B. Golden and W. Stewart, "Empiric analysis of heuristics," in The traveling salesman problem, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy-Kan, D.B. Shmoys (Eds.), Wiley and Sons: New York, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....probability 0 0.01 0.02 0.03 0.04 0.05 0.06 0102030405060708090100 Cooperating ants Noncooperating ants Fig. 7. Cooperation changes the probability distribution of first finishing times: cooperating ants have a higher probability to find quickly an optimal solution. Test problem: CCAO [21]. The number of ants was set to m=4. Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: CCAO [21] Average on 25 runs. The ....

....solution. Test problem: CCAO [21] The number of ants was set to m=4. Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: CCAO [21]. Average on 25 runs. The number of ants was set to m=4. D. The importance of the pheromone and the heuristic function Experimental results have shown that the heuristic function is fundamental in making the algorithm find good solutions in a reasonable time. In fact, when =0 ACS performance ....

B. Golden and W. Stewart, "Empiric analysis of heuristics," in The traveling salesman problem, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy-Kan, D.B. Shmoys (Eds.), Wiley and Sons: New York, 1985.

....of probability 0 0.01 0.02 0.03 0.04 0.05 0.06 0 10 20 30 40 50 60 70 80 90 100 Cooperating ants Noncooperating ants Fig. 7. Cooperation changes the probability distribution of first finishing times: cooperating ants have a higher probability to find quickly an optimal solution. Test problem: CCAO [21]. The number of ants was set to m=4. Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: CCAO [21] Average on 25 runs. The number of ants ....

....an optimal solution. Test problem: CCAO [21] The number of ants was set to m=4. Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: CCAO [21]. Average on 25 runs. The number of ants was set to m=4. D. The importance of the pheromone and the heuristic function Experimental results have shown that the heuristic function h is fundamental in making the algorithm find good solutions in a reasonable time. In fact, when b=0 ACS performance ....

B. Golden and W. Stewart, "Empiric analysis of heuristics," in The traveling salesman problem, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy-Kan, D.B. Shmoys (Eds.), Wiley and Sons: New York, 1985.

....we informally discuss why and how the AS paradigm functions. Conclusions are in Section IX. II. The Ant System In this section we introduce the AS. We decided to use the well known traveling salesman problem [26] as benchmark, in order to make the comparison with other heuristic approaches easier [20]. Although the model definition is influenced by the problem structure, we will show in Section VII that the same approach can be used to solve other optimization problems. Given a set of n towns, the TSP can be stated as the problem of finding a minimal length closed tour that visits each town ....

....the average node branching of the problem s graph (Oliver30) Typical run. IEEE Transactions on Systems, Man, and Cybernetics Part B, Vol.26, No.1, 1996, pp.1 13 11 The same process can be observed in the graphs of Fig. 6, where the AS was applied to a very simple 10 cities problem (CCA0, from [20]) and which depict the effect of ant search on the trail distribution. In the figure the length of the edges is proportional to the distances between the towns; the thickness of the edges is proportional to their trail level. Initially (Fig. 6a) trail is uniformly distributed on every edge, and ....

[Article contains additional citation context not shown here]

B.Golden, W.Stewart, "Empiric analysis of heuristics," in The Travelling Salesman Problem, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy-Kan, D. B. Shmoys eds., New York:Wiley, 1985.

....we informally discuss why and how the AS paradigm functions. Conclusions are in Section IX. II. The Ant System In this section we introduce the AS. We decided to use the well known traveling salesman problem [26] as benchmark, in order to make the comparison with other heuristic approaches easier [20]. Although the model definition is influenced by the problem structure, we will show in Section VII that the same approach can be used to solve other optimization problems. 6 Given a set of n towns, the TSP can be stated as the problem of finding a minimal length closed tour that visits each ....

....30 Average node branching 0 1000 500 1500 2000 2500 3000 Cycles 13 Fig. 5. Evolution of the average node branching of the problem s graph (Oliver30) Typical run. The same process can be observed in the graphs of Fig. 6, where the AS was applied to a very simple 10cities problem (CCA0, from [20]) and which depict the effect of ants search on the trail distribution. In the figure the length of the edges is proportional to the distances between the towns; the thickness of the edges is proportional to their trail level. Initially (Fig. 6a) trail is uniformly distributed on every edge, and ....

[Article contains additional citation context not shown here]

B.Golden, W.Stewart, "Empiric analysis of heuristics," in The Travelling Salesman Problem, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy-Kan, D. B. Shmoys eds., New York:Wiley, 1985. 31

....of the ant algorithms and then introduce three of them, called Ant density, Ant quantity and Antcycle. To test the ant algorithms, we decided to apply them to the well known travelling salesman problem (TSP) 13] to have a comparison with results obtained by other heuristic approaches [9]: the model definition is influenced by the problem structure, however we will hint in section 6 that the same approach can be used to solve other optimization problems. We stress that the choice of TSP is due to its ubiquity as a benchmark for heuristics: we are interested in the proposal of a ....

B.Golden, W.Stewart, "Empiric analysis of heuristics," in The Travelling Salesman Problem, E.L.Lawler, J.K.Lenstra, A.H.G.Rinnooy-Kan, D.B.Shmoys eds., New York:Wiley, 1985.

....of probability 0 0.01 0.02 0.03 0.04 0.05 0.06 0 10 20 30 40 50 60 70 80 90 100 Cooperating ants Noncooperating ants Fig. 7. Cooperation changes the probability distribution of first finishing times: cooperating ants have a higher probability to find quickly an optimal solution. Test problem: CCAO [21]. The number of ants was set to m=4. Dorigo and Gambardella Ant Colony System 12 26 Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: ....

....number of ants was set to m=4. Dorigo and Gambardella Ant Colony System 12 26 Cpu time (msec) Tour length 49.5 50 50.5 51 51.5 52 52.5 0 100 200 300 400 500 600 700 800 Cooperating ants Noncooperating ants Fig. 8. Cooperating ants find better solutions in a shorter time. Test problem: CCAO [21]. Average on 25 runs. The number of ants was set to m=4. D. The importance of the pheromone and the heuristic function Experimental results have shown that the heuristic function h is fundamental in making the algorithm find good solutions in a reasonable time. In fact, when b=0 ACS performance ....

B. Golden and W. Stewart, "Empiric analysis of heuristics," in The traveling salesman problem, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy-Kan, D.B. Shmoys (Eds.), New York: Wiley and Sons, 1985.

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