Discrete Sizing Optimization of Steel Structures Using Modified Fireworks Algorithm

Journal of Soft Computing in Civil Engineering 9-1 (2025) 39-63
How to cite this article: Seraji N, Babaei M. Discrete sizing optimization of steel structures using modified fireworks algorithm.
J Soft Comput Civ Eng 2025;9(1):39–63. https://doi.org/10.22115/scce.2024.396409.1642
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Discrete Sizing Optimization of Steel Structures Using Modified
Fireworks Algorithm
Navid Seraji 1
; Mehdi Babaei 2,*
1. Ph.D. Student, Department of Civil Engineering, Faculty of Engineering, University of Zanjan, Zanjan, Iran
2. Associate Professor, Department of Civil Engineering, Faculty of Engineering, University of Zanjan, Zanjan, Iran
* Corresponding author: mbabaei@znu.ac.ir
https://doi.org/10.22115/scce.2024.396409.1642
ARTICLE INFO ABSTRACT
Article history:
Received: 08 May 2023
Revised: 13 August 2023
Accepted: 22 January 2024
Fireworks algorithm (FWA) is an artificial intelligence algorithm
developed by emulating the burst process of fireworks. This paper
applies a modified version of fireworks algorithm called MoFWA
to design disparate steel trusses and planar frames. In this study,
the objective is to improve FWA’s performance by adding two
operators to it: 1) fly-back mechanism 2) duplicate spark remover
operator. Also, its amplitude is changed into a dynamic one to
enhance its compatibility with different optimization problems.
The function we are focusing on is the total weight of the structure.
This takes into account the requirements for serviceability and
strength as outlined by the American Institute for Steel
Construction's Load and Resistance Factor Design (LRFD)
standards. A total of six benchmark structures including a 10-bar
truss, a 25-bar truss, a 582-bar tower truss, a two-bay three-story
frame, a one-bay 10-story frame, and a three-bay 24-story frame
are chosen from previous studies for the optimization. In addition,
a comparison is presented between the results of MoFWA with
FWA and other optimization methods such as modified sine-cosine
algorithm (MSCA), newton meta-heuristic algorithm (NMA),
improved grey wolf optimizer (GWOM), enhanced whale
optimization algorithm (EWOA), switching teams algorithm
(STA), MHBMO, artificial bee colony (ABC), school-based
optimization (SBO), teaching-learning based optimization (TLBO),
design-driven harmony search (DDHS), and inscribed
hyperspheres (IHS). The results indicate that MoFWA is
completely better than FWA and can generate superior optimal
solutions compared to the other optimization algorithms.
Keywords:
Fireworks algorithm;
Discrete optimization;
Structural engineering;
Moment-resisting frames;
Truss structures.

40 N. Seraji; M Babaei/ Journal of Soft Computing in Civil Engineering 9-1 (2025) 39-63
1. Introduction
Researchers worldwide have grappled with the challenge of structural optimization, and
numerous methods and algorithms have been developed to improve various aspects of it. One of
the main challenges that researchers are faced with is that most of these algorithms have either
significant accuracy with weak computational cost or vice versa. The reason that FWA is chosen
in this study is that it has a high potential of outdoing other algorithms. However, it must be
noted that FWA has an excessive computational process which is not a favorable characteristic.
In this study, we try to modify FWA by adding some new strategies to it and also refining the
relationship of its primary amplitude size and coefficients with the size of search space so that it
could function better in different circumstances.
When it comes to optimization problems, there are two types: those with one objective and those
with multiple objectives, which are known as multi-objective problems. Structural optimization
has the primary goal of minimizing the weight of a structure while still meeting the strength and
drift constraints specified by code. Our study will focus on discrete variable optimization for
steel structures, which we will discuss shortly. For this type of optimization, the design variable
selected from steel section tables provided in steel construction manuals is the cross-sectional
area of structural members.
There are myriads of meta-heuristic methods for optimal design of engineering problems (e.g.,
simulated annealing (SA) by [1], teaching–learning based optimization (TLBO) by [2], [3]), ant
colony optimization [4], genetic algorithm (GA) by [5,6], particle swarm optimization (PSO) by
[7–9], cellular automata (CA) by [10], cuckoo search algorithm (CS) by [11], charged system
search (CSS) by [12], Design-driven harmony search (DDHS) by [13], mine blast algorithm
(MBA) by [14], artificial bee colony (ABC) by [15], differential evolution (DE) by [16]).
Most of meta-heuristic algorithms are population-based algorithms which means that in each
iteration, they produce some probable results that may lead them to the optimal solution. The
performance of these algorithms depends on two main features including diversification, and
intensification. Also, the interaction between these two has effects on their results. On one hand
there is diversification which certifies that all the search space will be covered. In other words,
weak diversification leads to poor exploration in search space. In contrast, strong diversification
causes getting stuck in local optima which is not a desirable phenomenon. On the other hand,
there is intensification that affects the computational process of optimization since it controls the
exploitation. The complexity of an optimization problem depends on two factors: 1) search space
2) number of design variables.
The fireworks algorithm (FWA) is a swarm intelligence algorithm that was first introduced by
[17]. The FWA is developed by emulating the burst process of fireworks at night. In fireworks
algorithm, fireworks are generated randomly all around the search space and each one of them
produces a group of sparks all over itself. These fireworks and explosion sparks represent
potential solutions for the optimization problem and FWA keeps generating them until a
termination criterion is satisfied. One of the main advantages of FWA against other meta-
heuristics is its simplicity and capability to easily balance between diversification and

