## 5 Numerical Study
### The CGA in the Inventory Policy Optimization
#### 5.2.2 Large Network (N2)  
Network N2 encompassed four external sources and ten controlled nodes. The considered interconnection structure is illustrated in Figure 9. The lead-time delays were randomly selected from the range 1-7. The holding cost from the initial, worst-case simulation performed for the maximum persistent demand amounted to \(7.73\times 10^{5}\) units. The corresponding domain for the exhaustive search encompassed \(10^{33}\) combinations, thus being infeasible for the common computing platforms. The comparison of the initial RSLs calculated for the controlled nodes using (15) and the optimal vector determined using CGA is visualized in Figure 10.  
Figure 8: Fitness function improvements.  
Figure 7: Fill rate vs. holding cost trade-off for network N1.  
Figure 11 depicts the performance measures obtained for the considered network topology and Table 9 groups the numerical data. When compared with the small-network case discussed in the previous section, larger differences can be noticed between the investigated policies, which were all in favor of the centralized strategy NOUT. In the analyzed case of the network from Figure 9, the cost savings for a given fill rate fell in the range of 5-43%. The biggest savings in the case of the better, centralized, policy, for both the full and partial customer satisfaction, are marked in bold in Table 9.  
These observations are quite general. As was investigated for different network sizes and structural complexities, the cost-related benefits of the NOUT policy relative to the standard one grew for larger and more intricate topologies. Figure 12 summarizes the policy performances under different fill rate requirements. The bars represent three different topologies, involving three, four, and five supply nodes connected to each controlled node, respectively. The stacked bars reflect the overall holding cost for full (100%) and high (90%) customer satisfaction levels. As expected, the holding costs decreased when relaxing the fill rate objective. The gain from using the centralized policy relative to the standard one grew for more complex topologies. Interestingly, the holding costs decreased for more involved connectivity structures. This was probably caused by an averaging effect, i.e., although encompassing  
Figure 10: Initial and optimal reference stock levels (RSLs) for network N2.  
Figure 9: Goods distribution network N2.  
more nodes and having a denser linkage structure, the topology became more regular with respect to the connection specifics, e.g., the mean lead-time delay for a given node.  
Figure 13 presents the changes in the fitness function in the CGA search process. The same business objectives as for network N1, parametrized using \(\gamma=1\) and \(\varphi=20\), were considered. Despite having initially better effectiveness in the case of the OUT policy, ultimately, in line with the theoretical  
Figure 11: Fill rate vs. holding costs trade-off for network N2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Fitness Function Coefficients**} & \multicolumn{2}{c}{**OUT Policy**} & \multicolumn{2}{c}{**NOUT Policy**} \\ \hline \(\gamma\) & \(\varphi\) & **Holding Cost** & **Fill Rate** & **Holding Cost** & **Fill Rate** \\ \hline  
& 100 & 63,704 & 99\% & 56,342 & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: Policy performance in network N2. The biggest savings for full and partial customer satisfaction are given in bold.  
assumptions, the search for the best RSLs for the NOUT policy continued to progress beyond the level established for the OUT one and approached the optimal selection more closely.