## 5 Numerical Experiments
For each problem instance, we report the number of the Pareto front approximation elements denoted by _NoS_, the value of _Area_, and the value of _gap_ computed in the following way. Let the symbol _AreaAp_ denote the _Area_ of the approximate Pareto front. Similarly, let _AreaEx_ denote the _Area_ of the exact Pareto front. Then the quality of the approximation can be evaluated by _gap_ expressed as a relative difference in percentage computed according to Formula (9). \[gap=100*\frac{Area_{Ap}-Area_{Ex}}{Area_{Ex}} \tag{9}\]  
The results of recently completed tests using the suggested algorithm are presented in the right part of Table 2. The first reported series of experiments was performed so that the number of permitted milestones _NPM_ was set to 20. The real number of milestones _RNM_ is reported in the last column of the table. Table 3 summarizes the results, in which the number of permitted milestones _NPM_ was set to 15, 10, and 5, respectively. The table structure follows the right part of Table 2. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Region** & \(m\) & \(p\) & _NoS_ & _Area_ & \(f_{1}\) **(MLM)** & \(f_{2}\) **(MLM)** & \(f_{1}\) **(MRM)** & \(f_{2}\) **(MRM)** \\ \hline BA & 87 & 14 & 34 & 569,039 & 42,912 & 0 & 26,649 & 280 \\ \hline NR & 350 & 27 & 106 & 736,846 & 59,415 & 557 & 48,940 & 996 \\ \hline TN & 276 & 21 & 98 & 829,155 & 45,865 & 223 & 35,274 & 567 \\ \hline TT & 249 & 18 & 64 & 814,351 & 48,964 & 450 & 41,338 & 921 \\ \hline ZA & 315 & 29 & 97 & 407,293 & 48,025 & 254 & 42,110 & 728 \\ \hline \end{tabular}
\end{table} 
Table 1: Basic studied properties of used benchmarks and the exact Pareto fronts of non-dominated solutions. The results reported in Tables 2 and 3 confirm that the suggested approximate algorithm is able to bring much better results than the former solving method published in [28]. Based on the achieved results, the following observation can be stated as a particular conclusion. The higher the number of permitted milestones (_NPM_) was set, the lower the value of _Area_ that was achieved, and thus, the better the approximation of the exact Pareto front was returned as the output of the suggested algorithm. From the viewpoint of solution accuracy, we have constructed a very useful tool for bi-criteria optimization problems. As far as the efficiency is concerned, it must be noted that the computational demands of the suggested solving approach are in order smaller than those required by the former exact method to Pareto front determination. Average computational times are summarized in Table 4. The computational times of the exact approach were taken from [18]. As far as the computational time is concerned, the result was obtained within several seconds to a few minutes, depending on the problem size. Even if the values of time are not directly reported, the suggested approach is able to find a good Pareto front approximation in a short time. This fact makes the method applicable to a wide spectrum of problems. \begin{table}
\begin{tabular}{c c c c c} \hline
**Region** & \(m\) & \(p\) & _Exact Approach_ & _Approximate Approach_ \\ \hline BA & 87 & 14 & 140.86 & 0.11 \\ \hline NR & 350 & 27 & 5497.58 & 2.07 \\ \hline TN & 276 & 21 & 1331.34 & 0.96 \\ \hline TT & 249 & 18 & 1831.31 & 0.82 \\ \hline ZA & 315 & 29 & 2127.19 & 1.92 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of average computational times in seconds of the exact and approximate approaches to Pareto front determination.