Abstract
To address the issues of low solution accuracy, premature convergence, and susceptibility to local optima in the Parrot Optimization (PO) algorithm, a Reinforced Multi-strategy Parrot Optimization (RMPO) algorithm has been designed. First, cubic mapping is introduced in PO to initialize the solution positions, which helps generate a more uniformly distributed initial population, avoiding the initial solution clustering caused by random initialization, thereby enhancing the algorithm’s global exploration capability. Second, drawing inspiration from the vigilance behavior of sparrow populations, the search strategy is adjusted based on individual fitness to reduce the probability of the algorithm getting stuck in local optima. A nonlinear convergence factor is introduced to balance the algorithm’s exploitation ability. To prevent the convergence factor from converging too quickly in later iterations, an adaptive t-distribution operator and differential mutation are employed to enhance global exploration and local exploitation capabilities. Finally, to systematically evaluate the performance of RMPO, tests were conducted on the CEC2019 and CEC2022 test suites, and comparisons were made with eight other intelligent optimization algorithms. RMPO achieved the optimal solution on 7/10 functions of CEC2019 and 8/12 functions of CEC2022, with the first overall statistical ranking (average ranking 1.50) on CEC2022 and the top ranking on CEC2019 for high-dimensional complex problems. To verify the application capability of RMPO in engineering problems, the algorithm was further applied to solve design problems for pressure vessels, welded beams, and stepped cone pulleys, and it achieved the highest optimization accuracy and stability in all three engineering problems.
Keywords
Introduction
Metaheuristic Algorithms (MAs) are a class of efficient and versatile optimization algorithms. Unlike traditional optimization methods, these algorithms do not rely on the specific mathematical properties of the problem but instead employ intelligent search strategies to find global optimal solutions or high-quality approximate optimal solutions in the solution space. They are particularly well-suited to tackling complex issues such as multi-modal optimization and nonlinear optimization that are difficult to handle using traditional methods. 1 The design of MAs is primarily inspired by observations and simulations of natural phenomena, as well as the intelligent behavior of biological populations. Early algorithms were based on simple biological evolutionary mechanisms, such as the Particle Swarm Optimization (PSO) algorithm, 2 Genetic Algorithms (GA), 3 and Ant Colony Optimization (ACO) 4 ; With the deepening of research and the rapid development of computational technology, the MA field has seen the emergence of various new optimization methods. For example, the Moth-Flame Optimization (MFO) 5 and Seagull Optimization Algorithm (SOA) 6 simulate natural biological behaviors and demonstrate advantages in global exploitation and convergence speed; The Remora Optimization Algorithm (ROA) 7 further enhances the robustness of the algorithm through a symbiotic strategy; In recent years, a variety of novel meta-heuristic algorithms with excellent performance have been proposed, which further enrich the family of MAs and provide more solutions for complex optimization problems. For example, the Artificial Lemming Algorithm (ALA) 8 simulates the population migration and foraging behaviors of lemmings, which has the advantages of fast convergence speed and strong global exploration capability, but it is easy to appear the problem of population divergence in the later iteration, resulting in the decline of optimization accuracy. The joint opposite selection-based arithmetic artificial rabbits optimization algorithm (JOS-AARO) 9 combines opposite selection strategy and arithmetic crossover operator on the basis of the original artificial rabbits optimization algorithm, which effectively improves the population diversity and convergence accuracy, and has achieved good results in engineering optimization problems, but the algorithm structure is relatively complex, and the computational efficiency is reduced in high-dimensional scenarios. Wang et al. 10 innovatively proposed the Arctic Puffin Optimization (APO), a novel metaheuristic algorithm inspired by the survival and predation behaviors of Arctic puffins. The APO introduces mechanisms such as Levy flight, velocity factor, synergy strategy, and adaptive conversion factor to balance global exploration and local exploitation. Experimental results on multiple CEC test functions and engineering optimization problems verify the competitiveness and effectiveness of APO. However, the algorithm still has room for improvement in dealing with high-dimensional multimodal problems and complex constrained engineering scenarios.
However, due to the limitations of classical algorithms in terms of iteration speed, search range, and convergence, 11 recent studies have proposed various improved MAs to address the limitations of classical algorithms and improve their accuracy and performance in practical applications. For instance, Wang et al. 12 addressed the problem of low-quality candidate solutions during the exploration process of the Yin–Yang-pair Optimization (YYPO) algorithm and proposed an improved algorithm OOYO integrated with orthogonal opposition-based learning. The proposed algorithm improves the global optimization performance without additional complexity, yet it still suffers from insufficient stability when tackling ultra-high-dimensional multimodal optimization and complex engineering constrained problems. Abualigah et al. 13 introduced a novel hybrid optimization algorithm—the Improved Prairie Dog Optimization Algorithm (IPDOA). By incorporating the core update mechanism of the Dwarf Mongoose Optimization Algorithm (DMOA), addresses three key deficiencies present in the original Prairie Dog Optimization Algorithm (PDOA): slow convergence speed, imbalance between the exploration and exploitation phases, and susceptibility to premature convergence. To validate the algorithm’s performance, 23 standard benchmark functions were selected for experimental testing, and the results were compared and analyzed with existing optimization methods in the literature. The results demonstrate that the IPDOA exhibits significant advantages in solving optimization problems of varying complexity, with its overall performance notably superior to that of the comparison algorithms. Qin et al. 14 addressed the issues of insufficient convergence accuracy and significant fluctuations in optimization results in the original Gazelle Optimization Algorithm (GOA) by proposing a multi-strategy improved algorithm that integrates chaotic mappings—the Multi-strategy Particle Swarm Optimization with Gazelle Optimization Algorithm (MPSOGOA). This algorithm introduces segmented chaotic mappings during the population initialization phase to produce a well-distributed and high-quality initial population, effectively enhancing population diversity. Additionally, a global disturbance strategy further improves the algorithm’s convergence rate during the early iterations and the precision of convergence in the later stages. To validate its performance, comparative experiments were conducted on 35 standard benchmark functions, and statistical significance analysis was performed using the Friedman test and Wilcoxon signed-rank test. The experimental results show that MPSOGOA exhibits superior performance in convergence accuracy and stability compared to other MAs. Additionally, the application of this algorithm in engineering optimization problems further validates its excellent optimization performance.
