Flower Pollination Algorithm on optimal design of space trusses

Shaymaa Alsamia; Edina Koch; Majid Movahedi Rad

doi:10.1556/1848.2025.00958

Jump to Content

International Review of Applied Sciences and Engineering

Volume/Issue:: Accepted Manuscript / Online First

Flower Pollination Algorithm on optimal design of space trusses

Authors:

Shaymaa Alsamia

Shaymaa Alsamia Department of Structural and Geotechnical Engineering, Széchenyi István University, Hungary
Faculty of Engineering, University of Kufa, Iraq

Search for other papers by Shaymaa Alsamia in
Current site
Google Scholar
PubMed

shaymaam.alsameea@uokufa.edu.iq https://orcid.org/0000-0002-7012-6499

Edina Koch

Edina Koch Department of Structural and Geotechnical Engineering, Széchenyi István University, Hungary

Search for other papers by Edina Koch in
Current site
Google Scholar
PubMed

, and

Majid Movahedi Rad

Majid Movahedi Rad Department of Structural and Geotechnical Engineering, Széchenyi István University, Hungary

Search for other papers by Majid Movahedi Rad in
Current site
Google Scholar
PubMed

View More View Less

Online Publication Date:: 27 Mar 2025

Publication Date:: 27 Mar 2025

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2025.00958

Keywords:: Flower Pollination Algorithm; Particle Swarm Optimization; Dynamic Differential Annealed Optimization; Firefly Algorithm; truss optimization; structural design; nature-inspired algorithms

Open access

Download PDF

Abstract

This study assesses the performance of four nature-inspired optimization algorithms—Dynamic Differential Annealed Optimization (DDAO), Flower Pollination Algorithm (FPA), Firefly Algorithm (FF), and Particle Swarm Optimization (PSO) for achieving optimal space truss design. The aim is to minimize the structural weight of three benchmark trusses (10-bar, 25-bar, and 72-bar) while meeting stress and displacement constraints. The key contribution of this work is the first systematic evaluation of FPA in space truss optimization, demonstrating its greater effectiveness in obtaining optimal or near-optimal solutions with faster convergence and higher stability compared to PSO and FF. The results also highlight the limitations of DDAO in handling constrained engineering problems. Findings confirm that FPA and FF are highly effective for structural optimization, offering robust solutions with minimal computational cost. These insights contribute to advancing metaheuristic-based structural design, supporting the adoption of FPA in large-scale optimization problems.

Abstract

1 Introduction

Engineering and computer science often grapple with highly complex optimization problems [1–3], many of which are nonlinear and constrained by intricate conditions [4, 5]. Identifying optimal solutions in these situations might be like locating a needle in a haystack, especially when conventional optimization methods fail to address multimodal or high-order nonlinear problems1 [6, 7]. Many techniques are used to handle nonlinear problems, including machine learning [8, 9], Artificial intelligence [10–12], and metaheuristics [13, 14] as well as the classical numerical methods [15, 16]. In the 1970s, researchers began utilizing nature-inspired algorithms, known as metaheuristics, to address these challenges effectively [17, 18]. Metaheuristic algorithms rely on problem descriptions characterized by objective functions and variable domains. They are adept at exploring search spaces to locate global optima, surpassing local minima and maxima that typically confound traditional optimization techniques [19–22]. Optimization can be categorized as continuous, combinatorial, or multiobjective based on the variables' characteristics. Various metaheuristic algorithms have been created and enhanced over time [23]. Kennedy and Eberhart introduced the Particle Swarm Optimization (PSO) algorithm [24, 25] in 1995 to mimic fish and avian cooperative foraging behavior [26]. Particles inside a swarm traverse the search space, affected by their optimal placements and those of their neighbors, directed by mathematical models that adjust their velocities and positions. Artificial bee colony ABC [27] and its variant adaptive exploration artificial bee colony AEABC are another examples of swarm intelligence [28]. Grey wolf optimizer [29] is another swarm intelligence technique which is used in wide range of applications [30–32]. Optimization algorithms are usually a state of art that are inspired by natural or human-made phenomena An example of such an approach is the Whale optimization algorithm [33], pelican optimization algorithm [34], imperialist competitive algorithm [35], and genetic algorithm [36]. The Firefly Algorithm (FF) is another significant algorithm [37], and flower fertilization optimization [38]. Inspired by the bioluminescent communication of fireflies, the FF uses light intensity as a metaphor for solution quality. Less bright fireflies are attracted to brighter ones, facilitating a search process that efficiently navigates multimodal landscapes. Recent advancements also include the dynamic differential annealed optimization algorithm [39], a novel approach that combines random search with principles from simulated annealing. DDAO has demonstrated high performance across a variety of test functions and practical engineering problems, outperforming several well-established algorithms in certain cases. Amidst these developments, the Flower Pollination Algorithm (FPA) [40, 41] appears as a viable metaheuristic approach for addressing complicated optimization challenges. Drawing inspiration from the pollination processes of flowering plants, FPA simulates pollen transmission through both biotic and abiotic means to effectively explore and optimize the search space. The program equilibrates global exploration and local exploitation using a probabilistic mechanism that simulates the shift between global pollination (long-distance transfer) and local pollination (neighborhood-based transfer). This paper examines the implementation of the Flower Pollination Algorithm in optimizing space truss design, concentrating on the 10-bar, 25-bar, and 72-bar truss configurations. These benchmark issues are renowned in structural optimization for their complexity and the existence of several local optima. By applying FPA to these examples, we aim to demonstrate its efficacy in navigating complex design spaces and achieving optimal or near-optimal solutions. Several studies have explored metaheuristic algorithms for truss optimization, with PSO, FF, and other swarm intelligence techniques commonly applied to structural design problems. However, existing research has mainly focused on individual algorithms rather than direct comparative evaluations across multiple approaches on standardized truss problems. Additionally, while FPA has shown promising results in other engineering domains, its potential for space truss optimization has not been systematically evaluated. Furthermore, DDAO, a relatively recent optimization algorithm, lacks performance assessments in constrained structural applications. This study fills these gaps by conducting a comprehensive performance comparison of FPA, PSO, FF, and DDAO on well-known benchmark truss problems, offering new insights into their efficiency, convergence behavior, and robustness.

