# Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher
## 5 Multi-Objective Optimization Algorithm Example  
The armature electromagnetic thrust \(f_{1}\) of the electromagnetic orbital launcher is directly related to the exit velocity of the projectile; the higher the \(f_{1}\), the higher the exit velocity. The maximum current density at the contact surface of the armature track \(f_{2}\) reflects the armature ablation; the smaller the \(f_{2}\), the less likely it is that the armature will ablate [30]. Therefore, \(f_{1}\) and \(f_{2}\) are used as the optimization objective functions. One of the main problems encountered in the design of electromagnetic railguns is transition, which refers to the transition from solid-solid contact to solid-arc solid contact on the contact surface of the pivot rail. When transition occurs, there must be an intermediate gap filled by plasma at the contact surface of the pivot rail, and this intermediate gap is caused by friction and wear, resulting in material loss on the contact surface of the pivot rail, and thus in transition. The armature has to bear not only high stress but also high temperatures caused by Joule heat and friction heat in the process of dynamic emission with a large current load and ultra-high speed. The contact surface of the armature rail will wear and corrode, and then the thickness of the armature tail will change [31]. Therefore, the armature tail thickness is treated as an uncertainty variable at \(U\) to better match the actual operating conditions of the electromagnetic orbital launcher. Thus, the uncertainty optimization problem can be established as follows:  
\[ \begin{split}&\max_{x}f_{1}(X,U)\\ &\min_{x}f_{2}(X,U)\\ & s.t.\end{split}  \tag{19}\]  
To facilitate the creation of multi-objective optimization problems, this paper optimizes the opposite of \(f_{1}\), i.e., finds the minimum value of \((-f_{1}(X,U))\). The optimization problem can be converted as follows:  
\[ \begin{split}&\min_{x}-f_{1}(X,U)\\ &\min_{x}f_{2}(X,U)\\ & s.t.\end{split}  \tag{20}\]  
This paper seeks to find reasonable design parameter variables that make the electromagnetic thrust value as large as possible and the maximum current density at the contact surface as small as possible (considering the average design performance, i.e., the midpoint value of the objective function) and cause the electromagnetic thrust and the maximum current density at the contact surface to have the smallest radius values (i.e., with the smallest performance fluctuation interval). Thus, the optimization model can be described as follows:\[ \begin{split}&\min_{x}\left(-f_{1}^{c}(X),f_{2}^{c}(X)\right)\\ &\min_{x}\left(f_{1}^{w}(X),f_{2}^{w}(X)\right)\\ & s.t.\quad\left\{\begin{array}{l}7\leq X_{1}\leq 10\\ 9\leq X_{2}\leq 12\\ 5\leq X_{3}\leq 8\\ 6\leq X_{4}\leq 8\\ 4.0\leq U\leq 5.5\end{array}\right.\end{split}  \tag{21}\]  
A multi-objective evaluation function is applied to the above equation with the following parameters: \(\beta\) is 0.5, \(\zeta\) is 0, and \(\phi\) and \(\psi\) are 1.0. Then, this optimization problem can be transformed into  
\[ \begin{split}&\min_{x}\left(f_{1d}=0.5\times(-f_{1}^{c}(X))+0.5f_{ 1}^{c}(X)\right)\\ &\min_{x}\left(f_{2d}=0.5\times(-f_{2}^{w}(X))+0.5f_{2}^{w}(X)\right) \\ & s.t.\quad\left\{\begin{array}{l}1.28\leq X_{1}\leq 2.28\\ 4.2\leq X_{2}\leq 7.2\\ 29\leq X_{3}\leq 32\\ 4.8\leq X_{4}\leq 8\\ 33.5\leq X_{5}\leq 36.5\\ 0.67\leq U\leq 0.82\end{array}\right.\end{split}  \tag{22}\]  
In this complex multi-objective optimization scenario, a dual-layer model, featuring a deep belief networks (DBN) and deep neural network (DNN), known as DBN-DNN, is deployed. The architecture comprises an inner layer and an outer layer, both utilizing the DBN-DNN model, with the strategic selection of three hidden layers. Each layer is equipped with a distinct number of neurons: 30, 24, and 18, respectively. We use PyTorch, a Python-based machine learning library, to build the DBN-DNN model, and we verify the reliability of the model with the PyCharm software.