Fireworks algorithm with elitism-based selection strategy and optimal particle guidance mechanism

Abstract

With the increasing complexity and difficulty of numerical optimization problems in the real world, many efficient meta-heuristic optimization methods have been proposed to solve these problems. An improved Fireworks Algorithm (FWA) with elitism-based selection and optimal particle guidance strategies (EO-FWA) was proposed to address the limitations of the traditional FWA in terms of optimization accuracy and convergence speed, which not only improves the efficiency of the searching agent but also accelerates its convergence speed. In addition, by adopting boundary-based mapping rules, EO-FWA eliminates the randomness of traditional modulo operation mapping rules, which improves its stability and reliability. Twelve benchmark functions in CEC-BC-2022 are used to test the performance of EO-FWA, and the welded beam design problem is optimized at the end. The results show that EO-FWA exhibits stronger competitiveness than other algorithms in dealing with high-dimensional optimization problems and engineering optimization problem, and it can balance exploitation and exploration effectively so as to prevent the algorithm from falling into local optimal solutions.

Keywords

Fireworks algorithm elitist-based selection strategy optimal particle guidance optimization problem

1 Introduction

The demand for solving complex problems has been increasing over time. In recent years, bionic-based meta-heuristic algorithms have emerged as a research hotspot, providing faster methods for solving complex optimization problems [1]. Genetic Algorithm (GA), proposed by Prof. Holland in 1975, is the pioneer of bionic algorithms [2]. Since then, algorithms such as Particle Swarm Algorithm (PSO) proposed by Dorigo in 1995 [3], Artificial Bee Colony Algorithm (ABC) proposed by Karabog in 2005 [4], and Cat Swarm Algorithm (CSO) [5] proposed by Shu-An Chu in 2006 are typical bionic algorithms for solving optimization problems.In 2009, Xin-She Yang and Suash Deb proposed the Cuckoo Search (CS) [6] inspired by the unique breeding and Lévy flight characteristics of cuckoos. In 2015, Mirjalili et al. proposed Ant-Lion Optimizer (ALO) [7], which shows superiority in solving complex optimization problems. In 2016, Mirjalili proposed Sine Cosine Algorithm (SCA) [8], which utilizes the sine and cosine functions to perform the individual updating, and the parameter settings are less. In 2023, Shoyab Ali et al. proposed a new hybrid meta-heuristic algorithm combining the marine predator algorithm and the sine-cosine algorithm to optimize the hybrid active power filter parameters and improve power filtering for power generators [9]. In the same year, Rabia et al. fused GA and CS to develop a deep learning model to effectively detect fraudulent transactions and improve detection efficiency and performance [10].

Although there are many complex optimization problems for which it is still difficult to find the best solutions, it is this challenge that motivates researchers and scholars to keep exploring real-world optimization problems [11 –13]. The wide application of swarm intelligence algorithms, such as in the fields of combinatorial optimization [14], function optimization [15], multi-objective function optimization [16], three-bar truss problem [17], cantilever beam design problem [18], and gear train design problem [19], fully proves the great contribution of such algorithms in solving optimization problems. However, these algorithms still show certain limitations and unreliability when dealing with practical problems. These limitations have prompted scholars to carry out more in-depth research and improvement to enhance the efficiency and reliability of the algorithms [20].

In order to solve the common problem of falling into local optimal in meta-heuristic algorithms, many scholars have conducted in-depth studies and proposed innovative improvement schemes [21 –23]. Among them, Firework Algorithm (FWA) [24], proposed by Mirjalili et al. in 2010, is particularly good at solving function optimization problems. FWA is inspired by the phenomenon of numerous sparks produced when fireworks explode, and an efficient search mechanism is established by simulating this process. With the development of FWA, researchers have proposed a variety of improvement methods. Li et al. optimized the explosion effect of FWA by adaptive amplitude method and proposed the adaptive fireworks algorithm (AFWA) [25]. Zheng et al. on the other hand, improved its search capability by utilizing dynamic explosion amplitude through dynamic search in FWA (dynFWA) [26]. In order to enhance the directionality of the search operator and to utilize the location information of the optimal individuals, Tao et al. proposed an improved FWA with directional function [27]. In addition, to address the inefficiency and uneven resource allocation problem of cloud computing resource task scheduling, Chen et al. fused the improved FWA with ABC to form IFWA-ABC, which effectively improves the allocation efficiency of cloud computing resources [28]. Aiming at the problem of slow convergence in the late stage of FWA, Teng et al. proposed an improved method based on the μ-law characteristic curve, which balances the local and global searching ability and enhances the searching efficiency through the information exchange strategy [29]. For the problem of coagulant dosage control in waterworks, Li et al. proposed an improved FWA (IFWA) to optimize the weights and thresholds of BP neural network, which improves its search ability and optimization accuracy [30].

In this paper, an improved FWA (EO-FWA) was proposed, whose design concept is based on elitism-based selection strategy and optimal particle guidance mechanism, aiming to address the limitations of traditional FWA in terms of optimization accuracy and convergence speed. Although previous improved algorithms, such as Adaptive Fireworks Algorithm (AFWA) and Dynamic Search Fireworks Algorithm (dynFWA), have achieved remarkable results in improving the search efficiency and reinforcing the explosion effect, they are still deficient in maintaining the balance between the search accuracy and efficiency. EO-FWA effectively achieves such a balance by integrating the elitist-based selection strategy and the optimal particle guidance mechanism, thus enhancing the overall performance of the algorithm. In order to comprehensively evaluate the performance of EO-FWA, it was compared with some recent state-of-the-art algorithms, including Arithmetic Optimization Algorithm (AOA), Sparrow Search Algorithm (SSA), Student Psychological Optimization Algorithm (SPBO) and Trapped Swarm Optimization Algorithm (TSA). During the comparison process, 12 test functions from CEC-BC-2022 are selected, which are widely recognized as comprehensive tests of the convergence ability of optimization algorithms. This series of tests aim to verify the effectiveness and efficiency of EO-FWA in dealing with complex optimization problems.

2 Fireworks algorithm

2.1 Inspiration of fireworks algorithm

Simple principles that exist in nature and life can be used as inspiration for heuristic algorithms, as are fireworks algorithms. The moment firework explodes in the air creates sparks, and FWA is inspired by this phenomenon. The process of fireworks explosions has the characteristics of instantaneous, concurrent and widely distributed. Regarding the sky as the search space for optimization problems and the sparks generated by the explosion as search agents, the process of fireworks explosion can be simply converted into the mathematical model of FWA. Thus, random factors and selection strategies are introduced to generate a firework algorithm with a parallel search method. The inspiration of FWA is shown in Fig. 1. It shows that fireworks will explode within a certain range to produce sparks. FWA has the advantages of instantaneity and concurrent by simulating the explosion process.

Fig. 1

Inspiration of firework algorithm.

2.2 Mathematical model of firework algorithm

In order to create a mathematical model based on the phenomenon of fireworks explosion, FWA is designed with four components, namely the explosion operator, Gaussian mutation operation, modulus operation mapping, and selection strategy. The composition of the firework algorithm is shown in Fig. 2.

Fig. 2

The composition of the firework algorithm.

Firstly, the explosion operator is the core of FWA, which includes the explosion radius, the number of explosion sparks, and displacement operations. Fireworks can generate a certain number of sparks within the radius of the explosion through the operation of the explosion operator. Secondly, the Gaussian mutation operation makes the distribution of search agents more ergodic. Thirdly, if the search agent is beyond the scope of the search space, the modulo operation can map the search agent back into the search space. Finally, the algorithm selects eligible search agents as the next-generation population to continue execution by selecting strategies. The selection strategy of FWA is a distance-based selection strategy, which guarantees the diversity of the population.

