Enhanced Red-billed Blue Magpie Optimizer for engineering optimization problems

Abstract

The design of efficient and robust metaheuristic algorithms remains a fundamental challenge in addressing complex optimization problems, particularly those involving high dimensionality, nonlinearity, and multimodal landscapes where traditional methods often struggle. To overcome these difficulties and enhance search effectiveness and adaptability, this paper proposes the Enhanced Red-billed Blue Magpie Optimizer (ERBMO), an improved variant of the recently introduced Red-billed Blue Magpie Optimizer (RBMO). ERBMO integrates three synergistic enhancement mechanisms: Diversity-adaptive weight updating, Periodic pattern search, and Evolutionary probabilistic combinatorial mutation. These components are specifically designed to strengthen population diversity, mitigate premature convergence, and achieve a dynamic balance between exploration and exploitation throughout the optimization process. Comprehensive evaluations demonstrate the superior performance of ERBMO. On the CEC2017 benchmark suite, ERBMO achieves the highest Friedman rankings across all tested dimensions, with average ranks of 1.667 (30D), 1.433 (50D), and 1.133 (100D), consistently outperforming the original RBMO which ranked second. On the CEC2022 benchmark suite, ERBMO again secures the top overall rankings for both 10D (1.833) and 20D (1.833), surpassing nine state-of-the-art algorithms including ESC and RBMO. Ablation study results confirm the effectiveness of each proposed strategy, as the complete ERBMO achieves superior Friedman rankings (2.282 for 10D, 2.140 for 20D) compared to variants with any single strategy removed. Furthermore, when compared against classical algorithms (PSO, DE, CMA-ES) and their advanced variants (SaDE, LSHADE) on CEC2022, ERBMO obtains the best overall rankings (3.550 for 10D, 3.089 for 20D). When applied to four real-world engineering design problems—speed reducer, pressure vessel, step-cone pulley, and hydrostatic thrust bearing—ERBMO consistently ranks first, achieving optimal or near-optimal solutions with superior robustness. The superior performance across both benchmark and practical problems highlights the effectiveness and reliability of the proposed improvements. This work presents a framework for engineering optimization and metaheuristic algorithm design. The source code of ERBMO is publicly available at: https://github.com/x5865/Enhanced-Red-billed-Blue-Magpie-Optimizer-ERBMO-.

Introduction

In the realm of science and engineering, optimization problems are ubiquitous, ranging from structural design and mechanical systems to energy management and industrial scheduling. These real-world challenges often exhibit high dimensionality, nonlinearity, multimodality, and constraint complexity, rendering traditional gradient-based and deterministic methods inefficient or inapplicable—particularly in rugged, discontinuous, or high-dimensional search spaces¹. Consequently, there is a pressing need for intelligent optimizers capable of exploring complex design landscapes without relying on gradient information or extensive parameter tuning. This demand has catalyzed the rapid development of metaheuristic algorithms, which draw inspiration from natural phenomena, biological behaviors, and evolutionary processes to achieve global optimization with remarkable flexibility and resilience^2–5.

Driven by the need for robust, assumption-free search, metaheuristics have evolved from simple stochastic operators into sophisticated, nature-inspired frameworks that coordinate population dynamics through adaptive exploration and exploitation strategies. Metaheuristic algorithms can be categorized into four types: evolutionary algorithms, swarm intelligence algorithms, physics-based algorithms, and human behavior-based algorithms. Evolutionary algorithms include Genetic Algorithms (GA)⁶, Differential Evolution (DE)⁷, Evolution Strategy with Covariance Matrix Adaptation (CMA-ES)⁸, and the recently emerged Graph Neural Evolution (GNE)^9,10. Swarm intelligence algorithms include classic algorithms such as the Whale Optimization Algorithm (WOA)¹¹, Grey Wolf Optimizer (GWO)¹², and Slime Mould Algorithm (SMA)¹³, as well as recently emerged ones like Bighorn Sheep Optimization¹⁴, and the Cape Lynx Optimizer¹⁵. Physics-based algorithms include RIME Optimization Algorithm(RIME)¹⁶, Fata Morgana Algorithm (FATA)¹⁷, Polar Lights Optimizer(PLO)¹⁸, and Wave Optics Optimizer(WOO)¹⁹. Human behavior-based algorithms include Teaching–learning-based Optimization (TLBO)²⁰, Escape (ESC)²¹, and Love Evolution Algorithm (LEA)²². In addition to these, mathematically inspired algorithms can also be considered another category^23,24. As a class of simple and easy-to-implement black-box optimization methods, metaheuristic algorithms are also employed to solve a wide range of real-world application problems, including engineering design optimization, photovoltaic parameter identification, UAV path planning, and signal processing^25–31.

The No Free Lunch Theorem³² asserts that no single algorithm can universally outperform all others across all possible problem domains, implying that performance gains on one class of problems come at the expense of losses on another. This fundamental principle underscores the necessity of continuously refining metaheuristics to enhance their adaptability and robustness in complex scenarios. In this context, the Red-billed Blue Magpie Optimizer (RBMO)³³, a nature-inspired algorithm based on the foraging behavior of Urocissa erythroryncha, has emerged as a promising heuristic. It has demonstrated competitive performance on benchmark suites such as CEC2014 and CEC2017, as well as in engineering design and UAV path planning. Since its introduction, RBMO has attracted growing research interest and has been applied to diverse real-world tasks, including high-efficiency solar thermal design, sales forecasting, energy supply-demand matching, trajectory tracking control for USVs and multi-target path planning^28,34–37. Table 1 summarizes selected improved variants of RBMO.

Table 1.

Some improved metaheuristic algorithms.

Author	Algorithms	Improvement strategies
Wang et al.38	MORBO	Chaos initialization Levy flight strategy Grid-based global opti-mal solution extraction strategy
Zhang et al.39	ES-RBMO	Elite strategy Jump search
Zhou et al.40	LRBMO	Logistic-tent chaotic Adaptive control factor L-BFGS-B strategy
Ye et al.41	MRBMO	Dynamic boundary constraint handling mechanism Elite guidance strategy
Qi et al.42	E-RBMO	Sinusoidal algorithm Adaptive t-distribution mutation operator Dynamic merit mechanism
Zhu et al.43	IRBMO	Elite search behavior Collaborative hunting behavior Memory storage behavior

Nevertheless, these existing RBMO variants still face challenges in balancing exploration and exploitation and in escaping local optima when applied to complex engineering problems. To address these limitations and further enhance performance in demanding applications, this study proposes an Enhanced Red-billed Blue Magpie Optimizer (ERBMO), incorporating three core innovations:

• Diversity-adaptive weight updating: A dynamic weight adjustment mechanism that balances exploration and exploitation based on both population diversity and iterative performance. This strategy promotes exploration when the algorithm’s improvement is insufficient or population diversity is high, and encourages more exploitation otherwise, thereby adaptively responding to stagnation risks.

• Periodic pattern search: A periodically activated search module that enhances algorithmic adaptability by switching between different search patterns. By introducing structured variation at regular intervals, it improves exploitation efficiency without compromising global exploration capability.

• Evolutionary probabilistic combinatorial mutation: A hybrid mutation strategy that combines Gaussian mutation and uniform mutation with dynamically adjusted mutation rates. This mechanism increases search diversity and enables the algorithm to escape local optima in the later stages of iteration.

The primary contributions of this study are summarized as follows:

• Proposal of ERBMO: Integrates diversity-adaptive weight updating, periodic pattern search, and evolutionary probabilistic mutation to enhance search efficiency and robustness in complex optimization problems.

• Ablation Study Validation: Confirms the individual contribution of each proposed mechanism, as the complete ERBMO achieves superior Friedman rankings compared to variants with any single strategy removed.

• Comprehensive Comparison: Demonstrates ERBMO’s superiority over classical algorithms (PSO, DE, CMA-ES) and their advanced variants (e.g., SaDE, LSHADE) on CEC2022 benchmarks, securing the best overall rankings.

• Practical Engineering Validation: Achieves top Friedman rankings on CEC2017/2022 benchmarks and four engineering problems, with source code publicly released to ensure reproducibility.

The remainder of this paper is organized as follows. Section  2 reviews the original RBMO and presents the mathematical formulation of ERBMO. Section  3 analyzes the evolutionary dynamics of ERBMO. Section  4 reports experimental results on the CEC2017 and CEC2022 benchmark suites with statistical analysis. Section  5 evaluates ERBMO on four engineering optimization problems and compares its performance with several competitive metaheuristic algorithms. Section  6 discusses the limitations of the study. Section  7 concludes the paper and suggests future work.

Enhanced Red-billed Blue Magpie Optimizer

Original Red-billed Blue Magpie optimizer

The Red-billed Blue Magpie Optimizer (RBMO) is a metaheuristic algorithm inspired by the social behavior and foraging strategies of red-billed blue magpies. These birds exhibit sophisticated group behaviors, including cooperative foraging, adaptive predation, and food caching, which are abstracted into mathematical operators to solve complex optimization problems. The core philosophy of RBMO lies in balancing global search and exploitation through simulating the dynamic transition between small-group and large-group behaviors.

The operational framework of RBMO consists of four sequential phases: population initialization, food searching, prey attacking, and food caching.

Population initialization

The algorithm starts by generating a random population of search agents within the feasible solution space. For a problem with dimension and population size , the initial position of the agent in the dimension is initialized as:

1

where and denote the upper and lower bounds of the j-th dimension, respectively; is a uniformly distributed random number in the interval [0,1], ensuring uniform coverage of the solution space.

Food searching phase

This phase simulates the exploratory behavior of red-billed blue magpies when scouting for food sources, employing two strategies to enhance search diversity: small-group searching and cluster searching.

Small-group searching (group size ): Agents adjust their positions based on the average position of a randomly selected small group and a randomly chosen individual in the population, facilitating localized exploration:

2

where represents the position of the agent at iteration ; denotes positions of randomly selected agents in the small group; is the position of a randomly selected agent from the entire population; is a uniform random number in [0,1], introducing stochasticity to avoid premature convergence.

Cluster searching (cluster size ): A larger group size enables broader exploration by integrating information from more agents:

3

where (cluster size) is significantly larger than , allowing the algorithm to explore distant regions of the solution space.

Prey attacking phase

This phase models the exploitative behavior of red-billed blue magpies when converging on detected prey, focusing on refining solutions around promising regions through small-group and cluster attacking strategies.

Small-group attacking: Agents move toward the current optimal solution (denoted as ) using localized group information to enhance precision:

4

Cluster attacking: Larger-scale collaboration strengthens local exploitation by aggregating information from a broader group:

5

In the above equations, is the position of the global optimal solution at iteration t; (cooperation factor) is a dynamically adjusted parameter that balances exploration and exploitation, .where is the current iteration, and is the maximum number of iterations. decreases from 1 to 0 as iterations progress, promoting extensive exploration in early stages and focused exploitation in later stages. and are random numbers following a standard normal distribution, introducing controlled randomness to prevent stagnation in local optima.

Food caching phase

To preserve high-quality solutions, agents retain their new positions only if they yield improved fitness, mimicking the food caching behavior of red-billed blue magpies:

6

where denotes the objective function value, with lower values indicating better solutions for minimization problems.

The mathematical model of the proposed enhanced Red-billed Blue Magpie Optimizer

To address the potential limitations of the original RBMO, such as insufficient adaptability to complex optimization landscapes and over-reliance on iteration progress for parameter adjustment, three improved strategies are integrated into the algorithm framework. These strategies—diversity-adaptive weight updating, periodic pattern search, and evolutionary probabilistic combinatorial mutation—are designed to enhance the balance between exploration and exploitation, improve the algorithm’s ability to escape local optima, and strengthen its robustness in solving engineering optimization problems. Each strategy is mathematically modeled to complement the original RBMO’s mechanisms, as detailed in the following subsections.

Diversity-adaptive weight updating

In optimization algorithms, weight parameters are crucial for balancing conflicting factors like exploration and exploitation. In the original RBMO, the cooperation factor , which functions as a critical weighting parameter, is initialized to 0.8. However, its update in the original algorithm depends solely on the iteration ratio (), without considering real-time algorithm performance, such as solution improvement degree, or population diversity, which may lead to mismatches between the search strategy and the problem’s characteristics. To overcome this, a diversity-adaptive weight updating mechanism is proposed, where is dynamically adjusted using three components reflecting the algorithm’s current state, as shown in Eq. (7):

7

The formula integrates three components with specific parameters and roles:

• : is a constant between 0 and 1, acting as the decay coefficient for the current value. This term retains part of the from the previous iteration which serves as the basis for subsequent updates, to maintain continuity in the search strategy and prevent abrupt changes that could disrupt convergence trends.

• : is a weight constant adjusting the contribution of the improvement term, quantifies the difference between the current and previous global best solutions, is a random number introduced to avoid excessive sensitivity to minor improvements. This component adapts to improvement efficiency by reducing its value when the algorithm achieves significant progress and increasing its value when progress stagnates, thereby encouraging a shift from exploitation to exploration as required.

• : is another weight constant modulating the influence of population diversity, is a metric reflecting the spatial distribution spread of the population. This term adjusts according to the level of diversity such that higher diversity increases the term to promote exploration, whereas lower diversity decreases it to focus on exploitation.

When the algorithm achieves significant consecutive improvements, decreases to enhance exploitation of promising regions. If improvements stagnate, increases to stimulate broader exploration. For population diversity, a high diversity level elevates to maintain exploration, while low diversity reduces to strengthen local refinement. This mechanism enables to dynamically match the algorithm’s performance and population state, improving adaptive capability in complex optimization scenarios.

Periodic pattern search

A critical challenge in metaheuristic algorithms lies in avoiding stagnation in local optima, which often stems from fixed search patterns that fail to adapt to the dynamic characteristics of complex optimization landscapes. In the original RBMO, the food searching and prey attacking phases rely on guidance from group-average positions, leading to a tendency for search agents to converge prematurely to local optimal regions. To address this limitation, a periodic pattern search strategy is proposed, which modifies the search mode at fixed iteration intervals to enhance the algorithm’s ability to explore unvisited regions and escape local optima.

This strategy is triggered periodically: specifically, it is executed when the current iteration count satisfies (i.e., every 100 iterations). Its core logic consists of three sequential steps—exploratory moves, pattern moves, and step size adjustment—each mathematically modeled to adapt to the local landscape of the objective function, as elaborated below.

