On not-first/not-last conditions in disjunctive scheduling

This paper is concerned with the development of constraint propagation techniques for the characterization of feasible solutions in disjunctive scheduling. In disjunctive scheduling, a set of uninterruptible tasks is to be performed on a set of resources. Each task has a release date, a deadline, and a fixed processing time; each resource can handle only one task at a time. Some of these propagation techniques are implemented by rules that deduce either mandatory or forbidden sequences between tasks or sets of tasks. For instance, certain rules indicate whether a given task must or cannot be performed before or after a set of other competing tasks. We focus our attention on the latter problem, known as the ‘Not-First/Not-Last’ (NF/NL) problem. The genericity of propagation rules is a question of major importance. It induces that the result of the overall propagation must not depend on the order in which the inference rules are applied. Hence, one must search for completeness in the time-windows narrowing, in order to ensure the convergence of the propagation towards a unique fix-point. An efficient algorithm is proposed. It guarantees the completeness of time-windows narrowing due to NF/NL conditions. It has been integrated in a branch and bound procedure to solve job-shop instances. It has also been tested within several lower bounding procedures. Computational results are reported and the power and complementarity of NF/NL rules with other classical inference rules are discussed.