Improved Chebyshev Distance FOX Algorithm for Solving Numerical and Engineering Optimization Problems
DOI:
https://doi.org/10.31530/cjnst.2026.2.1.8Keywords:
CDFOX algorithm, metaheuristic, exploration, optimization, exploitationAbstract
Background: FOX optimizer, inspired by the smart hunting behavior of red foxes, is a powerful optimization algorithm used to solve difficult engineering and real-world problems. However, constrained exploration often affects the conventional FOX algorithm, leading to premature convergence and agent accumulation at local optima.
Aim: The aim of this research is to improve the performance of the FOX optimization algorithm by optimizing the exploration and exploitation processes of the FOX algorithm. The proposed changes aim to improve the global search process in the initial phase of the search process and ensure a good local search process in the later phase of the search process. The purpose of this research is to improve the balance between exploration and exploitation and to reduce the risk of premature convergence and entrapment in local optima.
Methodology: For this, the Chebyshev Distance FOX algorithm (CDFOX) is proposed, which effectively facilitates exploration and exploitation through a well-structured, multi-step process. First, Chebyshev distance is used to calculate how far two points are by taking the largest absolute difference in any single dimension. Second, using a chaos map in both time and exploration improves the algorithm by adding controlled randomness. This helps the search agents avoid getting stuck, explore the search space more widely, and reach better solutions faster. Finally, adaptive scaling of the search performance expands the search radius to preserve diversity and enable wider, circular exploration of the search space.
Result: CDFOX was tested rigorously for its performance against 10 established and competitive optimizers with 63 benchmarks, comprising 23 traditional test problems, 10 benchmarks from CEC2019, and 30 benchmarks from CEC2017. From these results, it was confirmed that CDFOX performed better than or equally as well as other optimizers for 82% of classical benchmarks, 80% of CEC2019 benchmarks, and 43% of CEC2017 benchmarks. Moreover, results from real-world engineering problems like pressure vessel design, gear train design, and cantilever beam problems further support its effectiveness and its efficiency. and merits in addressing hard problems in various domains.
Conclusion: CDFOX is an improvement over the original FOX optimizer in the sense that the exploration and exploitation of the search space are enhanced through a multi-strategy approach. The adaptive nature of the CDFOX helps in the avoidance of premature convergence and enhances the global searchability. It is evident from the results that CDFOX is a powerful optimizer for solving complex optimization problems.
Downloads
References
[1] K. SM, “Cuckoo Search and Firefly Algorithms-Biologically Inspired Algorithms,” Open Access Biostat. Bioinforma., vol. 1, pp. 1–4, 2018, doi: 10.31031/oabb.2018.01.000514.
[2] S. Zhang, “Real-life Application of Optimization Problem,” Theor. Nat. Sci., vol. 5, no. 1, pp. 53–57, 2023, doi: 10.54254/2753-8818/5/20230271.
[3] D. Liñán Romero, G. Contreras-Zarazua, E. Sánchez Ramírez, J. Segovia-Hernandez, and L. Ricardez-Sandoval, “A Hybrid Deterministic-Stochastic Algorithm for the Optimal Design of Process Flowsheets with Ordered Discrete Decisions,” Comput. Chem. Eng., vol. 180, p. 108501, Nov. 2023, doi: 10.1016/j.compchemeng.2023.108501.
[4] W. G. Alshanti, I. M. Batiha, M. A. Hammad, and R. Khalil, “A novel analytical approach for solving partial differential equations via a tensor product theory of Banach spaces,” Partial Differ. Equations Appl. Math., vol. 8, no. June, p. 100531, 2023, doi: 10.1016/j.padiff.2023.100531.
[5] M. Dehghani, E. Trojovská, and P. Trojovský, “A new human-based metaheuristic algorithm for solving optimization problems on the base of simulation of driving training process,” Sci. Rep., vol. 12, no. 1, pp. 1–21, 2022, doi: 10.1038/s41598-022-14225-7.
[6] Z. Marszałek, “Parallelization of modified merge sort algorithm,” Symmetry (Basel)., vol. 9, no. 9, 2017, doi: 10.3390/sym9090176.
