1. Introduction
Optimization problems are widespread [
1], defined as the process of identifying the optimal solution from feasible solutions to minimize or maximize a given objective [
2]. Optimization problems are generally characterized by large scale, numerous constraints, complex parameter control, and high computational cost [
3]. In many optimization problems, it is necessary to find an optimal solution to a particular problem with highly complex constraints in a reasonable time period [
4]. Effective methods are required to solve optimization problems involving numerous decision variables, complex nonlinear constraints, and objective functions [
5]. Traditional optimization methods, represented by gradient descent and Lagrange multiplier methods, derive their efficacy strictly from mathematical properties, such as the differentiability and convexity of the objective function, along with the explicit formulation of constraint conditions [
6]. However, optimization problems commonly encountered in real-world production and daily life are typically characterized by large-scale, high-dimensional, and nonlinear features [
7], thus conventional methods often struggle to obtain acceptable solutions within a feasible time frame [
8]. Specifically, the problem presents the following challenges:
Multimodal: Many optimization problems exhibit multiple local optima, causing algorithms to easily become trapped in local optima and to fail to obtain satisfactory solutions.
High-dimensional: As the number of decision variables increases, the problem’s dimensionality grows accordingly. The search space expands exponentially with increasing dimensions, leading to significantly higher problem complexity.
Nonlinear: The objective functions of many problems are frequently nonlinear and may even be non-differentiable, rendering certain algorithms inapplicable due to their strict requirements on objective function properties.
Multi-objective: Some problems require simultaneous optimization of multiple objectives that often exhibit conflicting relationships, making it difficult or impossible to find solutions that satisfy all objectives concurrently.
In contrast to conventional algorithms, heuristic algorithms [
9] demonstrate the capability to identify a feasible suboptimal solution for optimization problems within a reasonable timeframe or under constrained computational resources. This methodology achieves computational feasibility and suboptimality through a systematic trade-off between solution accuracy/precision and operational efficiency [
10]. However, these approaches frequently encounter convergence limitations toward local optima due to their relatively simplistic search patterns, which restrict their exploration capacity in complex solution spaces. The metaheuristic algorithm further combines the stochastic algorithm with the local search of the traditional heuristic algorithm to enhance the ability of the algorithm to jump out of the local optimal solution. It combines the laws of nature and biological laws for better solution results compared to heuristic algorithms [
11].
Swarm Intelligence (SI) [
12] is a significant branch of Artificial Intelligence (AI) that is grounded in the intelligent collective behavior observed in social groups in nature [
13]. Researchers draw inspiration from various species and natural phenomena, leading to the development of various metaheuristic optimization algorithms based on SI [
14]. The Particle Swarm Optimization Algorithm (PSO) [
15] conceptualizes feasible solutions to optimization problems as particles within a search space. Each particle possesses a specific velocity and position, updating these attributes based on its historical optimal position and the best position identified within the group, thereby facilitating the search for superior solutions. In this context, each particle represents a candidate solution within the solution space. Ant Colony Optimization (ACO) [
16] is another metaheuristic optimization algorithm inspired by the foraging behavior of natural ant colonies. Its fundamental principle involves addressing combinatorial optimization problems by simulating the collaborative behavior of ants as they release and perceive pheromones during food-searching activities. This bio-inspired algorithm, governed by simple rules that emulate biological swarm intelligence, provides an effective means of tackling complex optimization challenges. The Firefly Algorithm (FA) [
17] is a metaheuristic driven by swarm intelligence inspired by the bioluminescent attraction mechanisms of fireflies. It utilizes brightness-mediated interactions to guide individuals toward optimal solutions, achieving a balance between global exploration and local exploitation through adaptive attraction dynamics and stochastic movement. Grey Wolf Optimization (GWO) [
18] establishes an optimization search framework based on biological group intelligence by simulating the hierarchy, collaborative predation strategies, and group decision-making mechanisms of grey wolf populations in nature. The algorithm categorizes individual grey wolves into a four-level hierarchical structure with a strict social division of labor:
wolves serve as population leaders responsible for global decision-making;
wolves act as secondary coordinators assisting in decision execution;
wolves form the base population unit and engage in local searches; and
wolves function as followers, completing comprehensive explorations of the solution space. The Whale Optimization Algorithm (WOA) [
19] mimics the feeding behavior of whales, particularly humpback whales, in the ocean, addressing complex optimization problems through strategies of distributed search, autonomous decision-making, and adaptive adjustment. The Sparrow Search Algorithm [
20] simulates the role differentiation of sparrows during foraging, distinguishing between the leader (the sparrow that locates food) and the follower (the sparrow that trails the leader). The leader primarily focuses on finding food, while the follower remains within a certain range of the leader to assist in locating food. Additionally, sparrows make positional adjustments to evade predators, and these behavioral characteristics are abstracted into key steps within the algorithm to optimize the objective function.
