A Fuzzy Numerical Simulation-based Heuristic Method for Fully Fuzzy Systems of Linear Equations

Yıl 2023, , 1361 - 1376, 15.12.2023

Hande Günay Akdemir

https://doi.org/10.31466/kfbd.1275692

Öz

In this paper, a new method is proposed to find the approximate solutions to fully fuzzy systems of linear equations (FFSLEs). The technique integrates a bisection method with Fuzzy Numerical Simulation (FNS). The procedure starts with generating single values of fuzzy parameters and solving the resulting crisp problems repeatedly to determine the lower and upper bounds of the solutions. After computing the mean lower and upper bound values, the obtained supremum and infimum values are considered to be the lower and upper bounds of the solutions, respectively. It is attempted to improve solutions by considering an error function related to the sum of the absolute differences between the corresponding lower and upper bounds of the left and right sides of the equalities. When very large intervals are obtained for the solutions, the bisection algorithm is applied to reduce the error value. The method intends to solve square systems of large dimensions for arbitrary fuzzy numbers (FNs) by removing non-negativity confinements of the variables and/or coefficients to be more realistic. After the computational method is presented thoroughly, some benchmark examples are finally provided.

Anahtar Kelimeler

Approximate solution, Bisection method, Fully fuzzy systems of linear equations, Fuzzy numerical simulation

Tam Bulanık Lineer Denklem Sistemleri için Bulanık Sayısal Simülasyon Tabanlı Sezgisel Bir Yöntem

Öz

Bu çalışmada, tam bulanık lineer denklem sistemlerine yaklaşık çözümler bulmak için yeni bir yöntem önerilmiştir. Teknik, bir ikiye bölme yöntemini bulanık sayısal simülasyon ile bütünleştirir. Prosedür, bulanık parametrelerin tek değerlerinin üretilmesi ve çözümlerin alt ve üst sınırlarının belirlenmesi için ortaya çıkan net problemlerin tekrar tekrar çözülmesiyle başlar. Ortalama alt ve üst sınır değerleri hesaplandıktan sonra, elde edilen supremum ve infimum değerler sırasıyla çözümlerin alt ve üst sınırları olarak kabul edilir. Eşitliklerin sağ ve sol taraflarının karşılık gelen alt ve üst sınırları arasındaki mutlak farkların toplamına ilişkin bir hata fonksiyonu ele alınarak çözümler geliştirilmeye çalışılmıştır. Çözümler için çok büyük aralıklar elde edildiğinde, hata değerini azaltmak için ikiye bölme algoritması uygulanır. Yöntem, daha gerçekçi olmak için değişkenlerin ve/veya katsayıların negatif olmayan sınırlamalarını kaldırarak keyfi bulanık sayılar için büyük boyutlu kare sistemleri çözmeyi amaçlar. Hesaplama yöntemi kapsamlı bir şekilde sunulduktan sonra, son olarak bazı kıyaslama örnekleri verilmektedir.

Anahtar Kelimeler

Yaklaşık çözüm, İkiye bölme yöntemi, Tam bulanık lineer denklem sistemleri, Bulanık sayısal simülasyon