As the core part of FWA, the explosion operator can initially realize the phenomenon of fireworks explosion. Explosion operator include explosion radius, number of sparks, and displacement operations. The explosion radius defines the scope of the fireworks explosion, and the number of sparks defines the number of sparks generated by the explosion. The equations for calculating the explosion radius and the number of sparks are described as follows:

$A_{i} = a \frac{f_{i} - f_{min} + ɛ}{\sum_{i = 1}^{N} (f_{i} - f_{min}) + ɛ}$ (1)

$S_{i} = b \frac{f_{max} - f_{i} + ɛ}{\sum_{i = 1}^{N} (f_{max} - f_{i}) + ɛ}$ (2) where, A_i indicates the explosion radius of the i-th firework; S_i indicates the number of sparks of the i-th firework; f_i is the fitness value of the i-th firework; f_min is the optimal fitness value of the current population; f_max is the worst fitness value of the current population; a and b are constants that limit the size of the explosion radius and the number of sparks, generally a = 5 and b = 10; ɛ is a very small constant to prevent the numerator and denominator from being zero.

After the above operations, fireworks can generate sparks within the explosion range. However, the locations of sparks are not randomly distributed, and the distribution of search agents may be too concentrated or scattered. Therefore, the displacement operation is used to make the sparks randomly distributed in the explosion range, which increases the traversal of the population. The operation is defined as: $X_{i}^{k} = X_{i}^{k} + rand (0, A_{i})$ (3) where, $X_{i}^{k}$ is the position of the i-th spark in the j-th dimension; rand (0, A_i) is a random number within the explosion range. In order to further improve the traversal of the population, FWA introduced the Gaussian mutation operation. Select L sparks randomly in the population, and perform Gaussian mutation on the position of the L sparks. The Gaussian mutation operation is described as follows: $X_{i}^{k} = X_{i}^{k} \cdot g$ (4) where, g is a random number that obeys a Gaussian distribution. The mean and variance of this Gaussian distribution are both 1, that is to say g∼N(0,1).

In general, the optimization problems define constraints that divide the search space into feasible and unfeasible domains. Optimization problems are ineffective in the unfeasible domain. The sparks may exceed the search space through the explosion operator and Gaussian mutation operation. Sparks in the unfeasible domain are invalid, so a method needs to be taken to pull them back into the search space. FWA adopts modulo operation mapping rules to handle sparks outside the feasible region, which can be described as: $X_{i}^{k} = | X_{i}^{k} | % (l b_{k} - u b_{k}) + u b_{k}$ (5) where, % indicates the modulo operation, ub_k is the upper bound of the k-th dimension, and lb_k is the lower bound of the k-th dimension.

The can select eligible search agents to continue execution by selecting strategies. FWA first saves the search agent with the best fitness value, and the rest of the search agents adopt the distance-based selection strategy. The distance-based selection strategy needs to calculate the distance between the search agent and other agents, that is to say: $R_{i} = \sum_{j = 1}^{M} | X_{i} - X_{j} |$ (6) where, R_i is the sum of the distance between the i-th spark and other sparks, and M is the set of generated sparks. The probability that sparks will be selected as the next generation adopts the roulette method, which can be calculated by: $P_{i} = \frac{R_{i}}{\sum_{j = M} R_{j}}$ (7)

It can be seen from Equation (6)-(7) that the farther a spark is from other sparks, the greater the probability of being selected as the next generation. The distance-based selection strategy enables search agents to remain in the search space as widely as possible so as to ensure the traversal of the population.

Local exploitation and global exploration are two indispensable stages of the optimization algorithm. In the exploitation phase, the search agent searches towards the most promising area in the solution space in order to ensure the optimization accuracy of the algorithm. However, an excessively high exploitation capability will make the algorithm miss the global optimal value and fall into the local optimal. In the exploration stage, search agents are distributed as widely as possible in the search space so as to ensure the traversal of search agents and avoiding falling into local optimal values. However, an excessively high global exploration capability will reduce the convergence speed of the algorithm. Finding the right balance between local exploitation and global exploration can take into account the optimization accuracy and convergence speed of the algorithm. FWA also designed the adaptive balancing strategies for exploitation and exploration. It can be seen from Equation (1)-(2) that the explosion radius is inversely proportional to the fitness value of the fireworks. The number of sparks is directly proportional to the fitness value of the fireworks. In other words, if the fitness value of the fireworks is excellent, the explosion radius is small and the number of sparks is large. A careful search within a small range reflects the exploitative capabilities of the algorithm. The comparison of the fireworks explosion effect is shown in Fig. 3. The explosion effect produced by a excellent fitness value is shown Fig. 3 (a). If the firework fitness value is poor, the explosion radius is large, and the number of sparks is small. The extensive search reflects the exploration capabilities of the algorithm. The explosion effect caused by poor fitness value is shown in Fig. 3 (b). It shows that the explosion radius and the number of sparks can be adaptively changed according to the fitness value in order to ensure a balance between exploitation and exploration.

Fig. 3

Comparison of fireworks explosion effects.

It can be known from the mathematical model and pseudo code that FWA initializes a set of random solutions through an explosion operator, and then continuously improves this set of random solutions through Gaussian mutation and selection strategy. The fireworks algorithm has a simple structure and is easy to implement. At the same time, FWA can produce different explosion effects according to the fitness value, which can balance the exploration and exploitation stages. It saves the best search agent in each iteration. Even if the quality of other search agents becomes worse, it can ensure that the current optimal solution is in the population. However, the basic strategy of FWA has flaws, and it is necessary to introduce improved strategies to improve the optimization ability of the algorithm. In addition, the influence of the main parameters (a, b, L) on the performance of the algorithm also needs further study. The flow chart of FWA is shown in Fig. 4.

Fig. 4

The flow chart of FWA.

3 Improved fireworks algorithm based on elitism-based selection strategy and optimal particle guidance

3.1 Elite-based selection strategy

Generally, after iterative searching, the quality and location of search agents in the solution space are different. The optimization algorithm saves the eligible search agents through selection strategy and continues to execute. FWA uses a distance-based selection strategy. The farther an individual is from other individuals, the greater the probability that it will be selected as the next generation. The purpose of the distance-based selection strategy is to allow search agents to be distributed as widely as possible in the solution space so as to increase the global explorability of the algorithm. However, calculating the distance between every two individuals increases the computational cost and may reduce the convergence speed of the algorithm. Genetic algorithm (GA) [2] is inspired by Darwin’s theory of evolution. Through combination, crossover, and mutation, the best individuals are retained and the poor individuals are eliminated. This mechanism ensures that the new population is always better than the previous generation. In order to ensure the quality of the next generation population and improve the convergence speed, inspired by genetic algorithm, this paper proposes a selection strategy based on elitism. The idea of selection strategy based on elitism is to save only the first N elite individuals with the best fitness value, so the probability of an individual being selected as the next generation is only related to the fitness value based on Equation (8). $P_{i} = \frac{f_{max} - f_{i}}{f_{max} - f_{min}}$ (8) where, f_i is the fitness value of the i-th search agent, f_min indicates the optimal fitness value of the current population and f_max indicates the worst fitness value of the current population.

It can be seen from Equation (8) that the better the individual’s fitness value, the greater the probability of being selected as the next generation. This mechanism guarantees that the first N elite individuals with the best fitness value can be used as the next generation. Elitism-based selection strategy not only improves the quality of the population through selective selection, but also overcomes the shortcomings of the original algorithm, which results in slow convergence due to the large amount of computation. It should be noted that there are usually a large number of local optimal values over the optimization problems. The selection strategy based on elitism only selects the first N optimal individuals, which may destroy the diversity of the population and cause the algorithm to fall into the local optimal value. Increasing parameters a and b appropriately can avoid the above problems. Increasing the explosion radius and the number of sparks can enhance the effect of fireworks explosions and prevent search agents from falling into local optimums. The discussion of parameters a and b will be discussed in details in Section 4.2.

