Improving branch and bound for Jobshop scheduling with constraint propagation

Abstract

Task intervals were defined in [CL94] for disjunctive scheduling so that, in a scheduling problem, one could derive much information by focusing on some key subsets of tasks. The advantage of this approach was to shorten the size of search trees for branch&bound algorithms because more propagation was performed at each node.

In this paper, we refine the propagation scheme and describe in detail the branch&bound algorithm with its heuristics and we compare constraint programming to integer programming. This algorithm is tested on the standard benchmarks from Muth & Thompson, Lawrence, Adams et al, Applegate & Cook and Nakano & Yamada. The achievements are the following:

• Window reduction by propagation: for 23 of the 40 problems of Lawrence, the proof of optimality is found with no search, by sole propagation; for typically hard 10×10 problems, the search tree has less than a thousand nodes; hard problems with up to 400 tasks can be solved to optimality and among these, the open problem LA21 is solved within a day.

• Lower bounds very quick to compute and which outperform by far lower bounds given by cutting planes. The lower bound to the open 20×20 problem YAM1 is improved from 812 to 826