# Predictive Scheduling with Markov Chains and ARIMA Models
## 4 Experimental Verification of the Proposed Solution
### Experimental Results  
The first of the verification objectives is to compare the nominal and robust production schedules in terms of evaluation criteria. The values of the evaluation indicators of the schedules have been determined and are summarized below (Table 7). From the presented data, it can be seen that the implementation of service time buffers increases the completion times of all jobs. As a consequence, in each of the analyzed cases, one additional shift is required to complete the production process. This effect is not at all unexpected, given that incorporating service time buffers is inseparably connected with elongation. It should be noted, however, that in the robust schedule the time spent in the production system is not extended owing to the fact that the mean flow time is subject to slight elongation. Figure 2 presents the visual interpretation of the nominal and robust SPT schedules. Service time buffers are represented by crossed white blocks. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Machine No. (Technology)** & **MTTF * & **MTTR *** \\ \hline \(M_{1}\) & Uniform(0, 16.763) & Weibull(0.88, 1.28) \\ \(M_{2}\) & Uniform(0, 8.673) & Weibull(0.75, 1.51) \\ \(M_{3}\) & Uniform(0, 15.247) & Weibull(0.679, 1.72) \\ \(M_{6}\) & Uniform(0, 22.083) & Weibull(0.769, 1.43) \\ \(M_{7}\) & Uniform(0, 8.34) & Weibull(0.973, 1.58) \\ \(M_{8}\) & Uniform(0, 19.24) & Weibull(0.877, 1.45) \\ \hline \hline \end{tabular}  
* the parameters are expressed in hours.  
\end{table} 
Table 6: Machine failure parameters defined in the simulation environment. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Dispatching Rule**} & \multicolumn{6}{c}{**Evaluation Criterion [h]**} \\ \cline{2-9}  & \multicolumn{3}{c}{\(\overline{F}\)} & \multicolumn{3}{c}{\(\overline{C}\)} & \multicolumn{3}{c}{\(C_{\text{max}}\)} \\ \cline{2-9}  & **Nominal** & **Sched** & **Robust** & **Elong. [\%]** & **Nominal** & **Sched** & **Robust** & **Elong. [\%]** & **Nominal** & **Robust** & **Sched** & **Elong. [\%]** \\ \hline LPT & 23.34 & 23.86 & 2.3\% & 31.94 & 36.49 & 14.2\% & 46.93 & 53.14 & 13.2\% \\ SPT & 18.33 & 19.94 & 8.8\% & 20.95 & 23.42 & 11.8\% & 47.26 & 55.68 & 17.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation criteria in the nominal and robust schedules. A further indicator of robustness that is of great importance in scheduling is the number of critical operations. Scheduling should minimize its value because the stability of the executed process is compromised with the rising number of critical operations. In the analyzed example, the number of critical operations is considered in relation to individual jobs (\(Y_{KJ}\)) and machines (\(Y_{KM}\)). The robust scheduling results with respect to the number of critical operations are presented in Table 8.  
The incorporation of service time buffers is shown to have a positive effect on the considered parameters. The number of critical operations is reduced by up to 20%. This confirms the legitimacy of implementing service time buffers, which generate additional space in the schedule and thus can prove to be beneficial in the event of machinery failure or other process disruptions. Figure 2: Schedules obtained using the SPT (shortest processing time) rule: (**a**) nominal, (**b**) robust. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Number of Critical Operations [–]**} \\ \cline{2-7}
**Dispatching Rule** & & \(Y_{KJ}\) & & & \(Y_{KM}\) & \\ \cline{2-7}  & **Nominal Sched.** & **Robust Sched.** & **Reduction [\%]** & **Nominal Schedule** & **Robust Sched.** & **Reduction [\%]** \\ \hline LPT & 30 & 24 & \(-\)20.0\% & 26 & 21 & \(-\)19.2\% \\ SPT & 32 & 27 & \(-\)15.6\% & 25 & 21 & \(-\)16.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The number of critical operations in the nominal and robust schedules. Simulation tests are conducted to indicate which of the schedules features a makespan closer to the simulated completion time of all jobs.