## 2 Introduction to Multidimensional Interval Arithmetic  
Currently, there are many types of interval arithmetic (IA), for an overview see [45]. Examples are: standard interval arithmetic (SIA) of [PERSON], extended IA of [PERSON], non-standard IA of [PERSON], generalized Hukuhara IA of [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], affine IA of [PERSON] and [PERSON]. The common feature of these IAs is the type of the result: it is always the interval - a 1D-mathematical object. Hence, above types of IA can be called 1-dimensional. Meanwhile, as it will be shown in Example 1, IA requires a multidimensional approach. **Example 1**. : _In the slightly deformed cylindrical tank \(A\) there is a certain amount a of water which can not be accurately measured. It can only be roughly estimated from the tank dimensions and the water height as \(a\in[4,6]\)\([m^{3}]\). Water from tank \(A\) was poured into tank \(B\) (also deformed) and then tap water was added to \(B\) in an unknown amount \(x\)\([m^{3}]\). Basing on measurements of tank \(B\) (inner diameter, height) it was estimated that there is now the amount of water \(b\in[7.5,8.5]\)\([m^{3}]\) in it. Question: what amount of water \(x\) was added to the tank \(B\) from the tap? The problem of tanks can be presented in the form of Equation (3)  
In Formula (7) it is easy to see that the variable \(x\) is not 1-dimensional but that it is a function \(x(\gamma_{a},\gamma_{b})\) existing in 3D space. The set \((A,B,X)\) of possible states of the tank system is given by Equation (8). \[(a(\gamma_{a}),b(\gamma_{b}),x(\gamma_{a},\gamma_{b}))=(4+2\gamma_{a},7.5+\gamma _{b},3.5-2\gamma_{a}+\gamma_{b}),\ \ \gamma_{a},\gamma_{b}\in[0,1] \tag{8}\]  
Using all possible values of RDM variables \(\gamma_{a}\), \(\gamma_{b}\) we can generate all possible triples \((a,b,x)\), i.e., all possible states of the tank system. We can also define various simplified information about this set, such as the span \(SP_{x}\) of the set of possible values of the variable \(x\), the distribution of cardinality measure \(\text{Mcard}_{x}\), the center of gravity \(\text{Coff}G_{x}\)[46, 47, 48, 49]. Performing arithmetic operations with MIA is quite simple. If we have the interval knowledge of the true values of 2 variables \(a\) and \(b\), then the arithmetic operation \(*\in\{+,-,\cdot,\cdot\}\) is performed on mathematical models of these values shown in (9) to calculate the result variable value \(x\) (10). \[a=\underline{a}+\gamma_{a}(\overline{a}-\underline{a}),\ \ \gamma_{a}\in[0,1]\] \[b=\underline{b}+\gamma_{b}(\overline{b}-\underline{b}),\ \ \gamma_{b}\in[0,1] \tag{9}\]  
\[x=x(\gamma_{a},\gamma_{b}))=a(\gamma_{a})*b(\gamma_{b})=[\underline{a}+\gamma _{a}(\overline{a}-\underline{a})]*[\underline{b}+\gamma_{b}(\overline{b}- \underline{b})],\ \ \gamma_{a},\gamma_{b}\in[0,1] \tag{10}\]  
After performing the operation (10) we get 3D result set \((A,B,X)\):  
\[(A,B,X\mid X=A*B)=\{(a,b,x)\mid x=a*b,a\in A,b\in B,x\in X\}. \tag{11}\]  
In this set, \(X\) means the set of possible values of variable \(x(\gamma_{a},\gamma_{b})\). Span of \(X\) can be calculated with (12). \[SP_{X}=[\min_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b})],\ \ \gamma_{a},\gamma_{b}\in[0,1] \tag{12}\]  
The method of calculating the distribution of cardinality measure \(\text{Mcard}_{x}\) and of the center of gravity is given in [46,48,50] and will be shown in further examples.