Meta-heuristic algorithms (MAs) have been widely used in complex optimization problems, but most classical MAs face the problems of slow convergence in high-dimensional scenarios and easy premature convergence. The newly proposed Parrot Optimization (PO) algorithm 15 simulates the foraging, staying, communicating and fear behaviors of parrots, which has the advantages of simple structure, strong randomness and good population diversity compared with traditional MAs. Unlike other MAs that separate exploration and exploitation into distinct stages, PO enables each agent to randomly select behaviors in each iteration, which is more suitable for solving multi-modal and high-dimensional optimization problems. Therefore, the PO algorithm is selected as the research basis in this study, and its performance is further improved to make it more applicable to complex practical engineering optimization problems. However, the original PO algorithm still has obvious limitations in practical application: the random initialization of the population leads to uneven distribution of initial solutions, which weakens the global exploration capability of the algorithm; meanwhile, its staying and fear behaviors lack adaptive adjustment mechanisms, making the algorithm easy to fall into local optima when dealing with complex multi-modal functions. In addition, the fixed convergence trend cannot balance the global exploration and local exploitation capabilities of the algorithm in different iteration stages, and the lack of effective perturbation strategies leads to the decline of population diversity in the later iteration, which is difficult to escape from local optima. For the newly proposed PO algorithm, relevant scholars have also made further refinements to address its limitations. For example, Huang et al. 16 proposed an Improved Parrot Optimization Algorithm for Optimizing Random Forests (IPORF) and applied it to fault diagnosis in Autonomous Underwater Vehicle (AUV). This algorithm optimizes the behavior selection mechanism of PO, and the diagnostic accuracy is as high as 99.59%, which is significantly better than the original PO algorithm, but it only focuses on the improvement of a single link, and the performance improvement is limited in high-dimensional and multi-modal complex optimization problems. At the same time, the current PO variants are mostly applied to specific fields (such as fault diagnosis), and there is a lack of systematic improvement and verification for the inherent limitations of PO in global optimization, and the generalization ability of the algorithm needs to be further improved.
Given this, this study designs targeted improvement mechanisms: first, cubic mapping is adopted for population initialization to solve the problem of uneven distribution of initial solutions, which lays a foundation for global exploration; second, the scouter mechanism of Sparrow Search Algorithm (SSA) is introduced into the staying and fear behaviors of PO to realize adaptive adjustment of search strategies and avoid falling into local optima; third, a nonlinear convergence factor is added to dynamically adjust the convergence speed of the algorithm, which strengthens global exploration in the early stage and local exploitation in the later stage; finally, the hybrid perturbation strategy of t-distribution and differential mutation is designed to maintain population diversity in the whole iteration process, and solve the problem of population clustering or divergence in different iteration stages. All improvement mechanisms are designed for the inherent limitations of PO, and the multi-strategy fusion is realized to ensure the complementarity of each mechanism and maximize the optimization performance of the algorithm. Furthermore, to evaluate the optimization capability and robustness of RMPO, benchmark test functions from CEC2019 17 and CEC2022 18 were selected for testing, and a comparative analysis was conducted with eight mainstream intelligent optimization algorithms. Additionally, RMPO was applied to pressure vessel, welded beam, and stepped cone pulley problems to assess its applicability in real-world engineering problems.
The main contributions of this paper are as follows:
(1) A reinforced multi-strategy parrot optimization algorithm (RMPO) is proposed by fusing four targeted improvement mechanisms for the specific limitations of the original PO algorithm, which realizes the complementary advantages of each strategy and effectively improves the global exploration, local exploitation and anti-premature convergence capabilities of the algorithm.
(2) Systematic performance verification of RMPO is carried out on CEC2019 and CEC2022 benchmark test suites, and comprehensive comparison with eight mainstream intelligent optimization algorithms is conducted from the aspects of convergence curve, statistical results, radar chart and Wilcoxon rank-sum test, which fully proves the superiority of RMPO in optimization accuracy, convergence speed and stability.
(3) An ablation study is designed to verify the effectiveness of each single improvement strategy and the necessity of multi-strategy fusion, which provides a clear design idea for the improvement of PO algorithm and other meta-heuristic algorithms.
(4) The proposed RMPO algorithm is applied to three classical practical engineering optimization problems (pressure vessel design, welded beam design, stepped cone pulley design), and it achieves the highest optimization accuracy and stability in all problems, which verifies the practical engineering application value and generalization ability of RMPO.
The rest of this paper is organized as follows: Section 2 details the basic principles of the original PO algorithm, the design of each improvement strategy and the overall framework of RMPO, and gives the pseudo-code and time complexity analysis of RMPO. Section 3 carries out the ablation study of improvement strategies and the comprehensive performance verification of RMPO on CEC2019 and CEC2022 benchmark test suites. Section 4 applies RMPO to three practical engineering optimization problems and verifies its application effect by comparing with other algorithms. Section 5 summarizes the full text, points out the limitations of the proposed algorithm and prospects the future research directions. Finally, the acknowledgments and references are presented at the end of the paper.
Reinforced multi-strategy parrot optimization algorithm (RMPO)
Parrot optimization algorithm (PO)
Inspired by the collective behavior of parrots, the PO achieves an efficient balance between global optimization and local exploration by simulating four behaviors: foraging, staying, communicating, and fear of strangers. Each of these behaviors is embedded in all phases of the algorithm iteration. Unlike conventional MAs, which often isolate exploration and exploitation into distinct stages, every agent in PO randomly selects and performs one of the four behaviors at each iteration. This mechanism considerably enhances randomness and enhances population diversity, thus improving the ability to locate the global optimum.
Initialization phase
In PO, the optimization process commences with the initialization of the population. A population of candidate solutions is randomly generated, with each solution representing a potential solution within the search space. Their positions are typically generated within given boundaries using a uniform distribution or a specific probability distribution. Key parameters include:
(1) Population size (N): Determines the breadth of the search, affecting the algorithm’s global exploitation capabilities;
(2) Maximum number of iterations (
(3) Problem dimension (dim): Number of corresponding optimization variables;
(4) Behavioral choice probability: The basis for the random allocation of four behaviors (foraging, staying, communicating, fear) in subsequent iterations.
Among them,
Foraging behavior
By simulating the process of parrots randomly searching for food in their natural environment, combined with Levy Flight, global exploration capabilities are enhanced. The mathematical formulation representing this behavior is provided below:
As defined in the equation:
Staying behavior
The parrot randomly perches on different parts of the keeper’s body and remains stationary for periods of time. This behavior is encapsulated in the formula:
Among them, X best ·Levy(dim) represents the movement process toward the keeper, and rand(0, 1)·ones(1, dim) denotes the process of the parrot randomly staying on the keeper’s body parts.
Communicating behavior
As a typical gregarious animal, parrots exhibit significant sociality and cooperation in their group behavior. In their natural environment, individual parrots coordinate tasks such as foraging and vigilance through frequent information exchange. This cooperative mechanism manifests itself in two typical behavioral patterns: one is moving toward the center of the group to share information; the other is maintaining contact through specific means when away from the group. Inspired by this, PO developed a mathematical model of parrot communication behavior, which is mathematically expressed as follows:
The algorithm assumes that these two behavior patterns occur randomly with equal probability (50% each) and uses the current population’s average position
(1) Group consensus guidance: Operations that converge toward the average position promote information sharing within the population and accelerate convergence;
(2) Active exploration mechanism: The behavior of the principal group enhances randomness through exponential terms to avoid premature convergence;
(3) Dynamic balance: As iterations proceed, the weighting coefficient
Fear of strangers’ behavior
PO designed a dynamic perturbation mechanism by simulating the vigilant behavior of parrots toward strangers. This behavior is modeled in the algorithm as follows:
The escape direction is generated randomly using a cosine function to ensure diversity in the perturbation directions. Simultaneously, a decay factor tied to the iteration count is adopted to dynamically regulate the escape intensity: a larger perturbation is applied during initial stages to strengthen global exploration, which gradually decreases and stabilizes in later stages.
