A Novel Center of Mass Optimization (CMO) Algorithm for Truss Design Problems

Journal of Soft Computing in Civil Engineering 8-2 (2024) 119-142
How to cite this article: Varaee A. A novel center of mass optimization (CMO) algorithm for truss design problems. J Soft Comput
Civ Eng 2024;8(2):119–142. https://doi.org/10.22115/scce.2023.398542.1649
2588-2872/ © 2024 The Authors. Published by Pouyan Press.
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
A Novel Center of Mass Optimization (CMO) Algorithm for Truss
Design Problems
Hesam Varaee 1*
1. Assistant Professor, Department of Engineering, Ale Taha Institute of Higher Education, Tehran, Iran
Corresponding author: varaee.hesam@aletaha.ac.ir
https://doi.org/10.22115/SCCE.2023.398542.1649
ARTICLE INFO ABSTRACT
Article history:
Received: 22 May 2023
Revised: 15 August 2023
Accepted: 20 November 2023
This paper introduces a novel optimization algorithm rooted
in the mass center equations of particle systems. The
proposed Center of Mass Optimization (CMO) algorithm is
distinguished by its easy implementation, parameter
independence, and rapid, accurate solutions. In the proposed
CMO, a random walk operator is introduced to enhance the
exploitation capability of the CMO and help the search
agents jump out of the local optimal. Mutation and elitism
selection operators are also used to boost the overall
performance of the proposed algorithm. Some mathematical
benchmark optimization problems and two engineering truss
optimization examples are investigated to evaluate the
performance of the proposed method. The results are
compared with those of well-known optimization algorithms
such as DE, ABC, CBO, PSO, EO, LHHA, and SMA. The
results of Wilcoxon rank-sum and ANOVA tests indicate
that the performance of the proposed algorithm is robust and
reliable for a wide range of complex mathematical and
engineering optimization problems.
Keywords:
Center of mass;
Population-based algorithms;
Mathematical and engineering
optimization problems;
Wilcoxon rank-sum test;
ANOVA test.

120 H. Varaee/ Journal of Soft Computing in Civil Engineering 8-2 (2024) 119-142
1. Introduction
Reducing material consumption and construction costs while meeting standard requirements has
always been a critical challenge in civil and structural engineering [1–4]. In recent years, the
development of optimization algorithms and their increasing application in civil engineering
have made structural optimization popular for solving complex optimization problems [5–9].
Numerous studies have shown the exceptional performance of modern metaheuristic algorithms
in solving highly nonlinear structural optimization problems, effectively reducing the weight and
cost of structures [10,11]. These studies utilized various groups of metaheuristic optimization
algorithms, including evaluation-based, swarm-based, physics-based, and human-based
algorithms [12–14]. These algorithms have been successfully employed to solve complex
engineering optimization problems, such as optimizing the cost of steel and concrete frames,
determining the optimal weight of truss structures, performance-based design optimization
(PBDO), reliability-based design optimization (RBDO), and life cycle cost (LCC) optimization
of structures [15].
Evolution-based algorithms (EAs), which stimulate natural evolution such as migration,
chemotaxis, and reproduction elimination and dispersal, have played a significant role in
structural optimization. Among these, the genetic algorithm (GA) proposed by Holland is a
widely used and well-known EA in the field of structural optimization [16,17]. The GA stands
out for its ability to perform optimization in the search space without requiring explicit
derivation, a characteristic often present in mathematical optimization approaches. The GA
emulates the concept of the survival of the fittest in nature to evolve a population. Over the
years, the GA has been improved, and numerous variants have been developed, enabling it to
avoid local optima and provide efficient solutions. Other proposed evolutionary algorithms, such
as harmony search (HS), differential evolution (DE), and stochastic fractal search (SFS), have
also been successfully applied to solve engineering problems. Physics-based algorithms mimic
physical laws in the universe and have found wide application in engineering problems. One
such algorithm is simulated annealing (SA), which is based on thermodynamic principles of
thermodynamics [18–20]. SA imitates the annealing process in materials, were heating and
subsequent cooling leads to crystallization and energy minimization. Several novel physics-
inspired techniques have been developed, including water wave optimization (WWO), vortex
search algorithm (VSA), ideal gas molecular movement (IGMM) algorithm, and gravitational
search algorithm (GSA). Swarm-based algorithms simulate social behaviors in species, such as
the division of labor and self-organization. Notable instances encompass ant colony optimization
(ACO) and particle swarm optimization (PSO). PSO mimics the flocking patterns of birds to
iteratively update each agent within a population, drawing upon both the top-performing global
agent and the most successful individual agent [21]. Meanwhile, ACO takes inspiration from the
foraging conduct of ant colonies, where ants discover the most efficient route to food sources by
responding to pheromone concentrations.

H. Varaee/ Journal of Soft Computing in Civil Engineering 8-2 (2024) 119-142 121
Other swarm-inspired algorithms include African buffalo optimization (ABO), joint operations
algorithm (JOA), coyote optimization algorithm (COA), emperor penguin colony (EPC), gray
wolf optimization (GWO), and shark smell optimization (SSO). In recent years, new
metaheuristic algorithms, inspired by social behaviors and human ideologies, have emerged.
These include political optimizer (PO), heap-based optimizer (HBO), social engineering
optimizer (SEO), artificial human optimization (AHO), and parliamentary optimization
algorithm (POA), to name a few.
This study presents a new algorithm that uses a novel approach. The proposed algorithm is
inspired by the physical equation of the center of mass of a particle system. In addition, the
algorithm incorporates a random walk operator, mutation operator, and elitism selection to
enhance the local search and overall performance. A notable feature of the proposed algorithm is
its parameter independence, which alleviates the challenge and time-consuming nature of
parameter adjustment while significantly impacting the performance. To assess the performance,
twenty-three mathematical benchmark functions were evaluated [22,23], and each problem was
independently solved 20 times to reduce the uncertainty of the results. The results are compared
with those of well-known and powerful metaheuristic algorithms, such as DE, ABC, CBO, PSO,
EO, LHHA and SMA. The Wilcoxon rank-sum and ANOVA tests indicate that the proposed
CMO algorithm outperforms other algorithms [10,12]. Two examples of engineering truss
problems, one in two dimensions and the other in three dimensions, were also examined to
evaluate the performance of the algorithm in complex engineering problems. Statistical results
from 20 independent runs for each truss problem demonstrated the satisfactory performance of
the CMO algorithm. Solving highly nonlinear engineering problems often entails high
computational costs and requires a substantial number of analyses. Compared with other
algorithms, the CMO algorithm exhibits a slight standard deviation and requires fewer analyses
to achieve the optimal solution, indicating its suitability for complex and highly nonlinear
engineering problems.
The remainder of this paper is organized as follows. Section 2 presents the procedure of the
proposed CMO algorithm. Sections 3 and 4 evaluate mathematical and engineering truss
problems, respectively. Finally, Section 5 concludes the paper and highlights the key findings.
2. New proposed CMO algorithm
2.1. Initialization
In this study, a group of particles with different masses, each representing a potential solution in
the problem domain, was produced randomly with a uniform distribution within the allowable
range of the problem variables. To generate the initial population, Eq. 1 is used.
𝑋
̅ = 𝑢𝑛𝑖𝑓𝑟𝑛𝑑 (𝑋𝑚𝑖𝑛. 𝑋𝑚𝑎𝑥. 𝑑) (1)

