Fibonacci Kodlaması ve k-Zeckendorf Gösterimleri ile Yeni Bir Genetik Algoritma Modeli

Year 2025, Volume: 7 Issue: 2, 228 - 244, 31.08.2025

Yunus Emre Göktepe , Fikri Köken , Halime Ergun

Abstract

Bu çalışma, karmaşık optimizasyon problemlerini çözmenin verimliliğini artırmak için Fibonacci kodlamasını ve k-Zeckendorf gösterimlerini kullanan yeni bir genetik algoritma (GA) modeli sunmaktadır. Geleneksel GA, kromozomları temsil etmek için ikili veya sayısal kodlamayı kullanır, ancak bu çalışma Fibonacci dizisine ve Zeckendorf teoremine dayanan alternatif bir yaklaşım önermektedir. Bu gösterimleri GA çerçevesine dahil ederek, model arama sürecini iyileştirmeyi ve daha etkili çözümlere yol açmayı amaçlamaktadır. Bu yaklaşımın önemi, GA'ların performansını etkileyen kritik bir faktör olan kromozom gösterimini iyileştirme yeteneğinde yatmaktadır. Yeni kodlama şemaları, algoritmanın keşif ve yararlanma aşamalarını geliştirerek, optimum çözümlere doğru daha verimli bir yakınsama sağlar. Model iki zorlu problemde test edilmiştir: yüksek dereceli bir polinomun optimum değerini bulma ve dikdörtgen prizmanın hacmini optimize etme. Sonuçlar, önerilen yöntemin standart GA'dan daha başarılı olduğunu göstermektedir. Bu araştırma, alternatif kromozom gösterimlerinin GA performansını önemli ölçüde etkileyebileceğini göstermektedir. Ayrıca, özellikle karmaşık polinom denklemleri ve geometrik kısıtlamaları içeren optimizasyon problemleri için yeni bir çerçeve sağlar. GA'ların daha faklı optimizasyon problemlerine uygulanabileceğini gösterir ve hesaplamalı matematik ve mühendislik tasarımındaki potansiyelini vurgular.

Keywords

Fibonacci sayıları , Genetik algoritma , Optimizasyon , Zeckendorf gösterimleri

A Novel Genetic Algorithm Model with Fibonacci Encoding and k-Zeckendorf Representations

Abstract

This study presents a new genetic algorithm (GA) model that uses Fibonacci coding and k-Zeckendorf representations to improve the efficiency of solving complex optimization problems. Traditional GA uses binary or numerical coding to represent chromosomes, but this study proposes an alternative approach based on Fibonacci sequence and Zeckendorf theorem. By incorporating these representations into the GA framework, it aims to improve the model search process and lead to more efficient solutions. The importance of this approach lies in its ability to improve the chromosome representation, which is a critical factor affecting the performance of GAs. The new coding schemes improve the exploration and exploitation phases of the algorithm, allowing for more efficient convergence towards optimal solutions. The model is tested on two challenging problems: finding the optimum value of a high-degree polynomial and optimizing the volume of a rectangular prism. The results show that the proposed method outperforms the standard GA. This research shows that alternative chromosome representations can significantly affect GA performance. It also provides a new framework for optimization problems, especially those involving complex polynomial equations and geometric constraints. It shows that GAs can be applied to more diverse optimization problems and highlights their potential in computational mathematics and engineering design.

Keywords

Fibonacci numbers , Genetic algorithm , Optimization , Zeckendorf representations

