## 3 Methodology
### Output Validation  
S-Curve displays accumulated efforts or costs of a project and represents the function of time or cost. In fact, this curve was employed to control and check progress in a set of projects related to each other. Moreover, it is showed that this is possible to characterize the progression of the overall cost of any project [35]. The results of the study inductively concluded that the rising slope in the S-curve, and the point of time when half of the total funds have been consumed, are two numbers that represent this curve.  
In the absence of data, we can characterize an equation to illustrate project S-curves with three parameters:  
Figure 3: Project disciplines and their relations to project resources.  
* The total cost of the project (Y\({}_{1}\));
* The total life time of the project (t\({}_{1}\));
* The point of time when the project has spent half of its total funds (t\({}_{1/2}\)).  
Estimating the three above parameters, an S-curve equation that can specify in what way labor or funds could be represented during the project at the start of any project can be established for project management purposes.  
Define \(\beta\) as the proportion \(\beta=t/t_{1}\), and term the specific \(\beta\) where Y = 0.5Y\({}_{1}\) as \(\beta_{1/2}\) where Y indicates the project cost, given the condition that 0 \(<\beta_{1/2}<\) 1. If a project whose distribution is front-loaded, where the funds are spent earlier rather than later, \(\beta_{1/2}\) will be nearer to the beginning of the project. Other projects tend to have this time closer to the middle (\(\beta_{1/2}=0.5\)). For back-loaded projects, in which funds are disbursed nearer to the end of the project, \(\beta_{1/2}\) would be greater than 0.7.  
A gentle rise, a steep slope and a gradual path to the asymptote are the three parts of a typical S-curve. Suppose that the steep slope in the middle portion includes two-thirds of the rise from point Y = 0 to Y = \(Y_{\infty}\), where \(Y_{\infty}\) is the highest level of the project cost. One-third of the rise remaining takes place over the two gently rising portions. Factor \(r\) is defined as a parameter that displays the large size of the rise in the third middle and enables the project manager to freely select an approximation to this central slope, and \(r_{0.67}\) is defined through:  
\[r=\frac{2}{3}r_{0.67}\]  
Assume that the project manager selected desired \(\beta\) and \(r_{0.67}\), so the value of \(\gamma\) through Equation (1) can be calculated. The project manager can then plot the project's S-curve using Equation (2).  
\[\ln\left(\gamma+2\right)=8\ \beta_{1/2}r_{0.67} \tag{1}\]  
Finally, the S-curve equation, in regard to the ratio y = Y/Y\({}_{1}\), is defined as follows:  
\[\mathrm{y}(\beta)=\mathrm{y}_{\infty}\frac{1-\exp\left(-8r_{0.67}\beta\right) }{1+\gamma\mathrm{exp}\left(-8r_{0.67}\beta\right)} \tag{2}\]  
where \(y_{\infty}=Y_{\infty}/Y_{1}>\) 1. Theoretically, y = 1 shows that the project is complete, and at \(\beta=\beta_{1/2}\), y = \(\frac{Y_{\infty}}{2}\).  
Figure 4: S-curve technique flowchart.  
If the manager is willing to say that y = 1/2 at \(\beta\) = 1/2, conducting the \(\hat{\beta}\) calculation will obtain a new \(\beta_{1/2}\) that will meet the desired situation. Figure 4 represents the flowchart for using this technique to plot a project's S-curve.  
In the next section, the behavior of simulator results is compared to project life cycle trends, using an S-curve trend to verify these results.