However, as asserted by the “No Free Lunch” (NFL) theorem [
21], every algorithm has inherent limitations, and no single algorithm can solve all optimization problems. Consequently, many researchers are dedicated to proposing new algorithms or enhancing existing ones. For instance, Zhu Fang et al. [
22] introduced a good point set strategy during the initialization phase of the dung beetle optimization algorithm to increase population diversity. They also proposed a new nonlinear convergence factor to balance exploration and exploitation within the algorithm, as well as a dynamic balancing strategy between the number of dung beetles spawning and foraging. Additionally, they introduced a strategy based on perturbations from quantum computation and t-distributions to promote the algorithm’s ability of optimization search. Ya Shen et al. [
23] proposed an improved whale optimization algorithm based on multiple population evolution to address the defects of the whale optimization algorithm, which has a slow convergence speed and easily falls into a local optimum. The population is further divided into three sub-populations based on the fitness value: exploratory population, exploitative population, and fitness population, and each sub-population performs a different updating strategy, and then it is experimentally verified that this multiple swarms co-evolutionary strategy can effectively enhance the algorithm’s optimization search capability. Yiying Zhang et al. [
24] proposed an EJaya algorithm, whose local exploitation strategy enhances the local search capability by defining upper and lower bound local attraction points (combining the current optimal, worst, and population mean information), while the global exploration strategy expands the search scope by using random perturbations of historical population information to avoid stagnation at the later stage of the algorithm. Dhargupta et al. [
25] proposed a Selective Opposition-based Grey Wolf Optimizer (SOGWO) that applies dimension-selective opposition learning to enhance exploration efficiency. By identifying
-wolves through Spearman’s correlation analysis, the algorithm strategically targets low-status individuals for opposition learning, thereby reducing redundant search efforts and improving convergence speed while maintaining exploration–exploitation balance. Shuang Liang et al. [
26] proposed an Enhanced Sparrow Search Swarm Optimizer (ESSSO); firstly, ESSSO introduces an adaptive sinusoidal walking strategy (SLM) based on the von Mises distribution, which enables individuals to dynamically adjust the learning rate to improve the evolutionary efficiency; secondly, a learning strategy with roulette wheel selection (LSR) is adopted to maintain the population diversity and prevent premature convergence; furthermore, a two-stage evolutionary strategy (TSE) is designed, which includes a mutation mechanism (AMM) to enhance the local searching ability and accelerate the algorithm’s convergence rate through a selection mechanism (SMS). Through the study, it is shown that the appropriately improved intelligent algorithm has better adaptability and effectiveness in some complex applications, and also has certain advantages compared with other heuristic algorithms.
The Red-billed Blue Magpie Optimizer (RBMO) [
27] is a novel group intelligence-based metaheuristic algorithm inspired by the hunting behavior of red-billed blue magpies, which rely on community cooperation for activities such as searching for food, attacking prey, and food storage. The original paper applied the RBMO algorithm to various domains, including numerical optimization, engineering design problems, and UAV path planning. While the RBMO algorithm boasts a simple structure, minimal parameter requirements, and effective optimization performance, it still faces challenges in achieving optimal results for complex optimization problems. To address these challenges, we propose an improved version of the algorithm, termed the Multi-Strategy Enhanced Red-billed Blue Magpie Optimizer (MRBMO). This enhancement involves the design of a new boundary constraint and the establishment of an innovative prey attack model, which collectively improve the algorithm’s exploration and exploitation balance as well as its overall performance. The primary contributions of this paper are outlined as follows:
A new boundary constraint processing method is designed and verified to accelerate the convergence speed and improve the exploitation ability of the algorithm. The method can also be used for other algorithms that requiring boundary processing.
