Introduction
In the realm of engineering design, optimizing parameters is a crucial endeavor. While the Slime Mould Algorithm (SMA) has proven effective in parameter discovery under constrained conditions, it faces challenges in achieving global convergence and avoiding local optima traps when tackling complex tasks. To address these limitations, this article introduces an enhanced variant of SMA, termed the Chaotic Covariance Harris Slime Mould Algorithm (CCHSMA).
CCHSMA integrates a Chaotic Local Search (CLS) mechanism to improve initial population diversity, and combines Covariance Matrix Adaptation (CMA) and Harris Hawks Optimization (HHO) strategies to enhance global search efficiency. By incorporating these synergistic techniques, CCHSMA aims to improve search quality and reduce the likelihood of getting trapped in local optima.
The key contributions of this work are:
- Development of CCHSMA: An advanced version of the SMA, demonstrating enhanced search efficiency, local search capabilities, and the ability to avoid local optima.
- Performance Evaluation: Comparative analysis of CCHSMA against leading swarm intelligence algorithms using the IEEE CEC2017 benchmarks, showcasing its superior efficacy.
- Real-World Applications: Successful implementation of CCHSMA in engineering design optimization, effectively translating real-world constraints into mathematical models, with applications in Tension/Compression Spring Design (TCSD), Pressure Vessel Design (PVD), and Three-Bar Truss Design (TBTD).
The experimental results validate CCHSMA’s effectiveness in engineering optimization, highlighting its reliability and applicability across diverse scenarios.
Background: Slime Mould Algorithm (SMA)
The Slime Mould Algorithm (SMA) is modeled on the foraging behavior of slime moulds, particularly their complex movement patterns. This algorithm draws an analogy between slime mould foraging dynamics and optimization processes. During foraging, both the spatial distribution and the quantity of food significantly influence the slime mould’s path.
The distribution is influenced not only by proximity but also by the quantity of food, with greater abundance exerting a stronger pull on the slime mould, effectively increasing its ‘weight’ in the decision-making process. Slime moulds autonomously determine their actions by evaluating the food’s quantity and density. If highly concentrated food sources are far from the slime mould, their influence on its behavior diminishes.
Three key processes encapsulate slime mould behavior, providing a basis for mathematical modeling of their foraging patterns:
- Pheromone Selection: The slime mould selects its foraging path by analyzing pheromones, leading to varied weighting in its decision-making process.
- Food Concentration: The slime mould’s movement is influenced by the concentration of food sources, with higher concentrations exerting a stronger pull.
- Foraging Dynamics: The slime mould’s foraging behavior is dynamic, adapting to changes in food distribution and quantity over time.
These behavioral characteristics are mathematically represented within the SMA framework, allowing for the optimization of complex engineering problems.
Proposed CCHSMA Method
To enhance the performance of the original SMA, the CCHSMA framework integrates several strategies, including Chaotic Local Search (CLS), Covariance Matrix Adaptation (CMA), and Harris Hawks Optimization (HHO).
Chaotic Local Search (CLS)
Effective initialization is a crucial aspect of optimization algorithms. Chaotic mapping, known for its unpredictability and non-repetitiveness, is a widely used method for this purpose. It generates random numbers between 0 and 1, facilitating the exploration of diverse regions within the search space and accelerating convergence toward optimal solutions.
In CCHSMA, we employ logistic mapping as a classic example of chaotic initialization, which is defined as:
X(t+1) = μ * X(t) * (1 - X(t))
where μ is the most important parameter, set to 4, t denotes the number of iterations, and X(t) represents the current solution.
The CLS mechanism’s initialization significantly enhances experimental efficiency. However, due to reduced search capability in later stages, it necessitates the integration with other mechanisms, such as CMA, to further improve performance.
Covariance Matrix Adaptation (CMA)
Covariance measures the correlation between two variables, and the covariance matrix represents these correlations across multiple dimensions of a multidimensional random variable. CMA constructs a mutation distribution with exponentially decreasing weights for selected mutations, leveraging past generations’ selections to inform the direction and step size of new mutations.
The role of CMA in the CCHSMA algorithm can be delineated in three distinct steps:
- Initial Population Generation: The algorithm begins by generating random initial solutions, forming an overall population through a normal distribution centered around these solutions.
- Parent Population Selection: A portion of the optimal solutions is selected for future populations, based on a weighted average operation.
- Covariance Matrix and Step Size Update: The covariance matrix and step size are updated to control the mutation distribution, enhancing the algorithm’s exploration and exploitation capabilities.
The CMA mechanism helps refine the step size and search direction more systematically, improving the population’s search capabilities and broadening the search scope.
Harris Hawks Optimization (HHO)
HHO is inspired by the cooperative hunting behavior of Harris Hawks in nature. It simulates their search and capture strategies, dynamically adapting to different environments. In the CCHSMA algorithm, the optimal solution represents the prey, with the iterative process of HHO mirroring the hawks’ pursuit and capture strategies.
HHO comprises two main phases:
- Exploration Phase: This phase involves two strategies for locating prey, allowing the algorithm to explore different regions of the search space.
- Exploitation Phase: This phase uses four strategies for collaborative capture, enabling the algorithm to exploit promising regions more effectively.
The integration of the HHO mechanism strengthens the mid-search process, enhancing CCHSMA’s global search efficiency and ability to avoid local optima.
CCHSMA Framework
The overall CCHSMA framework combines the strengths of the CLS, CMA, and HHO mechanisms. During the initialization phase, the CLS strategy is applied to the population to ensure a uniform distribution across the entire search space, significantly enhancing search efficiency.
In the early stages, the SMA demonstrates efficient search capabilities, making it well-suited for iterative search and adaptation. However, as the search progresses into middle and late stages, SMA’s efficiency declines. To address this, the CMA and HHO mechanisms are introduced to enhance exploration and exploitation, respectively.
