## 4 Results
### Numerical Results
#### 4.3.1 Distance of the Best Trade-Off Solution to the Ideal Vector  
In Table 2, the distance between the ideal vector, representing an unattainable solution characterized by the best objective values across all executions, and the best trade-off or compromising solution is presented for each instance. The best trade-off solution is the efficient solution that has the smallest (Euclidean) normalized distance to the ideal vector within the space of the objectives [62], as depicted in Figure 5. To find the best trade-off solution, distances to the ideal vector are computed using the expression of the overall deviation provided in Equation (15).  
\[\Delta=\sqrt{\sum_{o\in O}\left(\frac{value_{o}-ideal_{o}}{ideal_{o}}\right) ^{2}} \tag{15}\]  
Since Pareto fronts usually consist of multiple solutions, summary metrics are necessary. In this context, it is common to analyze the best trade-off solution as one of the suitable candidate solutions for implementation in the real problem, since it is the solution that better balances the fulfillment of all the objectives [63]. The expression computes the overall normalized Euclidean distance of a solution to the ideal vector. To obtain this value, each objective is normalized with the ideal value. Thus, the solution with the minimal deviation \(\Delta\) is chosen as the best trade-off solution.  
Moving from left to right, Table 2 reports the percentage difference from the ideal vector for the number of transfers (\(\delta_{t}\)), the percentage difference from the ideal vector for the cost of the system (\(\delta_{c}\)), and lastly, the comprehensive overall deviation \(\Delta\).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**Successful Transfers**} & \multicolumn{3}{c}{**Cost**} & \multirow{2}{*}{\(\Delta\)} \\ \cline{2-2} \cline{4-7}  & **Ideal** & **Trade-Off** & & \(\delta_{T}\) & **Ideal** & **Trade-Off** & \(\delta_{T}\) \\ \hline
37.100.1 & 312.77 & 308.30 & 1.43\% & 6346.24 & 6346.24 & 0.00\% & 1.43\% \\
37.30.1 & 309.44 & 301.37 & 2.61\% & 6348.56 & 6470.94 & 1.93\% & 3.24\% \\
37.50.1 & 311.76 & 303.17 & 2.76\% & 6346.57 & 6453.30 & 1.68\% & 3.23\% \\
37.70.1 & 310.81 & 305.60 & 1.68\% & 6344.95 & 6344.95 & 0.00\% & 1.68\% \\
37.90.1 & 312.60 & 308.21 & 1.40\% & 6339.61 & 6339.61 & 0.00\% & 1.40\% \\
40.100.0 & 246.14 & 245.01 & 0.46\% & 6188.41 & 6312.70 & 2.01\% & 2.06\% \\
40.100.4 & 273.10 & 268.17 & 1.81\% & 6106.75 & 6159.77 & 0.87\% & 2.00\% \\
40.30.0 & 242.72 & 236.93 & 2.39\% & 6193.28 & 6293.08 & 1.61\% & 2.88\% \\
40.30.4 & 264.70 & 255.06 & 3.64\% & 6103.68 & 6281.73 & 2.92\% & 4.67\% \\
40.50.0 & 247.69 & 240.08 & 3.07\% & 6193.28 & 6280.23 & 1.40\% & 3.38\% \\
40.50.4 & 272.71 & 266.12 & 2.42\% & 6106.75 & 6272.04 & 2.71\% & 3.63\% \\
40.70.0 & 247.62 & 243.23 & 1.77\% & 6188.41 & 6320.16 & 2.13\% & 2.77\% \\
40.70.4 & 274.18 & 266.86 & 2.67\% & 6103.68 & 6269.77 & 2.72\% & 3.81\% \\
40.90.0 & 249.03 & 244.10 & 1.98\% & 6191.48 & 6297.10 & 1.71\% & 2.61\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Distances of the best trade-off solution to the ideal vector.  
Figure 5: Ideal vector and best trade-off solution.  
Analyzing the results of the distance between the ideal solution and the solution with the best tradeoff is shown in Table 2, where it can be concluded that the MILP model is able to obtain, in general, accurate solutions for most of the instances. The average overall deviation is just 2.75%. Moreover, the average of the percentage difference for each particular objective is also low (i.e., 2.11% and 1.52% for the number of transfers and the cost of the system, respectively). When the instances are grouped according to an increasing \(\lambda\), it can be depicted that in general, \(\delta_{T}\), \(\delta_{C}\) and \(\Delta\) tend to be reduced. The exception is \(\delta_{C}\) when \(\lambda\) goes from 90 to 100. This information can be visually depicted in Figure 6.