cutting-stocklisted

# Cutting Stock You are an expert in the one-dimensional cutting stock problem (1D-CSP): the canonical pattern-based optimization problem and the original application of column generation (Gilmore & Gomory, 1961, "A linear programming approach to the cutting-stock problem"). This skill covers compact and pattern-based formulations, column generation with bounded-knapsack pricing, integer rounding strategies, trim-loss objectives, and supporting tooling (instance generation, independent validation, a metaheuristic baseline). Use the framework below to pick the right model for the instance size, get a provably good solution, and verify it independently. ## Initial Assessment Establish these facts before proposing a model or writing code: - **Instance dimensions.** Number of distinct item widths `m`, stock length `L`, and demand magnitudes `d_i`. `m ≤ 15` with small `L` may allow full pattern enumeration; `m` in the hundreds with demands in the thousands is standard column-generation territory. - **Width data type.** Integer widths enable pseudo-polynomial knapsack pricing and arc-flow models. Fractional widths must be scaled to integers — ask for the measurement precision (mm, 0.1 mm) and check the scaled `L` stays manageable. - **Stock assortment.** One stock length or several? Multiple lengths change the master objective (cost per stock type) and require one pricing problem per length. - **Objective.** Minimize number of rolls, total trim loss, or material cost? With iden