In this paper, we consider a multi-staged two-dimensional cutting stock problem (CSP) in a paper industry. Paper production relies on the sequential processes of various large machines (pulp preparation, paper formation, winding, and sheet cutting), which has various machine related constraints. In addition, there are the operational constraints from the real-world situation. The problem is mod...

In this paper we consider a simplified version of the stock cutting (two-dimensional bin packing) problem. We compare three meta-heuristic algorithms (genetic algorithm (GA), tabu search (TS) and simulated annealing (SA)) when applied to this problem. The results show that tabu search and simulated annealing produce good quality results. This is not the case with the genetic algorithm. The prob...

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed n-fold integer programming problems are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several appl...

We consider two integer linear programming models for the one-dimensional cutting stock problem that include various difficulties appearing in practical real problems. Our primary goals are the minimization of the trim loss or the minimization of the number of master rolls needed to satisfy the orders. In particular, we study an approach based on the classical column-generation procedure by Gil...

The nesting problem is an irregular two-dimensional cutting problem where the shapes of the pieces to cut and the master surfaces are irregular in shape and different in size. In particular, we consider nesting problems where the master surface could contain defects. Some of them can be accepted (i.e., incorporated) in certain types of pieces, while other defected areas must be avoided. The pro...

Let b ∈ Zd be an integer conic combination of a finite set of integer vectors X ⊂ Zd . In this note we provide upper bounds on the size of a smallest subset X̃ ⊆ X such that b is an integer conic combination of elements of X̃ . We apply our bounds to general integer programming and to the cutting stock problem and provide an NP certificate for the latter, whose existence has not been known so far.

The algorithmic aspects of the following problem are investigated: n (22) persons want to cut a cake into n shares so that every person will get at least l/n of the cake by his own measure and so that the number of cuts made on the cake is minimal. The cutting process is to be governed by a protocol (computer program). It is shown that no deterministic protocol exists which is fair (in a sense ...

Type–token ratio (TTR), or vocabulary size divided by text length (V/N), is a timehonoured but unsatisfactory measure of lexical diversity. The problem is that the TTR of a text sample is affected by its length. We present an algorithm for rapidly computing TTR through a moving window that is independent of text length, and we demonstrate that this measurement can detect changes within a text a...

A number of optimisation problems involve the optimal grouping of a finite set of items into a number of categories subject to one or more constraints. Such problems raise interesting issues in mapping solutions in genetic algorithms. These problems range from the knapsack problem to bin packing and cutting stock problems. This paper describes research involving cutting stock problems. Results ...

This paper presents a way to construct the Sylvester A-resultant matrix for three bi-degree (m; n) polynomials whose exponent set is cut oo by rectangles at the corners. The paper also shows that the determinant of this matrix does give the resultant of the three polynomials.