First, exploratory moves are designed to identify promising search directions through probing along the coordinate axes of the solution space. For each dimension , the algorithm evaluates the objective function value after applying a small perturbation in both the positive and negative directions of the axis. If a better fitness value is obtained, the current solution is updated accordingly. The mathematical expressions for exploratory moves are given in Eq. (8):

8

where is the current candidate solution, is the current step size, and denotes the column vector of the identity matrix. Second, pattern moves are executed if exploratory moves successfully find a better solution. This step leverages the direction from the initial solution to the improved solution and applies an acceleration factor to extend the search range, accelerating convergence toward the identified promising direction. The mathematical expressions for pattern moves are shown in Eqs. (9)–(11):

9

10

11

Here, is a temporary variable that stores the solution’s position before the exploratory move and is used to compute the direction of improvement. The variable is then updated to the improved solution obtained from the exploratory move, serving as the new reference point for the subsequent pattern move. The acceleration coefficient is set to 1.2 based on extensive experimental validation.

Third, step size adjustment is performed is found during the exploratory moves. In this case, the step size is reduced to enhance the algorithm’s ability to refine solutions in the local region. This adjustment prevents the strategy from wasting computational resources on ineffective large-step probing and ensures stable convergence toward the optimal solution. The step size adjustment is defined in Eq. (12):

12

In this equation, is the step size reduction rate, fixed at 0.5 to ensure a gradual decrease in step size; and denotes the minimum allowable step size, set to to avoid numerical instability caused by excessively small step sizes.

The core idea of the periodic pattern search lies in periodically updating the search mode during the algorithm’s iteration process to adapt to the local characteristics of the objective function; specifically, after each iteration, the algorithm evaluates the effectiveness of the current search mode and determines whether to switch to a new search mode based on preset rules or performance metrics. This periodic mode switching helps the ERBMO escape the trap of local optima and enables more efficient exploration of the solution space on a global scale.

Evolutionary probabilistic combinatorial mutation

For complex optimization problems with multiple local optima, a single mutation strategy in metaheuristic algorithms often fails to provide sufficient diversity for comprehensive exploration of the solution space, leading to premature convergence to suboptimal regions. To address this limitation of the original RBMO, an evolutionary probabilistic combinatorial mutation strategy is proposed. This strategy synergistically combines Gaussian mutation and uniform mutation within a dynamically adjusted probabilistic framework, thereby balancing global exploration and local exploitation throughout the evolutionary process.

The mutation probability is adaptively increased over iterations to shift the search emphasis from exploration in the early stages to exploitation in the later stages. Specifically, the current mutation probability is updated as:

13

where is the initial mutation rate, is the incremental step per generation, is the maximum allowable mutation rate, and denotes the current iteration number. The increasing trend ensures a higher degree of randomness and diversity in the initial phase, while gradually focusing on fine-tuning promising regions as the algorithm converges.

For each individual in the population, a random number is generated. If this value is less than , a mutation operation is executed. Within the mutation process, Gaussian mutation is applied with probability , defined as:

14

where is the standard deviation, and generates a vector of random numbers drawn from a standard normal distribution . This operator facilitates local refinement by introducing small perturbations around the current solution.

Conversely, with probability , uniform mutation is performed:

15

where produces a vector of uniformly distributed random numbers in [0,1], and , are the lower and upper bounds of the search space, respectively. This operation enables large-scale jumps across the search domain, effectively escaping local optima and enhancing global exploration.

Following mutation and boundary constraint handling, the fitness of the mutated individual is evaluated. If an improved solution is found, it replaces the original individual in the population.

By dynamically adjusting the mutation rate and combining Gaussian and uniform mutation operators, the evolutionary probabilistic combinatorial mutation strategy demonstrates enhanced adaptability and flexibility in the tested optimization scenarios. This helps achieve a better balance between exploration and exploitation, contributing to improved robustness and a stronger capability to locate high-quality solutions in complex multimodal engineering problems. The pseudocode of ERBMO is presented in Algorithm 1.

Computational complexity

In this section, we analyze the computational complexity of the proposed ERBMO. The computational complexity of ERBMO depends on the initialization process, the main loop operations, and the additional enhancement mechanisms. Assume that the number of search agents is n, T denotes the maximum number of iterations, and dim represents the dimension size of the problem. The computational complexity of the solutions’ initialization process is . Within each iteration, the exploration and exploitation phases each require operations for position updates and boundary checks. Additionally, the fitness evaluation for all solutions is performed multiple times per iteration, contributing each. The food storage mechanism involves comparisons and updates with a complexity of . The adaptive parameter update, including diversity calculation, has a complexity of . The combined mutation operation adds . Furthermore, the periodic pattern search, executed every 100 iterations, has a complexity of per execution, which is negligible in the overall complexity. Therefore, the total computational complexity of ERBMO is (Fig. 1).

Fig. 1.

Flowchart of ERBMO.

Convergence behavior analysis

To gain deeper insights into the optimization dynamics and convergence behavior of ERBMO, Fig. 2 presents a convergence analysis across six representative CEC 2017 benchmark functions. Each row represents a test function, while each column highlights a specific characteristic of the search process.

Fig. 2.

Convergence behaviors of ERBMO in the search process.

In each sub-figure, the first column visualizes the two-dimensional landscape of the benchmark function, providing an intuitive understanding of its modality and complexity. The second column displays the final positions of the search agents after optimization, with the global optimum marked by a red dot. Despite their initial dispersion across the search space, the agents converge tightly around the optimal region, indicating strong solution concentration and high refinement intensity in the final stages.

The third column depicts the evolution of the population’s average fitness. Initially high values indicate a diverse initial distribution. As the search progresses, the curve follows a consistent downward trend and stabilizes near iteration 150, reflecting ERBMO’s rapid and stable convergence. The fourth column shows the trajectory of a representative search agent along the first dimension. Early iterations exhibit large fluctuations, indicative of extensive search activity, which gradually diminishes as the agent approaches the vicinity of the optimal solution.

Finally, the fifth column presents the convergence curve, defined as the best fitness value recorded at each iteration. Across all selected functions, covering multimodal and hybrid compositions, the curves exhibit a distinct stepwise pattern, characterized by periods of stagnation followed by abrupt improvements in solution quality. These plateaus indicate temporary entrapment in local optima, while the sudden drops reflect successful escape events, facilitated by the algorithm’s diversity-adaptive mechanisms and combinatorial mutation strategy. The consistent ability of ERBMO to recover from stagnation highlights its robustness in navigating complex and deceptive search landscapes.

Overall, the convergence analysis confirms ERBMO’s rapid convergence, effective escape from local optima, and consistent performance across diverse problem landscapes—demonstrating its strong optimization capability.

Experimental results analysis on benchmark functions

To validate the performance of the proposed ERBMO, extensive experiments are conducted on standardized benchmark suites, comparing ERBMO against several high-performing metaheuristic algorithms. The evaluation focuses on solution accuracy and robustness across optimization problems with diverse characteristics and varying dimensionalities. For this purpose, a rigorous statistical analysis is carried out using the non-parametric Wilcoxon rank-sum test for pairwise comparisons and the Friedman mean rank test for overall algorithm ranking, ensuring an objective and comprehensive assessment of comparative performance.

Experimental setup

To evaluate the performance of the proposed ERBMO, two widely used benchmark suites, CEC 2017⁴⁴ and CEC 2022⁴⁵, are selected for experimental testing. These test functions are commonly adopted in the evolutionary computation community to assess the effectiveness of optimization algorithms. The CEC 2017 suite includes 30 functions (F1–F30) with different characteristics, such as unimodal, multimodal, hybrid, and composite types. The CEC 2022 suite contains 12 complex functions (F1–F12), most of which combine multiple function types to increase difficulty. Detailed information about these functions is provided in Tables 2 and 3.

Table 2.

Review the CEC 2017 test suite.

Type	ID	Description	Dim
Unimodal	CEC2017-F1	Shifted and rotated bent cigar function	30/50/100	100
	CEC2017-F2	Shifted and rotated sum of different power function	30/50/100	200
	CEC2017-F3	Shifted and rotated Zakharov function	30/50/100	300
Multimodal	CEC2017-F4	Shifted and rotated Rosenbrock’s function	30/50/100	400
	CEC2017-F5	Shifted and rotated Rastrigin’s function	30/50/100	500
	CEC2017-F6	Shifted and rotated expanded Scaffer’s F6 function	30/50/100	600
	CEC2017-F7	Shifted and rotated Lunacek Bi_Rastrigin function	30/50/100	700
	CEC2017-F8	Shifted and rotated non-continuous Rastrigin’s function	30/50/100	800
	CEC2017-F9	Shifted and rotated Levy function	30/50/100	900
	CEC2017-F10	Shifted and rotated Schwefel’s function	30/50/100	1000
Hybrid	CEC2017-F11	Hybrid function 1 (N = 3)	30/50/100	1100
	CEC2017-F12	Hybrid function 2 (N = 3)	30/50/100	1200
	CEC2017-F13	Hybrid function 3 (N = 3)	30/50/100	1300
	CEC2017-F14	Hybrid function 4 (N = 4)	30/50/100	1400
	CEC2017-F15	Hybrid function 5 (N = 4)	30/50/100	1500
	CEC2017-F16	Hybrid function 6 (N = 4)	30/50/100	1600
	CEC2017-F17	Hybrid function 6 (N = 5)	30/50/100	1700
	CEC2017-F18	Hybrid function 6 (N = 5)	30/50/100	1800
	CEC2017-F19	Hybrid function 6 (N = 5)	30/50/100	1900
	CEC2017-F20	Hybrid function 6 (N = 6)	30/50/100	2000
Composition	CEC2017-F21	Composition function 1 (N = 5)	30/50/100	2100
	CEC2017-F22	Composition function 2 (N = 5)	30/50/100	2200
	CEC2017-F23	Composition function 3 (N = 5)	30/50/100	2300
	CEC2017-F24	Composition function 4 (N = 5)	30/50/100	2400
	CEC2017-F25	Composition function 5 (N = 3)	30/50/100	2500
	CEC2017-F26	Composition function 6 (N = 3)	30/50/100	2600
	CEC2017-F27	Composition function 7 (N = 5)	30/50/100	2700
	CEC2017-F28	Composition function 8 (N = 5)	30/50/100	2800
	CEC2017-F29	Composition function 9 (N = 5)	30/50/100	2900
	CEC2017-F30	Composition function 10 (N = 3)	30/50/100	3000

Search Range: [− 100,100]

Table 3.

Review the CEC 2022 test suite.

Type	ID	Description	Dim
Unimodal Function	CEC2022-F1	Shifted and full Rotated Zakharov Function	10/20	300
Basic Functions	CEC2022-F2	Shifted and full Rotated Zakharov Function	10/20	400
	CEC2022-F3	Shifted and full Rotated Expanded Schaffer’s F6 Function	10/20	600
	CEC2022-F4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	10/20	800
	CEC2022-F5	Shifted and full Rotated Levy Function	10/20	900
Hybrid Functions	CEC2022-F6	Hybrid Function 1 (N = 3)	10/20	1800
	CEC2022-F7	Hybrid Function 2 (N = 6)	10/20	2000
	CEC2022-F8	Hybrid Function 3 (N = 5)	10/20	2200
Composition Functions	CEC2022-F9	Composition Function 1 (N = 5)	10/20	2300
	CEC2022-F10	Composition Function 2 (N = 4)	10/20	2400
	CEC2022-F11	Composition Function 3 (N = 5)	10/20	2600
	CEC2022-F12	Composition Function 4 (N = 6)	10/20	2700

Search Range: [− 100,100]

Experiments are conducted on CEC 2017 functions F1–F30 with dimensions 30, 50, and 100, and on CEC 2022 functions F1–F12 with dimensions 10 and 20. For each function, all algorithms are independently run 30 times with a maximum of 500 iterations per run to ensure consistent and comparable results.

ERBMO is compared with nine well-known or recently developed metaheuristic algorithms. These include Harris Hawks Optimization (HHO)⁴⁶, Whale Optimization Algorithm (WOA)¹¹, Dung Beetle Optimizer (DBO)⁴⁷, Young’s Double-Slit Experiment Optimizer (YDSE)⁴⁸, Great Wall Construction Algorithm (GWCA)⁴⁹, Newton-Raphson-based Optimizer (NBRO)⁵⁰, Beluga Whale Optimization (BWO)⁵¹, Escape Algorithm (ESC)²¹, and the original Red-billed Blue Magpie Optimizer (RBMO)³³. The parameter settings for all algorithms are listed in Table 4, using default or recommended values from their original studies to ensure a fair comparison. The parameters of ERBMO are empirically set during the tuning process. Among them, α, β, and γ are adaptive weights that control the update process of CF. A larger α tends to keep CF closer to its original value, a larger β increases the degree to which fitness improvement reduces CF, and a larger γ amplifies the increase of CF when population diversity improves. Additionally, larger values of δ, b, and a enhance the exploration capability of ERBMO during its periodic pattern search.

Table 4.

Parameter settings of the chosen algorithms.

Algorithms	Name of the parameter	Value of the parameter
HHO		[-1,1]	30	500
WOA		2	30	500
DBO		0.2	30	500
YDSE		,1	30	500
GWCA		9,6,8.3,3,exp(3),exp(2)	30	500
NBRO		0.6	30	500
BWO		[0.1 0.05]	30	500
ESC		0.15,0.15,0.35,0.5,5	30	500
RBMO		[2,5], [10,n], [0,1],0.5	30	500
ERBMO		0.95,0.05,0.01,0.5,1.2,0.5	30	500

All experiments are implemented in MATLAB R2023a, running on a laptop equipped with an Intel(R) Core(TM) i7-12700 H processor (2.30 GHz) and 16 GB of RAM.

Experimental results analysis on the CEC2017 benchmark suite

Mean and variance analysis of the CEC2017 benchmark suite

The performance of the proposed ERBMO is evaluated against nine competing algorithms on the CEC 2017 benchmark suite across dimensions of 30, 50, and 100. The mean and standard deviation (Std) of the final fitness values, obtained from 30 independent runs with a maximum of 500 iterations per run, are recorded and presented in Tables 5, 6 and 7, respectively. The best result for each test instance is highlighted in bold.

Table 5.

Comparison of results on CEC2017 (Dim = 30).

