## 3 Materials and Methods
### General Notation  
In this subsection, we introduce the most relevant variables, subscripts, and sets. Let \(i=1,2\ldots n_{i}\in I\) be the set of _all activities_ in the project schedule. Let \(k=1,2\ldots n_{k}\in K\) be the set of all _critical activities_ in the project schedule, that is, those whose total slack is zero (noted later as \(s_{i}=0\)). Let \(j=1,2\ldots n_{j}\in J\) be the set of all _critical and subcritical activities_ in the project schedule (those whose slack \(0\leq s_{i}<\tau\), with \(\tau\) being a cut-off slack value named slack admissibility threshold defined later). From the definitions above, it is clear that \(K\) is always a subset of \(J\), whereas \(J\) is always a subset of \(I\), that is, \(K\subset J\subset I\). Additionally, \(d_{i}\), \(d_{k}\), and \(d_{j}\) represent the duration of activities \(i\) (all), \(k\) (critical), or \(j\) (critical and subcritical), respectively. In deterministic schedules, they are assumed as static (constant) (generally, \(d_{i}=\mu_{i}\)), even though they may actually have some variability \(\sigma_{i}\). \(s_{i}\), \(s_{k}\), and \(s_{j}\) represent the _total slack_ (or floats) of an activity \(i\) (all), \(k\) (critical), or \(j\) (critical or subcritical) obtained from a deterministic schedule. The activity _free slacks_ is not used. Furthermore, it is worth noting that, although PERT does not directly use the slack values in its calculations, they must be calculated anyway; otherwise, the critical activities cannot be identified. Now, due to its practical interest, we need to take a closer look at the activity duration standard deviations. Like the activity duration means (\(\mu_{i}\), \(\mu_{k}\), \(\mu_{j}\)), the activity duration standard deviations (\(\sigma_{i}\), \(\sigma_{k}\), \(\sigma_{j}\)) can be estimated in real contexts from the scheduler's past experience, from the PERT three-point estimates, or even by analogy to similar projects. With the three-point estimates, for example, the values of the first two moments of each activity duration distribution (\(\mu_{i}\) and \(\sigma_{i}\)) can be calculated with the following simple expressions:  
\[\mu_{i}=\frac{O_{i}+4L_{i}+P_{i}}{6}, \tag{2}\]  
\[\sigma_{i}=\frac{P_{i}-O_{i}}{6}\sqrt{\frac{5}{7}+\frac{16}{7}.\frac{(L_{i}-O _{i})(P_{i}-L_{i})}{\left(P_{i}-O_{i}\right)^{2}}}, \tag{3}\]  
where \(O_{i}\), \(L_{i}\), and \(P_{i}\) represent the optimistic (minimum), most likely (the mode), and pessimistic (maximum) duration of activity \(i\), respectively, to be elicited by the project scheduler. As can be seen, Equation (3) already includes a correction (the square root term) recently proposed by [PERSON] et al. [29]. This extra term makes the \(\sigma_{i}\) estimate more accurate and also unbiased. Regarding the possibility of estimating \(\sigma_{i}\) values by analogy, [PERSON] et al. [30] established that activity duration variability in construction projects is closely modeled by lognormal distributions with median zero and coefficients of variation (\(CV_{i}\)) of 0.10 (for activities with low variability), 0.20 (for medium variability), and 0.3 (for high variability). The coefficient of variation is the ratio of the activity duration standard deviation by the duration mean, that is, \(CV_{i}=\sigma_{i}\mu_{i}\). However, [PERSON] et al. [30] measured the _log_ ratios of actual versus planned activity durations. If we want to calculate any activity \(i\)'s duration in the first two moments in the natural scale (\(\mu^{\prime}\)/ and \(\sigma^{\prime}\)/), we must perform the following calculations (derived from the first two moments of a lognormal distribution):  
\[\mu^{\prime}_{i}=\mu_{i}e^{\frac{(2302 CV_{i})^{2}}{2}} \tag{4}\]  
\[\sigma^{\prime}_{i}=\sqrt{\left(e^{(2302 CV_{i})^{2}}-1\right)}e^{(2 LN(\mu_{i} )+(2.302 CV_{i})^{2})} \tag{5}\]  
where \(CV_{i}\) = 0.10, 0.20, and 0.30 for low, medium, and high activity duration variability, respectively. The coefficient 2.302 is added to change the base from 10 (the one used in [PERSON] et al.