the relations produced in two
consecutive dimensions follow the equation of a circle. It means that the relations in the even
dimension depend on the odd dimension. It should be noted that in the construction of the nest,
the MPF accumulates fine sand in peripheral dents (valleys). After spawning, the sand
accumulated in dents is moved by the water flow and covers the eggs [39]. Therefore, in the
algorithm provided and developed, some points are created on the imaginary circle perimeter as
the valleys. Also, some search points are created around the center of the nest inside the
imaginary circle. The valleys are created using the following relations
 
2 1 2 1
cos
new center
d m d m
i i i
d
x x r a
   
   (2)
 
2 2
sin
new center
d m d m
i i i
d
x x r a
 
   (3)

Where, 2m-1 shows the odd direction and 2m indicates the even direction. a is the angle of the
intended valley. It alters from the [0,2π] interval as a division of the number of valleys. d
i
r is the
radius of the nest reduced linearly in each iteration as follows:
 
0 0.5 d d
max mi
d
n
r x x
  (4)
0 0
0 0
1
0.5
0.5
0.5 1
0.5
0.5
d d d
d
i max
max
max
i m
m x
d
x
d
a
a
i
r r r i i
i
i i
r r r i i
i


    
 


  
     
 

(5)
Where, d
min
x and d
max
x are, respectively, the minimum and maximum values extractable for the
variables in direction d, 0
d
r is the initial radius of the nest, d
i
r is the circle radius in iteration i ,
and max
i is the maximum number of iterations. Exploration and exploitation are two necessary
operations that must be performed in order to improve the performance of meta-heuristic
algorithms. As observed in Eq. (5), the radius of the nest reaches zero at the middle of the total
iterations and increases in the next iteration. In fact, the exploration is performed in the first
relation of Eq. (5), while the exploitation is performed in the second relation.
The number of search points within the nest in the MOA is a submultiple of four. Using four
random numbers ( 1, 2, 3, 4
a a a a ) from the interval [0, 1], the following relations produce these
four points in the four quarters of the circle (see Fig. 2):
1 1
2 1 2 1 2 2 2 2
4 ( 1 ), 4 ( ( 1 ) )
new center new center
d m d m d m d m
i i i i
x x a a r x x a r a r
     
         (6)
2 2
2 1 2 1 2 2 2 2
3 ( 2 ), 3 ( ( 2 ) )
d m d m d m d m
i i i i
     
         (7)
3 3
2 1 2 1 2 2 2 2
2 ( 3 ), 2 ( ( 3 ) )
d m d m d m d m
i i i i
     
         (8)
4 4
2 1 2 1 2 2 2 2
1 ( 4 ), 1 ( ( 4 ) )
d m d m d m d m
i i i i
     
         (9)
r equals d
i
r , which is calculated using equation (5). The volume of fine sand accumulated in
valleys and search points is evaluated using EF.
Fig. 2. Produce search points as the spawning nest in step 1.

2.2. Step 2: Calculating velocity and direction of water flow
As illustrated in Fig. 3, the water flow into the nest is radially directed. It means the water enters
the nest through half of the valleys and leaves the nest through the other half of the valleys [39].
The fine sand on the seabed is moved by the water flow. Logically, the valleys in which the flow
direction is output should have a higher volume of fine sand. So, the algorithm identifies fifty
percent of adjacent valleys with the greatest volume of fine sand (minimum EF for minimization;
see Fig. 4). The orientation of the outlet flow is then considered for these valleys.
Fig. 3. Flow direction inside the nest.
Fig. 4. Detecting flow direction inside the nest.
One of the factors that causes the sand movements in the seabed is water flow. Dyer (1980) has
presented a relation to determine the velocity profile as follows [41]:
*
0
( / )
w
s
u
V ln y Z
K
 (10)
Where,
*
u is the friction velocity, s
K is the bed shear constant, 0
Z is the bed roughness length,
and w
y is the desired height.

