## 3 Optimization of the Hybrid Last-Stage Launch Vehicle
The objectives are:  
\[\underset{\forall s\in\mathbb{R}}{\text{max}}I_{t}/w_{0}(s)\text{ and } \underset{\forall s\in\mathbb{R}}{\text{min}}\lambda\gamma(s),\Delta t(s), \Delta V(s) \tag{8}\]  
To maintain consistent consideration of objectives throughout the optimization process, the percentage errors in target orbit parameters, specifically flight path angle, altitude, velocity (\(\gamma,h,V\)), and (\(1/(I_{t}/w_{0})\)), are scaled by a factor of 10 to align the magnitudes of these errors. For example, in this study, the value of (\(1/(I_{t}/w_{0})\)) is of an order of magnitude approximately 1/10 lower than the other objectives. Therefore, the value of (\(1/(I_{t}/w_{0})\)) is multiplied by 10 in the optimization process to bring all objectives to the same order of magnitude and to achieve the most optimal solution [57]. This incorporation of scaled terms \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Stage 1** & **Stage 2** & **Stage 3** & **Stage 4** \\ \hline \(L\), m & 7.49 & 4.12 & 3.07 & 1.34 \\ \(D\), m & 1.67 & 1.33 & 1.28 & 0.97 \\ \(m_{g}\), kg & 23,077 & 7032 & 4036 & 872.3 \\ \(T\), kN (Vacuum) & 792 & 267.7 & 194.4 & 36.9 \\ \(t_{burn}\), s & 61.3 & 66 & 71 & 66.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Minotaur-I launch vehicle characteristics. Figure 3: Flowchart of coupled analysis and multi-objective MDO approach. enables comparable effects of the objectives in the optimization process. The objectives are subjected to the following constraints and the subscript \(t\) denotes target parameters:  
\[g_{1}(s)=1-\frac{I_{sp}(s)}{287\,s}\leq 0\;\;g_{1}(s)e\mathbb{R} \tag{9}\]  
\[g_{2}(s)=\frac{h-h_{t}}{h}\times 100-1\leq 0\;\;g_{2}(s)e\mathbb{R} \tag{10}\]  
\[g_{3}(s)=\frac{V-V_{t}}{V}\times 100-1\leq 0\;\;g_{3}(s)e\mathbb{R} \tag{11}\]  
\[g_{4}(s)=\frac{\|\gamma\|}{0.2\text{deg}}-1\leq\;0\;g_{4}(s)e\mathbb{R} \tag{12}\]  
The specific impulse of the Minotaur-I launch vehicle's solid last-stage motor is 287 s, and it can tolerate a flight path angle of 0.2 degrees [56]. Therefore, these values were entered into the objective equations. The optimization parameters, represented as design variables, are continuous parameters that span between the lower limit \(\text{s}_{\text{L}}\) and upper limit \(\text{s}_{\text{U}}\). The range of these optimization parameters is defined as \(\text{s}_{\text{L}}\leq\text{s}\leq\text{s}_{\text{U}}\). The specific optimization parameters can be seen in Equation (13):  
\[\text{s}=\{\text{TVC}_{\text{first}},\text{TVC}_{\text{last}},\text{t}_{\text {burn}},p_{1},\text{ {MR}},\epsilon,\text{D}_{\text{t}}\} \tag{13}\]  
where **MR** represents the initial mass ratio, \(\text{t}_{\text{burn}}\) represents burn time, \(\text{D}_{\text{t}}\) represents throat diameter, \(p_{1}\) represents the initial chamber pressure, \(\epsilon\) represents the initial expansion ratio, and \(\text{TVC}_{\text{first}}\) and \(\text{TVC}_{\text{last}}\) represent the TVC angle of the first and last stage, respectively. The optimization design parameter primarily contributes to process and performance enhancement, while the launch vehicle design parameter plays a role in determining the physical structure and functionality. The optimization design parameters can play a dual role, serving as inputs both in the propulsion system and along the trajectory calculations, providing information about the launch vehicle's performance and efficiency. The optimization parameters were assigned lower and upper limits as follows: \(0\leq\text{TVC}_{\text{first}}\leq 6^{\circ}\), \(1\leq\text{TVC}_{\text{last}}\leq 3^{\circ}\), \(20\leq\text{t}_{\text{burn}}\leq 60\) s, \(2\leq p_{1}\leq 4\) MPa, \(1\leq\text{{MR}}\leq 3\), \(7.5\leq\epsilon\leq 11\), and \(0.065\leq\text{D}_{\text{t}}\leq 0.15\) m. The lower and upper limits were chosen to ensure physical feasibility. The trajectory properties of \(h\), \(V\), and \(\gamma\) serve as both objective and constraint parameters to optimize the operation effectively and efficiently.