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We propose an approximation algorithm for scheduling jobs with release and delivery times on a single machine with the objective to minimize the makespan. The algorithm is based on an implicit enumeration of the set of complete solutions in a search tree. By analyzing specific structural properties of the solutions created in each branch of the solution tree, a certain approximation factor of each solution from that branch is calculated. Our algorithm guarantees approximation factor 1+1/κ for a generated solution if there are κ jobs with a specified property in that solution (typically, the longer the length of the path from the root to the node representing that solution in the solution tree, the larger the κ value). We have carried out an extensive computational study to verify the practical performance of our algorithm and the effectiveness of the approximation factor provided by us. While our problem instances are generated randomly, we have discarded a considerable number of the instances, ones which were already solved optimally by the earlier known dominance rules. For the vast majority of the tested problem instances, within a short running time of our algorithm parameter κ becomes sufficiently large so that the approximation factor which the algorithm guarantees becomes better than that provided by the earlier known approximation algorithms.

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In this paper a coarse-grain execution model for evolutionary algorithms is proposed and used for solving numerical and combinatorial optimization problems. This model does not use migration as the solution dispersion mechanism, in its place a more efficient population-merging mechanism is used that dynamically reduces the population size as well as the total number of parallel evolving populations. Even more relevant is the fact that the proposed model incorporates an entropy measure to determine how to merge the populations such that no valuable information is lost during the evolutionary process. Extensive experimentation, using genetic algorithms over a well-known set of classical problems, shows the proposed model to be faster and more accurate than the traditional one.