N. Seraji; M Babaei/ Journal of Soft Computing in Civil Engineering 9-1 (2025) 39-63 41
intensification. In contrast, it has some deficiencies such as high computational process and low
consideration of the previous solutions. Over the past years, numerous adjustments have been
made to the original FWA; for instance, dynFWA by [18], IFWA by [19], FFWA by [20].
In our study, we made some modifications to the FWA (Fireworks Algorithm) by introducing a
dynamic amplitude and including two new operators called fly-back mechanism and duplicate
spark remover. These changes were aimed at reducing the computational cost of the algorithm
while also improving its ability to search for solutions in difficult and marginal areas. Our results
show that these modifications are effective and can lead to better performance of the FWA in
solving optimization problems.
2. Structural optimization problems
The main objective of structural optimization problems is to minimize the structure weight
according to the design variables vector 𝑉 = [𝑣1, 𝑣2, … , 𝑣𝑑] which contains the section number
of structure members for each grouping [21]. The structure weight is calculated as follows:
W = ∑ γiAiLi
Nm
i=1 (1)
Where W is the weight of structure; 𝑁𝑚 is the number of members; 𝛾𝑖 is the material density of
member i; 𝐴𝑖 is the cross-sectional area of member i and 𝐿𝑖 is the length of member i.
The original objective is penalized with the following formula:
𝐹 = 𝑊(1 + 𝐶)𝜀
(2)
Where ε is a number greater than 1 and C is defined as:
𝐶 = ∑ 𝐶𝑖
𝜎
+ ∑ 𝐶𝑖
𝛿
𝑁𝑛
𝑖=1
𝑁𝑚
𝑖=1 + ∑ 𝐶𝑖
𝐼
𝑁𝑐
𝑖=1 (1‌
)
Where 𝐶𝑖
𝜎
, 𝐶𝑖
𝛿
and 𝐶𝑖
𝐼
are penalty functions for strength constraint violation, drift constraint
violation and axial force-bending moment interaction, respectively. 𝑁𝑚, 𝑁𝑛 and 𝑁𝑐 are the
number of members, number of joints and the number of columns, respectively. The C factor is
defined as follows:
Ci = {
0 if ∝i< 0
∝i if ∝i> 0
(4)
Where ∝i is calculated separately from strength and drift constraints:
∝𝑖
𝜎
=
|𝜎𝑖|
|𝜎𝑖
𝑎
|
-1 , 𝑖 = 1, … , 𝑁𝑚 (5)
∝𝑖
𝛿
=
|𝛿𝑖|
|𝛿𝑖
𝑎
|
-1 , 𝑖 = 1, … , 𝑁n (6)

Where 𝜎𝑖 is the stress in member i and 𝜎𝑖
𝑎
is the allowable stress of member i; 𝛿𝑖 and 𝛿𝑖
𝑎
are the
inter-story drift and allowable inter-story drift of joint i;
∝𝑖
𝐼
is defined as:
For
𝑃𝑢
𝜑𝑐𝑃𝑛
< 0.2 ∶ ∝𝑖
𝐼
= [
𝑃𝑢
2𝜑𝑐𝑃𝑛
+
𝑀𝑢𝑥
𝜑𝑏𝑀𝑛𝑥
+
𝑀𝑢𝑦
𝜑𝑏𝑀𝑛𝑦
] − 1 ≤ 0 𝑖 = 1, … , 𝑁𝑒 (7)
For
𝑃𝑢
𝜑𝑐𝑃𝑛
≥ 0.2 ∶ ∝𝑖
𝐼
= [
𝑃𝑢
𝜑𝑐𝑃𝑛
+
8
9
(
𝑀𝑢𝑥
𝜑𝑏𝑀𝑛𝑥
+
𝑀𝑢𝑦
𝜑𝑏𝑀𝑛𝑦
)] − 1 ≤ 0 𝑖 = 1, … , 𝑁𝑒 (8)
Where 𝜑𝑐 is the resistance factor (𝜑𝑐 = 0.9 for tension, 𝜑𝑐 = 0.85 for compression); 𝜑𝑏 is the
flexural resistance reduction factor (𝜑𝑏 = 0.9); 𝑀𝑢𝑥 and 𝑀𝑢𝑦 are the required flexural strengths
in the x and y directions, respectively; 𝑀𝑛𝑥 and 𝑀𝑛𝑦 are the nominal flexural strengths in the x
and y directions, respectively; 𝑃
𝑢 is the required strength; and 𝑃
𝑛 denotes the nominal strength.
3. Fireworks algorithm
As previously mentioned, the Fireworks algorithm (FWA), was originally presented in a study by
Tan and Zhu named Fireworks Algorithm for Optimization 2010. During the past decade, a
myriad of researchers have proposed different versions of FWA and it is being used on different
branches of optimization such as spam detection, image recognition, digital filter design, and
power loss minimization.
A brief explanation of how the FWA algorithm works: First N locations are selected for N
fireworks to explode, then each one of them produces a group of explosion sparks around itself,
then the fitness values of both fireworks and explosion sparks are evaluated and the best location
is chosen to follow the next iteration, this process is continued until a termination criterion is
met. When a firework is well manufactured, it has plenty of explosion sparks within a small
range around it. In contrast, a bad firework is the one with a few explosion sparks within a large
radius. These two features are implemented in the FWA algorithm such that the firework in the
best location is the good one while the others are categorized in the bad fireworks group
depending on their fitness value.
3.1. Number of sparks
Suppose the FWA is designed for the general optimization problem:
Minimize 𝑓(𝑥) ∈ R , 𝑥𝑚𝑖𝑛 ≤ 𝑥 ≤ 𝑥𝑚𝑎𝑥 (9)
Where x = [𝑥1, 𝑥2, … , 𝑥𝑑] implies a location in potential space, 𝑑 is the dimension of design
variable, f(x) is the objective function, and 𝑥𝑚𝑖𝑛 and 𝑥𝑚𝑎𝑥 refer to the boundary of the potential
space. The number of sparks generated by each firework 𝑥𝑖 is calculated as follows:
𝑠𝑖 = 𝑀𝑒
𝑦𝑚𝑎𝑥−𝑓(𝑋𝑖)+𝜀
∑ (𝑦𝑚𝑎𝑥−𝑓(𝑋𝑖))+𝜀
𝑁
𝑖=1
(10)

