© J.C. Baltzer AG, Science Publishers
Single-machine scheduling to minimize maximum
tardiness with minimum number of tardy jobs
J.N.D. Gupta
a
, A.M.A. Hariri
b
and C.N. Potts
c
a
Department of Management, Ball State University, Muncie, IN 47306, USA
b
Department of Statistics, King Abdul-Aziz University, P.O. Box 9028,
Jeddah 21413, Saudi Arabia
c
Faculty of Mathematical Studies, University of Southampton,
Southampton SO17 1BJ, UK
E-mail: cnp@maths.soton.ac.uk
This paper develops a branch and bound algorithm for solving the single-machine sched-
uling problem with the objective of minimizing the maximum tardiness of any job, subject
to the constraint that the total number of tardy jobs is minimum. The algorithm uses a new
lower bounding scheme, which is based on due date relaxation. Various dominance rules are
used in the algorithm to limit the size of the search tree. Results of extensive computational
tests show that the proposed branch and bound algorithm is effective in solving problems
with up to 1000 jobs.
Keywords: scheduling, single machine, maximum tardiness, minimum number tardy, branch
and bound
AMS subject classification: 90B35
1. Introduction
This paper considers the problem of scheduling n jobs on a single machine, where
the primary objective is the minimization of the total number of tardy jobs and the
secondary objective is one of minimizing the maximum tardiness. Thus, it is required
to find a schedule for which the maximum tardiness is minimized, subject to the
constraint that no reduction in the number of tardy jobs is possible.
The need to consider multiple criteria in scheduling is widely recognized. Either
a hierarchical or a simultaneous approach can be adopted. Under a hierarchical
approach, the criteria are ranked in order of importance; the first criterion is optimized
first, the second criterion is then optimized, subject to achieving the optimum with
respect to the first criterion, and so on. For simultaneous optimization, there are two
approaches. First, all “efficient” schedules can be generated, where an efficient
Annals of Operations Research 92(1999)107–123 107