In the above equation, 𝑋
̅ represents the set of initial particles, 𝑋𝑚𝑖𝑛 and 𝑋𝑚𝑎𝑥 denote the upper
and lower bounds of the variables, respectively, and 𝑑 represents the dimension of the problem,
or in other words, the number of design variables. The 𝑢𝑛𝑖𝑓𝑟𝑛𝑑 function randomly generates
particles with a uniform distribution.
2.2. Mass assignment
After the particles were evaluated, a mass corresponding to the fitness of each particle was
assigned. Therefore, the fittest particles will have a larger mass and vice versa. The mass was
assigned to the particles using Eq. 2.
mi =
1
abs(Ci−best)/abs(worst−best)
(2)
In Eq. 2, 𝑚𝑖 represents the mass of the 𝑖-th particle, 𝐶𝑖 indicates the fitness of the 𝑖-th particle,
and 𝑏𝑒𝑠𝑡 and 𝑤𝑜𝑟𝑠𝑡 represent the best and worst particles found in the population, respectively.
2.3. Center of mass calculation
The center-of-mass relations of a set of particles are used as the main operator of the CMO
algorithm. The center of mass of a system of particles is the point at which the entire mass of the
system is concentrated when it moves, and all external forces affect that point. Fig. 1 illustrates a
system of 𝑛 particles with different masses and positions.
Fig. 1. Center of mass of a particle system.
m 1
m 4
m n m 2
m 3
r3
r2
rn
r1 r4
rc
COM
x
x
x
3
2
1

According to Fig. 1, the coordinates of the center of mass of a three-dimensional system
consisting of 𝑛 particles are obtained from Eq. 3:
𝑟𝑐𝑜𝑚 =
1
𝑀
∑ 𝑚𝑖𝑟𝑖
𝑛
𝑖=1 (3)
In the above equation, 𝑟𝑐𝑜𝑚 represents the position vector of the center of mass of the particle
system. 𝑚𝑖 and 𝑟𝑖 represent the mass and position vectors, respectively, of each particle. 𝑀 is the
sum of the masses of all the particles, that is, the mass of the entire system, and is obtained from
Eq. 4:
𝑀 = ∑ 𝑚𝑖
𝑛
𝑖=1 (4)
Similar equations can be written based on the coordinates of each particle in the 𝑑-Dimention
Cartesian system as follows:
(5)
𝑥𝑐𝑜𝑚
1
=
1
𝑀
∑ 𝑚𝑖𝑥𝑖
1
𝑛
𝑖=1
𝑥𝑐𝑜𝑚
2
=
1
𝑀
2
𝑛
𝑖=1
𝑥𝑐𝑜𝑚
3
=
1
𝑀
3
𝑛
𝑖=1
…
𝑥𝑐𝑜𝑚
𝑑
=
1
𝑀
𝑑
𝑛
𝑖=1
In Eq. 5, superscripts 1 to 𝑑 represent the first 𝑑-th dimension of the problem.
The main loop of the proposed CMO algorithm includes global and local searches, and is
enhanced by mutation and elitism operators. To search the entire problem space, in the first step
of the algorithm's main loop, the center of mass of each particle and the best particle found thus
far are calculated using Eq. 6.
𝑋𝑐𝑖 =
𝑟𝑎𝑛𝑑×𝑋𝑖×𝑚𝑖+𝑟𝑎𝑛𝑑×𝑋𝑏𝑒𝑠𝑡×𝑚𝑏𝑒𝑠𝑡
𝑟𝑎𝑛𝑑×(𝑚𝑖+𝑚𝑏𝑒𝑠𝑡)
(6)
where 𝑋𝑐𝑖 is the center of mass between the current particle and the best particle of the
population. 𝑟𝑎𝑛𝑑 represents a random number with a normal distribution between zero and one.
In the next step, by comparing the fitness of 𝑋𝑐𝑖 and the fitness of 𝑋𝑖 , the best particle is selected
and replaced instead of the current particle by comparing the fitness of Xci and Xi. It should be
noted that 𝑋𝑏𝑒𝑠𝑡 has the largest mass compared with the other particles, and 𝑋𝑐𝑖 is always

