# Predictive Scheduling with Markov Chains and ARIMA Models
## 4 Experimental Verification of the Proposed Solution  
### Historical Data  
In order to verify the solutions proposed in this publication, a process of robust job scheduling is performed on the production data and on historical data of technological machine failure. The production data describes the execution of 9 production jobs processed by 12 machines, constituting a manufacturing cell (Table 1). The parts are produced in batches of 50 elements (\(b=50\)) and the setup times of individual operations are not taken into account in the production scheduling process (uncertainty of setup times is a different factor and requires additional research). Therefore:  
\[pt_{ij}=b\cdot to_{ij}, \tag{14}\]  
where \(b\)--quantity of elements in the production batch, \(to_{ij}\)--operation time.  
The data describing the failure rate of machines in the scheduled production process have been obtained from the computer records of machine operation from a maintenance department (Table 2). The collected data describe the failure rate of six technological machines that are crucial for the performed production process. The numbers of observations are as follows:  
* Machine \(M_{1}\)--197 observations
* Machine \(M_{2}\)--166 observations  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Job** & **Operation** & **Machine** & **Type of Operation** & \(ts_{y}\) * **[min]** & \(ts_{y}\) * **[h]** & \(to_{ij}\) * **[min]** & \(to_{y}\) * **[h]** \\ \hline \multirow{5}{*}{2} & 10 & \(M_{1}\) & Laser1 & Laser-cutting sheets & 22 & 0.367 & 4 & 0.067 \\  & 20 & \(M_{4}\) & CNC saw & Band-saw cutting & 6 & 0.100 & 0.5 & 0.008 \\  & 30 & \(M_{3}\) & CNC press & Edge bending & 16 & 0.267 & 3 & 0.050 \\  & 40 & \(M_{8}\) & Drill & Drilling holes and threading & 12 & 0.200 & 1 & 0.017 \\  & 50 & \(M_{5}\) & Methowking & Metalworking & 5 & 0.083 & 1 & 0.017 \\  & 60 & \(M_{6}\) & MIG welder & MIG welding & 8 & 0.133 & 5.5 & 0.092 \\ \hline \multirow{5}{*}{6} & 10 & \(M_{1}\) & Laser1 & Laser-cutting sheets & 12 & 0.200 & 0.3 & 0.005 \\  & 20 & \(M_{2}\) & Laser2 & Laser-cutting profiles & 14 & 0.233 & 1 & 0.017 \\  & 30 & \(M_{5}\) & Metalworking & Metalworking & 5 & 0.083 & 1 & 0.017 \\  & 40 & \(M_{6}\) & MIG welder & MIG welding & 8 & 0.133 & 1 & 0.017 \\  & 50 & \(M_{10}\) & Turning & Turning & 11 & 0.183 & 2 & 0.033 \\ \hline \multirow{5}{*}{9} & 10 & \(M_{1}\) & Laser1 & Laser-cutting sheets & 20 & 0.333 & 5 & 0.083 \\  & 20 & \(M_{2}\) & Laser2 & Laser-cutting pipes and profiles & 12 & 0.200 & 2 & 0.033 \\  & 30 & \(M_{4}\) & CNC saw & Band-saw cutting & 6 & 0.100 & 1 & 0.017 \\ \cline{1-1}  & 40 & \(M_{3}\) & CNC press & Edge bending & 25 & 0.471 & 6.5 & 0.108 \\ \cline{1-1}  & 50 & \(M_{8}\) & Drill & Drilling holes and threading & 12 & 0.200 & 7 & 0.117 \\ \cline{1-1}  & 60 & \(M_{5}\) & Metalworking & Metalworking & 5 & 0.083 & 2 & 0.033 \\ \cline{1-1}  & 70 & \(M_{6}\) & MIG welder & MIG welding & 8 & 0.133 & 7.5 & 0.125 \\ \hline \hline \end{tabular}
* \(ts_{ij}\)—setup time, \(to_{ij}\)—operation time.  
\end{table}
Table 1: Production data implemented in the robust scheduling solution.  
The nominal and robust schedules are subsequently built based on the data presented above. This, however, must be preceded by the prediction of failure parameters with the application of Markov chain and ARIMA models, which is described in the next section.