## 6 Numerical Results
### Computing Time Study  
To analyse the maximum time spent in an iteration, in this study, we tested the maximum duration of an iteration (_MDI_) for the model resolution on different instances. The study has been performed on the solver FICO Xpress IVE, on a laptop of a Core i7 2.90 GHz. Ten different instances per problem size are tested. These are generated randomly, using the values of Section 5.1. Then, the maximums, the minimums, the averages and the standard deviations of _MDI_ in seconds are calculated. We have tested sets of instances containing respectively 5 and 7 initial jobs. Then, over a horizon of \(T=48\)\(ut\), we have disrupted these jobs by the arrival of new ones. \(\alpha\) is given by 0.8. Different values of \(p_{\theta}\) are tested (0.2; 0.5; 0.6; 0.7) to analyze its impact on the _MDI_. The obtained results are given in Table 13. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**MILP**} & \multicolumn{3}{c}{\(wSPT\)} & \multicolumn{3}{c}{**FIFO**} \\ \hline \(p_{\theta}\) **(TNAJ)** & **\(\alpha\)** & **(1)** & **(2)** & **(3)** & **(4)** & **(1)** & **(2)** & **(3)** & **(4)** & **(1)** & **(2)** & **(3)** & **(4)** \\ \hline  & 0.5 & 60.9 & 9.4 & 35.15 & 0.59 & 67.4 & 22.7 & 48.1 & 0.08 & 104.8 & 0 & 52.4 & 0.06 \\
0.2 (10 jobs) & 0.75 & 59.2 & 12.4 & 47.5 & 0.62 & 67.4 & 22.7 & 56.225 & 0.08 & 104.8 & 0 & 78.6 & 0.06 \\  & 1 & 58.4 & 16.8 & 58.4 & 0.76 & 67.4 & 22.7 & 61.9078 & 0.08 & 104.8 & 0 & 104.8 & 0.06 \\ \hline  & 0.5 & 305.3 & 79.4 & 192.35 & 247.8 & 286.7 & 162.9 & 224.8 & 0.08 & 596 & 0 & 298 & 0.07 \\
0.6 (28 jobs) & 0.75 & 277.3 & 121.7 & 238.5 & 242.1 & 286.7 & 162.9 & 255.75 & 0.09 & 596 & 0 & 447 & 0.08 \\  & 1 & 261.3 & 154.6 & 261.2 & 235.6 & 286.7 & 162.9 & 286.7 & 0.1 & 596 & 0 & 596 & 0.08 \\ \hline  & 0.5 & 770.42 & 212 & 491.21 & 1746.98 & 758 & 479.2 & 618.6 & 0.28 & 1523.8 & 0 & 761.9 & 0.11 \\
0.7 (33 jobs) & 0.75 & 665.28 & 379.71 & 593.89 & 9530.24 & 758 & 479.2 & 688.3 & 0.28 & 1523.8 & 0 & 1110.6 & 0.12 \\  & 1 & 646.42 & 445.85 & 646.42 & 1606.10 & 758 & 479.2 & 758 & 0.29 & 1523.8 & 0 & 1523.8 & 0.10 \\ \hline \hline \end{tabular} (1): _TWWT (2): TWCTD (3): Objective function (4): Computing time (in seconds) _TNAJ_: Total Number of Appearing jobs  
\end{table} 
Table 12: Comparison of the results for different solving methods. The set of jobs that must be executed, at each iteration, increases too much with \(p_{\theta}\), as well as the average of the _MDI_. This later increases also with the set of initial jobs. Thus, the average of _MDI_ depends on both, the appearance frequency of the jobs and the set of initial jobs. If these ones become large, the average and the standard deviation of the _MDI_ increase. This matter makes it difficult to estimate the necessary computing time that the MILP needs for solving the problem. In the dynamic environments, the decision makers must establish a planning after each disruption. This one should be constructed as soon as possible, in preference before the occurrence of another disruption. In our case, according to the discretization that we have assumed, the arrival of a new job can be possible at each \(\Delta t\). So, if the _MDI_ exceeds \(\Delta t\), it is considered as an unacceptable time. Based on the assumption assuming that \(\Delta t\) is equivalent to 10 min (600 s) and according to Table 13, the proposed MILP model is effective in solving the problems made up of 7 initial jobs subjected to disruptions which occur at each period with an appearance frequency up to 0.6. It is averagely equivalent