# Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher
## 5 Multi-Objective Optimization Algorithm Example
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimal** & \multirow{2}{*}{
\begin{tabular}{c} **Design Vector** \\ **X/mm** \\ \end{tabular} } & \multirow{2}{*}{\(f_{1d}(X)\)} & \multirow{2}{*}{\(f_{2d}(X)\)} & \multirow{2}{*}{\(f_{1}(X,U)\)} & \multirow{2}{*}{\(f_{2}(X,U)\)} \\
**Compromise** & & & & & & \\
**Solution** & & & & & & \\ \hline \multirow{3}{*}{1} & (9.44581, 9.75918, 5.98861, 7.92596) & \(-\)1.54101 & 0.66790 & [3.08201, 3.19037] & [1.30423, 1.33580] \\  & (9.55383, 9.60918, 5.86247, 7.90717) & \(-\)1.55151 & 0.66781 & [3.10302, 3.19253] & [1.30599, 1.33561] \\  & (9.38883, 11.09869, 7.61396, 7.49150) & \(-\)1.60515 & 0.66531 & [3.21029, 3.31429] & [1.31387, 1.33062] \\  & (9.53728, 11.09491, 7.49053, 7.12074) & \(-\)1.65271 & 0.67824 & [3.30542, 3.42124] & [1.33519, 1.35648] \\  & (9.89567, 0.86087, 7.54290, 6.47364) & \(-\)1.71869 & 0.70619 & [3.43738, 3.49947] & [1.39587, 1.41237] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Data before optimization of electromagnetic orbital launcher. Figure 10: Interval non-inferior solution set distribution. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Multi-** & \multirow{2}{*}{\(x_{1}\)**/mm**} & \multirow{2}{*}{\(x_{2}\)**/mm**} & \multirow{2}{*}{\(x_{3}\)**/mm**} & \multirow{2}{*}{\(x_{4}\)**/mm**} & **Predicted Value** & \multirow{2}{*}{**Actual Value**} & \multirow{2}{*}{**Error/\%**} \\
**Function** & & & & & & & \\ \hline \(f_{1d}(X)\) & 9.89567 & 0.86087 & 7.54290 & 6.47364 & 1.71869 & \(-\)1.69429 & 1.44 \\ \hline \(f_{2d}(X)\) & 9.89567 & 0.86087 & 7.54290 & 6.47364 & 0.70618 & 0.71744 & 1.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Armature optimization results of electromagnetic orbital launcher. uncertainty of the parameters; the non-inferiority solution sets obtained by the two methods are shown in Figure 11.  
As is visible from the figure, the deterministic optimization method, due to its disregard for the uncertainty factors of the electromagnetic orbital launcher in real-world conditions, bases its decision making on specific values of the objective function at each design vector. Consequently, the Pareto optimal solution set achieved is a series of Pareto points. Conversely, the methodology proposed in this paper acknowledges that the armature tail thickness is not constant owing to frictional wear. It employs interval numbers to denote the uncertainty of the armature tail thickness and integrates this into the optimization design process. As a result, the value of the objective function is also an interval. The upper and lower limits of this interval are the critical values ascertained when taking into account parameter uncertainty. The Pareto optimal solution set, in this case, is a series of interval solutions. Considering that the current paper addresses a minimization problem and that the Pareto front is concave, the optimization result is satisfactory. Our method offers a more fitting depiction of the performance variations of the electromagnetic orbital launcher due to uncertainty. Furthermore, it ensures the reliability of the optimization results, an attribute of paramount importance for the reliability estimation of the electromagnetic orbital launcher. The approach also provides a valuable reference point for designers when addressing real-world engineering problems.