# A Dynamic Adjusted Aggregate Load Method to Support Workload Control Policies
## 5 Numerical Results  
The simulation results are reported as a percentage difference compared to the relative benchmark. The figures report the results according the cases reported in Table 3:  
Figure 1 reports the performance of the average lateness of the models tested. Better results have been obtained for cases 3, 8 and 9 with three bottlenecks. These cases work with the adjusted workload. More generally, all cases with adjusted workload perform better for this performance. The number of bottlenecks has a different effect on the methods studied. The results highlight that the job shop with three bottlenecks is characterized by a higher fluctuation of this performance. The proposed method does not improve the average lateness compared to the classical approach. The benchmark works better on this performance, while all the other methods were worse on this performance (see Figure 2). Moreover, the fluctuation of this performance is very high over the methods studied. The number of bottlenecks is relevant for the cases with the dynamic workload and pre-shop queue management FIFO (First In First Out) and SPT (Shortest Processing Time). The proposed method improves this performance compared to the classical method. Figure 3 shows the average time of the jobs in the entire manufacturing system. The difference among the methods is low; the range of the percentage difference is about 5% (from \(-2\%\) to 3%). It can  
Figure 1: Average lateness. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cases** & **Benchmark** & **1** & **2** & **3** \\ \hline  & Workload: adjusted & Workload: adjusted & Workload: adjusted & Workload: adjusted \\  & Norm: machines & Norm: bottleneck & Norm: machines & Norm: bottleneck \\  & Pre-shop: FIFO & Pre-shop: FIFO & Pre-shop: SPT & Pre-shop: SPT \\ \hline
**Cases** & **4** & **5** & **6** & **7** \\ \hline  & Workload: dynamic & Workload: dynamic & Workload: dynamic & Workload: dynamic \\  & Norm: machines & Norm: bottleneck & Norm: machines & Norm: bottleneck \\  & Pre-shop: FIFO & Pre-shop: FIFO & Pre-shop: SPT & Pre-shop: SPT \\ \hline
**Cases** & **8** & **9** & **10** & **11** \\ \hline  & Workload: adjusted & Workload: adjusted & Workload: dynamic & Workload: dynamic \\  & Norm: machines & Norm: bottleneck & Norm: machines & Norm: bottleneck \\  & Pre-shop: balance & Pre-shop: balance & Pre-shop: balance & Pre-shop: balance \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cases studied. be observed that when the number of bottlenecks increases, the variation of the performance increases. The computation of the workload does not significantly affect this performance. Figure 4 shows the percentage of the jobs in delay. The better cases are 6, 7, 10 and 11; the number of the bottlenecks does not affect the results. These cases use the proposed dynamic workload computation. Therefore, the proposed approach has a relevant benefit on this performance. Figure 5 reports the shop time of the jobs; also, for this performance, the proposed approach leads to better results (cases 4, 5, 6, 7, 10 and 11) with about 10% of reduction. In this case, the number of bottlenecks changes the benefit obtained; in particular, when the number of bottlenecks increases, the benefit is higher. Then, the deviation of the average queue of the machines is studied (see Figure 6). The proposed method allows drastic reduction of the deviation of the jobs in queue among the machines. The reduction is from 20% with three bottlenecks to 30% with one bottleneck in the manufacturing system. The bottleneck shiftness (see Figure 7) evaluates the changes of the bottleneck during the manufacturing operations. The proposed approach increases this value; this means that the time in which the bottleneck is the real bottleneck of the system is lower. This leads to a better uniform workload among the machines due to the proposed approach. Figure 8 shows the total time of lateness accumulated by each model. Cases 6, 7 and 11 lead to better results, obtaining an improvement of about 30%.