Inspired by particle swarm optimization (PSO), a new update method is redesigned in the development stage of the RBMO algorithm, which mainly introduces the optimal information of individual history to guide the algorithm to deeply explore the solution space. The step size is controlled by Lévy Flight to avoid the algorithm falling into the local optimal solution.
The proposed MRBMO is evaluated on 41 benchmark functions. And its optimization performance is compared to eight other of the most advanced and high-performance algorithms. The MRBMO was successfully applied to four classical engineering design problems.
This paper is organized as follows.
Section 2 introduces the foundational principles of the Red-billed Blue Magpie Optimization (RBMO) algorithm. In
Section 3, we propose a multi-strategy enhanced version of the algorithm (MRBMO) to address the limitations of the original RBMO.
Section 4 presents a comprehensive experimental comparison between MRBMO and eight other state-of-the-art optimization algorithms. To further validate the practical utility of the proposed improvements,
Section 5 demonstrates the application of MRBMO in real-world engineering design problems. Finally,
Section 6 summarizes the key contributions and findings of this study.
4. Numerical Experiment
The CEC2017 [
33] test set is a set of benchmark functions for evaluating and comparing the performance of intelligent optimization algorithms. The test suite consists of 29 single-objective functions, covering a variety of complexities and diversity, designed to simulate real-world optimization problems and provide a fair evaluation platform for algorithmic researchers. This suite incorporates unimodal functions (F1–F2), which are characterized by a single global optimum devoid of local optima. These functions are primarily employed to evaluate the convergence velocity and exploration efficiency of optimization algorithms. Simple multimodal functions (F3–F9), containing multiple local optima, are used to evaluate the ability of the algorithm to avoid falling into local optima. Hybrid functions (F10–F19), which are composed of multiple basic functions, have a complex search space and test the performance of the algorithm when dealing with nonlinear and discontinuous problems. Composition functions (F20–F29), which are generated by multiple basic functions through rotation, migration and other operations, have higher complexity and are used to evaluate the performance of algorithms in high-dimensional and complex problems. The CEC2022 [
34] benchmark function set is similar to CEC2017, with a total of 12 test functions: Unimodal functions, Basic functions, Hybrid functions, and Composition functions. CEC2022 is further optimized in terms of function complexity, diversity, and real-world problem simulation, enabling a more comprehensive evaluation of algorithm performance in different types of optimization problems. CEC2017 and CEC2022 test functions are detailed in
Table A1 and
Table A2.
4.1. Parameters Sensitivity Testing
In the stage of attacking prey, Lévy Flight is introduced as the step size. The parameter
of Lévy Flight is a constant, and its value significantly influences the performance of the algorithm. To select the parameters
and
in the original algorithm, this paper designs a set of experiments for analysis. Specifically, we consider
and
. The CEC2022 benchmark was utilized in the test set, with the dimension of the vector X set to 20; the maximum number of iterations was 500, the population size was 30, and the experiment was repeated 30 times. The experimental results are summarized in
Table 1, which documents the average performance metrics and the corresponding rankings for each parameter configuration across 30 independent trials. To statistically evaluate the overall algorithmic performance, the Friedman rank sum test was applied to the ranked results, with lower average ranks indicating superior performance. In particular, the combination of parameters
and
consistently achieved optimal performance on multiple benchmark functions. Based on these findings, this configuration is proposed as the recommended parameter setting for the proposed algorithm.
4.2. Ablation Experiment
To ascertain the impact of the enhanced strategy introduced in this paper on the performance of the algorithm, an ablation experiment is meticulously designed in this section. Building upon the foundational RBMO algorithm, the first modification involves refining the out-of-bounds processing method, hereafter referred to as RBMO1. Subsequently, the hunting behavior within the original RBMO algorithm is revised, and a novel update mechanism, designated as RBMO2, is conceptualized. These two enhancements are then integrated to form MRBMO, which represents the ultimate refined algorithm presented in this study. A comparative analysis of RBMO1, RBMO2, and MRBMO against the original RBMO algorithm is conducted to comprehensively assess the magnitude of the improvements achieved.
