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| An uncertain single-machine scheduling problem is considered where the processing time of a job can take any real value from a given segment. The criterion is to minimize the total weighted completion time of the n jobs a weight being associated with each given job. We use the optimality box as a stability measure of the optimal schedule and derive an O(n)-algorithm for calculating the optimality box for a fixed permutation of the given jobs. We investigate properties of the optimality box using blocks of the jobs. If each job belongs to a single block then the largest optimality box may be constructed in O(n log n) time. For the general case we apply dynamic programming for constructing a job permutation with the largest optimality box. The computational results for finding a permutation with the largest optimality box show that such a permutation is close to an optimal one which can be determined after completing the jobs when their processing times became known. |
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