We consider the single-machine problem of scheduling n jobs to minimize the sum of the deviations of the job completion times from a given small common due date. For this NP-hard problem, we develop a branch-and-bound algorithm based on Lagrangian lower and upper bounds that are found in O(n log n) time. We identify conditions under which the bounds concur; these conditions can be expected to be satisfied by many instances with n not too small. In our experiments with processing times drawn from a uniform distribution, the bounds concur for ≥ 40. For the case where the bounds do not concur, we present a refined lower bound that is obtained by solving a subset-sum problem of small dimension to optimality. We further develop a 4/3-approximation algorithm based upon the Lagrangian upper bound.

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ERIM Top-Core Articles
Operations Research
Erasmus Research Institute of Management

Hoogeveen, H., Oosterhout, H., & van de Velde, S. (1994). New lower and upper bounds for scheduling around a small common due date. Operations Research, 102–110. Retrieved from http://hdl.handle.net/1765/12351