# Applying Game Theory to Teach the PERT Scheduling Method
## 3 Use of the PERT Method in Construction Planning and Scheduling
### PERT Method (Program Evaluation and Review Technique)  
For non-repetitive activities, the PERT method assumes that the time required is not known in advance, including probabilities in the time analysis and the concept of expected value to obtain the project's total duration [79]. This method represents the logical sequence and the interrelation of all project activities through a network flow chart, with initiation nodes and branches representing different tasks, which can be successive, simultaneous, or convergent. The PERT method uses probabilistic estimates of the durations of each activity, known as optimistic, pessimistic, and most probable. In this way, it is possible to calculate the probability of completing the project within a given time frame and find a critical path [80].  
Like tools such as the Gantt chart and the CPM Critical Path, PERT helps get closer to the desired optimal point in projects [81]. From an administrative perspective and the benefit provided to the decision-maker, PERT specifies and details how planning should be done. Among its advantages, the following stand out: Reduction of time and costs (it helps achieve the objective or complete a project with the minimum expenditure of time and cost) and optimization of resources (it minimizes the costs associated with the project and, in this way, achieves a more significant final utility and attractive returns) [79].  
In organizing project activities in an orderly and prioritized manner, tools and methods emerge that simplify planning. The PERT method (Program Evaluation and Review Technique) helps with scheduling control. It allows identifying which activities determine the duration of a project and which ones need to be controlled more to try to shorten their duration.  
The PERT method is a set of methods and techniques for effectively scheduling goal-oriented activities, establishing a baseline for scheduling, evaluating costs, controlling, and re-scheduling. It consists of arranging different activities in a network, which, due to their dependencies and links, contribute to the result of the project [82].  
In developing a project, there is often uncertainty about the timing of critical activities. These, being random variables, can be associated with a probability distribution. The PERT method takes this uncertainty into account, assuming that the estimated time for each activity is based on three different values: optimistic time (\(T_{o}\)) or the minimum period that elapses when the activity occurs in an ideal manner; pessimistic time (\(T_{p}\)) or the maximum period that elapses when probable but infrequent risks cause long delays; most probable time (\(T_{m}\)) or the most frequent duration, and finally, expected time (\(T_{e}\)) or the expected time for an activity.  
The most likely estimate Tm does not have to coincide with the midpoint of \((T_{o}+T_{p})/2\) because the duration of each activity is assumed to follow a Beta probability distribution.  
The standard deviation is represented by Equation (1), and the time variance will be represented by Equation (2):  
\[\sigma_{t}^{2}=\frac{T_{p}-T_{o}}{6} \tag{1}\]  
\[v_{t}=\sigma_{t}^{2} \tag{2}\]The midpoint weighs half of the most probable point \(T_{m}\); that is, the expected value \(T_{e}\) is the average of \(\left(T_{o}+T_{p}\right)/2\) and \(2T_{m}\), as can be seen in Equation (3) and is developed in Equation (4):  
\[T_{e}=\frac{1}{3}\left[2T_{m}+\frac{1}{2}\left(T_{o}+T_{p}\right)\right] \tag{3}\]  
\[T_{e}=\frac{4T_{m}+T_{o}+T_{p}}{6} \tag{4}\]  
It can also be assumed that the project's expected duration is a random variable approximating the Gaussian distribution, and probabilities can be calculated using a normal distribution table.  
On the other hand, Equation (5) represents the probability that the project is carried out in the desired duration (\(D\)):  
\[Z=\frac{D-T}{\sigma_{t}} \tag{5}\]  
where  
\(Z\): normalized variable N(0,1)  
\(T\): critical path time  
\(\sigma_{t}\): standard deviation of critical activities