[7] S. Saha, M. K.Bhaumik, and S. Das, “A New Modified Linear Search Algorithm,” Int. J. Math. Trends Technol., vol. 65, no. 12, pp. 148–152, 2019, doi: 10.14445/22315373/ijmtt-v65i12p516.
[8] Z. Montazeri, T. Niknam, J. Aghaei, O. P. Malik, M. Dehghani, and G. Dhiman, “Golf Optimization Algorithm: A New Game-Based Metaheuristic Algorithm and Its Application to Energy Commitment Problem Considering Resilience,” Biomimetics, vol. 8, no. 5, 2023, doi: 10.3390/biomimetics8050386.
[9] I. A. Falahah et al., “Frilled Lizard Optimization: A Novel Bio-Inspired Optimizer for Solving Engineering Applications,” Comput. Mater. Contin., vol. 79, no. 3, pp. 3631–3678, 2024, doi: 10.32604/cmc.2024.053189.
[10] C. Audet and W. Hare, “Genetic Algorithms,” Springer Ser. Oper. Res. Financ. Eng., pp. 57–73, 2017, doi: 10.1007/978-3-319-68913-5_4.
[11] A. Slowik, “Particle Swarm Optimization,” Ind. Electron. Handb. - Five Vol. Set, no. May 2011, 2011, doi: 10.1007/978-3-319-46173-1_2.
[12] R. S. Prado, R. C. P. Silva, F. G. Guimarães, and O. M. Neto, “A New Differential Evolution Based Metaheuristic for Discrete Optimization,” Int. J. Nat. Comput. Res., vol. 1, no. 2, pp. 15–32, 2010, doi: 10.4018/jncr.2010040102.
[13] E. Trojovska, M. Dehghani, and P. Trojovsky, “Zebra Optimization Algorithm: A New Bio-Inspired Optimization Algorithm for Solving Optimization Algorithm,” IEEE Access, vol. 10, pp. 49445–49473, 2022, doi: 10.1109/ACCESS.2022.3172789.
[14] M. C. Su, J. H. Chen, A. M. Utami, S. C. Lin, and H. H. Wei, “Dove Swarm Optimization Algorithm,” IEEE Access, vol. 10, pp. 46690–46696, 2022, doi: 10.1109/ACCESS.2022.3170112.
[15] T. S. L. V. Ayyarao et al., “War Strategy Optimization Algorithm: A New Effective Metaheuristic Algorithm for Global Optimization,” IEEE Access, vol. 10, pp. 25073–25105, 2022, doi: 10.1109/ACCESS.2022.3153493.
[16] O. N. Oyelade, A. E. S. Ezugwu, T. I. A. Mohamed, and L. Abualigah, “Ebola Optimization Search Algorithm: A New Nature-Inspired Metaheuristic Optimization Algorithm,” IEEE Access, vol. 10, pp. 16150–16177, 2022, doi: 10.1109/ACCESS.2022.3147821.
[17] J. Tang, G. Liu, and Q. Pan, “A Review on Representative Swarm Intelligence Algorithms for Solving Optimization Problems: Applications and Trends,” IEEE/CAA J. Autom. Sin., vol. 8, pp. 1627–1643, Oct. 2021, doi: 10.1109/JAS.2021.1004129.
[18] Z. K. A. Dler Hasan, Hardi M. Mohammed, “Modified FOX Optimizer for Solving optimization problems,” 2025, [Online]. Available: doi:10.48550/arXiv.2502.20027
[19] D. Wang, D. Tan, and L. Liu, “Particle swarm optimization algorithm: an overview,” Soft Comput., vol. 22, no. 2, pp. 387–408, 2018, doi: 10.1007/s00500-016-2474-6.
[20] H. Givi and M. Hubalovska, “Skill Optimization Algorithm: A New Human-Based Metaheuristic Technique,” Comput. Mater. Contin., vol. 74, no. 1, pp. 179–202, 2023, doi: 10.32604/cmc.2023.030379.
[21] R. V. Rao, V. J. Savsani, and D. P. Vakharia, “Teaching-learning-based optimization: A novel method for constrained mechanical design optimization problems,” CAD Comput. Aided Des., vol. 43, no. 3, pp. 303–315, 2011, doi: 10.1016/j.cad.2010.12.015.