3.2 Optimal particle guidance mechanism

In swarm intelligence-based algorithms, search agents usually communicate through information in the search space. For example, the Ant Colony Optimization (ACO) [31] simulates the ant colony to release information by releasing pheromones to find the shortest path to the food source in different environments. The Gray Wolf Optimizer (GWO) [32] assigns individual levels based on fitness values, and higher-level individuals guide lower-level individuals to perform optimization. The information exchange mechanism enables search agents to explore different regions based on the return value of the solution space, so the SI-based algorithm has better optimization performance. However, there is no information exchange mechanism in FWA. The development trend of search agents towards excellent areas is too slow, which will reduce the optimization accuracy and convergence speed of the algorithm. Inspired by the swarm intelligence-based algorithms, this paper proposes an optimal particle guidance mechanism to enable FWA to achieve information exchange between individuals. First, calculate the fitness value of each search agent, and use the search agent with the best fitness value as the optimal particle. Then, calculate the distance between other search agents and the optimal particle. Finally, the search agent moves closer to the optimal particle based on the distance. This strategy can be described as follows. $X_{i} = X_{i} \cdot c_{1} - D_{i} \cdot c_{2}$ (9) where, X_i indicates the position of the i-th spark, c₁ is a random number in [0,1], which is used to provide random weights for the position of the spark, c₂ is a random number in [0, π], which is used to strengthen (c₂ > π/2) or weaken (c₂ < π/2) the movement effect of the search agent, and D_i indicates the distance between the i-th spark and the optimal particle, which is defined as: $D_{i} = | F_{t} - X_{i} |$ (10) where, X_i indicates the position of the optimal particle in the t-th iteration. Equation (9) can make the search agent move closer to the optimal particle. This process is shown in Fig. 5.

Fig. 5

The process of the agents moving close to the optimal particle.

It can be seen from Equation (9) that the search agent guided by the optimal particle can develop towards a promising area in the search space. This mechanism increases the local exploitation capability of the algorithm. However, there are a large number of local optimal values in the search space. When the optimal particles fall into the local optimal values, it may mislead the entire population to move closer to the local optimal values. At this time, the global exploration capability and optimization accuracy of the algorithm will decrease. Therefore, this paper introduces the balance factor c₃ into Equation (9), so that the optimal particle guidance mechanism can balance local exploitation and global exploration. It can be defined as:

$c_{3} = sign (rand - 1 / - 2)$ (11) where, sign function indicates the sign (positive or negative) of a certain number. When x > 0, sign(x) = 1. When x < 0, sign(x) = 1. rand is a random number in [0,1]. After introducing the balance factor c₃, Equation (9) can be updated as: $X_{i} = X_{i} \cdot c_{1} - c_{3} \cdot D_{i} \cdot c_{2}$ (12) When c₃ = 1, Equation (12) is equivalent to Equation (9). The search agent can move closer to the optimal particle, which increases the local exploitation capability of the algorithm.When c₃ = –1, Equation (12) can be expressed as Equation (13): $X_{i} = X_{i} \cdot c_{1} + c_{3} \cdot D_{i} \cdot c_{2}$ (13)

At this time, the search agent can explore the area outside the optimal particle, so that the search agent reduces the probability of falling into the local optimal value, and increases the global exploration ability of the algorithm. The influence of the balance factor c₃ on the next search area of the search agent is shown in Fig. 6. It shows that the search agent can search different areas according to c₃. When c₃ = 1, the search agent moves closer to the area where the optimal particle is located. When c₃ = –1, the search agent explores the area outside the optimal particle. The introduction of the balance factor c₃ enables the optimal particle guidance mechanism to balance exploitation and exploration.

Fig. 6

Impact of parameter c₃ on searching agents.

3.3 Boundary-based mapping rule

During the optimization process, search agents may be out of the search space. Search agents that exceed the search space are invalid. Ineffective search agents will reduce the number of populations and reduce the search ability of the algorithm. Therefore, the optimization algorithm needs to process search agents outside the feasible domain. Generally, the algorithm can map search agents outside the feasible domain back to the feasible domain or newly generate the same number of search agents to ensure that the number of populations remains unchanged. For search agents outside the feasible domain, FWA adopts modulo operation mapping rules. However, it may map search agents outside the feasible region to near the optimal value, which will inadvertently increase the optimization precision of the algorithm. This advantage is not brought by the algorithm itself, but a coincidence. In order to reduce the convenience brought by accident to the algorithm, this paper proposes a boundary-based mapping rule shown in Equation (14) to deal with search agents outside the feasible region. $X_{i} = {\begin{matrix} min (X_{i}, ub) if X_{i} > ub \\ max (X_{i}, lb) if X_{i} < lb \end{matrix}$ (14) where, X_i indicates the position of the i-th search agent, ub indicates the upper bound of the search space and lb indicates the lower bound of the search space.

The modulo operation mapping rules of the original FWA are shown in Fig. 7(a). The boundary-based mapping rules are shown in Fig. 7(b). It can be seen from Fig. 7 that if the search agent exceeds the range of the search space, the modulo operation mapping rule may map the search agent to near the optimal value. The boundary-based mapping rules map search agents near the boundary, witch will avoid the impact of randomness on the algorithm. This mapping rule shows the optimization ability of the algorithm more objectively.

Fig. 7

Comparison of two mapping rules.

3.4 Procedure of improved FWA based on elitism-based selection strategy and optimal particle guidance

Aiming at the shortcomings of FWA, such as slow convergence speed and low optimization accuracy, this paper proposes an improved fireworks algorithm based on elitism-based selection strategy and optimal particle guidance (EO-FWA). It firstly introduced a selection strategy based on elitism. This selection strategy can not only improve the quality of search agents, but also reduce the amount of calculations and improve the convergence speed, which makes up for the shortcomings of distance-based selection strategies. Secondly, this paper proposes an optimal particle-guided information exchange mechanism that enables search agents to exploit and explore based on the location of the optimal particle. This mechanism guarantees the algorithm can balance between exploitation and exploration. Finally, EO-FWA improves the mapping rules of the original algorithm, and uses the boundary-based mapping rules to eliminate the contingency brought by the modulo operation mapping rules. The flowchart of EO-FWA is shown in Fig. 8.

Fig. 8

Flow chart of EO-FWA.

3.5 Time complexity analysis

Time complexity is the calculation workload required to execute the algorithm, and it has important reference value for evaluating the time cost of the algorithm. The time complexity is usually expressed in big O symbols. In meta-heuristic algorithms, the time complexity depends on the number of operating units of the algorithm. For FWA, the time complexity mainly depends on the number of initial populations, the number of iterations, and the composition of FWA. The time complexity of the improved firework algorithm EO-FWA depends not only on the structure of the original algorithm, but also on the improvement strategy. In order to evaluate the impact of the improved strategy on the time cost of the algorithm, the time complexity of the FWA and EO-FWA was analyzed.

The time complexity of each operation unit of the FWA is defined as:

Initialize N populations to be distributed in a D-dimensional search space, which needs to run ND times.

Calculate the explosion radius A_i and the number of sparks S_i of each firework, which needs to be run 2.ND times.

The explosion operator is executed to generate M sparks in the D-dimensional search space, which needs to be run M.D times.

Select L sparks to perform Gaussian mutation operation, which needs to be run L.D times.

The distance-based selection strategy saves N of the N + M search agents, which needs to be run log(N + M)!times.

Select the optimal value output from the current population, which needs to be run [N.(N–1)]/2times.

Each of the above operation units goes through T iterations, so the total time complexity of the FWA is $O (FWA) = T \cdot [ND + 2 ND + MD + LD + log (N + M)! + (\overset{2}{N} - N) / - 2]$

Pseudo-code:

Initialize fireworks position

Initialize parameters X_i(i = 1,2, . . . ,n), Max_iter, etc.

Calculate the Fitness Function for the candidate solutions (X)

t = 1

While (t < Max_iter) do

For each firework agent do

Update A, and S using Eq. (1)and (2)

Fireworks perform explosion operator to generate sparks

Perform Gaussian mutation on L sparks using Eq. (4)

If sparks out of search space then

Perform the boundary-based mapping rule by using Eq.(14)

End if

End for

Select the next-generation search agents by the Eq.(8)

Calculate the fitness of each search agent

Select the optimal particle F_t

Update the position of the search agents by the Eq.(12)

t = t+1

End while

Return the best solution (Best(X)).