The experimental integration of SMA with CMA and HHO proves effective, improving the population’s search capabilities and broadening the search scope. Additionally, it refines step size and search direction more systematically, allowing CCHSMA to effectively tackle complex engineering optimization problems.
Experiment Design and Results
To evaluate the performance of the CCHSMA method, extensive experiments were conducted using the IEEE CEC2017 benchmark functions. The CCHSMA was compared against six SMA variants, seven fundamental metaheuristics, and six advanced swarm intelligence algorithms.
Validation of the Effectiveness of Different Mechanisms
The first set of experiments examined the effects of various mechanisms on the SMA algorithm. The variants tested included CMAHHOSMA, CLSHHOSMA, CLSCMASMA, CLSSMA, CMASMA, and HHOSMA, each featuring a unique integration of mechanisms.
The results, presented in Table 2 and Figure 2, demonstrate that the CCHSMA outperforms its counterparts across most benchmark functions. The integration of the CLS, CMA, and HHO mechanisms in CCHSMA proves to be a synergistic approach, harnessing the strengths of each individual strategy and surpassing the sum of their separate impacts.
Qualitative Analysis of the Improved Algorithm
To further elucidate the fundamental characteristics of the CCHSMA, a qualitative analysis was conducted, focusing on three dimensions: search trajectory, average fitness, and population equilibrium.
The visual representations in Figure 3 reveal that CCHSMA exhibits efficient initialization, dynamic exploration, and consistent convergence towards the optimal solutions, outperforming the original SMA and its variants.
Comparison with Original Swarm Intelligence Algorithms
The CCHSMA was then compared against seven well-known metaheuristic algorithms: SSA, BA, PSO, ABC, FA, DE, and HHO. The results, presented in Table 4 and Figure 4, demonstrate that CCHSMA consistently outperforms its counterparts across the benchmark functions, showcasing its superior global search capability and ability to avoid local optima.
Comparison with Swarm Intelligence Algorithm Variants
The CCHSMA was further evaluated against six advanced swarm intelligence algorithm variants: OMGSCA, RDWOA, EOBLSSA, GBHHO, GOTLBO, and ALCPSO. The comparative analysis, summarized in Table 6 and Figure 5, validates CCHSMA’s robustness and versatility, as it achieves the highest average Friedman test result of 2.61333.
Comparative Analysis of Algorithms
To further demonstrate the robustness of the CCHSMA, additional experiments were conducted, expanding the comparison to include XMACO, LXMWOA, SCBA, and QCSCA. These experiments were performed across 30, 50, and 100 dimensions, with varying function evaluation counts.
The results, presented in Table 8 and Figures 6-7, confirm CCHSMA’s consistent performance, as it maintains its top ranking across different dimensional settings and function evaluation counts.
Engineering Problems
To validate the practical utility of the CCHSMA, it was applied to three engineering design optimization problems: Tension/Compression Spring Design (TCSD), Pressure Vessel Design (PVD), and Three-Bar Truss Design (TBTD).
Tension/Compression Spring Design (TCSD)
In the TCSD problem, the primary objective is to minimize the weight of the spring while ensuring its functional efficacy. CCHSMA demonstrated superior performance compared to other advanced algorithms, achieving the optimal solution of 0.012665233.
Pressure Vessel Design (PVD)
The PVD problem aims to minimize the total cost of cylindrical pressure vessel components, considering factors like material performance, force characteristics, and structural connections. CCHSMA outperformed other algorithms, obtaining the optimal solution of 6059.71445.
Three-Bar Truss Design (TBTD)
The TBTD problem focuses on minimizing the weight of a structural design while complying with constraints related to stress, deflection, and buckling. CCHSMA proved to be the most effective algorithm, delivering the optimal solution of 263.8958434.
The successful implementation of CCHSMA in these engineering design challenges highlights its reliability and applicability in translating real-world constraints into mathematical models, resulting in high-quality, cost-efficient designs.
Conclusion
This article introduced the Chaotic Covariance Harris Slime Mould Algorithm (CCHSMA), an enhanced variant of the Slime Mould Algorithm (SMA) designed to address the limitations of the original algorithm in achieving global convergence and avoiding local optima traps.
The integration of Chaotic Local Search (CLS), Covariance Matrix Adaptation (CMA), and Harris Hawks Optimization (HHO) mechanisms in the CCHSMA framework has proven effective in improving search efficiency, local search capabilities, and the ability to avoid local optima.
Extensive experiments using the IEEE CEC2017 benchmark functions and real-world engineering design optimization problems, such as Tension/Compression Spring Design, Pressure Vessel Design, and Three-Bar Truss Design, have demonstrated the superiority of CCHSMA over other leading swarm intelligence algorithms.
The key findings of this research include:
- CCHSMA Development: The CCHSMA framework combines the strengths of CLS, CMA, and HHO, resulting in an advanced algorithm that outperforms the original SMA and its variants.
- Benchmark Performance: CCHSMA consistently achieves superior results on the IEEE CEC2017 benchmark functions, showcasing its global search capability and ability to avoid local optima.
- Engineering Applications: CCHSMA’s successful implementation in engineering design optimization problems highlights its reliability and applicability in translating real-world constraints into effective solutions.
While the current experiments have achieved theoretical optima in some cases, there is still room for improvement in addressing local optimality and further enhancing global search capabilities. Future research will explore the integration of CCHSMA with multi-objective strategies and deep learning frameworks to address a wider range of engineering challenges.
The IT Fix blog is dedicated to providing practical tips and in-depth insights on technology, computer repair, and IT solutions. For more information, please visit https://itfix.org.uk/.