As a logical assumption the
*
u is considered 2
𝑐𝑚
𝑠
. According to the reference [41], the s
K is
equal to 0.4. The seabed sediment is moved like a blanket [41]. Due to the nature of sand
movement, the MOA considers the w
y equal to half of sand layer thickness. So, the w
y is
assumed 20 𝑐𝑚. The 0
Z is considered 0.01 𝑐𝑚. Because, for a flat bed of a uniform grain size,
the 0
Z is a thirtieth of the grain diameter [41]. Finally, the velocity profile is normalized by the
problem and use as follows:
 
0.033
d d d
max min
V x x
  (11)
Where, d
max
x and d
min
x are the maximum and minimum values that can be taken for x in direction
d .
2.3. Step 3: Creating and checking new points in the flow direction
Four points with the highest density of fine sand are selected from the valleys and search points
generated in the first step. Then, the average of d
i
x for the points on the circle perimeter that are
in the outflow direction (Figure 4) is calculated for all problem dimensions (the number of
problem variables). In order to generate new points in each dimension of the problem, we shift
the selected point's coordinates by d
x
 toward the stated average. If the EF value of the new
circumstance has improved, the coordinates of the point measured in that dimension will be
replaced with new coordinates. This process is repeated for each dimension of the examined
point, and a distinct point may be reported at the conclusion. During this process, a type of
sensitivity analysis has been conducted, which greatly aids in determining the optimal answer. It
is important to note that for each of the four points taken, a new point is created according to the
number of the problem's dimensions. The following mathematical relationships describe how the
aforementioned elements are formed:
d d
x V t
   (12)
( )
point
d
i
d
d
dir s x
ign X
  (13)
.
new point
d d d d
i i
x x x dir
   (14)
Where, point
d
i
x is the desired point's position in direction “d” and iteration “i”, and
d
X is the mean
of d
i
x for the points on the circle perimeter that are in the outflow direction in dimension “d” of
the problem. t
 is a random number between [0,p]. Fig. 5 explains how to construct points.
Before the next iteration is started, the nest's center is updated to the final point of the previous
iteration. The new iteration begins with the assumption that the nest of the previous iteration is
completely destroyed and the seabed is pristine. The iterations are continued until the
convergence criterion is satisfied. Finally, the best point of the last iteration has been the best
place on the seabed (the answer zone) for spawning. The best place for spawning mentioned
above is the global optimum of the problem. Fig. 6 is shown the flowchart of the MOA.

Fig. 5. Produce new points in step 3.
Fig. 6. Flowchart of the MOA.
3. Numerical result
In this section, three famous optimization problems are solved using MOA. In addition to the
number of iterations that must be specified at the beginning of the solution, the algorithm will
stop if there is not much change in the value of the EF in 10 consecutive iterations. The
following relation is presented for the stop condition:
10
10
100 0.5 , 20
i i
i
EF EF
i
EF



   (15)

Where, i
EF is the EF in iteration i . After 10 runs, the obtained results are compared with the
available methods in the literature and presented in separate tables.
3.1. Mathematical benchmark problems
The solutions to the 14 Functions of CEC in reference [42] are attempted here by MOA. The 3D
map for the 2D function of CEC functions is plotted in Fig. 7.
Fig. 7. The 3D map for the 2D CEC functions.

The CEC functions have already been attempted and solved by Yu et al. using FWA-DM [43].
The results of the MOA and the FWA-DM for the 10 and 30 dimension problems are presented
in tables 1 and 2, respectively. To assess the mobility and reliability of the proposed algorithm,
the program had to be run 50 times. Tables 1 and 2 show the MOA can reach acceptable answers.
So, in the results, the efficiency of the presented method is evident. Here, 0
p
r is adjusted to 5.
There are 200 search points in the algorithm at start and also, 201 search points is assumed in the
solving procedure. (1 in the center of the nest, 100 around the center, and 100 on the nest
perimeter).
Table 1
CEC2014 - 10 Dimension results.
Best Worst Average
Standard
deviation
Median
Discus Function MOA 0.0008 0.0764 0.0227 0.02702 0.0060
FWA-DM 0.0009 5.94e-8 1.88e-9 9.02e-9 5.38e-18
Rosenbrock’s Function MOA 0.00072 4.7806 2.8335 1.3156 2.7803
FWA-DM 0.0008 4.730 1.410 1.600 0.084
Ackley MOA 20.0001 20.0001 20.0001 0.0000 20.0001
FWA-DM 20.000 20.010 20.000 0.0417 20.000
Weierstrass MOA 0.0064 1.2728 0.5396 0.3369 0.5396
FWA-DM 0.0034 2.480 0.706 0.640 0.574
Griewank MOA 0.0087 0.2035 0.0761 0.0236 0.0761
FWA-DM 0.0172 0.224 0.0948 0.0492 0.0861
Shifted Rastrigin MOA 0.0890 1.9899 0.9899 0.0402 0.9899
FWA-DM 0.000 3.980 0.254 0.809 1.78e-15
Shifted and Rotated Rastrigin MOA 1.9849 15.9193 6.0592 2.9368 5.4672
FWA-DM 1.99 16.90 6.010 2.450 5.970
Shifted Schwefel MOA 0.0871 7.9320 2.2200 2.9712 0.6362
FWA-DM 9.09e-13 6.950 1.590 2.08 0.375
Shifted and Rotated Schwefel MOA 22.3239 185.8577 86.2017 38.4674 81.2763
FWA-DM 39.800 813.000 372.000 153.000 360.000
Shifted and Rotated Katsuura MOA 0.0355 0.0461 0.0408 0.0075 0.0408
FWA-DM 3.34e-6 0.284 0.043 0.0478 0.0282
HappyCat MOA 0.0199 0.0619 0.0609 0.0015 0.0609
FWA-DM 0.034 0.299 0.121 0.0718 0.104
HGBat MOA 0.0214 0.0217 0.0216 0.0002 0.0216
FWA-DM 0.039 0.552 0.214 0.120 0.186
Griewank + Rosenbrock MOA 0.1520 0.9410 0.5470 0.5585 0.5470
FWA-DM 0.321 1.470 0.775 0.263 0.743
Shifted and Rotated Expanded
Scaffer’s F6
MOA 0.6243 2.0093 0.9495 0.4700 0.9705
FWA-DM 0.796 2.82 1.760 0.468 1.730