This study makes the following key contributions to structural optimization and metaheuristic research:

First systematic evaluation of the FPA for space truss optimization – The study demonstrates the effectiveness of FPA in achieving optimal or near-optimal solutions with superior convergence speed and stability.
Comprehensive comparative analysis of four metaheuristic algorithms – The performances of FPA, PSO, DDAO, and FF are rigorously assessed on three benchmark trusses (10-bar, 25-bar, and 72-bar), providing valuable insights into their strengths and weaknesses.
Identification of DDAO's limitations in constrained structural problems – The results reveal that DDAO struggles with convergence and solution quality in truss optimization, highlighting its limitations for real-world structural applications.
Demonstration of FPA and FF as robust alternatives for structural design – The study confirms that FPA and FF consistently outperform PSO and DDAO regarding final weight reduction and computational efficiency, making them promising candidates for engineering applications.
Practical insights for applying metaheuristics in structural engineering – The findings support the broader adoption of FPA and FF for large-scale structural design optimization, encouraging further research and hybrid approaches.

2 Flower Pollination Algorithm

The Flower Pollination Algorithm (FPA) is a bio-inspired optimization technique designed to replicate the natural pollination process. Its search mechanism is based on pollination dynamics and can be mathematically formulated as follows:

Pollen may be considered a solution to the optimization issue, and the pollination process is comparable to discovering a proposed solution.
Levy flying behavior is used in cross and biotic pollination, which looks for a global answer.
Both abiotic and self-pollination correspond to a local search.
Reproduction likelihood, which depends on how similar a group of flowers (solution) are to one another, is represented by flower constancy.
The switch probability $p \in [0, 1]$ controls the transition between local and global pollination processes. The formulation of the global solution, combined with flower constancy, is represented as follows

X_{i}^{t + 1} = X_{i}^{t} + L (X_{i}^{t} - g *)

(1)

L is the pollination process's step size, g* is the optimal solution, and X is the index i solution during iteration t. The step size, which has the following definition, is represented by Levy's fight:

σ = {(\frac{Γ (1 + λ) \sin (\frac{π λ}{2})}{Γ (\frac{1 + λ}{2}) λ * 2^{\frac{λ - 1}{2}}})}^{\frac{1}{λ}}

(2)

u = r σ

(3)

v = r

(4)

s t e p = \frac{u}{{| v |}^{\frac{1}{λ}}}

(5)

L = 0.01 s t e p

(6)

Where r is a random integer, u and v are parameters,

λ

is a constant assumed to be 1.5, and

Γ

is the standard gamma function.

The representation of the local solution is

X_{i}^{t + 1} = X_{i}^{t} + ε (X_{j}^{t} - X_{k}^{t})

(7)

$S_{j}^{t} and S_{k}^{t}$ are several solutions (pollens) selected at random. The following pseudocode can be useful to implement FPA:

Where

initialize population (n, d): Initializes a list of n flowers with a random solution in the search space.
find_best_solution(population): Returns the best solution from the population based on the objective function f(x).
levy_flight(d): Generates a step vector L following a Lévy distribution for d dimensions.
f(x): The function that defines the objective as either minimized or maximized.
Random. Uniform (a, b): Generates a random float number between a and b.
random. Sample (population, k): Select k distinct elements randomly from the specified population and return them as a list.

3 Truss design problem

The main goal of this problem is to minimize the total weight while satisfying all stress and displacement constraints. This study examines trusses with a fixed geometry, where the design variables correspond to the cross-sectional areas of individual members. The optimization problem is formulated as follows [42]:

M i n i m i z e w = \sum_{e = 1}^{N} C_{e} L_{e} A_{e}

(8)

Subject to:

σ_{L} \leq σ_{e} \leq σ_{U}, δ_{L} \leq δ_{C} \leq δ_{U}, A_{L} \leq A_{e} \leq A_{U}

where: w is the overall weight of the truss, comprising (e = 1,2…,N) members.

