Sents the best value obtained from the 50 runs of each problem. In the “Average” column the average of the 50 runs is indicated and in the “Worst” column the higher obtained cost is indicated. Then, the “Gap” column is calculated as Gap ?jbest veragej ?100 . The sample standard deviation is prebest sented in the “Std. Dev.” column. The ” Best” and ” Best” columns indicate the total number of times and the percentage of times when the best value was reached, respectively. Finally, the “Time” column indicates the average time (in seconds) for solving 50 times each problem. From Table 4 it can be observed that for Benchmark 1 the algorithm reached the best value in more than half of the 50 runs. Moreover, the average from all the runs is very near from the best obtained value and the standard deviation indicates that the values are around the average; the small gap obtained (1.20 ) confirms the good performance of the developed algorithm in this problem. The consumed average time is 4.23 seconds. The Z-DEVD-FMK web results for Benchmark 2 indicate that despite the expected increase in the computational time (almost 14 seconds), the algorithm reached a very good gap between the best obtained value and the average of the 50 runs. This gap is lower than 2 . Also, the best value was obtained in almost the half of the experimentation (in 22 of the runs). Finally, the numerical experimentation conducted for Benchmark 3 was not as good as the previous ones but the results are still reasonable. The best value was reached in 18 of the 50 runs, while the gap increased to 3.18 and the consumed time was of 54.9 seconds. These results were clearly affected by the difficulty of finding the follower’s rational reaction.A comparison between the Stackelberg-Genetic and fnins.2015.00094 the Nash-Genetic HS-173MedChemExpress HS-173 algorithmsIn this pnas.1408988111 subsection, the solutions obtained by the Genetic algorithm developed in this paper (SG, hereafter) and by the Nash-Genetic algorithm (NG, hereafter) proposed in [14] are discussed.Table 4. Numerical results for the benchmark instances. Best Benchmark 1 Benchmark 2 Benchmark 3 493 1203 1602 Average 498.94 1226.30 1652.89 Worst 502 1282 1738 Gap 1.20 1.94 3.18 Std. Dev. 4.31 18.04 38.60 # Best 27 22 18 Best 54 44 36 Time 4.230 13.951 54.doi:10.1371/journal.pone.0128067.tPLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,14 /GA for the BLANDPThe solutions considered for the SG are the ones presented in subsection 4.1. On the other hand, for obtaining the solutions from the NG we emulated the algorithm described in [14] in order to solve the BLANDP. Next, a brief general description of the NG is shown. Let (y | x) be the string representing the potential solution for a bi-objective optimization problem. Then y denotes the subset of variables controlled by the leader and optimized accordingly to the connection costs. Similarly x denotes the subset of variables controlled by the follower and optimized with respect to the average delay in the network. According to the Nash perspective, the leader optimizes (y | x) with respect to his objective function by modifying y while x is fixed by ?the follower, i.e. the leader will find y (x). Symmetrically, the follower optimizes (y | x) with respect to his objective function by selecting x while y is fixed by the follower, i.e. the follower ?will find x (y). In the same way that in [14], two different populations are considered; the first one is named pop1 corresponds to the assignments y associated with the connection of the user.Sents the best value obtained from the 50 runs of each problem. In the “Average” column the average of the 50 runs is indicated and in the “Worst” column the higher obtained cost is indicated. Then, the “Gap” column is calculated as Gap ?jbest veragej ?100 . The sample standard deviation is prebest sented in the “Std. Dev.” column. The ” Best” and ” Best” columns indicate the total number of times and the percentage of times when the best value was reached, respectively. Finally, the “Time” column indicates the average time (in seconds) for solving 50 times each problem. From Table 4 it can be observed that for Benchmark 1 the algorithm reached the best value in more than half of the 50 runs. Moreover, the average from all the runs is very near from the best obtained value and the standard deviation indicates that the values are around the average; the small gap obtained (1.20 ) confirms the good performance of the developed algorithm in this problem. The consumed average time is 4.23 seconds. The results for Benchmark 2 indicate that despite the expected increase in the computational time (almost 14 seconds), the algorithm reached a very good gap between the best obtained value and the average of the 50 runs. This gap is lower than 2 . Also, the best value was obtained in almost the half of the experimentation (in 22 of the runs). Finally, the numerical experimentation conducted for Benchmark 3 was not as good as the previous ones but the results are still reasonable. The best value was reached in 18 of the 50 runs, while the gap increased to 3.18 and the consumed time was of 54.9 seconds. These results were clearly affected by the difficulty of finding the follower’s rational reaction.A comparison between the Stackelberg-Genetic and fnins.2015.00094 the Nash-Genetic algorithmsIn this pnas.1408988111 subsection, the solutions obtained by the Genetic algorithm developed in this paper (SG, hereafter) and by the Nash-Genetic algorithm (NG, hereafter) proposed in [14] are discussed.Table 4. Numerical results for the benchmark instances. Best Benchmark 1 Benchmark 2 Benchmark 3 493 1203 1602 Average 498.94 1226.30 1652.89 Worst 502 1282 1738 Gap 1.20 1.94 3.18 Std. Dev. 4.31 18.04 38.60 # Best 27 22 18 Best 54 44 36 Time 4.230 13.951 54.doi:10.1371/journal.pone.0128067.tPLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,14 /GA for the BLANDPThe solutions considered for the SG are the ones presented in subsection 4.1. On the other hand, for obtaining the solutions from the NG we emulated the algorithm described in [14] in order to solve the BLANDP. Next, a brief general description of the NG is shown. Let (y | x) be the string representing the potential solution for a bi-objective optimization problem. Then y denotes the subset of variables controlled by the leader and optimized accordingly to the connection costs. Similarly x denotes the subset of variables controlled by the follower and optimized with respect to the average delay in the network. According to the Nash perspective, the leader optimizes (y | x) with respect to his objective function by modifying y while x is fixed by ?the follower, i.e. the leader will find y (x). Symmetrically, the follower optimizes (y | x) with respect to his objective function by selecting x while y is fixed by the follower, i.e. the follower ?will find x (y). In the same way that in [14], two different populations are considered; the first one is named pop1 corresponds to the assignments y associated with the connection of the user.