The CEC2022 benchmark suite was utilized to evaluate algorithm performance, with metrics including minimum and average values, standard deviations, and Friedman rankings calculated for each test function (
Table 2). For the unimodal function F1, both RBMO1 and RBMO2 achieved superior values compared to RBMO, indicating that both strategies enhanced the performance of RBMO to varying degrees. A closer examination reveals that RBMO2 attained not only smaller values but also exhibited greater stability than RBMO1, suggesting that the second strategy contributed more significantly to the improvement of RBMO. For the hybrid function F6, analysis of average values reveals that implementing either strategy individually compromised the algorithm’s ability to escape local optima. However, their synergistic combination demonstrated superior performance, yielding solutions closest to the theoretical optimum. On composite functions F9–F12, the combination of the two strategies enhanced the convergence performance of RBMO. However, there remains a discrepancy from the theoretical optimum. Observing the final mean rank, both strategies enhance the overall performance of RBMO to varying degrees, with their synergistic combination yielding the most favorable outcomes.
4.3. Comparison of MRBMO with State-of-the-Art and Well-Established Algorithms
The comparison algorithms selected for this study encompass a diverse range of optimization techniques. These include the original Red-billed Blue Magpie Optimizer (RBMO), and two advanced algorithms: the Selective Opposition-based Grey Wolf Optimizer (SOGWO) [
25] and the Transient Trigonometric Harris Hawks Optimizer (TTHHO) [
35]. Additionally, the classical Particle Swarm Optimization (PSO) [
15] is included, along with more recently developed methods, such as the Sparrow Search Algorithm (SSA) [
20] and the RIME Optimization Algorithm (RIME) [
36]. Top-performing algorithms from the CEC competitions LSHADE_SPACMA [
37] are also incorporated. The parameter settings for each algorithm were strictly configured according to the recommendations provided in their respective original publications.
To further verify the effectiveness of the improved algorithm, the CEC2017 test function suite was used to fully verify the algorithm performance. First of all, the mean value, standard deviation, and ranking of the MRBMO and the comparison algorithm running 30 times with dimension 30, 50, and 100 were counted, respectively, as shown in
Table 3,
Table 4 and
Table 5.
When the dimension is 30, MRBMO achieved first-ranked average values for the unimodal functions F1 and F2. These results demonstrate the algorithm’s superior exploration capabilities. For simple multimodal functions, MRBMO achieved first-place average scores in F3–F9, outperforming all other algorithms. These results indicate that MRBMO possesses strong local-optimum avoidance capabilities. However, an examination of the standard deviation values reveals significant fluctuations in the results obtained by MRBMO. In Hybrid function F17, MRBMO (2.56E+03) is slightly inferior to RBMO (2.42E+03), but significantly superior to other algorithms. Looking at the standard deviation, it can be found that MRBMO has a lower standard deviation in most functions, and achieves the second or third standard deviation in very few functions. It shows that its performance is highly stable in different problems with little fluctuation, strong adaptability to problem characteristics, and better robustness.
Furthermore, we investigate the impact of dimensionality changes on the performance of MRBMO. Notably, in experiments with 50 and 100 dimensions, MRBMO consistently maintains a stable first-place average ranking across most test functions. This indicates that as the dimensionality increases, its search performance does not deteriorate. Collectively, MRBMO remains highly competitive in addressing complex high-dimensional optimization challenges.
A statistical analysis was conducted on the rankings of each algorithm across 29 benchmark functions in 30 dimensions. As illustrated in
Figure 2, the MRBMO algorithm demonstrates superior performance, securing first place on 26 test functions while attaining second place on three functions (F17, F21 and F28). This dominance can be attributed to its multi-strategy framework, which effectively balances exploration and exploitation through dynamic boundary constraint handling and elite-guided Lévy Flight mechanisms. In contrast, other algorithms, such as SSA and PSO, exhibit significant performance fluctuations, demonstrating competitive results on specific functions while showing limited effectiveness on others.
Furthermore, to rigorously evaluate the comprehensive performance of all algorithms, the Friedman rank-sum test was employed. A bar chart depicting the mean ranks for all algorithms is presented in
Figure 3, where lower numerical rankings indicate superior performance. The results demonstrate that across all tested dimensionalities, the MRBMO algorithm consistently attained the lowest average rank. This indicates the excellent robustness and insensitivity to dimension changes of MRBMO.