[22] L. Sharma, V. L. Chellapilla, and P. Chellapilla, “Socio-inspired evolutionary algorithms: a unified framework and survey,” Soft Comput., vol. 27, no. 19, pp. 14127–14156, 2023, doi: 10.1007/s00500-023-07929-z.
[23] B. A. Umit Can1, *, “Physics Based Metaheuristic Algorithms for Global Optimization,” 2015, doi: http://creativecommons.org/licenses/by-nc/4.0/.
[24] A. Kaveh and T. Bakhshpoori, “Metaheuristics: Outlines, MATLAB Codes and Examples,” Metaheuristics: Outlines, MATLAB Codes and Examples, pp. 1–190, 2019, doi: 10.1007/978-3-030-04067-3.
[25] H. Eskandar, A. Sadollah, A. Bahreininejad, and M. Hamdi, “Water cycle algorithm - A novel metaheuristic optimization method for solving constrained engineering optimization problems,” Comput. Struct., vol. 110–111, pp. 151–166, 2012, doi: 10.1016/j.compstruc.2012.07.010.
[26] S. Mirjalili, A. H. Gandomi, S. Z. Mirjalili, S. Saremi, H. Faris, and S. M. Mirjalili, “Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems,” Adv. Eng. Softw., vol. 114, pp. 163–191, 2017, doi: 10.1016/j.advengsoft.2017.07.002.
[27] J. Lian et al., “Parrot optimizer: Algorithm and applications to medical problems,” Comput. Biol. Med., vol. 172, 2024, doi: 10.1016/j.compbiomed.2024.108064.
[28] Y. Fang, S. Na, M. W. Mahoney, and M. Kolar, “Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems,” SIAM J. Optim., vol. 34, no. 2, pp. 2007–2037, 2024, doi: 10.1137/22M1537862.
[29] T. Strutynska, “Social and economic problems of regional TV news in the conditions of the Russian-Ukrainian War (on the example of the TV channel ‘3-Studio’ in Ivano-Frankivsk),” Commun. Commun. Technol., vol. 1813, no. 24, pp. 48–56, 2024, doi: 10.15421/292406.
[30] G. ATALI, I. PEHLIVAN, B. GÜREVIN, and H. I. SEKER, “Chaos in metaheuristic based artificial intelligence algorithms: A short review,” Turkish J. Electr. Eng. Comput. Sci., vol. 29, no. 3, pp. 1354–1367, 2021, doi: 10.3906/elk-2102-5.
[31] X. L. Y. S. Ke Xue∗, “Escaping Local Optima in Global Placement”, doi: https://arxiv.org/pdf/2402.18311.
[32] M. Banaie-Dezfouli, M. H. Nadimi-Shahraki, and Z. Beheshti, “R-GWO: Representative-based grey wolf optimizer for solving engineering problems,” Appl. Soft Comput., vol. 106, p. 107328, 2021, doi: 10.1016/j.asoc.2021.107328.
[33] T. A. R. *Hardi M. Mohammed1, “FOX: A Fox-inspired Optimization Algorithm,” 2022, [Online]. Available: https://doi.org/10.21203/rs.3.rs-1939478/v1
[34] G. Hu, K. Song, X. Li, and Y. Wang, “DEMFFA: a multi-strategy modified Fennec Fox algorithm with mixed improved differential evolutionary variation strategies,” J. Big Data, vol. 11, no. 1, 2024, doi: 10.1186/s40537-024-00917-6.
[35] Z. Fu, J. An, Q. Yang, H. Yuan, Y. Sun, and H. Ebrahimian, “Skin cancer detection using Kernel Fuzzy C-means and Developed Red Fox Optimization algorithm,” Biomed. Signal Process. Control, vol. 71, p. 103160, 2022, doi: https://doi.org/10.1016/j.bspc.2021.103160.
[36] Z. Zhang, X. Wang, and L. Cao, “FOX Optimization Algorithm Based on Adaptive Spiral Flight and Multi-Strategy Fusion,” Biomimetics, vol. 9, no. 9, p. 524, 2024, doi: 10.3390/biomimetics9090524.
