# Resilient Scheduling as a Response to Uncertainty in Construction Projects
## 3 Application of Resilient Scheduling on a Test Problem Instance
### Validating Resilient Scheduling Process
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & **Baseline** & **Duration** & **SM** & **Profit** \\ \hline Case 1 & 0, 5, 0, 0, 8, 5, 14, 10, 13, 18 & 18 & 0.487 & 904.12 \\ Case 2 & 0, 0, 3, 0, 5, 5, 14, 7, 10, 18 & 18 & 0.353 & 911.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Objective function values for two test instances. Equilibrium state was examined throughout 6 resilience dimensions: (_Eq?i_) probability of reaching baseline due date, (_Eq?ii_) probability of reaching baseline due date multiplied with coefficient 1.1, (_Eq?iii_) average tardiness amount for the proposed baseline, (_Eq?iv_) average tardiness amount for activities' start times, (_Eq?v_) the percentage of simulations where credit limit was broken, and (_Eq?vi_) probability of reaching or exceeding the baseline profit. The last equilibrium dimension (_Eq?vi_) is calculated by considering only those simulations for which credit limit, W, was not broken. From the simulation results, as shown in Table 8, it is evident that the first case (single optimal solution from the previous section) generates better results in all equilibrium calculations than the second case, for which profit maximization was considered as a more important objective than maximizing SM value. With an increase in _SM_ value, there is a constant rise in the probability of reaching both strict (_Eq?i_) and relaxed (_Eq?ii_) baseline duration, as well as a constant decrease of average project tardiness (_Eq?iii_), and a decrease of average tardiness for activities' start times (_Eq?iv_). Moreover, the percentage of simulations for which the credit limit is broken (_Eq?v_) is slightly lower in the first case, where the SM value was higher. Finally, the probability of reaching baseline profit or attaining even higher profit values (_Eq?vi_) is significantly better for the test case with a higher SM value. According to the simulation results, the initially suggested approach, where SM value maximization was taken as a more important objective function than maximizing the baseline profit, leads to better results considering the overall resilience of the baseline solution. Validation results indicate a positive correlation between increased SM value and resilience of baseline schedule with allocated resources. As a result of enhanced Monte Carlo analysis, probability distribution charts for makespan duration, considering both cases, are shown in Figure 5a. Probability distribution charts for final profit are shown in Figure 5b. Since the average deviation of the maximal cash gap is negligible in both baseline cases (average deviation from the negotiated CL was around 0.01% in the simulations where the CL was broken), the probability distribution for profit is shown by considering all simulations, including those where violation of credit limit has occurred. After simulating two different baseline schedules with the Monte Carlo approach, where additional resource  
Figure 4: Baseline schedules with additional resource constraints for (**a**) Case 1 with the highest SM value and lower profit than in Case 2, and (**b**) Case 2 with the highest profit and lower SM value than in Case 1. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Instance** & **SM Value** & **Profit** & **W** & _Eq?i_ & _Eq?ii_ & _Eq?iii_ & _Eq?iv_ & _Eq?v_ & _Eq?vi_ \\ \hline Case 1 & 0.487 & 904.12 & 280 & 0.3 & 0.9135 & 0.66 & 0.12 & 1.65\% & 0.3769 \\ Case 2 & 0.353 & 911.81 & 280 & 0.2046 & 0.8249 & 0.93 & 0.42 & 1.73\% & 0.0978 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Simulation results. relations were appended, it was shown that a schedule with a higher SM value produces a better response to uncertainty. The results of the equilibrium analysis showed improved performances in all equilibrium dimensions (\(Eq_{i}-Eq_{vi}\)) when comparing the baseline schedule with higher SM value to another solution in which profit maximization dominated SM improvement.