In this study, the Wilcoxon rank sum test [
38], a nonparametric statistical method, was utilized to evaluate whether the performance differences between the improved method and the comparative algorithms are statistically significant. The test was conducted at a significance level of 0.05. A
p-value lower than 0.05 indicates a significant difference between the two algorithms being compared, while a
p-value of 0.05 or higher suggests that the performance of the two algorithms is not significantly different and can be considered comparable. The test results are presented in
Table A3,
Table A4 and
Table A5, with data points exhibiting
p-values greater than 0.05 highlighted in bold font. The experimental results are summarized in
Table 6, where
denotes the number of comparisons showing statistically significant differences, and
represents the number of comparisons without statistically significant differences, with the final row presenting the aggregate results. It can be observed that the proposed method demonstrates statistically significant differences compared to competing algorithms in most test functions, and its superiority is substantiated.
4.4. Assessing Convergence Performance
To assess both the accuracy and the convergence speed of the algorithms, convergence curves were plotted for MRBMO and the other algorithms at dimension 30, as illustrated in
Figure 4 and
Figure 5. It is worth noting that in each subplot, the horizontal axis represents the number of iterations, while the vertical axis represents the average convergence curve over 30 runs.
During the optimization process of the unimodal test function F1, MRBMO exhibits faster convergence speed compared to other comparative algorithms. Notably, the MRBMO algorithm benefits from its enhanced global search capability, enabling it to obtain higher-quality feasible solutions. This advantageous characteristic is further validated in the optimization process of another unimodal function F2.
During the optimization processes of simple multimodal functions F3 and F8, the MRBMO algorithm did not demonstrate significant competitive advantages. However, when addressing functions F4, F5, F6, and F7, MRBMO exhibited unique optimization characteristics as other comparative algorithms became trapped in local optima stagnation: Not only did the algorithm avoid convergence stagnation, but it also obtained superior solutions through significantly accelerated convergence, thereby validating the advancement of its local optima avoidance mechanism. Notably, in the optimization scenario of function F9, although MRBMO showed lower convergence rates than the SSA, RIME, and PSO algorithms during the initial phase (iteration count < 300), these contrast algorithms were constrained by premature convergence due to their inability to escape local extremum constraints. In stark contrast, MRBMO demonstrated stronger sustained optimization capability in later stages (iteration count > 300), eventually achieving gradual approximation to the global optimum. This phenomenon highlights the algorithm’s superiority in long-term convergence performance. Such phased performance disparities validate the design advantages of MRBMO in maintaining an exploration–exploitation balance, enabling it to demonstrate enhanced global convergence characteristics when addressing complex optimization problems.
In the hybrid benchmark function tests, the MRBMO algorithm demonstrated significant performance advantages. Specifically, for the F11, F12, F14, F15, and F19 functions, this algorithm exhibited marked superiority over all reference algorithms in both key metrics: convergence rate and solution accuracy. Regarding the optimization of the F10 function, MRBMO showed the fastest convergence characteristics during the initial iteration phase (<100 iterations), with its solution quality rapidly approaching the theoretical optimum. During optimization processes for the F13, F17, and F18 functions, MRBMO achieved comparable convergence performance with the RBMO and LSHADE_SPACMA algorithms, with all three outperforming other comparative algorithms. Regarding the F16 function, in the early iteration stage, MRBMO merely maintained convergence levels comparable to those of other algorithms, without showcasing any remarkable superiority. In the later iteration stage, however, through continuous exploration, MRBMO managed to discover higher-quality solutions.
The proposed MRBMO algorithm exhibits superior convergence rates for multiple composition functions. This is particularly evident in the test cases of F20, F21, F22, F23, F25, and F29. By contrast, for functions F24, F26, F27, and F28, the convergence performance of MRBMO is comparable to that of other comparative algorithms.
4.5. Population Diversity Analysis
Maintaining adequate population diversity throughout the optimization process is a critical factor influencing the performance of metaheuristic optimization algorithms, as highlighted in recent studies. Population diversity refers to the distribution of individuals within the search space and plays a pivotal role in balancing exploration and exploitation. When diversity is relatively high, individuals are more broadly distributed across the search space. This enhances the algorithm’s ability to explore various regions, thereby reducing the likelihood of becoming trapped in local optima. Conversely, lower levels of diversity may result in premature convergence, where the algorithm stagnates around suboptimal solutions due to insufficient exploration.
