# A Hybrid JADE-Sine Cosine Approach for Advanced Metaheuristic Optimization
## 6 Application in Engineering Design
### Speed Reducer Design Problem  
The speed reducer design problem is an optimization task focused on minimizing the weight of a speed reducer subject to several constraints. This problem involves finding optimal dimensions for the various components of the reducer to achieve the minimum weight while satisfying mechanical and geometric constraints. The schematic of the speed reducer is shown in Figure 13.  
The objective function to be minimized is given by Equation (42):  
\[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\] \[\quad-1.508x_{1}\Big{(}x_{6}^{2}+x_{7}^{2}\Big{)}+7.4777\Big{(}x_ {6}^{3}+x_{7}^{3}\Big{)}+0.7854\Big{(}x_{4}x_{6}^{2}+x_{5}x_{7}^{2}\Big{)}, \tag{42}\]  
where \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\), \(x_{5}\), \(x_{6}\), and \(x_{7}\) are the design variables representing various dimensions and components of the speed reducer. The constraints of the problem are defined as follows in Equation (53):  
\[c_{1} =\frac{27}{x_{1}x_{2}^{2}x_{3}}-1, \tag{43}\] \[c_{2} =\frac{397.5}{x_{1}x_{2}^{2}x_{3}^{2}}-1,\] (44) \[c_{3} =\frac{1.93x_{4}^{3}}{x_{2}x_{3}x_{6}^{4}}-1,\] (45) \[c_{4} =\frac{1.93x_{5}^{3}}{x_{2}x_{3}x_{7}^{4}}-1,\] (46) \[c_{5} =\frac{1}{110x_{6}^{3}}\Bigg{(}\bigg{(}\frac{745x_{4}}{x_{2}x_{3} }\bigg{)}^{2}+16.9\times 10^{6}\Bigg{)}^{0.5}-1,\] (47) \[c_{6} =\frac{1}{85x_{7}^{3}}\Bigg{(}\bigg{(}\frac{745x_{5}}{x_{2}x_{3}} \bigg{)}^{2}+157.5\times 10^{6}\Bigg{)}^{0.5}-1,\] (48) \[c_{7} =\frac{x_{2}x_{3}}{40}-1,\] (49) \[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]0 (50) \[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]1 (51) \[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]2 (52) \[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]3  
\[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]4 Schematic of the speed reducer design problem. The penalty function used to handle constraint violations is defined as Equation (54):  
\[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]5  
where \(c\) is the vector of constraint violations. The final objective function, incorporating penalties for constraint violations, is given by the following:  
\[z =0.7854x_{1}x_{2}^{2}\Big{(}3.333x_{3}^{2}+14.9334x_{3}-43.0934 \Big{)}\]6  
where \(v\) is an indicator function that is 0 if a constraint is satisfied and 1 otherwise. As shown in Equation (55), the objective function \(f\) includes a large penalty for any constraint violations, ensuring that the optimizer seeks feasible solutions that also minimize the weight of the speed reducer. As shown in Table 12, the results from the speed reducer design problem reveal that JADESCA is the most effective optimizer, achieving the lowest minimum and mean values with a small standard deviation, indicating consistent performance and securing the first rank. MFO also performed well, matching the minimum value of JADESCA but with slightly higher mean and maximum values, placing it second. DOA, while achieving the same minimum, showed higher variability and ranked third. RIME and COA offered competitive and consistent results, ranking fourth and fifth, respectively. Lower-ranked optimizers like AVOA, ZOA, SCSO, and SHIO demonstrated higher mean and maximum values, indicating less optimal and more variable performance. In contrast, optimizers like HHO, WOA, TTHHO, ROA, FLO, CPO, and SSOA performed poorly, with high variability and suboptimal outcomes, making them unsuitable for this problem. As shown in Table 13, in the speed reducer design problem, JADESCA and MFO both achieved the best score of 2994.471066, with identical variable values across all dimensions, indicating their high effectiveness and consistency in finding optimal solutions. DOA also achieved a very close best score, differing only slightly, with minor variations in variables 5 and 7, suggesting high precision with some divergence from the ideal. RIME