## 6 Numerical Results
### Performance of OSNR-LA-PSA vs. MIP Model  
To evaluate the quality of solutions generated by the OSNR-LA-PSA method, we compared these solutions with the results obtained by solving the MIP model presented in Section 4 using the Cplex solver. The analysis was performed in different routing, traffic, and XT scenarios in network DT12. Namely, we considered \(k\in\{1,3\}\) candidate routing paths, \(|\mathcal{D}|\in\{50,80,100\}\) demands, and reference XT levels \(XT^{\text{REF}}\in\{-51,-57,-\infty\}\) dB/km. Cplex was run with a 1 h runtime limit, while \(10^{5}\) iterations were performed in OSNR-LA-PSA. Our main focus was on the objective function values \(z^{\text{MIP}}\) and \(z^{\text{OSNR-LA-PSA}}\) as well as on computation times \(T^{\text{MIP}}\) and \(T^{\text{OSNR-LA-PSA}}\) obtained, respectively, using the MIP model and the OSNR-LA-PSA method. The obtained results are presented in Table 6. Apart from the above-mentioned metrics, we report the MIP optimality gaps (\(\Delta^{\text{MIP}}\)) and the performance gaps between both methods (\(\Delta^{\text{rel}}\)). Namely, \(\Delta^{\text{MIP}}\) expresses a relative difference between \(z^{\text{MIP}}\) and Cplex lower bounds after the 1 h computation period, while \(\Delta^{\text{rel}}\) represents a relative difference between the obtained values of \(z^{\text{MIP}}\) and \(z^{\text{OSNR-LA-PSA}}\). First of all, we can see that the presence of crosstalk complicates extremely solving the MIP model. While Cplex is able to find optimal solutions (\(\Delta^{\text{MIP}}=0\%\)) in a few seconds in the XT-less scenario (\(-\infty\) dB/km), it does not achieve the goal in most cases for higher values of \(XT^{\text{REF}}\). Indeed, even for one candidate routing path (\(k=1\)) and \(|\mathcal{D}|=80\) demands, the MIP optimality gap is above 20%, and it increases for higher values of \(k\) and \(|\mathcal{D}|\); for \(k=3\) and \(|\mathcal{D}|=100\), a feasible solution could not be found within the 1 h runtime limit. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k\)} & \multicolumn{2}{c}{**DT12**} & \multicolumn{2}{c}{**TEL30**} & \multicolumn{2}{c}{**EURO16**} \\ \cline{2-9}  & \(z\) & \(T\) & \(z\) & \(T\) & \(z\) & \(T\) \\ \hline
1 & 58.3 & 5 & 44.7 & 6 & 79.1 & 7 \\
2 & 53.3 & 9 & 35.9 & 10 & 63.1 & 12 \\
3 & 48.7 & 13 & 32.8 & 14 & 61.1 & 19 \\
4 & 45.4 & 18 & 32.2 & 18 & 61.1 & 25 \\
5 & 44.4 & 22 & 31.1 & 23 & 60.8 & 31 \\
6 & 43.9 & 26 & 30.5 & 28 & 60.8 & 37 \\
8 & 43.8 & 36 & 29.8 & 35 & 60.9 & 50 \\
10 & 43.8 & 44 & 29.7 & 42 & 61.0 & 63 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: An average value of objective function (\(z\)) and algorithm processing times (\(T\)) for different number of candidate paths (\(k\)); \(T\) in seconds. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\rho\)} & \multicolumn{4}{c}{\(\tau\) (DT12)} & \multicolumn{4}{c}{\(\tau\) (TEL30)} & \multicolumn{4}{c}{\(\tau\) (EURO16)} \\ \cline{2-13}  & **0.1** & **0.5** & **1** & **5** & **10** & **0.1** & **0.5** & **1** & **5** & **10** & **0.1** & **0.5** & **1** & **5** & **10** \\ \hline
0.8 & 55.0 & 55.0 & 55.1 & 55. At the same time, OSNR-LA-PSA solutions are obtained in very short times, not exceeding about 10 seconds, for all evaluated problem instances. The OSNR-LA-PSA results (\(z^{\text{OSNR-LA-PSA}}\)) are either the same or better (lower) than the MIP results (\(z^{\text{MIP}}\)). In particular, whenever MIP solutions are optimal (i.e., \(\Delta^{\text{MIP}}=0\%\)), the solutions found using OSNR-LA-PSA have the same objective value and, thus, are also optimal. Moreover, in several cases when the optimality of MIP solutions is not confirmed (\(\Delta^{\text{MIP}}>0\%\)), OSNR-LA-PSA achieves much better results than MIP (within its runtime limit), where the relative difference \(\Delta^{\text{rel}}\) between the feasible solutions found reaches up to \(-30\%\). Note that a negative \(\Delta^{\text{rel}}\) signifies that \(z^{\text{OSNR-LA-PSA}}<z^{\text{MIP}}\).