The 1-dimensional Cutting Stock Problem (1-CSP) seeks to fulfill with a minimum trim loss the demand of parts of various lengths by cutting them out from a given stock size. In the Pattern Minimization Problem (1-PMP) one wants an optimal 1-CSP solution that uses a minimum number of distinct cutting patterns. We discuss two formulations of 1-PMP: (1) introduced by Vanderbeck, and (2) obtained b...

One of the factors that add to the complexity of the cutting stock problem is the large number of the cutting patterns that may be encountered. When the cutting stock problem is expressed as an integer-programming problem, the large number of cutting patterns involved generally makes computation infeasible. However, if the linear programming formulation of the cutting stock problem is free of i...

In the classical Gilmore-Gomory model for the one-dimensional cutting stock problem (1D-CSP) we have to deal implicitly with a huge number of variables representing all feasible patterns. The advantage is a strong relaxation and absence of symmetries. To reduce the number of variables, we can restrict them to subpatterns, i.e., partial patterns which are combined to produce whole patterns. Each...

The primary objective in cutting and packing problems is trim loss or material input minimization (in stock cutting) or value maximization (when packing into a knapsack). However, in real-life production we usually have many other objectives (costs) and constraints. Probably the most complex auxiliary criteria of a solution are the number of different cutting patterns (setups) and the maximum n...

Research addressing two-dimensional irregular shape packing has largely focused on the strip packing variant of the problem. However, it can be argued that this is a simplification. Materials from which pieces are required to be cut will ultimately have a fixed length either due to the physical dimensions of the material or through constraints on the cutting machinery. Hence, in order to cut al...

This paper investigates the one-dimensional cutting stock problem considering two conflicting objective functions: minimization of both the number of objects and the number of different cutting patterns used. A new heuristic method based on the concepts of genetic algorithms is proposed to solve the problem. This heuristic is empirically analyzed by solving randomly generated instances and also...

We compare two branch-and-price approaches for the cutting stock problem. Each algorithm is based on a different integer programming formulation of the column generation master problem. One formulation results in a master problem with 0–1 integer variables while the other has general integer variables. Both algorithms employ column generation for solving LP relaxations at each node of a branch-...

Branch-and-price is a well established technique for solving large scale integer programming problems. This method combines the standard branch-and-bound framework of solving integer programming problems with Column Generation. In each node of the branch-and-bound tree, the bound is calculated by solving the LP relaxation. The LP relaxation is solved using Column Generation. In this report, we ...

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 ...