Table 2
CEC2014 - 30 Dimension results.
best worst average
standard
deviation
median
Discus Function MOA 0.0008 0.0955 0.0226 0.0010 0.0114
FWA-
DM
2.35e-17 2.23e-15 4.42e-16 4.74e-16 2.48e-16
Rosenbrock’s Function MOA 0.0099 43.9844 8.1410 5.9840 9.4008
FWA-
DM
0.0012 74.400 20.400 19.100 15.800
Ackley MOA 20.0000 20.1017 20.0161 0.0272 20.0002
FWA-
DM
20.400 20.600 20.500 0.0536 20.500
Weierstrass MOA 0.0842 19.1568 8.8030 3.3227 8.7647
FWA-
DM
0.0812 20.900 12.900 8.250 17.500
Griewank MOA 0.0001 0.0265 0.0061 0.0044 0.0062
FWA-
DM
0.000 0.0369 0.0086 0.0098 0.0074
Shifted Rastrigin MOA 0.0000 0.0899 0.0388 0.0040 0.0310
FWA-
DM
0.000 2.69e-12 1.13e-13 4.51e-13 1.78e-15
Shifted and Rotated
Rastrigin
MOA
31.8387 57.4553 40.1121 7.1157 40.6520
FWA-
DM
33.200 78.200 56.600 10.800 56.200
Shifted Schwefel MOA 2.2933 16.3716 4.6237 1.4665 4.9200
FWA-
DM
4.650 15.400 8.530 2.420 8.090
Shifted and Rotated
Schwefel
MOA
132.2288 545.2928 93.8210 87.5645 91.6793
FWA-
DM
2000.000 3030.000 2630.000 248.000 2650.000
Shifted and Rotated
Katsuura
MOA
0.0150 0.2877 0.0894 0.0570 0.0726
FWA-
DM
0.208 0.521 0.371 0.0666 0.370
HappyCat MOA 0.1291 0.3636 0.2499 0.0572 0.2633
FWA-
DM
0.288 0.490 0.389 0.0551 0.400
HGBat MOA 0.0967 0.2802 0.1919 0.0419 0.1947
FWA-
DM
0.178 0.740 0.269 0.0776 0.259
Griewank + Rosenbrock MOA 2.6004 8.9873 4.9918 1.4635 4.9010
FWA-
DM
5.640 9.050 7.370 0.846 7.330
Shifted and Rotated
Expanded Scaffer’s F6
MOA 10.0825 10.9790 10.4483 0.5238 10.4677
FWA-
DM
10.300 11.400 11.000 0.271 11.000

The convergence history for the CEC functions attempted by MOA is presented in Fig. 8.
Fig. 8. The convergence history for the CEC functions.
3.2. Structural benchmark problem
The 25-bar truss is a popular engineering problem. This problem's primary objective is to reduce
the truss's weight, which can be computed as follows:
1
n
m m m
m
W A L