Where 𝑦𝑚𝑎𝑥 = max(𝑓(𝑥𝑖)) is the maximum amount of objective function among N fireworks,
𝑀𝑒 is a constant amount to control the number of explosion sparks, and 𝜀 is utilized to avoid
zero-division-error.
Also 𝑠𝑖 is limited to two boundaries as follows:
s
̂i = {
round(a . m) if si < 𝑎𝑚
round(b . m) if si > 𝑏𝑚 , 𝑎 < 𝑏 < 1
round(si) otherwise
(11)
Where a and b are constant parameters to control the boundaries.
3.2. Amplitude of explosion
The amplitude of a well generated firework explosion is smaller than a bad one and it has more
sparks. The amplitude of explosion for each firework is defined as follows:
𝐴𝑖 = 𝐴
̂ 𝑓(𝑋𝑖)−𝑦𝑚𝑖𝑛+𝜀
∑ (𝑓(𝑋𝑖)−𝑦𝑚𝑖𝑛)+𝜀
𝑁
𝑖=1
(12)
Where 𝑦𝑚𝑖𝑛 = min(𝑓(𝑥𝑖)) is the minimum (best) value of objective function among the N
fireworks, 𝐴
̂ is a constant for controlling the explosion amplitude.
3.3. Explosion sparks operator
Each firework generates 𝑠𝑖 explosion sparks around its current location. Fireworks with better
fitness can produce a larger population of explosion sparks within smaller explosion amplitude.
For each explosion spark, Algorithm 1 is performed as follows:
Algorithm 1. Spark locationing.
Algorithm
1: Initialization (locationing of the spark) 𝑥
̃𝑗 = 𝑥𝑖;
2: 𝑧 = 𝑟𝑜𝑢𝑛𝑑(𝑑. 𝑟𝑎𝑛𝑑(0,1));
3: Random selection z dimensions of 𝑥
̃𝑗;
4: Displacement calculation: h=𝐴𝑖 × 𝑟𝑎𝑛𝑑(−1,1) ;
5: for each dimension 𝑥
̃𝑘
𝑗
∈ {pre − selected 𝑧 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑥
̃𝑗} do
6: 𝑥
̃𝑘
𝑗
= 𝑥
̃𝑘
𝑗
+h
7: if 𝑥
̃𝑘
𝑗
out of bounds then
8: map 𝑥
̃𝑘
𝑗
to the potential space: 𝑥
̃𝑘
𝑗
= xk
min
+ |𝑥
̃𝑘
𝑗
|%(xk
max
− xk
min
);
9: end if
10: end for
Where d is the dimension of location x.
An additional way of generating sparks is shown in Algorithm 2:

Algorithm 2. A specific spark locationing.
Algorithm
1: Initialization of the spark location 𝑥
̃𝑗 = 𝑥𝑖;
2: 𝑧 = 𝑟𝑜𝑢𝑛𝑑(𝑑. 𝑟𝑎𝑛𝑑(0,1));
3: Random seceltion z dimensions of 𝑥
̃𝑗;
4: Calculation of Gaussian explosion coefficient: g = Gaussian (1, 1);
5: for each dimension 𝑥
̃𝑘
𝑗
∈ {pre − selected 𝑧 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑥
̃𝑗} do
6: 𝑥
̃𝑘
𝑗
= 𝑥
̃𝑘
𝑗
. 𝑔;
7: if 𝑥
̃𝑘
𝑗
out of bounds then
8: map 𝑥
̃𝑘
𝑗
to the potential space: 𝑥
̃𝑘
𝑗
= xk
min
+ |𝑥
̃𝑘
𝑗
|%(xk
max
− xk
min
);
9: end if
10: end for
3.4. Selection of locations
In the FWA algorithm, the location with the highest fitness value is retained as a reference for the
subsequent generation. Following that, n-1 locations are chosen according to their distance in
order to maintain a diverse range of solutions.
3.5. Dynamic fireworks algorithm
One of the improvements that have been made to FWA is called dynFWA which is proposed by
[18]. In dynFWA, the authors proved that Gaussian sparks could be removed from the algorithm
without affecting its performance. Also, fireworks were separated into two different groups in
dynFWA: 1) core fireworks (CF) 2) other fireworks. The CF’s responsibility is to explore the
vicinity of the local optimum location while other fireworks perform a global search. The
amplitude of the explosions for each group is calculated individually. Algorithm 3 demonstrates
the procedure of calculating the explosion amplitude for CF. The explosion amplitude of other
fireworks is calculated from Eq. (12).
Algorithm 3. Dynamic explosion amplitude update for CF.
Algorithm
Initialization: Define:
XCF (implies to the current location of the CF);
X
̂b (implies to the best location among all explosion sparks);
ACF (implies to the current explosion amplitude of the CF);
Ca (refers to the amplification coefficient);
Cr (refers to the reduction coefficient);
Iteration:
1: if 𝑓(X
̂b) − 𝑓(XCF) < 0 then
2: ACF ← ACF ∗ Ca ;
3: else
4: ACF ← ACF ∗ Cr ;
end if

