## 5 Numerical Study
### CGA vs. Monte Carlo (MC) Method  
The CGA performance was compared with two benchmark methods. The first one was a full-search algorithm, which can be utilized only in non-sophisticated structures due to the large computational effort. Another, which can verify more complex cases, assumed the performance of MC simulations based on the same parameters as the CGA. In this section, the results obtained through the GA-based optimization are compared with the MC simulations, as is typical in the initial examination of evolutionary algorithms [18]. The MC method operates on the stochastic RSL vectors established in each generation. In contrast to the GA, the fitness evaluation does not assume other evolutionary operations, i.e., selection, crossover, or mutation. The fitness function values are calculated for all the candidate solutions from the generated population. Hence, the subsequent populations do not arise from the previous ones, like in GA-based algorithms, but are arbitrary and independent of each other. The comparison performed in this section tracked the fitness function improvements in the process of the goods distribution using MC simulations and the CGA for \(1.5\times 10^{3}\) generations, without other stop conditions. In the numerical analysis, three topologies, assuming different network sizes and customer demands, were applied. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Phase** & **Parameter** & **Setting** \\ \hline \multirow{3}{*}{Initialization} & Population size & 10 individuals \\  & Stop criterion & \(10^{3}\)–\(10^{4}\) generations \\ \cline{2-3}  & Fitness function & \(\Psi=\)\(\left(1-\frac{HC_{max}}{HC_{max}}\right)^{\
u}FR^{\
u}\), \\ \hline \multirow{3}{*}{Evolution} & Selection & Four-way tournament selection \\  & Crossover & Two-point crossover \\ \cline{1-1}  & Mutation & 15\% mutation probability \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Continuous genetic algorithm (CGA) parameters for logistic network optimization. The simulation horizon was set to 30 periods. Each population of candidate solutions, both for the CGA and MC simulations, contained 10 individuals. The mutation probability for the CGA was set to 15%. The first examined topology encompassed five nodes (\(N=3,M=2\)), which led to \(1.7\times 10^{8}\) candidate solutions to examine the full-search algorithm with the same granularity set for all methods. The second topology comprised 14 nodes (\(N=10,M=4\)), with a search space of \(4.8\times 10^{29}\) solutions, which was no longer feasible for the full-search approach. Finally, the last network comprised 27 nodes (\(N=21,M=6\)), which translated to \(2\times 10^{69}\) possibilities for forming an RSL vector. Table 7 summarizes the results obtained for these three networks. The data gathered in Table 7 shows the effectiveness of the considered CGA and its superiority over the MC simulations. The evolutionary method achieved the goals set by the network managers in a short time. The conducted tests allow one to observe two significant aspects of GA-based optimization in the resource distribution environments under consideration. The first one concerns the size of the network topology: the bigger the interconnection structure, the greater the savings that are obtained using the GA-based optimization relative to the MC one. Owing to the relatively small search space, the results of the five-node network were similar for both methods. By exploring a large amount of the search space, the MC method had a bigger probability of finding a good solution than in the remaining two cases of more complex topologies. For the larger search spaces, correspondingly, for the fourteen- and twenty-seven-node topologies, the CGA found much better solutions. The CGA was more efficient at finding good solutions in the case of polarized priority in the objective function (either \(\gamma\) or \(\varphi\) set to be much larger than the other coefficient).