## 3 Materials and Methods
### Expression for Estimating the Average Project Duration \(\mu_{p}\)  
As anticipated earlier, the calculation of \(\mu_{p}\) involves screening out some initial activities and keeping only those activities which are either critical or subcritical. The explanations below illustrate the necessary calculations in the same exact order a practitioner should implement them. We consider as critical or subcritical those activities whose total slack (\(s_{i}\)) remains below a slack admissibility threshold (\(\tau\)), that is, \(i\
eq 0\leq s_{i}<\tau\). The slack admissibility threshold \(\tau\) is calculated from the critical activities (\(k\)), that is, those with \(s_{i}=0\), with the following expression:  
\[\tau=K_{0}(1-SP)\sigma_{K}, \tag{6}\]where \(\sigma_{K}\) is an estimate of the critical path(s) duration standard deviation. It can be calculated with the following straightforward (but approximated) expression:  
\[\sigma_{K}=\sqrt{\frac{PD}{\sum d_{k}}\sum\sigma_{k}^{2}}. \tag{7}\]  
When there is a single critical path, the sum of all critical activity durations is equal to the (deterministic) duration of the whole project, that is, \(\sum d_{k}=PD\). In this case, \(\sigma_{K}=\sqrt{\sum\sigma_{k}^{2}}\) and the value of \(\sigma_{K}\) is exact. Only in this case does \(\sigma_{K}\) coincide with the PERT estimate of the project duration standard deviation. \(SP\) is the serial-parallel topological indicator, which can be calculated as follows:  
\[SP=\frac{m-1}{n_{i}-1}, \tag{8}\]  
where \(m\) is the number of activities in the path with _more activities_ (not necessarily the longest in duration, that is, not necessarily the critical path); \(n_{i}\) is the total number of (initial) activities in the schedule. The \(SP\) measures how close a network is to a parallel network (\(SP=0\) when all activities are in parallel) or a serial network (\(SP=1\) when all activities are in series). Recent studies pointed out that most real-life project \(SP\) values range from 0.3 to 0.7 [31]. \(K_{0}\) is an empirical coefficient which represents the number of \(\sigma_{K}\) multiples that define the overall cut-off value of \(\tau\). Specifically, values of \(K_{0}=0\) would only accept critical activities (no subcritical activities, that is, no activities with \(s_{i}>0\)). Conversely, values of \(K_{0}>3\) (meaning three times \(\sigma_{K}\)) would identify as subcritical activities with too big slacks. After some experimentation, it was found that values of \(K_{0}\) within [1; 2] generally lead to higher \(R^{2}\) values when calculating \(\mu_{p}\) (Equation (9)) shown later). Consequently, most of the time, we just assume \(K_{0}=1.5\). The attentive reader would have noticed that the purpose of Equation (6) is just ruling out those activities with excessive slacks. This is simply because the paths to which those activities belong are more unlikely to extend the project duration and ultimately do not affect \(\mu_{p}\). Furthermore, the purpose of \(SP\) is to fine-tune the effectiveness of Equation (6) by considering the network macro-structure. Therefore, once the value of \(\tau\) is known, it is possible to know which activities are critical and subcritical (the \(J\) set). These are the activities whose total slack \(s_{i}\) remains below the threshold defined by \(\tau\). Only with those \(j\) activities can the average project duration distribution (\(\mu_{p}\)) then be easily obtained with  
\[\mu_{p}=PD\cdot\left[1+K_{1}\sqrt{\frac{N_{p}}{n_{j}}}\frac{\sum\sigma_{j}}{ \sum d_{j}}\cdot LN\left(S\cdot N_{p}\right)\right]. \tag{9}\]  
For calculating Equation (9), we need the total number of critical + subcritical activities \(n_{j}\), the (deterministic) activity durations \(d_{j}\), the activity duration standard deviations \(\sigma_{j}\), and the activity total slacks \(s_{j}\) (to calculate \(S\)), and the deterministic project duration \(PD\). This is the same input information that the PERT technique uses. Its components are outlined below.