The two-dimensional cutting stock problem within the roller blind production process

In this paper we consider a two-dimensional cutting stock problem encountered at a large manufacturer of window covering products. The problem occurs in the production process of made-to-measure roller blinds. We develop a solution method that takes into account the characteristics of the specific problem. In particular, we deal with the fact that fabrics may contain small defects that should end up with the waste. Comparison to previous practice shows significant waste reductions.cutting;trim loss;two-dimensional cutting stock problem

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Solving the two dimensional cutting problem using evolutionary algorithms with penalty functions

In this work a solution using evolutionary algorithms with penalty function for the non-guillotine cutting problem is presented. In this particular problem, the rectangular pieces have to be cut from an unique large object, being the goal to maximize the total value of cut pieces. Some chromosomes can hold pieces to be cut, but some pieces cannot be arranged into the object, generating infeasible solutions. A way to deal with this kind of solutions is to use a penalizing strategy. The used penalty functions have been originally developed for the knapsack problem and they are adapted for the cutting problem in this paper. Moreover, the effect on the algorithm performance to combine penalty functions with two different selection methods (binary tournament and roulette wheel) is studied. The algorithm uses a binary representation, one-point crossover, big-creep mutation and in order to evaluated the quality of solutions a placement routine is considered (Heuristic with Efficient Management of Holes). Experimental comparisons of the performance of the resulting algorithms are carried out using publicly available benchmarks to the non-guillotine cutting problem. We report on the high performance of the proposed models at similar (or better) accuracy with respect to existing algorithms.VI Workshop de Agentes y Sistemas Inteligentes (WASI)Red de Universidades con Carreras en Informática (RedUNCI

Two-Dimensional Cutting Problem

This paper deals with two-dimensional cutting problems. Firstly the complexity of the problem in question is estimated. Then, several known approaches for the regular (rectangular) and irregular (not necessarily rectangular) cutting problems are described. In the second part, a decision support system for cutting a rectangular sheet of material into pieces of arbitrary shapes, is presented. The system uses two earlier described methods which prefer different types of data and the user may decide which one is more suitable for the problem in question. After brief description of system data files and its manual, some experimental results are presented

A method for pricing American options using semi-infinite linear programming

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and e.g. the Greeks can be calculated. We apply the algorithm to (one- and) multidimensional diffusions and to L\'evy processes, and show it to be fast and accurate

The two-dimensional cutting stock problem within the roller blind production process

In this paper we consider a two-dimensional cutting stock problem encountered at a large manufacturer of window covering products. The problem occurs in the production process of made-to-measure roller blinds. We develop a solution method that takes into account the characteristics of the specific problem. In particular, we deal with the fact that fabrics may contain small defects that should end up with the waste. Comparison to previous practice shows significant waste reductions