ij (t) (t) j N k ij i

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if j Nik if j Nik

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(172)

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where Nik is the set of feasible nodes connected to node i, with respect to ant k If, for any node i and ant k, Nik = , then the predecessor to node i is included in Nik Note that this may cause loops to occur within constructed paths These loops are removed once the destination node has been reached In the equation above, is a positive constant used to amplify the in uence of pheromone concentrations Large values of give excessive importance to pheromone,

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17 Ant Algorithms

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especially the initial random pheromones, which may lead to rapid convergence to suboptimal paths Once all ants have constructed a complete path from the origin node to the destination node, and all loops have been removed, each ant retraces its path to the source node deterministically, and deposits a pheromone amount,

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k ij (t)

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1 Lk (t)

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(173)

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to each link, (i, j), of the corresponding path; Lk (t) is the length of the path constructed by ant k at time step t That is, ij (t + 1) = ij (t) +

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k=1 nk k ij (t)

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(174)

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where nk is the number of ants Using equation (173), the total pheromone intensity of a link is proportional to the desirability of the paths in which the link occurs, based on the length of the cork responding path The deposited pheromone, ij , calculated using equation (173), expresses the quality of the corresponding solution For SACO, the quality of a solution (the constructed path) is simply expressed as the inverse of the length of the path in terms of the number of hops in the path Any other measure can be used, for example the cost of traveling on the path, or the physical distance traveled In general, if xk (t) denotes a solution at time step t, then f (xk (t)) expresses the quality of the solution If k is not proportional to the quality of the solution and all ants deposit n 1 2 the same amount of pheromone (ie ij = ij = = ijk ), then it is only the di erential path length e ect that biases path selection towards the shortest path very similar to the observations of Deneubourg et al [199] This discussion leads to the two main forms of solution evaluation employed by ant algorithms, namely implicit evaluation, where ants exploit the di erential path length e ect to bias the search of other agents, and explicit evaluation, where pheromone amounts are proportional to some quality measure of constructed solutions If the amount of pheromone deposited is inversely proportional to the quality of the solution (as is the case in equation (173)), then the larger f (xk (t)) (that is, the worse the constructed solution), the smaller 1/f (xk (t)), hence the less the amount of pheromone deposited on the link Thus, a long path causes all the links of that path to become less desirable as a component of the nal solution This is the case for any quality measure, f , that needs to be minimized Any of a number of termination criteria can be used in Algorithm 172 (and for the rest of the ant algorithms discussed later), for example, terminate when a maximum number of iterations, nt , has been exceeded; terminate when an acceptable solution has been found, with f (xk (t)) ;

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171 Ant Colony Optimization Meta-Heuristic Algorithm 172 Simple ACO Algorithm Initialize ij (0) to small random values; Let t = 0; Place nk ants on the origin node; repeat for each ant k = 1, , nk do //Construct a path xk (t); xk (t) = ; repeat Select next node based on the probability de ned in equation (172); Add link (i, j) to path xk (t); until destination node has been reached; Remove all loops from xk (t); Calculate the path length f (xk (t)); end for each link (i, j) of the graph do //pheromone evaporation; Reduce the pheromone, ij (t), using equation (175); end for each ant k = 1, , nk do for each link (i, j) of xk (t) do 1 k = f (xk (t)) ; Update ij using equation (174); end end t = t + 1; until stopping condition is true; Return the path xk (t) with smallest f (xk (t)) as the solution;

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terminate when all ants (or most of the ants) follow the same path The initial experiments on the binary bridge problem [212] found that ants rapidly converge to a solution, and that little time is spent exploring alternative paths To force ants to explore more, and to prevent premature convergence, pheromone intensities on links are allowed to evaporate at each iteration of the algorithm before being reinforced on the basis of the newly constructed paths For each link, (i, j), let ij (t) (1 ) ij (t) (175)

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with [0, 1] The constant, , speci es the rate at which pheromones evaporate, causing ants to forget previous decisions In other words, controls the in uence of search history For large values of , pheromone evaporates rapidly, while small values of result in slower evaporation rates The more pheromones evaporate, the more random the search becomes, facilitating better exploration For = 1, the search is completely random At this point it is important to emphasize that solution construction is the result of

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