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...

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...

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 consider the problem of fair division of a two dimensional heterogeneous good among several agents. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols either consider a one-dimensional resource, or allocate each agent several disconnected pieces. In practice, however, the two dimensional shape of the allotted piece is of ...

A two-dimensional cutting stock problem can be defined as follows: A set of rectangular stock sheets of p different types is available. For each type of sheet Sp we know its length Lp and width Wp. From these sheets we have to cut smaller rectangular pieces of length li and width wi, i = 1, ...,m in order to satisfy a given demand for di pieces of each type. The objective is to minimize the tot...

The modi ed integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we e...

The modified integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we ...