# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 4 Case Study
### Results  
Table 1 shows the number of jobs in each of the nine samples used in the experiments. As can be seen, the length of the time window is not correlated with the number of jobs to be scheduled, which is due to the seasonal effect characteristic of the considered company. In the selected period, the smallest sample included approximately 3.5 thousand jobs to be scheduled (sample 2 of the 5-day period), and the largest sample included more than 8 thousand jobs (sample 2 of the 20-day period). ```
{Initial phase - assignment of jobs} for\(J\in\) Jobs do if\(Preds_{I}=\emptyset\)then  Calculate \(EST_{J}\) \(JF_{M_{J}}^{EST_{J}}=JF_{M_{J}}^{EST_{J}}\cup\{J\}\) else \(J_{M_{J}}=J_{M_{J}}\cup\{J\}\) endif endfor {Scheduling phase} while There are jobs to be scheduled do  Select \(M^{\prime}\) with \(MAT_{M^{\prime}}=\min\{MAT_{M}\mid M:JF_{M}\
eq\emptyset\}\)  Select \(J^{\prime}\) with \(PT_{J^{\prime}}=\max_{J\in I_{M^{\prime}}^{F}}PT_{J}\)  Schedule \(J^{\prime}\) on \(M^{\prime}\) for all\(J_{s}\in\mathit{Succs_{J}}\)do  Update feasibility of \(J_{s}\in\mathit{Succs_{J}}\) if\(J_{s}\) is feasible then  Calculate \(EST_{J_{s}}\) endif endfor for all\(J\in JF_{M_{J^{\prime}}}^{FT_{J_{s},J^{\prime}}}\)  Calculate \(EST_{J}\) endfor endwhile
```  
**Algorithm 1** Pseudo-code of EPO-forward algorithm  
Table 2 presents the makespan values obtained from the LPT dispatching rule (\(LPT_{T}\)) and the LPT with EPO extension (\(EPO_{T}\)), and also the actual makespan values, denoted as \(LPT_{A}\) and \(EPO_{A}\), respectively (the values were calculated using the three-step procedure described in the previous subsection). The results were scaled to anonymize sensitive data from the analyzed company. Each value presented in the table was divided by the same fixed factor (greater than 1); therefore, the proportions between the results have been preserved. To better illustrate the effects of using the proposed EPO extension, Table 3 presents the differences of the respective values from Table 2.  
As can be seen from Table 3, the theoretical results of the EPO extension are no better (except for one sample) than the results of the LPT dispatching rule. This is not surprising, taking into account the construction of the EPO extension. However, the actual makespan values confirm the effectiveness of the EPO extension. Indeed, only in two cases did the EPO extension produce the same result as the forward heuristics. In 1/3 of the considered \begin{table}
\begin{tabular}{c c c} \hline \hline
**Days** & **Sample** & **Jobs** \\ \hline
5 & 1 & 3804 \\
2 & 2 & 3542 \\
3 & 4056 \\ \hline
10 & 1 & 5510 \\
2 & 2 & 5582 \\
3 & 5890 \\ \hline
20 & 1 & 7635 \\
20 & 2 & 8083 \\  & 3 & 4889 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of jobs in samples of 5-day, 10-day, and 20-day periods. cases, the advantage of the EPO extension was very clear (exceeded 1 time unit). In the best case, the EPO extension allowed to save as much as 5% of the makespan compared to the forward method. Moreover, in each of the three groups of samples, there was a sample where the time savings exceeded 1 time unit. So, it seems that the length of the period does not affect the shortening of the actual makespan. Similarly, the difference between the theoretical makespan values obtained with the LPT and EPO extension also does not determine the actual time savings. Table 4 shows the difference between the actual and theoretical makespan for each sample. This difference can be used to compare schedules in terms of how much theoretical value differs from the actual value. Of course, in an ideal situation, these two values should be equal (the difference between the values should be 0). Based on the results in Table 4, it can be concluded that in one case only, the differences between the actual and theoretical times for LPT and the EPO extension are equal. In the remaining cases, the difference obtained with the EPO extension is smaller, which means that the theoretical makespan obtained with the EPO extension is closer to the actual makespan.