## 3 Optimization of the Hybrid Last-Stage Launch Vehicle
When objectives are also entered as constraints, especially because the targeted orbit parameters need to be met, it guides the algorithm to find solutions closest to these parameters. The optimization process aims to minimize or maximize the objective functions while satisfying the constraints and requirements to obtain a feasible solution. For this study, the modeFRONTIER 2019R1 optimization software is utilized as a multidisciplinary optimization solver, equipped with an automatic framework [57]. The flow scheme of the multi-objective multidisciplinary optimization process, along with the steps involved in solving the optimization problem, is illustrated in Figure 4. In Figure 4, \(\text{t}_{\text{burn}}\), \(\text{p}_{1}\), \(\text{TVC}_{\text{first}}\), \(\text{TVC}_{\text{last}}\), \(\text{{MR}}\), \(\text{D}_{\text{t}}\), and epsilon in the design parameters represent \(\text{t}_{\text{burn}}\), \(\text{p}_{1}\), \(\text{TVC}_{\text{first}}\), \(\text{TVC}_{\text{last}}\), \(\text{{MR}}\), \(\text{D}_{\text{t}}\), and \(\epsilon\), respectively. The process begins with \"SchedulingStart\", indicating the initiation of the optimization routine using multi-objective particle swarm optimization (MOPSO), a technique for solving problems with multiple objectives. Under the heading \"Design Parameters\", a compilation of variables amenable to modification during optimization is presented, encompassing parameters such as burn time, chamber pressure, TVC angle of the first and last stages, mass ratio, throat diameter, and expansion ratio. These parameters serve as inputs to the optimization process and are subject to iterative adjustments aimed at identifying the optimal solution. The workflow then moves to a script, suggesting a computational step, to calculate initial values and prepare data for optimization. At this stage, MATLAB scripts are employed, primarily for computations related to hybrid characteristics utilizing the optimization design parameters, thus aligning with the overarching objectives and constraints. The calculation steps involved in this study cover various aspects, including hybrid rocket propulsion, CATIA-based structural properties such as mass and inertia characteristics, aerodynamic characteristics, and trajectory determination, which are systematically executed. The initial phase of the optimization process involves the calculation of hybrid rocket propulsion characteristics. Subsequently, the obtained motor length and hybrid stage mass serve as inputs for the structural model, created using parametric shape design in the CATIA program. In this stage, the values for the center of gravity and moment of inertia, which vary as stages are ejected and propellant is consumed, are determined for the entire launch vehicle and its stages. All the pertinent information is then iteratively integrated into the flight mechanics 6 DOF model within the optimization main file until feasible solutions are achieved, satisfying the specified constraints and objectives. Within the 6 DOF simulation, the subsystems of aerodynamics, propulsion, structure, and trajectory are intricately interconnected. This interdependence signifies that the outputs produced by the propulsion subsystem serve as inputs for both the trajectory and structural subsystems. Downstream from the MATLAB step, the workflow branches into objectives and constraints. The objectives prefixed with \"Objective\", such as \"Objective_pa\", \"Objective_h\", and \"Objective_V\", represent the desired outcomes for the flight path angle, altitude, and velocity, respectively. The constraints prefixed with \"Constraint_\", such as \"Constraint_fpa\", \"Constraint_h\", and \"Constraint_V\", denote the limits and requirements that the optimization process must not break for the flight path angle, altitude, and velocity. The process concludes with an \"Exit\" node, signaling the end of the optimization routine, after which the best solution or set of solutions may be selected based on the objectives and constraints defined.