The Optimal Tolerance Solution of the Basic Interval Linear Equation and the Explanation of the Lodwick’s Anomaly

Abstract

Determining the tolerance solution (TS) of interval linear systems (ILSs) has been a task under consideration for many years. It seems, however, that this task has not been fully and unequivocally solved. This is evidenced by the multiplicity of proposed methods (which sometimes provide different results), the existence of many questions, and the emergence of strange solutions provided by, for example, Lodwick’s interval equation anomaly (LIEA). The problem of solving ILEs is probably more difficult than we think. The article presents a new method of ILSs solving, but it is limited to the simplest, basic equation [a̲,a¯]X=[b̲,b¯], which is an element of all more complex forms of ILSs. The method finds the optimal TS for this equation by using multidimensional interval arithmetic (MIA). According to the authors’ knowledge, this is a new method and it will allow researchers to solve more complex forms of ILSs and various types of nonlinear interval equations. It can also be used to solve fuzzy linear systems (FLSs). The paper presents several examples of the method applications (including one real-life case).