inclined towards the best particle of the system. Therefore, because of the higher chance of
finding better particles in the vicinity of the best particle, repeating the above process can cause
convergence of the system to the optimal region, and this process can be called a global search.
2.4. Random-walk operator
Then, a random-walk operator in the form of Eq. Seven was used to perform a local search. The
random walk operator generates a new particle adjacent to the current particle.
𝑋𝑙𝑖 = 𝑋𝑖 + 𝛿𝑖 (7)
where 𝑋𝑙𝑖 is the newly generated particle and 𝛿𝑖 is a partial random value with a uniform
distribution within the range of -1 to 1. The random-walk operator can be considered a simulator
for various natural phenomena. These include the Brownian motion of molecules in a liquid or
gas, the search for food-seeking behavior of animals, the volatility of stock prices, and the
investor’s financial condition. By comparing the fitness of the newly generated particle and the
fitness of the current particle, the best one was selected to continue the optimization process.
2.5. Mutation operator
A mutation operator can be helpful in avoiding local optima. Therefore, some particles (with a
probability of rand > 0.8) underwent mutations using Eq. 8:
𝑋𝑖 = 𝑋𝑖 + ∆ (8)
where ∆ is a random value with a uniform distribution within the range -2 to 2.
Finally, the elitism operator preserves the best particle ever found and carries it to the next
generation. Fig. 2 shows the flowchart of the proposed CMO algorithm.
3. Numerical and examples
3.1. mathematical benchmark functions
This study investigated 23 mathematical benchmark functions described in Tables 1-3 to evaluate
the performance of the proposed algorithm. Mathematical functions are divided into three
different groups: unimodal, multimodal, and multimodal functions with fixed dimensions. The
unimodal functions (𝐹1 to 𝐹6) have a single global optimum and are used to evaluate the global
search ability, exploration, and convergence speed of the algorithms. The multimodal functions
(𝐹7 to 𝐹14) have multiple local optima, with the best one being the global optimum. Algorithms
that can effectively balance exploration and exploitation can successfully handle these functions.
Functions 𝐹15 to 𝐹23 are multimodal functions with fixed dimensions. These functions are
typically simple examples with different variable ranges and can be used to assess the accuracy
and robustness of algorithms [24,25].

Start
Initialize Size, d, Xmin, Xmax, Iteration
Generate and evaluate initial population
Calculate best and worst
particle
Calculate mass for each
particle
Calculate Xci and
replace Xi
Calculate Xli and
replace Xi
Mutation
Elitism Selection
Is
stopping criteria
satisfied?
N
Report the best
particle
Y
End
Main
Operators
Fig. 2. The flowchart of the proposed CMO.
Yes
No

Table 1
Unimodal benchmark functions.
Type Test Function n S 𝐟𝐦𝐢𝐧
Unimodal F1(X) = ∑ xi
2
n
i=1
30 [−100.100]n 0
Unimodal F2(X) = ∑ |xi| + ∏ |xi|
n
i=1
n
i=1
30 [−10.10]n 0
Unimodal F3(X) = ∑ (∑ xj
i
j=1
)
n
i=1
2
30 [−100.100]n
0
Unimodal F4(X) = max
i
{|xi|. 1 ≤ i ≤ n} 30 [−100.100]n 0
Unimodal F5(X) = ∑ [100(xi+1 − xi
2
)2
+ (xi − 1)2]
n−1
i=1
30 [−30.30]n
0
Unimodal F6(X) = ∑ ([xi + 0.5])2
n
i=1
30 [−100.100]n
0
Table 2
Multimodal benchmark functions.
Multimodal F7(X) = ∑ ixi
4
n
i=1
+ random[0.1) 30 [−1.28.1.28]n
0
Multimodal
𝐹8(𝑋) = ∑ −𝑥𝑖 𝑠𝑖𝑛 (√|𝑥𝑖|)
𝑛
𝑖=1
30 [−500.500]𝑛
-418.98×30
Multimodal
𝐹9(𝑋) = ∑ [𝑥𝑖
2
− 10 𝑐𝑜𝑠(2𝜋𝑥𝑖) + 10]
𝑛
𝑖=1
30 [−5.12.5.12]𝑛 0
Multimodal
𝐹10(𝑋) = −20𝑒𝑥𝑝 (−0.2√
1
𝑛
∑ 𝑥𝑖
2
𝑛
𝑖=1
)
− 𝑒𝑥𝑝 (
1
𝑛
∑ 𝑐𝑜𝑠(2𝜋𝑥𝑖)
𝑛
𝑖=1
) + 20 + 𝑒
30 [−32.32]𝑛
0
Multimodal
𝐹11(𝑋) =
1
4000
∑ 𝑥𝑖
2
− ∏ 𝑐𝑜𝑠 (
𝑥𝑖
√𝑖
)
𝑛
𝑖=1
+ 1
𝑛
𝑖=1
30 [−600.600]𝑛
0
Multimodal
𝐹12(𝑋) =
𝜋
𝑛
{10 𝑠𝑖𝑛(𝜋𝑦1)
+ ∑ (𝑦𝑖 − 1)2[1 + 10 𝑠𝑖𝑛2(𝜋𝑦𝑖+1)]
𝑛−1
𝑖=1
+ (𝑦𝑛 − 1)2
} + ∑ 𝑢(𝑥𝑖. 10.100.4)
𝑛
𝑖=1
𝑦𝑖 = 1 +
𝑥𝑖 + 1
4
𝑢(𝑥𝑖. 𝑎. 𝑘. 𝑚) = {
𝑘(𝑥𝑖 − 𝑎)𝑚
𝑥𝑖 > 𝑎
0 −𝑎 < 𝑥𝑖 < 𝑎
𝑘(−𝑥𝑖 − 𝑎)𝑚
𝑥𝑖 < −𝑎
30 [−50.50]𝑛
0
Multimodal
𝐹13(𝑋) = 0.1 {𝑠𝑖𝑛2(3𝜋𝑥1)
+ ∑ (𝑥𝑖 − 1)2[1 + 𝑠𝑖𝑛2(3𝜋𝑥𝑖 + 1)]
𝑛
𝑖=1
+ (𝑥𝑛 − 1)2
[1 + 𝑠𝑖𝑛2(2𝜋𝑥𝑛)]}
30 [−50.50]𝑛
0
Multimodal
𝐹14(𝑋) = (
1
500
+ ∑
1
𝑗 + ∑ (𝑥𝑖 − 𝑎𝑖𝑗)6
2
𝑖=1
25
𝑗=1
)−1
2 [−65,65] 1

