A number of optimisation problems involve the optimal grouping of a finite set of items into a number of categories subject to one or more constraints. Such problems raise interesting issues in mapping solutions in genetic algorithms. These problems range from the knapsack problem to bin packing and cutting stock problems. This paper describes research involving cutting stock problems. Results ...

We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3. In fact, for constant d our alg...

Abstract Two-dimensional cutting and packing problems model a large number of relevant industrial applications.The literature on practical algorithms for such is very large. We introduce the , library two-dimensional orthogonal problems. The makes available, in unified format, 25 benchmarks from total over 3000 instances, provides direct links to surveys typologies, includes list links.

The paper deals with multi-dimensional cutting and packing and metaheuristic algorithms. A brief overview of this area is presented, and the general Tabu Search framework of Lodi, Martello and Vigo [LMV99a] is considered in detail. Computational results on multi-dimensional bin packing problems are reported, and possible methods for extending the framework to other packing problems are finally ...

We present an qpproximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm find,!; a packing of n rectangles whose total height is within a ,factor of (1 + E ) of optimal, and has running time polynomial both in n and in 1 / E . It is based on a reduction to fractional bin-packing, and ...

The nesting problem is a two-dimensional cutting and packing problem where the small pieces to cut have irregular shapes. The variant of the problem that we are going to deal with, which naturally arises in several industrial production processes (textile, garment, metalware, etc), considers only one big rectangular piece, the plate, with fixed width and infinite length. The objective will be t...

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of xed width and minimum height, a classical N P-hard cutting-stock problem. The algorithm nds a packing of n rectangles whose total height is within a factor of (1 +) of optimal (up to an additive term), and has running time polynomial both in n and in 1==. It is based on a ...

This paper is intended to give a concise understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope and gives cutting plane proofs for huge classes of valid inequalities of this polytope. We introduce generalized cycle, generalized clique, generalized antihole, generalized antiweb, gen...

The nesting problem is an irregular two-dimensional cutting problem where the shapes of the pieces to cut and the master surfaces are irregular in shape and different in size. In particular, we consider nesting problems where the master surface could contain defects. Some of them can be accepted (i.e., incorporated) in certain types of pieces, while other defected areas must be avoided. The pro...