Pile foundation design are known for their inherent uncertainties and complex calculations. The designing process involves non-convex and non-continuous objective functions, resulting in irregular solution patterns. An important consideration in this domain is to achieve the minimum cost with sufficient safety factor at the same time. Therefore, multi-objective optimization is employed to replace repetitive trial and error designs and address the optimization problem simultaneously. This study developed a new metaheuristic algorithm with the name of Multi-Objective Optical Microscope Algorithm (MOOMA) to search for optimal pile designs safety factors and costs. In order to help with the calculations, a widely used commercial geotechnical software PLAXIS 2D is utilized using the Application Programming Interface (API) for pile foundation safety factors assessment. Through a real-life case study involving pile design under multiple loads, the MOOMA algorithm is used to identify optimal pile designs with balanced safety factors and costs. The study also applies indifference curves to determine the optimal design from the Pareto front and compares it with the Taiwanese Geotechnical Standard for more effective design evaluations. The findings highlight the MOOMA algorithm as a valuable tool for designers seeking to strike a balance between safety factors and costs in pile design.

陳清山（2011）。考量多目標及規劃偏好之中小學校舍最適規劃模式。﹝博士論文。國立臺灣科技大學﹞臺灣博碩士論文知識加值系統。https://hdl.handle.net/11296/e49ve9。
Ali, M., Siarry, P., & Pant, M. (2012). An efficient Differential Evolution based algorithm for solving multi-objective optimization problems. European Journal of Operational Research, 217(2), 404-416. https://doi.org/https://doi.org/10.1016/j.ejor.2011.09.025
Audet, C., Cartier, D., Le Digabel, S., & Salomon, L. (2020). Performance indicators in multiobjective optimization. European Journal of Operational Research, 292. https://doi.org/10.1016/j.ejor.2020.11.016
Chan, C. M., Zhang, L., & Ng, J. T. (2009). Optimization of pile groups using hybrid genetic algorithms. Journal of Geotechnical and Geoenvironmental Engineering, 135(4), 497-505.
Chen, X., Zheng, B., & Liu, H. (2011). Optical and digital microscopic imaging techniques and applications in pathology. Anal Cell Pathol (Amst), 34(1-2), 5-18. https://doi.org/10.3233/acp-2011-0006
Chen, Z.-Y., & Shao, C.-M. (1988). Evaluation of minimum factor of safety in slope stability analysis. Canadian geotechnical journal, 25(4), 735-748.
Cheng, M.-Y., & Chen, C.-S. (2011). Optimal planning model for school buildings considering the tradeoff of seismic resistance and cost effectiveness: a Taiwan case study. Structural and Multidisciplinary Optimization, 43, 863-879.
Cheng, M.-Y., & Gosno, R. (2021). SOS 2.0: an evolutionary approach for SOS algorithm. Evolutionary Intelligence, 14, 1-19. https://doi.org/10.1007/s12065-020-00476-8
Cheng, M.-Y., & Sholeh, M. N. (2023). Optical Microscope Algorithm: A Novel Metaheuristic Inspired by Microscope Magnification for Solving Engineering Optimization Problems National Taiwan University of Science and Technology]. Taipei, Taiwan.
Cheng, Y., Wong, H., Leo, C. J., & Lau, C. (2016). Stability of Geotechnical Structures: Theoretical and Numerical Analysis (Vol. 1). Bentham Science Publishers.
Coello, C., Veldhuizen, D., & Lamont, G. (2007). Evolutionary Algorithms for Solving Multi-Objective Problems Second Edition. https://doi.org/10.1007/978-0-387-36797-2
Coello, C. A. C., & Lechuga, M. S. (2002, 12-17 May 2002). MOPSO: a proposal for multiple objective particle swarm optimization. Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600),
Deb, K., & Deb, K. (2014). Multi-objective Optimization. In E. K. Burke & G. Kendall (Eds.), Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques (pp. 403-449). Springer US. https://doi.org/10.1007/978-1-4614-6940-7_15
Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, 6(2), 182-197. https://doi.org/10.1109/4235.996017
Deng, Y., Zhang, K., Yao, Z., Zhao, H., & Li, L. (2023). Parametric analysis and multi-objective optimization of the coupling beam pile structure foundation. Ocean Engineering, 280, 114724. https://doi.org/https://doi.org/10.1016/j.oceaneng.2023.114724
Fonseca, C. M., & Fleming, P. J. (1993). Genetic algorithms for multiobjective optimization: formulationdiscussion and generalization. Icga,
Gandomi, A., Kashani, A., Mousavi, M., & Jalalvandi, M. (2017). Slope stability analysis using evolutionary optimization techniques. International Journal for Numerical and Analytical Methods in Geomechanics, 41(2), 251-264.