ID	Index	HHO	WOA	DBO	YDSE	GWCA	NBRO	BWO	ESC	RBMO	ERBMO
F1	Ave	4.596E + 08	5.342E + 09	2.657E + 08	1.546E + 09	9.930E + 06	1.751E + 10	5.349E + 10	2.329E + 07	1.776E + 04	1.000E + 02
	Std	2.596E + 08	1.773E + 09	1.537E + 08	5.129E + 08	2.220E + 07	5.075E + 09	4.248E + 09	1.226E + 08	2.494E + 04	4.502E-10
F2	Ave	1.197E + 34	1.787E + 36	5.445E + 31	1.366E + 27	2.682E + 33	2.234E + 36	4.948E + 46	3.380E + 25	9.021E + 20	1.545E + 16
	Std	5.647E + 34	5.395E + 36	1.994E + 32	4.605E + 27	6.675E + 33	6.413E + 36	2.531E + 47	1.677E + 26	4.922E + 21	8.322E + 16
F3	Ave	5.859E + 04	2.553E + 05	8.775E + 04	3.160E + 04	4.833E + 04	5.656E + 04	8.019E + 04	9.491E + 04	2.006E + 03	3.000E + 02
	Std	6.156E + 03	6.474E + 04	1.683E + 04	8.889E + 03	1.916E + 04	1.298E + 04	6.181E + 03	2.223E + 04	9.234E + 02	1.599E-05
F4	Ave	7.353E + 02	1.274E + 03	7.111E + 02	6.369E + 02	5.593E + 02	2.117E + 03	1.303E + 04	5.332E + 02	5.085E + 02	4.614E + 02
	Std	1.064E + 02	3.676E + 02	1.780E + 02	4.476E + 01	5.117E + 01	1.023E + 03	1.096E + 03	2.959E + 01	2.717E + 01	4.199E + 01
F5	Ave	7.626E + 02	8.539E + 02	7.585E + 02	7.183E + 02	6.765E + 02	8.359E + 02	9.338E + 02	5.976E + 02	5.738E + 02	5.743E + 02
	Std	3.692E + 01	5.238E + 01	4.982E + 01	1.565E + 01	4.615E + 01	4.245E + 01	2.146E + 01	2.605E + 01	1.720E + 01	1.695E + 01
F6	Ave	6.680E + 02	6.821E + 02	6.458E + 02	6.432E + 02	6.423E + 02	6.736E + 02	6.917E + 02	6.001E + 02	6.019E + 02	6.042E + 02
	Std	5.830E + 00	1.455E + 01	1.213E + 01	7.955E + 00	7.843E + 00	6.903E + 00	5.243E + 00	2.141E-01	2.466E + 00	2.244E + 00
F7	Ave	1.292E + 03	1.318E + 03	1.021E + 03	1.024E + 03	1.141E + 03	1.208E + 03	1.394E + 03	8.559E + 02	8.098E + 02	8.152E + 02
	Std	6.997E + 01	1.144E + 02	1.048E + 02	2.155E + 01	1.045E + 02	6.782E + 01	3.616E + 01	1.761E + 01	2.357E + 01	2.049E + 01
F8	Ave	9.895E + 02	1.071E + 03	1.019E + 03	1.005E + 03	9.403E + 02	1.081E + 03	1.149E + 03	8.973E + 02	8.730E + 02	8.705E + 02
	Std	1.935E + 01	5.602E + 01	6.060E + 01	2.389E + 01	3.613E + 01	2.602E + 01	1.374E + 01	2.349E + 01	1.943E + 01	1.684E + 01
F9	Ave	8.943E + 03	1.193E + 04	6.098E + 03	5.091E + 03	4.024E + 03	7.899E + 03	1.114E + 04	9.245E + 02	9.941E + 02	1.042E + 03
	Std	1.136E + 03	3.671E + 03	2.097E + 03	1.464E + 03	1.163E + 03	1.480E + 03	8.452E + 02	2.488E + 01	1.077E + 02	1.258E + 02
F10	Ave	6.150E + 03	7.527E + 03	6.516E + 03	7.417E + 03	5.210E + 03	8.042E + 03	8.812E + 03	7.102E + 03	5.190E + 03	5.512E + 03
	Std	7.834E + 02	7.587E + 02	9.455E + 02	3.701E + 02	7.094E + 02	5.864E + 02	3.175E + 02	4.684E + 02	7.292E + 02	7.905E + 02
F11	Ave	1.606E + 03	9.439E + 03	1.747E + 03	1.411E + 03	1.333E + 03	2.613E + 03	8.498E + 03	1.304E + 03	1.214E + 03	1.219E + 03
	Std	1.940E + 02	3.638E + 03	3.133E + 02	7.324E + 01	1.000E + 02	7.557E + 02	1.140E + 03	2.245E + 02	3.593E + 01	4.358E + 01
F12	Ave	9.002E + 07	4.544E + 08	6.407E + 07	1.592E + 07	2.028E + 06	1.647E + 09	1.140E + 10	2.211E + 06	1.569E + 05	2.084E + 04
	Std	9.697E + 07	2.588E + 08	7.360E + 07	9.637E + 06	2.562E + 06	7.831E + 08	1.930E + 09	2.095E + 06	1.797E + 05	1.607E + 04
F13	Ave	1.042E + 06	2.136E + 07	1.356E + 07	1.330E + 05	2.798E + 04	4.208E + 08	7.130E + 09	1.970E + 04	1.752E + 04	1.761E + 03
	Std	4.553E + 05	5.149E + 07	2.385E + 07	5.336E + 04	2.685E + 04	3.640E + 08	1.933E + 09	1.209E + 04	1.845E + 04	2.520E + 02
F14	Ave	1.173E + 06	2.290E + 06	2.708E + 05	1.587E + 03	4.238E + 04	1.195E + 05	4.476E + 06	2.920E + 05	1.504E + 03	1.509E + 03
	Std	1.237E + 06	2.568E + 06	3.199E + 05	2.486E + 01	4.098E + 04	2.526E + 05	2.484E + 06	3.653E + 05	3.199E + 01	3.166E + 01
F15	Ave	1.202E + 05	3.120E + 06	8.071E + 04	6.048E + 03	1.216E + 04	2.439E + 06	3.287E + 08	7.654E + 03	2.525E + 03	1.695E + 03
	Std	5.570E + 04	4.175E + 06	6.532E + 04	2.243E + 03	1.127E + 04	5.151E + 06	1.850E + 08	5.981E + 03	1.398E + 03	1.227E + 02
F16	Ave	3.632E + 03	4.063E + 03	3.359E + 03	3.241E + 03	2.845E + 03	3.929E + 03	5.647E + 03	2.494E + 03	2.573E + 03	2.387E + 03
	Std	5.199E + 02	6.235E + 02	3.616E + 02	2.041E + 02	3.367E + 02	3.408E + 02	4.422E + 02	2.709E + 02	2.939E + 02	2.976E + 02
F17	Ave	2.705E + 03	2.884E + 03	2.630E + 03	2.236E + 03	2.466E + 03	2.676E + 03	4.179E + 03	1.904E + 03	2.059E + 03	2.034E + 03
	Std	3.249E + 02	3.909E + 02	3.068E + 02	1.409E + 02	2.329E + 02	2.692E + 02	6.339E + 02	1.120E + 02	1.480E + 02	1.146E + 02
F18	Ave	2.085E + 06	1.768E + 07	5.074E + 06	1.116E + 04	4.092E + 05	1.952E + 06	5.234E + 07	8.477E + 05	2.318E + 03	2.117E + 03
	Std	2.524E + 06	1.670E + 07	7.310E + 06	4.824E + 03	3.912E + 05	2.206E + 06	2.787E + 07	7.421E + 05	2.900E + 02	1.250E + 02
F19	Ave	1.851E + 06	1.661E + 07	1.064E + 07	6.214E + 03	1.207E + 04	1.191E + 07	4.368E + 08	7.535E + 03	2.133E + 03	2.009E + 03
	Std	1.554E + 06	2.210E + 07	3.920E + 07	2.473E + 03	1.202E + 04	1.240E + 07	2.212E + 08	8.169E + 03	3.667E + 02	5.038E + 01
F20	Ave	2.817E + 03	2.976E + 03	2.768E + 03	2.669E + 03	2.715E + 03	2.782E + 03	3.043E + 03	2.285E + 03	2.338E + 03	2.395E + 03
	Std	1.912E + 02	2.067E + 02	2.262E + 02	1.334E + 02	2.333E + 02	1.509E + 02	1.433E + 02	1.113E + 02	1.625E + 02	1.406E + 02
F21	Ave	2.573E + 03	2.660E + 03	2.555E + 03	2.494E + 03	2.435E + 03	2.603E + 03	2.739E + 03	2.398E + 03	2.376E + 03	2.377E + 03
	Std	5.631E + 01	6.428E + 01	4.985E + 01	2.336E + 01	3.283E + 01	4.387E + 01	3.054E + 01	2.604E + 01	1.919E + 01	2.566E + 01
F22	Ave	7.252E + 03	8.204E + 03	4.551E + 03	8.064E + 03	5.269E + 03	5.982E + 03	8.996E + 03	3.526E + 03	4.819E + 03	4.245E + 03
	Std	1.444E + 03	1.865E + 03	2.458E + 03	1.730E + 03	2.161E + 03	2.249E + 03	4.654E + 02	2.452E + 03	2.018E + 03	2.175E + 03
F23	Ave	3.325E + 03	3.155E + 03	2.998E + 03	2.901E + 03	2.909E + 03	3.041E + 03	3.335E + 03	2.740E + 03	2.768E + 03	2.771E + 03
	Std	2.146E + 02	9.945E + 01	9.289E + 01	2.625E + 01	7.986E + 01	7.098E + 01	7.009E + 01	2.163E + 01	4.445E + 01	3.690E + 01
F24	Ave	3.494E + 03	3.291E + 03	3.161E + 03	3.054E + 03	3.070E + 03	3.212E + 03	3.601E + 03	2.943E + 03	2.942E + 03	2.930E + 03
	Std	1.434E + 02	1.028E + 02	7.828E + 01	2.262E + 01	9.248E + 01	7.869E + 01	8.072E + 01	1.916E + 01	4.320E + 01	3.740E + 01
F25	Ave	3.002E + 03	3.219E + 03	2.983E + 03	3.011E + 03	2.968E + 03	3.398E + 03	4.536E + 03	2.933E + 03	2.906E + 03	2.887E + 03
	Std	3.958E + 01	8.503E + 01	5.855E + 01	3.337E + 01	4.936E + 01	1.978E + 02	1.416E + 02	2.600E + 01	1.649E + 01	1.304E + 00
F26	Ave	8.196E + 03	8.691E + 03	6.897E + 03	5.906E + 03	6.067E + 03	7.908E + 03	1.084E + 04	4.430E + 03	4.805E + 03	4.708E + 03
	Std	8.203E + 02	1.347E + 03	8.918E + 02	3.522E + 02	1.367E + 03	1.193E + 03	6.644E + 02	2.995E + 02	3.222E + 02	4.981E + 02
F27	Ave	3.648E + 03	3.452E + 03	3.334E + 03	3.287E + 03	3.332E + 03	3.470E + 03	4.050E + 03	3.230E + 03	3.236E + 03	3.228E + 03
	Std	2.724E + 02	1.275E + 02	5.670E + 01	2.209E + 01	6.153E + 01	9.485E + 01	1.565E + 02	1.334E + 01	3.052E + 01	1.990E + 01
F28	Ave	3.520E + 03	4.012E + 03	3.491E + 03	3.444E + 03	3.323E + 03	4.193E + 03	6.590E + 03	3.329E + 03	3.264E + 03	3.151E + 03
	Std	1.343E + 02	6.820E + 02	2.643E + 02	5.364E + 01	4.883E + 01	4.075E + 02	2.985E + 02	4.933E + 01	5.128E + 01	6.561E + 01
F29	Ave	5.126E + 03	5.385E + 03	4.561E + 03	4.370E + 03	4.474E + 03	5.092E + 03	7.296E + 03	3.619E + 03	3.800E + 03	3.829E + 03
	Std	4.204E + 02	5.975E + 02	3.933E + 02	1.570E + 02	3.960E + 02	4.925E + 02	7.034E + 02	1.382E + 02	2.124E + 02	1.701E + 02
F30	Ave	9.246E + 06	6.885E + 07	4.251E + 06	5.968E + 05	1.718E + 04	7.289E + 07	1.161E + 09	3.440E + 04	2.208E + 04	1.067E + 04
	Std	7.316E + 06	6.995E + 07	5.738E + 06	4.783E + 05	1.047E + 04	3.827E + 07	4.557E + 08	2.615E + 04	3.425E + 04	4.024E + 03

Table 6.

Comparison of results on CEC2017 (Dim = 50).

