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Ant Colony Optimization (ACO) is a metaheuristic that has recently been applied to scheduling problems. We propose an ACO algorithm for the Single Machine Total Weighted Tardiness Problem and compare it to an existing ACO algorithm for the unweighted problem. The proposed algorithm has some novel properties that are of general interest for ACO optimization. A main novelty is that the ants are guided on their way through the decision space by global pheromone information instead of using only local pheromone information. It is also shown that the ACO optimization behaviour can be improved when priority scheduling heuristics are adapted so that they appropriately reflect absolute quality differences between the alternatives before they are used by the ants. Further improvements can be obtained by identifying situations where the ants can perform optimal decisions.