## 2 Preliminaries Required in IT2 FRAM
### Fuzzy Arithmetic Operations in IT2 FRAM  
A fuzzy set is defined as a set of elements with a degree of membership varying between 0 and 1. The elements of crisp sets, however, have membership degrees of either 1 (fully belong in the set) or 0 (do not belong in the set) [12; 33]. IT2 FRAM uses either the \(\alpha\)-cut technique (standard fuzzy arithmetic) or the extension principle based on different t-norms (extended fuzzy arithmetic) to perform fuzzy arithmetic operations. The standard fuzzy arithmetic is based on interval analysis and discretizes the input fuzzy numbers into several \(\alpha\)-cuts. Then, the \(\alpha\)-cut of the output is achieved by interval calculations on each \(\alpha\)-level cut of the inputs. Subsequently, the union of the \(\alpha\)-cuts is applied to gain the final fuzzy set based on the representation theorem. The mathematical representation of standard fuzzy arithmetic is illustrated in the following:  
\[C(z)=A(x)\oplus B(y)=sup_{\alpha\in[0,1]}\alpha((A_{\alpha}*B_{\alpha})(z)) \tag{1}\]  
where \(A(x)\) and \(B(y)\) are input fuzzy numbers and \(C(z)\) is an output fuzzy number. The \(\alpha\)-cuts of the input fuzzy numbers are represented by \(A\alpha\) and \(B\alpha\), and  represents the basic arithmetic operations. The accumulation of fuzziness results in the overestimation of uncertainty in a standard fuzzy arithmetic method [34]. Extended fuzzy arithmetic is preferred in recent applications because of its capability to reduce uncertainty overestimation problems using any t-norm other than min t-norm [35; 36; 37]. Extended fuzzy arithmetic, developed by [PERSON] [12; 13; 14], extends the domain of a function on fuzzy sets. It generalizes a common point-to-point mapping of a function to a mapping between fuzzy sets. As presented, in extended fuzzy arithmetic, the membership degree of each output is calculated by taking the supremum of the t-norms of the membership degrees of the inputs:  
\[C(z)=A(x)\oplus B(y)=sup_{z=xy}\left(t(A(x),B(y))\right) \tag{2}\]  
where \(t\) can be one of the common four t-norm operators on fuzzy sets, fuzzy number \(C(z)\) is the output, and fuzzy numbers \(A(x)\) and \(B(y)\) are the inputs. The t-norm \(t\) is a binary operation, \(T:[0,1]\times[0,1]\rightarrow[0,1]\), which is commutative, associative, and non-decreasing in each argument, and \(t(x,1)=x\) for each \(x\in(0,1)\). The strength and continuity of common fuzzy t-norms (minimum, algebraic product, Lukasiewicz, and drastic product) are different. In terms of strength, the minimum t-norm is the highest and the drastic product t-norm is the lowest [34]. Furthermore, the changes in output fuzzy numbers result in continuous t-norms, which are less sensitive to the changes in input fuzzy numbers compared to non-continuous t-norms.  
Various defuzzification methods are suggested in the literature. As illustrated in Figure 1, the single value (defuzzification) methods include the smallest of maximum (SOM), middle of maximum (MOM), largest of maximum (LOM), and the center of area (COA). The best representation of the shape of the output fuzzy number is the COA.  
The level of confidence associated with the range of output fuzzy number, represented by the confidence level, can be determined from the corresponding \(\alpha\)-cut level (or possibility degree) and ranges between 0 and 1. The possibility degree is the difference between 1 and the confidence level (1--confidence level).