## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objective**} & \multicolumn{6}{c}{**Problem-1**} & \multicolumn{6}{c}{**Problem-4**} \\ \cline{2-11}  & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**StDev**} & \multicolumn{2}{c}{**Percentage**} & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**StDev**} & \multicolumn{2}{c}{**Percentage**} \\ \cline{2-11}  & **RGA** & **2 SGA** & **RGA** & **2 SGA** & **Improvement \({}^{*}\)** & **RGA** & **2 SGA** & **RGA** & **2 SGA** & **Improvement \({}^{*}\)** \\ \hline \(Z_{1}\) & 6317 & 3767 & 972 & 412 & (40, 58) & 58,513 & 31,277 & 3293 & 748 & (47, 77) \\ \(Z_{2}\) & 5477 & 3221 & 1045 & 447 & (41, 57) & 57,435 & 30,524 & 3336 & 690 & (47, 79) \\ \(Z_{3}\) & 30,138 & 18,243 & 6116 & 2422 & (39, 60) & 4,574,999 & 2,476,978 & 299,643 & 89,360 & (46, 70) \\ \(Z_{4}\) & 5895 & 3483 & 1003 & 438 & (41, 56) & 57,957 & 30,346 & 3312 & 705 & (47, 79) \\ \(Z_{5}\) & 18,954 & 10,829 & 3420 & 1035 & (43, 70) & 2,081,854 & 1,122,475 & 119,664 & 17,273 & (46, 86) \\ \(Z_{6}\) & 2896 & 1491 & 1171 & 581 & (49, 50) & 21,250 & 10,607 & 4788 & 2282 & (50, 52) \\ \(Z_{7}\) & 6297 & 3084 & 2772 & 1232 & (51, 56) & 248,166 & 116,497 & 40,929 & 15,422 & (53, 62) \\ \(Z_{8}\) & 5058 & 3336 & 894 & 291 & (34, 67) & 37,728 & 29,216 & 2621 & 538 & (23, 79) \\ \(Z_{9}\) & 14,816 & 13,835 & 596 & 550 & (7, 8) & 682,856 & 667,035 & 6709 & 6442 & (2, 4) \\ \(Z_{10}\) & 3789 & 1197 & 1228 & 532 & (68, 57) & 19,471 & 5945 & 3411 & 1183 & (69, 65) \\ \hline \hline \end{tabular} \({}^{*}\) The percentage improvement in Mean and StDev (Mean, StDev) in the initial population achieved by 2 SGA.

\end{table} 
Table 13: The mean and standard deviation of the objective function terms in the initial population under RGA and 2 SGA. #### 5.2.2 Convergence Behaviors

The previous section illustrated that 2 SGA resulted in an improved initial population in all the objective function terms. In this section, we compare the convergence behavior of 2 SGA and RGA while solving large-size problems (Problems 4 to 7). The basic features of these problems are given in Table 14. The parameters of the GAs used in this numerical example are given in Table 15. Figure 12a-g shows the convergence along the objective function terms \(Z_{1}\) to \(Z_{10}\), respectively, of 2 SGA and RGA in solving Problem-4 while all these terms are simultaneously optimized with equal weight (\(W_{1}=W_{2}=\cdots=W_{10}=1\)). Each convergence curve is an average of 40 replications. From these convergence curves, we can see that 2 SGA was able to converge more rapidly than RGA along \(Z_{1}\) to \(Z_{5}\), \(Z_{8}\), and \(Z_{9}\). In terms of these objective function terms, 2 SGA was able to find better solutions in only a few hundred generations compared with those determined after more than 10,000 generations by RGA. In terms of \(Z_{6}\), \(Z_{7}\), and \(Z_{10}\), RGA was able to converge more rapidly than 2 SGA. However, 2 SGA was able to catch up with RGA only in a few hundreds of generations right after it changed the search stage, which occurred at 2500 generations. Figure 12h is the convergence of 2 SGA and RGA in \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & \(M\) & \(J\) & \(S_{j}\) **(max)** & \(O_{j}\) **(min, max)** & **NAMPJ (min, max) \({}^{*}\)** \\ \hline
4 & 25 & 40 & 4 & (8 15) & (3, 6) \\
5 & 30 & 60 & 4 & (8, 16) & (3, 6) \\
6 & 40 & 80 & 4 & (10, 18) & (2, 8) \\
7 & 50 & 100 & 4 & (10, 20) & (2, 8) \\ \hline \hline \multicolumn{5}{l}{\({}^{*}\)NAMPJ = Number of Alternative Routing per Operation.} \\ \end{tabular}
\end{table} 
Table 14: Basic features of the problems considered for performance evaluation of the proposed algorithm. Figure 11: The distribution of the objective function of the initial populations of 2 SGA and RGA in both Problem-1 and Problem-4.