# Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher
## 5 Multi-Objective Optimization Algorithm Example
At the same time, \(f_{1d}(X)\) hits a maximum value of \(-1.54101\), which corresponds with the \(f_{2d}(X)\) reaching its minimum value of 0.66790. These observations offer important insights into the dynamics and efficiency of the optimization process. For the analysis of the problem as stated in Equation (20), the set of interval Pareto optimal solutions obtained post-transformation is represented in Figure 10. Moreover, the analysis of the original problem, i.e., Equation (19), reveals that the maximum value of the midpoint of electromagnetic thrust is 3.46843 kN, and, concurrently, the midpoint of the maximum current density at the contact surface peaks at 1.40412 \(\times\) 10\({}^{10}\) A/m\({}^{2}\). Under the influence of uncertain parameters, the fluctuation range of electromagnetic thrust is [3.43738, 3.49947] kN, and for the maximum current density at the contact surface, it is [1.39587, 1.41237] \(\times\) 10\({}^{10}\) A/m\({}^{2}\). The minimum value of the electromagnetic thrust midpoint is 3.13619 kN, during which the contact surface maximum current density midpoint is at its lowest, 1.31852 \(\times\) 10\({}^{10}\) A/m\({}^{2}\). The fluctuation range of the electromagnetic thrust, influenced by uncertain parameters, is [3.08201, 3.19037] kN, and for the maximum current density at the contact surface, it is [1.30423, 1.33281] \(\times\) 10\({}^{10}\) A/m\({}^{2}\). Five solutions are uniformly chosen from the Pareto solution set, as outlined in Table 4. By comparing the data in Tables 1 and 4, it is apparent that the optimized values in the interval range of the armature electromagnetic thrust, after considering interval uncertainty, see an improvement of approximately 7.5\(\sim\)17.4%. Simultaneously, the values in the interval range of the pivot rail contact surface current density decrease by approximately 9.5\(\sim\)4.1% compared to the initial values, without significant alteration of the transmitter design parameters. Interval multi-objective optimization proves effective in enhancing both the electromagnetic thrust and the maximum current density of the contact surface of the electromagnetic orbital launcher. Moreover, it enables the efficient selection of the armature structure size to meet varying performance control requirements. For instance, if a designer prioritizes electromagnetic thrust, they can opt for the data set of group 5; if the maximum current density value of the contact surface is of more concern, they should choose the data from group 1; if the designer wishes to consider both the electromagnetic thrust and the maximum current density of the contact surface, the data sets from groups 2, 3, and 4 would be most suitable. Figure 9: Pareto solution sets with different iterations. Table 5 presents the comparative results regarding the objective evaluation function values derived from the optimization of the fifth Pareto solution and the actual evaluation function values computed using the finite element method. According to the data in the table, the discrepancies between the objective evaluation function values calculated by the surrogate model and the actual objective evaluation function values are 1.44% and 1.57%, respectively. This outcome serves to further confirm the practicality and validity of the surrogate model. For this optimization problem, the traditional optimization method is used to optimize it deterministically, without considering the influence of uncertain parameters, and then the method established in this paper is used to optimize its design by considering the