ID	Index	HHO	WOA	DBO	YDSE	GWCA	NBRO	BWO	ESC	RBMO	ERBMO
F1	Ave	6.105E + 09	2.154E + 10	1.255E + 10	1.262E + 10	3.011E + 09	5.275E + 10	1.069E + 11	2.447E + 09	3.257E + 07	1.000E + 02
	Std	2.240E + 09	5.451E + 09	2.111E + 10	2.795E + 09	2.701E + 09	8.451E + 09	4.350E + 09	1.788E + 09	2.421E + 07	1.687E-09
F2	Ave	3.976E + 65	1.634E + 80	6.608E + 66	1.643E + 54	3.398E + 69	2.244E + 65	7.306E + 79	1.145E + 52	1.338E + 44	2.430E + 37
	Std	1.513E + 66	7.817E + 80	2.470E + 67	5.589E + 54	1.279E + 70	8.119E + 65	1.533E + 80	6.136E + 52	4.718E + 44	1.169E + 38
F3	Ave	1.732E + 05	3.076E + 05	2.702E + 05	1.077E + 05	2.222E + 05	1.695E + 05	2.393E + 05	2.140E + 05	3.624E + 04	3.000E + 02
	Std	1.552E + 04	8.869E + 04	5.539E + 04	1.704E + 04	6.777E + 04	3.353E + 04	3.556E + 04	4.043E + 04	1.077E + 04	2.028E-02
F4	Ave	1.959E + 03	5.349E + 03	1.511E + 03	1.848E + 03	9.954E + 02	8.370E + 03	3.382E + 04	9.334E + 02	6.597E + 02	4.575E + 02
	Std	5.898E + 02	1.631E + 03	1.197E + 03	3.611E + 02	2.509E + 02	2.350E + 03	3.435E + 03	2.152E + 02	6.519E + 01	5.416E + 01
F5	Ave	9.347E + 02	1.133E + 03	9.533E + 02	9.860E + 02	8.235E + 02	1.106E + 03	1.195E + 03	7.459E + 02	7.008E + 02	6.721E + 02
	Std	3.289E + 01	8.150E + 01	1.046E + 02	3.194E + 01	6.285E + 01	3.874E + 01	2.098E + 01	4.843E + 01	3.050E + 01	4.824E + 01
F6	Ave	6.798E + 02	7.008E + 02	6.658E + 02	6.657E + 02	6.544E + 02	6.910E + 02	7.030E + 02	6.033E + 02	6.080E + 02	6.173E + 02
	Std	4.738E + 00	1.285E + 01	1.180E + 01	7.012E + 00	8.156E + 00	8.302E + 00	3.195E + 00	1.161E + 00	3.797E + 00	5.974E + 00
F7	Ave	1.871E + 03	1.915E + 03	1.405E + 03	1.466E + 03	1.704E + 03	1.804E + 03	1.988E + 03	1.081E + 03	1.004E + 03	1.009E + 03
	Std	9.442E + 01	1.409E + 02	1.375E + 02	7.662E + 01	2.100E + 02	1.060E + 02	4.084E + 01	3.668E + 01	6.469E + 01	5.781E + 01
F8	Ave	1.249E + 03	1.384E + 03	1.320E + 03	1.276E + 03	1.127E + 03	1.433E + 03	1.510E + 03	1.042E + 03	9.978E + 02	9.660E + 02
	Std	3.647E + 01	6.758E + 01	1.089E + 02	3.541E + 01	5.882E + 01	4.153E + 01	2.642E + 01	5.071E + 01	4.673E + 01	3.928E + 01
F9	Ave	3.135E + 04	3.664E + 04	2.775E + 04	2.181E + 04	1.130E + 04	2.936E + 04	4.016E + 04	2.076E + 03	2.891E + 03	3.313E + 03
	Std	2.945E + 03	1.076E + 04	7.208E + 03	3.582E + 03	2.337E + 03	3.074E + 03	2.660E + 03	6.705E + 02	9.975E + 02	1.379E + 03
F10	Ave	1.027E + 04	1.312E + 04	1.174E + 04	1.377E + 04	8.821E + 03	1.437E + 04	1.499E + 04	1.296E + 04	9.346E + 03	9.047E + 03
	Std	1.144E + 03	1.122E + 03	2.407E + 03	5.685E + 02	9.209E + 02	9.402E + 02	5.292E + 02	5.609E + 02	1.151E + 03	8.994E + 02
F11	Ave	3.079E + 03	9.540E + 03	5.319E + 03	3.094E + 03	2.668E + 03	1.055E + 04	2.295E + 04	2.418E + 03	1.454E + 03	1.318E + 03
	Std	6.548E + 02	2.485E + 03	3.492E + 03	8.826E + 02	1.445E + 03	2.321E + 03	1.827E + 03	8.800E + 02	1.022E + 02	6.124E + 01
F12	Ave	9.959E + 08	4.453E + 09	1.017E + 09	6.488E + 08	6.590E + 07	1.659E + 10	6.100E + 10	1.105E + 08	8.412E + 06	9.583E + 05
	Std	7.802E + 08	1.878E + 09	7.699E + 08	2.630E + 08	1.063E + 08	5.493E + 09	1.044E + 10	1.897E + 08	6.948E + 06	1.198E + 06
F13	Ave	4.595E + 07	4.955E + 08	1.901E + 08	1.353E + 07	3.745E + 05	4.233E + 09	3.650E + 10	7.261E + 05	1.923E + 04	2.384E + 03
	Std	8.583E + 07	2.777E + 08	3.019E + 08	5.225E + 06	1.291E + 06	2.602E + 09	6.638E + 09	2.374E + 06	1.086E + 04	4.615E + 02
F14	Ave	8.031E + 06	7.469E + 06	4.473E + 06	5.472E + 03	3.167E + 05	3.294E + 06	6.394E + 07	9.713E + 05	1.780E + 03	1.730E + 03
	Std	7.624E + 06	7.352E + 06	3.925E + 06	2.203E + 03	2.586E + 05	2.542E + 06	2.333E + 07	9.993E + 05	1.144E + 02	9.192E + 01
F15	Ave	4.516E + 06	6.880E + 07	5.580E + 07	2.621E + 05	1.118E + 04	3.661E + 08	6.522E + 09	1.235E + 04	1.285E + 04	1.954E + 03
	Std	9.248E + 06	9.520E + 07	1.909E + 08	1.548E + 05	6.515E + 03	2.064E + 08	1.466E + 09	5.955E + 03	6.880E + 03	1.721E + 02
F16	Ave	4.957E + 03	6.479E + 03	4.814E + 03	4.690E + 03	3.689E + 03	5.802E + 03	8.642E + 03	3.468E + 03	3.398E + 03	3.167E + 03
	Std	6.220E + 02	1.062E + 03	7.172E + 02	3.144E + 02	5.125E + 02	6.323E + 02	6.992E + 02	4.070E + 02	4.441E + 02	4.323E + 02
F17	Ave	3.912E + 03	4.735E + 03	4.158E + 03	3.709E + 03	3.800E + 03	4.653E + 03	7.129E + 03	2.977E + 03	3.069E + 03	3.247E + 03
	Std	3.997E + 02	8.017E + 02	4.522E + 02	1.987E + 02	3.565E + 02	5.313E + 02	1.406E + 03	2.008E + 02	3.398E + 02	3.709E + 02
F18	Ave	7.635E + 06	6.670E + 07	1.219E + 07	1.820E + 05	3.191E + 06	2.063E + 07	1.369E + 08	5.134E + 06	6.844E + 04	2.469E + 04
	Std	6.233E + 06	6.311E + 07	1.048E + 07	1.089E + 05	2.723E + 06	1.691E + 07	4.229E + 07	4.149E + 06	5.979E + 04	1.464E + 04
F19	Ave	2.304E + 06	1.739E + 07	8.344E + 06	3.174E + 05	1.894E + 04	2.611E + 08	3.460E + 09	2.454E + 04	1.370E + 04	2.090E + 03
	Std	2.037E + 06	1.443E + 07	9.385E + 06	1.838E + 05	1.260E + 04	2.657E + 08	9.352E + 08	1.386E + 04	1.360E + 04	5.391E + 01
F20	Ave	3.553E + 03	4.083E + 03	3.826E + 03	3.775E + 03	3.504E + 03	3.922E + 03	4.131E + 03	3.303E + 03	3.086E + 03	3.257E + 03
	Std	3.285E + 02	2.862E + 02	3.630E + 02	1.874E + 02	4.178E + 02	2.746E + 02	1.379E + 02	2.918E + 02	2.519E + 02	3.391E + 02
F21	Ave	2.973E + 03	3.046E + 03	2.872E + 03	2.755E + 03	2.649E + 03	2.977E + 03	3.205E + 03	2.548E + 03	2.508E + 03	2.476E + 03
	Std	1.180E + 02	1.104E + 02	7.624E + 01	3.593E + 01	6.013E + 01	6.416E + 01	5.538E + 01	4.141E + 01	4.549E + 01	3.482E + 01
F22	Ave	1.221E + 04	1.467E + 04	1.346E + 04	1.529E + 04	1.017E + 04	1.612E + 04	1.697E + 04	1.445E + 04	1.100E + 04	1.017E + 04
	Std	9.346E + 02	1.161E + 03	1.974E + 03	4.191E + 02	1.190E + 03	7.736E + 02	4.006E + 02	6.070E + 02	1.164E + 03	1.886E + 03
F23	Ave	4.048E + 03	3.915E + 03	3.562E + 03	3.286E + 03	3.372E + 03	3.626E + 03	4.104E + 03	2.952E + 03	3.062E + 03	3.005E + 03
	Std	1.918E + 02	2.074E + 02	1.483E + 02	4.716E + 01	1.358E + 02	1.095E + 02	8.628E + 01	4.998E + 01	8.870E + 01	7.990E + 01
F24	Ave	4.336E + 03	3.848E + 03	3.714E + 03	3.495E + 03	3.485E + 03	3.784E + 03	4.507E + 03	3.192E + 03	3.205E + 03	3.185E + 03
	Std	2.207E + 02	1.355E + 02	1.284E + 02	5.897E + 01	1.232E + 02	1.118E + 02	1.240E + 02	4.736E + 01	9.508E + 01	7.518E + 01
F25	Ave	3.791E + 03	5.349E + 03	3.427E + 03	4.094E + 03	3.517E + 03	7.152E + 03	1.458E + 04	3.564E + 03	3.145E + 03	3.029E + 03
	Std	1.996E + 02	6.776E + 02	1.449E + 02	2.907E + 02	2.375E + 02	1.127E + 03	6.169E + 02	2.620E + 02	4.123E + 01	4.097E + 01
F26	Ave	1.226E + 04	1.490E + 04	1.098E + 04	9.490E + 03	1.028E + 04	1.262E + 04	1.688E + 04	5.809E + 03	6.459E + 03	6.661E + 03
	Std	1.016E + 03	1.339E + 03	1.655E + 03	4.347E + 02	1.443E + 03	1.620E + 03	4.375E + 02	6.450E + 02	9.554E + 02	1.089E + 03
F27	Ave	5.089E + 03	4.808E + 03	4.096E + 03	3.857E + 03	3.845E + 03	4.432E + 03	6.250E + 03	3.562E + 03	3.499E + 03	3.472E + 03
	Std	5.928E + 02	4.673E + 02	4.727E + 02	1.317E + 02	1.692E + 02	3.098E + 02	3.289E + 02	8.238E + 01	1.214E + 02	1.283E + 02
F28	Ave	5.042E + 03	6.124E + 03	5.382E + 03	4.927E + 03	4.196E + 03	7.268E + 03	1.251E + 04	4.534E + 03	3.679E + 03	3.299E + 03
	Std	4.323E + 02	5.995E + 02	1.885E + 03	4.651E + 02	4.391E + 02	6.391E + 02	5.622E + 02	4.393E + 02	5.695E + 02	1.581E + 01
F29	Ave	7.620E + 03	9.115E + 03	6.336E + 03	6.075E + 03	5.568E + 03	8.437E + 03	3.380E + 04	4.017E + 03	4.629E + 03	4.417E + 03
	Std	1.090E + 03	1.670E + 03	8.538E + 02	3.816E + 02	4.931E + 02	1.064E + 03	1.680E + 04	2.381E + 02	4.512E + 02	4.455E + 02
F30	Ave	1.257E + 08	3.450E + 08	9.449E + 07	4.633E + 07	1.814E + 06	6.881E + 08	5.125E + 09	1.001E + 07	3.064E + 06	1.262E + 06
	Std	5.042E + 07	1.405E + 08	1.747E + 08	1.323E + 07	7.981E + 05	3.012E + 08	1.200E + 09	4.503E + 06	3.177E + 06	3.269E + 05

Table 7.

Comparison of results on CEC2017 (Dim = 100).

