## 2 Introduction to Multidimensional Type-1 Fuzzy Arithmetic (MT1 EFA)
In F-arithmetic, we most often use trapezoidal MFs, which, on the one hand, facilitate the execution of calculations, and on the other hand, include several frequently used cases of fuzzy numbers shown in Figure 2.  
Of course, other types of FNs as sinusoidal, polynomial, etc. can also be used. Classic FA uses vertical MFs. Formula (4) gives a mathematical model of the vertical trapezoidal MF from Figure 2.  
\[\mu(x_{A})=\left\{ \begin{array}{lcl}\mu^{L}(x_{A})=\frac{x_{A}-a_{1}}{a_{2}- a_{1}}&,&a_{1}\leq x_{A}\leq a_{2}\\ 1&,&a_{2}<x_{A}\leq a_{3}\\ \mu^{R}(x_{A})=\frac{a_{4}-x_{A}}{a_{4}-a_{3}}&,&a_{3}<x_{A}\leq a_{4}\\ \end{array} \right. \tag{4}\]  
Vertical MF is a model of only the borders of the fuzzy set and it results from the first definition of FS of [PERSON] [27]. However, the vertical MF does not model the interior of an FS. In MFA, horizontal MFs are used, which model not only the borders, but also the entire interior of the MF. The HMFs correspond to (follow from) the second definition of FS Zadeh [2]. **Definition 1**. : _A horizontal membership function type-1 \(A(\mu,\alpha_{xA})\) is based on a pair of functions \(x_{A}^{L}(\mu)\) and \(x_{A}^{R}(\mu)\), where \(x_{A}^{L}(\mu)\) is left border, bounded continuous nondecreasing function over \([0,1]\) and \(x_{A}^{R}(\mu)\) is right border, bounded continuous nonincreasing function over \([0,1]\). Functions \(x_{A}^{L}(\mu)=(\mu^{L}(x_{A}))^{-1}\) and \(x_{A}^{R}(\mu)=(\mu^{R}(x_{A}))^{-1}\) are horizontal representations of the left and right borders respectively, where \(\mu^{L}(x_{A})\) represents the left border of the vertical MF and \(\mu^{R}(x_{A})\) represents the right border. The type-1 horizontal multidimensional membership function type-1 (T1 HMF) in general notation has the form of (5)._  
\[ \begin{array}{l}A(\mu,\alpha_{xA}):x_{A}(\mu,\alpha_{xA})\in A(\mu,\alpha_ {xA})\\ x_{A}(\mu,\alpha_{xA})=x_{A}^{L}(\mu)+\alpha_{xA}[x_{A}^{R}(\mu)-x_{A}^{L}( \mu)],\ \mu,\alpha_{xA}\in[0,1]\\ \end{array}  \tag{5}\]  
**Definition 2**. : _Horizontal trapezoidal membership function type-1 (T1 HTMF) is formulated by Equation (6),_  
\[A(\mu,\alpha_{xA}):x_{A}(\mu,\alpha_{xA})=[a_{1}+\mu(a_{2}-a_{1})]+\alpha_{xA }[(a_{4}-a_{1})-\mu(a_{4}-a_{3}+a_{2}-a_{1})], \tag{6}\]  
_where \(\mu,\alpha_{xA}\in[0,1]\)._  
RDM means here relative-distance-measure, and can be denoted e.g., by \(\alpha_{xA}\in[0,1]\). It defines the relative position of the point \(x\) lying between the borders of MF. For the left border \(\alpha_{xA}=0\) and for the right border \(\alpha_{xA}=1\). T1 TMF is a mathematical model of the true value (epistemic case) of uncertain variable \(x_{A}\) in conditions when our knowledge about this value is expressed in the form of the trapezoidal MF, Figures 2 and 3. Figure 2: Trapezoidal fuzzy number and its special cases. **Definition 3**. : _(MTTITFN-arithmetic of normal trapezoidal FNs). Let \(A_{1}=(a_{11},a_{12},a_{13},a_{14})\) and \(A_{2}=(a_{21},a_{22},a_{23},a_{24})\) are two arbitrary trapezoidal T1 TFNs. Arithmetic operations \(\odot\in\{+,-,\cdot,/\}\) of TIFNs are defined as follows, \(0\
ot\in A_{2}\) for operation /._  
\[ \begin{array}{l}A_{1}(\mu,\alpha_{xA_{1}}):x_{A_{1}}(\mu,\alpha_{xA_{1}}) \in A_{1}(\mu,\alpha_{xA_{1}})\\ A_{2}(\mu,\alpha_{xA_{2}}):x_{A_{2}}(\mu,\alpha_{xA_{2}})\in A_{2}(\mu,\alpha_{ xA_{2}})\\ Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=A_{1}(\mu,\alpha_{xA_{1}})\odot A_{2}( \mu,\alpha_{xA_{2}})\\ Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}}):z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}}) \in Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})\\ z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=x_{A_{1}}(\mu,\alpha_{xA_{1}})\odot x _{A_{2}}(\mu,\alpha_{xA_{2}})\end{array}  \tag{7}\]  
_For trapezoidal T1 TFNs:_  
\[x_{A_{1}}(\mu,\alpha_{xA_{1}})=[a_{11}+\mu(a_{12}-a_{11})]+\alpha_{xA_{1}}[( a_{14}-a_{11})-\mu(a_{14}-a_{13}+a_{12}-a_{11})],\]  
\[x_{A_{2}}(\mu,\alpha_{xA_{2}})=[a_{21}+\mu(a_{22}-a_{21})]+\alpha_{xA_{2}}[( a_{24}-a_{21})-\mu(a_{24}-a_{23}+a_{22}-a_{21})],\]  
_where \(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}}\in[0,1]\).