## 1 Introduction  
The problem presented in this paper fits into a more general class of problems referred to in the research literature as interval-valued optimization problems (IVO-problems) [1, 2]. Although the optimization problems have been solved reasonably well for the crisp data case, the quality of their solutions for uncertain data is often not satisfactory. This is evidenced by the large number of studies conducted worldwide. Uncertain data is encountered in all types of systems: in static and dynamic systems, in technical systems (control of moving vehicles, flying or floating objects), and in medical, biological, ecological, economic, and other systems. To be able to satisfactorily solve IVO-problems in these diverse fields, new interval versions of many well-known methods in crisp data optimization, or methods that are completely new and previously unknown, must be developed. The authors of [1, 2] give examples of many new scientific tasks, e.g., concepts of convexity, invexity, generalization of convexity, Kuhn-Tucker-pseudoinvexity, control problems with multiple integrals, interval-valued nondifferentiable multi-objective fractional programming problems, vector interval-valued optimization problems with infinite interval constraints and others. All this indicates the enormity of the tasks waiting to be solved by using interval arithmetic. One such optimization task is the task mentioned in the title of this paper, aimed at finding the tolerance solution of the interval linear equation, which was formulated in the previous century and for which solution methods based on one-dimensional types of interval arithmetic have already been provided [3, 4, 5, 6, 7]. As will be shown further on, by applying the new multidimensional interval arithmetic (MIA), it will be possible to solve such tasks, which so far could not be solved. The method presented further also allows us to determine the optimal values of the control variable. The systems we want to control are characterized by various dependencies determining their operation. One of the simplest systems is a static system realizing the linear relationship \(ax=b\), where \(a\) is a gain coefficient, \(x\) is an input control (decision) variable, and \(b\) is an output variable that should have a certain value that we want. In practical tasks, the exact value of the coefficient \(a\) is often not known. We only know the interval range of its possible values \([\underline{a},\overline{a}]\). If we do not know the exact \(a\) value, then we cannot guarantee that the output \(b\) will be kept at the desired value. The only thing we can do is keep the output \(b\) in a certain interval called the tolerance corridor \([\underline{b},\overline{b}]\). This gives rise to the task of determining one \(x\) value or perhaps a set \([\underline{x},\overline{x}]\) of \(x\) values that will \"target\" the output \(b\) in the tolerance corridor. The problem of tolerance control is formulated by the Equation (1). \[[\underline{a},\overline{a}]X=[\underline{b},\overline{b}] \tag{1}\]  
The task of solving interval equations (IEs) and determining of the tolerance control (TC) has been dealt with by interval arithmetic for many years [6; 7; 8; 9; 10]. A big problem with solving IEs results from the fact that not one but many types of interval arithmetic (IA) exist. The following types of IA are given in [11]: standard IA (SIA), extended (generalized) IA of [PERSON], non-standard (inner) IA of [PERSON], generalized Hukuhara IA of [PERSON] and [PERSON], optimistic IA of [PERSON] and [PERSON], instantiated IA of [PERSON], constrained IA of [PERSON], single-level constrained IA of [PERSON], requisite constrained IA of [PERSON], gradual IA of [PERSON], [PERSON], [PERSON], [PERSON], affine IA of [PERSON] and [PERSON], and multidimensional IA of [PERSON], [PERSON], [PERSON]. In general, different IAs can give different results for one and the same problem, which is a paradox. However, new ways of performing interval calculations are still proposed.