Related optimization strategies
Cubic mapping strategy
The symmetry of the cubic mapping (
Scouter mechanism in SSA
To improve the performance of PO, this study introduces the scouter mechanism in the SSA 20 into the stay behavior and fear of strangers’ behavior in the PO algorithm.
Regarding the lingering behavior in the original PO algorithm, a scouter mechanism is introduced to enhance its adaptive capability and prevent the algorithm from becoming trapped in local optima. When the fitness of an individual surpasses the current optimum, the alert mechanism is activated, prompting the algorithm to conduct more effective exploration within the search space. This improvement transforms the previously passive lingering behavior into an adaptive response mechanism that balances exploitation and exploration.
Similarly, the behavior of parrots fearing strangers is adaptively adjusted through the scouter mechanism. This mechanism provides additional behavioral constraints for individual parrots, enabling them to respond more dynamically to environmental changes. When the fitness of an individual exceeds the known optimal solution, the scouter mechanism prompts the algorithm to adopt a more cautious search strategy, effectively avoiding premature convergence.
The mathematical expression is as follows:
Where
In summary,
Nonlinear Convergence Factor
The nonlinear convergence factor is an optimization strategy that enhances the adaptability and robustness of the algorithm by dynamically adjusting the convergence speed. 21 The mathematical definition of this strategy is as follows:
Where
This strategy converges rapidly in the early stages to expand the search range, and converges slowly in the later stages in combination with dynamic thresholds to enhance the ability to escape local optima; it is suitable for complex optimization problems such as high-dimensional and multi-peak problems. The innovation of this nonlinear convergence factor lies in the adjustable change trend, which can adaptively balance exploration and exploitation, and is more suitable for complex-constrained optimization problems with strict boundary requirements.
t-distribution and differential mutation perturbations
The t-distribution, also known as the Student’s t-distribution, 22 contains a degree of freedom parameter, and its probability density is shown in the formula:
Where Gam(x) is the gamma function.
Random numbers are generated using the t-distribution, as shown in the formula.
Here, X represents the original parameter value, and the number of iterations is used as the degrees of freedom parameter for the t-distribution, with t(C_iter) denoting a random variable drawn from this distribution. As the number of iterations increases, the degrees of freedom of the t-distribution also increase, causing its probability density to gradually shift from a Cauchy distribution toward a Gaussian distribution. In the early stages of the algorithm, the t-distribution is a Cauchy distribution, with a peak at 0 that is smaller than that of a Gaussian distribution and a flatter approach to 0, making it more suitable for global exploitation. In the later stages of the algorithm, as t increases, the t-distribution gradually approaches a Gaussian distribution, with a larger area around 0, which is beneficial for solving problems in the current domain and improves the algorithm’s local exploitation capabilities in the later stages.
The Differential Evolution (DE) algorithm is a meta-heuristic algorithm proposed by Storn and Price. 23 Among them, differential mutation is selected to perturb the population, and its mutation strategy is shown in the formula.
Where
To effectively balance exploitation and exploration while preventing the redundant activation of perturbation strategies, RMPO designs two distinct and mutually exclusive trigger mechanisms based on the algorithmic state:
(1) Improvement-Triggered Perturbation: During the inner iteration loop, if an agent successfully discovers a strictly better global optimal position (f i < f g ), the Improvement Trigger is immediately activated. The algorithm exploits this promising new region by generating a secondary search position based on the probability density function of the t-distribution. This mechanism acts as a local intensifier to accelerate convergence whenever a breakthrough is made.
(2) Stagnation-Triggered Mechanism: To prevent the algorithm from being permanently trapped in local optima, a rigorous stagnation-triggered condition is evaluated at the end of each iteration. The algorithm detects stagnation by comparing the real-time global best fitness (f g ), dynamically updated during the current iteration) with the historical convergence record from three iterations ago curvet-3.
The stagnation mechanism is triggered if and only if f g = curvet-3. This rigorous mathematical condition guarantees that if any improvement event occurs during the current iteration (meaning f g becomes strictly less than curvet-3), the stagnation condition is automatically invalidated.
Once true stagnation is detected (f
g
= curvet-3 for t > 3), the population state is assessed by calculating the average fitness of the population (
This adaptive hybrid perturbation mechanism, governed by strict mutually exclusive triggers, effectively solves the population degradation problem of PO in the later iterations of high-dimensional problems.
Synergistic mechanism of multi-strategy and targeted advantages for complex problems
Each improvement strategy in RMPO is designed for the inherent limitations of the original PO algorithm, and the multi-strategy fusion forms a complementary and synergistic optimization mechanism, which is highly targeted to the pain points of high-dimensional, multi-modal and complex-constrained optimization problems. The specific role, innovation and synergistic effect of each strategy are as follows:
Cubic mapping strategy: Different from the random initialization and traditional chaotic mapping (e.g. Logistic mapping) of PO, cubic mapping realizes symmetrical exploration of the solution space through its odd symmetry, and dynamically adjusts the search range by parameter tuning. For high-dimensional problems, it solves the problem of uneven distribution of initial solutions in high-dimensional space, enriches the initial population diversity, and avoids the algorithm falling into the local optimal region in the early stage of iteration due to the clustering of initial solutions.
Scouter mechanism of SSA: The original PO’s staying and fear behaviors lack adaptive adjustment and environmental perception capabilities. This study innovatively embeds the scouter mechanism of SSA into the two core behaviors of PO, and designs a fitness-based dynamic early warning and search strategy adjustment mechanism. For multi-modal problems with numerous local optima, this mechanism enables the algorithm to perceive the fitness change of the population in real time, activate the global exploration mode when approaching the local optimum, and effectively improve the ability to escape from local optima.
Nonlinear convergence factor: Compared with the fixed convergence rate of the original PO, the nonlinear convergence factor designed in this study realizes the dynamic adjustment of convergence speed with the iteration process (fast convergence in the early stage and slow convergence in the later stage). Its innovation lies in the introduction of the adjustment parameter k to flexibly control the change trend of the convergence factor, which can be adaptively adjusted according to the characteristics of the optimization problem. For complex-constrained problems, it balances the global exploration and local exploitation capabilities of the algorithm, avoids the algorithm crossing the constraint boundary due to excessive step size in the early stage, and improves the local exploitation precision of the feasible region in the later stage.
T-distribution and differential mutation hybrid perturbation: Different from the single perturbation strategy of most improved meta-heuristic algorithms, this study designs an adaptive hybrid perturbation mechanism based on the population state: t-distribution perturbation is adopted for population clustering, and differential mutation perturbation is adopted for population divergence. For high-dimensional and multi-modal problems, this mechanism maintains the population diversity throughout the iteration process, solves the problem of population degradation in the later stage of PO iteration, and further enhances the algorithm’s ability to explore the global optimal solution in the complex solution space.