C_e_ is the material density for member e.
σ_e the stress in member e.
L_e is the length of member e.
δ_c is the deflection at connection c.
A_e is the cross-sectional area of member e.
A_L and A_U are the lower and upper limits for the cross-sectional area, respectively.
σ_L and σ_U are the lower and upper limits of allowable stress.
δ_L and δ_U are the lower and upper limits of allowable deflection.

A penalty function addresses the infeasibility if the design fails to meet these constraints. The penalty for stress violation in each member is defined as follows:

\emptyset_{e}^{σ} = {\begin{array}{c} 0 & if σ_{L} \leq σ_{e} \leq σ_{U} \\ | \frac{σ_{e} - σ_{L, U}}{σ_{L, U}} | & if σ_{e} < σ_{L} o r σ_{e} > σ_{U} \end{array}

(9)

The penalty on the entire truss design k is:

\emptyset_{k}^{σ} = \sum_{e = 1}^{N_{m}} \emptyset_{e}^{σ}

(10)

Deflection penalties in the x, y, and z directions are determined as follows:

\emptyset_{c}^{δ (x, y, z)} = {\begin{array}{c} 0, & if δ_{c (x, y, z)} < δ_{L} o r δ_{c (x, y, z)} > δ_{U} \\ | \frac{δ_{c (x, y, z)} - δ_{L, U}}{δ_{L, U}} |, & if δ_{c (x, y, z)} \leq δ_{c (x, y, z)} \leq δ_{U} \end{array}

(11)

The total deflection penalty for design can be written as:

\emptyset_{k}^{δ} = \sum_{c = 1}^{N_{c}} (\emptyset_{c}^{δ x} + \emptyset_{c}^{δ y} + \emptyset_{c}^{δ z})

(12)

The overall penalty for design k is the penalty summation of the stress and deflection:

w_{k} = (1 + \emptyset_{k}^{σ} + \emptyset_{k}^{δ}) e

(13)

where the penalty exponent, e, is positive.

Next, the product of the truss weight and its total penalty is used to determine the penalized fitness function for each truss design k:

F_{K} = {w w}_{k}

(14)

This formulation allows for the optimal design of truss structures by iteratively minimizing the penalized fitness function, ensuring that the designs comply with all strength and displacement requirements.

4 Design examples

Three benchmark truss design issues are considered to evaluate the efficacy of the various optimization techniques: a 25-bar space truss with eight design variables, a 10-bar cantilever truss with 10 design variables, and a 72-bar multi-story truss with 16 design variables. Each example combines continuous and discrete variable designs to highlight the adaptability and efficacy of the algorithms.

4.1 Twenty-five bar space truss

The twenty-five bar space truss represents a transmission tower commonly utilized in structural optimization studies to assess several design processes and computational techniques, and it serves as the initial test case. Figure 1 displays this truss's configuration and node numbering. Each of the eight groups into which the structural members are divided shares the same cross-sectional characteristics and material.

Fig. 1.

Configuration of the 25-bar truss

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

The node coordinates are shown in Table 1. The truss material has a modulus of elasticity of 107 psi and a unit weight of 0.1 lb/in³.

Table 1.

Node definition of the 25-bar problem

Node	x (in.)	y (in.)	z (in.)
1	−37.5	0.0	200
2	37.5	0.0	200
3	−37.5	37.5	100
4	37.5	37.5	100
5	37.5	−37.5	100
6	−37.5	−37.5	100
7	−100	100	0.0
8	100	100	0.0
9	100	−100	0.0
10	−100	−100	0.0

The loading conditions on the truss are indicated in Table 2. Maximum permitted displacements on nodes in x, y, and z directions are ±0.35 inches, and Table 3 lists the stress limitations for each member group. Cross-sectional areas range from 0.01 to 3.40 in², and the design variables are handled as continuous.

Table 2.

The 25-bar truss under various loading conditions

Node	P_x (kips)	P_y (kips)	P_z (kips)
1	100	10.0	−5.0
2	0.0	10.0	−5.0
3	5.00E-01	0.0	0.0
6	5.00E-01	0.0	0.0

Table 3.

Allowable stresses on 25-bar design

Element group	Members	Compression (ksi)	Tension (ksi)
1	1	3.51E+01	3.50E+01
2	2,3,4,5	1.16E+01	3.50E+01
3	6,7,8,9	1.73E+01	3.50E+01
4	10,11	3.51E+01	3.50E+01
5	12,13	3.51E+01	3.50E+01
6	14,15,16,17	6.76E+00	3.50E+01
7	18,19,20,21	6.96E+00	3.50E+01
8	22,23,24,25	1.11E+01	3.50E+01

Table 4 shows the FPA generated a truss design weighing 559.405 lb. This is slightly better than the best feasible solutions provided by FF and PSO. The FPA also demonstrated a reduction in the standard deviation of analyses required to converge compared to other methods. Figure 6 shows a typical algorithmic convergence for this case.

Table 4.

