Single Machine Scheduling with Discrete Earliness and Tardiness

Abstract

This paper considers the problem of scheduling a given set of jobs on a single machine in order to minimize the total weighted earliness and tardiness costs. The scheduling horizon is divided into elementary periods; jobs have due-dates at the end of these periods. All jobs are assumed initially available. Jobs have unique (weighted) early and tardy staircase penalty functions. No preemption of jobs is permitted. and idle time may be inserted. We prove that this problem is NP-complete. Some results relating to job priorities and completion times in an optimal solution are presented. A Mixed Integer Linear Programming (MILP) formulation of this problem is developed. A branch-and-bound scheme that solves the above problem optimally is also presented. Heuristics, derived from simple priority rules, provide an initial upperbound to the search. We develop two lower bound procedures for the remaining jobs to be scheduled at any partial solution state. Numerical results relating to the performance of the branch-and- bound scheme are also presented.