The four improvement strategies of RMPO are not simply superposed, but form a closed-loop optimization system covering the initialization phase, behavioral evolution phase, convergence control phase and population perturbation phase. The cubic mapping lays a good foundation for global exploration in the initialization phase; the scouter mechanism optimizes the behavioral evolution of the algorithm and solves the problem of local optima; the nonlinear convergence factor dynamically controls the iteration speed and balances exploration and exploitation; the hybrid perturbation mechanism timely adjusts the population state and avoids population degradation. The synergy of the four strategies makes RMPO form a targeted optimization capability for high-dimensional, multi-modal and complex-constrained problems, which is the core reason why it significantly outperforms the original PO and other comparison algorithms.
Pseudo-code of RMPO
The comprehensive structure of RMPO is presented in Algorithm 1 and Figure 1, which provides a detailed roadmap of the entire improvement process. Specifically, Algorithm 1 explicitly defines the non-overlapping boundaries among the Identity-based Trigger, the Improvement-Triggered Condition, and the Stagnation-Triggered Condition, strictly aligning with the mathematical equations discussed in Section 2.2. The complete parameter list of RMPO is shown in Table 1.
Flowchart of RMPO.
Complete parameter list of RMPO.
Time complexity analysis of RMPO and PO
The time complexity analysis is still based on the CEC2022 test function set, with each test function run independently under 10- and 20-dimensional conditions. The time complexity of meta-heuristic algorithms is dominated by three core factors: population size (N), problem dimension (dim), and maximum iterations (Tmax). For the original PO algorithm, the time complexity is O(Tmax × N × dim), as its four core behaviors all involve linear operations on the population and dimensions. RMPO integrates four improvement strategies, and each module’s time complexity is explicitly stated as follows: the cubic mapping initialization generates initial populations through formula operations, requiring sequential calculations for dim dimensions of N individuals, resulting in a time complexity of O(N × dim); fitness evaluation involves computing the fitness of each individual based on the objective function, with a time complexity of O(N × dim); core behavior updates (including foraging, staying, communicating, and fear) involve linear operations such as optimal position calculation, step size generation, and fitness ratio computation, maintaining a time complexity of O(N × dim); the nonlinear convergence factor adjustment only requires coefficient multiplication for individual position update steps, with a time complexity of O(N × dim); the hybrid perturbation mechanism (t-distribution + differential mutation) performs position updates for clustered and divergent individuals respectively, with a worst-case time complexity of O(N × dim). Since all strategies maintain linear time complexity without introducing higher-order operations, the asymptotic time complexity of RMPO remains O(Tmax × N × dim), consistent with the original PO.
The running times and their ratios for RMPO and PO at different iteration counts are compared as shown in Table 2. The slight increase in actual runtime stems from the linear computational overhead of the additional strategies, which is an acceptable trade-off between performance and efficiency. Notably, as the dimension increases to 20D, RMPO’s runtime growth tends to flatten, demonstrating better scalability than PO in high-dimensional problems, as the hybrid perturbation mechanism avoids invalid iterations caused by population degradation in high-dimensional spaces. This phenomenon fully demonstrates that RMPO has significant computational advantages when handling high-dimensional complex optimization problems.
Test results of the Avg_Time.
Performance verification of RMPO
To validate that the RMPO algorithm possesses stronger optimization capabilities compared to other intelligent optimization algorithms, this study selected the CEC2019 and CEC2022 benchmark test functions for evaluation, which effectively test the performance of MAs. This study tested and compared eight algorithms: PO, Pigeon-Inspired Optimization (PIO), 24 Osprey Optimization Algorithm (OOA), 25 Black-winged Kite Algorithm (BKA), 26 Tornado Optimizer with Coriolis force (TOC), 27 GOOSE Algorithm (GOOSE), 28 Coati Optimization Algorithm (COA) 29 and Modified Beluga Whale Optimization (MBWO), 30 aiming to better validate the development and exploration capabilities of the RMPO. Secondly, the performance of the RMPO in local development and global exploration processes is intuitively evaluated by plotting convergence curves. Finally, the Wilcoxon rank-sum test 31 was employed to further validate the efficacy and superiority of the proposed RMPO algorithm. All algorithm parameters are strictly set according to the original literature, and every test function was executed for 30 independent runs to ensure empirical reliability.
To better understand the difficulty of optimizing these test functions, three-dimensional graphs of some functions and iteration curves of individual functions are shown in Figure 2. The maximum number of iterations is set to 300, corresponding to a fixed function evaluation budget of 30 × 300 = 9000 FEs per run, which is consistent across all algorithms to ensure fair comparison. All benchmark experiments are independently executed 30 times. Fixed random seeds are used to guarantee full reproducibility. The clamping strategy is adopted consistently for all algorithms, which maps out-of-bound solutions directly to the nearest boundary. All parameter settings of the compared algorithms are directly adopted from their original published literature without any additional tuning, ensuring objectivity and fairness. All experiments in this section were conducted under the same simulation environment and hardware specifications: a computer equipped with an Intel(R) Xeon(R) E3-1240 processor (3.50 GHz), 16 GB of RAM, and the Windows 10 (64 b) operating system, with MATLAB R2024a serving as the experimental platform.
Three-dimensional plots of some test functions and iterative curves of individual functions in CEC2019 and CEC2022.
Ablation study of improvement strategies
CEC2022 comprises 12 single-objective test functions subject to boundary constraints, which consist of unimodal, multi-modal, hybrid, and composite function types, covering 2D, 10D, and 20D test dimensions. All test functions are designed to solve minimization problems and are one of the most widely used tests sets currently available.
To evaluate the effectiveness of different improvement strategies, simulation tests were conducted using MATLAB software. The RMPO algorithm was compared with three variants of the PO algorithm improved by single strategies: MPO1, MPO2, MPO3, and MPO4, on the CEC2022 test functions. Among these, MPO1, MPO2, MPO3, and MPO4 enhance the original PO algorithm by introducing the cubic mapping, the scouter mechanism from the SSA, a non-linear convergence factor, and t-distribution and differential mutation perturbation strategies, respectively. In terms of parameter settings, the dimension was fixed at 20, the population size was uniformly set to 30, and the maximum number of iterations was established as 300. To ensure statistical reliability, each algorithm was executed independently 30 times on each test function. Through the above experimental design, the impact of each improved strategy on algorithm performance can be systematically evaluated, providing reliable experimental evidence for future algorithm improvements.
The average convergence curves, radar charts, and ranking charts for RMPO and three PO variant algorithms are shown in Figures 3 and 4. As can be seen from the figures, the area of the radar chart for RMPO is significantly smaller than that of the comparison algorithms (MPO1, MPO2, MPO3 and MPO4), and the range of fluctuations in each dimension is the narrowest, indicating that its performance is highly balanced and stable. Among the 12 benchmark functions (F1–F12), RMPO achieved optimal performance on 9 functions, demonstrating particular advantage on multi-peaked functions. On the remaining three functions, it ranked second or third with a minimal gap to the optimal result. Regarding iteration curves, RMPO maintained a lower average fitness at equivalent iteration counts in the vast majority of cases. In later iterations, RMPO consistently achieved final fitness below other algorithms, demonstrating superior escape capability from local optima.