Continuous variable design for a 25-bar truss

Variables	Algorithm
Variables	PSO	DDAO	FF	FPA
Weight (lb)	559.953	702.041	559.857	559.405
W_avg(lb)	579.148	959.095	560.314	559.411
W_Stdv(lb)	49.358	118.610	0.241	0.00299

Figure 2 reveals the convergences curves of the competitive optimization algorithms on 25 bar space truss optimization problem.

Fig. 2.

Convergence curves on 25 bar design

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

4.2 10-Bar truss design example

One common benchmark issue in structural optimization is the 10-bar cantilever truss. Figure 3 shows the 10-bar space truss design problem.

Fig. 3.

Configuration of ten-bar truss

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

The cross-sectional areas are permitted to change between 0.1 and 35.0 in² in the continuous variable case. As in the discrete situation, the material characteristics are unchanged. In comparison to alternative technologies, the FPA created a design that weighed 5060.854 lb. Table 5 shows specific data, and Fig. 4 illustrates the convergence curves.

Table 5.

Continuous variable design for a ten-bar problem

Variables		Cross-sectional areas (in.²)
Element Group	Members	PSO	DDAO	FF	FPA
1	1	32.558	22.661	30.447	30.522
2	2	0.1	0.1	0.1	0.1
3	3	23.243	24.193	23.22	23.199
4	4	14.24	16.3168	14.985	15.216
5	5	0.1	0.1	0.1	0.1
6	6	0.454	0.1	0.53	0.552
7	7	21.25	22.595	21.145	21.037
8	8	7.57	15.99	7.475	7.457
9	9	0.1	0.1	0.1	0.1
10	10	20.668	21.776	21.632	21.53
Weight (lb)		5069.803	5750.075	5061.452	5060.854
W_avg(lb)		5103.984	6208.882	5062.582	5060.856
W_Stdv(lb)		40.403	191.542	0.968	0.0011

Fig. 4.

Convergence curves on 10 bar space design problem

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

4.3 72-Bar space truss

Numerous optimization techniques have been used in the detailed study of the 72-bar space truss. Figure 5 shows the 72-bar truss arrangement, including nodes and connections.

Fig. 5.

72-bar truss geometry and element definitions: (a) dimensions and node numbering system; (b) first-story element numbering pattern

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

As shown in Table 6, the truss is subjected to two separate load scenarios. The material's modulus of elasticity is 107 psi, and its unit weight is 0.1 lb/in³. The highest joints can have a maximum displacement of 0.25 inches in all directions and a maximum stress of 25 ksi. The upper and lower limits of the cross-sectional areas is 0.1–3.0 in².

Table 6.

Various loading conditions for the 72-bar truss

Node	P_x (kips)	P_y (kips)	P_z (kips)
17	0.0	0.0	−5.0
18	0.0	0.0	−5.0
19	0.0	0.0	−5.0
20	0.0	0.0	−5.0

The FPA achieved a minimum weight of 382.102 lb, slightly better than other published results. A summary of the results is provided in Table 7 and the convergence curves are shown in Fig. 6.

Table 7.

Seventy-two-bar truss designs under various load conditions

Variables		Cross-sectional areas (in.²)
Element Group	Members	PSO	DDAO	FF	FPA
1	1–4	1.170	2.280	1.747	1.898
2	5–12	0.704	0.489	0.552	0.479
3	13–16	0.1	2.21	0.1	0.1
4	17,18	0.1	0.504	0.1	0.1
5	19–22	0.558	1.839	1.216	1.357
6	23–30	0.611	0.502	0.516	0.514
7	31–34	0.167	0.781	0.1	0.101
8	35,36	0.1	0.1	0.1	0.1
9	37–40	0.156	2.051	0.525	0.575
10	41–48	0.474	0.38	0.466	0.533
11	49–52	0.369	0.743	0.102	0.115
12	53,54	0.360	0.1	0.1	0.103
13	55–58	1.170	0.474	0.156	0.157
14	59–66	0.704	0.506	0.578	0.565
15	67–70	0.1	0.356	0.409	0.323
16	71,72	0.102	0.648	0.685	0.618
Weight (lb)		397.186	680.418	382.044	382.102
W_avg(lb)		565.385	1177.548	384.459	385.393
W_Stdv(lb)		245.204	296.037	0.934	2.096

Fig. 6.

