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

TL;DR

This paper analyzes the one-dimensional cutting stock problem, classifies instances into equivalence classes, and identifies the smallest instances with proper non-IRUP properties, providing computational and theoretical insights.

Contribution

It introduces a classification of 1CSP instances into equivalence classes and identifies the minimal instances with proper non-IRUP properties, combining computational and theoretical methods.

Findings

All instances with n ≤ 9 are proper IRUP.

Proper non-IRUP instances exist at n=10.

The gap exceeds 1 for n=11, reaching 1.0625.

Abstract

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.