Table 3
Multimodal benchmark functions with fixed dimensions.
Multimodal-Fixed
dimensions 𝐹15(𝑋) = ∑ [
11
𝑖=1
𝑎𝑖−
𝑥1(𝑏𝑖
2
+ 𝑏𝑖𝑥2)
𝑏𝑖
2
+ 𝑏𝑖𝑥3 + 𝑥4
]2 4 [−5,5] 0.00030
Multimodal-Fixed
dimensions
𝐹16(𝑋) = 4𝑥1
2
− 2.1𝑥1
4
+
1
3
𝑥1
6
+ 𝑥1𝑥2 − 4𝑥2
2
+ 4𝑥2
4
2 [−5,5] -1.0316
Multimodal-Fixed
dimensions
𝐹17(𝑋) = (𝑥2 −
5.1
4𝜋2
𝑥1
2
+
5
𝜋
𝑥1 − 6)2
+ 10 (1 −
1
8𝜋
) 𝑐𝑜𝑠𝑥1 + 10
2 [−5,5] 0.398
Multimodal-Fixed
dimensions
𝐹18(𝑋) = [1 + (𝑥1 + 𝑥2 + 1)2
(19 − 14𝑥1 + 3𝑥1
2
−
14𝑥2 + 6𝑥1𝑥2 + 3𝑥2
2
)] ×[30+(2𝑥1 − 3𝑥2)2
×
(18 − 32𝑥1 + 12𝑥1
2
+ 48𝑥2 − 36𝑥1𝑥2 + 27𝑥2
2
)]
2 [−2,2] 3
Multimodal-Fixed
dimensions 𝐹19(𝑋) = − ∑ 𝑐𝑖 𝑒𝑥𝑝(
4
𝑖=1
− ∑ 𝑎𝑖𝑗(
3
𝑗=1
𝑥𝑗 − 𝑝𝑖𝑗)
2
) 3 [1,3] -3.86
Multimodal-Fixed
dimensions 𝐹20(𝑋) = − ∑ 𝑐𝑖 𝑒𝑥𝑝(
4
𝑖=1
− ∑ 𝑎𝑖𝑗 (
6
𝑗=1
𝑥𝑗 − 𝑝𝑖𝑗)2
) 6 [0,1] -3.32
Multimodal-Fixed
dimensions 𝐹21(𝑋) = − ∑ [(𝑋 − 𝑎𝑖)(𝑋 − 𝑎𝑖)𝑇
+ 𝑐𝑖]−1
5
𝑖=1
4 [0,10] -10.1532
Multimodal-Fixed
+ 𝑐𝑖]−1
7
𝑖=1
4 [0,10] -10.4028
Multimodal-Fixed
+ 𝑐𝑖]−1
10
𝑖=1
4 [0,10] -10.5363
To compare the results of the CMO algorithm with those of other well-known algorithms, each
problem was evaluated 20 times independently using the DE, ABC, CBO, PSO, EO, LHHA and
SMA algorithms in addition to the CMO. To ensure fair comparisons and identical conditions
among all algorithms, the number of particles for all algorithms was set to 50, and the maximum
number of optimization cycles was set to 100. A distinguishing feature of the CMO algorithm is
its lack of dependence on initial parameter adjustments. The parameter settings for the other
optimization algorithms, as listed in Table 4, were established based on the values proposed in
the original references for each algorithm.

Table 4
Parameter settings of optimization algorithms.
Algorithm Parameters
DE [26] Beta-min=0.1, Beta-max=0.2, PCR=0.2
ABC [27] FS=NB/2, ne=NB/2, no=NB/2, Scout=1, Limit=ne×D
CBO [28] COR=1-(iter/maxIt)
PSO [29] Vmax=5, Wmax, Wmin=0.4-0.9, C1,C2=2
EO [30] V=1, α1=2, α2=1, GP=0.5
LHHA [31] E0=2×rand()-1, E1=2× (1-(iter/maxIt))
SMA [32] z=0.03, a = tan−1
(-(iter/ maxIt)+1, b = 1-(iter/ maxIt)
The comparison results for each algorithm are presented in Tables 5–7. The results demonstrate
that the CMO algorithm outperforms the other algorithms in 16 of the 23 functions. In addition,
the CMO algorithm exhibits a small standard deviation, highlighting its high robustness and
stability. This characteristic is particularly crucial for solving complex and large-scale problems
that cannot be solved repeatedly.
Table 5
Statistical results for the unimodal benchmark functions.
Func. CMO DE ABC CBO PSO EO LHHA SMA
F1
Best 5.07E-151 1.21E+03 2.87E+00 3.19E+02 2.15E+03 7.74E-08 1.17E-49 2.19E-81
Average 2.30E-83 1.68E+03 6.28E+02 9.83E+02 4.92E+03 9.95E-07 5.61E-41 1.52E-58
Worst 4.13E-82 2.60E+03 3.15E+03 1.77E+03 1.22E+04 3.04E-06 9.28E-40 3.05E-57
Std. 9.24E-83 3.52E+02 8.78E+02 4.03E+02 3.00E+03 8.71E-07 2.09E-40 6.82E-58
F2
Best 1.69E-80 1.27E+01 5.47E-01 6.34E+00 3.55E+01 4.74E-05 3.43E-27 1.81E-45
Average 1.04E-48 1.50E+01 9.44E-01 9.83E+00 7.38E+01 9.64E-05 1.46E-21 3.52E-17
Worst 1.95E-47 1.84E+01 2.47E+00 2.28E+01 1.17E+02 1.70E-04 1.25E-20 7.04E-16
Std. 4.36E-48 1.49E+00 4.30E-01 3.93E+00 2.11E+01 3.20E-05 2.97E-21 1.57E-16
F3
Best 1.14E-126 3.60E+04 2.12E+04 7.75E+03 2.98E+04 2.53E-01 4.21E-40 5.05E-96
Average 1.00E-46 4.72E+04 3.31E+04 1.51E+04 4.39E+04 5.62E+00 1.80E-27 5.38E-60
Worst 2.01E-45 6.41E+04 4.73E+04 2.33E+04 6.84E+04 2.04E+01 1.81E-26 1.07E-58
Std. 4.49E-46 7.37E+03 5.35E+03 3.87E+03 9.75E+03 6.22E+00 4.72E-27 2.38E-59
F4
Best 1.16E-64 4.91E+01 5.76E+01 3.14E+01 5.13E+01 2.01E-02 2.69E-25 2.12E-45
Worst 4.00E-39 6.54E+01 7.84E+01 6.49E+01 7.68E+01 2.01E-01 4.23E-20 3.73E-30
Std. 1.30E-39 3.81E+00 5.30E+00 1.07E+01 6.25E+00 4.59E-02 9.41E-21 8.33E-31
F5
Best 1.17E-01 3.58E+05 1.94E+02 9.33E+04 9.79E+05 2.70E+01 3.84E-06 4.63E-06
Average 1.51E+01 6.54E+05 8.50E+02 5.44E+05 4.60E+06 2.75E+01 3.22E+00 5.24E-04
Worst 2.89E+01 1.16E+06 3.00E+03 2.04E+06 1.08E+07 2.85E+01 2.87E+01 2.75E-03
Std. 1.23E+01 2.10E+05 9.09E+02 4.20E+05 2.75E+06 3.91E-01 8.78E+00 6.67E-04
F6
Best 3.95E-02 9.39E+02 3.10E+00 5.33E+02 1.98E+03 3.12E-02 2.27E-04 9.64E-08
Worst 1.33E+00 2.22E+03 4.25E+03 1.63E+03 1.50E+04 1.07E+00 2.13E-02 1.19E-04
Std. 3.79E-01 3.58E+02 1.29E+03 2.92E+02 4.53E+03 2.68E-01 4.75E-03 2.63E-05