Convergence curves on 25 bar space truss

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00958

5 Results and discussion

Table 5 illustrates the findings from the four algorithms (FPA, PSO, DDAO, and FF) for the 10-bar truss issue in terms of cross-sectional areas and matching weights. Figure shows the convergence curves for this problem. The algorithm that produced the lowest end weight, Flower Pollination Algorithm (FPA), weighed 5060.854 lb, closely followed by Firefly Algorithm (FF) with 5061.452 lb. PSO also performed well, yielding a weight of 5069.803 lb. On the other hand, DDAO performed poorly with a significantly higher weight of 5750.075 lb. The average weight results further confirm the superiority of FPA and FF, with FPA achieving the lowest (5060.856 lb), closely matching its best performance. DDAO's average is much higher (6208.882 lb), indicating higher variability and suboptimal convergence behavior. The standard deviation values highlight the stability of FPA (0.0011) and FF (0.968). In contrast, DDAO has a much larger deviation (191.542), suggesting inconsistent performance across multiple runs. As shown in the convergence curve for the 10-bar truss problem in Fig. 2, FPA rapidly decreases the weight within the first 200 iterations and continues to gradually improve, outperforming the other algorithms. FF and PSO also converge quickly, while DDAO lags behind with a much slower rate of convergence and higher final weight. FPA demonstrates excellent performance in terms of both final weight and consistency (low WStdv). FF also performs comparably but with slightly higher variation. DDAO, while effective in some other problems, struggles in this case, both in convergence speed and final solution quality. PSO performs reasonably well but fails to match the precision of FPA and FF. For the 25-bar truss optimization problem, the results are presented in Table 4. Figure 4 displays the convergence behavior of the algorithms. FPA once again provides the best performance with the lowest final weight (559.405 lb), followed by FF (559.857 lb) and PSO (559.953 lb). DDAO, however, yields a much higher weight (702.041 lb). The average weight shows a similar trend, with FPA (559.411 lb) and FF (560.314 lb) being the most consistent. DDAO's average (959.095 lb) again reflects its suboptimal performance. FPA shows exceptional stability, with the lowest standard deviation of 0.00299, followed by FF (0.241). DDAO shows significant variability with a value of 118.610. The convergence curves for the 25-bar truss problem in Fig. 4 show that FPA converges rapidly, reaching its optimal weight in fewer iterations compared to the other algorithms. FF and PSO show similar convergence behaviors but with slightly higher final weights. DDAO, on the other hand, takes longer to converge and settles at a much higher weight. The FPA continues to outperform the other algorithms in terms of both accuracy and stability. FF and PSO are competitive, but FPA achieves the best balance between fast convergence and low standard deviation. DDAO's performance is notably inferior, with slower convergence and a final solution far from the optimal. For the most complex case, the 72-bar truss problem, the results are given in Table 7. Figure 6 shows the convergence curves for this problem. FF achieves the lowest final weight (382.044 lb), closely followed by FPA (382.102 lb). PSO performs worse, with a final weight of 397.186 lb, while DDAO yields the highest weight (680.418 lb). The average weight values mirror this trend, with FF (384.459 lb) and FPA (385.393 lb) showing the best performance. DDAO's average weight (1177.548 lb) is much higher, reflecting its poor performance. The standard deviation reveals the high consistency of FF (0.934) and FPA (2.096). PSO shows more variability (245.204), and DDAO has the highest variability (296.037). The convergence curves for the 72-bar truss problem in Fig. 6 show that FF and FPA both converge to optimal solutions very quickly, with minimal difference in their final weights. PSO takes longer to converge and settles at a higher weight. DDAO struggles once again, with slow convergence and a much higher final weight. In this most complex problem, FF slightly outperforms FPA in terms of final weight, but both algorithms show strong performance with fast convergence and low variability. PSO is less competitive, with a slower convergence rate and higher final weight. DDAO performs poorly once again, indicating its limitations in handling this type of optimization problem. Across all three truss design problems, the Flower Pollination Algorithm (FPA) consistently shows the best or near-best performance in terms of final weight, convergence speed, and stability (low standard deviation). FF also performs strongly in all cases, often providing comparable results to FPA, especially in the 72-bar truss problem. PSO is generally effective but shows higher variability and slower convergence compared to FPA and FF. DDAO, while useful in some contexts, performs poorly in these truss design problems, with higher weights, slower convergence, and greater variability. The convergence curves clearly illustrate the rapid convergence of FPA and FF, particularly in the early iterations, which is essential in practical applications where computation time is critical. DDAO's poor performance is evident from its slower convergence and suboptimal solutions, indicating that it may not be well-suited for these specific truss design problems.

These results suggest that FPA and FF are the most suitable for structural optimization due to their stability, convergence speed, and solution accuracy. Future research could explore hybridizing FPA with PSO to further optimize the balance between exploration and exploitation, potentially improving performance in larger-scale engineering applications.

