Cutting patterns on demand: column generation

Most sawmills choose how to cut a log from a fixed catalogue of cutting patterns: ready-made layouts saying which central cant and which side boards come from a log of a given diameter. The Chilean paper by Mena-Reyes, Soto-Concha, Gatica and Linfati (Mathematics, 2026; authors at Chilean universities) argues that fixed, historically derived patterns can misalign with demand and raw material, causing overproduction, poor log use and residues. It proposes generating new patterns on demand.

Scope warning. The portal could read only the abstract, introduction and part of the methods. Results beyond the abstract (instance sizes, run times, the baseline behind the 31% figure) are unverified. Treat everything below as abstract-level.

Two linked models

1. Master (primary) model. A mixed-integer linear program. In its baseline form it minimises the number of logs processed. For each diameter class the logs assigned to compatible patterns cannot exceed supply; for each product the output must cover demand, and any excess counts as overproduction. It can be steered towards other objectives: overproduction, residue volume, total production.
2. Pricing model. A mixed-integer quadratically constrained program that creates one new pattern for a chosen diameter class. It uses the dual prices (marginal values of the supply and demand constraints) from the relaxed master model to look for a pattern with favourable reduced cost, and a pattern is added only if its reduced cost is negative.

The geometry is enforced with quadratic constraints based on the Pythagorean theorem: the central cant plus lateral boards must fit inside the circular log cross-section, with a kerf allowance.

The loop (standard column-generation scheme)

The step-by-step loop is the standard scheme and lies partly beyond the readable text:

1. Start from a base set of patterns (here an industrial benchmark of 160 patterns).
2. Solve the relaxed master model and read the dual prices.
3. Solve the pricing model to find a pattern with negative reduced cost.
4. Add it and repeat until no improving pattern is found.

Reported results (abstract only)

• Log consumption falls as patterns are generated, by up to 31%.
• Strategies that minimise log use and residues reach near-optimal solutions with only about 20 to 25 patterns; more patterns give marginal gains but increase set-up time and complexity.
• Near-optimal performance with a moderate number of active patterns.

Assumptions that limit transfer

• Logs within a diameter class are geometrically homogeneous.
• Cant-sawing configuration and kerf are fixed.
• At most one central product and symmetric lateral boards.
• No taper or defects, no processing times or schedules.
• Supply and demand are deterministic over the planning horizon (weekly in the data).

Worked example: reading "up to 31%"

"Up to 31% fewer logs" is a maximum under the model's conditions. Suppose, purely as an illustration, a baseline plan needed 1,000 logs. A 31% reduction would mean 1,000 x (1 - 0.31) = 690 logs. The paper's baseline is not known to us, so the real saving of any mill depends on how wasteful its current catalogue is and on whether its saw line can change cant configurations. The 1,000 logs is invented.

A portal reading for a mill

Interpretation, not a claim of the paper:

• Review your pattern catalogue periodically against current demand and log mix; drop rarely useful patterns to reduce set-up complexity.
• Collect the inputs the model expects: log supply by diameter class, product demand and product dimensions.
• Use the result as a planning guide, not as an instruction for the sawing line, because defects, taper and scheduling are ignored.

Checklist: is my mill a candidate?

• Do I have a weekly demand picture by product?
• Do I know log availability by diameter class?
• Can the line change cant configurations?
• Do I know how many patterns are actually in use?

Key points

• Column generation creates new cutting patterns on demand, guided by dual prices, instead of picking from a fixed catalogue (Chilean study, abstract-level evidence).
• Reported: log consumption down by up to 31%; near-optimal results with about 20 to 25 patterns.
• The model ignores defects, taper and scheduling, so it is a planning tool.
• More patterns add set-up complexity for marginal gains.
• Most results beyond the abstract could not be verified; the baseline behind 31% is unknown.