Complete and robust no-fit polygon generation for the irregular stock cutting problem

The no-fit polygon is a construct that can be used between pairs of shapes for fast and efficient handling of geometry within irregular two-dimensional stock cutting problems. Previously, the no-fit polygon (NFP) has not been widely applied because of the perception that it is difficult to implement and because of the lack of generic approaches that can cope with all problem cases without specific case-by-case handling. This paper introduces a robust orbital method for the creation of no-fit polygons which does not suffer from the typical problem cases found in the other approaches from the literature. Furthermore, the algorithm only involves two simple geometric stages so it is easily understood and implemented. We demonstrate how the approach handles known degenerate cases such as holes, interlocking concavities and jigsaw type pieces and we give generation times for 32 irregular packing benchmark problems from the literature, including real world datasets, to allow further comparison with existing and future approaches.

[1]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[2]  K. A. Dowsland,et al.  Jostling for position: local improvement for irregular cutting patterns , 1998, J. Oper. Res. Soc..

[3]  B. Chazelle,et al.  Optimal Convex Decompositions , 1985 .

[4]  William B. Dowsland.  On a Research Bibliography for Cutting and Packing Problems , 1992 .

[5]  G. Scheithauer,et al.  Modeling of packing problems , 1993 .

[6]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[7]  Yu. G. Stoyan,et al.  $\Phi $-functions for complex 2D-objects , 2004, 4OR.

[8]  Joseph O'Rourke,et al.  Computational geometry in C (2nd ed.) , 1998 .

[9]  Joseph ORourke,et al.  Computational Geometry in C Second Edition , 1998 .

[10]  Kamineni Pitcheswara Rao,et al.  Quick and precise clustering of arbitrarily shaped flat patterns based on stringy effect , 1997 .

[11]  Kurt Mehlhorn,et al.  Fast Triangulation of Simple Polygons , 1983, FCT.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

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

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

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

[16]  Nina Amenta,et al.  Computational geometry software , 1997 .

[17]  Pijush K. Ghosh,et al.  An algebra of polygons through the notion of negative shapes , 1991, CVGIP Image Underst..

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

[19]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[20]  Victor J. Milenkovic,et al.  Rotational polygon containment and minimum enclosure using only robust 2D constructions , 1999, Comput. Geom..

[21]  Richard Carl Art.  An approach to the two dimensional irregular cutting stock problem. , 1966 .

[22]  Ray Cuninghame-Green.  Cut out waste! , 1992 .

[23]  Kathryn A. Dowsland,et al.  An algorithm for polygon placement using a bottom-left strategy , 2002, Eur. J. Oper. Res..

[24]  Jean-Daniel Boissonnat,et al.  Simultaneous containment of several polygons , 1987, SCG '87.

[25]  K. Dowsland,et al.  A tabu thresholding implementation for the irregular stock cutting problem , 1999 .

[26]  Paul E. Sweeney,et al.  Cutting and Packing Problems: A Categorized, Application-Orientated Research Bibliography , 1992 .

[27]  Harald Dyckhoff,et al.  A typology of cutting and packing problems , 1990 .

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

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

[30]  Jean-Daniel Boissonnat,et al.  Polygon Placement Under Translation and Rotation , 1988, RAIRO Theor. Informatics Appl..

[31]  Yu. G. Stoyan,et al.  Phi-functions for primary 2D-objects , 2002, Stud. Inform. Univ..

[32]  Pijush K. Ghosh,et al.  A unified computational framework for Minkowski operations , 1993, Comput. Graph..

[33]  Raimund Seidel,et al.  A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons , 1991, Comput. Geom..

[34]  T. M. Cavalier,et al.  A new algorithm for the minimal-area convex enclosure problem , 1995 .

[35]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[36]  Pankaj K. Agarwal,et al.  Polygon decomposition for efficient construction of Minkowski sums , 2000, Comput. Geom..