Models and Bounds for Two-Dimensional Level Packing Problems

We consider two-dimensional bin packing and strip packing problems where the items have to be packed by levels. We introduce new mathematical models involving a polynomial number of variables and constraints, and show that their LP relaxations dominate the standard area relaxations. We then propose new (combinatorial) bounds that can be computed in O(nlog n) time. We show that they dominate the other bounds, and establish their absolute worst-case behavior. The quality of models and bounds is evaluated through extensive computational experiments.

[1]  Paolo Toth,et al.  Lower bounds and reduction procedures for the bin packing problem , 1990, Discret. Appl. Math..

[2]  Robert E. Tarjan,et al.  Performance Bounds for Level-Oriented Two-Dimensional Packing Algorithms , 1980, SIAM J. Comput..

[3]  J. B. G. Frenk,et al.  Hybrid next-fit algorithm for the two-dimensional rectangle bin-packing problem , 2006, Computing.

[4]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[5]  Zhenyu Li,et al.  Multiple Containment Methods , 1994 .

[6]  An LP-based approach to a two-stage cutting stock problem , 1995 .

[7]  Daniele Vigo,et al.  Recent advances on two-dimensional bin packing problems , 2002, Discret. Appl. Math..

[8]  R. Gomory,et al.  Multistage Cutting Stock Problems of Two and More Dimensions , 1965 .

[9]  Gerhard Wäscher,et al.  Cutting and packing , 1995, Eur. J. Oper. Res..

[10]  Hidetoshi Onodera,et al.  Branch-and-bound placement for building block layout , 1991, 28th ACM/IEEE Design Automation Conference.

[11]  D. S. Johnson,et al.  On Packing Two-Dimensional Bins , 1982 .

[12]  François Vanderbeck,et al.  A Nested Decomposition Approach to a Three-Stage, Two-Dimensional Cutting-Stock Problem , 2001, Manag. Sci..

[13]  Nicos Christofides,et al.  An exact algorithm for orthogonal 2-D cutting problems using guillotine cuts , 1995 .

[14]  J. M. Valério de Carvalho,et al.  A Computer Based Interactive Approach To A Two-Stage Cutting Stock Problem , 1994 .

[15]  Daniele Vigo,et al.  Heuristic and Metaheuristic Approaches for a Class of Two-Dimensional Bin Packing Problems , 1999, INFORMS J. Comput..

[16]  Nicos Christofides,et al.  An exact algorithm for general, orthogonal, two-dimensional knapsack problems , 1995 .

[17]  J. O. Berkey,et al.  Two-Dimensional Finite Bin-Packing Algorithms , 1987 .

[18]  S. Martello,et al.  Exact Solution of the Two-Dimensional Finite Bon Packing Problem , 1998 .

[19]  John E. Beasley,et al.  An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure , 1985, Oper. Res..

[20]  Chin-Sheng Chen,et al.  An analytical model for the container loading problem , 1995 .

[21]  Brenda S. Baker,et al.  Shelf Algorithms for Two-Dimensional Packing Problems , 1983, SIAM J. Comput..