Cutting Stock Problem

In this paper, we propose approximate and exact algorithms for the double constrained two-dimensional guillotine cutting stock problem (DCTDC). The approximate algorithm is a two-stage procedure. The first stage attempts to produce a starting feasible solution to DCTDC by solving a single constrained two dimensional cutting problem, CDTC. If the solution to CTDC is not feasible to DCTDC, the second stage is used to eliminate non-feasibility. The exact algorithm is a branch-and-bound that uses efficient lower and upper bounding schemes. It starts with a lower bound reached by the approximate two-stage algorithm. At each internal node of the branching tree, a tailored upper bound is obtained by solving (relaxed) knapsack problems. To speed up the branch and bound, we implement, in addition to ordered data structures of lists, symmetry, duplicate, and non-feasibility detection strategies which fathom some unnecessary branches. We evaluate the performance of the algorithm on different problem instances which can become benchmark problems for the cutting and packing literature.

A method of solving a version of the two-dimensional cutting-stock problem is presented. In this version of the problem one is given a number of rectangular sheets and an order for a specified number of each of certain types of rectangular shapes. The goal is to cut the shapes out of the sheets in such a way as to minimize the waste. However, in many practical applications computation time is also an important economic consideration. For such applications the goal may be to obtain the best solution possible without using excessive amounts of computation time. The method of solution described here avoids exhaustive search procedures by employing an approach utilizing a constrained dynamic programming algorithm to lay out groups of rectangles called strips. This paper also describes the results obtained when the algorithm was tested with some typical rectangular layout problems.

In this paper we propose an algorithm for the constrained two-dimensional cutting stock problem (TDC) in which a single stock sheet has to be cut into a set of small pieces, while maximizing the value of the pieces cut. The TDC problem is NP-hard in the strong sense and finds many practical applications in the cutting and packing area. The

Rita Macedo a, Elsa Silva a, Cláudio Alves ab , Filipe Pereira e Alvelos ab, JM Valério de Carvalho ab, Claudio Arbib c , Fabrizio Marinelli d, Ferdinando Pezzella d,Luigi De Giovannid, Luca Gambella d ... Università Politecnica delle Marche Via Brecce Bianche, I-60131, Ancona, Italy

Despite its great applicability in several industries, the combined cutting stock and lot-sizing problem has not been sufficiently studied because of its great complexity. This paper analyses the trade-off that arises when we solve the cutting stock problem by taking into account the production planning for various periods. An optimal solution for the combined problem probably contains non-optimal solutions for the cutting stock and lot-sizing problems considered separately. The goal here is to minimize the trim loss, the storage and setup costs. With a view to this, we formulate a mathematical model of the combined cutting stock and lot-sizing problem and propose a solution method based on an analogy with the network shortest path problem. Some computational results comparing the combined problem solutions with those obtained by the method generally used in industry—first solve the lot-sizing problem and then solve the cutting stock problem—are presented. These results demonstrate that by combining the problems it is possible to obtain benefits of up to 28% profit. Finally, for small instances we analyze the quality of the solutions obtained by the network shortest path approach compared to the optimal solutions obtained by the commercial package AMPL.

In this paper, we propose a cooperative solution procedure for the two-staged two-dimensional cutting stock problem (2TDC). We solve 2TDC by considering two key features: a search strategy and a fast filling procedure. The search strategy uses a beam-search method and the second strategy uses a local filling procedure for improving the quality of the obtained solutions

Cutting stock and bin packing problems are common in several industrial sectors, such as paper, glass, furniture, steel industry, construction, transportation, among others. The classical cutting stock problem (CSP) ignores the production planning and scheduling of multiple customer orders. Nevertheless, in real industrial settings customer orders have to be planned and scheduled over time so as to meet demand and required due dates. We propose an integer linear programming model for the one-dimensional cutting stock and scheduling problem with heterogeneous orders. As this problem is a generalization of the classical single-period one-dimensional CSP, which is known to be NP-hard, it is difficult to solve real-sized instances to optimality. Thus we propose a novel matheuristic algorithm based on a fix-and-optimize strategy hybridized with a random local search. The proposed matheuristic was tested on a set of 80 synthetic problem instances based on a real-world problem and compared with CPLEX solver. In larger instances, the proposed matheuristic performed better than CPLEX, with average relative percentage deviation (RPD) regarding objective values as high as 72.2%. On the other hand, in small instances CPLEX showed a marginal advantage, with best average RPD of 19.61% with relation to the matheuristic. We also performed a paired t-test with significance level 0.05 and null hypothesis: no difference between the proposed matheuristic and CPLEX. In small test instances, the performance of the proposed matheuristic was statistically indistinguishable from CPLEX, while in larger instances the matheuristic outperformed CPLEX in most cases with p-value < 0.05.

Rita Macedo a, Elsa Silva a, Cláudio Alves ab , Filipe Pereira e Alvelos ab, JM Valério de Carvalho ab, Claudio Arbib c , Fabrizio Marinelli d, Ferdinando Pezzella d,Luigi De Giovannid, Luca Gambella d ... Università Politecnica delle Marche Via Brecce Bianche, I-60131, Ancona, Italy

Rita Macedo a, Elsa Silva a, Cláudio Alves ab , Filipe Pereira e Alvelos ab, JM Valério de Carvalho ab, Claudio Arbib c , Fabrizio Marinelli d, Ferdinando Pezzella d,Luigi De Giovannid, Luca Gambella d ... Università Politecnica delle Marche Via Brecce Bianche, I-60131, Ancona, Italy

Rita Macedo a, Elsa Silva a, Cláudio Alves ab , Filipe Pereira e Alvelos ab, JM Valério de Carvalho ab, Claudio Arbib c , Fabrizio Marinelli d, Ferdinando Pezzella d,Luigi De Giovannid, Luca Gambella d ... Università Politecnica delle Marche Via Brecce Bianche, I-60131, Ancona, Italy

This paper addresses a constrained two-dimensional (2D), non-guillotine restricted, packing problem, where a fixed set of small rectangles has to be placed into a larger stock rectangle so as to maximize the value of the rectangles packed. The algorithm we propose hybridizes a novel placement procedure with a genetic algorithm based on random keys. We propose also a new fitness function to drive the optimization. The approach is tested on a set of instances taken from the literature and compared with other approaches. The experimental results validate the quality of the solutions and the effectiveness of the proposed algorithm.