An Optimal Constraint Programming Approach to the Open-Shop Problem

This is a summary of the journal article (Malapert et al. 2012) published by Journal on Computing entitled “An Optimal Constraint Programming Approach to the Open-Shop Problem”. The article presents an optimal constraint programming approach for the Open-Shop scheduling problem, which integrates recent constraint propagation and branching techniques with new upper bound heuristics. Randomized restart policies combined with nogood recording allow to search diversification and learning from restarts. This approach is compared with the best-known metaheuristics and exact algorithms, and shows better results on a wide range of benchmark instances. Open-Shop problems are at the core of many scheduling problems involving unary resources such as Job-Shop or Flow-Shop problems, which have received an important amount of attention because of their wide range of applications. Among the many techniques proposed in the literature, Constraint Programming (CP) is among the most successful. In shop problems, n jobs, consisting each of m tasks, must be processed on m machines. A machine can process only one task at a time. The processing orders of tasks which belong to a job can vary: global order (flowshop); order per job (job-shop); no order (open-shop). In Open-Shop problems, the tasks of a job can be processed in any order, but only one at a time. The processing times are known in advance and constant. We consider the construction of non-preemptive schedules of minimal makespan Cmax which is NP-Hard for m ≥ 3. The study and classification of models and search algorithms show that one of the major challenge of solving optimally these problems is to provide good solutions as quick as the metaheuristics. Copyright c © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. The most famous exact method (Brucker et al. 1997) consists of fixing precedences on the critical path of heuristic solutions computed at each node. It has been improved since by using intelligent backtracking (Guéret, Jussien, and Prins 2000) and consistency techniques (Dorndorf, Pesch, and Huy 2001). More recently, (Laborie 2005) applied a method for cumulative scheduling on Open-Shop problems and (Tamura et al. 2006) applied a method that encodes Constraint Satisfaction/Optimization problems with integer linear constraints into a Boolean Satisfiability Testing problem. Many metaheuristics algorithms have been developed in the last decade to solve the Open-Shop problem. The most recent and successful metaheuristics are: Genetic Algorithm (Prins 2000), Construction and Repair (Chatzikokolakis, Boukeas, and Stamatopoulos 2004), Ant Colony Optimization (Blum 2005) and Particle Swarm Optimization (Sha and Hsu 2008). Constraint programming techniques have been widely used to solve scheduling problems. A Constraint Satisfaction Problem (CSP) consists of a set V of variables defined by a corresponding set of possible values (the domains D) and a set C of constraints. A solution of the problem is an assignment of a value to each variable such that all constraints are simultaneously satisfied. Constraints are handled through a propagation mechanism which allows the reduction of the domains of variables and the pruning of the search tree. The propagation mechanism coupled with a backtracking scheme allows the search space to be explored in a complete way. Scheduling is probably one of the most successful areas for CP thanks to specialized global constraints, which allow modelling resource limitations and temporal constraints. Constraint programming models in scheduling usually represent a non-preemptive task by a triplet of non-negative integer variables representing the start, processing time and end of the task. We now present our constraint programming model to tackle Open-Shop problems. First, we state disjunctive global constraints which model the fact that a single machine or job Proceedings of the Twenty-Third International Conference on Automated Planning and Scheduling