Scheduling Projects with Labour Constraints

In this paper we consider a labour constrained scheduling problem (LCSP) which is a simpliication of a practical problem arising in industry. Jobs are subject to precedence constraints and have speciied processing times. Moreover, for each job the labour requirement varies as the job is processed. Given the amount of labour available in each period, the problem is to nish all the jobs as soon as possible, that is, to minimize makespan, subject to the precedence and labour constraints. Several Integer Programming (IP) formulations for this problem are discussed and valid inequalities for these diierent models are introduced. It turns out that a major drawback in using the IP approach is the weakness of the lower bound relaxations. However, we report computational experiments showing how the solution of the linear relaxation of the IP models can be used to provide good schedules. Solutions arising from these LP-based heuristics are considerably improved by local search procedures. We further exploit the capabilities of local search for LCSP by designing 1 1 a Tabu Search algorithm. The computational experiments on a benchmark data set show that the Tabu algorithm generates the best known upper bounds for almost all these instances. We also show how IP can be used to provide reasonably good lower bounds for LCSP when the makespan is replaced by suitably modiied objective functions. Finally some directions for further investigations which may turn IP techniques into a more interesting tool for solving such a problem are suggested.