## 3 Optimization of the Hybrid Last-Stage Launch Vehicle
The flowchart outlining the multi-objective MDO approach and the interaction of the subsystems employed in this study for an example application of the Minotaur-I launch vehicle is depicted in Figure 3. This flowchart is not only applicable for the Minotaur-I launch vehicle but can also be utilized and generalized for other launch vehicles. In the flowchart, the optimization process begins with the determination of optimization design parameters to maximize the performance of the launch vehicle. This process starts with the identification of key elements, such as initial expansion ratio, initial chamber pressure, throat diameter, and thrust vector control of the first and last stages, which are optimization design parameters for the problem. Subsequently, the parameters relevant to these are calculated for the hybrid propulsion system. Then, the structural and aerodynamic features of the launch vehicle, whose geometric properties are computed, are calculated using software such as DATCOM Version 6 and CATIA V5R19. The trajectory of the launch vehicle is determined using calculated aerodynamic, structural, and propulsion subsystems along with environmental models such as gravity and atmosphere. Results obtained after each optimization step are stored, and an optimization algorithm is employed to optimize the performance of the launch vehicle. The obtained solution is determined through a fitness check to ascertain whether it meets design constraints such as targeted orbit parameters. If not, the optimization design parameters are adjusted, and the optimization process is iterated. Ultimately, upon completion of this comprehensive process, the launch performance of the launch vehicle is successfully optimized. For the optimization study, the Minotaur-I launch vehicle was selected to determine the hybrid propulsion characteristics and TVC angle required to achieve the desired trajectory. This launch vehicle consists of four solid stages. Detailed information regarding the launch vehicle, including its geometry, stage masses, and structural and propulsion characteristics, can be found in Figure 1 and Table 2[55]. The first stage is 7.49 m in length, with a diameter of 1.67 m and a mass of 23,077 kg. This stage has a thrust power of 792 kN in a vacuum, and the burn time of its motor is 61.3 s. The second stage, with a shorter length of 4.12 m, a diameter of 1.33 m, and a mass of 7032 kg, is equipped with a motor that produces 267.7 kN of thrust, and its burn time is 66 s. The third stage measures 3.07 m in length and 1.28 m in diameter, with a mass of 4036 kg. With a thrust of 194.4 kN, this stage has a motor burn time of 71 s. The final stage is 1.34 m in length, has a diameter of 0.97 m, and a mass of 872.3 kg. Its motor produces a thrust of 36.9 kN, and the burn time is 66.8 s.  
The optimization process focuses on modifying the last stage of the Minotaur-I launch vehicle by incorporating a hybrid propulsion system. This modification aims to improve performance and achieve an efficient design. The target mission requires orbit parameters of \(h=741.2\) km and related inclination for target orbit altitude, \(i=98.3^{\circ}\), for a sun-synchronous orbit (SSO), with a PM of 302 kg [56]. During the optimization process, the input parameters are adjusted within the defined limits of the design space. Mathematically, the optimization problem can be defined as shown in Equation (8). The objective is to determine the optimal values for the propulsion characteristics of the hybrid stage and the TVC angles of the first and last stages. These values should maximize the total impulse-to-weight ratio (\(I_{t}/w_{0}\)) while adhering to \(I_{sp}\) constraints and ensuring the target orbit parameters are met, as stated in Equations (9)-(12).