## 5 Application  
As an example application of the proposed methodology, a hypothetical engine upgrade scenario is examined. This considers a replacement of the aircraft's original J79 turbojet engine with the EJ200 low-bypass turbofan, which has a similar design air mass flow rate. Under this scope, the climb performance of three aircraft-engine configurations is examined using the proposed multi-objective climb path optimization methodology. A summary of their specifications is given in Table 4. Configuration C1 is the original aircraft configuration, while configurations C2 and C3 correspond to EJ200-equipped variants. Configuration C2 shares the same AUM with configuration C1, assuming that, as a result of the reduced weight of the EJ200 engines, the airframe's weight is allowed to increase with the addition of extra equipment or internal fuel. Configuration C3 shares the shame airframe and internal fuel weight with Configuration C1, resulting in a reduced aircraft AUM. To assess the performance of the above configurations, two test cases are evaluated, their details being provided in Table 5. Case A examines a typical subsonic mission climb scenario, where an aircraft, after takeoff, uses a military (maximum, non-afterburning) thrust setting to climb to the optimum cruise altitude. Case B, on the other hand, considers a maximum power supersonic climb, to be encountered in a supersonic, point-intercept-type mission. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Configuration**} & \multirow{2}{*}{**Engine Type**} & \multicolumn{3}{c}{**Mass [kg]**} \\ \cline{3-5}  & & **Airframe + Fuel** & **Engines** & **Total** \\ \hline C1 & \(J79\) & \(15,512\) & \(3500\) & \(19,012\) \\ C2 & \(E/200\) & \(17,052\) & \(1960\) & \(19,012\) \\ C3 & \(E/200\) & \(15,512\) & \(1960\) & \(17,472\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Specifications of the examined configurations. Figure 11: Pareto front produced by the proposed two-level MOPSO vs. standard MOPSO after 100 iterations, using a 20-particle population. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Thrust Setting**} & \multicolumn{2}{c}{**Start**} & \multicolumn{2}{c}{**End**} \\ \cline{3-5}  & & **Mach** & **Alt [m]** & **Mach** & **Alt [m]** \\ \hline \(A\) & \(Military\) & 0.3 & 0 & 0.9 & \(10,000\) \\ \(B\) & \(Maximum\) & 0.3 & 0 & \(1.8\) & \(15,000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Test case specifications. A population of 20 particles was selected and run for 300 low-level and 100 high-level iterations in all test cases. Four control points were employed for the test runs of Case A and six for the runs of Case B, corresponding to eight and twelve design variables, respectively. In both cases, PSO constants were set at \(c_{1}=c_{2}=1.7\) and \(w=0.3\). Results are presented in Figures 12-15 for Case A and Figures 16-19 for Case B. For reasons of clarity, only flight paths corresponding to minimum-time and minimum-fuel solutions are shown, intermediate flight paths being bounded within them. A qualitative assessment of the resulting trajectories indicates that these are in agreement with related theoretical estimates and published results [2; 3; 4; 5; 9]: All paths begin with a level acceleration at sea level where the aircraft has its maximum acceleration capability. In the subsonic climb case, this is followed by an accelerated climb that follows the peaks of the Specific Excess Power (for the minimum-time climb) and Energy Efficiency (for the minimum-fuel climb) contours up to the specified end conditions; this is a good indication for the accuracy of the generated solutions, since the optimizer, by definition, uses no information about the gradients of the respective functions. In the supersonic climb case, the tracking of contours usually results in a dive occurring in the transonic  
Figure 12: Case A; comparison of fronts of non-dominated solutions obtained for Configurations C1, C2 and C3.