by Gafarov, E.R., Lazarev, A.A., Werner, F..

**Series:** 2009-38, Preprints

- MSC:
- 90B35 Scheduling theory, deterministic

**Abstract:**

In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized

in contrast to usual minimization problems.

We propose exact algorithms for the single machine problem of maximizing total tardiness $1||\max \sum T_j$ and for the problem of maximizing the number of tardy jobs $1||\max \sum U_j$. In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem $1||\max \sum U_j$ is polynomially solvable. For several special cases of problem $1||\max \sum T_j$, we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.

**Keywords:**

Scheduling; Single machine problems; Maximization problems; Total tardiness; Number of Tardy Jobs

**This paper was published in:**

Automation and Remote Control, Vol. 71, No. 10, 2010, 2070 - 2084 (under the title: Algorithms for Some Maximization Problems on a Single Machine; Russian version in Avtomatika i Telemekhanika, No. 10, 2010, 63 - 79).