4 Simulation experiments and result analysis

In order to verify the effectiveness of the improved FWA in optimizing performance, 12 test functions of CEC-BC-2022 are used for detailed comparative analysis. The experimental design is described as follows. The experiments of each test function are carried out independently for 30 times, and the average value is taken to ensure the reliability and consistency of the results. In each experiment, the number of iterations of the improved algorithm was limited to 1000, while the size of the population was set as 40. In order to fully evaluate the performance of the improved algorithm, it was compared with several known state-of-the-art optimization algorithms, including Arithmetic Optimization Algorithm (AOA), Sparrow Search Algorithm (SSA), Student Psychological Optimization Algorithm (SPBO) and Tucked Swarm Optimization Algorithm (TSA). In this part of the experiment, each algorithm was run under the same conditions. 30 experiments were executed independently, the population size was set as 40, and the number of iterations was capped at 1000. Such a design aims to ensure fairness and consistency in the comparison. In addition, this study further explores the application of the improved FWA to real engineering optimization problem. By finding optimal solutions to these problems, we further verify the effectiveness and practicality of the improved FWA based on elitism and optimal particle guidance strategy in practical applications.

4.1 CEC-BC-2022

In this study, the experiments were conducted by using 12 benchmark functions from the CEC-BC-2022 suite, as detailed in Table 1. These functions are categorized into different types, each serving a specific purpose in evaluating the algorithm’s capabilities. Function F1 is a unimodal function, characterized by a single global optimum within its solution space. This type of function is crucial for assessing the algorithm’s search precision and convergence speed. Functions F2 to F5 are multimodal, featuring multiple local optima. This characteristic makes algorithms prone to getting trapped in local optima, thus serving as a stringent test of the algorithm’s optimization ability. Functions F6 to F8 are hybrid functions, and Functions F9 to F12 are composition functions. The inclusion of these diverse function types provides a comprehensive evaluation of the algorithm’s performance across different problem landscapes, illustrating its efficacy in various optimization scenarios.

Table 1
Properties and summary of the CEC2022 test functions

Type No. Functions Dim Range Fmin

Unimodal Function F1 Shifted and full Rotated Zakharov Function 10 [–100,100] 300

Multimodal Functions F2 Shifted and Rotated Rosenbrocks Function 10 [–100,100] 400

F3 Shifted and full Rotated Expanded Schaffer’s f6 Function 10 [–100,100] 600

F4 Shifted and full Rotated Non-Continuous Rastrigin’s Function 10 [–100,100] 800

F5 Shifted and full Rotated Levy Function 10 [–100,100] 900

Hybrid Functions F6 Hybrid Function 1 (N = 3) 10 [–100,100] 1800

F7 Hybrid Function 2 (N = 6) 10 [–100,100] 2000

F8 Hybrid Function 3(N = 5) 10 [–100,100] 2200

Composition Functions F9 Composition Function 1 (N = 5) 10 [–100,100] 2300

F10 Composition Function 2 (N = 4) 10 [–100,100] 2400

F11 Composition Function 3 (N = 5) 10 [–100,100] 2600

F12 Composition Function 4 (N = 6) 10 [–100,100] 2700

Type	No.	Functions	Dim	Range	Fmin
Unimodal Function	F1	Shifted and full Rotated Zakharov Function	10	[–100,100]	300
Multimodal Functions	F2	Shifted and Rotated Rosenbrocks Function	10	[–100,100]	400
	F3	Shifted and full Rotated Expanded Schaffer’s f6 Function	10	[–100,100]	600
	F4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	10	[–100,100]	800
	F5	Shifted and full Rotated Levy Function	10	[–100,100]	900
Hybrid Functions	F6	Hybrid Function 1 (N = 3)	10	[–100,100]	1800
	F7	Hybrid Function 2 (N = 6)	10	[–100,100]	2000
	F8	Hybrid Function 3(N = 5)	10	[–100,100]	2200
Composition Functions	F9	Composition Function 1 (N = 5)	10	[–100,100]	2300
	F10	Composition Function 2 (N = 4)	10	[–100,100]	2400
	F11	Composition Function 3 (N = 5)	10	[–100,100]	2600
	F12	Composition Function 4 (N = 6)	10	[–100,100]	2700

4.2 Analysis of parameters affecting EO-FWA performance

In exploring the performance optimization of the FWA, three key parameters were focused on: the frequency at which Gaussian mutations are performed (L), the parameter controlling the radius of the explosion (a), and the parameter for the number of sparks (b). A detailed comparative analysis of these parameters was performed by comparing tests by using 12 test functions of the CEC-BC-2022 functional test set, with each set of experiments run independently for 30 runs and averaged, and the results of the experiments are shown in Table 2-3. For the Gaussian mutation frequency parameter L, three different settings of L = 10, L = 15 and L = 20 were tested. The Gaussian mutation serves to provide excellent localized search capability by enabling the search process to focus on the local region around the current solution. Experimental results show that FWA performs best in terms of optimization accuracy and convergence speed at L = 15. We infer that an appropriate increase in the value of L can improve the local search efficiency of the algorithm without compromising the global convergence. For parameter a, which controls the blast radius, and parameter b, which controls the number of sparks, three sets of settings were compare, that is, a = 5, b = 10; a = 10, b = 20; a = 15, b = 30. Different combinations of parameters a and b have a significant effect on the explosion effectiveness and search range of the algorithm. If a is too large and b is too small, it will lead to the explosion effect being too sparse, which in turn reduces the algorithm’s ability to utilize locally. On the contrary, if a is too small and b is too large, the explosion effect is too centralized, which may weaken the global search ability of the algorithm. Through our experiments, we find that the setting of a = 15, b = 30 outperforms other combinations in terms of optimization accuracy and convergence speed, suggesting that the appropriate explosion intensity and number of sparks can help to balance the exploitation and exploration capabilities of the algorithm.