[37] M. A. Jumaah, Y. H. Ali, and T. A. Rashid, “Improved FOX Optimization Algorithm,” pp. 1–34, 2025, [Online]. Available: http://arxiv.org/abs/2504.09574
[38] A. K. Feda, M. Adegboye, O. R. Adegboye, E. B. Agyekum, W. Fendzi Mbasso, and S. Kamel, “S-shaped grey wolf optimizer-based FOX algorithm for feature selection,” Heliyon, vol. 10, no. 2, p. e24192, 2024, doi: 10.1016/j.heliyon.2024.e24192.
[39] D. P. Nikezić, D. S. Radivojević, N. S. Mirkov, I. M. Lazović, and T. A. Miljojčić, “Symmetric U-Net Model Tuned by FOX Metaheuristic Algorithm for Global Prediction of High Aerosol Concentrations,” 2024.
[40] Z. Huo, S. Liu, and H. Ebrahimian, “Aircraft Energy Management System Using Chaos Red Fox Optimization Algorithm,” J. Electr. Eng. Technol., vol. 17, Aug. 2021, doi: 10.1007/s42835-021-00884-5.
[41] L. Xu and M. Mohammadi, “Brain tumor diagnosis from MRI based on Mobilenetv2 optimized by contracted fox optimization algorithm,” Heliyon, vol. 10, no. 1, 2024, doi: 10.1016/j.heliyon.2023.e23866.
[42] T. T. & A. F. K. Fahrettin Burak Demir, “A chaotic optimization method based on logistic-sine map for numerical function optimization,” pp. 14227–14239, 2022, [Online]. Available: https://doi.org/10.1007/s00521-020-04815-9
[43] A. E. Haryati, “Implementation of Minkowski-Chebyshev Distance in Fuzzy Subtractive Clustering,” vol. 2, no. 2, pp. 82–87, 2021, doi: 10.20885/EKSAKTA.vol2.iss2.art1.
[44] V. S. D. M. Sahu, P. Samal, and C. K. Panigrahi, “Tyrannosaurus optimization algorithm: A new nature-inspired meta-heuristic algorithm for solving optimal control problems,” e-Prime - Adv. Electr. Eng. Electron. Energy, vol. 5, no. August, p. 100243, 2023, doi: 10.1016/j.prime.2023.100243.
[45] O. Al-Baik et al., “Pufferfish Optimization Algorithm: A New Bio-Inspired Metaheuristic Algorithm for Solving Optimization Problems,” Biomimetics, vol. 9, no. 2, 2024, doi: 10.3390/biomimetics9020065.
[46] A. Qi, D. Zhao, A. A. Heidari, L. Liu, Y. Chen, and H. Chen, “FATA: An efficient optimization method based on geophysics,” Neurocomputing, vol. 607, 2024, doi: 10.1016/j.neucom.2024.128289.
[47] C. Yuan et al., “Artemisinin optimization based on malaria therapy: Algorithm and applications to medical image segmentation,” Displays, vol. 84, no. May, 2024, doi: 10.1016/j.displa.2024.102740.
[48] H. A. Shehadeh, I. Ahmedy, and M. Y. I. Idris, “Sperm swarm optimization algorithm for optimizing wireless sensor network challenges,” ACM Int. Conf. Proceeding Ser., no. February, pp. 53–59, 2018, doi: 10.1145/3193092.3193100.
[49] Y. Fu, D. Liu, J. Chen, and L. He, Secretary bird optimization algorithm: a new metaheuristic for solving global optimization problems, vol. 57, no. 5. Springer Netherlands, 2024. doi: 10.1007/s10462-024-10729-y.
[50] K. K. Mishra, S. Tiwari, and A. K. Misra, “A bio inspired algorithm for solving optimization problems,” 2011 2nd Int. Conf. Comput. Commun. Technol. ICCCT-2011, pp. 653–659, 2011, doi: 10.1109/ICCCT.2011.6075211.
[51] P. Sarangi and P. Mohapatra, “Chaotic-Based Mountain Gazelle Optimizer for Solving Optimization Problems,” Int. J. Comput. Intell. Syst., vol. 17, no. 1, 2024, doi: 10.1007/s44196-024-00444-5.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Charmo Journal of Natural Sciences and Technologies
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