Kaynakça

• Ahlatcioglu, M., Albayrak, I., Kocken, H. G., and Ozkok, B. A. (2016). A mixed integer programming approach to a square fully fuzzy linear equation. Journal of Intelligent and Fuzzy Systems, 31(3), 2009-2015. https://doi.org/10.3233/JIFS-16227
• Akdemir, H. G. (2023). Approximate Fuzzy Inverse Matrix Calculation Method using Scenario-based Inverses and Bisection. Fundamental Journal of Mathematics and Applications, 6(1), 42-50. https://doi.org/10.33401/fujma.1195121
• Akdemir, H. G., and Kocken, H. G. (2022). A new fuzzy linear regression algorithm based on the simulation of fuzzy samples and an application on popularity prediction of Covid-19 related videos. Journal of Statistics and Management Systems, 1 17. https://doi.org/10.1080/09720510.2021.2016988
• Albayrak, I. (2017). On fuzzy solutions of the nonsquare fully fuzzy linear equation system with arbitrary triangular fuzzy numbers. Journal of Intelligent and Fuzzy Systems, 33(6), 3929-3938. https://doi.org/10.3233/JIFS-17774
• Allahviranloo, T., Hosseinzadeh, A. A., Ghanbari, M., Haghi, E., and Nuraei, R. (2014). On the new solutions for a fully fuzzy linear system. Soft Computing, 18, 95-107. https://doi.org/10.1007/s00500-013-1037-3
• Allahviranloo, T., Kiani, N. A., Barkhordary, M., and Mosleh, M. (2008). Homomorphic solution of fully fuzzy linear systems. Computational Mathematics and Modeling, 19, 282-291. https://doi.org/10.1007/s10598-008-9004-z
• Allahviranloo, T., and Mikaeilvand N. (2011). Non zero solutions of the fully fuzzy linear systems. Applied and Computational Mathematics, 10(2), 271-282.
• Allahviranloo, T., Salahshour, S., Homayoun-Nejad, M., and Baleanu, D. (2013, January). General solutions of fully fuzzy linear systems. In Abstract and Applied Analysis (Vol. 2013). Hindawi. https://doi.org/10.1155/2013/593274
• Allahviranloo, T., Salahshour, S., and Khezerloo, M. (2011). Maximal-and minimal symmetric solutions of fully fuzzy linear systems. Journal of Computational and Applied Mathematics, 235(16), 4652-4662. https://doi.org/10.1016/j.cam.2010.05.009
• Babbar, N., Kumar, A., and Bansal, A. (2013a). Linear programming approach to find the solution of fully fuzzy linear systems with arbitrary fuzzy coefficients. Journal of Intelligent and Fuzzy Systems, 25(3), 747-753. https://doi.org/10.3233/IFS-120681
• Babbar, N., Kumar, A., and Bansal, A. (2013b). Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers. Soft Computing, 17(4), 691-702. https://doi.org/10.1007/s00500-012-0941-2
• Behera, D., and Chakraverty, S. (2017). A note on “A new method for solving an arbitrary fully fuzzy linear system”. Soft Computing, 21, 7117-7118. https://doi.org/10.1007/s00500-016-2254-3
• Chanas, S., and Nowakowski, M. (1988). Single value simulation of fuzzy variable. Fuzzy Sets and Systems, 25(1), 43-57. https://doi.org/10.1016/0165-0114(88)90098-X
• Dehghan, M., and Hashemi, B. (2006). Solution of the fully fuzzy linear systems using the decomposition procedure. Applied Mathematics and Computation, 182(2), 1568-1580. https://doi.org/10.1016/j.amc.2006.05.043
• Dehghan, M., Hashemi, B., and Ghatee, M. (2006). Computational methods for solving fully fuzzy linear systems. Applied Mathematics and Computation, 179(1), 328-343. https://doi.org/10.1016/j.amc.2005.11.124
• Dehghan, M., Hashemi, B., and Ghatee, M. (2007). Solution of the fully fuzzy linear systems using iterative techniques. Chaos, Solitons and Fractals, 34(2), 316-336. https://doi.org/10.1016/j.chaos.2006.03.085
• Ezzati, R., Khezerloo, S., Mahdavi-Amiri, N., and Valizadeh, Z. (2014). Approximate Nonnegative Symmetric Solution of Fully Fuzzy Systems Using Median Interval Defuzzification. Fuzzy Information and Engineering, 6(3), 331-358. https://doi.org/10.1016/j.fiae.2014.12.005
• Ezzati, R., Khezerloo, S., Valizadeh, Z., and Mahdavi-Amiri, N. (2012). New models and algorithms for solutions of single-signed fully fuzzy LR linear systems. Iranian Journal of Fuzzy Systems, 9(3), 1-26.
• Guo, X., Wei, Y., and Li, Z. (2018). Further investigation to approximate fuzzy inverse. Journal of Intelligent and Fuzzy Systems, 35(1), 1161-1168. https://doi.org/10.3233/JIFS-18027
• Jeswal, S. K., and Chakraverty, S. (2019). Connectionist model for solving static structural problems with fuzzy parameters. Applied Soft Computing, 78, 221-229. https://doi.org/10.1016/j.asoc.2019.02.025
• Kocken, H. G., Ahlatcioglu, M., and Albayrak, I. (2016). Finding the fuzzy solutions of a general fully fuzzy linear equation system. Journal of Intelligent and Fuzzy Systems, 30(2), 921-933. https://doi.org/10.3233/IFS-151813
• Kumar, A., Bansal, A., and Babbar, N. (2013). Fully fuzzy linear systems of triangular fuzzy numbers (a, b, c). International Journal of Intelligent Computing and Cybernetics, 6(1), 21-44. https://doi.org/10.1108/17563781311301508
• Kumar, A., Neetu, and Bansal, A. (2012). A new computational method for solving fully fuzzy linear systems of triangular fuzzy numbers. Fuzzy Information and Engineering, 4(1), 63-73. https://doi.org/10.1007/s12543-012-0101-5
• Moloudzadeh, S., Allahviranloo, T., and Darabi, P. (2013). A new method for solving an arbitrary fully fuzzy linear system. Soft Computing, 17(9), 1725-1731. https://doi.org/10.1007/s00500-013-0986-x
• Mosleh, M. (2013). Evaluation of fully fuzzy matrix equations by fuzzy neural network. Applied Mathematical Modelling, 37(9), 6364-6376. https://doi.org/10.1016/j.apm.2013.01.011
• Mosleh, M., and Otadi, M. (2015). A discussion on “Calculating fuzzy inverse matrix using fuzzy linear equation system”. Applied Soft Computing, 28, 511-513. https://doi.org/10.1016/j.asoc.2014.11.035
• Otadi, M., and Mosleh, M. (2012). Solving fully fuzzy matrix equations. Applied Mathematical Modelling, 36(12), 6114-6121. https://doi.org/10.1016/j.apm.2012.02.005
• Otadi, M., Mosleh, M., and Abbasbandy, S. (2011). Numerical solution of fully fuzzy linear systems by fuzzy neural network. Soft Computing, 15(8), 1513-1522. https://doi.org/10.1007/s00500-010-0685-9
• Rao, S. S., and Chen, L. (1998). Numerical solution of fuzzy linear equations in engineering analysis. International Journal for Numerical Methods in Engineering, 42(5), 829-846. https://doi.org/10.1002/(SICI)1097-0207(19980715)42:5%3C829::AID-NME386%3E3.0.CO;2-G
• Ziqan, A., Ibrahim, S., Marabeh, M., and Qarariyah, A. (2022). Fully fuzzy linear systems with trapezoidal and hexagonal fuzzy numbers. Granular Computing, 7(2), 229-238. https://doi.org/10.1007/s41066-021-00262-6