In this article, a combined method for the solution of the General one-Dimensional Cutting Stock Problem (G1D-CSP) is proposed. The main characteristic of G1D-CSP is that all stock lengths can be different. The new approach combines two existing methods: Sequential Heuristic Procedure, and branch-and-bound. The algorithm based on the proposed method leads to almost optimal solutions, which are ...

The conventional assignment-based first/best fit decreasing algorithms (FFD/BFD) are not polynomial in the cutting stock input size in its most common format. Therefore, even for small instances with large demands, it is difficult to compute FFD/BFD solutions. We present pattern-based methods that overcome the main problems of conventional heuristics in cutting stock problems by representing th...

Cutting stock problems and bin packing problems are basically the same problems. They differ essentially on the variability of the input items. In the first, we have a set of items, each item with a given multiplicity; in the second, we have simply a list of items (each of which we may assume to have multiplicity 1). Many approximation algorithms have been designed for packing problems; a natur...

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...

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.

There are many different kinds of cutting stock problems (CSPs) occurring in practice and in theory having in common that they ask for a set of patterns, where each pattern is a collection of elements, to satisfy given element demands while minimizing the costs of the patterns. The classical CSP only considers fixed costs for each individual pattern, but in many practical applications an additi...

In case of the one-dimensional cutting stock problem (CSP) one can observe for any instance a very small gap between the integer optimal value and the continuous relaxation bound. These observations have initiated a series of investigations. An instance possesses the integer roundup property (IRUP) if its gap is smaller than 1. It is well-known that there exist instances of the CSP having a gap...

This paper presents the application of the one new approach using Genetic Algorithm in solving OneDimensional Cutting Stock Problems in order to minimize two objectives, usually conflicting, i.e., the number of processed objects and setup while simultaneously treating them as a single goal. The model problem, the objective function, the method denominated SingleGA10 and the steps used to solve ...