Average convergence curves of RMPO and four PO variant algorithms.
Radar chart (a) and ranking chart (b) of RMPO and four PO variant algorithms.
The test results show that removing any strategy will lead to a decline in performance, verifying the necessity of multi-strategy design. The iteration diagram further shows that RMPO is significantly superior to PO’s single-strategy variant in terms of convergence speed and stability, especially in complex functions.
CEC2019 benchmark suite testing and comparative analysis
Algorithm convergence curve on the CEC2019
The CEC2019 benchmark includes 10 optimization problems with different characteristics. F1, a unimodal function, is employed to evaluate the algorithm’s global convergence capability; F2 is a simple multi-peak function with a few local optima, designed to test the algorithm’s capability to escape from local extrema; F3–F5 are hybrid functions composed of multiple sub-functions, creating a complex search space; F6–F10 are composite functions that further increase the optimization difficulty by weighting and combining multiple benchmark functions.
The average convergence curves of RMPO and eight other algorithms on the F1-F10 functions are shown in Figure 5. As shown in the figure, for the majority of test functions, RMPO achieves significantly better convergence performance than the original PO algorithm and other comparative algorithms. Specifically: RMPO’s fitness curve exhibits steeper descent, indicating higher optimization efficiency; during early iterations, RMPO rapidly escapes local optima, approaching the global optimum much faster than the blind gradual search of PO; On multiple test functions (such as F5, F6, and F10), the final fitness attained by RMPO (curve endpoints) are generally lower than those of other algorithms, approaching theoretical optima more closely. This indicates superior convergence precision, whereas PO and other algorithms may remain trapped at local optima due to premature convergence or insufficient local exploration.
Average convergence curves of nine algorithms on the CEC2019 test function.
Statistical results of comparative tests on the CEC2019
The minimum values, standard deviations, and averages obtained by RMPO and eight other algorithms on the CEC2019 are shown in Table 3. The results show that RMPO demonstrated significant performance advantages, as it stably converges to the theoretical optimum for both unimodal and multi-modal functions. For instance, in the F4 function, RMPO achieves min = 6.97E+00, substantially outperforming BKA and COA, demonstrating its superior gradient optimization capability in high-dimensional nonlinear problems. For composite functions, such as the F9 function, RMPO’s min = 1.08E+00 and avg = 1.28E+00 are both superior to TOC (min = 1.15E+00, avg = 1.75E+00), demonstrating its potential in constrained optimization problems. The low average values of RMPO in most functions (such as F3 and F6) indicate its ability to converge quickly, while algorithms like OOA suffer from significant deterioration in avg due to premature convergence. RMPO demonstrated significant advantages in global convergence, robustness, and computational efficiency in the CEC2019 test, particularly excelling in multi-modal and mixed functions optimization.
Test results of CEC2019.
The bold values in Table 3 indicate the best results achieved by the proposed RMPO algorithm among all compared algorithms for each test function.
Algorithm comparison radar chart on the CEC2019
The radar chart comparing the performance of algorithms such as RMPO, PO, and PIO across 10 core metrics (F1–F10) is shown in Figure 6. As shown in the figure, RMPO demonstrates significant advantages in the following key dimensions: The total area covered by the radar chart corresponding to RMPO is smaller compared to other algorithms (such as TOC, GOOSE, etc.), indicating higher solution accuracy. The fluctuation range of RMPO’s radar chart boundaries is extremely minimal, demonstrating that this algorithm maintains relatively stable performance when confronted with complex and variable external disturbances, exhibiting smaller data fluctuations and greater robustness. RMPO consistently approaches the radar chart center across most dimensions, demonstrating superior operational efficiency during iteration and significantly enhancing system response speed. When comprehensively considering these multi-dimensional metrics, RMPO has no obvious performance shortcomings, demonstrates better overall efficiency in the corresponding evaluation system, and is better suited to meet the requirements of tasks with multiple performance demands in actual application scenarios, possessing more significant advantages.
Radar chart of nine algorithms on the CEC2019 test function.
The outcomes of the Wilcoxon rank-sum test and Friedman test on the CEC2019
The results of the rank-sum test on the CEC2019 benchmark are presented in Table 4. It can be observed that in the comparison between RMPO and PO, BKA, and MBWO, PO and MBWO each exhibited p-values exceeding 0.05 for four test functions, while BKA showed p-values above 0.05 for five functions. This indicates that for these functions, RMPO’s performance did not demonstrate significant differences compared to these three optimization algorithms. However, RMPO performed notably better for the remaining functions, and its overall performance still held an advantage. When comparing RMPO with GOOSE, p-values were below 0.05 for all ten functions, indicating RMPO’s performance significantly outperformed GOOSE. For PIO, OOA, TOC, and COA, only function F1 exhibited a p-value above 0.05. Regarding function F1, RMPO’s p-value exceeded 0.05 relative to all other optimization algorithms except TOC and GOOSE, implying that RMPO performs comparably to most algorithms on this function. Overall, RMPO demonstrated outstanding performance across most functional tests, exhibiting significant differentiation from other algorithms. This highlights RMPO’s advantages in addressing the CEC2019 benchmark problems.
The rank-sum test results of nine algorithms on the CEC 2019 benchmark.
The bold values in Table 4 indicate that the p-values are less than 0.05 in the Wilcoxon rank-sum test, which means the performance difference between RMPO and the corresponding compared algorithm is statistically significant at the 95% confidence level.
To address the issue of multiple pairwise comparisons, we first conducted a Friedman test to detect overall significant differences among all compared algorithms. The median of the best fitness was adopted as the test statistic, as it is more robust to outliers and non-normality than the mean in evolutionary optimization. After confirming a significant difference via the Friedman test, we performed the Nemenyi post-hoc test at a significance level of α = 0.05 to conduct pairwise comparisons with correction for multiple comparisons. The Friedman mean ranks are presented in Figure 7, showing that RMPO achieves the lowest mean rank across most test functions, indicating its superior overall performance. The Nemenyi critical difference (CD) diagram (CD = 2.193) further confirms that RMPO is significantly better than all other compared algorithms, as the distance between their mean ranks exceeds the critical difference threshold.
Friedman mean ranks and Nemenyi critical difference diagram of all compared algorithms on CEC2019.
Overall radar chart and mean ranking plot on the CEC2019
Figure 8 presents the radar chart and average ranking of RMPO alongside eight other intelligent algorithms on the CEC2019 benchmark. As can be seen from the figures, the radar chart of RMPO exhibits the smallest area fluctuation, indicating its stable performance across multiple test functions. Specifically, it achieved first place on seven functions, and second on two, and third on one, demonstrating highly stable results across multiple independent runs with minimal susceptibility to random factors. Overall, RMPO continues to perform exceptionally well on high-dimensional complex problems. It achieved the highest average ranking on the stable 10-dimensional case, undoubtedly offering a superior solution for tackling high-dimensional complex challenges.
