Solving the Two-dimensional single stock size Cutting Stock Problem with SAT and MaxSAT

TL;DR

This paper introduces a SAT-based framework for the 2D-CSSP, improving optimality certification and solution quality over existing solvers by leveraging novel encoding and MaxSAT techniques.

Contribution

The paper presents a new SAT and MaxSAT approach for 2D-CSSP, including demand expansion, sheet assignment variables, and an infeasible-orientation elimination rule, outperforming traditional solvers.

Findings

Best SAT configurations certify 2-3 times more instances as optimal.

SAT approaches achieve lower optimality gaps than OR-Tools, CPLEX, and Gurobi.

Incremental SAT is more effective without rotation, non-incremental SAT excels with rotation.

Abstract

Cutting rectangular items from stock sheets to satisfy demands while minimizing waste is a central manufacturing task. The Two-Dimensional Single Stock Size Cutting Stock Problem (2D-CSSP) generalizes bin packing by requiring multiple copies of each item type, which causes a strong combinatorial blow-up. We present a SAT-based framework where item types are expanded by demand, each copy has a sheet-assignment variable and non-overlap constraints are activated only for copies assigned to the same sheet. We also introduce an infeasible-orientation elimination rule that fixes rotation variables when only one orientation can fit the sheet. For minimizing the number of sheets, we compare three approaches: non-incremental SAT with binary search, incremental SAT with clause reuse across iterations and weighted partial MaxSAT. On the Cui--Zhao benchmark suite, our best SAT configurations certify two to three times more instances as provably optimal and achieve lower optimality gaps than OR-Tools, CPLEX and Gurobi. The relative ranking among SAT approaches depends on rotation: incremental SAT is strongest without rotation, while non-incremental SAT is more effective when rotation increases formula size.