## 6 Discussion
Starting from the fourth cluster, it is apparent that the solutions of the cluster from the original approach were fully dominated by those of the corresponding cluster from new approach. This behavior was similar in Case C (Figure A3);
4. Considering the situation of Case B (Figure A2) and Case D (Figure A4), the improvement of the new algorithm with respect to the original one was less relevant. To obtain a further understanding of these results, the optimal solutions of the additional run from both the approaches were clustered. The IP criterion with the investment cost per part was introduced and was calculated as:  
\[\text{IP}=\frac{\text{C}}{\text{PR}\times\text{T}_{\text{A}}(4800\text{hours}/ \text{year})\times\text{P}_{\text{exp}}(1\text{year})}\]  
Therefore, the best solution (leader) for each cluster would be with the smallest IP value. The results of Case A are shown in Table 4 (new algorithm) and Table 5 (original algorithm), while the results of Case B, Case C, and Case D are shown in Tables A6-A11 in Appendix G. Moreover, an example of a detailed solution (the leader solution of cluster 5 in Table 4) is illustrated in Table A12. Figure 8: Comparison for Case A: results from the optimal sets of the additional runs. As shown in these tables, the results of Case A obtained by the new approach showed a much better spread of the production level (38.090, 43.008, 47.816, 52.719, 57.546, 62.334 parts per year) from the leading solution of the six clusters, and properly fulfilled the required interval of Case A (35.000, 60.000). Moreover, the best solutions from the new approach in Table 4 provided a quite similar and low investment cost per part IP (around 630 CNY/part), while those solutions from the original approach in Table 5 started from 633 CNY/part and gradually increased to 830 CNY/part. Apparently, the new approach proposed in this paper showed a better performance in Case A. Similarly, a better performance was obtained by this approach in Case C. The results from this approach provided a better spread of the production level (31.660, 35.209, 38.692, 42.273, 45.761, 49.028 parts per year) from the best solution of six clusters, meeting the required demand interval of Case C (28.000, 45.000). The best solutions from this approach in Table A8 gave the lower investment cost per part IP (around 857 CNY/part), while the solutions from the original approach in Table A9 started from 858 CNY/part and gradually grew to 983 CNY/part. As for Case B and Case D, the two approaches showed a similar performance on the clusters with a lower production rate, while the new approach showed a better performance on the clusters with a higher production rate (especially the cluster with highest production rate). In addition, as shown in the tables, the new approach always provided a quite stable investment cost per part IP in all the four cases, which means that the new approach had a better searching ability within the solution space. Therefore, the proposed approach showed great advantages and provided much better solutions in Case A and Case C, and also obtained better solutions in Case B and Case D.  
In addition, to test the stability of both the algorithms, we ran 100 simulations for each solution of the 120 optimal sets and considered the median value for each solution as the reference point (namely PR\({}_{\text{med}}\)). The percentage of deviation between the original production rate PR\({}_{0}\) and the reference point PR\({}_{\text{med}}\) was calculated as \(\frac{\text{PR}_{0}-\text{PR}_{\text{med}}}{\text{PR}_{\text{med}}}\). The results of both the algorithms are shown in Figure A5. As shown in the figure, considering the 6934 solutions of the 120 optimal sets of the new approach, the median value of the deviation percentage was 0.58%, which was lower than the 0.89% obtained considering the 5678 solutions of the 120 optimal sets of the original approach. Thereforethis shows that the final solutions from the new approach were more stable than those from the original one.