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<p>Shallow foundations are crucial elements for ensuring the stability of structures. Due to financial constraints and structural requirements, the design of this structural member poses a significant challenge. In recent years, researchers have managed to provide effective approaches for foundation design using metaheuristic algorithms. This study presents a novel metaheuristic optimization algorithm, named the K-means-based Gazelle Optimization Algorithm (KMGO), for the economical and safe design of foundations. This is achieved by enhancing the initial population selection of the standard MGO using the K-means clustering method, which improves its accuracy and convergence rate in the geometric optimization of foundations. The model was validated by evaluating its performance on the optimization of a reinforced concrete foundation in comparison with nine other metaheuristic algorithms. Subsequently, a probabilistic assessment of the obtained optimal design, based on the Monte Carlo Simulation (MCS) method, demonstrated the necessity of a probabilistic approach for safe design. The results revealed that the cost and carbon-optimal design was susceptible to an 85% probability of failure under uncertainty. Specifically, the proposed KMGO achieved a global minimum construction cost of $1450.24 with a standard deviation of 1.86 × 10<sup>−13</sup> and a global minimum CO<sub>2</sub> emission of 1190&#xa0;kg with zero standard deviation, demonstrating perfect convergence across all ten independent runs. In contrast, the standard MGO failed to converge in three and four runs for the respective objective functions. Under geotechnical and structural uncertainty, the cost-optimal and CO<sub>2</sub>-optimal designs exhibited failure probabilities of 85 and 63%, respectively. In the second phase of the study, a probabilistic approach was implemented by generating uncertainty scenarios to achieve optimal designs using the proposed algorithm. This approach yielded a wide range of solutions, offering balanced trade-offs between safety levels and construction costs. The findings indicate that the proposed algorithm is capable of being effectively utilized in real-world civil engineering projects for the safe and economical design of foundations.</p>

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A novel K-means–based gazelle optimization algorithm for optimal and safe design of shallow foundations: a deterministic and probabilistic study

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Aref Azizian,
  • Sajad Tavakoli,
  • Ahmad Maleki,
  • Hadi Azizian

摘要

Shallow foundations are crucial elements for ensuring the stability of structures. Due to financial constraints and structural requirements, the design of this structural member poses a significant challenge. In recent years, researchers have managed to provide effective approaches for foundation design using metaheuristic algorithms. This study presents a novel metaheuristic optimization algorithm, named the K-means-based Gazelle Optimization Algorithm (KMGO), for the economical and safe design of foundations. This is achieved by enhancing the initial population selection of the standard MGO using the K-means clustering method, which improves its accuracy and convergence rate in the geometric optimization of foundations. The model was validated by evaluating its performance on the optimization of a reinforced concrete foundation in comparison with nine other metaheuristic algorithms. Subsequently, a probabilistic assessment of the obtained optimal design, based on the Monte Carlo Simulation (MCS) method, demonstrated the necessity of a probabilistic approach for safe design. The results revealed that the cost and carbon-optimal design was susceptible to an 85% probability of failure under uncertainty. Specifically, the proposed KMGO achieved a global minimum construction cost of $1450.24 with a standard deviation of 1.86 × 10−13 and a global minimum CO2 emission of 1190 kg with zero standard deviation, demonstrating perfect convergence across all ten independent runs. In contrast, the standard MGO failed to converge in three and four runs for the respective objective functions. Under geotechnical and structural uncertainty, the cost-optimal and CO2-optimal designs exhibited failure probabilities of 85 and 63%, respectively. In the second phase of the study, a probabilistic approach was implemented by generating uncertainty scenarios to achieve optimal designs using the proposed algorithm. This approach yielded a wide range of solutions, offering balanced trade-offs between safety levels and construction costs. The findings indicate that the proposed algorithm is capable of being effectively utilized in real-world civil engineering projects for the safe and economical design of foundations.