Single-machine scheduling with inserted idle time to minimise a weighted quadratic function of job lateness

This study is concerned with the problem of finding an optimal sequence that minimises the weighted sum of a quadratic function of job lateness on a single machine where an idle time is allowed to be inserted only before the processing of the first job begins. We first prove that the solution method of Sen et al. (1995) for the problem does not work because their ordering rule for adjacent jobs is not applicable to nonadjacent jobs. We then develop an exact algorithm for this Non-deterministic Polynomial-time hard (NP-hard) problem based on a precedence relation structure among adjacent jobs. The algorithm considers the lateness cost of an incumbent sequence as the initial upper bound, and incorporates some improved lower bounds and a number of effective branching rules into a Branch-and-Bound (B&B) algorithm to discard dominated branches. Our computational results demonstrate that the algorithm can solve large problem instances quickly. [Submitted 06 December 2008; Revised 02 February 2009; Accepted 07 February 2009]