The overall radar chart (a) and mean ranking chart (b) of nine algorithms on the CEC2019 benchmark functions.
CEC2022 benchmark suite testing and comparative analysis
Algorithm convergence curve on the CEC2022
The CEC2022 single-objective bounded-constraint optimization benchmark suite consists of 12 test functions spanning unimodal, multimodal, hybrid, and composite categories. The experiments are conducted at two problem dimensions: 10D and 20D, with the search space for all functions constrained to the interval [−100, 100]. The optimization objective is to minimize the objective function value, and the evaluation budget is set to 9000 function evaluations per run (30 independent runs × 300 maximum iterations). Compared to the test sets from CEC2019 and earlier versions (such as CEC2005 and CEC2017), CEC2022 primarily focuses on dynamic parameter optimization in constraint optimization, enabling more effective evaluation of optimization algorithms’ performance in complex scenarios and posing significant challenges to the algorithms’ real-time adaptability. By comprehensively comparing these two test sets, the robustness and adaptability of the RMPO across diverse optimization problems can be more thoroughly validated.
The average convergence curves of RMPO and eight other algorithms on functions F1–F12 are presented in Figure 9. It can be observed that RMPO exhibits a distinct downward trend and the lowest final average fitness on most test functions (such as F2, F3, F4, F7, F9, F11, and F12), demonstrating its excellent convergence speed and accuracy. For function F5, although the initial convergence behavior of all algorithms was similar, RMPO exhibited a faster decline in the later stages and achieved the optimal average fitness, highlighting its superior global optimization capability. For function F6, RMPO’s convergence curve rapidly declined during the initial iterations. Compared to algorithms such as PO and PIO, it more swiftly escaped higher mean fitness and pioneered entry into the local exploration phase. As the number of iterations increases, the convergence curve of RMPO transitions from a rapid decline to a steady plateau, indicating that the algorithm implements a strategic shift from global exploitation to local exploration. This enables it to balance search efficiency with convergence accuracy, forming one of its core characteristics that surpasses the comparison algorithms. In summary, when confronted with relatively complex test functions, RMPO retains robust global exploitation capabilities alongside effective local exploitation. It demonstrates an ability to escape local optima while maintaining strong robustness, thereby enhancing the algorithm’s overall solution capacity.
Average convergence curves of nine algorithms on the CEC2022 test function.
Statistical results of comparative tests on the CEC2022
The minimum, standard deviation, and average values obtained by RMPO and eight other algorithms on the CEC2022 test functions are shown in Table 5. For function F2, RMPO achieved min = 4.06E + 02 and avg = 4.67E + 02. This algorithm not only attained the optimal minimum value but also exhibited the smallest standard deviation, demonstrating superior optimization performance and stability on this function. On the multi-modal function F3, where avg = 6.17E + 02, it outperforms all other comparison algorithms, demonstrating robust stability in complex multi-modal problems. On function F5, RMPO simultaneously attained the best minimum value and a low standard deviation, indicating its high precision and robust stability in function optimization. On the composite function F10, although the minimum values obtained by all algorithms were relatively close, RMPO gained an advantage through its more reasonable standard deviation, demonstrating more stable convergence performance that renders it more competitive in long-term optimization tasks. For both F11 and F12, RMPO once again achieved the best average values, further confirming its robust overall performance and strong average optimization capability in composite function optimization. Overall, RMPO demonstrated high convergence accuracy, robust stability, and excellent global convergence capability across the CEC2022 test functions, exhibiting significantly superior performance to comparison algorithms particularly when handling multi-modal and composite functions.
Test results of CEC2022
The bold values in Table 5 indicate the best results achieved by the proposed RMPO algorithm among all compared algorithms for each test function.
Algorithm comparison radar chart on the CEC2022
Figure 10 presents a radar chart comparison of the performance of RMPO, PO, PIO, and other algorithms across 12 core metrics (F1–F12). As illustrated, RMPO demonstrates significant advantages in the following key aspects: The radar map corresponding to the RMPO covers the smallest total area, indicating more concentrated values across various metrics and higher solution accuracy. Concurrently, the RMPO exhibits minimal boundary fluctuation, demonstrating its capacity to maintain relatively stable performance and stronger robustness even within complex and dynamic environments. Overall, across these 12 multi-dimensional indicators, RMPO exhibits balanced and outstanding performance in all aspects, with no noticeable weaknesses. This comprehensive and well-rounded performance enables RMPO to adapt to diverse optimization needs and demonstrates superior overall effectiveness within the established evaluation framework.
Radar chart of nine algorithms on the CEC2022 test function.
The outcomes of the Wilcoxon rank-sum test on the CEC2022
The results of the Wilcoxon rank-sum test on the CEC2022 benchmark are presented in Table 6. Analysis of the data in the table reveals that when comparing RMPO with OOA, TOC, and COA, yielded p-values below 0.05 across all 12 test functions, demonstrating that the RMPO optimization algorithm markedly outperforms the aforementioned three algorithms. For functions F1, F6, F7, and F8, the p-value for RMPO relative to BKA exceeded 0.05, indicating comparable performance between RMPO and BKA on these functions. For PO, PIO, GOOSE, and MBWO, only one or two functions exhibited p-values exceeding 0.05. Overall, RMPO demonstrates significant performance in the majority of functional tests, revealing substantial differences compared to other algorithms and outperforming them considerably. This underscores the strong competitiveness of RMPO in solving CEC2022 benchmark problems.
The rank-sum test results of nine algorithms on the CEC 2022 benchmark.
The bold values in Table 6 indicate that the p-values are less than 0.05 in the Wilcoxon rank-sum test, which means the performance difference between RMPO and the corresponding compared algorithm is statistically significant at the 95% confidence level.
Overall radar chart and mean ranking plot on the CEC2022
Figure 11 presents the overall radar chart and an average ranking plot, comparing the performance of RMPO with eight other algorithms on the CEC2022. As observed, the radar chart boundary corresponding to RMPO exhibits the smallest fluctuations amplitude. Compared to the extreme variations of other algorithms on certain metrics, RMPO maintains high performance across multiple critical dimensions, including global optimization capability, convergence rate, and precision of solutions, demonstrating the unique advantages of its multi-strategy integration. RMPO achieved first place on eight functions, second place on three functions, and fourth place on one function, indicating superior performance on the vast majority of test functions. The average ranking of RMPO is 1.50, securing the top position, which further validates its leading overall optimization capability. Owing to its high stability, balanced optimization performance, and exceptional ranking, RMPO demonstrates significant advantages in the CEC2022 benchmark tests. Its outstanding comprehensive performance across multiple test functions makes it a preferred choice for multi-objective optimization problems.
