# Predictive Scheduling with Markov Chains and ARIMA Models
## 4 Experimental Verification of the Proposed Solution
### Evaluation Criteria  
To verify the effectiveness of the robust scheduling solutions, as well as for the sake of comparative analysis against the nominal schedules, the following assessment criteria are applied:  
* makespan \(C_{\text{max}}\)--total production time,
* mean completion time \(\overline{C}\) given by: \[\overline{C}=\frac{1}{n}\sum\
olimits_{i=1}^{n}(C_{i}),\] (15) where \(C_{i}\)--the completion time of job \(i\).
* mean flow time \(\overline{F}\) given by: \[\overline{F}=\frac{1}{n}\sum\
olimits_{i=1}^{n}(F_{i}),\] (16) where \(F_{i}\)--the flow time of job \(i\).
* the number of critical operations \(Y_{K}\) is derived from: \[Y_{K}=\sum\
olimits_{i=1}^{n}\sum\
olimits_{j=1}^{m}\big{(}y_{ij}\big{)},\] (17) \[y_{ij}=\left\{\begin{array}{l}1,\;when\;\big{(}tz_{ij}-tr_{ij+1})=0\big{)} \\ 0,when\;\big{(}tz_{ij}-tr_{ij+1})\
eq 0\big{)}\end{array}\right.,\] (18) where \(Y_{K}\)--the number of critical operations, \(tz_{ij}\)--the completion time of operation \(o_{ij}\) (current), \(tr_{ij+1}\)--the start time of operation \(o_{ij}+1\) (subsequent).  
The verification of the obtained schedules is performed during the online stage (production execution), modelled with the Enterprise Dynamics software in a series of simulation tests. The computations serve to determine total completion times of production jobs under strategic machinery failure. The modelling tool used in the study allows detecting machine failure times by setting the MTTF and MTTR indicators and selected probability distributions (Table 6). The MTTF values are specified for the uniform probability distribution (i.e., the machine failure can occur at any time--from the start of the job on the machine until its completion). On the other hand, the MTTR values are determined using the Weibull distribution, obtained for machine repair times from the Cullen-Frey graph [6].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Machine No.** & **Elements of Set \(FT_{Ml}\) [h]** & **Elements of Set \(TB_{Ml}\) [h]** \\ \hline \(M_{1}\) & \(FT_{M1}=\) [8] & \(TB_{M1}=\) [0.646, 0.702, 0.698, 0.699, 0.699] \\ \(M_{2}\) & \(FT_{M2}=\) [8] & \(TB_{M2}=\) [0.663, 0.663, 0.663, 0.663, 0.663] \\ \(M_{3}\) & \(FT_{M3}=\) [8] & \(TB_{M3}=\) [0.613, 0.680, 0.667, 0.669, 0.669] \\ \(M_{6}\) & \(FT_{M6}=\) [8] & \(TB_{M6}=\) [0.802, 0.626, 0.720, 0.630, 0.702] \\ \(M_{7}\) & \(FT_{M7}=\) [8] & \(TB_{M7}=\) [0.913, 0.903, 0.895, 0.889, 0.884] \\ \(M_{8}\) & \(FT_{M8}=\) [8] & \(TB_{M8}=\) [0.864, 0.831, 0.831, 0.831] \\ \hline \hline \end{tabular}
\end{table}
Table 5: The data implemented into the robust production schedule.  
Twenty-five simulations of the production process are performed for each of the LPT or SPT schedules. The indicators employed in the assessment of the results from simulations are:  
* Increase of completion time of all jobs \(\Delta C_{\text{max}}\) given by: \[\Delta C_{\text{max}}=C_{\text{max}}-C^{\prime}_{\text{max}}\,\] (19) where \(\Delta C_{\text{max}}\)--increase of completion time of all jobs, \(C_{\text{max}}\)--nominal schedule makespan, \(C^{\prime}_{\text{max}}\)--actual (executed) schedule makespan.
* Relative increase of makespan \(E_{C_{\text{max}}}\) given by: \[E_{C_{\text{max}}}=\frac{C_{\text{max}}}{C^{\prime}_{\text{max}}},\] (20) where \(E_{C_{\text{max}}}\)--relative increase of makespan, \(C_{\text{max}}\)--nominal schedule makespan, \(C^{\prime}_{\text{max}}\)--actual (executed) schedule makespan.