## 3 Materials and Methods
### Crossover Operator DPPX
Figure 2: Initial form of application for experimental testing. Mk10) compared to other GAs (the average number of machines is the third number in first row of every tkt file from [PERSON]'s problem instances). This is a consequence of using the DPPX sequential genetic operator, which, despite modifications compared to the classic PPX operator (where the deviations compared to the best lower limit are greater, proving the effectiveness of the DPPX operator), cannot ensure a sufficient degree of crossing between individuals. There is a need to change individuals \"more\" in order to achieve a better makespan, and this can be accomplished by increasing the mutation rate [39], because a mutation usually corresponds to the generation of a neighbor in some neighborhood if permutation coding is used. In this case, the mutation rate denotes the probability of applying a mutation to the current individual, and in these cases, the mutation rate is typically much larger than the five percent used in a first series of experiments. The authors performed another test of the presented algorithm five times for every instance where the negative deviation was greater than three percent and tested different values of the algorithm parameters. The experimental results show that the following parameters are the most effective:

Small- and medium-scale problems (up to 100 operations on five machines):

* Population size: 250;
* Number of parents 40% of population size: 100;
* Mutation probability: 50%;
* Number of generations: 150. * Large-scale problems (up to 240 operations on 15 machines):
* Population size: 500;
* Number of parents 50% of population size: 250;
* Mutation probability: 50%;
* Number of generations: 600. The results of the second series of experiments are shown in Table 8.

The results of both series of experiments clearly show that RMGA is very effective on small- and medium-scale problems, and its effectiveness decreases slightly with large-scale problems. In seven of the ten instance problems considered, RMGA outperforms the best-known lower bound. For small- and medium-scale problems, the speed of RGMA execution was 10-20 s, but for large-scale problems, the speed was much higher, also because the application simultaneously creates Gantt charts and graphically displays the convergence of GA solutions in real time. In order to accurately determine the execution speed of RMGA, it is necessary to test only the algorithm itself without graphical representations of solution convergence or Gantt charts. Speed measurement and speed improvement of the RMGA will be part of future research. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Problem Instance**} & \multirow{2}{*}{\(\mathbf{n}\times\mathbf{m}\)} & **BS** & **RMGA (PPX)** & **RMGA (DPPY0)** & **RMGA (PPX)** & **RMGA (DPPY0)** \\ \cline{3-6}  & & \(\mathbf{C_{m}}\) & \(\mathbf{C_{m}}\) & \(\mathbf{C_{R}}\) & **Deviation** & **Deviation** \\ \hline Mk01 & \(10\times 6\) & 37 & 38 & 38 & \(-2.70\%\) & \(-2.70\%\) \\ \hline Mk02 & \(10\times 6\) & 26 & 34 & 33 & \(-30.77\%\) & \(-26.92\%\) \\ \hline Mk03 & \(15\times 8\) & 204 & 199 & 191 & 2.45\% & 6.37\% \\ \hline Mk04 & \(15\times 8\) & 60 & 62 & 61 & \(-3.33\%\) & \(-1.67\%\) \\ \hline Mk05 & \(15\times 4\) & 172 & 173 & 172 & \(-0.58\%\) & 0.00\% \\ \hline Mk06 & \(10\times 15\) & 57 & 71 & 70 & \(-24.56\%\) & \(-22.81\%\) \\ \hline Mk07 & \(20\times 5\) & 139 & 168 & 158 & \(-20.86\%\) & \(-13.67\%\) \\ \hline Mk08 & \(20\times 10\) & 523 & 506 & 494 & 3.25\% & 5.54\% \\ \hline Mk09 & \(20\times 10\) & 307 & 356 & 344 & \(-15.96\%\) & \(-12.05\%\) \\ \hline Mk10 & \(20\times 15\) & 189 & 326 & 316 & \(-72.49\%\) & \(-67.20\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Deviation of RMGA experimental results compared to best-known lower bound. By changing the numbers in Table 9, the lower limit of the average hourly cost of operations scheduling can be lowered or raised. This informs the decision-maker how much it will cost the SME to create that operation schedule.