Table 2
Performance comparison results of CEC-2022 function optimization

Functions FWA

L = 5 L = 10 L = 15 L = 20

F1 Ave 8.922E+03 5.814E+03 1.958E+03 5.937E+03

Std 2.056E+03 1.491E+03 1.147E+03 1.316E+03

Rank 4 2 1 3

F2 Ave 1.313E+03 6.799E+02 4.733E+02 5.406E+02

Std 3.689E+02 2.225E+02 3.325E+01 8.383E+01

Rank 4 3 1 2

F3 Ave 6.519E+02 6.378E+02 6.336E+02 6.339E+02

Std 6.940E+00 8.512E+00 7.081E+00 7.675E+00

Rank 4 3 1 2

F4 Ave 8.576E+02 8.378E+02 8.265E+02 8.458E+02

Std 6.472E+00 6.159E+00 4.658E+00 5.661E+00

Rank 4 2 1 3

F5 Ave 1.506E+03 1.346E+03 1.186E+03 1.281E+03

Std 1.466E+02 1.818E+02 1.192E+02 1.364E+02

Rank 4 3 1 2

F6 Ave 9.589E+07 8.849E+04 5.390E+03 1.991E+06

Std 6.320E+07 1.721E+05 2.140E+03 2.393E+06

Rank 4 2 1 3

F7 Ave 2.109E+03 2.078E+03 2.071E+03 2.076E+03

Std 2.205E+01 1.454E+01 1.942E+01 1.536E+01

Rank 4 3 1 2

F8 Ave 2.278E+03 2.234E+03 2.228E+03 2.237E+03

Std 4.141E+01 6.524E+00 2.856E+00 4.186E+00

Rank 4 2 1 3

F9 Ave 2.743E+03 2.619E+03 2.639E+03 2.639E+03

Std 4.556E+01 5.033E+01 3.449E+01 4.539E+01

Rank 4 1 2 2

F10 Ave 2.611E+03 2.566E+03 2.564E+03 2.508E+03

Std 7.260E+01 7.292E+01 7.156E+01 4.650E+00

Rank 4 3 2 1

F11 Ave 3.263E+03 2.918E+03 2.742E+03 2.824E+03

Std 2.316E+02 1.388E+02 3.902E+01 4.310E+01

Rank 4 3 1 2

F12 Ave 3.066E+03 2.924E+03 2.942E+03 2.885E+03

Std 5.088E+01 3.346E+01 3.761E+01 1.156E+01

Rank 4 2 3 1

Average Ranking 4 3 1 2

Functions	FWA
F1	Ave	8.922E+03	5.814E+03	1.958E+03	5.937E+03
	Std	2.056E+03	1.491E+03	1.147E+03	1.316E+03
	Rank	4	2	1	3
F2	Ave	1.313E+03	6.799E+02	4.733E+02	5.406E+02
	Std	3.689E+02	2.225E+02	3.325E+01	8.383E+01
	Rank	4	3	1	2
F3	Ave	6.519E+02	6.378E+02	6.336E+02	6.339E+02
	Std	6.940E+00	8.512E+00	7.081E+00	7.675E+00
	Rank	4	3	1	2
F4	Ave	8.576E+02	8.378E+02	8.265E+02	8.458E+02
	Std	6.472E+00	6.159E+00	4.658E+00	5.661E+00
	Rank	4	2	1	3
F5	Ave	1.506E+03	1.346E+03	1.186E+03	1.281E+03
	Std	1.466E+02	1.818E+02	1.192E+02	1.364E+02
	Rank	4	3	1	2
F6	Ave	9.589E+07	8.849E+04	5.390E+03	1.991E+06
	Std	6.320E+07	1.721E+05	2.140E+03	2.393E+06
	Rank	4	2	1	3
F7	Ave	2.109E+03	2.078E+03	2.071E+03	2.076E+03
	Std	2.205E+01	1.454E+01	1.942E+01	1.536E+01
	Rank	4	3	1	2
F8	Ave	2.278E+03	2.234E+03	2.228E+03	2.237E+03
	Std	4.141E+01	6.524E+00	2.856E+00	4.186E+00
	Rank	4	2	1	3
F9	Ave	2.743E+03	2.619E+03	2.639E+03	2.639E+03
	Std	4.556E+01	5.033E+01	3.449E+01	4.539E+01
	Rank	4	1	2	2
F10	Ave	2.611E+03	2.566E+03	2.564E+03	2.508E+03
	Std	7.260E+01	7.292E+01	7.156E+01	4.650E+00
	Rank	4	3	2	1
F11	Ave	3.263E+03	2.918E+03	2.742E+03	2.824E+03
	Std	2.316E+02	1.388E+02	3.902E+01	4.310E+01
	Rank	4	3	1	2
F12	Ave	3.066E+03	2.924E+03	2.942E+03	2.885E+03
	Std	5.088E+01	3.346E+01	3.761E+01	1.156E+01
	Rank	4	2	3	1
Average Ranking	4	3	1	2

Table 3

Performance comparison results of CEC-2022 function optimization

Functions		FWA
		a = 5 b = 10	a = 10 b = 20	a = 15 b = 30
F1	Ave	6.713E+03	4.461E+03	3.213E+02
	Std	1.321E+03	1.741E+03	1.155E+02
	Rank	3	2	1
F2	Ave	1.137E+03	5.894E+02	4.191E+02
	Std	2.346E+02	1.132E+02	2.993E+01
	Rank	3	2	1
F3	Ave	6.472E+02	6.357E+02	6.241E+02
	Std	6.700E+00	8.532E+00	1.057E+01
	Rank	3	2	1
F4	Ave	8.519E+02	8.343E+02	8.226E+02
	Std	6.338E+00	7.011E+00	6.279E+00
	Rank	3	2	1
F5	Ave	1.372E+03	1.289E+03	1.012E+03
	Std	1.371E+02	1.471E+02	1.142E+02
	Rank	3	2	1
F6	Ave	8.570E+07	9.898E+04	5.475E+03
	Std	1.994E+08	3.853E+05	1.668E+03
	Rank	3	2	1
F7	Ave	2.100E+03	2.077E+03	2.040E+03
	Std	1.792E+01	1.268E+01	1.013E+01
	Rank	3	2	1
F8	Ave	2.260E+03	2.232E+03	2.225E+03
	Std	1.652E+01	5.799E+00	4.931E+00
	Rank	3	2	1
F9	Ave	2.703E+03	2.600E+03	2.534E+03
	Std	3.206E+01	3.915E+01	5.433E+00
	Rank	3	2	1
F10	Ave	2.562E+03	2.562E+03	2.533E+03
	Std	4.294E+01	6.890E+01	6.017E+01
	Rank	2	2	1
F11	Ave	2.562E+03	2.562E+03	2.533E+03
	Std	4.294E+01	6.890E+01	6.017E+01
	Rank	2	2	1
F12	Ave	3.025E+03	2.922E+03	2.901E+03
	Std	3.159E+01	2.576E+01	3.389E+01
	Rank	3	2	1
Average Ranking		3	2	1

4.3 Effectiveness analysis of improvement strategies

The Fireworks Algorithm (FWA), as an emerging swarm intelligence optimization algorithm, has demonstrated excellent performance in various fields. Nevertheless, the potential of FWA remains to be further explored in the face of complex optimization problems. In this study, four improvement schemes are proposed around the performance enhancement of FWA: FWA Change parameters, E-FWA, O-FWA and EO-FWA, focusing on enhancing its efficiency and accuracy in dealing with high-dimensional and multi-peak problems. Firstly, FWA with variable parameters improves the flexibility and adaptability of the algorithm by adjusting its parameters. Immediately after, E-FWA employs an elitist-based selection strategy aimed at enhancing the algorithm’s global search capability. O-FWA introduces an optimal particle guidance mechanism to strengthen the directionality and efficiency of the search. Ultimately, EO-FWA combines elitism and optimal particle guidance, and is committed to achieving a comprehensive performance optimization.

In order to verify the effectiveness of these strategies, 30 test functions of CEC-BC-2022 are selected for comparative testing in this study. Each group of experiments was run independently for 30 times, and the performance was measured by the average value, setting the maximum number of iterations to 1000 and the number of populations to 40. The experimental results are shown in Fig. 9 and Table 4. The experimental results show that all the improved strategies effectively improve the performance of FWA. Particularly noteworthy is that O-FWA outperforms FWA with only changing parameters on most of the tested functions, which indicates that the optimal particle-guided selection strategy has a significant advantage in improving the performance of the algorithm. Compared to O-FWA, E-FWA demonstrates the effectiveness of the elite-based selection strategy in compensating for the inadequacy of the distance-based selection strategy, while improving the search accuracy and efficiency. Among all the strategies, EO-FWA performs the best in both optimization accuracy and convergence speed, which fully validates the importance of combining multiple improvement strategies. The comprehensive advantages of EO-FWA emphasize the importance of introducing optimal particle guidance and elitist strategies in FWA. This combined strategy not only improves the global search capability of the algorithm, but also balances the dynamics of the algorithm between exploration and exploitation, ensuring an efficient and accurate optimization process. In addition, the excellent performance of EO-FWA demonstrates that when solving complex optimization problems, the combination of multiple strategies can significantly improve the performance of FWA more than a single strategy. These findings provide valuable guidance for future research and applications in FWA and lay the foundation for solving more complex optimization problems.

Fig. 9

Convergence curves on optimization functions.