Gunantara, N. (2018). A review of multi-objective optimization: Methods and its applications. Cogent Engineering, 5(1), 1502242. https://doi.org/10.1080/23311916.2018.1502242
He, Z., & Yen, G. G. (2014). Comparison of many-objective evolutionary algorithms using performance metrics ensemble. Advances in Engineering Software, 76, 1-8. https://doi.org/https://doi.org/10.1016/j.advengsoft.2014.05.006
Hoang, N.-D., & Pham, A.-D. (2016). Hybrid artificial intelligence approach based on metaheuristic and machine learning for slope stability assessment: A multinational data analysis. Expert Systems with Applications, 46, 60-68.
Hong, W.-C., Dong, Y., Chen, L.-Y., & Wei, S.-Y. (2011). SVR with Hybrid Chaotic Genetic Algorithms for Tourism Demand Forecasting. Applied Soft Computing, 11, 1881-1890. https://doi.org/10.1016/j.asoc.2010.06.003
Huang, V. L., Suganthan, P. N., Qin, A. K., & Baskar, S. (2008). Multiobjective Differential Evolution with External Archive and Harmonic Distance-Based Diversity Measure. In.
Kumar, M., Nallagownden, P., & Elamvazuthi, I. (2016). Advanced Pareto Front Non-Dominated Sorting Multi-Objective Particle Swarm Optimization for Optimal Placement and Sizing of Distributed Generation. Energies, 9, 982. https://doi.org/10.3390/en9120982
Leung, Y. F., Klar, A., Soga, K., & Hoult, N. A. (2017). Superstructure–foundation interaction in multi-objective pile group optimization considering settlement response. Canadian geotechnical journal, 54(10), 1408-1420. https://doi.org/10.1139/cgj-2016-0498
Liu, W., Moayedi, H., Nguyen, H., Lyu, Z., & Bui, D. T. (2021). Proposing two new metaheuristic algorithms of ALO-MLP and SHO-MLP in predicting bearing capacity of circular footing located on horizontal multilayer soil. Engineering with Computers, 37(2), 1537-1547. https://doi.org/10.1007/s00366-019-00897-9
Mankiw, N. G. (2014). Principles of economics. Cengage Learning.
May, R. M. (2004). Simple mathematical models with very complicated dynamics. In B. R. Hunt, T.-Y. Li, J. A. Kennedy, & H. E. Nusse (Eds.), The Theory of Chaotic Attractors (pp. 85-93). Springer New York. https://doi.org/10.1007/978-0-387-21830-4_7
Moayedi, H., Nguyen, H., & Rashid, A. S. A. (2021). Novel metaheuristic classification approach in developing mathematical model-based solutions predicting failure in shallow footing. Engineering with Computers, 37, 223-230.
Öser, C., & Temür, R. (2018). Optimization of pile groups under vertical loads using metaheuristic algorithms. In Handbook of research on predictive modeling and optimization methods in science and engineering (pp. 276-298). IGI Global.
Qi, C., & Tang, X. (2018). Slope stability prediction using integrated metaheuristic and machine learning approaches: A comparative study. Computers & Industrial Engineering, 118, 112-122.
Ravichandran, N., Wang, L., & Rahbari, P. (2022). Robust Optimization for Stability of I-Walls and Levee System Resting on Sandy Foundation. KSCE Journal of Civil Engineering, 26(1), 57-68.
Storn, R. M., & Price, K. (1997). Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization, 11, 341–359.
Tran, D.-H., Cheng, M.-Y., & Prayogo, D. (2016). A novel Multiple Objective Symbiotic Organisms Search (MOSOS) for time–cost–labor utilization tradeoff problem. Knowledge-Based Systems, 94, 132-145. https://doi.org/https://doi.org/10.1016/j.knosys.2015.11.016
Veldhuizen, D. A. V., & Lamont, G. B. (2000, 16-19 July 2000). On measuring multiobjective evolutionary algorithm performance. Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512),
Verma, S., Pant, M., & Snasel, V. (2021). A Comprehensive Review on NSGA-II for Multi-Objective Combinatorial Optimization Problems. IEEE Access, 9. https://doi.org/10.1109/ACCESS.2021.3070634
Wang, Y.-N., Wu, L.-H., & Yuan, X.-F. (2010). Multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 14(3), 193-209. https://doi.org/10.1007/s00500-008-0394-9
Wang, Y., Wu, L., & Yuan, X. (2010). Multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure. Soft Comput., 14, 193-209. https://doi.org/10.1007/s00500-008-0394-9
Wang, Z., Pei, Y., & Li, J. (2023). A Survey on Search Strategy of Evolutionary Multi-Objective Optimization Algorithms. Applied Sciences, 13(7).
Wood, K. L., & Antonsson, E. K. (1989). Computations with imprecise parameters in engineering design: background and theory.
Wulandari, P. S., & Tjandra, D. (2015). Analysis of piled raft foundation on soft soil using PLAXIS 2D. Procedia Engineering, 125, 363-367.
Zimmermann, H. J. (2001). Fuzzy Control. In H. J. Zimmermann (Ed.), Fuzzy Set Theory—and Its Applications (pp. 223-264). Springer Netherlands. https://doi.org/10.1007/978-94-010-0646-0_11
Zitzler, E., & Thiele, L. (1999). Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, 3(4), 257-271. https://doi.org/10.1109/4235.797969