## 3 Fuzzy Approach to Modeling Construction Duration Using Risk Matrix
### Case Study
#### 3.2.3 Risk Analysis and Assessment  
The construction project managers or planners analyzed and estimated the possibility of risk occurrence by using their own experience, data from subcontractors, and other methods of expert assessments for every risk event. The possibility of a risk event is shown in Table 4. In addition, the impact of risk events has been estimated. The level of impact is calculated in days and shows how many days of work can be increased if a risk event occurs. The level of impact for this example and the determined level of impact as a percentage using Equation (19) are summarized in Table 5. The project duration is determined using PDM methods and is presented in Figure 11. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activity** & **Risk Event** & **Probability, \%** \\ \hline (A) & Bad weather & 38 \\ Site Investigation and Preparation (B) & & \\ Foundation Works & Unreliable soil information & 46 \\ on Section 1 & & \\ (C) & & \\ Foundation Works & Unreliable soil information & 29 \\ on Section 2 & & \\ (D) & & \\ Construction of Monolithic Building Frame on Section 1 & Tower crane failure & 84 \\ (E) & Misunderstanding of technical & \\ Construction of Monolithic Building Frame on Section 2 & documentation by workers & 70.5 \\ (F) & Worker absenteeism & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The activities and their possible risk events with the level of probability of occurrence. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description of Activity** & **Risk Event** & **Risk Factor** \\ \hline (A) & Bad weather & The amount of rainfall \\ Site Investigation and Preparation (B) & & \\ Foundation Works & Unreliable soil information & Error from survey team \\ on Section 1 & & Unexpected underground objects \\ (C) & & \\ Foundation Works & Unreliable soil information & Error from survey team \\ on Section 2 & & Unexpected underground objects \\ (D) & & \\ Construction of Monolithic Building Frame on & Tower crane failure & Lack of maintenance \\ Section 1 & & \\ (E) & Misunderstanding of technical & \\ Construction of Monolithic Building Frame on & documentation by workers & Lack of technical education \\ Section 2 & & Worker illness \\ Masonry Works & Worker absenteeism & Rule and regulation \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The activities and their possible risk events and factors. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activity** & **Normal Duration, Days** & **Impact of Risk Event** & **Impact of Risk Event** \\  & & **(Lag Time), Days** & **(IoRE\%), \%** \\ \hline (A) & 24 & 12 & 50 \\ Site Investigation and Preparation (B) & & \\ Foundation Works & 45 & 13 & 29 \\ on Section 1 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Lag Time is associated with activities and risks. The next step is the fuzzification of the input data sets, such as the probability of risk events and the level of impact. The degree of the membership of the individual fuzzy sets defined by linguistic variables is determined. The processes of fuzzification, calculation, and defuzzificaton have utilized MATLAB (ver. R2023) software. The membership functions are shown in Figures 12 and 13. The input and output data sets are related based on risk matrices. Using these relations creates the rule base. The rule base contains logical rules defining cause-effect relationships between the fuzzy input and output sets. The developed base includes nine rules for each risk matrix. The basic rule for matrix type 1 is as follows: If the level of probability is unlikely and the level of impact is low, then the level of threat is low. Mamdani fuzzy inference is used as a method to create a control system by synthesizing a set of linguistic control rules obtained from experienced operators. The output of each rule is a fuzzy set. The output of each rule is a fuzzy set derived from the output