We investigate a new application of DC (Difference of Convex functions) programming and DCA (DC Algorithm) in solving the constrained two-dimensional non-guillotine cutting problem. This problem consists of cutting a number of rectangular pieces from a large rectangular object. The cuts are done under some constraints and the objective is to maximize the total value of the pieces cut. We reform...

This paper presents a greedy randomized adaptive search procedure (GRASP) for the constrained two-dimensional non-guillotine cutting problem, the problem of cutting the rectangular pieces from a large rectangle so as to maximize the value of the pieces cut. We investigate several strategies for the constructive and improvement phases and several choices for critical search parameters. We perfor...

In this paper, six different approaches using genetic algorithms (GA) and/or simulated annealing (SA) with improved bottom left (I-BL) algorithm [1] were applied for solution of two dimensional non-guillotine cutting problems. As examples, test problems including 29 individual rectangular pieces were used [2]. Performances of hybrid approaches on solutions of cutting problems were compared. Due...

The MIRUP (Modified Integer Round-Up Property) leads to an upper bound for the gap between the optimal value of the integer problem and that of the corresponding continuous relaxation rounded up. This property is known to hold for many instances of the one-dimensional cutting stock problem but there are not known so far any results with respect to the two-dimensional case. In this paper we inve...

The MIRUP (Modiied Integer RoundUp Property) leads to an upper bound for the gap between the optimal value of the integer problem and that of the corresponding continuous relaxation rounded up. This property is known to hold for many instances of the one-dimensional cutting stock problem but there are not known so far any results with respect to the two-dimensional case. In this paper we invest...

The d-dimensional bin packing problem (OBPP-d) is the problem of finding the minimum number of containers needed to contain a set of orthogonally packed d-dimensional rectangular boxes. In OBPP-d solvers two subproblems are crucial: Calculating lower bounds and solving the decision problem (OPP-d) of determining if a set of boxes can be orthogonally packed into a single container. This thesis f...

The Backtracking Heuristic (BH) methodology consists in to construct of items by combination between heuristics that solve mathematical programming models, and backtrack search algorithm to figure out the best heuristics and their best ordering. BH was firstly introduced in the literature in order to solve three-dimensional Knapsack Loading Problems, showing promising results. In this present w...

Gilmore and Gomory's algorithm is one of the better actually known exact algorithms for solving unconstrained guillotine two-dimensional cutting problems. Herz's algorithm is more effective, but only for the unweighted case. We propose a new exact algorithm adequate for both weighted and unweighted cases, which is more powerful than both algorithms. The algorithm uses dynamic programming proced...

In this study we are concerned with the non-exact two-stage two-dimensional guillotine cutting problem considering usable leftovers, in which stock plates remainders of the cutting patterns (non-used material or trim loss) can be used in the future, if they are large enough to fulfill future demands of items (ordered smaller plates). This cutting problem can be characterized as a residual bin-p...