Table 6
Statistical results for the multimodal benchmark functions.
F7
Best 7.27E-05 5.16E-01 5.08E-01 2.71E-01 7.73E-01 1.75E-03 1.43E-05 5.98E-03
Average 6.66E-04 7.50E-01 1.07E+00 6.73E-01 5.09E+00 4.70E-03 5.80E-04 3.64E-02
Worst 2.07E-03 9.98E-01 2.02E+00 2.17E+00 2.59E+01 8.72E-03 1.70E-03 1.10E-01
Std. 5.39E-04 1.40E-01 3.71E-01 4.64E-01 5.58E+00 2.06E-03 5.88E-04 2.45E-02
F8
Best -12569.48 -6851.74 -8593.80 -5417.67 -9293.97 -8959.75 -12569.49 -9913.27
Average -12562.00 -5977.60 -8098.83 -4598.40 -8226.00 -7821.54 -12554.33 -5207.09
Worst -12544.81 -5426.06 -7644.57 -3782.71 -6957.59 -6380.28 -12330.16 -3131.94
Std. 6.92E+00 3.64E+02 3.10E+02 5.60E+02 5.16E+02 6.57E+02 5.30E+01 2.21E+03
F9
Best 0.00E+00 1.66E+02 7.54E+01 7.70E+01 1.65E+02 1.87E-06 0.00E+00 0.00E+00
Average 0.00E+00 1.83E+02 9.55E+01 1.43E+02 2.58E+02 7.73E-01 0.00E+00 0.00E+00
Worst 0.00E+00 1.99E+02 1.14E+02 2.25E+02 3.09E+02 6.81E+00 0.00E+00 0.00E+00
Std. 0.00E+00 9.33E+00 1.32E+01 4.33E+01 4.12E+01 1.60E+00 0.00E+00 0.00E+00
F10
Best 8.88E-16 8.84E+00 1.13E+01 5.39E+00 9.00E+00 7.75E-05 8.88E-16 8.88E-16
Worst 8.88E-16 1.11E+01 1.59E+01 9.47E+00 1.82E+01 4.42E-04 8.88E-16 4.44E-15
Std. 0.00E+00 5.79E-01 1.07E+00 9.58E-01 2.83E+00 8.68E-05 0.00E+00 7.94E-16
F11
Best 0.00E+00 8.90E+00 9.87E-01 4.02E+00 1.67E+01 5.99E-07 0.00E+00 0.00E+00
Average 0.00E+00 1.58E+01 7.13E+00 9.03E+00 4.13E+01 8.60E-04 0.00E+00 0.00E+00
Worst 0.00E+00 2.21E+01 1.89E+01 1.38E+01 1.17E+02 1.72E-02 0.00E+00 0.00E+00
Std. 0.00E+00 3.17E+00 6.39E+00 2.67E+00 2.35E+01 3.84E-03 0.00E+00 0.00E+00
F12
Best 1.20E-04 2.19E+02 2.59E-03 3.62E+01 5.94E+04 8.95E-04 2.65E-05 4.36E-08
Worst 1.08E-01 1.88E+05 2.13E-01 6.81E+05 1.02E+07 2.62E-02 3.04E-03 1.12E-05
Std. 3.50E-02 4.10E+04 6.56E-02 1.66E+05 2.73E+06 8.40E-03 6.91E-04 2.45E-06
F13
Best 2.47E-05 6.96E+04 3.30E-02 4.77E+04 5.05E+05 9.72E-02 5.11E-06 2.87E-08
Worst 2.81E+00 1.96E+06 1.29E+00 3.51E+06 1.33E+07 6.25E-01 5.27E-02 1.50E-04
Std. 7.82E-01 4.19E+05 2.91E-01 8.39E+05 4.38E+06 1.40E-01 1.64E-02 3.34E-05
F14
Best 9.98E-01 9.98E-01 9.98E-01 9.98E-01 9.98E-01 9.98E-01 9.98E-01 9.98E-01
Average 9.98E-01 9.98E-01 9.98E-01 3.80E+00 9.98E-01 9.98E-01 1.10E+00 2.78E+00
Worst 9.98E-01 9.98E-01 9.98E-01 8.88E+00 9.98E-01 9.98E-01 1.99E+00 9.80E+00
Std. 5.59E-05 7.20E-17 6.91E-13 2.31E+00 6.99E-11 2.60E-16 3.06E-01 2.31E+00

