# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 5 Discussion and Conclusions  
Automatic scheduling, tracking, and production control are an important aspects of modern production systems, including those implementing the concept of Industry 4.0. The concept of digital twins enables a more accurate monitoring of production and the collection of more reliable data about the production process, including the production time itself. Actual production times are essential for creating more reliable production schedules. One of the most popular and efficient approaches to automatic production scheduling is the dispatching rules. The main drawback of dispatching rules is that they do not take into account the uncertainty of the job processing times, which can cause machine downtime due to production delays. The EPO extension proposed in this paper extends dispatching rules to include the uncertainty in the job processing times. The important feature of the proposed extension is that it does not increase the computational time of fast dispatching rules, i.e., it does not negatively affect their main advantage over other production scheduling algorithms. Furthermore, the EPO extension can be integrated with the digital twin of the production process, which is essential in an Industry 4.0 setting. This means that real-time data \begin{table}
\begin{tabular}{c c c c} \hline
**Days** & **Numb.** & \(\boldsymbol{LPT_{T}}\)**-\(\boldsymbol{EPO_{T}}\)** & \(\boldsymbol{LPT_{A}}\)**-\(\boldsymbol{EPO_{A}}\)** \\ \hline \multirow{3}{*}{5} & 1 & 0.1 & 0.2 \\  & 2 & 0 & \(-\)0.3 \\  & 3 & 0.1 & 1 \\ \hline \multirow{3}{*}{10} & 1 & 0.2 & 0.2 \\  & 2 & 0.1 & \(-\)0.2 \\  & 3 & \(-\)0.1 & 0.4 \\ \hline \multirow{3}{*}{20} & 1 & 0.5 & 1.8 \\  & 2 & \(-\)1.3 & \(-\)0.3 \\ \cline{1-1}  & 3 & 0.2 & 1.8 \\ \hline \end{tabular}
\end{table} 
Table 6: Differences between theoretical and actual average completion time obtained from LPT and LPT with EPO extension. \begin{table}
\begin{tabular}{c c c c} \hline
**Days** & **Numb.** & \(\boldsymbol{LPT_{A}}\)**-\(\boldsymbol{LPT_{T}}\)** & \(\boldsymbol{EPO_{A}}\)**-\(\boldsymbol{EPO_{T}}\)** \\ \hline \multirow{3}{*}{5} & 1 & 3 & **2.9** \\  & 2 & **2.3** & 2.6 \\  & 3 & 6.9 & **6** \\ \hline \multirow{3}{*}{10} & 1 & 4.9 & 4.9 \\  & 2 & **4.4** & 4.7 \\  & 3 & 6 & **5.5** \\ \hline \multirow{3}{*}{20} & 1 & 6.5 & **5.2** \\  & 2 & 6.8 & **5.8** \\ \cline{1-1}  & 3 & 5.9 & **4.3** \\ \hline \multicolumn{2}{c}{AVG} & 5.2 & **4.7** \\ \hline \end{tabular}
\end{table} 
Table 7: Difference between theoretical and actual values for analyzed dispatching rules (\(AC\)). Best values are marked in bold. collected from machines (e.g., actual operation times) can be used to modify the EPO estimates for jobs within any chosen time window, enabling dynamic schedule adjustments and enhancing resilience to disruption. The proposed EPO extension eliminates the above-mentioned drawback of dispatching rules, and what follows enables us to obtain schedules that are robust to production delays, machine failures, and other factors that have an impact on the job processing times. The good operation of the EPO extension has been confirmed by experiments on real data coming from the company, which are characterized by a very complex production profile. The performed experiments have confirmed that the differences between the theoretical (computed based on job and machine parameters) and actual job processing times had a smaller impact (measured using makespan and AC) on the schedule obtained, with the EPO extension than with the LPT dispatching rule. For both of the objective functions analyzed, the proposed EPO extension gives better results when dealing with data representing actual production times. In the case of makespan, in none of the samples analyzed did the LPT scheduling rule give better results than LPT with the EPO extension for actual data. For both makespan and AC, the difference between the results obtained by LPT and LPT with EPO extension was very significant in three out of nine samples.