## 6 Numerical Results
### The Weight Increasing Effect  
As mentioned above, the jobs with low weights are always placed in the last place in the sequence. Thus, it is consistent to proceed with dynamic weights. Thus, the weights become in function of time, release date \(r_{j}\) and the weight index \(\rho\). The proposed formula is given by: \(w_{jt}~{}=~{}w_{j}(t-r_{j}+1)^{\rho}\). \(\rho\) allows regulating the weight increasing effect. It is assumed that its value is between 0 and 1 for obtaining relatively small augmentation of the weight values. The case when \(\rho=0\) signifies that the jobs' weights are static and no variation of the weight occurs over the time. Otherwise, when \(\rho=1\), the weight variation becomes too large and may impact the schedule efficiency. Choosing a value of \(\rho\) that exceeds 1 becomes unrealistic. Therefore, the selected range for \(\rho\) is [0,1]. Thanks to this new formula, the weight becomes in function of the time \(t\). \((t-r_{j}+1)\) is the difference between the current time and the release date of the job \(j\). If \(t=r_{j}\), we would obtain a weight equal to zero, which is not the aimed effect. So, we have added plus one. Hence, the new objective function will be:  
\[Min\,\alpha\sum_{j~{}=~{}1}^{n^{\prime}}w_{jt}W_{j}+(1-\alpha)\sum_{j~{}=~{}1 }^{n^{\prime}-z}w_{jt}(C_{j}-Co_{j})\]  
This new weight expression helps the low weight earliest jobs to not wait too long time in the system, through increasing their weights in function of the time. The dates of jobs arrival are defined by their release dates \(r_{j}\) and the dates of their execution by their completion times \(C_{j}\). Thus, to discuss the trade-off involved while increasing the weight in function of time, we analyzed two criteria. The first one is the Mean Flow Time (_MFT_), \(MFT~{}=~{}1/n^{\prime}\sum\limits_{j~{}=~{}1}^{n^{\prime}}(C_{j}-r_{j})\), which measures the sum of the jobs flowtimes divided by \(n^{\prime}\). The flow time of a job measures the duration between its release date and its completion time, \(F_{j}~{}=~{}C_{j}-r_{j}\). The second one is the standard deviation of the flowtime. It assesses the deviations of the jobs' flowtimes from the \(MFT\). Table 10 presents the obtained results for the same instance as the one studied in Section 5.2. The variation of \(\rho\) is from 0 to 1 and \(\alpha\) from 0.5 to 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 0.5} & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 0.6} & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 0.7} & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 0.8} & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 0.9} & \multicolumn{2}{c}{\(\mathbf{\kappa}\) = 1} \\ \hline \multirow{2}{*}{\(\rho\)} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} & \multirow{2}{*}{MFT} & \multicolumn{2}{c}{_Std_} \\  & & _Dev_ & & & & & & & & & & & \\ \hline
0 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.9 & 6.59 \\
0.1 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 & 7.8 & 6.5 \\
0.2 & 7.8 &  
As can be seen, the increase of \(\rho\) increases the value of \(MFT\). However, the standard deviation of the flowtimes decreases. This means that the dispersion around the \(MFT\) decreases. In other words, the values of \(F_{j}\) approximate to each other. Thus, the increase of \(\rho\) helps the low weight jobs to be scheduled. On the other hand, the increase of \(\alpha\) makes it difficult to decrease the standard deviation of the flowtimes. For example, the standard deviation of the flowtimes with \(\rho\)=0.6 is equal to 5.5 when \(\alpha=1\), instead of 5.21 when \(\alpha=0.5\). When \(\alpha\) decreases, the stability criterion is more considered. Systemically, the new arriving jobs are placed at the end of the schedule, so as not to disrupt the established schedule.