Table 7
Statistical results for the multimodal with fixed dimension benchmark functions.
F15
Best 4.20E-04 7.98E-04 7.40E-04 7.27E-04 7.18E-04 3.37E-04 3.08E-04 3.20E-04
Average 7.78E-04 1.01E-03 1.52E-03 2.05E-03 5.11E-03 6.67E-04 6.39E-04 4.49E-04
Worst 1.23E-03 1.26E-03 2.19E-03 1.65E-02 2.04E-02 1.22E-03 1.43E-03 7.30E-04
Std. 2.51E-04 1.37E-04 4.26E-04 3.49E-03 7.87E-03 2.27E-04 4.02E-04 1.31E-04
F16
Best -1.0316 -1.0316 -1.0316 -1.0316 -1.0316 -1.0316 -1.0316 -1.0316
Average -1.0316 -1.0316 -1.0316 -1.0311 -1.0316 -1.0316 -1.0316 -1.0311
Worst -1.0316 -1.0316 -1.0316 -1.0299 -1.0316 -1.0316 -1.0316 -1.0285
Std. 6.85E-08 1.43E-09 1.53E-12 4.98E-04 1.07E-12 1.44E-16 6.24E-10 6.76E-04
F17
Best 0.3979 0.3979 0.3979 0.3979 0.3979 0.3979 0.3979 0.3979
Average 0.3979 0.3979 0.3979 0.3996 0.3979 0.3979 0.3979 0.3981
Worst 0.3979 0.3979 0.3979 0.4170 0.3979 0.3979 0.3979 0.4001
Std. 7.48E-07 1.95E-06 4.72E-07 5.34E-03 4.73E-10 0.00E+00 4.01E-07 4.75E-04
F18
Best 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000
Average 3.0000 3.0000 3.0035 3.0000 3.0000 3.0000 3.0000 3.0281
Worst 3.0000 3.0000 3.0146 3.0000 3.0000 3.0000 3.0000 3.1154
Std. 2.97E-08 3.67E-13 5.69E-03 1.80E-15 1.87E-11 2.68E-15 2.42E-07 3.60E-02
F19
Best -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8625
Average -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8601 -3.7968
Worst -3.8627 -3.8628 -3.8628 -3.8628 -3.8628 -3.8628 -3.8489 -3.6008
Std. 2.42E-05 4.43E-15 1.04E-08 2.04E-15 2.09E-10 1.84E-15 3.98E-03 6.87E-02
F20
Best -3.3219 -3.3220 -3.3220 -3.3220 -3.3220 -3.3220 -3.2595 -3.1890
Average -3.2274 -3.3196 -3.3220 -3.3220 -3.2485 -3.2471 -3.1036 -2.9660
Worst -3.1244 -3.3048 -3.3218 -3.3220 -3.1299 -3.1376 -2.9227 -2.6053
Std. 5.85E-02 4.03E-03 3.79E-05 2.30E-07 7.10E-02 6.44E-02 9.16E-02 1.50E-01
F21
Best -10.1527 -9.9878 -10.1532 -10.1532 -10.1532 -10.1532 -10.1456 -10.1384
Average -10.1441 -8.7810 -10.1405 -5.1580 -6.7809 -7.9998 -8.2304 -10.0436
Worst -10.0547 -5.6056 -10.0827 -2.6829 -2.6305 -2.6829 -5.0365 -9.8898
Std. 2.24E-02 1.20E+00 2.00E-02 3.12E+0 3.82E+0 2.75E+00 2.40E+0 6.97E-02
F22
Best -10.4028 -10.4029 -10.4029 -10.4029 -10.4029 -10.4029 -10.4008 -10.3859
Average -9.8711 -9.6307 -10.3243 -10.0206 -7.5084 -9.4214 -7.3729 -10.3079
Worst -5.0873 -7.6669 -10.0223 -2.7659 -1.8376 -2.7659 -5.0766 -10.1440
Std. 1.63E+00 8.32E-01 1.07E-01 1.71E+0 3.69E+0 2.43E+00 2.60E+0 6.63E-02
F23
Best -10.5364 -10.5363 -10.5352 -10.5364 -10.5364 -10.5364 -10.5362 -10.5296
Average -10.5345 -9.6576 -10.1501 -9.8779 -6.7265 -9.8146 -8.2751 -10.4536
Worst -10.5293 -5.4537 -5.0981 -3.1809 -2.4217 -2.8066 -5.1230 -10.1355
Std. 1.76E-03 1.32E+00 1.20E+0 2.04E+0 3.65E+0 2.23E+00 2.64E+0 9.78E-02
Figs. 3-5 show the convergence curves of the problems. The significant difference between the
CMO results and other algorithms results in both the accuracy and speed of convergence.

Fig. 3. The convergence curve of the unimodal benchmark problems.
Fig. 4. The convergence curve of the multimodal benchmark problems.

Fig. 5. The convergence curve of the multimodal benchmark problems with fixed dimensions.
3.2. Wilcoxon rank-sum test
The Wilcoxon rank-sum test was employed to conduct a robust statistical comparison of the
outcomes. In adherence to a significance level of 5%, Table 8 showcases the comprehensive
results derived from this rigorous analysis. The discerned patterns solidify the position of the
CMO algorithm as the standout performer among all algorithms under scrutiny.
Moreover, Fig. 6 provides a visual representation in the form of a boxplot, capturing the
outcomes of the ANOVA test performed across the spectrum of the 23 mathematical benchmark
functions. This graphical depiction enhances our understanding of the CMO algorithm's
consistent efficacy across varied function landscapes.

