We address a 1-dimensional cutting stock problem where, in addition to trimloss minimization, we require that the set of cutting patterns forming the solution can be sequenced so that the number of stacks of parts maintained open throughout the process never exceeds a given s. For this problem, we propose a new integer linear programming formulation whose constraints grow quadratically with the...

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of xed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm nds a packing of n rectangles whose total height is within a factor of (1 +) of optimal (up to an additive term), and has running time polynomial both in n and in 1==. It is based on a r...

In this paper, an integer programming model for two-dimensional cutting stock problems is proposed. In the problems addressed, it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized. The two-stage and three-stage, exact and non-exact, problems are considered. Other issues are a...

In this work we consider the 3-staged 2-dimensional cutting stock problem, which appears in many real-world applications such as glass and wood cutting and various scheduling tasks. We suggest a variable neighborhood search (VNS) employing “ruin-and-recreate”based very large neighborhood searches (VLNS). We further present a polynomial-sized integer linear programming model (ILP) for solving th...

The two-dimensional Single Large Object Placement Problem (SLOPP) problem consists of determining a cutting pattern of a set of n small rectangular piece types (little object) on a rectangular stock plate (large object) of length L and width W, as to maximize the sum of the profits of the pieces to be cut. Each piece type i, i = 1, . . ., m, is characterized by a length li, a width wi, a profit...

Consider the feasibility problem in higher-dimensional orthogonal packing. Given a set I of d-dimensional rectangles, we need to decide whether a feasible packing in a d-dimensional rectangular container is possible. No item rotation is allowed and item edges are parallel to the coordinate axes. Typically, solution methods employ some bounds to facilitate the decision. Various bounds are known,...

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: Unbounded Knapsack, Cutting Stock and Strip Packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the Unbounded 3D Knapsack problem, we extend the recurrence formula proposed by Beasley for the Rectangular Knapsack Pr...

Nowadays, One-Dimensional Cutting Stock Problem (1D-CSP) is used in many industrial processes and re-cently has been considered as one of the most important research topic. In this paper, a metaheuristic algo-rithm based on the Simulated Annealing (SA) method is represented to minimize the trim loss and also to fo-cus the trim loss on the minimum number of large objects. In this method, the 1D-...