## 3 Type-2 Fuzzy Logic System
### RVS Modeling Based on Interval Type-2 Fuzzy System  
Figure 1 presents the schematic structure of a proposed RVS model based on IT2 FLS. The model consists of 6 input variables and one output variable as damage category (damage index). Following are the explanation of different parts of the model: Input variables represent structural and environmental characteristics, which affect the global seismic response of a structure. These characteristics are presented in Figure 1 as input variables and in the fuzzification component. Each characteristic is described in linguistic terms that indicate its state. These terms are attributed to fuzzy sets, which are defined over the variable's universe of discourse via MFs. As can be seen in Figure 1 the fuzzifier of maps a vector of six inputs \(x=(x^{\prime}_{1},...,x^{\prime}_{6})^{T}\) into six IT2 fuzzy sets \(\bar{X}_{i}\), \(i=1,2,...,6\). In this study a non-singleton fuzzifier has been considered because it is useful when the measurements are corrupted by non-stationary noise and has shown better performance [45]. Generally, there is no restriction on the number of MFs one could use; however, it is practically recommended to use less than 7 MFs in each input domain of a type-1 or type-2 fuzzy system to reduce computational cost, reduce number of rules, and make interpreting more straightforward [46]. The common MF shapes in type-2 are Gaussian, bell-shaped and piecewise linear. Figure 2 shows the  
Figure 1: The structure of the proposed RVS based on IT2 FLS  
Gaussian and generalized bell-shaped MFs for story number, plan irregularity, vertical irregularity, age, soil type, and PGV, respectively. While a few of these variables have numerical values, the others are all in linguistic forms. However, in the classical modeling the inputs, plan irregularity and vertical irregularity are quantified as 1 implying existence (yes) and 0 as nonexistence (no) of the effect according to the crisp two-valued logic. The linguistic terms used for the definition of each variable and the respective fuzzy sets are described as:  
* Story number: Depending on the number of stories, one of the following classes should be selected. The classifications on this study are for mid-rise buildings as the number of stories less than 3 (_Low_), between 3 and 6 (_Medium_), and more than 6 (_High_) as presented in Figure 2a. * Plan Irregularity: This parameter should be considered when any of the irregularities, for instance, buildings with re-entrant corners (L, T, U, E, + shape) and buildings with different lateral resistance in both directions, have been observed. Any asymmetrical plan and distribution of vertical elements can cause torsion to the building (Figure 2b). * Vertical irregularity: If any of the irregularities such as steps in elevation view, inclined walls, buildings on a hill, soft story, buildings with short columns, and discontinuity in frames are identified, then this parameter should be considered (see Figure 2c). * Age of building: This parameter is classified into three different input variable membership function as _New_ (age \(<\)15 years), _Moderate_ (15 \(<\) age \(<\) 30), and _Old_ (age \(>\) 30) as presented in Figure 2d. * Soil type: The soil type is classified into three different input variable membership function as _A/B_ (rock and dense soil), \(C\) (stiff soil), and \(E\) (soft) as shown in Figure 2e. * Peak Ground Velocity (PGV): The velocity is used to characterize the amplitude of seismic motion at intermediate frequencies therefore, it is useful to indicate the potential damage for structures sensitive to the field of intermediate frequencies [9; 47]. In this paper, the PGV numerical values at any desired locations are based on the micro zoning studies by [PERSON] and [PERSON] [48], which are fuzzified into three MFs as _Low_, _Medium_, and _High_, which are illustrated in Figure 2f. Lastly, the output variable _damage index_ represents the possibility that a structure will suffer a particular damage type.