Table 8
P-value of the Wilcoxon rank-sum test for mathematical benchmark functions.
Function DE ABC CBO PSO EO LHHA SMA
F1 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08
F2 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08
F3 6.80E-08 6.80E-08 6.80E-08 6.80E-08 8.35E-03 6.80E-08 6.80E-08
F4 6.80E-08 6.80E-08 6.80E-08 6.80E-08 2.36E-06 6.80E-08 6.80E-08
F5 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 2.30E-05 0.29
F6 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 0.19
F7 6.80E-08 6.80E-08 6.80E-08 6.80E-08 6.80E-08 0.27 9.17E-08
F8 6.80E-08 6.80E-08 6.61E-08 6.80E-08 6.80E-08 0.08 6.80E-08
F9 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09
F10 8.01E-09 8.01E-09 8.01E-09 8.01E-09 0.34 8.01E-09 8.01E-09
F11 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09 8.01E-09
F12 6.80E-08 3.34E-03 6.80E-08 6.80E-08 6.80E-08 1.10E-05 0.82
F13 6.80E-08 3.15E-02 6.80E-08 6.80E-08 1.06E-07 3.38E-04 3.97E-03
F14 1.51E-08 1.57E-07 9.17E-08 6.58E-05 6.92E-07 2.07E-02 5.18E-08
F15 1.63E-03 5.87E-06 2.22E-04 7.11E-03 8.60E-06 4.99E-02 0.10
F16 1.69E-06 6.78E-08 1.38E-06 6.79E-08 6.80E-08 2.92E-07 3.96E-08
F17 4.64E-05 8.60E-06 6.04E-06 7.89E-08 9.17E-08 7.38E-05 8.01E-09
F18 6.79E-08 1.33E-02 5.35E-08 6.80E-08 6.80E-08 3.64E-03 6.71E-08
F19 6.22E-08 6.80E-08 4.14E-08 6.80E-08 6.80E-08 2.30E-05 2.94E-08
F20 1.78E-03 2.56E-07 6.80E-08 0.11 1.06E-07 9.75E-06 2.14E-03
F21 6.80E-08 0.74 6.56E-03 0.80 3.94E-07 2.56E-07 0.29
F22 5.09E-04 6.22E-04 1.60E-05 0.54 1.60E-05 3.50E-06 1.61E-04
F23 1.20E-06 1.38E-06 1.60E-05 0.30 7.90E-08 1.05E-06 1.44E-04

Fig. 6. ANOVA test results for mathematical benchmark functions.

3.3. Engineering truss problems
Trusses are widely used in engineering because of their design flexibility and fast construction
[33]. Finding methods to reduce the weight and cost of trusses while adhering to the standard
rules and regulations is highly practical. In this section, the performance of the proposed CMO
algorithm is evaluated using two benchmark examples of two- and three-dimensional truss
optimization problems. The objective function of truss optimization is to minimize the structural
weight. Design constraints typically include limitations on element stress and nodal
displacements. The optimization problem formulation for truss structures can be expressed as
follows:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝑊 = ∑ 𝛾𝑖𝑙𝑖𝑋𝑖
𝑛
𝑖=1 , 𝑖 = 1,2, … , 𝑛 (9)
Subject to:
gj
d
=
dj
d
̅j
− 1, j = 1,2, … , m
𝑔𝑘
𝑠
=
𝜎𝑘
𝜎
̅𝑘
− 1, 𝑗 = 1,2, … , 𝑛 (10)
𝑋𝑖
𝐿
≤ 𝑋𝑖 ≤ 𝑋𝑖
𝑈
In the equations above, 𝑔𝑘
𝑠
and 𝑔𝑗
𝑑
represent the stress and displacement constraints, respectively.
𝑊 represents the structural weight, 𝑋𝑖, 𝛾𝑖, and 𝑙𝑖 are the cross-sectional area, density of the
material, and length of the 𝑖th element, respectively. 𝜎𝑘 and 𝑑𝑗 represent the stress and
displacement of the 𝑘th element and 𝑗th node, respectively; 𝜎
̅𝑘 and 𝑑̅𝑗 are the allowable values
for stress and displacement, respectively. 𝑚 and 𝑛 denote the numbers of nodes and elements,
respectively.
To handle the constraints in the optimization problem, a multistage assignment penalty function
was utilized as an exterior penalty function (EPF). The penalty function is defined as follows:
𝐹(𝑥) = 𝑓(𝑥) + ℎ(𝑘)𝐻(𝑥), 𝑥 ∈ 𝑆 ⊂ 𝑅𝑛
(11)
where 𝑘 represents the current iteration number, ℎ(𝑘) is the dynamically modified penalty value,
𝐻(𝑥) is the penalty factor, and 𝑓(x) is the original objective function. 𝐻(𝑥) is defined as:
𝐻(𝑥) = ∑ 𝜃 (𝑞𝑖(𝑥)) 𝑞𝑖(𝑥) 𝛾 (𝑞𝑖(𝑥))
𝑚
𝑖=1 (12)
In equation (12), 𝑞𝑖(𝑥) takes the values of either 0 or 𝑔𝑖(𝑥), where 𝑔𝑖(𝑥) represents the
constraint functions and 𝜃(𝑞𝑖(𝑥) is a multi-segment assignment function. The penalty function
power is denoted as 𝛾(𝑞𝑖(𝑥)).
The penalty value is initially set to the lowest possible value, but increases with each iteration as
the algorithm progresses [34]. The selection of the penalty coefficient and initial value is crucial
in penalty-function methods. If the penalty value is extremely small, the algorithm may generate
solutions outside the feasible region. However, if the penalty value is too large, reaching the

boundary of the feasible region becomes challenging and the boundary may not be properly
explored. It is common for at least one active constraint to exist at the optimum point,
emphasizing the need to search for the entire feasible region, including its boundaries [35].
In this study, following the suggestions in [36], the penalty parameters are set as follows: if
𝑞𝑖(𝑥) < 1, then 𝛾 (𝑞𝑖(𝑥)) = 1; otherwise, 𝛾 (𝑞𝑖(𝑥)) = 2. Additionally, if 𝑞𝑖(𝑥) < 0.001, then
𝜃 (𝑞𝑖(𝑥)) = 10. For the range 0.001 < 𝑞𝑖(𝑥) < 0.1, 𝜃 (𝑞𝑖(𝑥)) = 20. For 0.1 < 𝑞𝑖(𝑥) < 1,
𝜃 (𝑞𝑖(𝑥)) = 100. otherwise, 𝜃 (𝑞𝑖(𝑥)) = 300. ℎ(𝑘) is set to 𝑘√𝑘, where 𝑘 is the current
iteration number.
3.3.1. 10-bar truss example
The 10-bar truss problem is one of the most extensively studied truss problems. The geometrical
data for the problem are presented in Fig. 7. The Young's modulus of the material is E = 104
ksi
and the specific weight is 0.1 lb/in3
. The cross-sectional area of each bar varies between 0.1 and
35 °in2
while the stress in each bar did not exceed ± 25 ksi. The displacement in each free node
in each direction should not exceed 2 inches. As shown in Fig. 7, the vertical load in nodes 2 and
4 is 105
lb.
Fig. 7. Schematic presentation of 10-bar truss problem.
During the optimization procedure, a total of 50 particles took part, while the maximum iteration
limit was defined at 1000. The optimization outcomes achieved through the CMO approach were
juxtaposed with the results of alternative algorithms in Table 9. These findings validate that the
CMO algorithm stands out with the lowest essential count of function evaluations (NFEs) and
exhibits a narrow standard deviation, ultimately leading to the most optimal solution.
Table 9 presents a comparison of the CMO optimization results with those of other available
studies by addressing the number of function evaluations (NFE), optimal design variables,
standard deviation, and best and average weights. It is worth mentioning that the maximum
displacements, which were assessed using HPSACO in the optimum designs, exceeded the 2.0-
inch displacement limits. The suggested CMO converged to a completely feasible design. The
CMO algorithm provides an optimized design that can compete with the feasible designs
obtained from the ABC-AP and HPSSO.

