# A Fuzzy-Based Decision Support Model for Risk Maturity Evaluation of Construction Organizations
## 3 Research Methodology
### Data Analysis and Evaluation
#### 3.4.2 Fuzzy Set Theory  
The fuzzy set theory is a frequently used method for managing construction risks [20; 21; 22]. The fuzzy representation of linguistic terms can capture the vagueness and imprecision associated with the input data provided by the participants [23]. In the second part of the questionnaire, the participants were asked to rate the degree of implementation corresponding to each risk maturity attribute. The input data collected from the qualified individuals was in a linguistic form, where values of 1 = very low, 2 = low, 3 = medium, 4 = high and 5 = very high. In order to transfer the linguistic term to a number, each linguistic value was represented with a fuzzy number as shown in Figure 4.  
As shown in Figure 4, each fuzzy number has an overlap with its neighboring sets. It was pointed out by [24] that in most of the cases the overlap for triangle-to-triangle fuzzy regions varies between 25% and 50% of the fuzzy set base. In this study, the overlap was considered as 50% of fuzzy set base and the expected value of the organization's risk maturity can be calculated using Equations (1)-(5).  
The fuzzy implementation level associated with attribute (\(i\)) can be calculated according to the following equation [10]:  
\[F_{i}=(f_{1},f_{2},f_{3})=\frac{1}{n}\times\sum_{p=1}^{n}F_{ip}, \tag{1}\]  
Figure 4: Membership functions of linguistic values [10].  
Figure 3: ANP entry-matrices: (**a**) the network supermatrix, (**b**) the network sub-matrix [19].  
where (\(F_{i}\)) is the triangular fuzzy number of the attribute (\(i,f_{i1},f_{i2},f_{i3}\)), which are the lower bound, strongest membership degree, and upper bound of (\(F_{i}\)), respectively; (\(n\)) is the number of questions associated with attribute (\(i\)); and (\(F_{ip}\)) is the triangular fuzzy number associated with (\(p\)th) question of attribute (\(i\)). (\(F_{ip}\)) is computed utilizing Equation (2):  
\[F_{ip}=\frac{1}{k}\times\sum\
olimits_{j=1}^{k}F_{i\,pj}, \tag{2}\]  
where (\(k\)) is number of individuals who assess the implementation level of each question; (\(F_{ip}\)) is the triangular fuzzy number of (\(p\)th)question of attribute (\(i\)) collected from (\(j\)th) individual. The triangular fuzzy number of the overall risk maturity of the organization (\(M\)) can be computed according to the Equations (3) and (4):  
\[M=(m_{1},m_{2},m_{3})=\sum\
olimits_{i=1}^{6}(F_{i}\times W_{i}\,), \tag{3}\]  
where (\(m_{1}\)), (\(m_{2}\)), (\(m_{3}\)) are the lower bound, the strongest membership degree, and the upper bound of (\(M\)), respectively; (\(W_{i}\)) is the importance weight of the attribute (\(i\)) calculated by the ANP. The fuzzy number of the overall risk maturity is transferred to a crisp number employing the center of the area method, which is a commonly used method for defuzzification [25]. The expected value (\(EV\)) represents the defuzzified value of a fuzzy number according to Equation (4) [26]:  
\[EV=\frac{(a+2b+c)}{4}, \tag{4}\]  
where, (\(a\)), (\(b\)), and (\(c\)) are triples of a triangular membership function. Therefore, in this study, the expected value of the organization risk maturity is calculated as Equation (5):  
\[EV=\frac{(m_{1}+2m_{2}+m_{3})}{4} \tag{5}\]