  (16)
Where,  is the density of materials, A is the cross-sectional area of each bar element, L is the
truss bar length, m is the counter of the truss members, and n is the total number of members in
the truss.
In this problem, the density of materials is set to 0.1
𝑙𝑏
𝑖𝑛3
(2768
𝑘𝑔
𝑚3
) and the elasticity modulus is
10 𝑚𝑠𝑖 (68.95 𝐺𝑃𝑎). The node displacements are limited to ±0.35 𝑖𝑛 (±0.889 𝐶𝑚). Eight group
divisions are considered for the members. The maximum and minimum values for allowable
stresses for each group are expressed in Table 3. The load conditions that must be applied to the
structure are shown in Table 4. The maximum and minimum cross-sectional areas of the
members are 4 𝑖𝑛2
and 0.01 𝑖𝑛2
, respectively. Fig. 9 shows the geometry of the problem.

Table 3
member stress limitations - 25-bar truss (ksi).
member number compression limit tension limit
1 35.092 40.000
2-5 11.59 40.000
6-9 17.305 40.000
10-11 35.092 40.000
12-13 35.092 40.000
14-17 6.759 40.000
18-21 6.959 40.000
22-25 11.082 40.000
Table 4
load conditions – 25-bar truss (kips).
Direction
load case node x y Z
1 1 1.0 10.0 -5.0
2 0.0 10.0 -5.0
3 0.5 0.0 0.0
2 5 0.0 20.0 -5.0
6 0.0 -20.0 -5.0
Fig. 9. Geometry of the 25-bar truss.
The 25-bar truss problem has already been attempted and solved by many researchers, such as
Lee and Geem using HS [3], Sonmez using ABC [44], Kaveh et al. using CSP [45], and Varaee
and Ghasemi using IGMM [46]. The obtained results by the MOA here are compared with those
results and are expressed in Table 5, which shows the superiority of the MOA. The convergence
history of the proposed method is presented in Fig. 10. Here, 0
p
r
is adjusted to 1.1. There are 40
search points in the algorithm at start and also, 9 search points is assumed in the solving
procedure. (1 in the center of the nest, 4 around the center, and 4 on the nest perimeter).

Table 5
comparing results for 25-bar truss.
member
number
HS ABC CSP IGMM
present
work
1 0.047 0.011 0.01 0.01 0.0122
2-5 2.022 1.979 1.91 2.006 1.8514
6-9 2.95 3.003 2.798 2.961 3.0353
10-11 0.01 0.01 0.01 0.01 0.0100
12-13 0.014 0.01 0.01 0.01 0.0100
14-17 0.688 0.69 0.708 0.687 0.6762
18-21 1.657 1.679 1.836 1.676 1.7607
22-25 2.663 2.652 2.645 2.668 2.6410
weight(lb) 544.38 545.19 545.09 545.19 545.12
NFEs 15,000 300,000 17,500 5,240 4000
C.V. 0.0715 0.058 0.0202 0 0
Fig. 10. Convergence history of 25-bar truss.
In order to better comprehend the enhancement of results obtained by the MOA, Reducing the
number of function evaluations and the constraint violations are plotted in Fig. 11.
Fig. 11. (a) Comparing results for NFEs.- 25-bar truss; (b) Comparing results for C.V.- 25-bar truss.

3.3. Mechanical benchmark problem
The welded beam optimum design that is introduced by Rao [47] is one of the most challenging
constrained problems in engineering. In this problem, the purpose is to reduce the construction
cost considering the stress and strain constraints. Fig. 12 shows the geometry of the problem. The
design variables are the weld's thickness (h=x1), the welded joint's length (l=x2), the beam's
width (d=x3), and the beam's thickness (b=x4). The goal function and the constraints are as
follows:
   
2
1 2 3 4 2
1.10471 0.04811 14
minf x x x x x
  
X (17)
  2 2
2
1 13600 0
x
g
D

 
    
X (18)
 
2 2
4 3
504000
30000 0
g
x x
  
X (19)
 
3 1 4 0
g x x
  
X (20)
   
2
4 1 3 4 2
0.10471 0.04811 14 5 0
g x x x x
    
X (21)
 
5 1
0.125 0
g x
  
X (22)
 
6 3
4 3
65856
0.25 0
30000
g
x x
  
X (23)
 
3 3
6 3 4
7
30
48
6000 0.61423 10 1 0
6 28
x
x x
g
 
 
 
    
 
 
 
X (24)
2
6000 14
2
x
Q
 
 
 
 
(25)
 