Table 4

Performance comparison results of CEC-2022 function optimization

Functions		FWA	FWA (Change	E-FWA	O-FWA	EO-FWA
			parameters)
F1	Ave	7.396E+03	5.941E+03	3.506E+03	4.859E+03	3.129E+02
	Std	2.310E+03	1.458E+03	1.954E+03	1.709E+03	6.691E+01
	Rank	5	4	2	3	1
F2	Ave	1.167E+03	7.189E+02	5.907E+02	4.937E+02	4.277E+02
	Std	3.463E+02	1.522E+02	1.332E+02	3.928E+01	3.431E+01
	Rank	5	4	3	2	1
F3	Ave	6.486E+02	6.427E+02	6.320E+02	6.313E+02	6.250E+02
	Std	6.497E+00	6.624E+00	8.051E+00	6.835E+00	1.069E+01
	Rank	5	4	3	2	1
F4	Ave	8.531E+02	8.492E+02	8.283E+02	8.414E+02	8.232E+02
	Std	5.956E+00	6.726E+00	6.259E+00	7.253E+00	7.122E+00
	Rank	5	4	2	3	1
F5	Ave	1.454E+03	1.310E+03	1.228E+03	1.224E+03	1.009E+03
	Std	1.519E+02	1.221E+02	1.663E+02	1.367E+02	1.062E+02
	Rank	5	4	3	2	1
F6	Ave	4.427E+07	2.331E+07	1.765E+04	8.756E+05	4.401E+03
	Std	2.637E+07	1.603E+07	4.460E+04	8.195E+05	1.661E+03
	Rank	5	4	2	3	1
F7	Ave	2.107E+03	2.091E+03	2.064E+03	2.071E+03	2.039E+03
	Std	1.826E+01	1.591E+01	1.619E+01	1.505E+01	1.324E+01
	Rank	5	4	3	2	1
F8	Ave	2.254E+03	2.247E+03	2.229E+03	2.236E+03	2.226E+03
	Std	1.494E+01	1.028E+01	3.577E+00	3.918E+00	6.741E+00
	Rank	5	4	2	3	1
F9	Ave	2.720E+03	2.682E+03	2.611E+03	2.610E+03	2.530E+03
	Std	3.289E+01	2.867E+01	4.157E+01	4.340E+01	1.189E+00
	Rank	5	4	3	2	1
F10	Ave	2.576E+03	2.527E+03	2.564E+03	2.507E+03	2.543E+03
	Std	5.131E+01	1.013E+01	6.955E+01	3.691E+00	6.188E+01
	Rank	5	3	4	2	1
F11	Ave	3.236E+03	2.969E+03	2.916E+03	2.807E+03	2.641E+03
	Std	2.291E+02	1.000E+02	1.907E+02	2.923E+01	6.733E+01
	Rank	5	4	3	2	1
F12	Ave	3.029E+03	2.950E+03	2.917E+03	2.881E+03	2.884E+03
	Std	3.298E+01	2.787E+01	2.433E+01	9.225E+00	1.986E+01
	Rank	5	4	3	1	2
Average Ranking		5	4	3	2	1

4.4 Comparison of EO-FWA with other algorithms

Figure 10 shows the convergence of the EO-FWA compared to FWA, AOA [33], SSA [34], SPBO [35] and TSA [36]. Table 5 lists the mean and standard deviation of these algorithms. Each algorithm is averaged over 30 runs, with a maximum number of 1000 iterations and a population size of 40. The multi-peak function has multiple local optima, which makes the algorithms easily fall into the local optima, and the test results of the multi-peak function can be a good indication of the algorithm’s ability to optimize the function. Compared with other algorithms, the improved FWA based on elitism and optimal particle bootstrapping is very effective, and the mean value of 11 out of 12 tested functions achieves the best result, and all the remaining ones achieve the best result except for the hybrid function F6, so the proposed algorithm has a very obvious advantage in solving function optimization. Meanwhile, nearly half of the EO-FWA was optimized with very good stability, except for F3, F6 and F12, the algorithms perform very well in terms of solution stability. Finally, this paper provides a comprehensive ranking of the optimization performance of each algorithm on the test functions based on the number of times the average value appears. By comparing their average rankings, it can be intuitively seen that EO-FWA has the best optimization performance and ranks first, AOA and FWA rank second and third, respectively.

Fig. 10

Convergence curves of optimization functions.

Table 5

Performance comparison results of CEC-2022 function optimization

Functions		EO-FWA	FWA	AOA	SSA	SPBO	TSA
F1	Ave	3.176E+02	7.165E+03	7.757E+03	8.469E+03	3.037E+04	1.004E+04
	Std	9.244E+01	1.871E+03	3.326E+03	3.357E+03	8.520E+03	4.320E+03
	Rank	1	2	3	6	5	4
F2	Ave	3.176E+02	7.165E+03	7.757E+03	8.469E+03	3.037E+04	1.004E+04
	Std	9.244E+01	1.871E+03	3.326E+03	3.357E+03	8.520E+03	4.320E+03
	Rank	1	2	3	4	6	5
F3	Ave	6.251E+02	6.463E+02	6.352E+02	6.473E+02	6.693E+02	6.400E+02
	Std	1.239E+01	5.858E+00	7.611E+00	1.330E+01	8.097E+00	1.281E+01
	Rank	1	4	2	5	6	3
F4	Ave	8.250E+02	8.527E+02	8.294E+02	8.394E+02	9.051E+02	8.327E+02
	Std	6.498E+00	7.509E+00	1.113E+01	1.446E+01	1.254E+01	8.612E+00
	Rank	1	5	2	4	6	3
F5	Ave	9.745E+02	1.425E+03	1.331E+03	1.493E+03	3.964E+03	1.448E+03
	Std	8.538E+01	1.542E+02	1.269E+02	1.051E+02	6.844E+02	1.427E+02
	Rank	1	3	2	5	6	4
F6	Ave	4.764E+03	7.678E+07	3.799E+03	3.986E+03	4.071E+08	3.637E+07
	Std	1.910E+03	1.629E+08	1.413E+03	1.949E+03	3.149E+08	1.963E+08
	Rank	3	5	1	2	6	4
F7	Ave	2.039E+03	2.102E+03	2.098E+03	2.122E+03	2.172E+03	2.101E+03
	Std	1.064E+01	1.965E+01	2.912E+01	3.589E+01	4.544E+01	3.262E+01
	Rank	1	4	2	5	6	3
F8	Ave	2.225E+03	2.269E+03	2.266E+03	2.268E+03	2.454E+03	2.283E+03
	Std	6.748E+00	4.867E+01	6.843E+01	4.888E+01	6.272E+02	8.433E+01
	Rank	1	2	3	4	6	5
F9	Ave	2.530E+03	2.738E+03	2.691E+03	2.563E+03	2.769E+03	2.697E+03
	Std	1.180E+00	4.199E+01	4.162E+01	4.109E+01	1.136E+02	5.507E+01
	Rank	1	5	3	2	6	4
F10	Ave	2.527E+03	2.584E+03	2.637E+03	2.781E+03	2.576E+03	2.685E+03
	Std	5.348E+01	5.957E+01	1.345E+02	4.520E+02	4.465E+01	3.481E+02
	Rank	1	3	4	6	2	5
F11	Ave	2.621E+03	3.220E+03	3.467E+03	2.875E+03	3.788E+03	3.252E+03
	Std	5.182E+01	2.785E+02	3.900E+02	2.892E+02	4.526E+02	4.549E+02
	Rank	1	3	5	2	6	4
F12	Ave	2.885E+03	3.049E+03	3.001E+03	2.990E+03	2.887E+03	2.924E+03
	Std	1.766E+01	5.120E+01	5.744E+01	1.055E+02	7.243E+00	4.283E+01
	Rank	1	6	5	3	2	4
Average Ranking		1	3	2	4	6	4

4.5 Solving engineering optimization problem

In order to verify the effectiveness of EO-FWA in engineering optimization problem, this study applies it to the optimization of welded beam design. Engineering optimization problems usually involve a variety of constraints, so it is crucial to choose an appropriate constraint handling method. Among the many methods, penalty functions are widely adopted due to their low cost and easy operation. There are numerous types of penalty functions, including static, dynamic, annealing, adaptive, co-evolutionary and death penalty functions. In this study, the death penalty function was used to handle the constraints because it can effectively eliminate infeasible solutions in the optimization process. The main objective of the welded beam optimization is to minimize its fabrication cost while satisfying four important optimization constraints: shear stress, bending stress of the beam, buckling load and end deflection of the beam. The optimized variables include weld thickness, welded splint beam length, beam material height and beam material thickness. By setting these variables reasonably, not only the cost optimization can be achieved, but also the structural strength and functionality of the beam can be ensured. The objective function, constraints and optimization variables of the welded beam optimization problem can be expressed in Equation (15).