The whole dynFWA framework is shown in the following algorithm:
Algorithm 4. dynFWA framework.
Algorithm
1: Initialization (of N fireworks and evaluation the quality)
2: Initialization (of the explosion amplitude for CF)
3: while termination criteria are not satisfied do
4: Calculation the number of explosion sparks
5: Calculation of the explosion amplitude for non-CF
6: for each firework do
7: Generation of explosion sparks
8: Mapping sparks at invalid locations back to search space
9: Evaluation of the quality of explosion sparks
10: end for
11: Updating the explosion amplitude of CF
12: Selecting the N fireworks for next iteration
13: end while
3.6. Modified fireworks algorithm
In this study, we used the FWA and changed its amplitude into a dynamic one. Also, two new
operators were added to it to reduce its computational cost as well as enhancing its searching
capabilities in marginal locations. In addition, after testing the MoFWA, it was observed that the
existence of only one core firework is adequate for optimization. Also, we set the number of
explosion sparks for each example, depending on the number of design variables and the size of
search space to maintain the functionality of the algorithm. Moreover, the initial bounds of core
firework were set so that the exploration capability of the algorithm be preserved.
3.6.1. Fly-back mechanism
Solutions to most mechanical problems are usually located in the vicinity of the feasible region’s
boundaries [29], [30]. Meanwhile, artificial algorithms face some challenges when they try to
produce new solutions near those boundaries. In other words, when it comes to optimization
close to the boundaries, some of the newly produced solutions may be located outside the
feasible space. The fly-back mechanism ensures that all of the solutions are located within the
feasible region by relocating the outer ones to their previous position [31]. Although flying back
to the former location will guarantee an acceptable solution, it was observed that restoring half of
the range between the initial location and the infeasible position is more effective than the
previous method. This mechanism was implemented in the MoFWA to enhance its searching
capabilities near the boundaries.

Fig. 1. Fly-back mechanism.
3.6.2. Duplicate sparks removal operator
When MoFWA reaches the global optima, its location could not get better much, so the explosion
amplitude keeps multiplying to the reduction coefficient in every iteration. As a result, the
explosion amplitude keeps shrinking. If the explosion amplitude is significantly reduced, it's
possible that some of the sparks created may be the same as previous ones. This can result in
increased computational cost and decreased speed for the MoFWA. To tackle this issue, we
introduce the DSR operator that saves the sparks location of last two iterations to compare them
with new sparks and to determine whether they are authentic or not. This procedure is explained
in algorithm 5:
Algorithm 5. removing duplicate sparks.
Algorithm
1: survey s generated sparks
2: remove n-1 sparks from n similar sparks
3: compare s-(n-1) remaining sparks with last two generations
4: remove m similar sparks from current generation
5: evaluate remaining sparks
6: save the current and last generation for next iteration
3.6.3. Initial CF amplitude
In dynFWA, the initial CF amplitude was set manually. However, in MoFWA a new formula is
introduced to calculate the initial CF amplitude automatically. After testing dynFWA numerous
times under different circumstances, we concluded that the best size for the initial CF amplitude
is around 60 percent of the total number of variables. The initial CF amplitude is calculated from
Eq (13):

𝐼𝐶𝐹_𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = 0.6*length(variables) (13)
Where ICF_amplitude is the initial core firework amplitude
Fig. 2. MoFWA algorithm flowchart.

4. Design examples
In this part, six steel structures including three trusses and three frames are presented to make a
comparison between MoFWA and other algorithms. The design examples are:
 A 25-bar truss
 A 10-bar truss
 A 582-bar tower truss
 A two-bay three-story frame
 A one-bay 10-story frame
 A three-bay 24-story frame
The amplification (𝐶𝑎) and reduction coefficient factors (𝐶𝑟) were set to 1.2 and 0.9 for all of the
examples. For each design example, 40 individual runs were carried out and the results of
MoFWA were compared with FWA and also a wide range of other heuristic algorithms.
4.1. 25-bar space truss
The 25-bar transmission tower is frequently studied by numerous researchers with different
methods. Fig. 3 shows topology and nodal numbering of the 25-bar space truss structure. The
material density and modulus of elasticity were considered as 0.1 lb/in.3
(2767.99 kg/m3
) and
10,000 ksi (68,950 MPa), respectively. The loading condition is shown in Table 1. Maximum
displacement for nodes 1 and 2 was limited to ±0.35 in (±8.89 mm) in every direction, also the
stress limit was considered as ±40 ksi (275.6MPa). Truss members were divided into 8 groups,
Table 2 lists the grouping. The sections available for this problem were: {0.1, 0.2, 0.3, …, 2.5,
2.6, 2.8, 3.0, 3.2, 3.4} (in.2
). The sparks generation factor (𝑀𝑒) was set to 150 for this example.
Table 1
Single load case for the 25-bar truss.
Directions
Node
Z
y
x
-10.0
-10.0
1.0
1
-10.0
-10.0
0
2
0
0
0.5
3
0
0
0.6
6
Table 2
Element information for the 25-bar truss.
Group 1 2 3 4 5 6 7 8
Elements 1 2-5 6-9 10,11 12,13 14-17 18-21 22-25

Fig. 3. Configuration of the first example.
Table 3
Optimal results of the 25-bar truss.
Design Variables Optimum cross section area (𝒊𝒏.𝟐
)
No Des.Var. MSCA [22] NMA [23] GWOM1 [24] FWA MoFWA
1 𝐴1 0.1 0.1 0.1 0.1 0.1
2 𝐴2~𝐴5 0.3 0.3 0.6 0.4 0.4
3 𝐴6~𝐴9 3.4 3.4 3.4 3.4 3.4
4 𝐴10~𝐴11 0.1 0.1 0.1 0.1 0.1
5 𝐴12~𝐴13 2.1 2.1 2.3 1.8 1.8
6 𝐴14~𝐴17 1.0 1.0 1 0.9 0.9
7 𝐴18~𝐴21 0.5 0.5 0.2 0.6 0.6
8 𝐴22~𝐴25 3.4 3.4 3.4 3.4 3.4
Best (lb) 484.85 484.85 482.21 484.56 484.56
Average (lb) 484.94 484.94 N/A 498.39 488.73
Worst (lb) N/A N/A N/A 546.24 495.03
Evaluation (#) 5000 250 N/A 8200 6000
Constraint violation None None None None None

