# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 4 Case Study
### Results
A non-parametric Wilcoxon signed-rank test was used to analyse the statistical significance of the differences obtained. At a significance level of 0.1, the results were found to differ statistically from each other. However, it is also important to compare the results in terms of machine availability. For this purpose, the AC (average completion time) objective function is used, which complements the makespan. The makespan value is crucial from the perspective of the total time required to complete production tasks, while the AC value provides information \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Days**} & \multirow{2}{*}{**Sample**} & \multirow{2}{*}{\(\boldsymbol{LPT_{T}}\)} & \multirow{2}{*}{\(\boldsymbol{EPO_{T}}\)} & \multirow{2}{*}{\(\boldsymbol{LPT_{A}}\)} & \multirow{2}{*}{\(\boldsymbol{EPO_{A}}\)} \\ \cline{5-6}  & & & & & \\ \hline \multirow{3}{*}{5} & 1 & 39.3 & 39.2 & 43.8 & 42.5 \\  & 2 & 43.3 & 44.3 & 46.0 & 45.7 \\  & 3 & 40.6 & 40.8 & 49.4 & 49.2 \\ \hline \multirow{3}{*}{10} & 1 & 53.8 & 53.9 & 58.9 & 56.0 \\  & 2 & 65.2 & 65.4 & 68.2 & 68.2 \\  & 3 & 48.0 & 48.0 & 62.6 & 62.1 \\ \hline \multirow{3}{*}{20} & 1 & 72.4 & 72.4 & 76.7 & 76.6 \\  & 2 & 74.4 & 74.4 & 79.1 & 78.0 \\ \cline{1-1}  & 3 & 57.4 & 57.4 & 64.5 & 64.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Theoretical and actual makespan (\(C_{\text{max}}\)) obtained from LPT dispatching rule and LPT with EPO extension for samples of 5-day, 10-day, and 20-day periods. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Days**} & \multirow{2}{*}{**Sample**} & \multirow{2}{*}{\(\boldsymbol{LPT_{T}}\)-\(\boldsymbol{EPO_{T}}\)} & \multirow{2}{*}{\(\boldsymbol{LPT_{A}}\)-\(\boldsymbol{EPO_{A}}\)} \\ \hline \multirow{3}{*}{5} & 1 & 0.1 & 1.3 \\  & 2 & \(-1.0\) & 0.3 \\  & 3 & \(-0.2\) & 0.2 \\ \hline \multirow{3}{*}{10} & 1 & \(-0.1\) & 2.9 \\  & 2 & \(-0.2\) & 0 \\  & 3 & 0 & 0.5 \\ \hline \multirow{3}{*}{20} & 1 & 0 & 0.1 \\  & 2 & 0 & 1.1 \\ \cline{1-1}  & 3 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Differences between theoretical and actual makespan obtained from LPT and LPT with EPO extension. about the capacity to process new jobs on the given machines. The results obtained using the LPT rule and the LPT with EPO extension for the AC objective function are presented in Tables 5-7. In two out of nine cases, based on theoretical data, the result of the LPT rule without the EPO extension proved to be more favorable, while in one case, the results were identical. It should be noted that for observation 2 of the 20-day sample, the LPT rule saved an average of 1.3 time units compared to the EPO extension. In fact, in seven out of nine cases, the differences between the theoretical values were between \(-0.1\) and \(0.2\). These values confirm that, for theoretical data, the efficiency of the LPT rule and LPT with the EPO extension are at a comparable level. However, it may be surprising that in 2/3 of the cases, the EPO extension gave better results. Considering that the EPO extension includes time buffers to minimize potential delays, such a result is not trivial to predict. For actual data, the differences between the results are significantly greater than for theoretical data. The LPT rule gave better results in three cases, with differences ranging from \(-0.2\) to \(-0.3\). In six of nine cases, LPT with EPO extension gave better results, and in three of these cases, the difference was substantial, exceeding the value of 1. Based on the results in Table 7, it is also worth emphasizing that the EPO extension allows obtaining lower prediction errors compared to when the extension is not applied. In 2/3 of the cases, the EPO extension led to a lower error and a significantly lower average prediction error. The presented analysis of the results confirms the effectiveness of the EPO extension. Depending on the adopted evaluation factor, the results obtained for individual samples