## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
The results of ANOVA for Problem-4 are presented in Table 16 and Figure 13.0 The \(p\)-values corresponding to the main effects of mutation and crossover probabilities are zero, thus, implying that these two factors have statistically significant effects on the final solution quality. On the other hand, the \(p\)-value for the interaction effect is high (compared to a typical significance level \(\alpha=0.05\)), which indicates the absence of interaction between these two factors. This lack of interaction simplifies parameter tuning, thereby, allowing the user to optimize them independently. The plots of the main effects in Figure 19 show that the mutation probability needs to be set close to 0.35, and the crossover needs to be set at higher values between 0.90 and 1.00. The residual plots do not indicate unusual patterns, thus, confirming the adequacy of the ANOVA. The analysis also rendered similar results on several other problems of a varying size considered in this paper. Hence, the recommended values of the mutation and crossover probabilities can be used to solve different sets of problems using the proposed algorithm. Figure 13.1 The convergence of 2 SGA under three different selection operators while solving Problem-4. ## 6 Discussion, Conclusions, and Future Research

### Discussion and Conclusions

The recent trend in manufacturing scheduling is in developing efficient algorithms for complex and comprehensive scheduling problems. Apparently following this trend, a group of authors recently developed an efficient two-stage genetic algorithm for a comprehensive flexible job shop scheduling problem (FJSP) that incorporated (1) sequence-dependent setup time, (2) attached and detached nature of setups, (3) machine release date, and (4) lag-time. The authors demonstrated the superiority of the developed two-stage genetic algorithm in solving large-size problems, which motivated our current research. In this paper, we expand the application of the two-stage genetic algorithm to solve a comprehensive flexible job shop lot streaming problem that incorporates many objective functions. Several empirical investigations were conducted on the proposed model and the two-stage genetic algorithm through which the following observations and conclusions were made. * The magnitude of the severity of a single objective optimization on the objective function terms that are not incorporated increased as the problem size increased. The result emphasizes the need for multi-objective optimization in real industrial scheduling problems that are typically large in size. * Optimizing the subdit flowtime is more desirable than optimizing the job flowtime. However, optimizing both terms simultaneously can also result in favorable solutions with respect to the overall flowtime performance. * In lot streaming, one sublot of a given job may be finished much sooner than the other sublot of the same job. This may increase the work-in-process inventory. The newly proposed objective function terms (to minimize the maximum sublot finish-time separation and total sublot finish-time separation) can alleviate this problem with minimal impacts on the sublot and job flowtime. * Instead of minimizing the maximum or the total sublot flowtime, it is advantageous to minimize both its maximum and total values simultaneously. The same is true with the other performance measures (the job flowtime, sublot finish-time separation, and machine workload). * Workload balancing in FJSP may not be fully achieved by minimizing the maximum or the total workload or both. A newly proposed objective function term (minimizing