## 6 Discussion  
As we discussed in Section 2.3, referring to the industrial case study in [27], the simultaneous optimization of TLBP and BAP leads to a better overall solution (global optimization) with respect to the sequential optimization of the two problems (local optimization). To the best of our knowledge, reference [4] provides the only existing approach to the simultaneous optimization of the two problems. Therefore, we tried to compare this approach  
Figure 6: Possible fixture devices and part orientations for the four parts. with the approach in [4] on the case study of four parts and the benefits of this approach are discussed in this section. Based on the starting information in Section 5, the proposed algorithm with its tuned parameters was applied and compared with the approach in [4]. In order to ensure the robustness of the comparison, 30-time runs of both algorithms on each of the four engine blocks were considered. Therefore, \(30\times 4=120\) optimal sets (Pareto fronts) were obtained for each of the approaches. Firstly, we compared these results at the level of optimal sets. Every optimal set from one algorithm was compared with the 30 optimal sets from the other one based on the set coverage function in Equation (28). Figure 7 reports the results. As shown in the figure, the optimal sets of the new approach showed a significantly higher average value on their coverage over the ones from the original approach. Among the four cases, the new approach provided a quite high average value of the coverage on both Case A (78.71%) and Case C (83.27%), and a relatively high average value of the coverage on Case B (59.06%) and Case D (62.94%). On the contrary, the optimal sets of the original approach showed a very low average value on their coverage over the ones from the new approach (Case A 14.05%; Case B 27.04%; Case C 9.33%; Case D 22.01%). Therefore, the new algorithm presented in this paper showed great advantages on the quality level of the solution sets. To further investigate the solutions inside the optimal set and the robustness of the algorithms, we performed an additional run on each of the four cases through both of the algorithms. The obtained optimal sets of the two algorithms applied on Case A are shown in Figure 8 (Figures A2-A4 show similar results for Case B, Case C, and Case D, respectively). The horizontal and vertical axes of the figure represent the two objective functions: production rate and total investment cost, whose units were parts per hour and Million Yuan, respectively. Each single point represents a solution, where the red points are generated by the new algorithm, while the blue ones are obtained by the original algorithm: a point closer to the origin (left bottom) represents a better solution. Figure 7: Interval plot of the coverage (data) of 30 times \(\times\) 4 case results of the two algorithms. As shown in the figure:  
1. Solutions in the optimal sets were distributed in groups and could be clustered. Clusters mainly differed with respect to the number and type of machine tools, while the solutions inside a cluster mainly differed with respect to buffer capacity. Of course, the allocation of the operations also influenced the results;
2. In the comparison, the overall solutions from the new approach were better than those from the original approach in all four cases;
3. Considering the details of Case A, there were six clusters in the two optimal sets from both of the algorithms. The first two clusters with a lower production rate from the new algorithm showed slightly better solutions. Starting from the third cluster, the difference between the solutions from the two optimal sets was significantly larger. Considering the third clusters from both of the algorithms, the investment cost of the two clusters was quite similar and around 3.000 Million Yuan, while the production rates of solutions (around 9.9 parts/hour) from the new approach were relatively higher than those from the original approach (around 9.5 parts/hour).