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the best path [216] The pheromone update equation changes to
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e ij (t + 1) = ij (t) + ij (t) + ne ij (t)
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(1715)
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(1716)
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and e is the number of elite ants In equation (1716), x(t) is the current best route, with f ( (t)) = mink=1,,nk {f (xk (t))} The elitist strategy has as its objective directx ing the search of all ants to construct a solution to contain links of the current best route
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The ant colony system (ACS) was developed by Gambardella and Dorigo to improve the performance of AS [77, 215, 301] ACS di ers from AS in four aspects: (1) a di erent transition rule is used, (2) a di erent pheromone update rule is de ned, (3) local pheromone updates are introduced, and (4) candidate lists are used to favor speci c nodes Each of these modi cations is discussed next The ACS transition rule, also referred to as a pseudo-random-proportional action rule [301], was developed to explicitly balance the exploration and exploitation abilities of the algorithm Ant k, currently located at node i, selects the next node j to move to using the rule, j=
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arg maxu Nik (t) { iu (t) iu (t)} if r r0 J if r > r0
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(1717)
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where r U (0, 1), and r0 [0, 1] is a user-speci ed parameter; J Nik (t) is a node randomly selected according to probability pk (t) = iJ Nik (t) is a set of valid nodes to visit The transition rule in equation (1717) creates a bias towards nodes connected by short links and with a large amount of pheromone The parameter r0 is used to balance exploration and exploitation: if r r0 , the algorithm exploits by favoring the best edge; if r > r0 , the algorithm explores Therefore, the smaller the value of r0 , the less best links are exploited, while exploration is emphasized more It is important to note that the transition rule is the same as that of AS when r > r0 Also note that the ACS transition rule uses = 1, and is therefore omitted from equation (1718) Unlike AS, only the globally best ant (eg the ant that constructed the shortest path, x+ (t)) is allowed to reinforce pheromone concentrations on the links of the corresponding best path Pheromone is updated using the global update rule, ij (t + 1) = (1 1 ) ij (t) + 1 ij (t) (1719)
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(1718)
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171 Ant Colony Optimization Meta-Heuristic where ij (t) =
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(1720)
with f (x+ )(t) = |x+ (t)|, in the case of nding shortest paths The ACS global update rule causes the search to be more directed, by encouraging ants to search in the vicinity of the best solution found thus far This strategy favors exploitation, and is applied after all ants have constructed a solution Gambardella and Dorigo [215, 301] implemented two methods of selecting the path, x+ (t), namely iteration-best, where x+ (t) represents the best path found during the current iteration, t, denoted as x(t), and global-best, where x+ (t) represents the best path found from the rst iteration of the algorithm, denoted as x(t) For the global-best strategy, the search process exploits more by using more global information Pheromone evaporation is also treated slightly di erently to that of AS Referring to equation (1719), for small values of 1 , the existing pheromone concentrations on links evaporate slowly, while the in uence of the best route is dampened On the other hand, for large values of 1 , previous pheromone deposits evaporate rapidly, but the in uence of the best path is emphasized The e ect of large 1 is that previous experience is neglected in favor of more recent experiences Exploration is emphasized While the value of 1 is usually xed, a strategy where 1 is adjusted dynamically from large to small values will favor exploration in the initial iterations of the search, while focusing on exploiting the best found paths in the later iterations In addition to the global updating rule, ACS uses the local updating rule, ij (t) = (1 2 ) ij (t) + 2 0 (1721)
with 2 also in (0, 1), and 0 is a small positive constant Experimental results on di erent TSPs showed that 0 = (nG L) 1 provided good results [215]; nG is the number of nodes in graph G, and L is the length of a tour produced by a nearestneighbor heuristic for TSPs [737] (L can be any rough approximation to the optimal tour length [215]) ACS also rede nes the meaning of the neighborhood set from which next nodes are selected The set of nodes, Nik (t), is organized to contain a list of candidate nodes These candidate nodes are preferred nodes, to be visited rst Let nl < |Nik (t)| denote the number of nodes in the candidate list The nl nodes closest (in distance or cost) to node i are included in the candidate list and ordered by increasing distance When a next node is selected, the best node in the candidate list is selected If the candidate list is empty, then node j is selected from the remainder of Nik (t) Selection of a non-candidate node can be based on equation (1718), or alternatively the closest non-candidate j Nik (t) can be selected