ID	Index	HHO	WOA	DBO	YDSE	GWCA	NBRO	BWO	ESC	RBMO	ERBMO
F1	Ave	5.007E + 10	1.101E + 11	7.930E + 10	9.006E + 10	4.279E + 10	1.746E + 11	2.572E + 11	5.442E + 10	5.793E + 09	1.000E + 02
	Std	7.590E + 09	1.510E + 10	5.920E + 10	8.707E + 09	1.181E + 10	1.080E + 10	6.731E + 09	1.286E + 10	3.171E + 09	1.528E-09
F2	Ave	8.818E + 152	4.329E + 181	5.015E + 156	6.624E + 143	7.095E + 163	5.843E + 153	7.296E + 171	5.624E + 138	3.277E + 126	4.159E + 106
	Std	6.554E + 04	6.554E + 04	6.554E + 04	3.627E + 144	6.554E + 04	6.554E + 04	6.554E + 04	3.078E + 139	1.795E + 127	2.278E + 107
F3	Ave	3.652E + 05	9.611E + 05	6.468E + 05	3.014E + 05	6.739E + 05	3.818E + 05	3.783E + 05	5.712E + 05	2.265E + 05	3.018E + 02
	Std	9.672E + 04	1.595E + 05	2.989E + 05	3.035E + 04	1.459E + 05	7.291E + 04	3.003E + 04	6.962E + 04	3.300E + 04	1.891E + 00
F4	Ave	9.637E + 03	2.131E + 04	1.379E + 04	1.119E + 04	6.260E + 03	3.115E + 04	9.949E + 04	6.292E + 03	1.632E + 03	5.961E + 02
	Std	1.636E + 03	4.136E + 03	1.179E + 04	2.283E + 03	2.157E + 03	5.509E + 03	9.315E + 03	1.460E + 03	2.974E + 02	1.014E + 02
F5	Ave	1.685E + 03	1.938E + 03	1.731E + 03	1.751E + 03	1.476E + 03	2.008E + 03	2.124E + 03	1.243E + 03	1.171E + 03	9.944E + 02
	Std	7.031E + 01	1.212E + 02	2.167E + 02	6.144E + 01	9.774E + 01	5.013E + 01	2.218E + 01	1.055E + 02	7.472E + 01	6.462E + 01
F6	Ave	6.917E + 02	7.055E + 02	6.825E + 02	6.921E + 02	6.679E + 02	7.051E + 02	7.126E + 02	6.213E + 02	6.322E + 02	6.397E + 02
	Std	3.366E + 00	1.087E + 01	1.027E + 01	7.076E + 00	6.026E + 00	5.001E + 00	2.617E + 00	3.425E + 00	6.192E + 00	5.373E + 00
F7	Ave	3.757E + 03	3.824E + 03	2.965E + 03	3.193E + 03	3.891E + 03	3.717E + 03	3.894E + 03	2.091E + 03	1.886E + 03	1.839E + 03
	Std	1.079E + 02	1.216E + 02	2.545E + 02	1.675E + 02	5.003E + 02	1.374E + 02	5.333E + 01	1.833E + 02	1.683E + 02	1.642E + 02
F8	Ave	2.134E + 03	2.426E + 03	2.225E + 03	2.114E + 03	1.875E + 03	2.459E + 03	2.609E + 03	1.521E + 03	1.474E + 03	1.300E + 03
	Std	6.809E + 01	1.274E + 02	1.901E + 02	6.476E + 01	1.147E + 02	6.953E + 01	2.721E + 01	8.733E + 01	1.092E + 02	9.598E + 01
F9	Ave	6.966E + 04	8.229E + 04	7.658E + 04	7.041E + 04	2.975E + 04	7.003E + 04	7.931E + 04	1.964E + 04	1.745E + 04	1.397E + 04
	Std	4.184E + 03	1.862E + 04	1.281E + 04	8.526E + 03	4.212E + 03	4.501E + 03	3.297E + 03	4.872E + 03	5.077E + 03	2.940E + 03
F10	Ave	2.454E + 04	2.960E + 04	2.907E + 04	3.054E + 04	1.953E + 04	3.114E + 04	3.241E + 04	2.917E + 04	2.311E + 04	1.688E + 04
	Std	1.984E + 03	1.527E + 03	4.456E + 03	1.081E + 03	1.470E + 03	8.865E + 02	6.365E + 02	7.155E + 02	2.057E + 03	1.668E + 03
F11	Ave	1.512E + 05	3.464E + 05	2.423E + 05	7.955E + 04	6.877E + 04	1.214E + 05	3.524E + 05	1.097E + 05	1.197E + 04	2.044E + 03
	Std	3.429E + 04	1.532E + 05	5.070E + 04	1.266E + 04	3.124E + 04	2.649E + 04	6.854E + 04	2.662E + 04	3.494E + 03	2.427E + 02
F12	Ave	1.218E + 10	3.029E + 10	7.936E + 09	1.424E + 10	4.376E + 09	7.146E + 10	1.935E + 11	9.253E + 09	4.525E + 08	1.095E + 07
	Std	3.463E + 09	8.662E + 09	3.098E + 09	2.744E + 09	4.317E + 09	1.254E + 10	8.045E + 09	4.396E + 09	2.251E + 08	1.514E + 07
F13	Ave	3.219E + 08	2.607E + 09	2.998E + 08	8.465E + 08	8.683E + 07	1.512E + 10	4.465E + 10	8.075E + 08	5.836E + 04	4.034E + 03
	Std	1.747E + 08	1.110E + 09	1.678E + 08	3.015E + 08	2.464E + 08	4.073E + 09	3.306E + 09	6.431E + 08	2.753E + 04	7.037E + 02
F14	Ave	9.559E + 06	2.371E + 07	2.005E + 07	8.849E + 05	4.299E + 06	2.591E + 07	8.151E + 07	9.719E + 06	2.638E + 05	5.056E + 04
	Std	3.124E + 06	1.219E + 07	1.052E + 07	4.573E + 05	3.856E + 06	1.216E + 07	1.974E + 07	5.683E + 06	1.956E + 05	3.338E + 04
F15	Ave	1.896E + 07	5.245E + 08	9.104E + 07	6.587E + 07	4.232E + 05	5.471E + 09	2.292E + 10	1.383E + 08	2.076E + 05	2.068E + 03
	Std	1.046E + 07	3.271E + 08	1.382E + 08	3.385E + 07	1.340E + 06	2.957E + 09	2.484E + 09	3.883E + 08	1.066E + 06	1.094E + 02
F16	Ave	1.001E + 04	1.694E + 04	9.191E + 03	1.042E + 04	7.283E + 03	1.511E + 04	2.314E + 04	7.506E + 03	6.531E + 03	5.469E + 03
	Std	9.628E + 02	2.848E + 03	1.787E + 03	5.560E + 02	7.422E + 02	1.794E + 03	1.112E + 03	9.283E + 02	8.309E + 02	7.353E + 02
F17	Ave	8.590E + 03	2.777E + 04	9.426E + 03	7.464E + 03	7.388E + 03	1.168E + 05	4.846E + 06	5.572E + 03	5.768E + 03	5.304E + 03
	Std	2.334E + 03	3.506E + 04	1.172E + 03	5.315E + 02	2.407E + 03	1.633E + 05	2.636E + 06	4.645E + 02	6.363E + 02	6.785E + 02
F18	Ave	9.500E + 06	1.648E + 07	2.451E + 07	1.313E + 06	5.787E + 06	4.565E + 07	2.112E + 08	1.096E + 07	7.495E + 05	9.997E + 04
	Std	5.477E + 06	1.116E + 07	1.607E + 07	9.504E + 05	4.709E + 06	2.085E + 07	5.778E + 07	6.494E + 06	5.221E + 05	5.955E + 04
F19	Ave	4.058E + 07	5.117E + 08	8.522E + 07	7.096E + 07	2.639E + 06	3.639E + 09	2.213E + 10	1.189E + 08	6.889E + 04	2.348E + 03
	Std	2.176E + 07	2.743E + 08	6.260E + 07	2.800E + 07	1.110E + 07	2.014E + 09	3.070E + 09	1.839E + 08	1.848E + 05	9.688E + 01
F20	Ave	6.117E + 03	7.517E + 03	7.046E + 03	7.173E + 03	5.496E + 03	7.105E + 03	7.743E + 03	6.267E + 03	5.414E + 03	5.384E + 03
	Std	5.941E + 02	5.917E + 02	7.399E + 02	3.400E + 02	6.699E + 02	5.370E + 02	2.446E + 02	3.464E + 02	4.490E + 02	6.366E + 02
F21	Ave	4.437E + 03	4.433E + 03	4.068E + 03	3.644E + 03	3.488E + 03	4.108E + 03	4.755E + 03	3.052E + 03	3.107E + 03	2.917E + 03
	Std	1.538E + 02	1.912E + 02	1.940E + 02	6.311E + 01	1.260E + 02	1.843E + 02	1.199E + 02	1.102E + 02	1.070E + 02	9.600E + 01
F22	Ave	2.789E + 04	3.207E + 04	2.706E + 04	3.307E + 04	2.182E + 04	3.367E + 04	3.478E + 04	3.160E + 04	2.534E + 04	2.022E + 04
	Std	1.286E + 03	1.005E + 03	4.680E + 03	6.957E + 02	1.767E + 03	1.303E + 03	4.506E + 02	8.498E + 02	2.452E + 03	1.628E + 03
F23	Ave	5.915E + 03	5.232E + 03	4.806E + 03	4.377E + 03	4.581E + 03	4.994E + 03	6.130E + 03	3.422E + 03	3.858E + 03	3.668E + 03
	Std	4.742E + 02	2.274E + 02	2.075E + 02	8.182E + 01	2.338E + 02	1.972E + 02	1.474E + 02	7.761E + 01	1.883E + 02	1.556E + 02
F24	Ave	8.660E + 03	6.807E + 03	6.175E + 03	5.224E + 03	5.959E + 03	6.380E + 03	9.418E + 03	4.167E + 03	4.685E + 03	4.551E + 03
	Std	6.858E + 02	4.830E + 02	3.803E + 02	1.584E + 02	4.307E + 02	4.279E + 02	3.401E + 02	1.307E + 02	2.744E + 02	2.865E + 02
F25	Ave	6.983E + 03	1.087E + 04	1.078E + 04	1.025E + 04	7.110E + 03	1.483E + 04	2.809E + 04	7.631E + 03	4.415E + 03	3.308E + 03
	Std	5.978E + 02	1.191E + 03	6.808E + 03	1.155E + 03	1.412E + 03	1.781E + 03	9.045E + 02	9.767E + 02	2.991E + 02	6.672E + 01
F26	Ave	3.215E + 04	3.915E + 04	2.728E + 04	2.361E + 04	2.883E + 04	3.741E + 04	5.080E + 04	1.582E + 04	1.635E + 04	1.545E + 04
	Std	1.902E + 03	4.248E + 03	3.073E + 03	1.151E + 03	4.322E + 03	3.073E + 03	1.279E + 03	1.737E + 03	2.331E + 03	2.571E + 03
F27	Ave	7.058E + 03	6.306E + 03	4.681E + 03	5.045E + 03	4.537E + 03	6.459E + 03	1.252E + 04	4.097E + 03	3.734E + 03	3.631E + 03
	Std	1.055E + 03	1.086E + 03	4.141E + 02	3.290E + 02	4.620E + 02	7.382E + 02	8.958E + 02	1.497E + 02	1.013E + 02	1.093E + 02
F28	Ave	9.865E + 03	1.492E + 04	1.829E + 04	1.442E + 04	1.001E + 04	2.041E + 04	2.778E + 04	1.159E + 04	6.054E + 03	3.412E + 03
	Std	8.758E + 02	1.081E + 03	6.101E + 03	1.652E + 03	1.656E + 03	1.992E + 03	6.837E + 02	1.117E + 03	1.140E + 03	4.153E + 01
F29	Ave	1.353E + 04	2.084E + 04	1.270E + 04	1.218E + 04	9.867E + 03	2.563E + 04	4.625E + 05	7.708E + 03	8.108E + 03	7.108E + 03
	Std	1.508E + 03	4.797E + 03	5.722E + 03	8.749E + 02	8.916E + 02	8.310E + 03	1.912E + 05	7.216E + 02	7.230E + 02	5.881E + 02
F30	Ave	8.241E + 08	2.754E + 09	2.961E + 08	5.128E + 08	3.590E + 07	1.115E + 10	4.044E + 10	7.083E + 08	1.270E + 06	2.467E + 04
	Std	4.423E + 08	1.046E + 09	1.912E + 08	1.306E + 08	7.899E + 07	3.754E + 09	4.175E + 09	6.711E + 08	1.087E + 06	2.501E + 04

Analysis of the mean values demonstrates the strong optimization capability of ERBMO. Specifically, ERBMO achieves the best mean performance on 16 out of 30 test functions in the 30-dimensional case. This advantage grows with increasing dimensionality: ERBMO obtains the top mean value on 21 functions at 50 dimensions and on 27 functions at 100 dimensions. This trend indicates that ERBMO scales effectively to higher-dimensional spaces and consistently locates high-quality solutions in complex and challenging search landscapes.

The standard deviation results further confirm the algorithm’s stability. As shown in Fig. 3, which displays boxplots of the final results for a representative subset of functions, ERBMO exhibits the narrowest interquartile ranges and the least dispersion of data points in most cases. This low variance across independent runs indicates that ERBMO produces highly stable outcomes, confirming its strong reliability and robustness.

Fig. 3.

Comparison of boxplot of different algorithms on CEC 2017 test suite.

Furthermore, convergence speed is assessed by comparing the fitness evolution curves in Fig. 4. The results show that ERBMO converges faster than the competing algorithms and consistently reaches superior solutions. In many cases, ERBMO approaches or achieves optimal performance within a relatively small number of iterations, demonstrating an efficient search mechanism. This rapid convergence is particularly beneficial for optimization tasks with limited computational budgets.

Fig. 4.

Comparison of iteration curves of different algorithms on CEC 2017 test suite.

Wilcoxon signed-rank test on the CEC2017 benchmark suite

To rigorously assess the performance differences observed in Sect.  4.2.1, the non-parametric Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is conducted at a 5% significance level⁵². This test evaluates whether ERBMO’s superior performance, as reflected in the mean fitness values, is significant when compared to each of the nine competing algorithms on the CEC 2017 benchmark functions in 30, 50, and 100 dimensions.

The test results are summarized in Table 8, with each entry formatted as +/−/=, indicating the number of functions for which ERBMO is significantly better (+), significantly worse (−), or statistically equivalent (=) to the corresponding algorithm.

Table 8.

CEC2017 Wilcoxon signed rank test results.

ERBMO VS.	CEC 2017 (Dim = 30)	CEC 2017 (Dim = 50)	CEC 2017 (Dim = 100)
HHO	30/0/0	30/0/0	30/0/0
WOA	30/0/0	30/0/0	30/0/0
DBO	30/0/0	30/0/0	30/0/0
YDSE	30/0/0	30/0/0	30/0/0
GWCA	29/0/1	28/0/2	29/0/1
NBRO	30/0/0	30/0/0	30/0/0
BWO	30/0/0	30/0/0	30/0/0
ESC	20/7/3	22/6/2	25/3/2
RBMO	13/1/16	17/2/11	26/1/3
Overall (+/=/-)	242/8/20	247/8/15	260/4/6

In the 30-dimensional case, ERBMO shows a significant advantage over HHO, WOA, DBO, YDSE, NBRO, and BWO on all 30 functions, with no instances of significantly worse performance. Compared to GWCA and ESC, ERBMO performs significantly better on 29 and 20 functions, respectively, while being significantly worse on only 1 and 3 functions. When compared to its predecessor RBMO, ERBMO achieves significantly better results on 13 functions and is outperformed on 16, with one function showing no significant difference. At this dimension, although ERBMO wins fewer times than RBMO, the difference between them is small. Meanwhile, the analysis in Sect.  4.2.3 shows that ERBMO still achieves a higher average Friedman ranking, demonstrating that it remains highly competitive at this dimension as well.

As the dimensionality increases, the superiority of ERBMO becomes increasingly evident. In the 50-dimensional scenario, ERBMO maintains strong performance against all competitors. Notably, it outperforms RBMO on 17 functions and is outperformed on 11, with two functions showing equivalent results—marking a shift toward improved competitiveness.

The performance advantage of ERBMO is most pronounced in the 100-dimensional case. ERBMO demonstrates near-universal superiority, achieving significantly better results on 25 out of 30 functions against ESC and 26 out of 30 against RBMO, with only 3 and 1 significantly worse outcomes, respectively. Against all other algorithms, ERBMO is significantly better on either all 30 functions or 29 (in the case of GWCA). Aggregating the results across all dimensions and functions, the overall comparison yields a total of 260/4/6(+/−/=). This result provides conclusive evidence that ERBMO outperforms the competing algorithms on the CEC 2017 benchmark suite, with its advantage growing substantially in high-dimensional search spaces.

Friedman mean rank test on the CEC2017 benchmark suite

The Friedman mean rank test⁵³ is conducted to provide a systematic comparison of the relative performance of the algorithms across the CEC 2017 benchmark functions in 30, 50, and 100 dimensions. Figure 5 illustrates the average ranks obtained by each algorithm, while the numerical results are summarized in Table 9.

Fig. 5.

Average Friedman ranking of different algorithms on CEC 2017.

Table 9.

Friedman mean rank test on CEC2017.

