## 3 Fuzzy Approach to Modeling Construction Duration Using Risk Matrix  
### Theoretical Aspects of the Proposed Fuzzy Approach  
#### 3.1.1 Critical Path Method (CPM)  
The Critical Path Method (CPM) is a tool for project planning and schedule management [53]. The central aspect of the CPM method is the critical path. The critical path is the longest sequence of tasks upon which the entire project depends. It is a chain of activities where the next task can only be started once the previous one is completed. Using the CPM algorithm, a minimum planned duration project can be determined, and for individual project activity, both the earliest start and the latest finish time are then calculated. In this study, the CPM has been applied with the precedence diagramming method (PDM), a strategy for developing a project schedule network diagram that utilizes nodes to represent activities and associates them with projectiles that illustrate the dependencies. This method is the activity-on-node (AON). The term AON pertains to a methodology in project management known as precedence diagramming, wherein rectangular nodes represent scheduled activities. These nodes are interconnected by arrows, delineating the sequential dependencies among the activities. Each node is assigned a unique identifier, typically a letter or number, corresponding to a specific activity in the project schedule. Primarily, an activity-on-node diagram illustrates the prerequisites for commencing subsequent activities. This arrangement is commonly known as the \"finish-to-start\" precedence, implying that an activity must conclude before the succeeding one can begin. The PDM can be described into four basic types of dependencies or logical relationships between activities: finish-to-start (FS), start-to-start (SS), finish-to-finish (FF), and start-to-finish (SF). FS is an activity that cannot start before a previous activity has ended. SS is a defined relationship between the start of activities. FF is a defined relationship between the end dates of activities. SF is a defined relationship between the start of one activity and the end date of a successor activity. Every activity has four states: early start (ES) is the earliest time an activity can be started; late start (LS) is the latest time an activity can be started. If the activity is started beyond this time, it will affect the critical path; early finish (EF) is the earliest time an activity is completed; late finish (LF) is the latest time the activity can be completed. If the activity crosses this time, the project will be delayed. An example of a precedence diagram is presented in Figure 3.  
The ES for the first activity after the start is always equal to 0. The ES, EF, LS, and LF for subsequent activities are determined using the following Equations (1)-(7):  
\[\text{ES}_{\text{i}}=\text{EF}_{\text{i}-1}, \tag{1}\]  
where ES\({}_{\text{i}}\)--early start for activity i (in days), EF\({}_{\text{i}-1}\)--early finish for activity i\(-1\) (in days). \[\text{EF}_{\text{i}}=\text{ES}_{\text{i}}+\text{D}_{\text{i}}, \tag{2}\]  
where EF\({}_{\text{i}}\)--early finish for activity i (in days), ES\({}_{\text{i}}\)--early start for activity i (in days), D\({}_{\text{i}}\)--duration for activity i (in days). \[\text{ES}_{\text{i}+1}=\text{max}\{\text{EF}_{\text{i}};\text{EF}_{\text{j}}\}, \tag{3}\]  
where ES\({}_{\text{i}+1}\)--early start for activity i+1 (in days), EF\({}_{\text{i}}\)--early finish for activity i (in days), EF\({}_{\text{j}}\)--early finish for activity j (in days). For the last activity in the sequence before the finish, for example, for activity i+1 as shown in Figure 3, Equation (4) is used. \[\text{EF}_{\text{i}+1}=\text{LF}_{\text{i}+1}, \tag{4}\]  
where EF\({}_{\text{i}+1}\)--early finish for activity i+1 (in days), LF\({}_{\text{i}+1}\)--late finish for activity i+1 (in days). After determining all the values for EF, we proceed to determine LF.