In the production process for wooden furniture, raw material costs account more than 50% of furniture costs, and utilization rate materials depends mainly on layout scheme. Therefore, a reasonable is an important measure to reduce costs. This paper investigates solid wood board cutting stock problem (CSP) establishes optimization model, with goal highest possible original boards. An ant colony-...

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting — dividing a heterogeneous resource among agents with different preferences. Resourceand population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all o...

We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily consider a one-dimensional resource, or allocate each player multiple infinitesimally small “pieces”. In practice, however, the two dimensional shape of the allo...

The G12 project is developing a software environment for stating and solving combinatorial problems by mapping a high-level model of the problem to an efficient combination of solving methods. Model annotations are used to control this process. In this paper we explain the mapping to branch-and-price solving. G12 supports the selection of specialised sub-problem solvers, the aggregation of iden...

The cutting stock problem it is of great interest in relation with several real world problems. Basically it means that there are some smaller pieces that have to be cut from a greater stock piece, in such a way, that the remaining part of the stock piece should be minimal. The classical solution methods of this problem generally need a great amount of calculation. In order to reduce the comput...

with A ∈ IRm×n, b ∈ IR, and x, c ∈ IR. We will assume that the number of columns n of matrix A is so large that it is impossible to generate and store the matrix A in memory. The revised simplex method in Chapter 7 of Chvátal can be appropriately modified to solve linear programs of the form (1); the resulting method is known as the column generation method where columns of A are generated as n...

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their wid...

There appear to be two versions of the Dual Bin Packing problem in the literature. In addition, one of the versions has a counterpart in the cutting stock literature, known as the Skiving Stock Problem. This paper outlines branch-andprice algorithms for both. We introduce combinatorial upper bounds and well-performing heuristics from the literature in the branch-and-price framework. Extensive c...

Cutting and packing problems occur in many real-world applications such as industrial glass, paper or steel cutting, container loading and VLSI design [1]. In this work we consider in particular the two-dimensional cutting stock problem with variable sheet size (2CSV) in which we are given a set of nE rectangular element types E = {1, . . . , nE}, each i ∈ E specified by a height hi ∈ N, a widt...