## 3 A Brief Introduction to Multidimensional Interval Arithmetic  
The MIA concept was developed in 2010-2011, and the first publication [22] on this subject appeared in 2012. Then other publications appeared, e.g., [23; 24]. The authors of MIA used this arithmetic to create multidimensional fuzzy arithmetic (MFA Type 1) by using the \(\alpha\)-cuts principle [25]. In turn, the MFA Type 1 was used to develop the MFA Type 2, e.g., [26; 27; 28]. The research team of [PERSON], [PERSON], [PERSON] and others by the beginning of 2022 published 47 articles on MIA, MFA Type 1 and 2. MFA has met with great interest from many scientists who have applied it to solve various problems. By the beginning of 2022, more than 40 application publications had been published, e.g., [29; 30]. The main feature that distinguishes MIA from all other types of IA is the form of the computed result. In MIA, the result of an arithmetic operation on intervals is not an interval, i.e., a one-dimensional mathematical object, but is a multidimensional result (set) depending on the number of intervals involved in the operation. This is important in case of complex problems and prevents the loss of information during the computation. In MIA, the interval \([\underline{a},\overline{a}]\) is transformed into the form of relative-distance-measure (RDM), (22). \[[\underline{a},\overline{a}]\rightarrow\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\quad\gamma_{a}\in[0,1] \tag{22}\]  
The model (22) is epistemic [7], i.e., it is a model of a single, true value of \(a\)-variable, which, however, we do not know. The method of performing the interval addition operation is presented in (23). \[a(\gamma_{a})+b(\gamma_{b})=c(\gamma_{a},\gamma_{b}),\quad\gamma_{a}, \gamma_{b}\in[0,1] \tag{23}\] \[[\underline{a}+\gamma_{a}(\overline{a}-\underline{a})]+[ \underline{b}+\gamma_{b}(\overline{b}-\underline{b})]=(\underline{a}+ \underline{b})+\gamma_{a}(\overline{a}-\underline{a})+\gamma_{b}(\overline{b} -\underline{b})\]  
The interpretation of the addition operation is as follows: we know the real value of neither the variable \(a\) nor \(b\), therefore it is impossible to determine the exact value of the result \(c=a+b\). What can we do in this situation? We can only, on the basis of our knowledge, define a set of possible conditional hypotheses. For example, if \((a=2)\) and \((b=3)\) then \((c=5)\). Each of the hypotheses can be presented generally in the form of triple (24). \[(a(\gamma_{a}),b(\gamma_{b}),c(\gamma_{a},\gamma_{b}))=a(\gamma_{a})+b(\gamma_ {b})) \tag{24}\]  
If \([\underline{a},\overline{a}]=[1,2]\) and \([\underline{b},\overline{b}]=[3,5]\) then examples of possible hypotheses are as follows:  
\[(1.00,3.00,4.00),(1.01,3.01,4.02),\dots. \tag{25}\]  
The set \(S_{A+B}\) of all possible solutions \(\{(a(\gamma_{a}),b(\gamma_{b}),c(\gamma_{a},\gamma_{b}))\}\) is a 3-dimensional set and the result \(c(\gamma_{a},\gamma_{b})\) depends on \(a(\gamma_{a})\) and \(b(\gamma_{b})\). Figure 3 illustrates the operation of adding 2 interval sets \([a]\) and \([b]\) containing all possible values of these variables and shows the set of possible results of adding \(c(\gamma_{a},\gamma_{b})\) in 2D-space \(A\times B\). The 3D-result of adding \(S_{A+B}=\{(a,b,c)\}\) is the set of all hypothetical states of the addition system. In this set, we can distinguish many states \((a,b,c)\) containing the same result \(c\) with different values of \(a\) and \(b\), e.g., \((1,4,5)\), \((1.5,3.5,5)\), \((2,3,5)\), etc. This means that all the \(c\) values in the set \(S_{A+B}\) form a bag (they contain repeating values). The bag \(BG_{c}\) can be defined by Formula (26). \[BG_{C}=\{c(\gamma_{a},\gamma_{b})=a(\gamma_{a})+b(\gamma_{b})\mid\forall\gamma _{a}\in[0,1],\forall\gamma_{b}\in[0,1]\} \tag{26}\]  
However, for practical calculations the span of the bag \(SP_{BG_{C}}\) will be used (27). The span does not contain the same repeated values of \(c\).