Minimal proper non-IRUP instances of the one-dimensional Cutting Stock Problem

We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed demand n, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property we computationally show that all 1CSP instances with $n\le 9$ are proper IRUP, while we give examples of a proper non-IRUP instances with $n=10$. A gap larger than 1 occurs for $n=11$. The worst known gap is raised from 1.003 to 1.0625. The used algorithmic approaches are based on exhaustive enumeration and integer linear programming. Additionally we give some theoretical bounds showing that all 1CSP instances with some specific parameters have the proper IRUP.Comment: 14 pages, 2 figures, 2 table

An Improved Arcflow Model for the Skiving Stock Problem

Because of the sharp development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a viable tool for solving cutting and packing problems in recent years. Constituting a natural (but independent) counterpart of the well-known cutting stock problem, the one-dimensional skiving stock problem (SSP) asks for the maximal number of large objects (specified by some threshold length) that can be obtained by recomposing a given inventory of smaller items. In this paper, we introduce a new arcflow formulation for the SSP applying the idea of reflected arcs. In particular, this new model is shown to possess significantly fewer variables as well as a better numerical performance compared to the standard arcflow formulation

On the MAXGAP Problem for Cutting Stock Problems

The MAXGAP problem of a linear integer optimization problem P consists in determining the maximum difference (gap) \Delta(P ) between the optimal value z (E) of an instance E 2 P and the LP bound z c (E) with respect to all E 2 P . In the case of the one-dimensional cutting stock problem (1D CSP) it is known that \Delta(1D CSP) 1 + 5 132 but there is also a conjecture that \Delta(1D CSP) 2. In this paper we show that in the case of the restricted exact 2-stage two-dimensional cutting stock problem (RE2 CSP) the maximum gap increases at least linearly in the number m of pieces. It holds \Delta(RE2 CSP) ? m 3 . For the (non-restricted) exact 2-stage two-dimensional cutting stock problem (E2 CSP) an example is given such that \Delta(E2 CSP) 2(1 + 5 132 ) ? 2:075 follows which yields also a lower bound for the general twodimensional cutting stock problem. Furthermore some consequences for higher-dimensional cutting stock problems and with respect to applications are discussed..

The Solution of Packing Problems With Pieces of Variable Length and Additional Allocation Constraints

In this paper one-dimensional partitioning and packing problems of the following type are investigated. The pieces, which are to be packed, are assumed to have a variable length in a given range. Furthermore, the placement of any packed piece has to fulfill additional allocation constraints. The value of a piece is dependent affine linearly on its length. Find a feasible partition or packing, resp., with maximum total value. For these problems mixed-integer optimization models with linear constraints are developed and suitable solution methods are discussed. Especially, a branch&bound algorithm and an algorithm based on the so-called forward state strategy are proposed. AMS 1991 Subject Classification: Primary 90 C 11, 90 B 30, Secondary 05 B 40 Key words: packing problem, partitioning problem, cutting problem, mixedinteger optimization, branch&bound, dynamic programming 1 Introduction The following one-dimensional packing problem will be considered: Pieces T i , (i = 1; : : : ; m) ..

A branch&bound algorithm for solving one-dimensional cutting stock problems exactly

Many numerical computations reported in the literature show only a small difference between the optimal value of the one-dimensional cutting stock problem (1CSP) and that of the corresponding linear programming relaxation. Moreover, theoretical investigations have proven that this difference is smaller than 2 for a wide range of subproblems of the general 1CSP

Modelling of Packing Problems

this paper packing problems of the following type are investigated: Let be given a closed and bounded region B with B ae

NEW THEORETICAL INVESTIGATIONS ON THE GAP OF THE SKIVING STOCK PROBLEM

ABSTRACT The one-dimensional skiving stock problem is a combinatorial optimization problem being of high relevance whenever an efficient and sustainable utilization of given resources is intended. In the classical formulation, a given supply of (small) item lengths has to be used to build as many large objects (specified by some target length) as possible. For this ����-hard (discrete) optimization problem, we investigate the quality of the continuous relaxation by considering the additive integrality gap, i.e., the difference between the optimal values of the integer problem and its LP relaxation. In a first step, we derive an improved upper bound for the gap by focusing on the concept of residual instances. Moreover, we show how further upper bounds can be obtained if all problem-specific input data are considered. Additionally, we constructively prove the integer round-down property for two new classes of instances, and introduce several construction principles to obtain gaps greater than or equal to one