## 3 Model and Methodology
### Solution Procedure
#### 3.2.4 Three-Stage Procedure Execution Time-Frame  
To determine a project's execution time-frame, the pseudo-code Algorithm 1 is shown below, and the details is discussed in the following subsections. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Priority Rule** & **Description** \\ \hline SFM (Shortest Feasible Mode) & Find the feasible mode combination for which the makespan is minimal \\ \hline LRP (Least Resource Proportion) & Choose the mode that leads to the smallest value of the criterion, \(\max\frac{\int_{i,k}^{\mathcal{P}_{ik}}}{\left\{\frac{\mathcal{P}_{ik}}{ \mathcal{P}_{i}}\right\}}\vee\mu\) \\ \hline LPSRD (Least Product Sum of Resource and Duration) & For each activity, choose the execution mode that has the minimum product sum of non-renewable resource usage and its corresponding mode duration, \(\min\sum_{k=1}^{\mathcal{K}^{v}}\left(\tau_{i,ik}^{v}\ *\ d_{ij}\right)\ \forall\ \mu\) \\ \hline LTRU (Least Total Resource Usage) & Choose the execution mode that requires the least total non-renewable usage, \(\min\sum_{k=1}^{\mathcal{K}^{v}}\tau_{i,ik}^{v}\ \forall\ \mu\) \\ \hline LRS (Least sum of Non-renewable Resource) & Choose the execution mode that requires the least sum of the ratio of the non-renewable consumption to its corresponding resource limitation, \(\min\sum_{k=1}^{\mathcal{K}^{v}}\frac{\tau_{i,ik}^{v}}{\mathcal{K}_{i}^{v}}\ \forall\ \mu\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mode selection rules. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Priority Rule** & **References** & **Description** \\ \hline max ACTIM & [51] & \(\mathit{CPM}-\mathit{LST}_{i}\) \\ \hline max GCUMRD & [52] & The sum of the renewable resource demand of the activity considered and the renewable resource demands of all its immediate successors \\ \hline max MTS & [53] & \(\left|\overline{S}_{i}\right|\) the total number of successors for activity \(i\) \\ \hline max MIS & [53] & \(\left|S_{i}\right|\) the number of immediate successors for activity \(i\) \\ \hline max ROT & [54] & \(\frac{\sum_{i=1}^{\mathcal{K}^{v}}\frac{\tau_{i,ik}^{v}}{\mathcal{P}_{ik}}}{ \mathcal{P}_{ik}}\) sum of the ratio of the renewable resource requirement over the resource availability divided by the activity duration for activity \(i\) \\ \hline max WRUP & [55] & \(0.7*\left|S_{i}\right|\ +0.3*\ \frac{\sum_{k=1}^{\mathcal{K}^{v}}\frac{\tau_{i,ik}^{v}}{ \mathcal{P}_{ik}}}{\mathcal{P}_{ik}}\), a weighted sum of the number of immediate successors and an average resource use over all renewable resource types \\ \hline min EFT & [56] & \(\mathit{LET}_{i}\) \\ \hline min LFT & [56] & \(\mathit{LET}_{i}\) \\ \hline min SLK & [56] & \(\mathit{LET}_{i}-\mathit{EFT}_{i}\) \\ \hline min LNRJ & [52] & \(\left|\overline{\text{NS}_{i}}\right|\) the total number of activities that are not precedence related with activity \(i\) \\ \hline min RSM & [57] & \(d_{ij}=\max\{0,(\mathit{EFT}_{i}-\mathit{LST}_{j})\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Activity priority rules. ```
Stage 1:Minimize Makespan (Upper Bound Makespan, \(MS_{1}\))  Initialization Phase  While \(i<\) population  Evaluate Mode Selection Rules (MSR)  Evaluate Activity Priority Rules (APR)  End  Repeat until MNC  Employed Bees Phase  Onlooker Bees Phase  Scout Bees Phase  End  Stage 2:Compute schedule's entropy (Upper Bound Makespan) Stage 3:Maximize Robustness  Initialization Phase  While \(i<\) population  Evaluate Mode Selection Rules (MSR)  Evaluate Activity Priority Rules (APR)  End  Repeat until MNC  Employed Bees Phase  Onlooker Bees Phase  Scout Bees Phase  End
```  
**Algorithm 1**Repeat until all instances are solved  
After selecting the execution modes and determining the activity list, the third step is to generate an actual schedule. Recall that schedules must be time feasible (no cycles of positive length), as well as resource feasible. A Serial Generation Scheme (SGS), which means that at every time period, only one activity is considered for scheduling, is employed.