Combining AI/OR techniques for solving Open-Shop problems

In the Open-Shop problem, a set J of n jobs, consisting each of m operations (or tasks) must be processed on a set M of m machines. The operations of a job can be processed in any order, but only one at a time. We consider the construction of non-preemptive schedules of minimal makespan, which is NPHard for m 3 [Gonzalez and Sahni, 1976]. In the following, the processing time of a task i is denoted by pi, its head (least feasible starting time) by ri and its tail (symmetric notion from the end of the scheduling) by qi. Few branch and bound methods for that problem have been published so far. One of the best [Brucker et al., 1997] consists, in each node, in xing disjunctions (adding new tentative precedence constraints) on the critical path of a heuristic solution. This branch and bound method uses immediate selections [Carlier and Pinson, 1989] in each node in order to x additional disjunctions by propagation. Although that method is the best one as far as we know, some problems from size 7 7 remain unsolved. In this paper, we present the integration of two techniques in that branch and bound method in order to improve the search. The rst one is an AI intelligent backtracking technique. The second one is a new propagation rule developed thanks to OR algorithms. We tested those two techniques on OpenShop problems from the literature. The search is de nitely improved: on some square problems of size 10, the number of backtracks is reduced by more than 90%.