Solving Fuzzy Equations

Introduction

We start in the next Section 10.2 with looking at possible solutions to the simple fuzzy linear equation $\overline{A} \cdot \overline{X} + \overline{B} = \overline{C}$. We discuss three different types of solution which we have studied before in solving fuzzy equations. Then we present a fourth type of solution, based on our fuzzy Monte Carlo method, in Section 10.2.2. This new solution is based on random fuzzy numbers. In Section 10.3 we look at only “classical” solutions to the fuzzy quadratic equation and apply our fuzzy Monte Carlo method to obtain new solutions. Then in Section 10.4 we consider the fuzzy matrix equation $\overline{A} \cdot \overline{X}=\overline{B}$ and a number of solution types for $\overline{X}$ and then another solution based on fuzzy Monte Carlo techniques. The last section contains a brief summary and our conclusions.

In this chapter ${\overline {M}}$ ≤ ${\overline {N}}$ will mean that ${\overline {M}}$ is a fuzzy subset of ${\overline {N}}$ (Section 2.2.3) and not that ${\overline {M}}$ is less than or equal to ${\overline {N}}$. Solving fuzzy equations has always been an active area of research. Some recent references on this topic are ([1]-[4],[16]-[18],[21],[22]).