Dimensions	30		50		100
Algorithms	Ave. Rank	Ove. Rank	Ave. Rank	Ove. Rank	Ave. Rank	Ove. Rank
HHO	7.067	7	6.467	7	5.733	5
WOA	8.567	9	8.400	9	8.100	8
DBO	5.933	6	6.067	6	5.933	7
YDSE	4.800	5	5.200	5	5.733	5
GWCA	4.333	4	4.033	4	4.133	4
NBRO	7.767	8	8.067	8	8.133	9
BWO	9.867	10	9.933	10	9.867	10
ESC	2.967	3	3.233	3	3.967	3
RBMO	2.033	2	2.167	2	2.267	2
ERBMO	1.667	1	1.433	1	1.133	1

The test results establish ERBMO as the top-ranked algorithm in all dimensional settings. ERBMO achieves the lowest average rank—1.67 at 30 dimensions, 1.43 at 50 dimensions, and 1.13 at 100 dimensions—demonstrating consistent superiority across all test cases. This trend of improving performance with increasing dimensionality highlights ERBMO’s strong scalability and robust search capability in complex, high-dimensional landscapes. The proposed enhancements effectively enable ERBMO to maintain a leading position as problem complexity increases.

Among the competing algorithms, RBMO and ESC achieve the next best performances. At 30 and 50 dimensions, RBMO ranks second (2.03 and 2.17), followed by ESC in third (2.97 and 3.23). At 100 dimensions, RBMO maintains its second position (2.27), while ESC drops to seventh (3.97), indicating a decline in its effectiveness in higher dimensions. GWCA consistently ranks fourth across all dimensions. The remaining algorithms—HHO, WOA, DBO, YDSE, NBRO, and BWO—exhibit significantly higher average ranks, reflecting inferior performance across the benchmark suite.

The overall ranking, based on all 30 functions across three dimensions, confirms ERBMO as the top performer, followed by RBMO and ESC in second and third places, respectively, with the remaining algorithms ranked from fourth to tenth. These results provide a global evaluation, underscoring the effectiveness of ERBMO’s enhanced search strategy.

Experimental results analysis on the CEC2022 benchmark suite

Mean and variance analysis of the CEC2022 benchmark suite

All algorithms were evaluated on the CEC 2022 benchmark functions in 10 and 20 dimensions, with the mean and standard deviation of 30 independent runs reported in Tables 10 and 11, respectively. In the 10-dimensional case, ERBMO achieves the best mean performance on five functions, outperforming other algorithms. YDSE and ESC each achieve the best results on three functions, while RBMO leads on only one function. In the 20-dimensional tests, ERBMO again achieves the best mean on five functions, maintaining a significant lead over most competitors in terms of solution quality. RBMO performs best on only one function, whereas ESC achieves top performance on six functions. However, the performance gap between ERBMO and ESC is relatively small on these functions, suggesting highly competitive behavior. Nonetheless, the Wilcoxon signed-rank test and Friedman test still confirm the superior performance of ERBMO, as detailed in Sect.  4.3.2 and 4.3.3.

Table 10.

Comparison of results on CEC2022 (Dim = 10).

ID	Index	HHO	WOA	DBO	YDSE	GWCA	NBRO	BWO	ESC	RBMO	ERBMO
F1	Ave	1.132E + 03	2.539E + 04	1.223E + 03	3.025E + 02	1.204E + 03	1.895E + 03	7.233E + 03	5.011E + 02	3.000E + 02	3.000E + 02
	Std	4.714E + 02	1.176E + 04	1.396E + 03	1.757E + 00	9.785E + 02	1.486E + 03	1.697E + 03	1.994E + 02	2.187E-06	0.000E + 00
F2	Ave	4.531E + 02	4.734E + 02	4.305E + 02	4.063E + 02	4.178E + 02	4.518E + 02	8.689E + 02	4.062E + 02	4.058E + 02	4.056E + 02
	Std	5.086E + 01	7.204E + 01	3.236E + 01	2.251E + 00	2.598E + 01	2.780E + 01	1.595E + 02	3.164E + 00	3.252E + 00	3.421E + 00
F3	Ave	6.396E + 02	6.441E + 02	6.099E + 02	6.069E + 02	6.005E + 02	6.240E + 02	6.480E + 02	6.000E + 02	6.001E + 02	6.000E + 02
	Std	1.302E + 01	1.379E + 01	7.220E + 00	2.355E + 00	1.233E + 00	8.655E + 00	6.416E + 00	5.874E-05	2.867E-01	4.470E-02
F4	Ave	8.295E + 02	8.416E + 02	8.323E + 02	8.228E + 02	8.226E + 02	8.328E + 02	8.483E + 02	8.118E + 02	8.103E + 02	8.089E + 02
	Std	8.434E + 00	1.661E + 01	1.108E + 01	6.458E + 00	9.135E + 00	8.306E + 00	5.415E + 00	4.389E + 00	4.706E + 00	3.548E + 00
F5	Ave	1.445E + 03	1.573E + 03	9.923E + 02	9.322E + 02	9.033E + 02	1.084E + 03	1.462E + 03	9.000E + 02	9.000E + 02	9.001E + 02
	Std	1.859E + 02	3.684E + 02	1.141E + 02	1.701E + 01	1.007E + 01	1.200E + 02	1.195E + 02	1.635E-02	1.162E-01	1.887E-01
F6	Ave	6.669E + 03	5.498E + 03	4.113E + 03	1.826E + 03	6.261E + 03	3.585E + 03	2.933E + 06	3.051E + 03	1.810E + 03	1.805E + 03
	Std	4.542E + 03	4.608E + 03	2.276E + 03	1.121E + 01	1.972E + 03	2.037E + 03	1.481E + 06	1.354E + 03	1.841E + 01	6.490E + 00
F7	Ave	2.081E + 03	2.085E + 03	2.042E + 03	2.037E + 03	2.023E + 03	2.059E + 03	2.104E + 03	2.007E + 03	2.019E + 03	2.020E + 03
	Std	3.011E + 01	3.260E + 01	2.149E + 01	7.465E + 00	2.658E + 00	1.927E + 01	1.393E + 01	9.186E + 00	6.991E + 00	4.970E + 00
F8	Ave	2.232E + 03	2.238E + 03	2.229E + 03	2.225E + 03	2.221E + 03	2.235E + 03	2.242E + 03	2.217E + 03	2.216E + 03	2.217E + 03
	Std	1.017E + 01	1.125E + 01	7.966E + 00	3.930E + 00	6.355E + 00	2.175E + 01	6.252E + 00	7.624E + 00	8.359E + 00	8.294E + 00
F9	Ave	2.606E + 03	2.621E + 03	2.556E + 03	2.529E + 03	2.534E + 03	2.582E + 03	2.701E + 03	2.529E + 03	2.534E + 03	2.529E + 03
	Std	5.143E + 01	5.017E + 01	4.749E + 01	7.302E-02	2.683E + 01	5.590E + 01	2.806E + 01	1.895E-08	2.683E + 01	0.000E + 00
F10	Ave	2.624E + 03	2.753E + 03	2.545E + 03	2.501E + 03	2.554E + 03	2.551E + 03	2.563E + 03	2.544E + 03	2.524E + 03	2.508E + 03
	Std	1.816E + 02	4.101E + 02	6.310E + 01	1.391E-01	8.880E + 01	6.610E + 01	6.544E + 01	5.389E + 01	6.179E + 01	2.735E + 01
F11	Ave	2.801E + 03	2.879E + 03	2.748E + 03	2.630E + 03	2.782E + 03	2.866E + 03	3.224E + 03	2.665E + 03	2.761E + 03	2.748E + 03
	Std	1.716E + 02	1.862E + 02	1.962E + 02	1.208E + 01	1.874E + 02	1.801E + 02	2.256E + 02	9.407E + 01	1.543E + 02	1.659E + 02
F12	Ave	2.920E + 03	2.898E + 03	2.870E + 03	2.863E + 03	2.863E + 03	2.867E + 03	2.908E + 03	2.865E + 03	2.864E + 03	2.866E + 03
	Std	4.062E + 01	3.644E + 01	7.773E + 00	1.130E + 00	1.462E + 00	2.992E + 00	1.486E + 01	1.004E + 00	1.673E + 00	1.736E + 00

Table 11.

Comparison of results on CEC2022 (Dim = 20).

ID	Index	HHO	WOA	DBO	YDSE	GWCA	NBRO	BWO	ESC	RBMO	ERBMO
F1	Ave	2.659E + 04	4.136E + 04	3.920E + 04	3.930E + 03	2.950E + 04	1.537E + 04	8.154E + 04	1.523E + 04	3.094E + 02	3.000E + 02
	Std	9.597E + 03	1.849E + 04	1.153E + 04	2.042E + 03	1.014E + 04	4.088E + 03	2.137E + 04	6.330E + 03	1.885E + 01	2.790E-12
F2	Ave	5.577E + 02	6.422E + 02	5.230E + 02	4.716E + 02	4.651E + 02	7.350E + 02	2.140E + 03	4.557E + 02	4.571E + 02	4.421E + 02
	Std	6.431E + 01	8.904E + 01	9.414E + 01	1.116E + 01	3.201E + 01	1.291E + 02	3.046E + 02	1.164E + 01	1.956E + 01	1.763E + 01
F3	Ave	6.624E + 02	6.663E + 02	6.354E + 02	6.286E + 02	6.084E + 02	6.600E + 02	6.826E + 02	6.000E + 02	6.003E + 02	6.005E + 02
	Std	8.502E + 00	1.101E + 01	1.060E + 01	5.194E + 00	4.315E + 00	1.142E + 01	6.888E + 00	1.820E-03	4.406E-01	7.058E-01
F4	Ave	8.894E + 02	9.389E + 02	9.216E + 02	9.003E + 02	8.830E + 02	9.382E + 02	9.747E + 02	8.601E + 02	8.318E + 02	8.374E + 02
	Std	1.595E + 01	4.339E + 01	3.307E + 01	1.468E + 01	2.998E + 01	1.711E + 01	1.004E + 01	1.033E + 01	1.149E + 01	1.428E + 01
F5	Ave	2.933E + 03	4.403E + 03	2.235E + 03	1.695E + 03	2.360E + 03	2.733E + 03	3.686E + 03	9.014E + 02	9.051E + 02	9.019E + 02
	Std	3.341E + 02	1.612E + 03	5.621E + 02	4.749E + 02	1.100E + 03	4.610E + 02	2.325E + 02	4.121E + 00	6.307E + 00	1.974E + 00
F6	Ave	2.484E + 05	5.109E + 06	3.264E + 05	1.678E + 04	1.314E + 04	3.124E + 07	1.306E + 09	5.262E + 03	1.969E + 03	1.875E + 03
	Std	1.409E + 05	4.857E + 06	7.930E + 05	8.152E + 03	8.941E + 03	2.772E + 07	5.014E + 08	4.180E + 03	8.335E + 01	4.448E + 01
F7	Ave	2.187E + 03	2.249E + 03	2.135E + 03	2.117E + 03	2.080E + 03	2.177E + 03	2.233E + 03	2.037E + 03	2.051E + 03	2.065E + 03
	Std	5.390E + 01	7.474E + 01	4.833E + 01	2.037E + 01	3.131E + 01	5.401E + 01	2.710E + 01	1.241E + 01	1.879E + 01	2.593E + 01
F8	Ave	2.310E + 03	2.308E + 03	2.313E + 03	2.239E + 03	2.251E + 03	2.300E + 03	2.289E + 03	2.231E + 03	2.243E + 03	2.225E + 03
	Std	1.054E + 02	7.541E + 01	7.091E + 01	3.644E + 00	4.020E + 01	5.806E + 01	2.393E + 01	1.892E + 00	3.959E + 01	4.551E + 00
F9	Ave	2.557E + 03	2.606E + 03	2.521E + 03	2.486E + 03	2.481E + 03	2.638E + 03	3.082E + 03	2.483E + 03	2.481E + 03	2.481E + 03
	Std	5.906E + 01	4.829E + 01	3.620E + 01	2.137E + 00	1.826E + 00	6.518E + 01	1.657E + 02	2.282E + 00	3.453E-04	2.327E-10
F10	Ave	4.058E + 03	4.956E + 03	3.335E + 03	3.627E + 03	4.044E + 03	5.145E + 03	3.993E + 03	2.561E + 03	3.733E + 03	2.896E + 03
	Std	7.302E + 02	1.117E + 03	1.194E + 03	1.391E + 03	7.620E + 02	1.612E + 03	1.210E + 03	1.213E + 02	7.298E + 02	8.116E + 02
F11	Ave	3.559E + 03	4.085E + 03	3.194E + 03	3.390E + 03	2.948E + 03	5.116E + 03	8.209E + 03	2.908E + 03	2.946E + 03	2.913E + 03
	Std	6.396E + 02	9.577E + 02	3.792E + 02	1.012E + 02	1.489E + 02	7.954E + 02	4.626E + 02	7.915E + 01	1.202E + 02	9.732E + 01
F12	Ave	3.197E + 03	3.097E + 03	3.021E + 03	2.972E + 03	2.951E + 03	3.053E + 03	3.366E + 03	2.949E + 03	2.968E + 03	2.965E + 03
	Std	1.674E + 02	1.118E + 02	4.233E + 01	1.097E + 01	1.284E + 01	5.773E + 01	9.976E + 01	8.899E + 00	3.369E + 01	1.882E + 01

As shown in Fig. 6, the box plots demonstrate that ERBMO produces consistently close results across multiple runs, with narrow interquartile ranges and few outliers, reflecting low variability in performance. The convergence curves in Fig. 7 further illustrate that ERBMO steadily improves solution quality over iterations, gradually approaching high-precision solutions without premature stagnation (Fig. 8).

Fig. 6.

Comparison of boxplot of different algorithms on CEC 2022 test suite.

Fig. 7.

Comparison of iteration curves of different algorithms on CEC 2022 test suite.

Fig. 8.

Average Friedman ranking of different algorithms on CEC 2022.

Wilcoxon signed-rank test on the CEC2022 benchmark suite

The results of the Wilcoxon signed-rank test on the CEC2022 benchmark suite are summarized in Table 12. The test compares the performance of ERBMO against nine competing algorithms across 12 functions in both 10 and 20 dimensions.

Table 12.

CEC2022 Wilcoxon signed rank test results.

