## 3 Materials and Methods
### Crossover Operator DPPX
The effectiveness of the proposed algorithm is compared with the existing algorithms of older dates from the available literature, like M&G Mastrolilli and [PERSON] [32], [PERSON] [33], [PERSON] et al. [34], and [PERSON] et al. [35], and with the existing algorithms of newer dates such as Hybrid Genetic Tabu Search (HGTS) defined by [PERSON] et al. [36] and [PERSON] et al. [37] and Teaching-Learning-Based Optimization (TLBO) proposed by [PERSON] et al. [38] and [PERSON] et al. [24]. In order to obtain meaningful results, the authors

\begin{table}
\begin{tabular}{l l l l l l l l l l l l} \hline
**RnIndeks** & **Olndeks** & **M1** & **M2** & **M3** & **M4** & **M5** & **M6** & **M7** & **M8** & **M9** & **M10** & **M11** \\ \hline
4 & 1 & 25 & 11 & 45 & 33 & 0 & 0 & 0 & 32 & 11 & 0 & \\ \hline
4 & 2 & 16 & 0 & 0 & 25 & 22 & 33 & 28 & 0 & 0 & 0 & \\ \hline
4 & 3 & 0 & 36 & 25 & 0 & 0 & 36 & 25 performed optimization five times for each instance. Table 6 shows the experimental results compared to the previously mentioned algorithms. The first column reports the instance name, and the second column reports the number of machines and number of working orders for each instance, where m is the number of machines and n is the number of working orders. The third column reports the best-known lower bound, which was taken from [PERSON] et al. (2018) and [PERSON] et al. (2017). It should be noted that the best lower bound for the Mk01 problem instance is taken from [PERSON] et al. (2017). From the fourth to the eleventh columns are the best results of the previously mentioned algorithms. The twelfth column shows the results of executing our algorithm named Relative Makespan GA (RMGA) with the classic PPX crossover, and the thirteenth shows the results of executing with the DPPX crossover. Table 7 shows the deviation of RMGA results in relation to the best lower bound (the third column from Table 6). The deviation was calculated according to the formula (CM-CMRMGA)/CM in %, where CM is the best-achieved makespan of other GAs and CMRMGA is the best-achieved makespan with RMGA. The experimental results and deviation calculation show that RMGA achieves quite good (and in some cases excellent) results when the average number of machines per work order is less than or equal to two (instances Mk01, Mk03, Mk05, and Mk08). Its efficiency is significantly lower when the average number of machines is greater than two (instances Mk02, Mk06, Mk07, Mk09, and \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem Instance**} & \multirow{2}{*}{\(\mathbf{n\times m}\)} & **BS** & **M \& G** & **GENACE** & \begin{tabular}{c} **[PERSON]** \\ **et al.**[34] \\ \end{tabular} & \begin{tabular}{c} **[PERSON]** \\ **et al.**[35] \\ \end{tabular} & **HGTS** & \begin{tabular}{c} **[PERSON]** \\ **et al.**[37] \\ \end{tabular} & **TLBO** & **DIGWO** & \
\begin{tabular}{c} **RMGA** \\ **(PPX)** \\ \end{tabular} &
\begin{tabular}{c} **RMGA** \\ **(DPPY)** \\ \end{tabular} \\ \hline Mk01 & \(10\times 6\) & 37 & 40 & 40 & 40 & 40 & 37 & 40 & 40 & 38 & 38 \\ \hline Mk02 & \(10\times 6\) & 26 & 26 & 32 & 26 & 26 & 26 & 28 & 26 & 34 & 33 \\ \hline Mk03 & \(15\times 8\) & 204 & 204 & \(\mathrm{N/A}\) & 204 & 204 & 204 & 204 & 204 & 199 & 191 \\ \hline Mk04 & \(15\times 8\) & 60 & 60 & 67 & 60 & 60 & 60 & 63 & 60 & 62 & 61 \\ \hline Mk05 & \(15\times 4\) & 172 & 173 & 176 & 173 & 173 & 172 & 173 & 172 & 173 & 173 & 172 \\ \hline Mk06 & \(10\times 15\) & 57 & 58 & 67 & 63 & 58 & 57 & 67 & 65 & 62 & 71 & 60 \\ \hline Mk07 & \(20\times 5\) & 139 & 144 & 147 & 139 & 144 & 139 & 148 & 144 & 140 & 168 & 158 \\ \hline Mk08 & \(20\times 10\) & 523 & 523 & 523 & 523 & 523 & 523 & 523 & 523 & 523 & 506 & 494 \\ \hline Mk09 & \(20\times 10\) & 307 & 320 & 320 & 311 & 307 & 307 & 307 & 311 & 307 & 356 & 344 \\ \hline Mk10 & \(20\times 15\) & 189 & 229 & 229 & 212 & 198 & 198 & 212 & 214 & 211 & 326 & 316 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Experimental results of GA execution and comparison with other GAs.