Algorithms for Special Single Machine Total Tardiness Problems

The scheduling problem of minimizing total tardiness on a single machine is known to be NP -hard in the ordinary sense. In this paper, we consider the special case of the problem when the processing times pj and the due dates dj of the jobs j, j ∈ N = {1, 2, . . . , n}, are oppositely ordered: p1 ≥ p2 ≥ . . . ≥ pn and d1 ≤ d2 ≤ . . . ≤ dn. It is shown that already this special case is NP -hard in the ordinary sense, too. The set of jobs N is partitioned into k, 1 ≤ k ≤ n, subsets M1,M2, . . . ,Mk, Mν ⋂Mμ = ∅ for ν = μ, N = M1 ⋃M2 ⋃ . . .⋃Mk, such that maxi,j∈Mν |di − dj | ≤ minj∈Mν pj for each ν = 1, 2, . . . ,k. We propose algorithms which solve the problem: in O(kn ∑ pj) time if 1 ≤ k < n; in O(n2) time if k = n; and in O(n2) time if maxi,j∈N |di − dj| ≤ 1. The polynomial algorithms do neither require the conditions p1 ≥ p2 ≥ . . . ≥ pn mentioned above nor integer processing times to construct an optimal schedule. Finally, we apply the idea of the presented algorithm for the case k = 1 to the even-odd partition problem.