# A Hybrid JADE-Sine Cosine Approach for Advanced Metaheuristic Optimization
## 6 Application in Engineering Design
### Spring Design Problem  
The spring design problem focuses on optimizing the dimensions of a compression spring to minimize its volume while satisfying various constraints related to the spring's functionality and physical limitations. The schematic of the spring is shown in Figure 12.  
The objective function to be minimized is given by Equation (35):  
\[z=(x_{3}+2)x_{2}x_{1}^{2}, \tag{35}\]  
where \(x_{1}\), \(x_{2}\), and \(x_{3}\) are the design variables representing the wire diameter, the mean coil diameter, and the number of active coils, respectively. The constraints of the problem are defined as follows in Equation (39):  
\[c_{1} =-\frac{x_{2}^{3}x_{3}}{71785x_{1}^{4}}+1, \tag{36}\] \[c_{2} =\frac{4x_{2}^{2}-x_{1}x_{2}}{12566(x_{2}x_{1}^{3}-x_{1}^{4})}+ \frac{1}{5108x_{1}^{2}}-1,\] (37) \[c_{3} =1-\frac{140.45x_{1}}{x_{2}^{2}x_{3}},\] (38) \[c_{4} =\frac{x_{1}+x_{2}}{1.5}-1. \tag{39}\]  
The penalty function used to handle constraint violations is defined as Equation (40):  
\[g=\max(0,c), \tag{40}\]  
where \(c\) is the vector of constraint violations. The final objective function, incorporating penalties for constraint violations, is given by Equation (41):  
\[f=z+10^{5}(\sum v+\sum g), \tag{41}\]  
where \(v\) is an indicator function that is 0 if a constraint is satisfied and 1 otherwise. As shown in Equation (41), the objective function \(f\) includes a large penalty for any constraint violations, ensuring that the optimizer seeks feasible solutions that also minimize the volume of the spring. As shown in Table 10, the comparison of various optimization algorithms for the spring design problem reveals that MPA achieved the best performance, with the lowest minimum value (0.012665) and an exceptionally low standard deviation (\(2.1\times 10^{-8}\)), indicating remarkable consistency and precision. POA closely follows, also achieving the same minimum value but with slightly more variability. FDA and AHA also performed well but showed more variability and less consistency compared to MPA and POA. GTO  
Figure 12: Schematic of the spring design problem. ranked fifth, showing good performance with a small standard deviation but a longer computational time. In contrast, lower-ranking algorithms like MFO, MGO, and HLOA demonstrated greater variability and less precision in finding optimal solutions. AOA and MVO exhibited the poorest performance, with the highest minimum, mean, and maximum values, as well as notable standard deviations, indicating substantial inefficiency in converging to optimal solutions. As shown in Table 11, the comparison of optimization algorithms for the spring design problem indicates that JADESCA and POA are the top performers, achieving the best scores (0.012665237 and 0.012665268, respectively) with very similar values for critical design variables. FDA and AHA also performed well, with scores nearly identical to the best performers, but with slight variations in X3, suggesting minor differences in their approach. GTO achieved a slightly higher best score (0.012666223) and showed more variation in X3, indicating a different trade-off in design parameters. In contrast, algorithms like MFO, MGO, and HLOA demonstrated higher best scores and more significant deviations in variables 2 and 3, indicating less effective optimization. AOA and MVO performed the worst, with AOA showing a considerably higher best score (0.014104043) and significant deviation in X3, while MVO exhibited extreme values for all variables, suggesting poor convergence and substantial inefficiency in optimizing the design.