# Enhancing Production Efficiency Through Digital Twin Simulation Scheduling
## 2 Materials and Methods
\[Z_{o}=\Sigma S_{pk}\;\times\;k_{pk}+S_{od}\;\times\;k_{ad}+S_{ mod}\;\times\;k_{mod}+N_{o}\;\times\;k_{no}+P_{o}\;\times\;k_{po}+z_{o}\;\times\;k_{zo}+ CR_{o}\;\times\;k_{c\sigma_{o}}+S_{o}\;\times\;k_{So} \tag{3}\]  
where  
\(Z_{o}\)--order sorting index;  
\(S_{pk}\)--meeting the number of pieces in the order;  
\(k_{pk}\)--coefficient when the required number of pieces is met;  
\(S_{od}\)--meeting the completion date;  
\(k_{ad}\)--coefficient when the completion deadline is met;  
\(S_{nod}\)--not meeting the completion date;  
\(k_{nod}\)--coefficient when the completion deadline is not met;  
\(N_{o}\)--number of pieces in the order;  
\(k_{no}\)--coefficient for the number of pieces in the order;  
\(P_{o}\)--estimated time for the execution of the order;  
\(k_{po}\)--coefficient for the expected time to execute the order;  
\(z_{o}\)--priority of the order assigned to the sales department;  
\(k_{zo}\)--coefficient for the acceptance of the priority from the sales department;  
\(CR_{o}\)--critical ratio of the given order;  
\(k_{cro}\)--importance factor for the critical ratio;  
\(S_{o}\)--time reserve of the order;  
\(k_{So}\)--coefficient for time reserve. After running a model with different coefficient setups (see Table 1) and comparing these variants, we can choose the optimal schedule. Considering the need for the shortest makespan and number of delayed orders, the critical ratio-oriented schedule is the optimal variant in this case. However, it is not possible to assert that a CR-oriented solution is the optimal setup for other models with different sets of orders. Thus, it is always necessary to run several order execution experiments with different prioritization scenarios and choose the best solution for execution in the real system. The simulation model of the production system should mimic the real production system to such an extent that deviations are minimized as much as possible. The key elements of the simulation model are as follows. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Variant** & **k\_CR\({}_{0}\)** & **k\_P\({}_{0}\)** & **k\_S\({}_{0}\)** & **k\_pk** & **k\_od** & **k\_no** & **k\_zo** & **k\_mod** & **Makespan** & **Delayed** & **Notes** \\ \hline A & 1 & 0.2 & 0.25 & 0.3 & 0.1 & 0.17 & 0.7 & 0 & 8:34:01 & 3 & Balanced setup \\ \hline B & 2 & 0.2 & 0 & 0.4 & 0.15 & 0.5 & 0.1 & 0 & 8:18:02 & 1 & CR-focused \\ \hline C & 0.5 & 1.5 & 1 & 1 & 0.15 & 0.1 & 0.25 & 0 & 8:35:34 & 5 & \begin{tabular}{c} Focuses on longest execution time \\ \end{tabular} \\ \hline D & 1 & 0.5 & 0.5 & 0.45 & 0.15 & 0.75 & 1.5 & 0 & 8:43:02 & 4 & \
\begin{tabular}{c} With sales-assigned priority \\ \end{tabular} \\ \hline E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 09:15:00 & 8 &
\begin{tabular}{c} No prioritization \\ (control) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different coefficient setups. * Production equipment: This requires a 2D and 3D mock-up and the key parameters of individual devices, such as availability, failure rates, algorithms to control incoming tasks, etc. * Service: This contains a calendar of changes, qualifications, and the basic characteristics of each worker. * Products: These represent the production tasks themselves. The product is linked to the relevant technological process, based on which it manages its path through the production process. * Handling units: These are another limitation of the production system, not only in terms of size but also in terms of their capacity. * Handling equipment. * The logistics network. For the simulation model to be as close as possible to the real production system during its creation, it is necessary to verify it by comparing the simulation behavior with the real system, and employees directly from production should be involved in this process. During verification, the following questions must be answered:  
* Were all the necessary questions included in the model? * Is the correct logic of the model maintained?