The overall radar chart (a) and mean ranking chart (b) of nine algorithms on the CEC2022 benchmark functions.
Analysis of RMPO performance in different iterative stages
To further reveal the optimization mechanism of RMPO, this section divides the iteration process into early stage (1–100), middle stage (101–200) and late stage (201–300) according to the convergence characteristics of the algorithm, and takes the 20D multi-modal function F5 (CEC2019) and complex-constrained function F11 (CEC2022) as examples to analyze the performance changes of RMPO and PO in each stage, and expound how the multi-strategy collaboration of RMPO overcomes the limitations of PO. The iteration stage performance comparison curve is shown in Figure 12.
Performance comparison curve of RMPO and PO in different iterative stages. (a) F5 in CEC2019. (b) F11 in CEC2022.
At the early iteration stage, PO adopts random initialization, resulting in uneven distribution of initial solutions in the high-dimensional space. The fixed behavioral selection probability and convergence speed make the algorithm conduct blind global search, and the fitness decreases slowly. Some individuals are clustered in the local optimal region, and the phenomenon of premature convergence tendency appears at the end of the early stage (80–100 iterations). In RMPO, the cubic mapping optimizes the initial population distribution, realizes symmetrical and uniform exploration of the high-dimensional solution space, and lays a foundation for global search. The nonlinear convergence factor adopts a fast convergence rate in the early stage, which accelerates the fitness decline. The scouter mechanism monitors the population fitness in real time, and activates the global exploration mode when individual fitness is close to the local optimum, effectively avoiding the premature convergence tendency of PO. At the end of the early stage, RMPO fitness consistently underperformed those of the PO, demonstrating a pronounced advantage in global exploration.
At the middle iteration stage, PO’s population diversity decreases rapidly due to the lack of effective perturbation mechanism, and the algorithm is completely trapped in the local optima at the beginning of the middle stage. The fitness stops decreasing and remains stable, which is the most prominent limitation of PO in solving multi-modal and complex-constrained problems. However, RMPO’s scouter mechanism detects that the population fitness stops changing in real time, and adjusts the staying and fear behaviors of the algorithm, switching from local exploitation to global exploration to escape the local optimum. The adaptive hybrid perturbation mechanism judges the population clustering state, and adopts t-distribution perturbation to re-diversify the population, breaking the population degradation of PO. The nonlinear convergence factor slows down the convergence rate in the middle stage, giving the algorithm sufficient time to explore the new solution space. The fitness of RMPO continues to decrease rapidly in the middle stage, and remains lower than that of PO by the end of the middle phase.
At the late iteration stage, PO remains trapped in the local optimum, the fitness has no change, and the optimization performance is completely stagnant. In RMPO, the scouter mechanism confirms that the population approaches the global optimal region, and switches the algorithm to the local exploitation mode. The nonlinear convergence factor adopts the slowest convergence rate, and the differential mutation perturbation is used to conduct precise local search in the feasible region, further improving the optimization accuracy. The hybrid perturbation mechanism maintains the moderate diversity of the population, avoiding the algorithm from falling into the secondary local optimum. The fitness of RMPO converges to the global optimal value in the late stage, and the final fitness is lower than that of PO.
The multi-strategy collaboration of RMPO forms a dynamic adaptive optimization mechanism covering the whole iteration process, which fundamentally overcomes the two core limitations of the original PO: premature convergence and local optima. The cubic mapping and nonlinear convergence factor optimize the early global exploration; the scouter mechanism solves the problem of local optima in the middle stage; the hybrid perturbation mechanism and nonlinear convergence factor realize the precise local search in the late stage. The four strategies complement each other and form a closed-loop optimization, which is the core reason for the significant performance improvement of RMPO.
Analysis of practical engineering applications
To evaluate the effectiveness of RMPO in addressing practical complex engineering challenges, it was tested on function-based design optimization problems, such as pressure vessel design, welded beam design, and stepped cone pulley design. Comparative analyses were conducted with eight other algorithms under identical conditions. All algorithms were executed using a population size of 30 for 300 iterations, and each run was independently repeated 20 times. A feasibility rule-based constraint handling approach is consistently applied to all algorithms for the three mechanical design optimization problems to ensure solution feasibility and comparison fairness. The core rules are: (1) A feasible solution is always superior to an infeasible one regardless of the objective function value; (2) For two feasible solutions, the one with a smaller objective function value is superior; (3) For two infeasible solutions, the one with a smaller total constraint violation is superior. The total constraint violation is calculated as the sum of normalized excess values of all violated constraints.
Pressure vessel design problem
The pressure vessel model is depicted in Figure 13, featuring a cylindrical container sealed at both ends with hemispherical heads. The pressure vessel optimization problem aims to minimize the total cost, comprising the costs of material, forming, and welding.
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This problem is defined by four design variables: shell thickness
Pressure vessel problem.
The optimization solutions obtained using the proposed improved algorithm and comparative algorithms are illustrated in Figure 14, which presents the convergence curves of the iterative process and the corresponding boxplots. It can be observed that the box plot corresponding to RMPO occupies a lower position with a smaller median fitness, indicating its superior overall performance and ability to converge toward higher-quality solutions. Concurrently, its shorter box length reflects its commendable stability. The outcomes of the pressure vessel design optimization are summarized in Table 7. By combining the convergence curve in Figure 14(a) and the corresponding quantitative results, it is observed that RMPO exhibits a relatively gradual convergence trend during the initial iteration phase, which is due to the additional computational cost of the cubic mapping strategy. Nevertheless, the investment in the initial exploration phase of the solution space significantly enhances the algorithm’s ability to conduct a broad global exploitation. This is manifested in the rapid decline of the fitness during the 20th–30th iterations, followed by stable convergence after the 50th iteration, ultimately achieving the optimal fitness (6.52E + 03). The superiority of RMPO can be ascribed to the synergistic integration of multiple enhancement mechanisms. Although the cubic chaotic mapping strategy incurs certain computational costs in early iterations, it greatly enriches the solution space diversity and avoids premature convergence. By incorporating the scouter mechanism from the SSA, the algorithm enhances the population’s capability to escape local optima, whereas the differential mutation strategy provides robust local exploitation capabilities. Furthermore, the t-distribution perturbation strategy helps maintain population diversity throughout the optimization process, and the non-linear convergence factor ensures an adaptive balance between exploration and exploitation. Overall, although RMPO exhibits slower progress in early iterations, its attainment of the optimal solution within 50 iterations remains efficient. This demonstrates the viability of the proposed multi-strategy integrated algorithm in tackling complex constrained optimization problems. For the pressure vessel design problem, the optimal solution obtained by the proposed RMPO algorithm is x* = [13.472, 7.397, 42.098, 176.637] inches, with a maximum constraint violation of 0, confirming that it is a strictly feasible solution that satisfies all design constraints.
Iteration curves and box plots of nine algorithms for the pressure vessel design problem: (a) Iteration curves and (b) Box plots.
Statistical results of different algorithms for the pressure vessel problem.
Welded beam design problem
The optimization of welded beams is to minimize material usage and manufacturing costs.