This section employs Equation (
12) [
39] to quantify population diversity. In this equation, the parameter
represents the degree of dispersion of the population relative to its center of mass during each iteration
t. Specifically,
denotes the value of the
d-th dimension of the
i-th individual at iteration
t. Here, n represents the population size, and D is the total number of dimensions in the search space. By calculating the distance of each individual from the center of mass, this metric provides a numerical measure of how dispersed the population is at any given point in the optimization process.
Figure 6 compares the population diversity dynamics of RBMO and MRBMO on CEC2017 benchmark functions on 30 dimensions. The figure reveals that population diversity generally exhibits a declining trend throughout the optimization process. This decline is expected, as the optimization progresses and the algorithm increasingly focuses on refining solutions near the optimal region. However, a notable difference emerges between RBMO and MRBMO: MRBMO demonstrates consistently lower diversity compared to RBMO. This phenomenon can be attributed to the boundary constraint strategy implemented in MRBMO. This strategy is designed to guide individuals toward feasible regions of the search space, potentially accelerating convergence toward the current optimal solution. While this approach enhances exploitation by focusing the search around promising areas, it simultaneously reduces the breadth of exploration.
4.6. Exploration and Exploitation Evaluation
During evolutionary optimization processes, metaheuristic algorithms employ distinct coordination mechanisms to balance exploration and exploitation among population members. This section uses a methodology [
40] to quantify algorithmic search behavior through tracking dynamic variations in individual dimensional components, thereby enabling quantitative assessment of exploration–exploitation tendencies throughout the evolutionary process [
41]. In this section, the exploration–exploitation balance of the algorithm is evaluated through the formulation defined in Equation (
13). Where
X denotes the population matrix at the
t-th iteration (structured as an
matrix),
quantifies the dimensional diversity metric,
corresponds to the median value of the
d-th dimension across all individuals at iteration
t, while
indicates the
d-th dimensional value of the
i-th individual. The exploration-to-exploitation ratio (ERP/ETP) corresponds to percentage-based metrics derived from these measurements. The experimental findings are comprehensively illustrated in
Figure 7. It can be observed that as the iterations proceed, the algorithm’s exploitation curve progressively ascends towards 100%, while the exploration curve correspondingly diminishes to zero. This pattern indicates that the algorithm predominantly engages in exploratory behaviors during the initial stages, shifting its emphasis to exploitation in subsequent phases.
6. Conclusions
This study proposes a Multi-Strategy Enhanced Red-billed Blue Magpie Optimizer (MRBMO) to address the limitations of convergence efficiency and solution precision inherent in the original RBMO algorithm. The primary innovations of MRBMO include the design of a novel dynamic boundary constraint processing mechanism, the introduction of an elite guidance strategy during the predation stage, and the implementation of a Lévy Flight strategy to adjust the search step size. These synergistic modifications effectively reconcile the exploration–exploitation trade-off in metaheuristic optimization.
Comprehensive experimental evaluations using the standardized CEC2017 and CEC2022 benchmark suites demonstrate that MRBMO outperforms classical optimizers (SOGWO, TTHHO, PSO) and achieves competitive results against a state-of-the-art algorithm (LSHADE_SPACMA). Nonparametric statistical analyses (Friedman rank-sum test and the Wilcoxon signed-rank test) statistically validate MRBMO’s superiority across multiple dimensional configurations (30D, 50D, 100D), confirming its algorithmic robustness. To systematically evaluate the contribution of each methodological enhancement, a comprehensive ablation study was conducted to quantify the individual effects of the proposed algorithmic components on overall performance. Furthermore, an experiment was carried out to assess the method’s capacity for balancing exploration and exploitation mechanisms, while a thorough analysis was performed on the algorithm’s population diversity characteristics to evaluate its evolutionary dynamics.
To validate the performance of the proposed method in practical applications, MRBMO was applied to four engineering design optimization problems: extension/compression spring design, reducer design, welded beam design, and gear train design. The experimental results demonstrate that MRBMO not only achieves superior convergence accuracy and speed but also exhibits enhanced stability and robustness.