## 3 Type-2 Fuzzy Logic System  
[PERSON] has introduced the initial concept of fuzzy set (type-1) in 1965 [31] and later in 1975 developed the extension of it as the type-2 fuzzy set [32]. In this paper, it has been assumed that the reader is familiar with the basics of the fuzzy logic system and type-1 Fuzzy Logic Systems (T1 FLS) so that here the focus is entirely on the IT2 FLS and its advantages over T1 FLS. The main limitations of the T1 FLS is that it cannot adequately handle the linguistic, measurement and parameter uncertainties [33] due to [5]: 1--the expressions and words that are used in the antecedents and consequences of rules can be uncertain, and 2--consequences may have a histogram of values associated with them, mainly when knowledge is obtained from a group of experts who do not all agree. In this regard, IT2 FLS, characterized by MFs that are themselves fuzzy, therefore, in case there are difficulties in the determination of membership grade even as a crisp number in [0, 1], type-2 fuzzy sets are then adequate to use. So far, IT2 FLSs have been used in different areas to deal with high uncertainty, non-linearity and time-varying behavior [34], including computing with words [35], intelligent controllers [36], pattern recognition [37]. A typical IT2 FLS consists of five parts as fuzzifier, rule base, inference engine, type-reducer, and defuzzifier. Moreover, in T2 FLS, at least one of the fuzzy sets (membership functions) in the rule base must be type-2. All parts of T2 FLS are similar to T1 FLS except the type-reducer, which is introduced to convert the type-2 membership functions into a type-1 before defuzzification. The process of type reduction is usually performed by the most popular computationally intensive Karnik-Mendel (KM) iterative algorithms proposed by [PERSON] and [PERSON] [38]. A type-2 fuzzy set \(\bar{A}\) can be defined by its type-2 membership function \(\mu_{\bar{A}}(x,u)\) as:  
\[\bar{A}=\int_{x\in X}\int_{u\in I_{x}}\frac{\mu_{\bar{A}}(x,u)}{(x,u)} \tag{1}\]  
where \(x\in X\), \(u\in J_{x}\subseteq[0,1]\), and \(X\) represents the universe of the primary variable \(x\) of \(\bar{A}\). Here \(\iint\) denotes all the admissible \(x\) and \(u\). The point-value representation of \(\bar{A}\) is as:  
\[\bar{A}=\left\{((x,u),\mu_{\bar{A}}(x,u))\mid\forall x\in X,\forall u\in[0,1]\right\} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Author(s)** & **Year** & **Studied Data** & **Purpose** & **Parameters** & **Fuzzy Inference** \\ \hline [PERSON] and & 2008 & 101 and 454 buildings & Proposed a fuzzy logic & Same as below & Hierarchical \\ Dritsos [29] & & in Athens, Greece & rapid visual screening & parameters & fuzzy \\  & & & procedure based on & & rule-based,Sugano \\  & & & Greece method to & & \\  & & & improve the screening & & \\  & & & procedures & & \\ \hline Demartinos & 2006 & 102 buildings in & Fuzzy logic–based & Seismic hazard (ground & Hierarchical \\ and Dristos & Athens, Greece & rapid visual screening & motion, soil quality, & fuzzy \\
[30] & & & procedure for & building height), & rule-based, \\  & & & categorization of & structural strength & Sugano \\  & & & buildings into five & (building height, infill & \\  & & & different types of & wall layout, soft story, & \\  & & possible damage with & short columns, design & \\  & & respect to the potential & code), regularity (plan & \\  & & occurrence of a major & regularity, torsion & \\  & & seismic event. & possibility, height & \\  & & & regularity, bounding & \\  & & & possibility, plan & \\  & & & regularity), structure’s & \\  & & & condition (previous & \\  & & damage, maintenance) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. The secondary MF of \(\tilde{A}\) is also called a vertical slice of \(\mu_{\tilde{A}}(x,u)\):  
\[\mu_{\tilde{A}}(x=x^{\prime},u)\equiv\mu_{\tilde{A}}(x^{\prime})=\int_{u\in I_{ x^{\prime}}}\frac{f_{x^{\prime}}(u)}{u} \tag{3}\]  
where \(0\leq f_{x^{\prime}}(u)\leq 1\), and \(\mu_{\tilde{A}}(x^{\prime})\) denotes the secondary MF of \(\tilde{A}\).