Mathematical models and decomposition algorithms for cutting and packing problems

Bin packing and cutting stock problems: Mathematical models and exact algorithms

We review the most important mathematical models and algorithms developed for the exact solution of the one dimensional bin packing and cutting stock problems, and experimentally evaluate, on state-of-the art computers, the performance of the main available software tools.

LP models for bin packing and cutting stock problems

We review several linear programming (LP) formulations for the one-dimensional cutting stock and bin packing problems, namely, the models of Kantorovich, Gilmore–Gomory, onecut models, as in the Dyckhoff–Stadtler approach, position-indexed models, and a model derived from the vehicle routing literature. We analyse some relations between the corresponding LP relaxations, and their relative stren...

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: Unbounded Knapsack, Cutting Stock and Strip Packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the Unbounded 3D Knapsack problem, we extend the recurrence formula proposed by Beasley for the Rectangular Knapsack Pr...

Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm —for staged pat...

Combinatorial algorithms for packing and scheduling problems