6 Conclusion

This study conducted a comparative analysis of four nature-inspired optimization algorithms—Flower Pollination Algorithm (FPA), Particle Swarm Optimization (PSO), Dynamic Differential Annealed Optimization (DDAO), and Firefly Algorithm (FF)—for the optimal design of space trusses. The results demonstrated that FPA consistently achieved the best or near-best solutions in terms of weight minimization, convergence speed, and solution stability. Across the benchmark truss problems, FPA outperformed PSO, FF, and DDAO in final weight minimization. Specifically, for the 25-bar truss, FPA reduced the weight to 559.405 lb, improving upon PSO by 0.09%, FF by 0.08%, and significantly surpassing DDAO, which yielded a weight of 702.041 lb (a 25.5% higher weight). Similarly, in the 10-bar truss problem, FPA achieved a final weight of 5060.854 lb, outperforming PSO (5069.803 lb) and DDAO (5750.075 lb, 13.6% higher). For the 72-bar truss, FF achieved the lowest weight (382.044 lb), closely followed by FPA (382.102 lb), with PSO performing worse (397.186 lb) and DDAO significantly underperforming (680.418 lb, 78% higher weight). Regarding convergence speed, FPA reached near-optimal solutions in significantly fewer iterations compared to PSO and DDAO. For the 10-bar truss, FPA converged within 200 iterations, while PSO required approximately 350 iterations. The standard deviation of FPA's results was also the lowest across all cases, with 0.00299 for the 25-bar truss and 0.0011 for the 10-bar truss, demonstrating high stability and consistency. These findings confirm that FPA and FF are the most effective algorithms for space truss optimization, providing fast convergence, reduced structural weight, and improved reliability over multiple runs. PSO remains competitive but exhibits higher variability, while DDAO proves unsuitable for constrained truss optimization due to its slow convergence and suboptimal weight minimization. Future research could explore the integration of FPA with hybrid optimization techniques or its application to real-world structural engineering problems. Despite the promising results, this study has some limitations. First, the experiments were conducted on benchmark truss structures, which may not fully represent the complexity of real-world structural design problems. Second, parameter tuning was not extensively explored, meaning some algorithms (such as PSO and DDAO) might perform better with problem-specific adjustments. Lastly, while FPA demonstrated strong performance, its computational cost for large-scale truss designs remains an area for further investigation.

