## 6 Numerical Results
### Impact of the Efficiency-Stability Coefficient \(\alpha\) on the Schedules  
In this study, the value of \(\alpha\) is varied from 0.5 to 1 to study its impact on the optimal schedule. It is assumed that the _TWWT_ is a principal criterion and the _TWCTD_ is a secondary one. Thus, the values of \(\alpha\) inferior to 0.5 are not tested. If \(\alpha\) becomes too small, the schedule efficiency will be negligible and the situation will be unrealistic in real-life problems. The study has been conducted and analyzed for 10 different instances, generated accordingly to the parameters presented in Section 5.1. We present, in the following example, the results obtained on the instance presented in Table 1.  
**Example 3**. : _After resolution of the initial problem (see Table 2), it is assumed that new jobs occur disrupting the system. A total of 20 jobs are treated online. The obtained results are given in Table 7._  
In Table 7, \(p_{\theta}=0.5\) (20 jobs appeared in a time horizon of 48 \(ut\)). At each disruption, the jobs are rescheduled and three values are extracted, the value of _TWWT_, the value of _TWCTD_ and the value the objective function combining _TWWT_ and _TWCTD_ thanks to \(\alpha\). We have also numbered the schedules with their step numbers. When \(\alpha=1\), the objective function only considers the efficiency criterion, _TWWT_. In this case, when the release dates are equal, _wSPT-rule_ is optimal for this problem. Otherwise, the jobs have been scheduled according to its weight and its processing time. When \(\alpha\) decreases, the schedule stability is more considered. Hence, a diminution of a stability criterion _TWCTD_ values is observed, as well as an increase on the values of _TWWT_. This increase becomes larger in the last steps when the number of jobs is larger (see Figure 6). Figure 6: _TWWT_ according to \(\alpha\). The more disruptions, the higher the value of _TWWT_. In fact, the increase of _TWWT_ in the last steps signifies that there are some jobs that stay for a long time in the system. This increase becomes large when \(\alpha\) decreases. To illustrate this phenomenon, we present hereafter the first five jobs disposition in the schedule (see Table 8). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(\alpha\) & **(1)** & **(2)** & **(3)** & **(1)** & **(2)** & **(3)** & **(1)** & **(2)** & **(3)** & **(1)** & **(2)** & **(3)** \\ \hline \multicolumn{12}{c}{2 nd Schedule} & \multicolumn{4}{c}{3 rd Schedule} & \multicolumn{4}{c}{4 th Schedule} & \multicolumn{4}{c}{5 th Schedule} \\ \hline  
The increase of the _TWWT_ at each rescheduling step is due to the low-weight jobs. These jobs move forwards after each disruption, since the new arriving jobs are included in the middle of the sequence, when they have a high weight. However, when \(\alpha\) decreases, the stability criterion is more considered. Thus, the existing jobs make less movement, so as not to disrupt the previous sequence. Therefore, the new arriving jobs are placed lastly which also impacts the value of _TWWT_ in the case they have a high weight. This explains the increase of the _TWWT_ with the small \(\alpha\) values. **Remark 1**. : _The high weight jobs are always prioritized. Consequently, the jobs with low weights are delayed, even if they arrive early (the case of job \(B\) in the example). In fact, the low-weight jobs move forwards at every rescheduling step. There are few chances for these jobs to be executed. To deal with this problem, an innovative concept is conceived. It consists in increasing the jobs weight in function of the time. This is the purpose of Section 5.3._  
To analyze the behavior of the criteria with respect to the number of jobs, we studied the variation of the final values of _TWWT_, _TWCD_ and the objective function for different values of \(\alpha\) and \(p_{\theta}\). Ten different instances have been generated, according to the parameters of Section 5.1. Each instance starts by 5 initial jobs.