https://www.mansci.ovgu.de/mansci_media/publikationen/2007/typology-EGOTEC-5t0pvr6fjifln4r4oav60tt612.pdf

An improved typology of cutting and packing problems

The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.

The Three-Dimensional Bin Packing Problem

We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.

An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure

This paper updates a survey [53] written about 3 years ago. All of the results mentioned there are covered here as well. However, as a major justification for this second edition we shall be presenting many new results, some of which represent important advances. As a measure of the impressive amount of research in just 3 years, the present reference list more than doubles the list in [53].

Approximation Algorithms for Bin-Packing — An Updated Survey

Random-key genetic algorithms were introduced by Bean (ORSA J. Comput. 6:154---160, 1994) for solving sequencing problems in combinatorial optimization. Since then, they have been extended to handle a wide class of combinatorial optimization problems. This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems. Biased random-key genetic algorithms are a variant of random-key genetic algorithms, where one of the parents used for mating is biased to be of higher fitness than the other parent. After introducing the basics of biased random-key genetic algorithms, the paper discusses in some detail implementation issues, illustrating the ease in which sequential and parallel heuristics based on biased random-key genetic algorithms can be developed. A survey of applications that have recently appeared in the literature is also given.

Biased random-key genetic algorithms for combinatorial optimization

We present a tree-search algorithm for two-dimensional cutting problems in which there is a constraint on the maximum number of each type of piece that is to be produced The algorithm limits the size of the tree search by deriving and imposing necessary conditions for the cutting pattern to be optimal A dynamic programming procedure for the solution of the unconstrained problem and a node evaluation method based on a transportation routine are used to produce upper bounds during the search The computational performance of the algorithm is illustrated by tests performed on a large number of randomly generated problems with constraints of varying "tightness" The results indicate that the algorithm is an effective procedure for solving cutting problems of medium size

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Charles Whitlock

Papers

An Algorithm for Two-Dimensional Cutting Problems