Ankara Hacı Bayram Veli Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi 2667-405X Ankara Hacı Bayram Veli University 10.26745/ahbvuibfd.1190046 Operation Yöneylem Doğrusal Olmayan Gri Bernoulli Model için Parametre ve Model Yapısı Optimizasyonu Parameter and Model Structure Optimization for the Nonlinear Grey Bernoulli Model https://orcid.org/0000-0002-0889-9191 Taştan Serkan SİVAS CUMHURİYET ÜNİVERSİTESİ, İKTİSADİ VE İDARİ BİLİMLER FAKÜLTESİ, YÖNETİM BİLİŞİM SİSTEMLERİ BÖLÜMÜ 04 11 2023 25 1 77 94 10 16 2022 01 13 2023 Copyright © 1999, Ankara Hacı Bayram Veli University Journal of the Faculty of Economics and Administrative Sciences 1999 Ankara Hacı Bayram Veli University Journal of the Faculty of Economics and Administrative Sciences

Anlaşılması ve tahmin edilmesi kolay geleneksel gri modeller yaygın olarak kullanılmaktadırlar. Ancak, bu modeller mevcut kestirim ve öngörü hassasiyeti arttırılmak istenildiği zaman ihtiyaç duyulan uyarlamalar için gereken esneklikten yoksundurlar. Diğer taraftan, oldukça esnek olan doğrusal olmayan gri Bernoulli model tek parametresi ayarlanarak, birikim üretim operatörü uygulanmış zaman serisine uyan eğriyi etkin bir şekilde uydurulabilmektedir. Yine de, bu modelinin parametreleri ve yapısı çerçevesinde yapılabilecek iyileştirmeler mevcuttur. Bu yüzden, bu çalışmada doğrusal olmayan gri Bernoulli model için önerilen başlangıç koşulunu optimizasyonu, gri modellerin öngörü performanslarını yükseltmek adına önerilen kayan pencere yöntemi ve sezgisel algoritmalar ile model parametrelerinin optimizasyonu yaklaşımları bir arada kullanılmıştır. Doğrusal olmayan gri Bernoulli model kayan pencere yöntemine dayalı olarak tahmin edilmiştir. Diferansiyel denklemin çözümünde başlangıç koşulu olarak birinci dereceden birikim üretim operatörü uygulanmış serinin düzeltilmiş son elemanı kullanılmıştır. Geçmiş değer ve kuvvet katsayısının en iyi değerleri ise salp sürüsü optimizasyon algoritması kullanılarak bulunmuştur. Dolayısıyla, model yapısının ve parametrelerinin birlikte değerlendirildiği yeni bir optimize edilmiş doğrusal olmayan gri Bernoulli model önerilmiştir. Çalışmada, parametre tahmin yöntemi ve/veya model yapısı açısından birbirinden farklı sekiz gri model değerlendirilmiştir. Ulaşılan sonuçlar önerilen modelin diğer gri modellere göre daha başarılı olduğunu göstermektedir. Sonuç olarak, Türkiye’nin gayrisafi yurt içi hasıla hacim endeksi önerilen gri model ile daha iyi modellenmiştir.

Traditional grey models, easy to understand and estimate, are widely used. However, these models lack the flexibility with regard to the adaptations required to increase the current prediction and forecast accuracy. However, the highly flexible non-linear grey Bernoulli model can effectively fit the curve for the accumulated generating operation series by adjusting its single parameter. Nevertheless, there are still improvements that can be made within the framework of the structure and parameters of this model. Therefore, in this study, initial condition optimization proposed for the non-linear grey Bernoulli model, the rolling window method proposed to improve the forecasting performance of the grey models and, optimization of the model parameters with heuristic algorithm combined. The non-linear grey Bernoulli model was estimated by using the rolling window method. The corrected last element of the accumulated generating operation series was used as the initial condition in the solution of the differential equation. The optimal values of the background value and power index were determined by using the salp swarm optimization algorithm. Therefore, a new optimized non-linear grey Bernoulli model is proposed by considering the structure and parameters of the model together. In the study, eight different grey models were evaluated in terms of parameter estimation method and/or model structure. The results showed that the proposed model outperformed the other grey models. Consequently, gross domestic product volume index of Turkey was better modeled with the proposed grey model.

