Improving search for job-shop scheduling with CLP(FD)

Abstract

Constraint logic programming can be effectively applied to solve realistic job-shop scheduling problems. The role of constraints is twofold: they model dependencies among tasks and resources (e.g. temporal relations and capacities of machines), and they are used to actively prune the search space during the computation of a schedule. Since the job-shop problem is N P-complete, constraint solving techniques alone do not suffice to get efficient schedules for problems with 100 tasks and more. In order to judge a scheduling method, one has to investigate two questions: how good are the solutions in comparison to the optimum and how much search is required to find them.

This paper reports on achievable improvements with respect to both aspects by applying three methods: an efficient encoding of capacity constraints, a new semi-dynamic variable selection heuristics, and an algorithm for enforcing global capacity constraints. The first method transfers choices inside capacity constraints entirely to the disposition of the solver, which optimally supports the available pruning capabilities. The second method orders variables in a way that prefers tasks that must be placed early or that occur in predicted machine bottlenecks. The third method detects inconsistencies at a early stage of search and is also able to enforce partial orderings among tasks.

Tests on randomly generated problems have shown that the combination of these methods yields an average approximation of the optimum of 7–13% within a few hundred search steps, whereas without them the approximation had been worse than 120%.

This work has been financially supported by the Bundesminister für Forschung und Technologie under grant nr. 01 IW 206. The authors are responsible for their results.