## 6 Numerical Results
### Comparative Analysis  
As mentioned in the introduction section, some papers consider different dispatching rules and heuristics to implement the reactive approach for rescheduling the jobs in response to disruptions. Some frequently used dispatching rules and heuristics are mentioned in [80]. Among them, we select FIFO (first in first out) to compare with the proposed MILP, since it is commonly used in industrial companies, when not any rescheduling strategy is implemented [43]. In addition, it is also interesting to compare with some heuristic considering the jobs' weights. In [81], the authors compared their proposed heuristics based on MILP with \(wSPT\), since they also considered the jobs' weights in a flowshop  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{\(p_{\theta}\) = 0.2} & \multicolumn{3}{c}{\(p_{\theta}\) = 0.5} & \multicolumn{3}{c}{\(p_{\theta}\) = 0.6} \\ \hline \(\rho\) & **(1)** & **(2)** & **(3)** & **(4)** & **(1)** & **(2)** & **(3)** & **(4)** & **(1)** & **(2)** & **(3)** & **(4)** \\ \hline
0 & 62 & 8 & 51.2 & 0\% & 226 & 128 & 206.4 & 0\% & 349 & 210 & 321.2 & 0\% \\
0.1 & 62 & 8 & 51.2 & 0\% & 268 & 104 & 235.2 & 14\% & 377 & 185 & 338.6 & 5\% \\
0.2 & 62 & 8 & 51.2 & 0\% & 277 & 87 & 239 & 16\% & 416 & 151 & 363 & 13\% \\
0.3 & 62 & 8 & 51.2 & 0\% & 297 & 59 & 249.4 & 21\% & 477 & 110 & 403.6 & 26\% \\
0.4 & 62 & 8 & 51.2 & 0\% & 362 & 53 & 330.2 & 60\% & 517 & 84 & 430.4 & 34\% \\
0.5 & 62 & 8 & 51.2 & 0\% & 393 & 43 & 323 & 56\% & 625 & 72 & 514.4 & 60\% \\
0.6 & 68 & 6 & 55.6 & 9\% & 443 & 39 & 362.2 & 75\% & 727 & 52 & 592 & 84\% \\
0.7 & 68 & 6 & 55.6 & 9\% & 493 & 23 & 399 & 93\% & 809 & 37 & 654.6 & 104\% \\
0.8 & 68 & 6 & 55.6 & 9\% & 513 & 17 & 413.8 & 100\% & 848 & 25 & 683.4 & 113\% \\
0.9 & 68 & 6 & 55.6 & 9\% & 513 & 17 & 413.8 & 100\% & 848 & 25 & 683.4 & 113\% \\
1 & 68 & 6 & 55.6 & 9\% & 516 & 16 & 416 & 102\% & 848 & 25 & 683.4 & 113\% \\ \hline \hline \end{tabular} (1): \(TWWT\) (2): \(TWCTD\) (3): The value of the objective function (4): percentage deviation  
\end{table}
Table 11: Variation of \(TWWT\), \(TWCTD\) and objective function in function of \(\rho\), with different values of \(p_{\theta}\).  
rescheduling problem. Therefore, we adapted these two heuristics, FIFO and \(wSPT\), to match our problem. Ten different instances are tested. Each instance starts with 5 initial jobs, then it will be disrupted, at each step, by the arrival of a new job with the appearance frequency \(p_{\theta}\) respectively equal to 0.2, 0.6 and 0.7. Averages are presented in Table 12.  
The proposed MILP always provides better solution than \(wSPT\) and FIFO. \(wSPT\) which consists in sequencing the jobs by increasing order of \(p_{i}/w_{i}\) misses some optimal solutions since, in the presence of the release dates, this heuristic becomes not able to optimally solve the problem. In addition, it does not take into account the stability aspect. However, FIFO which consists in placing, at each rescheduling step, the new job in the last position, provides bad solutions. Even if FIFO heuristic gives stable sequences, it is weak in terms of efficiency since it does not consider the weights to construct the schedules. Indeed, when \(\alpha\) increases, the schedule efficiency becomes more important. In this case, FIFO solution moves away from MILP solution, this gap becomes wider when the number of jobs is important. Figure 7 depicts this behavior.  
In terms of computation time, the two heuristics, \(wSPT\) and FIFO, are very fast compared to the MILP method. Using this latter heuristic, a break is observed when \(p_{\theta}=0.7\). In the following, a thorough study of the computing time is presented.