Fig. 4. Convergence history of 25-bar truss.
Table 3 lists the optimal designs of modified sine-cosine algorithm, newton meta-heuristic
algorithm, improved grey wolf optimizer, FWA, and MoFWA. The FWA and MoFWA obtained
the second-best weight of 484.56 lbs. The number of fitness function evaluations (FE) of
MoFWA is 6000 which is less than FWA. Fig. 4 shows the convergence history of MoFWA for
the twenty-five-bar truss reaching the optimal design in the 28th
iteration.
4.2. Ten-bar planar truss
The second example is a 10-bar truss. The material density and modulus of elasticity are
considered as 0.1 lb/in.3
(2767.99 kg/m3
) and 10,000 ksi (68,950 MPa), respectively. A single
load P1 = 100 kips is applied to nodes 4 and 2. The available cross-sectional areas for all
members are mentioned in the following bracket:
A= {1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.63, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84,
3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.5, 13,5 13.9, 14.2, 15.5,
16.0, 16.9, 18.8, 1.99, 22.0, 22.9, 26.5, 30.0, 33.5} (in.2
).
Fig. 5. Configuration of the 10-bar truss.

The allowable stress and nodal displacement are limited to ±25 ksi (±172.25 MPa) and ±2 in.
(±50.8 mm). In this example, the sparks generation factor (𝑀𝑒) is set to 150.
Table 4
Optimal results of the ten-bar truss.
Design Variables Optimum cross section area (𝒊𝒏.𝟐
)
No Des.Var. MSCA [22] NMA [23] GWOM1 [24] FWA MoFWA
1 𝐴1 33.50 33.50 33.50 30.00 33.5
2 𝐴2 1.62 1.62 1.62 1.62 1.62
3 𝐴3 22.90 22.90 22.90 24.5 22
4 𝐴4 14.20 14.20 14.20 15.5 15.5
5 𝐴5 1.62 1.62 1.62 1.62 1.62
6 𝐴6 1.62 1.62 1.62 1.62 1.62
7 𝐴7 7.97 7.97 7.97 9.3 8.53
8 𝐴8 22.90 22.9 22.9 22.0 22.0
9 𝐴9 22.00 22.0 22.0 22.0 22.0
10 𝐴10 1.62 1.62 1.62 1.62 1.62
Best (lb) 5,490.74 5,490.74 5,490.74 5,490.74 5476.32
Average (lb) 5492.64 5,490.91 N/A 5,502.4 5493.615
Worst (lb) N/A N/A N/A 5,534.7 5525.026
Evaluation (#) 10000 2880 N/A 18,000 6560
Constraint violation None None None None None
Fig. 6. Convergence history of 10-bar truss.

Table 4 lists the optimal design of mentioned algorithms. In this example, MoFWA obtained the
best weight of 5476.32 lbs with a total evaluation of 6560, which outranks FWA and other
algorithms. Fig. 6 demonstrates the convergence history of MoFWA for the ten-bar truss
example. All in all, it is obvious that MoFWA has the best performance among all of the
mentioned algorithms.
4.3. 582-bar tower truss
The last truss structure in this study is a 582-bar tower truss which can be seen in Fig. 7. The
material density and modulus of elasticity were considered as 0.283 lb/in.3
(7,833.4 kg/m3
) and
29,000 ksi (200 GPa), respectively. The members of this truss were divided into 32 groups of
design variables. A single load case was considered consisting of lateral loads of 5.0 KN (1.12
kips) applied in both x and y directions and a vertical load of 30 KN (6.74 kips) applied in the z-
direction to all nodes of the tower. According to Table 5, a set of 140 standard steel sections are
considered for this problem. The maximum nodal displacement is limited to ±8.0 cm (3.15 in.)
in every direction, and stress limitations are defined as provisions of ASD-AISC. In this
example, the sparks generation factor (𝑀𝑒) is set to 200.
Fig. 7. Configuration of the 582-bar tower truss.