In order to comprehensively evaluate the performance of EO-FWA in solving complex engineering optimization problems, a variety of state-of-the-art optimization algorithms were selected for comparative analysis. These algorithms include Whale Optimization Algorithm (WOA) [33], Gravitational Search Algorithm (GSA) [34], Genetic Algorithm (GA) [35, 36] and Harmony Search (HS) [37]. In addition, some traditional optimization methods, such as stochastic, simplex, Davidon-Fletcher-Powell and continuous linear approximation methods [38] were introduced to increase the diversity and comprehensiveness of the comparison. In conducting the experiments for the welded beam optimization problem, the performance metrics of each algorithm are recorded in details. These metrics include convergence speed, solution quality, stability of the algorithms and adaptability to different problem sizes. $Consider \vec{x} = [x_{1} x_{2} x_{3} x_{4}] = [h l t b]$ (15) $\begin{matrix} Minimize f (\vec{x}) = 1.10471 x_{1}^{2} x_{2} \\ + 0.04811 x_{3} x_{4} (14.0 + x_{2}) \end{matrix}$ $\begin{matrix} Subject \underset{1}{g} (\vec{x}) = τ (\vec{x}) - \underset{max}{τ} ⩽ 0 \\ \underset{2}{g} (\vec{x}) = θ (\vec{x}) - \underset{max}{θ} ⩽ 0 \\ \underset{3}{g} (\vec{x}) = δ (\vec{x}) - \underset{max}{δ} ⩽ 0 \\ \underset{4}{g} (\vec{x}) = \underset{1}{x} - \underset{4}{x} ⩽ 0 \\ \underset{5}{g} (\vec{x}) = p - \underset{c}{p} (\vec{x}) ⩽ 0 \\ \underset{6}{g} (\vec{x}) = \underset{1}{0.125 - x} ⩽ 0 \\ \underset{7}{g} (\vec{x}) = 1.10471 x_{1}^{2} + 0.04811 \underset{3}{x} \underset{4}{x} (14.0 + \underset{2}{x}) ⩽ 0 \end{matrix}$ $\begin{matrix} Variable range 0.1 ⩽ \underset{1}{x} ⩽ 2 \\ 0.1 ⩽ \underset{2}{x} ⩽ 10 \\ \underset{3}{0.1 ⩽ x} ⩽ 10 \\ 0.1 ⩽ \underset{4}{x} ⩽ 2 \end{matrix}$

The results presented in Table 6 clearly show that EO-FWA excels in most of the metrics, and its performance is second only to that of WOA.This finding is particularly important because it indicates that EO-FWA is highly effective and reliable in dealing with complex real-world engineering problems. Further analyzing the performance of EO-FWA, it is particularly good at avoiding local optimal stagnation. This feature is especially critical for complex optimization problems, which are often filled with a large number of local optimal solutions, easily causing the algorithm to fall into local optimality instead of reaching the global optimum. EO-FWA effectively overcomes this problem through its unique search mechanism and adaptive tuning strategy, and is able to maintain a good balance of exploration and exploitation in a wide range of search spaces. Moreover, the superiority of EO-FWA is further demonstrated by comparing it with traditional optimization methods. Compared with traditional methods, such as the simplex method and the continuous linear approximation method, EO-FWA not only has a significant improvement in solution accuracy, but also is more efficient and flexible in dealing with complex, multivariate optimization problems. Overall, the experimental results show that EO-FWA is a powerful and reliable optimization tool. It is not only capable of solving engineering optimization problems efficiently, but also demonstrates excellent performance in a wide range of performance metrics. This is clearly demonstrated in comparisons with a wide range of modern and traditional optimization algorithms. Therefore, EO-FWA is a valuable choice for solving complex engineering optimization problems.

Table 6

Simulation performance comparison

Algorithms	Optimum variables				Optimum cost
	h	l	t	b
EO-FWA	0.189004	3.723388	8.798582	0.226878	1.849045
WOA	0.205396	3.484293	9.037426	0.206276	1.730499
GSA	0.182129	3.856979	10	0.202376	1.879952
GA	0.2489	6.173	8.1789	0.2533	2.4331
HS	0.2442	6.2231	8.2915	0.2443	2.3807
Random	0.4575	4.7313	5.0853	0.66	4.1185
Simplex	0.2792	5.6256	7.7512	0.2796	2.5307
David	0.2434	6.2552	8.2915	0.2444	2.3841
APPROX	0.2444	6.2189	8.2915	0.2444	2.3815

5 Conclusions

As a novel meta-heuristic algorithm, FWA has demonstrated excellent development capabilities in optimization problems in numerous domains. The improved fireworks algorithm (EO-FWA) proposed in this paper makes important innovations and optimizations based on the original algorithm. Firstly, an elitism-based selection strategy was introduced, which is centered on improving the quality and convergence speed of search agents. With the elitist selection strategy, the algorithm is able to efficiently filter out the dominant solutions and use them as the basis for subsequent iterations, thus accelerating the convergence process. Although this strategy increases the computational cost, the performance improvement it brings is obvious. Secondly, to address the lack of effective communication among search agents in traditional FWA, a novel information exchange mechanism was designed, that is, the optimal particle guidance mechanism. This mechanism allows the search agent to refer to and utilize known optimal solutions when exploring the space as a way to guide the search direction. This not only improves the search efficiency, but also maintains the balance between exploration and exploitation, and avoids the algorithm from over-focusing on the local region and neglecting the global search. In addition, in order to solve the instability that may be brought by random mapping in traditional FWA, EO-FWA improves the mapping rules and adopts a boundary-based mapping method. This approach reduces the uncertainty brought by randomness and enhances the robustness of the algorithm.

In experimental simulations, the algorithm was tested from multiple angles and verified the effects of the number of Gaussian mutations, explosion radius and number of sparks on the performance of the algorithm. The results show that increasing these parameters appropriately can significantly enhance the explosion effect and thus improve its exploitation. Through these experiments, it can be observed that diverse improvement strategies not only improve the optimization accuracy of FWA, but also significantly accelerate its convergence speed. Finally, the application of EO-FWA to high-dimensional function optimization and engineering optimization problem shows that it can effectively avoid falling into local optimums and exhibit strong performance in complex optimization problems. These experimental results not only enrich the theoretical study of FWA, but also provide a solid theoretical and empirical foundation for its promotion in practical applications.

In summary, the improved fireworks algorithm (EO-FWA) proposed in this paper shows significant advantages in both theory and practice, and provides an efficient and reliable new tool for solving complex optimization problems.

Footnotes

Acknowledgments

This work was supported by the Project by Liaoning Provincial Natural Science Foundation of China (Grant No. 20180550700).

References

Tansel

, Ender

, Tayfun

, Ahmet

, et al. A survey on new generation metaheuristic algorithms[J], Computers & Industrial Engineering (2019), 137.

David Goldberg

and John Holland

, Genetic Algorithms and Machine Learning[J], Machine Learning 3(2-3) (1988), 95–99.

Eberhart

, Kennedy

, A new optimizer using particle swarm theory [C]// Proceedings of the Sixth International Symposium on Micro Machine and Human Science. Nagoya, Japan: [s.n.], (1995), 39–43. DOI: 10.1109/MHS.1995.494215.

Karaboga

and Basturk

, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm[J],459, Journal of Global Optimization 39(3) (2007), 0–471.

Shu-Chuan

, Pei-wei

and Jeng-Shyang

, Cat Swarm Optimization[C], Pacific Rim International Conference on Artificial Intelligence 4099 (2006), 854–858.

Yang

X.-S.

, Deb

, Cuckoo Search via Lévy flights[C], Nature and Biologically Inspired Computing (2009), 210–214.

Mirjalili

, The Ant Lion Optimizer[J], Advances in Engineering Software 83(C) (2015), 80–98.

Mirjalili

, SCA: A Sine Cosine Algorithm for Solving Optimization Problems[J], Knowledge-Based Systems 96 (2016), 120–133.

Ali

, Bhargava

, Saxena

and Kumar

, A Hybrid Marine Predator Sine Cosine Algorithm for Parameter Selection of Hybrid Active Power Filter[J], Mathematics 11(3) (2016), 598.

10.

