## 5 Results and Discussion  
Following the methodology described in the previous section, we calculated and summarized the results listed in Tables 2-4. Each table compares the conventional algorithm, the new modified algorithms, and the theoretical bounds of each problem using two metrics. The first metric is the associated probabilities of improving, equaling, or worsening the local optimum result of each MS-FAP problem. For example, the probability of improving the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Problem**} & **Compatibility** & \multirow{2}{*}{**Demand Vector**} \\  & **Matrix** & & \\ \hline P1 & & \(\mathbf{D}_{1}:=[20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20]\) \\ P2 & & \(\mathbf{D}_{2}:=[5,5,5,8,12,25,30,25,30,40,40,45,20,30,25,15,15,30,20,20,25]\) \\ P3 & \(\mathbf{C}_{1}\) & \(\mathbf{D}_{3}:=[8,25,8,8,8,15,18,52,77,28,13,15,31,15,36,57,28,8,10,13,8]\) \\ P4 & & \(\mathbf{D}_{4}:=2\times\mathbf{D}_{3}\) \\ P5 & & \(\mathbf{D}_{5}:=4\times\mathbf{D}_{3}\) \\ \hline P6 & & \(\mathbf{D}_{1}\) \\ P7 & \(\mathbf{C}_{2}\) & \(\mathbf{D}_{2}\) \\ P8 & & \(\mathbf{D}_{3}\) \\ \hline P9 & \(\mathbf{C}_{3}\) & \(\mathbf{D}_{3}\) \\ P10 & \(\mathbf{C}_{4}\) & \(\mathbf{D}_{3}\) \\ P11 & \(\mathbf{C}_{5}\) & \(\mathbf{D}_{3}\) \\ P12 & \(\mathbf{C}_{6}\) & \(\mathbf{D}_{3}\) \\ \hline P13 & \(\mathbf{C}_{7}\) & \(\mathbf{D}_{2}\) \\ P14 & \(\mathbf{C}_{8}\) & \(\mathbf{D}_{3}\) \\ \hline P15 & \(\mathbf{C}_{8}\) & \(\mathbf{D}_{2}\) \\ P16 & \(\mathbf{C}_{9}\) & \(\mathbf{D}_{3}\) \\ \hline P17 & \(\mathbf{C}_{10}\) & \(\mathbf{D}_{2}\) \\ P20 & \(\mathbf{C}_{11}\) & \(\mathbf{D}_{6}:=[10,11,9,5,9,4,5,7,4,8,8,9,10,7,7,6,4,5,5,7,6,4,5,7,5]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark description, where the compatibility matrices \(\mathbf{C}_{1},\cdots,\mathbf{C}_{10}\) and \(\mathbf{C}_{11}\) are described in [43, 56, 57] and [58, 59], respectively. result was calculated by dividing the number of times that the new algorithm discovered a better solution than the conventional algorithms by the total number of solutions (i.e., 100). To simplify the tables, the probability of matching the conventional algorithm's result was not displayed, but it can be calculated using the reported probabilities. The second metric compares the MS-FAP problem bounds to the best algorithm result, which is the lowest value among the 100 optimum values found by the modified algorithms. It is essential to emphasize that the theoretical bound column does not imply that the minimum result is that value but rather that no algorithm can find a result lower than that value as a theoretical limit. Furthermore, to aid in data comparison, results that are equal to or better than the F/DR algorithm are highlighted in bold, with an underline, while results that are very close to the theoretical limits are highlighted with a gray background. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Conventional Approach**} & \multicolumn{3}{c}{**Random Permutation of the DR List**} \\ \hline P1 & 239 & 259 & 262 & 0\% & 100\% & 1.0 \\ P2 & 257 & 292 & 315 & 0\% & 100\% & 1.4 \\ P3 & 426 & 448 & 536 & 0\% & 100\% & 1.9 \\ P4 & 855 & 895 & 1082 & 0\% & 100\% & 9.4 \\ P5 & 1713 & 1801 & 2137 & 0\% & 100\% & 62.4 \\ P6 & 179 & 240 & **222** & 58\% & 42\% & 0.8 \\ P7 & 252 & 269 & 294 & 0\% & 100\% & 1.1 \\ P8 & 426 & 476 & 520 & 0\% & 100\% & 1.6 \\ P9 & 524 & 593 & 612 & 0\% & 100\% & 2.7 \\ P10 & 426 & 467 & 493 & 0\% & 100\% & 2.1 \\ P11 & 426 & 472 & 513 & 0\% & 100\% & 2.2 \\ P12 & 532 & 533 & 538 & 0\% & 100\% & 1.8 \\ P13 & 308 & 310 & **309** & 27\% & 56\% & 0.8 \\ P14 & 532 & 533 & 534 & 0\% & 100\% & 1.2 \\ P15 & 308 & 315 & 319 & 0\% & 100\% & 1.1 \\ P16 & 532 & 533 & 562 & 0\% & 100\% & 1.6 \\ P17 & 220 & 222 & 225 & 0\% & 100\% & 0.8 \\ P18 & 380 & 381 & 392 & 0\% & 100\% & 1.1 \\ P19 & 528 & 533 & **529** & 46\% & 44\% & 1.4 \\ P20 & 72 & 74 & **73** & 4\% & 90\% & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between F/DR and F/PER.