Table 5
Profile list from AISC code for the 582-bar tower truss.
No. section cm2 No. section cm2 No. section cm2
1 W8X21 39.74 48 W12X58 109.68 95 W21X111 210.97
2 W12X22 41.81 49 W8X58 110.32 96 W10X112 212.26
3 W14X22 41.87 50 W18X60 113.55 97 W27X114 216.13
4 W10X22 41.87 51 W10X60 113.55 98 W24X117 221.94
5 W8X24 45.68 52 W14X61 115.48 99 W18X119 226.45
6 W10X26 49.10 53 W24X62 118.06 100 W14X120 227.74
7 W12X26 49.35 54 W21X62 118.06 101 W12X120 227.74
8 W16X26 49.55 55 W18X65 123.23 102 W21X122 231.61
9 W14X26 49.61 56 W12X65 123.23 103 W24X131 248.39
10 W8X28 53.16 57 W8X67 127.10 104 W21X132 250.32
11 W12X30 56.71 58 W21X68 129.03 105 W14X132 250.32
12 W10X30 57.03 59 W16X67 129.03 106 W12X136 257.42
13 W14X30 57.10 60 W14X68 129.03 107 W14X145 275.48
14 W8X31 58.84 61 W10X68 129.03 108 W24X146 277.42
15 W16X31 58.90 62 W24X68 129.68 109 W27X146 278.06
16 W10X33 62.65 63 W18X71 134.19 110 W21X147 278.71
17 W14X34 64.52 64 W12X72 136.13 111 W12X152 288.39
18 W18X35 66.45 65 W21X73 138.71 112 W14X159 301.29
19 W12X35 66.45 66 W14X74 140.64 113 W27X161 307.10
20 W8X35 66.45 67 W18X76 143.87 114 W24X162 307.74
21 W16X36 68.39 68 W24X76 144.52 115 W12X170 322.58
22 W14X38 72.26 69 W10X77 145.81 116 W14X176 334.19
23 W10X39 74.19 70 W16X77 147.74 117 W27X178 338.71
24 W12X40 75.48 71 W12X79 149.68 118 W12X190 360.00
25 W8X40 75.48 72 W14X82 154.84 119 W14X193 366.45
26 W18X40 76.13 73 W21X83 156.77 120 W12X210 398.71
27 W16X40 76.13 74 W24X84 159.35 121 W14X211 400.00
28 W14X43 81.29 75 W27X84 160.00 122 W12X230 436.77
29 W21X44 83.87 76 W18X86 163.23 123 W14X233 441.93
30 W12X45 84.52 77 W12X87 165.16 124 W12X252 477.42
31 W16X45 85.81 78 W10X88 167.10 125 W14X257 487.74
32 W10X45 85.81 79 W16X89 170.32 126 W12X279 528.39
33 W18X46 87.10 80 W14X90 170.97 127 W14X283 537.42
34 W14X48 90.97 81 W21X93 176.13 128 W12X305 578.06
35 W8X48 90.97 82 W27X94 178.71 129 W14X311 589.68
36 W10X49 92.90 83 W24X94 178.71 130 W12X336 637.42
37 W12X50 94.19 84 W12X96 181.94 131 W14X342 651.61
38 W21X50 94.84 85 W18X97 183.87 132 W14X370 703.22
39 W18X50 94.84 86 W14X99 187.74 133 W14X398 754.84
40 W16X50 94.84 87 W10X100 189.68 134 W14X426 806.45
41 W14X53 100.64 88 W16X100 191.61 135 W14X455 864.51
42 W12X53 100.64 89 W21X101 192.26 136 W14X500 948.39
43 W10X54 101.94 90 W27X102 193.55 137 W14X550 1045.16
44 W18X55 104.52 91 W24X104 197.42 138 W14X605 1148.38
45 W24X55 105.16 92 W18X106 200.64 139 W14X665 1264.51
46 W21X57 107.74 93 W12X106 201.29 140 W14X730 1387.09
47 W16X57 108.39 94 W14X109 206.45

Table 6
Optimization results of the 582-bar truss.
variables Optimal W-shaped sections
No STA [25] ABC [15] CBO [26] FWA MoFWA
1 W8x21 W8x21 W8x21 W8x24 W8x21
2 W10x68 W18x86 W12x79 W24x84 W24x84
3 W8x21 W8x24 W8x28 W8x28 W8x21
4 W10x77 W10x60 W10x60 W18x71 W18x71
5 W8x21 W8x24 W8x24 W8x24 W8x21
6 W8x21 W8x21 W8x21 W8x24 W8x21
7 W10x60 W10x49 W10x68 W10x49 W10x60
8 W8x21 W8x24 W8x24 W8x24 W8x21
9 W8x21 W8x21 W8x21 W8x24 W8x21
10 W14x48 W12x53 W14x48 W14x48 W14x48
11 W8x21 W8x24 W12x26 W12x30 W8x21
12 W14x74 W21x62 W21x62 W24x76 W24x76
13 W16x67 W27x84 W18x76 W27x84 W18x76
14 W12x65 W10x45 W12x53 W10x49 W14x48
15 W12x65 W27x84 W14x61 W27x84 W10x100
16 W8x21 W8x31 W8x40 W8x31 W8x21
17 W12x65 W8x21 W10x54 W14x61 W14x34
18 W8x21 W12x53 W12x26 W12x53 W12x40
19 W8x21 W8x24 W8x21 W8x21 W8x21
20 W10x68 W10x22 W14x43 W12x40 W14x34
21 W8x21 W16×36 W8x24 W16x36 W14x38
22 W8x21 W8×24 W8x21 W8x24 W8x21
23 W10x22 W8x21 W10x22 W8x31 W14x22
24 W8x21 W10×22 W8x24 W8x28 W12x26
25 W12x40 W6×25 W8x21 W8x21 W8x21
26 W6x25 W8x21 W8x21 W8x21 W8x21
27 W10x22 W8x21 W8x24 W8x24 W8x21
28 W8x21 W8x24 W8x21 W8x28 W8x21
29 W8x28 W8x21 W8x21 W16x36 W12x22
30 W10x22 W10x22 W6x25 W8x24 W8x21
31 W16x36 W8x24 W10x33 W8x24 W8x21
32 W12x53 W6x25 W8x28 W8x24 W8x21
Weight (lb) 364,143.43 368,484.1 376,344.74 401,895.6 360,456.22
Evaluation
(#)
6000 50000 6400 58,000 41,000
1 lb = 0.45359 Kg

Fig. 8. convergence history of 582-bar truss.
Table 6 illustrates the optimum W-shaped sections and minimum weights that are derived from
MoFWA and other algorithms. MoFWA’s design weighs 360,456.22 lbs which is the best among
all of the algorithm’s results. Fig. 8 demonstrates the convergence curve of MoFWA algorithm
for the 582-bar truss.
4.4. 2-bay, three-story frame
In Fig. 9, the topology and loading conditions of a 2-bay three-story frame are depicted. This
frame consists of 15 members, which can be categorized into two groups. We have chosen 267
standard W-shape profiles for the beams and limited the columns to W10 sections. Table 7 lists
all of the available profiles. This frame is optimized according to the AISC-LRFD specifications.
Also, displacement constraints are not considered. In this example, the modulus of elasticity of
steel is taken as 𝐸 = 200 𝐺 𝑃 𝑎 (29,000 𝑘 𝑠 𝑖 ), the yield stress is considered as 𝐹 𝑦 = 248.2
𝑀 𝑃 𝑎 (36 𝑘 𝑠 𝑖 ) and the material unit weight of 𝛾 = 7861 kg/m3
(0.284 lb/in.3
) is used. Each
column is considered unbraced along its length, and the unbraced length of each beam member is
specified as one-sixth of span length. In this example, the sparks generation factor (𝑀𝑒) is set to
50.