ERBMO VS.	CEC 2022 (Dim = 10)	CEC 2022 (Dim = 20)
HHO	12/0/0	12/0/0
WOA	12/0/0	12/0/0
DBO	11/0/1	12/0/0
YDSE	9/1/2	11/0/1
GWCA	10/1/1	11/1/0
NBRO	11/0/1	12/0/0
BWO	12/0/0	12/0/0
ESC	7/2/3	7/4/1
RBMO	4/1/7	8/1/3
Overall (+/=/-)	88 / 5 / 15	97 / 6 / 5

In the 10-dimensional case, ERBMO outperforms HHO, WOA, and BWO on all 12 functions. It also achieves better results than DBO, YDSE, GWCA, and NBRO, with favorable outcomes on the majority of functions and only a few instances of inferior performance. When compared to ESC, ERBMO performs better on 7 functions, worse on 2, and shows no significant difference on 3. Against RBMO, it performs better on 4 functions, worse on 1, and equivalent on 7, indicating a modest advantage.

In the 20-dimensional case, ERBMO maintains strong performance, consistently outperforming HHO, WOA, DBO, YDSE, GWCA, NBRO, and BWO across nearly all functions. Compared to RBMO, ERBMO demonstrates superior performance on 8 functions, inferior performance on only 1, and equivalent results on 3. Against ESC, ERBMO is superior on 7 functions, inferior on 4, and equivalent on 1, reflecting a favorable outcome despite close competition.

Overall, ERBMO achieves better performance in 88 out of 108 comparisons in 10 dimensions and 97 out of 108 in 20 dimensions, with only 5 and 6 comparisons, respectively, showing inferior results. The results show that ERBMO performs competitively on most CEC2022 benchmark functions. It is noteworthy that ERBMO does not exhibit a significant performance advantage over ESC or RBMO on the CEC2022 test suite. This can be attributed to the algorithm’s design, which emphasizes exploitation. While beneficial for complex landscapes, this strong exploitation bias may limit the algorithm’s exploratory capabilities. On lower-dimensional problems, where the function landscape is relatively simple, the reduced need for exploration curtails the algorithm’s opportunity for trial and error. This lack of exploration can increase the risk of premature convergence, thereby limiting ERBMO’s performance gains on these simpler functions.

Friedman mean rank test on the CEC2022 benchmark suite

The Friedman test results for the CEC2022 benchmark functions in 10 and 20 dimensions are summarized in Table 13.

Table 13.

Friedman mean rank test on CEC2022.

Dimensions	10		20
Algorithms	Ave. Rank	Ove. Rank	Ave. Rank	Ove. Rank
HHO	7.583	8	7.083	7
WOA	8.917	9	8.417	9
DBO	6.000	6	5.917	6
YDSE	3.667	4	5.083	5
GWCA	4.583	5	4.333	4
NBRO	7.250	7	8.000	8
BWO	9.750	10	9.583	10
ESC	3.083	3	2.333	2
RBMO	2.333	2	2.417	3
ERBMO	1.833	1	1.833	1

In 10 dimensions, ERBMO is ranked first with an average rank of 1.83, indicating consistent and competitive performance across the test functions. RBMO and ESC follow in second and third positions (2.33 and 3.08), demonstrating competitive performance across the benchmark set. The remaining algorithms—YDSE, GWCA, DBO, NBRO, HHO, and BWO—yield higher values, ranging from 4.58 to 9.75, indicating relatively weaker effectiveness.

In 20 dimensions, ERBMO maintains its leading position with an average rank of 1.83, reflecting stable behavior as problem dimensionality increases. ESC improves to second place (2.33), slightly outperforming RBMO (2.42), which suggests better scalability in high-dimensional landscapes. The performance ordering of the other algorithms remains largely unchanged, with values between 4.33 and 9.58.

ERBMO consistently achieves the best ranking across both dimensions, with only RBMO and ESC approaching its performance level. This consistent top ranking confirms ERBMO’s superior capability in handling complex optimization tasks under the CEC2022 suite.

Additional experiment for performance verification

In this section, three additional analyses are conducted on the CEC2022 benchmark suite to further validate the performance of ERBMO. These include: (1) a comprehensive comparison between ERBMO and a range of classical and advanced algorithms, namely Particle Swarm Optimization (PSO)⁵⁴, Gaussian Quantum-Behaved Particle Swarm Optimization (GQPSO)⁵⁵, Terminal Crossover and Steering-based Particle Swarm Optimization (TCSPSO)⁵⁶, PSO variant for single-objective numerical optimization(PSO_sono)⁵⁷, Differential Evolution(DE)⁷, Improving Differential Evolution through Bayesian Hyperparameter Optimization (MadDE)⁵⁸, Differential Evolution Algorithm With Strategy Adaptation(SaDE)⁵⁹, Evolution Strategy with Covariance Matrix Adaptation (CMA-ES)⁶⁰, and Improving the Search Performance of SHADE Using Linear Population Size Reduction (LSHADE)⁶¹; (2) an ablation study to investigate the individual contributions of the three improvement strategies incorporated into ERBMO; and (3) a runtime analysis comparing ERBMO with its competitor algorithms.

Comparison with classic and advanced optimization algorithms

We compared ERBMO with three classical algorithms, PSO, DE, and CMAES, as well as improved versions of PSO and DE, including GQPSO, TCSPSO, PSO sono, MadDE, SaDE, and LSHADE. Experiments were conducted on the CEC2022 benchmark suite with dimensions 10 and 20. Each algorithm was run independently 30 times, with a population size of 30 and a maximum number of 500 iterations.

From Fig. 9; Table 14, it can be observed that compared to PSO and its variants, CMA-ES, as well as DE and its variants, ERBMO achieves the highest overall Friedman rankings on both the 10-dimensional and 20-dimensional problems. Specifically, on the 10-dimensional problems, ERBMO ranks first with an average rank of 3.550, significantly outperforming the second-ranked SaDE (3.806) and the third-ranked LSHADE (4.146). On the 20-dimensional problems, ERBMO further extends its advantage to 3.089, leading over LSHADE (3.919) and SaDE (4.111). In contrast, traditional PSO algorithms and their improved versions (e.g., GQPSO, TCSPSO) generally rank lower, placing between 5th and 10th on both dimensional settings. Among the DE family, although SaDE and LSHADE demonstrate relatively strong performance, they still fall short of ERBMO’s leading position. CMA-ES ranks 4th on the 20-dimensional problems, slightly better than its 8th-place performance on the 10-dimensional problems, yet a noticeable gap remains when compared to ERBMO. These results fully demonstrate that ERBMO possesses excellent global search capability and stability when solving the CEC2022 test functions. Its comprehensive performance not only surpasses classical algorithms but also significantly outperforms several advanced variants, thoroughly validating the effectiveness and superiority of its design strategies.

Fig. 9.

Friedman rank comparison: ERBMO vs. classical and advanced algorithms.

Table 14.

Average Friedman Rankings of ERBMO and Classical/Advanced Algorithms on CEC2022.

Dimensions	10		20
Algorithms	Ave. Rank	Ove. Rank	Ave. Rank	Ove. Rank
PSO	5.544	6	5.875	6
GQPSO	9.461	10	9.306	10
TCSPSO	6.671	9	6.594	9
PSO_sono	5.250	5	5.222	5
DE	5.696	7	5.972	8
MadDE	4.761	4	5.908	7
SaDE	3.806	2	4.111	3
CMA_ES	6.115	8	5.003	4
LSHADE	4.146	3	3.919	2
ERBMO	3.550	1	3.089	1

Ablation study of ERBMO

In this section, we investigate the performance of algorithms obtained by removing three specific strategies from ERBMO: Diversity-adaptive weight updating, Periodic pattern search, and Evolutionary probabilistic combinatorial mutation. The resulting algorithms are referred to as ERBMO-1, ERBMO-2, and ERBMO-3, respectively, and are compared with the original ERBMO. The experiments are conducted on the CEC2022 test suite with dimensions of 10 and 20. Each algorithm is independently run 30 times, with a population size of 30 and the number of iterations set to 500.

From Fig. 10; Table 15, it can be observed that in the ablation study, ERBMO achieves the highest Friedman rankings compared to ERBMO-1, ERBMO-2, and ERBMO-3, which are obtained by removing specific strategies. Specifically, on the 10-dimensional problems, ERBMO ranks first with an average rank of 2.282, outperforming ERBMO-1 (2.364), ERBMO-3 (2.589), and ERBMO-2 (2.765). On the 20-dimensional problems, ERBMO further extends its advantage with an average rank of 2.140, again surpassing ERBMO-1 (2.308), ERBMO-3 (2.608), and ERBMO-2 (2.943). Among the ablated variants, ERBMO-1, which removes the Diversity-adaptive weight updating strategy, achieves the second-highest rankings on both dimensions, indicating that this strategy contributes positively but is not the most critical component. ERBMO-3, which removes the Evolutionary probabilistic combinatorial mutation, ranks third, suggesting that this mutation mechanism plays a notable role in enhancing search performance. ERBMO-2, which removes the Periodic pattern search, consistently obtains the lowest rankings on both dimensions, demonstrating that the Periodic pattern search strategy is the most crucial component among the three, making the most significant contribution to the overall performance of ERBMO. These results collectively validate the effectiveness of each proposed strategy, with the Periodic pattern search identified as the key factor driving the superior performance of ERBMO.

Fig. 10.

Comparison of Friedman rankings of different algorithms in ablation study.

Table 15.

Average Friedman rankings of different algorithms in the ERBMO ablation study.

Dimensions	10		20
Algorithms	Ave. Rank	Ove. Rank	Ave. Rank	Ove. Rank
ERBMO-1	2.364	2	2.308	2
ERBMO-2	2.765	4	2.943	4
ERBMO-3	2.589	3	2.608	3
ERBMO	2.282	1	2.140	1

Runtime analysis of different algorithms

In this section, we analyze the runtime comparison of ERBMO and its competitors in the CEC2022 experiments with dimensions of 10 and 20 in Sect.  4.3.

From Fig. 11, it can be observed that ERBMO exhibits moderate computational efficiency compared to other algorithms in the runtime analysis on the CEC2022 test suite. Specifically, on 10-dimensional problems, ERBMO records a runtime of 0.4533 s, which is higher than HHO (0.0494s), WOA (0.0624s), DBO (0.0797s), YDSE (0.1206s), GWCA (0.1155s), NRBO (0.1183s), RBMO (0.2628s), and ESC (0.3543s), while being significantly lower than BWO (1.1337s). On 20-dimensional problems, ERBMO requires 0.9035 s, again surpassing HHO (0.0817s), WOA (0.0935s), DBO (0.1246s), YDSE (0.1642s), GWCA (0.1474s), NRBO (0.1348s), RBMO (0.3252s), and ESC (0.4672s), while still substantially outperforming BWO (1.1495s). Notably, although ERBMO exhibits higher runtime compared to most lightweight algorithms such as HHO, WOA, and even its base version RBMO, its computational cost remains within an acceptable range given its superior optimization performance demonstrated in previous sections. The increase in runtime compared to RBMO and ESC can be attributed to the additional computational overhead introduced by the three enhancement strategies: Diversity-adaptive weight updating, Periodic pattern search, and Evolutionary probabilistic combinatorial mutation. These results indicate that while the incorporation of multiple enhancement strategies inevitably increases computational cost, ERBMO maintains a reasonable trade-off between solution quality and computational efficiency, achieving state-of-the-art optimization performance at the expense of moderate additional runtime.

Fig. 11.

Computational time comparison: ERBMO vs. competitors.

Applications in engineering optimization problems

In this section, ERBMO is applied to four engineering design optimization problems to evaluate its practical effectiveness. The selected problems include the weight minimization of a speed reducer⁶², pressure vessel design⁶³, step-cone pulley design⁶⁴, and hydrostatic thrust bearing design⁶⁵, which are characterized by nonlinear objective functions, multiple inequality and equality constraints, and mixed variable types (continuous, discrete, and integer). Each problem is solved with a maximum of 500 iterations, and the algorithm is independently run 30 times to ensure statistical reliability. The results are used to assess ERBMO’s ability to handle complex constraints and produce high-quality solutions in realistic engineering scenarios. This study employs a penalty term method to handle equality and inequality constraints.

Weight minimization of a speed reducer

The speed reducer design problem aims to minimize the weight of a two-stage gear train. The design involves seven key parameters: the face width (), the module of gears (), the number of teeth on the pinion (, integer), the lengths of the first and second shafts between bearings ( and ), and the diameters of the two shafts ( and ). These parameters are subject to constraints related to gear bending stress, surface stress, shaft deflection, and geometric limitations. A schematic representation of the problem is illustrated in Fig. 12. The problem is formulated as follows:

Fig. 12.

Speed reducer design problem.

Consider:

Minimize:

Subject to:

16

Variable bounds:

The optimization results are presented in Table 16. ERBMO achieves the best overall performance among all compared algorithms, securing the top rank in the Friedman test. It obtains the lowest best solution, mean, and median, with values of 2.993E + 03, 2.994E + 03, and 2.994E + 03, respectively. The worst solution reached by ERBMO is 2.994E + 03, the second-best among all algorithms, slightly higher than that of GWCA; its variance of 2.828E-01 is also relatively low, demonstrating high solution consistency across independent runs. GWCA ranks second, achieving the smallest worst solution of 2.994E + 03 and the lowest variance of 1.155E-09 among all algorithms. However, its best and mean solutions are marginally higher than those of ERBMO, resulting in a narrow performance gap. RBMO and ESC rank third and fourth, respectively, exhibiting competitive performance although slightly inferior to GWCA and ERBMO. YDSE, DBO, and NRBO are capable of identifying high-quality solutions in certain runs. However, their higher variances of 2.625E + 00, 5.981E + 01, and 4.914E + 01 indicate unstable search behavior, leading to rankings from fifth to seventh. BWO, WOA, and HHO show limited effectiveness, with higher values across all statistical measures, placing them at the lowest ranks (Fig. 13).

Table 16.

Comparison of optimization results among algorithms for the Speed reducer design problem.

Algorithm	Worst	Best	Mean	Median	Std	Friedman ranking
HHO	4.989E + 03	3.012E + 03	3.625E + 03	3.366E + 03	5.914E + 02	10
WOA	4.975E + 03	3.032E + 03	3.440E + 03	3.180E + 03	4.966E + 02	9
DBO	3.222E + 03	2.994E + 03	3.046E + 03	3.034E + 03	5.981E + 01	6
YDSE	3.008E + 03	2.996E + 03	2.998E + 03	2.997E + 03	2.625E + 00	5
GWCA	2.994E + 03	2.994E + 03	2.994E + 03	2.994E + 03	1.155E-09	2
NRBO	3.177E + 03	3.003E + 03	3.063E + 03	3.052E + 03	4.914E + 01	7
BWO	3.266E + 03	3.035E + 03	3.111E + 03	3.104E + 03	5.493E + 01	8
ESC	2.995E + 03	2.994E + 03	2.995E + 03	2.994E + 03	7.256E-02	4
RBMO	2.994E + 03	2.994E + 03	2.994E + 03	2.994E + 03	7.820E-06	3
ERBMO	2.994E + 03	2.993E + 03	2.994E + 03	2.994E + 03	2.828E-01	1

Fig. 13.

