## 5 Results and Discussion

Table 3: Comparison between F/DR and F/DR-k.  
algorithm's result. In Table 5, we compare results for instances P1-P5 from the literature, including the optimal value, the well-established greedy heuristic algorithm developed in [60], Tabu Search and Simulated Annealing as discussed in [61], the F/DR algorithm [43], and our modified algorithms. We refrain from comparing our algorithm to Bio-inspired algorithms due to their distinct characteristics, namely, differences in implementation complexity and simulation time. In fact, Tabu Search and Simulated Annealing, for instance, enhance the performance of greedy algorithms, yielding results closely approaching the optimal value. However, it is worth noting that greedy algorithms exhibit significantly faster execution time. For instance, in the case of the largest network evaluated (i.e., P5), the F/DR-D and F/DR-D-k algorithms require only 1 minute of simulation time and they achieve an error rate of less than 3% for the MS-FAP solution. Based on Tables 3 and 4, the PoOBR and the PoOBE were calculated and are summarized in Table 6. Therefore, F/DR-D-10 is characterized by the best results, which are as follows: (1) In 100% of the benchmarks used, at least one local optimum value (calculated by the modified algorithm) outperforms the conventional algorithm's result, (2) in 95% of the benchmarks used, the probability of discovering a local optimum value (calculated by the modified algorithm) that is better than or equal to the one discovered by the conventional algorithm is greater than 50%. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Conventional Approach**} & \multicolumn{2}{c}{**F/DR-D-10**} & \multicolumn{2}{c}{**F/DR-D-50**} & \multicolumn{2}{c}{**F/DR-D-90**} \\ \hline P1 & 239 & 259 & **252** & 69\% & 14\% & **251** & 74\% & 17\% & **251** & 69\% & 22\% & 1.2 \\ P2 & 257 & 292 & **284** & 86\% & 8\% & **282** & 96\% & 3\% & **283** & 97\% & 3\% & 1.6 \\ P3 & 426 & 448 & **438** & 66\% & 8\% & **437** & 93\% & 5\% & **438** & 75\% & 17\% & 2.0 \\ P4 & 855 & 895 & **878** & 87\% & 4\% & **882** & 82\% & 15\% & **882** & 24\% & 71\% & 9.3 \\ P5 & 1713 & 1801 & **1761** & 100\% & 0\% & 1771 & 100\% & 0\% & **1786** & 29\% & 67\% & 60.6 \\ P6 & 179 & 240 & **231** & 52\% & 6\% & **229** & 70\% & 13\% & **231** & 82\% & 13\% & 1.0 \\ P7 & 252 & 269 & **267** & 21\% & 25\% & **265** & 20\% & 58\% & **266** & 9\% & 81\% & 1.1 \\ P8 & 426 & 476 & **461** & 72\% & 5\% & **450** & 98\% & 1\% & **451** & 100\% & 0\% & 1.7 \\ P9 & 524 & 593 & **589** & 15\% & 77\% & **592** & 1\% & 99\% & **592** & 1\% & 99\% & 3.6 \\ P10 & 426 & 467 & **464** & 44\% & 20\% & **462** & 28\% & 58\% & **463** & 13\% & 84\% & 2.8 \\ P11 & 426 & 472 & **468** & 33\% & 26\% & **463** & 84\% & 8\% & **459** & 86\% & 11\% & 2.6 \\ P12 & 532 & 533 & **533** & 0\% & 0\% & **533** & 0\% & 0\% & **533** & 0\% & 0\% & 1.9 \\ P13 & 308 & 310 & **310** & 0\% & 0\% & **309** & 1\% & 14\% & 309 & 8\% & 18\% & 0.9 \\ P14 & 532 & 533 & **533** & 0\% & 0\% & **533** & 0\% & 0\% & **533** & 0\% & 0\% & 1.1 \\ P15 & 308 & 315 & **312** & 1\% & 0\% & **310** & 11\% & 0\% & **310** & 24\% & 0\% & 1.1 \\ P16 & 532 & 533 & **533** & 0\% & 0\% & **533** & 0\% & 0\% & **533** & 0\% & 0\% & 1.7 \\ P17 & 220 & 222 & **221** & 1\% & 1\% & **221** & 2\% & 9\% & **221** & 7\% & 20\% & 0.9 \\ P18 & 380 & 381 & **381** & 0\% & 0\% & **381** & 0\% & 0\% & **381** & 0\% & 0\% & 1.2 \\ P19 & 528 & 533 & **530** & 3\% & 0\% & **530** & 8\% & 2\% & **530** & 16\% & 13\% & 1.4 \\ P20 & 72 & 74 & **73** & 2\% & 0\% & **73** & 40\% & 1\% & **73** & 89\% & 1\% & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between F/DR and F/DR-D-k.