Solving Cutting Stock Problems via an Extended Ryan-Foster Branching Scheme and Fast Column Generation
Renan F. F. da Silva, Rafael C. S. Schouery
TL;DR
This work addresses the Cutting Stock Problem (CSP) by integrating a branch-cut-and-price framework built on strong Set Covering/Set Packing relaxations. It introduces an Extended Ryan-Foster branching scheme for non-binary problems, a fast diversified column generation process, and numerically safe lower bounds to reliably converge under floating-point arithmetic. Key contributions include a Multiple Pattern Generation pricing approach, Subset-Row Inequalities, bespoke primal heuristics (Rounding, Relax-and-Fix variants), and model-leaning techniques (splay, reduced-cost cleaning, waste control). The framework achieves state-of-the-art performance for CSP and related problems (SSP, IPMS, OOEBPP, CCBPP), including a new challenging CSP benchmark, while preserving numerical safety and scalability. Together, these advances offer a robust, extensible methodology for strong-relaxation problems in combinatorial optimization with practical impact on industrial packing and scheduling applications.
Abstract
We present a branch-cut-and-price framework to solve Cutting Stock Problems with strong relaxations using Set Covering (Packing) Formulations, which are solved by column generation. The main contributions of this paper include an extended Ryan-Foster scheme, which allows us to use this powerful branching scheme even in non-binary problems by using a conflict propagation lemma; a fast column generation process based on a diversification strategy; custom primal heuristics, enabling us to find optimal solutions for several open instances; and a technique to use a smaller feasibility tolerance in floating-point linear programming solvers, combined with numerically safe methods to produce stronger and safer lower bounds. Additional performance-improving strategies include a technique that controls the height of the branch-and-bound tree; a variable selection algorithm based on branching history; a new set of dual inequalities; insights to obtain a lean model; and the subset-row inequalities. By employing this comprehensive framework, we overcame the current state-of-the-art concerning the following problems: Cutting Stock, Skiving Stock, Ordered Open-End Bin Packing, Class-Constrained Bin Packing, and Identical Parallel Machines Scheduling with Minimum Makespan. Additionally, a new challenging benchmark for Cutting Stock is introduced.