## 3 Fuzzy Approach to Modeling Construction Duration Using Risk Matrix
### Theoretical Aspects of the Proposed Fuzzy Approach
#### 3.1.3 Fuzzy Approach to Estimate the Risk Lag Time Using the Level of Influence of the Risk Factors  
The conceptual framework of the model was developed to establish a reliable estimate of the total duration of a construction project. Figure 8 illustrates the model to estimate the total duration of a construction project.  
For the new construction project, the construction project manager or planner begins by generating a work breakdown structure, which is a systematic division of projects into smaller sub-projects in a hierarchical order to achieve the project objectives [20]. The work breakdown structure consists of work packages, work units, and activities. Then, relationships between activities are established, and the normative duration of each activity is determined. The ES, EF, LS, and LF are determined for every activity, critical path, and total duration of the construction project using PDM and Equations (1)-(8). The next step is a risk identification for each construction activity. After that, the construction project manager or planners should analyze and assess the identified risks for each activity. The risk assessment for each activity will be divided into two major parts: part A and part B. In part A, the project manager should estimate the level of the probability for each activity based on their own experience or collected data from experts. The impact assessment should indicate the severity or damage that will occur to the project. If the risk event  
Figure 8: Approach flowchart for modeling construction duration using a risk matrix.  
influences the duration of the activity, the impact of this risk event may be estimated in days. The impact of a risk event as a percentage is determined by Equation (19):  
\[\text{IoRE}_{\%}=\text{IoRE}_{\text{d}}\times 100/\text{ND}_{\text{d}}, \tag{19}\]  
where IoRE\({}_{\%}\)--the impact of a risk event (in percentage), IoRE\({}_{\text{d}}\)--the impact of a risk event (in days), ND\({}_{\text{d}}\)--normal (average) duration (in days).  
The next step is fuzzification. Fuzzification is the process of decomposing a system input and/or output into one or more fuzzy sets using Equations (11)-(13). Finally, the construction project manager should establish the dependencies between input data, such as probability and impact, and output data, such as the level of threat for activity, and represent them in the risk matrix form. An example of a risk matrix is presented in Table 1. Thus, the risk matrix dependencies will be used to build the rule base using Equation (14).  
The authors' approach applies the Mamdani FIS (Figure 5) [50] and Equations (15) and (16) for output fuzzy sets. After that, the crisp value of the risk lag time is determined using a defuzzification method such as centroid, bisector, middle of maximum, smallest of maximum, and largest of maximum. A description of each defuzzification method is presented in Figures 6 and 7 and Equations (17) and (18).  
The output obtained after defuzzification is the calculated value of the risk lag time in a percentage (RLT\({}^{\text{c}}\)%). The calculated value of the risk lag time is an additional reserve time that depends on the probability of the risk, the magnitude of its impact, and the level of threat determined based on the risk matrix. To convert the risk lag time from percentage to days, Equation (20) is applicable:  
\[\text{RLT}^{\text{c}}_{\text{d}}=\text{IoRE}_{\text{d}}\times\text{RLT}^{ \text{c}}\text{\%}/100, \tag{20}\]  
where RLT\({}^{\text{c}}_{\text{d}}\)--the calculated value of the risk lag time (in days), IoRE\({}_{\text{d}}\)--the impact of risk event (in days), RLT\({}^{\text{c}}\)%--the calculated value of the risk lag time (in percentage).