References

[1]↑
B. S. Yıldız, N. Pholdee, N. Panagant, S. Bureerat, A. R. Yildiz, and S. M. Sait, “A novel chaotic Henry gas solubility optimization algorithm for solving real-world engineering problems,” Eng. Comput., pp. 1–13, 2021.
- Search Google Scholar
- Export Citation
[2]
H. N. Ghafil and K. Jármai, “Optimum dynamic analysis of a robot arm using flower pollination algorithm,” 2019.
- Search Google Scholar
- Export Citation
[3]
X.-S. Yang, “Nature-inspired optimization algorithms: challenges and open problems,” J. Comput. Sci., vol. 46, 2020, Art no. 101104.
- Search Google Scholar
- Export Citation
[4]↑
F. Meng, X. Shen, and H. R. Karimi, “Emerging methodologies in stability and optimization problems of learning‐based nonlinear model predictive control: a survey,” Int. J. Circuit Theor. Appl., vol. 50, no. 11, pp. 4146–4170, 2022.
- Search Google Scholar
- Export Citation
[5]↑
H. N. Ghafil and K. Jármai, “Optimization algorithms for inverse kinematics of robots with MATLAB source code,” in Vehicle and Automotive Engineering, 2020, pp. 468–477.
- Search Google Scholar
- Export Citation
[6]↑
R. Wang, K. Hao, L. Chen, T. Wang, and C. Jiang, “A novel hybrid particle swarm optimization using adaptive strategy,” Inf. Sci. (Ny), vol. 579, pp. 231–250, 2021.
- Search Google Scholar
- Export Citation
[7]↑
H. Albedran and K. Jármai, “Evolutionary control system of asymmetric quadcopter,” Int. Rev. Appl. Sci. Eng., vol. 14, no. 3, pp. 374–382, 2023.
- Search Google Scholar
- Export Citation
[8]↑
S. R. Vadyala, S. N. Betgeri, J. C. Matthews, and E. Matthews, “A review of physics-based machine learning in civil engineering,” Results Eng., vol. 13, 2022, Art no. 100316.
- Search Google Scholar
- Export Citation
[9]↑
S. Alsamia and E. Koch, “Random forest regression on pullout resistance of a pile,” Pollack Period., 2024.
- Search Google Scholar
- Export Citation
[10]↑
A. Kaveh, “Applications of Artificial neural networks and machine learning in Civil Engineering,” Stud. Comput. Intell., vol. 1168, 2024.
- Search Google Scholar
- Export Citation
[11]
S. Alsamia and E. Koch, “Evaluation the behavior of pullout force and displacement for a single pile: experimental validation with plaxis 3D,” Kufa J. Eng., vol. 14, no. 2, pp. 105–116, 2023.
- Search Google Scholar
- Export Citation
[12]
A. I. Lawal and S. Kwon, “Application of artificial intelligence to rock mechanics: an overview,” J. Rock Mech. Geotech. Eng., vol. 13, no. 1, pp. 248–266, 2021.
- Search Google Scholar
- Export Citation
[13]↑
S. Alsamia, E. Koch, and H. S. Hamadi, “Comparative study of metaheuristics on optimal design of gravity retaining wall,” Pollack Period., 2023.
- Search Google Scholar
- Export Citation
[14]↑
D. Sattar and R. Salim, “A smart metaheuristic algorithm for solving engineering problems,” Eng. Comput., vol. 37, no. 3, pp. 2389–2417, 2021.
- Search Google Scholar
- Export Citation
[15]↑
S. M. Alsamia, M. S. Mahmood, and A. Akhtarpour, “Prediction of the contamination track in Al-Najaf city soil using numerical modelling,” IOP Conf. Ser. Mater. Sci. Eng., vol. 888, no. 1, 2020, Art no. 12050.
- Search Google Scholar
- Export Citation
[16]↑
C. Touzé, A. Vizzaccaro, and O. Thomas, “Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques,” Nonlinear Dyn., vol. 105, no. 2, pp. 1141–1190, 2021.
- Search Google Scholar
- Export Citation
[17]↑
E. Cuevas, F. Fausto, A. González, E. Cuevas, F. Fausto, and A. González, “An introduction to nature-inspired metaheuristics and swarm methods,” New Adv. Swarm Algorithms Oper. Appl., pp. 1–41, 2020.
- Search Google Scholar
- Export Citation
[18]↑
S. Alsamia, H. Albedran, and K. Jármai, “Comparative study of different metaheuristics on CEC 2020 benchmarks,” in Vehicle and Automotive Engineering 4: Select Proceedings of the 4th VAE2022, Miskolc. Hungary: Springer, 2022, pp. 709–719.
- Search Google Scholar
- Export Citation
[19]↑
B. F. Azevedo, A. M. A. C. Rocha, and A. I. Pereira, “Hybrid approaches to optimization and machine learning methods: a systematic literature review,” Mach. Learn., pp. 1–43, 2024.
- Search Google Scholar
- Export Citation
[20]
S. Alsamia, H. Albedran, and M. S. Mahmood, “Contamination depth prediction in sandy soils using fuzzy rule-based expert system,” Int. Rev. Appl. Sci. Eng., 2022.
- Search Google Scholar
- Export Citation
[21]
X.-S. Yang, Nature-inspired Optimization Algorithms. Academic Press, 2020.
- Search Google Scholar
- Export Citation
[22]
H. N. Ghafil, K. László, and K. Jármai, “Investigating three learning algorithms of a neural networks during inverse kinematics of robots,”, in Solutions for Sustainable Development. CRC Press, 2019, pp. 33–40.
- Search Google Scholar
- Export Citation
[23]↑
G. P. Rangaiah and S. Sharma, Differential Evolution in Chemical Engineering: Developments and Applications, vol. 6, World Scientific, 2017.
- Search Google Scholar
- Export Citation
[24]↑
N. Q. Yousif, A. F. Hasan, A. H. Shallal, A. J. Humaidi, and L. T. Rasheed, “Performance improvement of nonlinear differentiator based on optimization algorithms,” J. Eng. Sci. Technol., vol. 18, no. 3, pp. 1696–1712, 2023.
- Search Google Scholar
- Export Citation
[25]↑
M. A. Kamarposhti, I. Colak, H. Shokouhandeh, C. Iwendi, S. Padmanaban, and S. S. Band, “Optimum operation management of microgrids with cost and environment pollution reduction approach considering uncertainty using multi‐objective NSGAII algorithm,” IET Renew. Power Gener., 2022.
- Search Google Scholar
- Export Citation
[26]↑
K. C. Onyelowe, F. F. Mojtahedi, A. M. Ebid, A. Rezaei, K. J. Osinubi, A. O. Eberemu, and Z. U. Rehman, “Selected AI optimization techniques and applications in geotechnical engineering,” Cogent Eng., vol. 10, no. 1, 2023, Art no. 2153419.
- Search Google Scholar
- Export Citation
[27]↑
M. A. Kamarposhti, S. M. M. Khormandichali, and A. A. A. Solyman, “Locating and sizing of capacitor banks and multiple DGs in distribution system to improve reliability indexes and reduce loss using ABC algorithm,” Bull. Electr. Eng. Inform., vol. 10, no. 2, pp. 559–568, 2021.
- Search Google Scholar
- Export Citation
[28]↑
S. Alsamia, E. Koch, H. Albedran, and R. Ray, “Adaptive exploration artificial bee colony for mathematical optimization,” AI, vol. 5, no. 4, pp. 2218–2236, 2024.
- Search Google Scholar
- Export Citation
[29]↑
S. Alsamia, D. S. Ibrahim, and H. N. Ghafil, “Optimization of drilling performance using various metaheuristics,” Pollack Period., 2021.
- Search Google Scholar
- Export Citation
[30]↑
H. Shokouhandeh, M. Ahmadi Kamarposhti, I. Colak, and K. Eguchi, “Unit commitment for power generation systems based on prices in smart grid environment considering uncertainty,” Sustainability, vol. 13, no. 18, 2021, Art no. 10219.
- Search Google Scholar
- Export Citation
[31]
E. Eslami and M. Ahmadi Kamarposhti, “Optimal design of solar–wind hybrid system-connected to the network with cost-saving approach and improved network reliability index,” SN Appl. Sci., vol. 1, no. 12, p. 1742, 2019.
- Search Google Scholar
- Export Citation
[32]
S. Habib, M. A. Kamarposhti, H. Shokouhandeh, and I. Colak, “Economic dispatch optimization considering operation cost and environmental constraints using the HBMO method,” Energy Rep., vol. 10, pp. 1718–1725, 2023.
- Search Google Scholar
- Export Citation
[33]↑
Z. A. Waheed and A. J. Humaidi, “Design of optimal sliding mode control of elbow wearable exoskeleton system based on whale optimization algorithm,” J. Eur. Des Systèmes Autom., vol. 55, no. 4, pp. 459–466, 2022.
- Search Google Scholar
- Export Citation
[34]↑
R. Z. Khaleel, H. Z. Khaleel, A. A. Abdullah Al-Hareeri, A. S. Mahdi Al-Obaidi, and A. J. Humaidi, “Improved trajectory planning of mobile robot based on pelican optimization algorithm,” J. Eur. Des Systèmes Autom., vol. 57, no. 4, 2024.
- Search Google Scholar
- Export Citation
[35]↑
H. Shokouhandeh, M. Ahmadi Kamarposhti, F. Asghari, I. Colak, and K. Eguchi, “Distributed generation management in smart grid with the participation of electric vehicles with respect to the vehicle owners’ opinion by using the imperialist competitive algorithm,” Sustainability, vol. 14, no. 8, p. 4770, 2022.
- Search Google Scholar
- Export Citation
[36]↑
F. M. Makahleh, A. Amer, A. A. Manasrah, H. Attar, A. A. Solyman, M. A. Kamarposhti, and P. Thounthong, “Optimal management of energy storage systems for peak shaving in a smart grid,” Comput. Mater. Contin., vol. 75, no. 2, pp. 3317–3337, 2023.
- Search Google Scholar
- Export Citation
[37]↑
W. Wang, L. Xu, K. Chau, and D. Xu, “Yin-Yang firefly algorithm based on dimensionally Cauchy mutation,” Expert Syst. Appl., vol. 150, 2020, Art no. 113216.
- Search Google Scholar
- Export Citation
[38]↑
S. A. E. K. Hazim Albedran, “Flower fertilization optimization algorithm with application to adaptive controllers,” Sci. Rep., 2025. https://doi.org/10.1038/s41598-025-89840-1.
- Search Google Scholar
- Export Citation
[39]↑
H. N. Ghafil and K. Jármai, “Dynamic differential annealed optimization: new metaheuristic optimization algorithm for engineering applications,” Appl. Soft Comput., 2020, Art no. 106392.
- Search Google Scholar
- Export Citation
[40]↑
P. Rajesh, S. Muthubalaji, S. Srinivasan, and F. H. Shajin, “Leveraging a dynamic differential annealed optimization and recalling enhanced recurrent neural network for maximum power point tracking in wind energy conversion system,” Technol. Econ. Smart Grids Sustain. Energy, vol. 7, no. 1, p. 19, 2022.
- Search Google Scholar
- Export Citation
[41]↑
S. Alsamia and E. Koch, “Applying clustered artificial neural networks to enhance contaminant diffusion prediction in geotechnical engineering,” Sci. Rep., vol. 14, no. 1, 2024, Art no. 28750.
- Search Google Scholar
- Export Citation
[42]↑
C. V. Camp and M. Farshchin, “Design of space trusses using modified teaching–learning based optimization,” Eng. Struct., vol. 62, pp. 87–97, 2014.
- Search Google Scholar
- Export Citation