Angelova

, Roeva

, Vassilev

and Pencheva

, Multi-Population Genetic Algorithm and Cuckoo Search Hybrid Technique for Parameter Identification of Fermentation Process Models[J], Processes 11(2) (2023), 427.

11.

LeCun

, Bengio

and Hinton.

, Deep learning[J], Nature 521(7553) (2015), 436–444.

12.

Glorot

and Bengio

, Understanding the difficulty of training deep feedforward neural networks[J], Computer Science 9 (2010), 249–256.

13.

Karaboga

and Basturk

, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm[J], Journal of Global Optimization 39(3) (2007), 459.0–471.

14.

Bengio

, Lodi

and Prouvost

, Machine learning for combinatorial optimization: A methodological tour d’horizon[J], European Journal of Operational Research 290(2) (2021), 405–421.

15.

Yirui

, Shangce

, Mengchu

, Yang

, et al. A multi-layered gravitational search algorithm for function optimization and real-world problems[J], IEEE/CAA Journal of Automatica Sinica 8(1) (2021), 94–109.

16.

Chongshan

, Bin

, Jiaqi

, Qi

, Yuan

, et al. Multiobjective Function Optimization of 2-RPU&2-SPS Parallel Mechanism Based on Adams[J], 2021 6th IEEE International Conference on Advanced Robotics and Mechatronics (ICARM), (2021), 224–229.

17.

Hojat

, Mahdi

V.A.

, Saeed

, Seyedali

, et al. Flow Direction Algorithm (Fda): A Novel Optimization Approach For Solving Optimization Problems[J], Computers & Industrial Engineering 156 (2021), 107224.

18.

Yildiz

B.S.

, Pholdee

, Bureerat

, Yildiz

A.R.

and Sait

S.M.

, Enhanced grasshopper optimization algorithm using elite opposition-based learning for solving real-world engineering problems, Engineering with Computers 38(5) (2022), 4207–4219.

19.

Yildirim

A.E.

, Karci

, Application Of Three Bar Truss Problem Among Engineering Design Optimization Problems Using Artificial Atom Algorithm[C], International Conference on Artificial Intelligence (2018), 1–5.

20.

Andrea

, Raffaele

, S.A.

, Raimondo

, F. M D S, et al. Free vibrations of elastic beams by modified nonlocal strain gradient theory[J], International Journal of Engineering Science 133 (2018), 99–108.

21.

Zhao

, Wang

and Zhang

, Artificial ecosystem-based optimization: a novel nature-inspired meta-heuristic algorithm[J], Neural Computing and Applications 32(13) (2020), 9383–9425.

22.

Kanungo

, David

M.M.

, Nathan

S.N.

, Christine

D.P.

, Ruth

, Angela

Y.W.

, et al. An Efficient k-Means Clustering Algorithm: Analysis and Implementation[J], IEEE Transactions on Pattern Analysis and Machine Intelligence 24(7) (2002), 881–892.

23.

, Minimap2: pairwise alignment for nucleotide sequences[J], Bioinformatics 34(18) (2018), 3094–3100.

24.

Tan

and Zhu

, Fireworks Algorithm for Optimization[C], International Conference on Swarm Intelligence 6145 (2010), 355.

25.

, Zheng

, Tan

, Adaptive fireworks algorithm[C]//2014 IEEE Congress on evolutionary computation (CEC). ieee (2014), 3214–3221.

26.

Zheng

, Janecek

, Li

, et al. Dynamic search in fireworks algorithm[C]//2014 IEEE Congress on evolutionary computation (CEC). IEEE, (2014), 3222–3229.

27.

Tao

, Chen

J.-l.

and Xie

X.-l.

, Improved firework algorithm with directional function[J], Computer Engineering and Design 40(12) (2019), 3479–3486.

28.

Xuan

, Dawei

, Changliang

, Dan

, et al. Cloud computing task scheduling algorithm based on IFWA-ABC[J], Application Research of Computers 36(10) (2019), 3022–3026.

29.

Teng

, Li

, Wang

, Huangfu

, Shen

, et al. Improved firework algorithm based on μ-law explosion operator and information exchange mapping strategy[J], Journal of Shaanxi University of Science & Technology 40(2) (2022), 187–194.

30.

, Zhang

, Tian

, Mao

, Wang

, Ma

, et al. Prediction of coagulant dosage in waterworks based on improved FWA-BP neural network[J], Journal of Shaanxi University of Technology(Natural Science Edition) 37(6), 2021.

31.

Socha

and Dorigo

, Ant colony optimization for continuous domains[J], European Journal of Operational Research 185(3) (2008), 1155–1173.

32.

Mirjalili

, Mirjalili

S.M.

and Lewis

, Grey Wolf Optimizer[J], Advances in Engineering Software 69 (2015), 46–61.

33.

Mirjalili

and Lewis

, The whale optimization algorithm[J], Advances in Engineering Software 95 (2016), 51–67.

34.

Rashedi

, Nezamabadi-pour

and Saryazdi

, GSA: A Gravitational Search Algorithm[J], Information Sciences 179(13) (2009), 2232–2248.

35.

Deb

, Optimal design of a welded beam via genetic algorithms[J], AIAA Journal 29(11) (1991), 2013–2015.

36.

Deb

, An efficient constraint handling method for genetic algorithms[J], Computer Methods in Applied Mechanics and Engineering 186(2-4) (2000), 311–338.

37.

Lee

K.S.

and Geem

Z.W.

, A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice[J], Computer Methods in Applied Mechanics and Engineering 194(36-38) (2005), 3902–3933.

38.

Ragsdell

K.M.

, Phillips

D.T.

, Optimal design of a class of welded structures using geometric programming[J]. 1976.

Functions		FWA
		L = 5	L = 10	L = 15	L = 20
F1	Ave	8.922E+03	5.814E+03	1.958E+03	5.937E+03
	Std	2.056E+03	1.491E+03	1.147E+03	1.316E+03
	Rank	4	2	1	3
F2	Ave	1.313E+03	6.799E+02	4.733E+02	5.406E+02
	Std	3.689E+02	2.225E+02	3.325E+01	8.383E+01
	Rank	4	3	1	2
F3	Ave	6.519E+02	6.378E+02	6.336E+02	6.339E+02
	Std	6.940E+00	8.512E+00	7.081E+00	7.675E+00
	Rank	4	3	1	2
F4	Ave	8.576E+02	8.378E+02	8.265E+02	8.458E+02
	Std	6.472E+00	6.159E+00	4.658E+00	5.661E+00
	Rank	4	2	1	3
F5	Ave	1.506E+03	1.346E+03	1.186E+03	1.281E+03
	Std	1.466E+02	1.818E+02	1.192E+02	1.364E+02
	Rank	4	3	1	2
F6	Ave	9.589E+07	8.849E+04	5.390E+03	1.991E+06
	Std	6.320E+07	1.721E+05	2.140E+03	2.393E+06
	Rank	4	2	1	3
F7	Ave	2.109E+03	2.078E+03	2.071E+03	2.076E+03
	Std	2.205E+01	1.454E+01	1.942E+01	1.536E+01
	Rank	4	3	1	2
F8	Ave	2.278E+03	2.234E+03	2.228E+03	2.237E+03
	Std	4.141E+01	6.524E+00	2.856E+00	4.186E+00
	Rank	4	2	1	3
F9	Ave	2.743E+03	2.619E+03	2.639E+03	2.639E+03
	Std	4.556E+01	5.033E+01	3.449E+01	4.539E+01
	Rank	4	1	2	2
F10	Ave	2.611E+03	2.566E+03	2.564E+03	2.508E+03
	Std	7.260E+01	7.292E+01	7.156E+01	4.650E+00
	Rank	4	3	2	1
F11	Ave	3.263E+03	2.918E+03	2.742E+03	2.824E+03
	Std	2.316E+02	1.388E+02	3.902E+01	4.310E+01
	Rank	4	3	1	2
F12	Ave	3.066E+03	2.924E+03	2.942E+03	2.885E+03
	Std	5.088E+01	3.346E+01	3.761E+01	1.156E+01
	Rank	4	2	3	1
Average Ranking		4	3	1	2