# Deling with Uncertainty in the MRCPSP/Max Using Discrete Differential Evolution and Entropy
## 3 Methods
### Computational Results  
All experiments were carried out using an Intel i7 personal computer with 8 GB of RAM, and the problem was coded using MATLAB. The DDE iterates through the total problem instances available for each set and randomly selects a predetermined number of feasible instances. Table 6 shows the optimal solutions found and the average runtime for each benchmark when running each algorithm. For all algorithms, increasing the number of activities elevates the problem's complexity. Hence, the average run times increase, whereas the total numbers of optima decrease. Furthermore, the results indicate that the average runtime of DDE was slightly higher than that of ABC running all MRCPSP/max benchmarks. On average, the DDE algorithm takes 20.556 s to find a schedule with optimized robustness, whereas the takes 20.329 s. However, DDE obtains slightly higher numbers of optima than ABC. Thus, we can conclude that DDE is more effective than ABC in this regard. Furthermore, Table 7 presents the results obtained from evaluating all 270 instances of every benchmark set. The target, entropy-based, and resilient schedules in Stage I, II, and III are referred to as S1, S2, and S3, respectively. They were assessed based on two measures: the makespan (Avg. Dev.) and the robustness (Avg. RM.). Avg. Dev. refers to the average of all the deviations computed by (\(M_{s}-\) BKO)/BKO. \(M_{s}\) denotes the schedule makespan at the current stage, whereas the best-known makespan (BKO) represents the reference makespan, i.e., the optimal solution when comparing the target schedule of Stage I and the upper-bound schedule of Stage III. Furthermore, Avg. RM. is a measure of average robustness. Finally, the algorithmic performance was evaluated for three benchmark sets (i.e., MM30, MM50, and MM100) against the ABC. The three-stage procedure proved to be robust enough to produce results comparable to different optimization algorithms. These results are encouraging, given that practitioners can add anywhere between 8% to 15% of their original estimates as response time intervals (i.e., buffer times). Furthermore, both ABC and DDE algorithms performed better when considering that response time intervals used by practitioners rely primarily on intuition and experience. On the other hand, this three-stage procedure relies solely on information available to every project, including activity durations and precedence. Meanwhile, Figures 3-5 show the results of the average deviation of every benchmark instance when compared against the BKO for every stage, divided according to the optimization algorithm applied. In addition, Figures 6-8 show the robustness measures obtained at each stage using the DDE and ABC algorithms on different sizes of benchmark instances, i.e., MM30, MM50, and MM100. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Benchmark Set**} & \multicolumn{2}{c}{**Optima Found (No.)**} & \multicolumn{2}{c}{**Average Run Time (s)**} \\ \cline{2-5}  & **ABC** & **DDE** & **ABC** & **DDE** \\ \hline MM30 & 260 & 263 & 11.888 & 12.189 \\ MM50 & 123 & 124 & 17.063 & 17.223 \\ MM100 & 84 & 87 & 32.037 & 32.257 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The average runtime per MRCPSP/Max benchmark set for each algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Setting** \\ \hline Population Size (\(N_{p}\)) & 30 \\ Abandonment Limit & 5 \\ Maximum Number of Cycles (MNC) & 20 \\ \(\Delta_{t}\) & 1 \\ _frac_ & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter settings for the ABC algorithms.