Table 9
Results of optimization for the 10-bar truss problem.
Design
Variables
WEO [37] HPSSO [38] HPSACO [39] ABC-AP [40] Present Study
𝐴1 30.5755 30.5838 30.493 30.548 30.57
𝐴2 0.1 0.1 0.1 0.1 0.1
𝐴3 23.3368 23.15103 23.23 23.18 23.08
𝐴4 15.1497 15.20566 15.346 15.218 15.22
𝐴5 0.1 0.1 0.1 0.1 0.1
𝐴6 0.5276 0.548897 0.538 0.551 0.56
𝐴7 20.9892 21.06437 20.99 21.058 21.04
𝐴8 7.4458 7.465322 7.451 7.463 7.47
𝐴9 0.1 0.1 0.1 0.1 0.1
𝐴10 21.5236 21.52935 21.458 21.501 21.56
Best 5060.99 5060.86 5056.56 5060.88 5060.9
Average 5062.09 5062.28 5057.66 N/A 5061.24
Std. 2.05 4.325 1.42 N/A 0.55
NFE 19,540 14,118 10,650 500,000 12,500
3.3.2. 25-bar spatial truss problem
The second problem addressed in this section involves a 25-bar spatial truss structure, depicted
in Figure 8. The material density and elasticity modulus for this structure are set at 0.1 lb/in³ and
10,000 ksi, respectively. Each node's maximum allowable displacement in all three coordinate
directions is limited to 0.35 inches, while the permissible stress for the members ranges from -40
ksi to +40 ksi. The structural components are categorized into eight groups, denoted as follows:
A₁, A₂-A₅, A₆-A₉, A₁₀-A₁₁, A₁₂-A₁₃, A₁₄-A₁₇, A₁₈-A₂₁, and A₂₂-A₂₅.
Fig. 8. Schematic presentation of 25-bar spatial truss problem.

The tension and compression stress constraints for each group are presented in Table 10. The
cross-sectional areas of each group of elements are assumed variable and change continually
from 0.01 in2
to 3.4 in2
. There are 110 inequality constraints for this problem. The truss is
subjected to the loading conditions listed in Table 11, where 𝑃
𝑥, 𝑃
𝑦, and 𝑃
𝑧 are the loads along
𝑥 −, 𝑦 −, and 𝑧 −axes, respectively.
Table 10 compares the optimization results of the CMO algorithm, which used 50 particles and
1000 iterations, with those of the BB-BC, EBA, TLBO, and WEO for the 25-bar spatial truss
problem described earlier. The results show that all algorithms were able to converge to the best
or near-optimal design, but CMO required significantly fewer function evaluations and less
computational effort than the other algorithms.
Table 10
Results of optimization for the 25-bar truss.
Element Groups BB–BC [41] EBA [42] TLBO [43] WEO [44] Present Study
𝐴1 0.01 0.01 0.01 0.01 0.01
𝐴2 − 𝐴5 2.092 1.9789 2.0712 1.9814 1.977
𝐴6 − 𝐴9 2.964 3.0047 2.957 3.0023 3.006
𝐴10 − 𝐴11 0.01 0.01 0.01 0.01 0.01
𝐴12 − 𝐴13 0.01 0.01 0.01 0.01 0.01
𝐴14 − 𝐴17 0.689 0.6888 0.6891 0.6827 0.684
𝐴18 − 𝐴21 1.601 1.6783 1.6209 1.6778 1.680
𝐴22 − 𝐴25 2.686 2.6527 2.6768 2.6612 2.658
Weight (lb) 545.38 545.1688 545.09 545.166 545.18
Average (lb) 545.78 546.4464 545.41 N/A 545.36
SD (lb) 0.491 N/A 0.42 N/A 0.23
NFE 20,566 20,000 15,318 19,750 7,500
Table 11
Loading conditions acting on the 25-bar truss.
Load case Nodes
Loads
𝑷𝒙 (kips) 𝑷𝒚 (kips) 𝑷𝒛 (kips)
1 1 0.0 20.0 -5.0
2 0.0 -20.0 -5.0
2 1 1.0 10.0 -5.0
2 0.0 10.0 -5.0
3 5.0 0.0 0.0
6 5.0 0.0 0.0
4. Conclusions
The paper introduces the Center of Mass Optimization (CMO) algorithm, which is a physics-
based optimization method inspired by the center of mass of a system of particles. The CMO
algorithm incorporates a random walk operator, a mutation operator, and elitism selection to
improve local search capabilities and prevent becoming trapped in local optima. One of the

notable features of the CMO algorithm is its lack of adjustable parameters, which makes it easy
to implement and use in various engineering applications. The performance of the CMO
algorithm was evaluated on 23 mathematical benchmark functions, including unimodal,
multimodal, and low-dimensional functions, as well as two truss optimization problems. The
experimental results demonstrate that the CMO algorithm outperforms other popular
optimization algorithms in terms of efficiency and achieves optimal solutions. Future work can
focus on extending the CMO algorithm to handle binary and multi-objective optimization
problems, which would further enhance its applicability and versatility. Overall, the CMO
algorithm shows promising potential as an effective optimization technique, and its simplicity
and strong performance make it a valuable tool for solving engineering optimization problems.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Authors contribution statement
HV: Conceptualization; Data curation; formal analysis; Investigation; Methodology; Software;
Validation; Visualization; Roles/Writing – original draft; writing – review, and editing.
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