 Doğrusal olmayan gri Bernoulli model Salp sürüsü algoritması GSYH öngörüsü Non-linear grey Bernoulli model Salp swarm algorithm GDP forecast Aljarah, I., Mafarja, M., Heidari, A. A., Faris, H., Zhang, Y. ve Mirjalili, S. (2018). Asynchronous accelerating multi-leader salp chains for feature selection. Applied Soft Computing, 71, 964–979. https://doi.org/10.1016/j.asoc.2018.07.040 Chen, C.-I., Chen, H. L. ve Chen, S.-P. (2008). Forecasting of foreign exchange rates of Taiwan’s major trading partners by novel nonlinear Grey Bernoulli model NGBM(1,1). Communications in Nonlinear Science and Numerical Simulation, 13​(6), 1194–1204. https://doi.org/10.1016/j.cnsns.2006.08.008 Chen, C.-I., Hsin, P.-H. ve Wu, C.-S. (2010). Forecasting Taiwan’s major stock indices by the Nash nonlinear grey Bernoulli model. Expert Systems with Applications, 37​(12), 7557–7562. https://doi.org/10.1016/j.eswa.2010.04.088 Cheng, M., Liu, Y., Li, J. ve Liu, B. (2022). Nonlinear Grey Bernoulli model NGBM (1, 1)’s parameter optimisation method and model application. Journal of Industrial and Management Optimization, 18​(3), 2017. https://doi.org/10.3934/jimo.2021054 Chia-Nan, W. ve Van-Thanh, P. (2015). An Improved Nonlinear Grey Bernoulli Model Combined with Fourier Series. Mathematical Problems in Engineering, 2015, e740272. https://doi.org/10.1155/2015/740272 Comert, G., Begashaw, N. ve Huynh, N. (2021). Improved grey system models for predicting traffic parameters. Expert Systems with Applications, 177, 114972. https://doi.org/10.1016/j.eswa.2021.114972 Deng, J. L. (1982). Control Problems of Grey System. Systems and Control Letters, 5, 288-294. Hsu, L.-C. (2003). Applying the Grey prediction model to the global integrated circuit industry. Technological Forecasting and Social Change, 70​(6), 563–574. https://doi.org/10.1016/S0040-1625(02)00195-6 Hsu, L.-C. (2010). A genetic algorithm based nonlinear grey Bernoulli model for output forecasting in integrated circuit industry. Expert Systems with Applications, 37​(6), 4318–4323. https://doi.org/10.1016/j.eswa.2009.11.068 Jiang, J., Feng, T. ve Liu, C. (2021). An Improved Nonlinear Grey Bernoulli Model Based on the Whale Optimization Algorithm and Its Application. Mathematical Problems in Engineering, 2021, e6691724. https://doi.org/10.1155/2021/6691724 Kayacan, E., Ulutas, B. ve Kaynak, O. (2010). Grey system theory-based models in time series prediction. Expert Systems with Applications, 37​(2, 2), 1784–1789. https://doi.org/10.1016/j.eswa.2009.07.064 Lu, J., Xie, W., Zhou, H. ve Zhang, A. (2016). An optimized nonlinear grey Bernoulli model and its applications. Neurocomputing, 177, 206–214. https://doi.org/10.1016/j.neucom.2015.11.032 Mirjalili, S., Gandomi, A. H., Mirjalili, S. Z., Saremi, S., Faris, H. ve Mirjalili, S. M. (2017). Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Advances in Engineering Software, 114, 163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002 Ngo, H. A. ve Hoang, T. N. (2020). A Rolling Optimized Nonlinear Grey Bernoulli Model RONGBM (1, 1) and application in predicting total COVID-19 infected cases. https://arxiv.org/abs/2008.07581 Özcan, T. ve Tüysüz, F. (2018). Healthcare Expenditure Prediction in Turkey by Using Genetic Algorithm Based Grey Forecasting Models. C. Kahraman ve Y. I. Topcu içinde, Operations Research Applications in Health Care Management (159–190). Springer International Publishing. https://doi.org/10.1007/978-3-319-65455-3_7 Pao, H.-T., Fu, H.-C. ve Tseng, C.-L. (2012). Forecasting of CO2 emissions, energy consumption and economic growth in China using an improved grey model. Energy, 40​(1), 400–409. https://doi.org/10.1016/j.energy.2012.01.037 Rizk-Allah, R. M., Hassanien, A. E., Elhoseny, M. ve Gunasekaran, M. (2019). A new binary salp swarm algorithm: Development and application for optimization tasks. Neural Computing and Applications, 31​(5, 5), 1641–1663. https://doi.org/10.1007/s00521-018-3613-z Tien, T.-L. (2009). A new grey prediction model FGM(1, 1). Mathematical and Computer Modelling, 49​(7, 7), 1416–1426. https://doi.org/10.1016/j.mcm.2008.11.015 Wang, Z.-X., Li, Q. ve Pei, L.-L. (2018). A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors. Energy, 154, 522–534. https://doi.org/10.1016/j.energy.2018.04.155 Wu, L., Liu, S., Yao, L., Yan, S. ve Liu, D. (2013). Grey system model with the fractional order accumulation. Communications in Nonlinear Science and Numerical Simulation, 18​(7), 1775–1785. https://doi.org/10.1016/j.cnsns.2012.11.017 Wu, W.-Z., Zhang, T. ve Zheng, C. (2019). A Novel Optimized Nonlinear Grey Bernoulli Model for Forecasting China’s GDP. Complexity, 2019, e1731262. https://doi.org/10.1155/2019/1731262 Xia, M. ve Wong, W. K. (2014). A seasonal discrete grey forecasting model for fashion retailing. Knowledge-Based Systems, 57, 119–126. https://doi.org/10.1016/j.knosys.2013.12.014 Xie, N.-m. ve Liu, S.-f. (2009). Discrete grey forecasting model and its optimization. Applied Mathematical Modelling, 33​(2), 1173–1186. https://doi.org/10.1016/j.apm.2008.01.011 Zhou, J., Fang, R., Li, Y., Zhang, Y. ve Peng, B. (2009). Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization. Applied Mathematics and Computation, 207​(2), 292–299. https://doi.org/10.1016/j.amc.2008.10.045