2
2
2 1 3
1
2
D x x x
   (26)
 
2
2
1 3
2
1 2
2
6 2
x x
x
J x x
 

 
 
 
 
(27)
1 2
6000
2x x
  (28)
QD
J
  (29)
1 4 2 3
0 , 2 , 0 , 10
x x x x
    (30)

Fig. 12. Geometry of welded beam.
The welded beam optimum design has been solved before by Montes and Coello using GA [48],
He and Wang using CPSO [49], Mirjalili et al. using MVO [50], Kaveh and Talatahari using
ACO [51], and El-Kenawy et al. using SCMWOA [52]. The mentioned problem has been solved
using the proposed method, and the obtained results are presented in Table 6 and compared with
other research. Improvements in the responses of the proposed method are clearly visible. The
convergence history of the proposed method is presented in Fig. 13. Here, 0
p
r is adjusted to 1.1.
There are 60 search points in the algorithm at start and also, 81 search points is assumed in the
solving procedure. (1 in the center of the nest, 40 around the center, and 40 on the nest
perimeter).
Table 6
the welded beam - comparing results.
GA CPSO MVO ACO SCMWOA present work
x1 0.205986 0.202369 0.205463 0.2057 0.205604 0.20571
x2 3.471328 3.544214 3.473193 3.471131 3.479712 3.4709
x3 9.020224 9.04821 9.044502 9.036683 9.041001 9.0367
x4 0.20648 0.205723 0.205695 0.205731 0.205739 0.20573
g1 -0.103 -13.6555 0.0261 -0.0846 -25.5857 -0.0559
g2 -0.2317 -75.8141 -47.1961 -0.5907 -30.4048 -0.5577
g3 -0.0005 -0.0034 -0.0002 -3.10E-05 -0.0001 -2.00E-05
g4 -3.43 -3.4246 -3.4317 -3.4329 -3.4313 -3.4329
g5 -0.081 -0.0774 -0.0805 -0.0807 -0.0806 -0.0807
g6 -0.2355 -0.2356 -0.2356 -0.2355 -0.2356 -0.2355
g7 -58.6006 -4.427 -0.3596 -0.099 -2.6836 -0.0189
fmin 1.728223 1.731484 1.725898 1.724912 1.726738 1.724892
NFEs 80000 30000 15000 17600 9900 9360

Fig. 13. Convergence history of welded beam.
In order to better comprehend the enhancement of results obtained by the MOA, Reducing the
number of function evaluations is plotted in Fig. 14.
Fig. 14. Comparing results for NFEs. - welded beam.
4. Conclusion
In this paper, a new meta-heuristic, nature-inspired method is introduced. The proposed MOA is
inspired by the masterpiece of Japanese puffer fish. In fact, the masterpiece is the nest that is
constructed by MPF for spawning, a name given due to the high construction time and unique
design of the nest. In the reproduction of pufferfish, the volume of fine sand on the seabed plays
a key role. So, this method is focused on the volume of fine sand and its movement due to water
flow on the seabed. The point with the maximum volume of fine sand in the MPF activity area

(answer zone) is the best point for spawning. The MOA tries to find this key point in the answer
zone and reports it as the global optimum of the problem.
Each optimization method should find the optimal points through exploration and exploitation
operations. In the MOA, these operations operate perfectly. To prove this claim, the proposed
algorithm must reach accurate answers with a lower number of function evaluations. Also, the
algorithm must not trap at the local optimal point. So, the efficiency of the proposed method was
assessed through different benchmark problems when compared with available methods in the
literature. The following results can be concluded:
1. The MOA achieves more accurate answers.
2. The MOA reaches the answers with a lower number of function evaluations.
3. In constrained problems, the constraint violation was equal to zero.
4. The MOA was able to achieve acceptable answers for mathematical problems.
5. The MOA increases the convergence speed while reducing analysis time due to the reduced
number of function evaluations.
6. The MOA causes economic savings due to reduced analysis time.
Due to the high efficiency of the MOA, using this method is recommended for solving
engineering constrained and unconstrained optimization problems.
Authors’ contributions statement
N.H.A. proposed the idea, designed the research, did model analysis, data fitting, and made all
figures and analyzed the results. M.R.G. and H.G.A. supervised the manuscript.
Competing interests
The authors declare no competing interests.
Data availability
Some or all data, models, or code that support the findings of this study are available from the
corresponding author upon reasonable request.
Funding source
There is no funding source for doing this research and writing this paper.
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