References

• M. Mitchell, An Introduction to Genetic Algorithms, MIT Press, 1998.
• A.O. Faouri̇, P. Kasap, Maximum likelihood estimation for the Nakagami distribution using particle swarm optimization algorithm with applications, Necmettin Erbakan University Journal of Science and Engineering. 5(2) (2023), 209-218. doi:10.47112/neufmbd.2023.19.
• A. Pektaş, İ. Onur, Ağaç tohum algoritmasının kümeleme problemlerine uygulanması, Necmettin Erbakan University Journal of Science and Engineering. 4 (2022), 1-10.
• A. Ünlü, İ. İlhan, A novel hybrid gray wolf optimization algorithm with harmony search to solve multi-level image thresholding problem, Necmettin Erbakan University Journal of Science and Engineering. 5(2) (2023), 230-245. doi:10.47112/neufmbd.2023.21.
• K.R. Srimathi, A. Padmarekha, K.S. Anandh, Automated construction schedule optimisation using genetic algorithm, Asian Journal of Civil Engineering. 24 (2023), 3521-3528. doi:10.1007/s42107-023-00729-8.
• O. Ulkir, G. Akgun, Predicting and optimising the surface roughness of additive manufactured parts using an artificial neural network model and genetic algorithm, Science and Technology of Welding and Joining. 28 (2023), 548-557. doi:10.1080/13621718.2023.2200572.
• E. Singh, S.S. Afshari, X. Liang, Wind turbine optimal preventive maintenance scheduling using Fibonacci search and genetic algorithm, Journal of Dynamics, Monitoring and Diagnostics. 2 (2023), 157-169.
• M. Basu, M. Das, Uses of second order variant Fibonacci universal code in cryptography, Control and Cybernetics. 45 (2016), 239-251.
• A. Rehman, T. Saba, T. Mahmood, Z. Mehmood, M. Shah, A. Anjum, Data hiding technique in steganography for information security using number theory, Journal of Information Science. 45 (2019), 767-778. doi:10.1177/0165551518816303.
• L. Wu, H. Cai, Novel stream ciphering algorithm for big data images using Zeckendorf representation, Wireless Communications and Mobile Computing. 2021 (2021), 1-19. doi:10.1155/2021/4637876.
• M.S. Taha, M.S.M. Rahem, M.M. Hashim, H.N. Khalid, High payload image steganography scheme with minimum distortion based on distinction grade value method, Multimedia Tools and Applications. 81 (2022), 25913-25946. doi:10.1007/s11042-022-12691-9.
• Y. Wu, L. Wu, H. Cai, Cloud-edge data encryption in the internet of vehicles using Zeckendorf representation, Journal of Cloud Computing. 12 (2023), 39. doi:10.1186/s13677-023-00417-7.
• Z. Liang, Q. Qin, C. Zhou, An image encryption algorithm based on Fibonacci Q-matrix and genetic algorithm, Neural Computing and Applications. 34 (2022), 19313-19341.
• T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 2, John Wiley & Sons, 2019.
• C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly. 33 (1995), 3-8.
• C. Kimberling, Edouard Zeckendorf, Fibonacci Quarterly. 36 (1998) 416-418.
• P. Pooksombat, P. Udomkavanich, W. Kositwattanarerk, Multidimensional Fibonacci Coding, (2017). http://arxiv.org/abs/1706.06655 (erişim 09 Mart 2024).
• M. Bicknell-Johnson, The Zeckendorf-Wythoff Array Applied to Counting the Number of Representations of N as Sums of Distinct Fibonacci Numbers, in: F.T. Howard (Ed.), Applications of Fibonacci Numbers, Springer Netherlands, Dordrecht, 1999: ss. 53-60.
• W. Lang, ed., The Wythoff and the Zeckendorf Representations of Numbers Are Equivalent, Kluwer Academic Publishers, Dordrecht, 1996. doi:10.1007/978-94-009-0223-7.
• J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, The MIT Press, 1992. doi:10.7551/mitpress/1090.001.0001.
• J. McCall, Genetic algorithms for modelling and optimisation, Journal of Computational and Applied Mathematics. 184 (2005), 205-222. doi:10.1016/j.cam.2004.07.034.
• D. Dumitrescu, B. Lazzerini, L.C. Jain, A. Dumitrescu, Evolutionary Computation, CRC Press, 2000.
• T. DeAlwis, Maximizing or minimizing polynomials using algebraic inequalities, in: Proceedings of the 9th Asian Technological Conference on Mathematics, 2004: ss. 88-97. https://atcm.mathandtech.org/EP/2004/2004I328/fullpaper.pdf (erişim 21 Mart 2024).
• A. Mikhalev, I.V. Oseledets, Rectangular maximum-volume submatrices and their applications, Linear Algebra and its Applications. 538 (2018), 187-211. doi:10.1016/j.laa.2017.10.014.