Table 7
Profile list from ASIC code (2009).
W-shape profile list
W36X359
W27X258
W12X190
W21X132
W16X89
W21X57
W8X31
W6X8.5
W30X357
W36X260
W30X191
W14X132
W14X90
W16X57
W16X31
W6X9
W40X362
W30X261
W24X192
W30X132
W21X93
W12X58
W10X33
W8X10
W27X368
W44X262
W14X193
W36X135
W27X94
W8X58
W14X34
W10X12
W40X372
W33X263
W36X194
W12X136
W24X94
W18X60
W18X35
W6X12
W24X370
W40X264
W27X194
W33X141
W12X96
W10X60
W12X35
W4X13
W14X370
W40X277
W40X199
W18X143
W18X97
W14X61
W8X35
W8X13
W33X387
W40X278
W33X201
W14X145
W30X99
W24X62
W16X36
W12X14
W40X392
W12X279
W21X201
W24X146
W14X99
W21X62
W14X38
W10X15
W30X391
W24X279
W24X207
W27X146
W10X100
W18X65
W10X39
W8X15
W36X393
W36X280
W36X210
W21X147
W16X100
W12X65
W12X40
W6X15
W40X397
W27X281
W12X210
W30X148
W21X101
W8X67
W8X40
W12X16
W14X398
W14X283
W40X211
W40X149
W27X102
W16X67
W18X40
W5X16
W14X426
W33X291
W14X211
W36X150
W24X103
W21X68
W16X40
W6X16
W40X431
W44X290
W30X211
W12X152
W24X104
W14X68
W14X43
W10X17
W36X439
W30X292
W40X215
W33X152
W18X106
W10X68
W21X44
W8X18
W14X455
W40X297
W27X217
W18X158
W12X106
W24X68
W12X45
W5X19
W14X500
W36X300
W33X221
W14X159
W30X108
W18X71
W16X45
W12X19
W40X503
W12X305
W24X229
W36X160
W14X109
W12X72
W10X45
W10X19
W36X527
W24X306
W36X230
W27X161
W21X111
W21X73
W18X46
W6X20
W27X539
W27X307
W44X230
W24X162
W10X112
W14X74
W21X48
W8X21
W14X550
W14X311
W12X230
W21X166
W27X114
W18X76
W14X48
W12X22
W40X593
W33X318
W36X232
W40X167
W30X116
W24X76
W8X48
W14X22
W14X605
W40X324
W14X233
W33X169
W24X117
W10X77
W10X49
W10X22
W36X650
W30X326
W40X235
W12X170
W33X118
W16X77
W12X50
W8X24
W14X665
W40X327
W30X235
W36X170
W18X119
W12X79
W21X50
W6X25
W14X730
W36X328
W27X235
W30X173
W14X120
W14X82
W18X50
W10X26
W36X798
W40X331
W33X241
W18X175
W12X120
W21X83
W16X50
W12X26
W14X808
W44X335
W36X245
W24X176
W21X122
W24X84
W14X53
W16X26
W24X335
W40X249
W14X176
W30X124
W27X84
W12X53
W14X26
W12X336
W24X250
W27X178
W27X129
W18X86
W10X54
W8X28
W27X336
W12X252
W36X182
W18X130
W12X87
W21X55
W12X30
W14X342
W36X256
W21X182
W33X130
W10X88
W18X55
W10X30
W33X354
W14X257
W40X183
W24X131
W30X90
W24X55
W14X30
Fig. 9. Fig Configuration of the two-bay, three-story frame.

Table 8
Optimization results of the two-bay, three-story frame.
Variables Optimal W-shaped sections
No Des.Var. MHBMO [27] IHS [28] DDHS [13] FWA MoFWA
1 beams W24x62 W24x62 W24x62 W14x61 W14x61
2 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 W10x49 W10x60 W10x60 W10x49 W10x49
Weight (lb) 17,789 18,792 18,792 17,549.55 17,549.55
Evaluation (#) 650 N/A 270 480 270
Fig. 10. Convergence history of two-bay, three-story frame.
Table 8 lists the best designs developed by MoFWA, and other algorithms. MoFWA designed a
frame weighing 17,549.55 lbs which is lighter than the designs of other algorithms. Fig. 10
shows the convergence curve of MoFWA reaching the optimal weight in the 6th
iteration.
4.5. one-bay, ten-story frame
Fig. 11 demonstrates the topology and loading condition of the one-bay ten-story frame. This
structure has 30 members that are grouped into 9 groups.‌The beams are chosen from Table7
while column selection is limited to W12 and W14 sections. The frame is designed following the
AISC-LRFD specifications with a maximum inter-story drift constraint of story-height/300. In
addition, Young’s modulus is considered as E=29,000 ksi, and yield stress is considered as fy =
36 ksi. In this example, the sparks generation factor (𝑀𝑒) is set to 50.

