An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem

The irregular strip packing problem is a combinatorial optimization problem that requires to place a given set of two-dimensional polygons within a rectangular container so that no polygon overlaps with other polygons or protrudes from the container, where each polygon is not necessarily convex. The container has a fixed width, while its length can change so that all polygons are placed in it. The objective is to find a layout of the set of polygons that minimizes the length of the container. We propose an algorithm that separates overlapping polygons based on nonlinear programming, and an algorithm that swaps two polygons in a layout so as to find their new positions in the layout with the least overlap. We incorporate these algorithms as components into an iterated local search algorithm for the overlap minimization problem and then develop an algorithm for the irregular strip packing problem using the iterated local search algorithm. Computational comparisons on representative instances disclose that our algorithm is competitive with other existing algorithms. Moreover, our algorithm updates several best known results.

[1]  V. Milenkovic,et al.  Compaction and separation algorithms for non-convex polygons and their applications☆ , 1995 .

[2]  José Fernando Oliveira,et al.  TOPOS – A new constructive algorithm for nesting problems , 2000, OR Spectr..

[3]  Antonio Albano,et al.  Optimal Allocation of Two-Dimensional Irregular Shapes Using Heuristic Search Methods , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

[4]  Kathryn A. Dowsland,et al.  Hybridising Tabu Search with Optimisation Techniques for Irregular Stock Cutting , 2001, Manag. Sci..

[5]  Mark de Berg,et al.  Computational geometry: algorithms and applications, 3rd Edition , 1997 .

[6]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[7]  Leonidas J. Guibas,et al.  Penetration Depth of Two Convex Polytopes in 3D , 2000, Nord. J. Comput..

[8]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[9]  José Fernando Oliveira,et al.  A 2-exchange heuristic for nesting problems , 2002, Eur. J. Oper. Res..

[10]  Graham Kendall,et al.  A New Bottom-Left-Fill Heuristic Algorithm for the Two-Dimensional Irregular Packing Problem , 2006, Oper. Res..

[11]  G. D. Ramkumar,et al.  An algorithm to compute the Minkowski sum outer-face of two simple polygons , 1996, SCG '96.

[12]  Jens Egeblad,et al.  Fast neighborhood search for two- and three-dimensional nesting problems , 2007, Eur. J. Oper. Res..

[13]  Francis Y. L. Chin,et al.  Finding the Medial Axis of a Simple Polygon in Linear Time , 1995, ISAAC.

[14]  Leonidas J. Guibas,et al.  A linear-time algorithm for computing the voronoi diagram of a convex polygon , 1989, Discret. Comput. Geom..

[15]  E. Hopper,et al.  A Review of the Application of Meta-Heuristic Algorithms to 2D Strip Packing Problems , 2001, Artificial Intelligence Review.

[16]  Dinesh Manocha,et al.  Incremental penetration depth estimation between convex polytopes using dual-space expansion , 2004, IEEE Transactions on Visualization and Computer Graphics.

[17]  José Fernando Oliveira,et al.  Solving Irregular Strip Packing problems by hybridising simulated annealing and linear programming , 2006, Eur. J. Oper. Res..

[18]  Kathryn A. Dowsland,et al.  The irregular cutting-stock problem - a new procedure for deriving the no-fit polygon , 2001, Comput. Oper. Res..

[19]  A. M. Gomes,et al.  A New Constructive Algorithm for Nesting Problems , 2001 .

[20]  David G. Kirkpatrick,et al.  Computing the intersection-depth of polyhedra , 1993, Algorithmica.

[21]  Antonio Albano,et al.  NESTING TWO-DIMENSIONAL SHAPES IN RECTANGULAR MODULES , 1976 .