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The schematic of the welded beam design problem is illustrated in Figure 15. The mathematical model is described by equation (19), where
Welded beam design problem.
Figure 16 presents the convergence curves and boxplots of RMPO and other comparative algorithms for the welded beam design problem. As observed, the boxplot of RMPO exhibits the smallest median fitness, demonstrating optimal convergence performance and low data dispersion, indicating excellent stability. This implies that the RMPO algorithm exhibits a substantially faster convergence rate compared to the other algorithms evaluated. The outcomes of the welded beam design optimization are comprehensively presented in Table 8. Based on the iterative numerical outcomes presented in the table, the RMPO algorithm rapidly converges to the global optimum (1.73E + 00) during the early stages of iteration. Compared to the final convergence value of the original PO algorithm (1.87E+00), RMPO demonstrates markedly improved performance. This indicates that the multi-strategy enhanced RMPO algorithm not only maintains population diversity but also effectively addresses the two inherent challenges in welded beam design: complex stress and deflection constraints. Furthermore, these outcomes support the viability of the proposed strategy improvements and integrations. In contrast, the representative mainstream intelligent optimization algorithms BKA and TOC demonstrated slightly inferior performance on this problem. BKA converges to a local optimum (1.82E + 00) after slow iterations, while TOC escapes a local optimum around the 220th iteration and converges to a suboptimal solution (1.75E + 00). This outcome highlights the RMPO algorithm’s efficient global exploration, high stability, and superior optimization accuracy. In the welded beam optimization, RMPO yields the optimal design vector x* = [0.199, 3.337, 9.192, 0.199] inches. The maximum constraint violation for this solution is 0, indicating that all constraints on shear stress, bending stress, and deflection are fully satisfied.
Iteration curves and box plots of nine algorithms for the welded beam design problem: (a) Iteration curves and (b) Box plots.
Statistical results of different algorithms for the welded beam problem.
Stepped cone pulley design problem
The stepped cone pulley design is illustrated in Figure 17. The weight minimization of a four-step cone pulley constitutes the core objective of this design problem, with five variables defining its geometry: four diameters corresponding to the individual steps of the pulley, and one variable specifying its width. 34 The model involves 11 nonlinear constraints to guarantee that the transmitted power meets the requirement of 0.75 horsepower. The design’s mathematical model is detailed by equation (20).
Stepped cone pulley problem.
During the operation to address the stepped cone pulley design problem, the performance of RMPO was compared with other optimization algorithms. The convergence curves of the iterative process and the box plot are shown in Figure 18. From Figure 18(a), it can be observed that RMPO demonstrated the fastest convergence rate during the optimization process, enabling it to escape local optima rapidly while exhibiting optimal stability and achieving the lowest mean fitness. Figure 18(b) further illustrates that RMPO was the sole algorithm showing no significant outliers deviating markedly from the overall data, indicating its relative stability throughout the optimization process. The comparatively low optimal value achieved by RMPO underscores its robust capabilities in exploration, convergence, and escaping local optima. Concurrently, its minimal standard deviation reflects the algorithm’s commendable robustness. These findings demonstrate RMPO’s efficacy in handling complex test functions, thereby validating its effectiveness and reliability within a competitive algorithmic environment.
Iteration curves and box plots of nine algorithms for the stepped cone pulley design problem: (a) Iteration curves and (b) Box plots
The results of the stepped cone pulley design problem are summarized in Table 9. As shown in the table, during the early stages of iteration, both RMPO and BKA exhibited rapid convergence trends, swiftly locating local optima and significantly outperforming other algorithms. However, in the later iteration phase, BKA succumbed to premature convergence and became trapped in a suboptimal solution (8.64E + 04), whereas RMPO successfully escaped the current local optimum region after approximately 210 iterations and converged to the global optimum (1.78E + 01) after only a few additional iterations, achieving results ten orders of magnitude superior to other algorithms. This demonstrates that the multi-strategy fusion RMPO algorithm possesses stronger global exploration and adaptive capabilities than the original PO algorithm. The multiple inflection points observed during iteration further indicate RMPO’s ability to dynamically adjust optimization strategies across different exploration phases, thereby underscoring its strong feasibility for practical engineering applications. For the stepped cone pulley problem, RMPO achieves the optimal design x* = [40.917, 56.305, 75.068, 90.000, 90.000] mm. The maximum constraint violation is 0, verifying that the solution meets all 11 nonlinear power transmission constraints. In contrast, some comparative algorithms, produce solutions with maximum constraint violations exceeding 100, rendering their low fitness practically meaningless as they correspond to infeasible designs.
Statistical results of different algorithms for the stepped cone pulley problem.
In summary, the overall performance of the RMPO algorithm in addressing various classical practical engineering design problems validates its reliability and efficiency. Compared to many mainstream meta-heuristic algorithms such as BKA and TOC, the proposed algorithm achieves superior convergence accuracy, faster convergence rate, and enhanced exploration capability.
Conclusions
To address the shortcomings of the PO algorithm, including its susceptibility to local optima and limited convergence speed in complex optimization scenarios, this paper developed the RMPO algorithm by integrating multiple strategies. The initial population was enriched using the cubic mapping to enhance the uniformity of solution space distribution. A scouter mechanism from the SSA was introduced to improve global exploitation capability through a dynamic early-warning strategy. A nonlinear convergence factor was incorporated to perturb individuals, thereby increasing solution accuracy and speed. Furthermore, a hybrid strategy combining t-distribution and differential mutation was employed to perturb solution positions, broadening the search area. The performance of RMPO was systematically evaluated through ablation study, benchmark test suite verification and practical engineering application, and the results show that RMPO has significant advantages in optimization accuracy, convergence speed, stability and practical application value, and is a promising optimization tool for complex global optimization problems.
Although the proposed RMPO algorithm has achieved good performance in benchmark tests and engineering applications, it still has certain limitations:
(1) The multi-strategy fusion of RMPO increases the complexity of the algorithm to a certain extent, and the computational time is slightly longer than the original PO algorithm, which needs to be further optimized for the computational efficiency in ultra-high-dimensional (e.g. 100D above) problems.
(2) RMPO is currently applied to single-objective optimization problems, and its extension to multi-objective, multi-constraint and dynamic optimization problems needs further research.
In view of the above limitations, future research will focus on three aspects: simplifying the RMPO algorithm structure to design a lightweight multi-strategy fusion mechanism for higher computational efficiency in ultra-high-dimensional complex optimization while ensuring performance; extending RMPO to multi-objective, multi-constraint and dynamic optimization domains with tailored decision-making and response strategies to expand its application scope; and applying RMPO to cutting-edge engineering fields (e.g. new energy equipment design, intelligent manufacturing, aerospace engineering) to verify its practical value and generalization ability.
Footnotes
Handling Editor: Fatih Karpat
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is sponsored by State Key Laboratory of Clean and Efficient Turbomachinery Power Equipment (DEC8300CG202516944A1228183).
Declaration of conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.