## 4 Conclusions  
This research includes a preliminary design of a conceptual pump-fed LPRE cycle, including mass and dimension estimates of all sub-systems using mathematical models and not empirical models. Unlike previous research, the only inputs required for this model were the mission requirements (\(H\) and \(\Delta V\)) and the propellant pair. The model then calculates the optimum set of engine performance parameters. In addition, seven optimization techniques and four Objective Functions were analyzed in order to provide \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} **T/W** \\ **OF** \\ \end{tabular} }} & \multicolumn{8}{c}{**SC Cycle**} \\
**OF** & **NM** & **SA** & **CSA** & **PSO** & **PIO** & **GA** & **GA-Swarm** \\ \hline F (kN) & 271 & 269 & 272 & 267 & 269 & 269 & 272 \\ \(\Delta t\) (s) & 99.6 & 99.6 & 99.9 & 99.0 & 99.5 & 99.5 & 100.0 \\ \(p_{c}\) (bar) & 95.5 & 96.1 & 88.5 & 111.4 & 98.2 & 97.7 & 86.3 \\ \(l_{sp}\) (s) & 270.4 & 270.5 & 268.7 & 273.8 & 271.0 & 270.9 & 268.1 \\ GLOW (tonnes) & 12.2 & 12.1 & 12.3 & 11.8 & 12.1 & 12.1 & 12.4 \\ T/W & 2.26 & 2.26 & 2.25 & 2.30 & 2.27 & 2.27 & 2.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Values of Variables at max T/W for each technique. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} **GLOW** \\ **OF** \\ \end{tabular} }} & \multicolumn{8}{c}{**SC Cycle**} \\
**OF** & **NM** & **SA** & **CSA** & **PSO** & **PIO** & **GA** & **GA-Swarm** \\ \hline F (kN) & 265 & 270 & 264 & 267.3 & 269 & 264 & 274 \\ \(\Delta t\) (s) & 98.7 & 99.6 & 102.6 & 98.4 & 99.5 & 102.7 & 100.2 \\ \(p_{c}\) (bar) & 122.3 & 94.2 & 74.2 & 133.1 & 97.0 & 73.7 & 81.9 \\ \(l_{sp}\) (s) & 275.9 & 270.1 & 264.8 & 277.9 & 270.7 & 264.6 & 267.0 \\ GLOW (tonnes) & 11.7 & 12.2 & 12.5 & 11.5 & 12.1 & 12.5 & 12.5 \\ T/W & 2.32 & 2.59 & 2.16 & 2.34 & 2.27 & 2.16 & 2.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: SC Cycle: Values of Variables at min GLOW for each technique. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} **GLOW** \\ **OF** \\ \end{tabular} }} & \multicolumn{8}{c}{**SC Cycle**} \\
**OF** & **NM** & **SA** & **CSA** & **PSO** & **PIO** & **GA** & **GA-Swarm** \\ \hline F (kN) & 265 & 270 & 264 & 267.3 & 269 & 264 & 274 \\ \(\Delta t\) (s) & 98.7 & 99.6 & 102.6 & 98.4 & 99.5 & 102.7 & 100.2 \\ \(p_{c}\) (bar) & 122.3 & 94.2 & 74.2 & 133.1 & 97.0 & 73.7 & 81.9 \\ \(l_{sp}\) (s) & 275.9 & 270.1 & 264.8 & 277.9 & 270.7 & 264.6 & 267.0 \\ GLOW (tonnes) & 11.7 & 12.2 & 12.5 & 11.5 & 12.1 & 12.5 & 12.5 \\ T/W & 2.32 & 2.59 & 2.16 & 2.34 & 2.27 & 2.16 & 2.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: SC Cycle: Values of Variables at max \(l_{sp}\) for each technique. the maximum performance of the cycle for a given mission, propellant pair and payload. An Objective Function simultaneously maximizing both \(I_{sp}\) and \(T/W\) was used to validate all techniques. One of the optimization techniques, GA-Swarm, used was a novel technique. This research concludes that the best-performing OF for the GG cycle was T/W as the optimized engine performance values for all variables were better compared to using GLOW as an Objective Function. The best-performing Objective Function for SC was GLOW. However, when comparing two cycles, the benchmark OF to consider for both is GLOW. Looking at the optimization techniques, Nelder Mead is extremely susceptible to initial guess across both cycles and, therefore, cannot be used for an initial optimization. Simulated Annealing shows variance, provides a wide range of values and also returns results in clusters, highlighting that the results are immune to local minima. However, Simulated Annealing does not match the constraints on each run and, therefore, requires additional computational time to generate a data set of values for analysis. Cuckoo Search Algorithm technique works well for SC cycle only.