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.


Nesting Constraint programming Global constraints 


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  1. 1.
    Dowsland, K., Dowsland, W.: Packing problems. European Journal of Operational Research 56, 2–14 (1992)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dowsland, K., Dowsland, W.: Solution approaches to irregular nesting problems. European Journal of Operational Research 84, 506–521 (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    Dowsland, K., Dowsland, W., Bennell, J.: Jostling for position: Local improvement for irregular cutting patterns. Journal of the Operational Research Society 49, 647–658 (1998)zbMATHGoogle Scholar
  4. 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
  5. 5.
    Stoyan, Y., Yaskov, G.: Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints. International Transactions on Operational Research 5, 45–57 (1998)zbMATHGoogle Scholar
  6. 6.
    Milenkovic, V., Daniels, K.: Translational polygon containment and minimal enclosure using mathematical programming. International Transactions in Operational Research 6, 525–554 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bennell, J.A., Dowsland, K.A.: Hybridising tabu search with optimization techniques for irregular stock cutting. Management Science 47, 1160–1172 (2001)CrossRefGoogle Scholar
  8. 8.
    Gomes, A.M., Oliveira, J.F.: A 2-exchange heuristic for nesting problems. European Jornal of Operational Research 141, 359–370 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 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. 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
  11. 11.
    Carravilla, M.A., Ribeiro, C., Oliveira, J.F.: Solving nesting problems with nonconvex polygons by constraint logic programming. International Transactions in Operational Research 10, 651–663 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Milano, M., Ottosson, G., Refalo, P., Thorsteinsson, E.S.: The Role of Integer Programming Techniques in Constraint-Programming’s Global Constraints. INFORMS Journal on Computing 14, 387–402 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Swedish Institute of Computer Science: SICStus Prolog User’s Manual (1995) Google Scholar
  14. 14.
    Carlsson, M., Ottosson, G., Carlson, B.: An Open-Ended Finite Domain Constraint Solver. In: Glaser, H., Hartel, P., Kucken, H. (eds.) PLILP 1997. LNCS, vol. 1292, pp. 191–206. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Ottosson, G., Thorsteinsson, E.S., Hooker, J.N.: Mixed Global Constraints and Inference in Hybrid CLP-IP Solvers. Annals of Mathematics and Artificial Intelligence 34, 271–290 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Bockmayr, A., Kasper, T.: Branch-and-Infer: A Unifying Framework for Integer and Finite Domain Constraint Programming. INFORMS Journal on Computing 10, 287–300 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Art, R.: An Approach to the Two-Dimensional, Irregular Cutting Stock Problem. Technical Report 36.008, IBM Cambridge Centre (1966)Google Scholar
  18. 18.
    Mahadevan, A.: Optimization in Computer-Aided Pattern Packing. PhD thesis, North Carolina State University (1984)Google Scholar
  19. 19.
    Fernandéz, J., Cánovas, L., Pelegrín, B.: Algorithms for the decomposition of a polygon into convex polygons. European Journal of Operational Research 121, 330–342 (2000)CrossRefzbMATHGoogle Scholar
  20. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Cristina Ribeiro
    • 1
    • 2
  • Maria Antónia Carravilla
    • 1
    • 2
  1. 1.FEUP — Faculdade de Engenharia da Universidade do Porto 
  2. 2.INESC — PortoPortoPortugal

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