# Construction Schedule versus Various Constraints and Risks
## 4 Probabilistic Time Coupling Methods--PTCM
The computational segment of the PTCM assigned to the \(P_{j}\) process and the \(S_{i}\) sector is shown in Figure 1.  
Where:  
* \(t_{ij}\)--calculated implementation time of construction process \(j\) on plot \(i\);
* \(\sigma_{ij}\)--standard deviation of the completion time of construction process \(j\) on plot \(I\);
* \(\Sigma_{ij}\)--the sum of standard deviations of processes preceding the construction process \(j\) on plot \(i\) and the standard deviation \(\sigma(t_{ij})\) of current work, calculated by the formula for the sum of deviations of independent variables;
* \(t_{ij}\)--prognostic start time of the construction process \(j\) on plot \(i\);
* \(t_{ij}\)--prognostic time of completion of the construction process \(j\) on plot \(i\) (most probable);
* \(t_{ij}\)--prognostic minimum time for completing the construction process \(j\) on plot \(i\) (the most optimistic);
* \(t_{ij}\)--prognostic time of maximum completion of the construction process \(j\) on plot \(i\) (the most pessimistic). Figure 1: Graphical representation of the transformation of a single TCM computational segment into a single PTCM computational segment. A single calculation segment of the PTCM corresponds to the work of one working team (\(P_{j}\)) on one working plot (\(S_{i}\)). The size of the investment affects the number of plots/work sectors, and its complexity affects the number of processes. The number of calculation segments on the PTCM sheet depends on the designated number of sectors and processes on the construction site and results from the adopted technology and execution technique. General calculation formulas of the PTCM methodology, repeated in all variants of the methodology, are shown below (1-33):  
\[t_{1,1}^{r}=0\text{ or given initial value} \tag{1}\]  
\[\tilde{t}_{1,1}^{z}=t_{1,1}^{r}+t_{1,1} \tag{2}\]  
\[\Sigma_{1,1}=\sigma_{1,1} \tag{3}\]  
\[t_{1,1}^{\min}=t_{1,1}^{z}-\Sigma_{1,1} \tag{4}\]  
\[t_{1,1}^{\max}=t_{1,1}^{z}+\Sigma_{1,1} \tag{5}\]  
\[t_{i,1}^{z}=t_{i,1}^{r}+t_{i,1} \tag{6}\]  
\[t_{i,1}^{\min}=t_{i,1}^{z}-\Sigma_{i,1} \tag{7}\]  
\[t_{i,1}^{\max}=t_{i,1}^{z}+\Sigma_{i,1} \tag{8}\]  
Figure 1:0  
Figure 1:1  
Figure 1:2  
Figure 1:3  
Figure 1:4 General notation of the PTCM method. Figure 1:5 Example of using the method PTCM I.  
following: \(m\) sectors, \(n\) processes, \(t_{nm}\) work completion times, and \(\sigma_{mn}\) standard deviations of work completion times (see Figure 1). The output data obtained include the following: the most probable completion time, the minimum completion time and the maximum completion time for the implementation of construction processes in individual sectors. An example cyclogram resulting from calculations according to the PTCM II is shown in Figure 1:6 A detailed study of the PTCM II computational case has not been presented due to the extensive scope of the material [62]. Figure 1:7 Example of application of the PTCM II. Calculation formulas characterizing the PTCM II (46-57):  
\[t^{r}_{i,1}=t^{r}_{(i-1),1}+\max\left\{ \begin{array}{l}t_{(i-1),1}\\ t_{(i-1),1}+t_{(i-1),2}-t_{i,1}\\ \ldots\\ t_{(i-1),1}+t_{(i-1),2}+\ldots+t_{(i-1),n}-t_{i,1}-t_{i,2}-\ldots-t_{i,(n-1)} \end{array} \right. \tag{46}\]  
\[\Sigma_{i,1}=\sqrt{\Sigma^{2}_{(i-1),j}+\max\left\{ \begin{array}{l}t_{(i-1),1 }\Rightarrow\sigma^{2}_{(i-1),1}\\ t_{(i-1),1}+t_{(i-1),2}-t_{i,1}\Rightarrow\sigma^{2}_{(i-1),1}+\sigma^{2}_{( i-1),2}-\sigma^{2}_{i,1}\\ \ldots\\ t_{(i-1),1}+t_{(i-1),2}+\ldots+t_{(i-1),n}-t_{i,1}-t_{i,2}-\ldots-t_{i,(n-1)} \\ \Rightarrow\sigma^{2}_{(i-1),1}+\sigma^{2}_{(i-1),2}+\ldots+\sigma^{2}_{(i-1),n}-\sigma^{2}_{i,1}-\sigma^{2}_{i,2}-\ldots-\sigma^{2}_{i,(n-1)}\end{array} \right. \tag{47}\]  
\[t^{r}_{m,1}=t^{r}_{(m-1),1}+\max\left\{ \begin{array}{l}t_{(m-1),1}\\ t_{(m-1),1}+t_{(m-1),2}-t_{m,1}\\ \ldots\\ t_{(m-1),1}+t_{(m-1),2}+\ldots+t_{(m-1),n}-t_{mi,1}-t_{m,2}-\ldots-t_{m,(n-1)} \\ \Rightarrow\sigma^{2}_{(m-1),1}+\sigma^{2}_{(m-1),2}+\ldots+\sigma^{2}_{(m-1),n}-\sigma^{2}_{mi,1}-\sigma^{2}_{m,2}-\ldots-\sigma^{2}_{m,(n-1)}\end{array} \right.