The Unary Resource with Transition Times

Abstract

Transition time constraints are ubiquitous in scheduling problems. They are said to be sequence-dependent if their durations depend on both activities between which they take place. In this context, we propose to extend the $\varTheta$-tree and $\varTheta$-$\varLambda$-tree data structures introduced by Vilím in order to strengthen the bound computation of the earliest completion time of a set of activities, by taking into account the sequence dependent transition time constraints. These extended structures can be substituted seamlessly in the state-of-the-art Vilím’s filtering algorithms for unary resource constraints (Overload Checking, Detectable Precedences, Not-First/Not-Last and Edge-Finding algorithms) without changing their $\mathcal {O}(n\log (n))$ time complexities. Furthermore, this new propagation procedure is totally independent from additional constraints or the objective function to optimize. The proposed approach is able to reduce the number of nodes by several order of magnitudes on some instances of the job-shop with transition times problem, without introducing too much overhead on other instances for which it is less effective.