# Heuristics for a Two-Stage Assembly-Type Flow Shop with Limited Waiting Time Constraints
## 4 Computational Experiments
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(m\) & \(n\) & **Set A** & **Set B** & **Set C** \\ \hline \multirow{4}{*}{2} & 10 & \(0.181\)1  & 0.000 \\  & 20 & \(-1.091\) & \(0.000\) & \(0.048\) \\  & 30 & \(0.155\) & \(0.011\) & \(-0.024\) \\  & 40 & \(-0.720\) & \(0.000\) & \(0.000\) \\  & 50 & \(-1.638\) & \(0.000\) & \(0.000\) \\ \hline \multirow{4}{*}{5} & 10 & \(-0.935\) & \(0.000\) & \(0.000\) \\  & 20 & \(-1.418\) & \(0.000\) & \(-0.368\) \\  & 30 & \(0.053\) & \(-0.353\) & \(-0.178\) \\  & 40 & \(-0.988\) & \(-0.017\) & \(-0.369\) \\  & 50 & \(-0.909\) & \(-0.14 The second experiment examined the remaining parameters (\(P_{size}\), \(p_{c}\), \(p_{m}\), \(P_{l}\), \(P_{r}\)). The full factorial design of the parameters could make 3125 combinations. However, because testing these combinations requires significant time, we adopted the Taguchi L\({}_{25}\) orthogonal array design instead of a full factorial design. The L\({}_{25}\) design is provided in Table 6; that is, the 25 combinations provided in the table were tested. Note that the Taguchi design developed by [PERSON] can achieve parametric research and optimization while reducing the number of experiments or numerical tests [54]. Figure 8 shows that the lowest RDIs for the parameters were achieved at (\(P_{size}\), \(p_{c}\), \(p_{m}\), \(P_{l}\) and \(P_{r}\)) = (30, 0.2, 1, 60 and 60). Remarkably, the probabilities of crossover and mutation were opposite. With these probabilities, the GA occasionally executes crossover, but it always performs mutation operations. Considering the calibration results, i.e., OX1 and \(p_{c}\) = 0.2, the mutual interchange between two solutions may not help search for better solutions to this problem. In contrast, the mutation was more useful for improving the solution. The values of \(P_{l}\) and \(P_{r}\) were the same as 60. This indicates that the local search and restart procedures start when the best solution did not improve during 60 generations. Figure 8: GA calibration for \(P_{size}\), \(p_{C}\), \(p_{M}\), \(P_{l}\), and \(P_{r}\). Figure 7: GA calibration for selection, crossover, and mutation. The suggested IG has two parameters: \(d\) and \(p_{IS}\). This calibration considered five levels of each parameter, that is, \(d\) = (6, 8, 10, 12, and 14) and \(p_{IS}\) = (0, 0.25, 0.5, 0.75, and 1). Thus, we tested 25 combinations. A summary of the results is presented in Figure 9. As shown in the figure, RDI had the lowest at \(d\) = 10 and the difference in RDIs between \(d\) values was very clear. This value differed significantly from the original value of 4 which was experimentally found in [47]. It is likely because computing power improved compared to in the past. Note that the destruction size \(d\) is determined by considering the trade-off between the solution quality and computation time. That is, a large \(d\) can provide good solutions but requires a large amount of computation. Therefore, in a limited CPU time, if the \(d\) value is too large, IG may provide a poor solution. The effect of the \(p_{IS}\) was not significant. This may be because IG is based on the insertion mechanism. Consequently, the IG uses \(d\) = 10 and \(p_{IS}\) = 0.75. For calibration of the proposed SA, we considered five levels of each of the three parameters (\(\alpha\), \(L\), and \(p_{IS}\)). In this experiment, the epoch length \(L\) was defined as \(L=n\times l\), where \(l\) is a control parameter. The design of this calibration was also based on the Taguchi L\({}_{25}\) orthogonal array (Table 7). Figure 10 shows the calibration results. Three parameters (\(\alpha\), \(l\), and \(p_{IS}\)) had the lowest RDI values at (0.995, 15, and 0.25), respectively. In contrast to GA and IG, the results showed bath-curves for all the parameters. In other words, both higher and lower values from the best values resulted in worse outcomes. This is probably because SA generates a neighborhood solution using only a local search of insertion and interchange without any other methods.