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

Save
Cite
Email this content

Share Link

Copy this link, or click below to email it to a friend
Email this content
or copy the link directly:

https://akjournals.com/view/journals/1848/aop/article-10.1556-1848.2025.00958/article-10.1556-1848.2025.00958.xml
The link was not copied. Your current browser may not support copying via this button.

Link copied successfully

Collapse
Expand

International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
H-4028 Debrecen, Ótemető u. 2-4. Hungary
Email: irase@eng.unideb.hu

Indexing and Abstracting Services:

DOAJ
ERIH PLUS
Google Scholar
ProQuest
SCOPUS
Ulrich's Periodicals Directory

2024
Scopus
CiteScore
CiteScore rank
SNIP
Scimago
SJR index	0.261
SJR Q rank	Q2

2023
Scimago
Scimago H-index	11
Scimago Journal Rank	0.249
Scimago Quartile Score	Architecture (Q2) Engineering (miscellaneous) (Q3) Environmental Engineering (Q3) Information Systems (Q4) Management Science and Operations Research (Q4) Materials Science (miscellaneous) (Q3)
Scopus
Scopus Cite Score	2.3
Scopus CIte Score Rank	Architecture (Q1) General Engineering (Q2) Materials Science (miscellaneous) (Q3) Environmental Engineering (Q3) Management Science and Operations Research (Q3) Information Systems (Q3)
Scopus SNIP	0.751

International Review of Applied Sciences and Engineering
Publication Model	Gold Open Access Online only
Submission Fee	none
Article Processing Charge	1100 EUR/article
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
Further Discounts	Limited number of full waivers available. Editorial Board / Advisory Board members: 50% Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription Information	Gold Open Access

International Review of Applied Sciences and Engineering
Language	English
Size	A4
Year of Foundation	2010
Volumes per Year	1
Issues per Year	3
Founder	Debreceni Egyetem
Founder's Address	H-4032 Debrecen, Hungary Egyetem tér 1
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	2062-0810 (Print)
ISSN	2063-4269 (Online)