Fig 1.
A network example for the presentation of original PMM.
(a) The initial network, (b) The intermediate network evolved by the original PMM, and (c) The final network evolved by the original PMM.
Fig 2.
A network example for illustrating the proposed PMM with multi-pair of inlet/outlet nodes.
(a) The initial network and (b) The final network evolved by the proposed PMM.
Fig 3.
The illustration of working mechanism of iPM-MOACOs.
The food sources and tubes of Physarum network represent cities and paths in a road network, respectively.
Fig 4.
The framework of iPM-MOACOs, which implies our proposed strategies based on the PMM can optimize the initialization of MOACOs.
Table 1.
The algorithm of iPM-MOACOs for solving a bTSP.
Table 2.
Major parameters and their default values used in this paper.
Fig 5.
PFs returned by MOACOs and iPM-MOACOs in four bi-objective symmetric TSP instances.
(a) PACO and iPM-PACO, (b) MACS and iPM-MACS, (c) BIANT and iPM-BIANT. From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. Results show that most of solutions generated by iPM-MOACOs in four instances can dominate the solutions generated by MOACOs, which mean that iPM-MOACOs can obtain better PF than that of MOACOs. Specially, the distribution of solutions generated by iPM-BIANT is better than that of BIANT, as shown in (c).
Fig 6.
M1 metric comparison between MOACOs and iPM-MOACOs in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. Results show that each corresponding M1 values of optimized MOACOs are much lower than those of MOACOs in four instances, which means that solutions generated by the optimized MOACOs are much closer to the pseudo-optimal PFs.
Fig 7.
M2 metric comparison between MOACOs and iPM-MOACOs in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. Results show that the each corresponding M2 values of iPM-MOACOs are more reasonable than those of the corresponding original algorithms.
Fig 8.
M3 metric comparison between MOACOs and iPM-MOACOs in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. According to these results, we know that most of M3 metrics of iPM-MOACOs are better than those of original MOACOs, especially for BIANT.
Table 3.
C metric comparison results between MOACOs and PM-MOACOs.
Fig 9.
PFs returned by PACO, PM-PACO and iPM-PACO in four benchmark instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. The results show that most of solutions generated by iPM-PACO can dominate the solutions generated by PM-PACO and PACO. Since the distributions of solutions generated by PM-PACO are better than that of iPM-PACO, the solutions of PM-PACO are not dominated by solutions of iPM-PACO in the bottom-right regions of the PFs.
Fig 10.
M1 metric comparisons among PACO, PM-PACO and iPM-PACO in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. According to these results, each corresponding M1 values of iPM-PACO is the lowest, meanwhile, each corresponding M1 values of PACO is the highest. Results show that solutions generated by iPM-PACO are the closest to the pseudo-optimal PFs.
Fig 11.
M2 metric comparison between PACO, PM-PACO and iPM-PACO in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. Results show that each corresponding M2 values of PACO is the lowest. Meanwhile, box-plots of PM-PACO are longer than those of iPM-PACO. Meanwhile, the distribution of PACO is the narrowest, and the distribution of iPM-PACO is more stable than PM-PACO.
Fig 12.
M3 metric comparison between PACO, PM-PACO and iPM-PACO in four instances.
From left to right, the instances are euclidAB100, kroAB100, kroAB150 and kroAB200. This figure shows that M3 values of iPM-PACO and PACO are close, and those of PM-PACO are slightly higher. Results show that the extent solutions of three algorithms are approximate.
Table 4.
C metric comparison results among PACO, PM-PACO and iPM-PACO.
Table 5.
HV comparison results between MOACOs and iPM-MOACOs.
Fig 13.
PFs returned by MOACOs and iPM-MOACOs in euclidAC100 instances.
(a) shows that the PF generated by PACO always converges to the top-left regions, while the PF calculated by the optimized PACO converges to bottom-left regions. Meanwhile, (b) performs that most of intersecting points of PF obtained by the optimized BIANT aggregate in the bottom-left regions when compared with those obtained by the original BIANT. (c) displays that the PF returned by the optimized MACS is closed to the PF calculated by the original. The comparisons show that solutions obtained by the optimized algorithms are more reasonable.