A Global Constraint for Nesting Problems
Nesting problems are particularly hard combinatorial problems. They involve the positioning of a set of small arbitrarily-shaped pieces on a large stretch of material, without overlapping them. The problem constraints are bidimensional in nature and have to be imposed on each pair of pieces. This all-to-all pattern results in a quadratic number of constraints.
Constraint programming has been proven applicable to this category of problems, particularly in what concerns exploring them to optimality. But it is not easy to get effective propagation of the bidimensional constraints represented via finite-domain variables. It is also not easy to achieve incrementality in the search for an improved solution: an available bound on the solution is not effective until very late in the positioning process.
In the sequel of work on positioning non-convex polygonal pieces using a CLP model, this work is aimed at improving the expressiveness of constraints for this kind of problems and the effectiveness of their resolution using global constraints.
A global constraint “outside” for the non-overlapping constraints at the core of nesting problems has been developed using the constraint programming interface provided by Sicstus Prolog. The global constraint has been applied together with a specialized backtracking mechanism to the resolution of instances of the problem where optimization by Integer Programming techniques is not considered viable.
The use of a global constraint for nesting problems is also regarded as a first step in the direction of integrating Integer Programming techniques within a Constraint Programming model.
KeywordsNesting Constraint programming Global constraints
Unable to display preview. Download preview PDF.
- 4.Błażewicz, J., Hawryluk, P., Walkowiak, R.: Using tabu search approach for solving the two-dimensional irregular cutting problem in tabu search. In: Glover, F., Laguna, M., Taillard, E. (eds.) Tabu Search. Annals of Operations Research. J.C. Baltzer AG, vol. 41 (1993)Google Scholar
- 9.Ribeiro, C., Carravilla, M.A., Oliveira, J.F.: Applying constraint logic programming to the resolution of nesting problems. In: Workshop on Integration of AI and OR techniques in Constraint Programming for Combinatorial Optimization Problems (1999)Google Scholar
- 10.Ribeiro, C., Carravilla, M.A., Oliveira, J.F.: Applying constraint logic programming to the resolution of nesting problems. Pesquisa Operacional 19, 239–247 (1999)Google Scholar
- 13.Swedish Institute of Computer Science: SICStus Prolog User’s Manual (1995) Google Scholar
- 17.Art, R.: An Approach to the Two-Dimensional, Irregular Cutting Stock Problem. Technical Report 36.008, IBM Cambridge Centre (1966)Google Scholar
- 18.Mahadevan, A.: Optimization in Computer-Aided Pattern Packing. PhD thesis, North Carolina State University (1984)Google Scholar
- 20.Beldiceanu, N., Carlsson, M.: Sweep as a Generic Pruning Technique Applied to the Non-Overlapping Rectangles Constraint. In Walsh, T., ed.: CP’2001, Int. Conf. on Principles and Practice of Constraint Programming. Volume 2239 of Lecture Notes in Computer Science., Pisa, Springer-Verlag (2001)Google Scholar