Fig. 11. topology of the 0ne-bay, ten-story frame.
Table 9
Optimization results of one-bay, ten-story frame.
No Des.Var. SBO [29] MHBMO [27] CS [11] FWA MoFWA
1 𝐴1~𝐴4 W14x233 W14x233 W14x233 W14x233 W14x233
2 𝐴5~𝐴8 W14x176 W14x176 W14x176 W14x176 W14x176
3 𝐴9~𝐴12 W14x145 W14x145 W14x132 W14x145 W14x132
4 𝐴13~𝐴16 W14x99 W14x99 W14x109 W14x99 W14x109
5 𝐴17~𝐴20 W14x61 W14x61 W14x61 W12x65 W12x61
6 𝐴21~𝐴23 W30x108 W30x108 W33x118 W33x118 W33x118
7 𝐴24~𝐴26 W30x90 W30x90 W30x108 W30x99 W30x108
8 𝐴27~𝐴29 W27x84 W27x84 W24x55 W27x84 W24x55
9 𝐴30 W18x46 W21x44 W18x40 W14x53 W18x40
Best (lb) 62,430 61,617 62,074 64,408.2 62,074
Evaluation (#) 11,677 3600 4438 3200 1600

Fig. 12. Convergence history of one-bay, ten-story frame.
Table 9 shows the results of different algorithms optimizing the one-bay ten-story frame. The
convergence history can be seen in Fig. 12. MoFWA obtained the optimal weight of 62,074 lbs
which is second-best among other algorithms. However, MoFWA has fewer evaluations than
other algorithms.
Table 10
Optimization results of three-bay, 24-story frame.
No TLBO [30] MHBMO [27] SBO [29]
EWOA
[31]
FWA MoFWA
1 W30x90 W30×90 W30x90 W30x90 W30x90 W30x90
2 W8x18 W8×18 W8x18 W10x30 W6x20 W5x16
3 W24x62 W24×55 W21x48 W24x55 W21x44 W21x44
4 W6x9 W14×22 W6x8.5 W6x8.5 W6x9 W6x8.5
5 W14x132 W14×145 W14x152 W14x159 W14x159 W14x159
6 W14x120 W14×120 W14x120 W14x99 W14x145 W14x145
7 W14x99 W14×99 W14x109 W14x120 W14x132 W14x132
8 W14x82 W14×82 W14x74 W14x74 W14x99 W14x99
9 W14x74 W14×68 W14x82 W14x74 W14x68 W14x68
10 W14x53 W14×53 W14x43 W14x43 W14x61 W14x53
11 W14x34 W14×30 W14x34 W14x30 W14x43 W14x34
12 W14x22 W14×22 W12x19 W14x22 W14x22 W14x22
13 W14x109 W14×120 W14x109 W14x90 W14x109 W14x109
14 W14x99 W14×109 W14x109 W14x120 W14x109 W14x99
15 W14x99 W14×99 W14x99 W14x90 W14x90 W14x90
16 W14x90 W14×90 W14x99 W14x99 W14x82 W14x82
17 W14x68 W14×68 W14x68 W14x68 W14x74 W14x68
18 W14x53 W14×53 W14x61 W14x61 W14x43 W14x48
19 W14x34 W14×30 W14x34 W14x43 W14x30 W14x30
20 W14x22 W14×22 W14x22 W14x22 W14x26 W14x22
Weight (KN) 903.54 901.895 900.42 905.17 912.26 900.24
Evaluation (#) 12,000 6000 14,572 18,820 37,500 25,000
1 KN = 224.81 lb

4.6. three-bay, twenty-four-story frame
The last design example is a three-bay twenty-four-story frame which is shown in Fig. 13
consisting of 168 members that are divided into 20 group. The yield stress and modulus of
elasticity are considered as 230.3 MPa and 205 GPa, respectively. All beams are chosen from
267 W-shape sections from Table 7 and all of the columns are restricted to W14 sections. This
frame is optimized according to the AISC-LRFD specifications with inter-story drift constraints
of H/300. In this example, the sparks generation factor (𝑀𝑒) is set to 250.
Fig. 13. Configuration of the three-bay, 24-story frame.

Fig. 14. Convergence history of three-bay, 24-story frame.
Table 10 indicates that the MoFWA algorithm achieved the best weight of 900.24 KN for 3-bay,
24-story frame. However, it took more evaluations for MoFWA to find the best result. Fig. 14
illustrates the convergence history of MoFWA.
5. Conclusions
In this article, a modified version of the fireworks algorithm called MoFWA was introduced.
FWA had some deficiencies causing the algorithm to have higher computational cost. To tackle
this issue, some strategies were implemented into FWA to enhance its performance in finding the
global optimum and reducing its computational cost. The modifications made on FWA are: 1)
adding the fly-back mechanism 2) introducing the duplicate sparks removal operator, and 3)
Using a dynamic amplitude and setting a new formula for the calculation of the initial amplitude.
The dynamic amplitude, the initial CF amplitude, the reduction coefficient, the amplification
coefficient, the sparks generation factor, the duplicate sparks removal operator, and the number
of fireworks are the main controlling parameters of MoFWA and the accuracy of designs along
with computational efficiency of the algorithm depend on them.
In order to compare MoFWA with the original FWA and other metaheuristics, they were tested
on six steel structures. As mentioned before, the objective function of these problems was to
minimize the total weight of the structures and constraints were beam/column design strength
and maximum allowable inter-story drift specified in AISC-LRFD.

In five out of six examples, MoFWA performed brilliantly and attained lighter structures with
less computational cost than other algorithms, especially FWA. The results show the vast
applicability of the MoFWA algorithm. MoFWA produced comparably accurate and robustness
results, with low computational cost for both large and small structures. In addition, MoFWA
outperformed FWA in all of the examples, achieving better results with lower computational
cost. However, Overcoming the difficulty of getting stuck in a local minimum, like many other
algorithms, remains a challenge for MoFWA. It may be beneficial to introduce an appropriate
mutation operator in future studies.
To sum up, the changes made to MoFWA have proven to be highly beneficial, surpassing the
performance of the initial FWA, particularly in terms of function evaluations. Furthermore, the
inclusion of the fly-back mechanism has enhanced its ability to exploit marginal regions
effectively.
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