Generating sawmill cutting patterns on demand: column generation for leaner log use

Introduction

Most sawmills decide how to cut a log from a fixed catalogue of cutting patterns: ready-made layouts that say which central cant and which side boards to produce from a log of a given diameter. Such catalogues are built from experience and tend to age. Product demand changes from week to week, log supply arrives in a particular diameter mix, and a pattern that was sensible last year may now cause overproduction of some products, shortages of others and more residue than necessary.

A paper published in Mathematics (volume 14, issue 1, article 10; online December 2025, issue year 2026) by Jorge Felix Mena-Reyes, Raul Soto-Concha, Gustavo Gatica and Rodrigo Linfati proposes an optimisation approach that creates new cutting patterns on demand instead of choosing only from a fixed list. The method is framed as a tactical planning tool with a sustainability angle, motivated by the Chilean forestry sector, where pressure to use raw material efficiently is rising.

Important note on scope: for this article we could read the abstract, the introduction and most of the methods section of the paper, but not its results tables, discussion or conclusions. The numerical findings below are therefore limited to those stated in the abstract, and this summary is deliberately shorter than our usual articles.

Research methodology

Problem setting. The authors identify four gaps in the literature: many approaches rely on fixed pattern sets and lack adaptability; sustainability criteria such as residue reduction, material efficiency and overproduction control are rarely built into tactical models; few studies combine pattern generation with explicit supply-demand control through column generation; and few validate their methods against industrial pattern libraries with realistic supply and demand data.

Two linked models.

1. A primary (master) model, a mixed-integer linear program, in its baseline form minimises the number of logs processed. For each diameter class, the logs assigned to compatible patterns cannot exceed the available supply, and for each product the output must cover demand, with any excess counted as overproduction. The decision variables are the numbers of logs cut with each pattern. The model can also be steered towards other objectives such as reducing overproduction or residues.
2. A secondary (pricing) model, a mixed-integer quadratically constrained program, creates one new pattern for a chosen diameter class. It uses the dual prices (the marginal values of supply and demand constraints) from the relaxed primary model to look for a pattern with the most favourable reduced cost. Only one type of central product can be active in a pattern, and the central cant plus lateral boards must fit inside the circular log cross-section; this is enforced with quadratic constraints based on the Pythagorean theorem, with saw kerf margins included.

Column generation loop. The algorithm starts from a base set of industrial patterns, solves the relaxed primary model, reads the dual prices, solves the pricing model, and adds the new pattern if its reduced cost is negative (that is, if it can improve the current solution). The loop repeats until no beneficial pattern is found. According to the abstract, the approach is solved at the root node.

Data. The starting point is an industrial benchmark of 160 cutting patterns, which the introduction attributes to earlier work on small and medium sawmills, paired with log availability and weekly demand data.

Assumptions stated by the authors. Logs within a diameter class are geometrically homogeneous; the cant-sawing configuration and kerf are fixed; taper and defects are not represented; processing times and machine schedules are not modelled; and supply and demand are deterministic over the planning horizon.

Key findings

The abstract reports three computational experiments:

• Convergence. As new patterns are generated, total log consumption falls, with reductions of up to 31%.
• Comparison of objectives. Strategies that minimise log use and residues reach near-optimal solutions with only about 20 to 25 patterns. Adding more patterns gives marginal gains but increases set-up time and complexity.
• Limit on active patterns. Near-optimal performance can be reached with a moderate number of active patterns.

The authors compare four objective functions (log consumption, overproduction, residue volume and total production) and present the approach as a flexible, sustainability-oriented decision-support tool. We could not read the underlying tables, solver details, run times or instance sizes, so we cannot say under which conditions the 31% figure holds or relative to which baseline.

Practical applications for sawmills

From the paper (abstract and methods).

• A small set of well-chosen patterns, about 20 to 25, can be nearly as good as a much larger catalogue, which suggests that more patterns do not necessarily mean better planning.
• Pattern design can be tied to demand and log supply, and to sustainability measures such as residue and overproduction.

Portal interpretation (not claims of the paper).

• Sawmills with sawing lines that can change cant configurations could review their pattern catalogue periodically against current demand and log mix, and drop rarely useful patterns to reduce set-up complexity.
• The model ignores defects, taper and scheduling, so results would be a planning guide rather than an instruction for the sawing line.
• Mills would need the inputs the model expects: log supply by diameter class, product demand and product dimensions.

Limitations

• Access. Only the abstract, introduction and methods were readable for this summary. The results, discussion, authors' own limitations and conclusions sections were not available to us, so the paper's quantitative detail is not covered. This article is therefore based on abstract-level results ("abstract only" access).
• Model assumptions. As listed above: homogeneous logs per class, fixed cant configuration and kerf, no defects or taper, no processing-time or schedule modelling, deterministic demand and supply.
• Reported text inconsistencies. The description of the overproduction variable differs in places between continuous and integer in the text we saw; we have not relied on it.
• Licence. The MDPI page we reviewed did not state a Creative Commons licence explicitly, only an MDPI copyright notice and, in its metadata, CC BY 3.0.

Conclusions

Dynamic generation of cutting patterns through column generation is presented as a way to move sawmill planning away from static pattern catalogues. According to the abstract, it can reduce log consumption by up to 31% and reach near-optimal results with only 20 to 25 patterns, while incorporating sustainability goals. Pending a reading of the full results, these claims should be treated as promising but unverified in this summary, and as applying to the model's assumptions rather than directly to a particular mill.

Original research

• Authors: Jorge Felix Mena-Reyes, Raul Soto-Concha, Gustavo Gatica, Rodrigo Linfati
• Title: Dynamic Generation of Cutting Patterns in Sawmills for Sustainable Planning
• Journal: Mathematics (MDPI), volume 14, issue 1, article 10
• Year: 2026 (published online 20 December 2025)
• DOI: 10.3390/math14010010
• Link: https://www.mdpi.com/2227-7390/14/1/10
• Licence: MDPI open access; page metadata lists CC BY 3.0, but the page text itself shows only an MDPI copyright notice

This article is an independent summary prepared by the portal; the original paper should be consulted for full details and exact figures.