Comparison of iteration curves in speed reducer design problem.

Pressure vessel design

The pressure vessel design problem aims to minimize the total manufacturing cost of a cylindrical pressure vessel with hemispherical ends, while satisfying mechanical strength and geometric requirements. The design involves four key variables: the thickness of the shell , the thickness of the head , the inner radius , and the length of the cylindrical section . The schematic of the pressure vessel is illustrated in Fig. 14, and the mathematical formulation of the optimization problem is as follows.

Fig. 14.

Pressure vessel design problem.

Consider:

Minimize:

Subject to:

17

Variable bounds:

The results presented in Table 17 demonstrate that ERBMO exhibits strong performance on the pressure vessel design problem. In terms of the worst solution, ERBMO achieves a value of 6.371E + 03, which is the second lowest among all algorithms, only slightly higher than YDSE’s worst result of 6.122E + 03 by a small margin of 2.488E + 02, indicating good robustness across independent runs. ERBMO’s best and median solutions are slightly inferior to those of RBMO, with differences of just 1.000E-11 and 2.228E-01, respectively; however, ERBMO achieves a significantly smaller standard deviation of 5.693E + 01 compared to RBMO’s 4.728E + 02, reflecting superior convergence consistency. This high level of stability enables ERBMO to rank first in the overall performance comparison. YDSE ranks second due to its extremely low standard deviation of 1.486E + 01, which demonstrates excellent repeatability, while RBMO follows closely in third place. DBO, GWCA, NRBO, and ESC are able to obtain relatively good optimal solutions, but their larger variances lead to moderate rankings from fourth to seventh. In contrast, BWO, HHO, and WOA exhibit significantly higher mean values and greater solution dispersion, resulting in the lowest overall rankings—eighth, ninth, and tenth, respectively—indicating limited effectiveness in handling the nonlinear constraints and complex search landscape of this engineering problem. Overall, ERBMO achieves the top rank with relatively small gaps in best and median performance across test functions, indicating a good balance between solution quality and stability (Fig. 15).

Table 17.

Comparison of optimization results among algorithms for the Pressure vessel design problem.

Algorithm	Worst	Best	Mean	Median	Std	Friedman ranking
HHO	1.277E + 04	6.246E + 03	7.185E + 03	6.901E + 03	1.143E + 03	9
WOA	3.118E + 04	6.854E + 03	1.001E + 04	8.823E + 03	4.479E + 03	10
DBO	7.544E + 03	6.060E + 03	6.491E + 03	6.357E + 03	3.972E + 02	4
YDSE	6.122E + 03	6.063E + 03	6.079E + 03	6.073E + 03	1.486E + 01	2
GWCA	7.544E + 03	6.060E + 03	6.646E + 03	6.413E + 03	5.257E + 02	5
NRBO	7.876E + 03	6.091E + 03	6.711E + 03	6.529E + 03	5.056E + 02	6
BWO	7.821E + 03	6.354E + 03	7.031E + 03	7.052E + 03	3.620E + 02	8
ESC	7.497E + 03	6.115E + 03	6.641E + 03	6.572E + 03	3.528E + 02	7
RBMO	7.333E + 03	6.060E + 03	6.296E + 03	6.060E + 03	4.728E + 02	3
ERBMO	6.371E + 03	6.060E + 03	6.077E + 03	6.060E + 03	5.693E + 01	1

Fig. 15.

Comparison of iteration curves in pressure vessel design problem.

Step-cone pulley problem

The step-cone pulley design problem aims to minimize the weight of a four-step cone pulley system used in mechanical power transmission. The pulley consists of four stepped grooves with different diameters () to achieve multiple speed ratios, and a uniform width () for all steps. The design must satisfy constraints on belt length, speed ratios, and structural strength to ensure proper operation under load. The schematic of the step-cone pulley system is illustrated in Fig. 16, and the mathematical formulation of the optimization problem is as follows.

Fig. 16.

Step-cone pulley problem.

Consider:

Minimize:

Subject to:

18

where,

The results of the step-cone pulley design problem are presented in Table 18. ERBMO achieves the best overall performance, ranking first with an optimal solution of 1.609E + 01, a worst solution of 1.670E + 01, and mean and median values of 1.623E + 01 and 1.616E + 01, respectively. A standard deviation of 1.576E-01 indicates high solution stability. ERBMO outperforms all other algorithms in every metric except the worst solution, where it is surpassed by RBMO by 3.739E-02. RBMO follows in second place, achieving the best worst-case result while slightly trailing ERBMO in other metrics. GWCA ranks third, with a best solution of 1.714E + 01, a worst solution of 1.617E + 01, and a standard deviation of 2.844E-01, demonstrating competitive solution quality and moderate stability. ESC and YDSE exhibit small variances (7.883E + 01 and 1.467E + 02, respectively), indicating tightly clustered solutions across runs. However, their higher worst-case values limit their overall performance, resulting in fourth and fifth positions. DBO, HHO, and NRBO achieve relatively good optimal solutions (1.714E + 01, 1.687E + 01, and 1.653E + 01, respectively), but their larger variances lead to inconsistent results, placing them sixth, seventh, and ninth. In contrast, WOA and BWO deliver the poorest performance, with higher mean values and greater solution dispersion, ranking at the bottom. Overall, ERBMO delivers the most effective performance for this problem (Fig. 17).

Table 18.

Comparison of optimization results among algorithms for the Step-cone pulley problem.

Algorithm	Worst	Best	Mean	Median	Std	Friedman Ranking
HHO	1.787E + 08	1.687E + 01	1.423E + 07	1.483E + 06	4.102E + 07	7
WOA	1.222E + 11	4.449E + 01	7.434E + 09	1.343E + 04	2.821E + 10	8
DBO	8.857E + 10	1.714E + 01	7.433E + 09	1.386E + 07	2.219E + 10	6
YDSE	7.216E + 02	2.019E + 01	1.491E + 02	8.240E + 01	1.467E + 02	5
GWCA	1.714E + 01	1.617E + 01	1.694E + 01	1.709E + 01	2.844E-01	3
NRBO	8.293E + 10	1.653E + 01	2.556E + 10	2.613E + 09	3.361E + 10	9
BWO	9.059E + 10	4.833E + 08	2.186E + 10	1.741E + 10	2.054E + 10	10
ESC	4.496E + 02	1.672E + 01	3.613E + 01	1.738E + 01	7.883E + 01	4
RBMO	1.666E + 01	1.609E + 01	1.627E + 01	1.623E + 01	1.730E-01	2
ERBMO	1.670E + 01	1.609E + 01	1.623E + 01	1.616E + 01	1.576E-01	1

Fig. 17.

Comparison of iteration curves in step-cone pulley problem.

Hydro-static thrust bearing design problem

The objective of the hydro-static thrust bearing design problem is to minimize the total power loss during operation. The design involves four key variables: the bearing step radius , the recess radius , the lubricant flow rate , and the oil viscosity . The schematic diagram of the bearing is illustrated in Fig. 18. The mathematical model is formulated as follows.

Fig. 18.

Hydrostatic thrust bearing design problem.

Consider:

Minimize:

Subject to:

19

where,

Variable bounds:

The optimization results for the hydro-static thrust bearing design problem are summarized in Table 19. Among all algorithms, ERBMO achieves the highest performance ranking in the Friedman test, yielding the lowest worst solution of 2.818E + 03, mean of 2.058E + 03, and median of 1.984E + 03, along with a best solution of 1.793E + 03. Its standard deviation of 2.570E + 02—the second lowest among all competitors—further confirms consistent performance across runs. RBMO achieves the best optimal solution of 1.633E + 03, but its relatively high worst-case result and larger variance limit overall consistency, resulting in a second-place ranking. YDSE attains a less competitive best solution, yet secures the smallest standard deviation of 2.223E + 02 and a relatively low worst solution of 3.040E + 03, ranking third overall. NRBO, GWCA, and DBO achieve optimal solutions of 1.893E + 03, 1.903E + 03, and 2.108E + 03, respectively, but their higher worst-case values and larger variances place them in fourth to sixth positions. BWO, WOA, ESC, and HHO exhibit poor performance across all metrics, characterized by high mean and worst solutions as well as significant solution dispersion, failing to achieve consistent convergence and occupying the lowest ranks (Fig. 19).

Table 19.

Comparison of optimization results among algorithms for the Hydrostatic thrust bearing design problem.

Algorithm	Worst	Best	Mean	Median	Std	Friedman Ranking
HHO	8.267E + 19	2.998E + 03	2.871E + 18	1.203E + 09	1.507E + 19	10
WOA	1.163E + 09	2.046E + 03	2.700E + 08	4.162E + 03	4.979E + 08	8
DBO	4.698E + 03	2.108E + 03	3.286E + 03	3.432E + 03	7.694E + 02	6
YDSE	3.040E + 03	2.189E + 03	2.631E + 03	2.696E + 03	2.223E + 02	3
GWCA	1.141E + 09	1.903E + 03	7.569E + 07	3.063E + 03	2.880E + 08	5
NRBO	1.164E + 09	1.893E + 03	3.880E + 07	2.805E + 03	2.125E + 08	4
BWO	2.423E + 17	2.477E + 03	9.422E + 15	3.829E + 03	4.430E + 16	7
ESC	3.837E + 19	2.795E + 03	2.734E + 18	4.518E + 03	9.706E + 18	9
RBMO	6.474E + 03	1.633E + 03	2.303E + 03	1.899E + 03	1.206E + 03	2
ERBMO	2.818E + 03	1.793E + 03	2.058E + 03	1.984E + 03	2.570E + 02	1

Fig. 19.

Comparison of iteration curves in hydrostatic thrust bearing design problem.

Limitations

Although ERBMO demonstrates excellent performance, it still has some limitations. For instance, its superior performance comes at the cost of longer runtime. Additionally, the algorithm involves numerous parameters, making fine-tuning for specific problems relatively complex. On the other hand, when dealing with functions characterized by complex multimodal landscapes, ERBMO occasionally underperforms compared to RBMO and ESC, which can be attributed to excessive exploitation at the expense of sufficient exploration within its mechanism. To address these issues in future work, we plan to introduce strategies such as adaptive population decay to reduce runtime, conduct comprehensive parameter sensitivity analysis, and consider implementing adaptive parameter adjustment mechanisms to enhance the algorithm’s exploration capability.

Conclusion

This paper proposes an Enhanced Red-billed Blue Magpie Optimizer (ERBMO), an improved variant of the original RBMO. By integrating three key strategies—Diversity-Adaptive Weight Updating, Periodic Pattern Search, and Evolutionary Probabilistic Combinatorial Mutation—ERBMO aims to enhance solving performance for complex engineering optimization problems. The effectiveness of ERBMO is systematically validated through experiments on the CEC2017 and CEC2022 benchmark suites, as well as four classical engineering design problems.

First, in the experiments on the CEC2017 benchmark functions, ERBMO achieved the best average Friedman rankings across all tested dimensions (30D, 50D, and 100D), with scores of 1.667, 1.433, and 1.133, respectively, consistently outperforming the original RBMO which ranked second. This result clearly demonstrates the significant performance advantage of ERBMO over its base version and other compared algorithms such as HHO, WOA, and BWO. In the tests on the CEC2022 benchmark functions, ERBMO similarly exhibited excellent performance, obtaining the highest overall Friedman rankings for both the 10D and 20D problems (1.833 and 1.833). It not only surpassed advanced algorithms like ESC and RBMO but also secured the top position when compared with classical algorithms such as PSO, DE, and CMA-ES, and their advanced variants like SaDE and LSHADE. This further confirms the stable superiority of ERBMO across problems of varying dimensions.

Second, the results of the ablation study quantify the contribution of each improvement strategy. The experimental data show that the complete ERBMO achieved superior Friedman rankings on the CEC2022 10D and 20D problems (2.282 and 2.140) compared to all algorithm variants where a single strategy was removed (ERBMO-1, ERBMO-2, ERBMO-3). This strongly demonstrates that each of the three proposed strategies plays an indispensable and positive role in the overall performance of the algorithm.

Finally, when applied to four real-world engineering optimization problems with complex constraints, ERBMO consistently demonstrated superior optimization capabilities. For the Speed Reducer Design problem, ERBMO achieved the lowest mean cost (2.994E + 03). In the Pressure Vessel Design problem, ERBMO successfully found the global optimum (6.060E + 03) and exhibited the smallest standard deviation (5.693E + 01), indicating high solution accuracy and robustness. For the Step-Cone Pulley and Hydrostatic Thrust Bearing Design problems, ERBMO also obtained the best Friedman ranking among all compared algorithms. These results fully demonstrate the strong practical applicability and solution quality of ERBMO when handling complex, constrained real-world problems.

Future research will focus on enhancing the adaptability of ERBMO for a broader range of complex optimization scenarios. Key directions include: (1) Extending the algorithm to handle multi-objective and large-scale optimization problems to further assess its search behavior; (2) Exploring hybridization with local search or knowledge-based operators to strengthen its exploitation capability and accelerate convergence; and (3) Evaluating ERBMO’s performance in dynamic and constrained environments, particularly in emerging application domains such as renewable energy system design, smart manufacturing, and intelligent transportation systems, to comprehensively validate its robustness and practical potential in modern engineering contexts.

Author contributions

Conceptualization, H.W. and Z.L.; methodology, H.W., Z.X. and Z.L.; software, H.W. and Z.X.; validation, X.Q. and Q.Z.; formal analysis, X.Q. and J.L.; investigation, H.W., Z.X., X.Q., Q.Z. and J.L.; resources, Z.L.; data curation, H.W. and Z.X.; writing—original draft preparation, H.W. and Z.X.; writing—review and editing, H.W., and Z.L.; visualization, X.Q. and Q.Z.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request (Zhe Li, seven_lizhe@foxmail.com).

Declarations

Competing interests

The authors declare no competing interests.