The authors 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, they develop a branch-and-bound algorithm based on Lagrangean lower and upper bounds that are found in O(nlogn) time. The authors identify conditions under which the bounds concur; these conditions can be expected to be satisfied by many instances with n not too small. In the present experiments with processing times drawn from a uniform distribution, the bounds concur for n≥40. For the case where the bounds do not concur, the authors present a refined lower bound that is obtained by solving a subset-sum problem of small dimension to optimality. They further develop a 4/3-approximation algorithm based upon the Lagrangean upper bound.
