An Exact Algorithm for the Two-Dimensional Orthogonal Packing Problem with Unloading Constraints

This paper describes an exact algorithm for solving a two-dimensional orthogonal packing problem with unloading constraints, which occurs as a subproblem of mixed vehicle routing and loading problems. The packing considered in this work is basically a feasibility problem involving a single bin. The problem is addressed through a decomposition approach wherein a branch-and-cut algorithm is designed for solving a one-dimensional relaxation of the original problem. When an integer solution is found in the branching tree, a subsidiary problem is solved to identify a two-dimensional packing that does not lead to any overlap and satisfies the unloading constraints. Cuts are added when the subsidiary problem proves to be infeasible. Several preprocessing techniques aimed at reducing the size of the solution space and uncovering infeasibility are also described. A numerical comparison with the best known exact method is reported at the end based on benchmark instances.

[1]  Andreas Bortfeldt,et al.  Container Loading Problems - A State-of-the-Art Review , 2012 .

[2]  Antoine Jouglet,et al.  A new constraint programming approach for the orthogonal packing problem , 2008, Comput. Oper. Res..

[3]  Hiroshi Nagamochi,et al.  Exact algorithms for the two-dimensional strip packing problem with and without rotations , 2009, Eur. J. Oper. Res..

[4]  Alberto Caprara,et al.  Bidimensional packing by bilinear programming , 2005, Math. Program..

[5]  Daniele Vigo,et al.  An Exact Approach to the Strip-Packing Problem , 2003, INFORMS J. Comput..

[6]  G. Belov.  A Branch-and-Price Graph-Theoretical Algorithm for Orthogonal-Packing Feasibility , 2009 .

[7]  Zhenzhen Zhang,et al.  A meta-heuristic algorithm for heterogeneous fleet vehicle routing problems with two-dimensional loading constraints , 2013, Eur. J. Oper. Res..

[8]  Eitan Zemel,et al.  Easily Computable Facets of the Knapsack Polytope , 1989, Math. Oper. Res..

[9]  Ramón Alvarez-Valdés,et al.  A GRASP algorithm for constrained two-dimensional non-guillotine cutting problems , 2005, J. Oper. Res. Soc..

[10]  Mauro Dell'Amico,et al.  Combinatorial Benders' Cuts for the Strip Packing Problem , 2014, Oper. Res..

[11]  Gleb Belov,et al.  LP bounds in various constraint programming approaches for orthogonal packing , 2012, Comput. Oper. Res..

[12]  Jacques Carlier,et al.  A new exact method for the two-dimensional orthogonal packing problem , 2007, Eur. J. Oper. Res..

[13]  Daniele Vigo,et al.  An Exact Approach for the Vehicle Routing Problem with Two-Dimensional Loading Constraints , 2007, Transp. Sci..

[14]  Guenther Fuellerer,et al.  Ant colony optimization for the two-dimensional loading vehicle routing problem , 2009, Comput. Oper. Res..

[15]  Andreas Bortfeldt,et al.  Constraints in container loading - A state-of-the-art review , 2013, Eur. J. Oper. Res..

[16]  Hiroshi Nagamochi,et al.  An exact strip packing algorithm based on canonical forms , 2012, Comput. Oper. Res..

[17]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

[18]  Gleb Belov,et al.  LP Bounds in an Interval-Graph Algorithm for Orthogonal-Packing Feasibility , 2013, Oper. Res..

[19]  Emmanouil E. Zachariadis,et al.  A Guided Tabu Search for the Vehicle Routing Problem with two-dimensional loading constraints , 2009, Eur. J. Oper. Res..

[20]  Gerhard Wäscher,et al.  An improved typology of cutting and packing problems , 2007, Eur. J. Oper. Res..

[21]  Eduardo C. Xavier,et al.  Heuristics for the strip packing problem with unloading constraints , 2013, Comput. Oper. Res..

[22]  Adam N. Letchford,et al.  Separation algorithms for 0-1 knapsack polytopes , 2010, Math. Program..

[23]  Sándor P. Fekete,et al.  An Exact Algorithm for Higher-Dimensional Orthogonal Packing , 2006, Oper. Res..

[24]  Laurence A. Wolsey,et al.  Valid inequalities for 0-1 knapsacks and mips with generalised upper bound constraints , 1990, Discret. Appl. Math..

[25]  Sándor P. Fekete,et al.  A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing , 2003, Math. Oper. Res..

[26]  J. C. Herz,et al.  Recursive computational procedure for two-dimensional stock cutting , 1972 .

[27]  Manuel Iori,et al.  Routing problems with loading constraints , 2010 .

[28]  Michel Gendreau,et al.  A Tabu search heuristic for the vehicle routing problem with two‐dimensional loading constraints , 2008, Networks.

[29]  Ramón Alvarez-Valdés,et al.  A branch and bound algorithm for the strip packing problem , 2009, OR Spectr..

[30]  Matteo Fischetti,et al.  Combinatorial Benders' Cuts for Mixed-Integer Linear Programming , 2006, Oper. Res..

[31]  David Pisinger,et al.  An Adaptive Large Neighborhood Search Heuristic for the Pickup and Delivery Problem with Time Windows , 2006, Transp. Sci..

[32]  Philippe Lacomme,et al.  A multi-start evolutionary local search for the two-dimensional loading capacitated vehicle routing problem , 2011, Comput. Oper. Res..

[33]  Marco A. Boschetti,et al.  An Exact Algorithm for the Two-Dimensional Strip-Packing Problem , 2010, Oper. Res..