Minimal proper non-IRUP instances of the one-dimensional cutting stock problem

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 number n of demanded pieces, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property (IRUP) we use the proper LP relaxation of the Gilmore and Gomory model that allows us to consider the 1CSP as the bin packing problem (BPP). We computationally show that all 1CSP instances with n≤9 have the proper IRUP, while we give examples of proper non-IRUP instances with n=10 and proper gap 1. Proper gaps larger than 1 occur for n≥11. The largest known proper 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.