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

Year 2023, Volume: 13 Issue: 4, 1361 - 1376, 15.12.2023

Hande Günay Akdemir

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

Abstract

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.

Keywords

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

Abstract

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.

Keywords

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

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