Linear cutting optimization guide
How to fit required lengths into bars, pipes, timber, profiles or other repeated stock while accounting for kerf and waste.
Linear cutting is a one-dimensional packing problem
Unlike sheet cutting, a linear cut has only one geometric dimension to allocate: length. The objective is to assign requested cuts to stock lengths without exceeding the stock capacity, while using as few stock units—or as little waste—as practical.
The same model can represent timber lengths, aluminium extrusions, steel bar, pipe, cable tray, moulding, profiles and many other materials as long as the relevant planning problem is one-dimensional.
1. Establish a theoretical lower bound
Imagine 6000 mm stock lengths and six requested pieces of 1800 mm. Ignoring kerf for a moment, the total requested material is:
6 × 1800 = 10,800 mm
Dividing by the 6000 mm stock length gives a theoretical lower bound of two stock units. That lower bound is achievable geometrically because three 1800 mm cuts use 5400 mm. Including a 3 mm kerf between pieces and a final 3 mm separation cut, three finished pieces consume:
1800 + 3 + 1800 + 3 + 1800 + 3 = 5409 mm
So three pieces still fit in each 6000 mm stock and two stocks remain feasible.
2. Why longest-first is a useful starting point
If short cuts are packed first, they can leave several medium-size remnants that are too small for a later long cut. Sorting longer cuts first reduces that risk. For that reason, many practical cutting-stock heuristics use decreasing-length order as a baseline.
But one fixed rule is not always best. PackMize compares multiple deterministic orderings together with first-fit and best-fit decreasing placement. First-fit places a cut in the first stock that can accept it. Best-fit chooses the stock that would leave the smallest valid remainder after placement.
3. Understand the difference between stock quantity and waste
Using the minimum number of stock lengths is often the primary goal, but two plans using the same number of stocks can leave very different remnants. A production planner may prefer a pattern with one large reusable remnant instead of several small scraps even when total waste is similar.
| Metric | What it tells you |
|---|---|
| Stock lengths required | How many full stock units the selected plan consumes |
| Cuts placed | How many requested pieces were successfully assigned |
| Utilization | Requested finished length relative to stock length used |
| Total waste | Unused stock plus material reserved for kerf |
4. Model real allowances explicitly
A generic optimizer knows only what you enter. It does not automatically know that a saw needs 10 mm of end trim, a tube has a damaged section, a clamp requires 150 mm of clear stock or a mill forbids short remnants. If those constraints matter, reserve them in the available stock length or handle them in the production workflow.
For example, if a nominal 6000 mm profile always loses 20 mm from each end, enter 6000 mm stock and 20 mm trim per end. The usable length is 5960 mm; do not subtract the trim a second time.
5. Pattern repetition can simplify production
In larger jobs, the theoretically lowest-waste solution is not always the easiest to manufacture. Repeating a smaller number of patterns can reduce operator errors and setup complexity. PackMize currently focuses on material allocation and does not currently optimize for the number of distinct patterns or production changeovers.
This is one reason the output should be treated as a planning basis. Users can inspect each pattern visually and decide whether a slightly less material-efficient arrangement would be operationally simpler.
6. What the current engine does and does not solve
PackMize supports repeated equal stock lengths, requested cut lengths and quantities, kerf between pieces, terminal separation, end trim and reusable-offcut thresholds. It compares deterministic heuristics and returns the best pattern set found. It does not model angled cuts, two-dimensional cross-section constraints, defects, batch traceability, clamp zones, machine sequencing or stock of multiple lengths in the free version.
Practical workflow
- Measure or define the effective usable stock length.
- Enter realistic kerf for the cutting process.
- Add every requested cut length and quantity.
- Run the optimizer and confirm that no requested pieces remain unplaced.
- Review waste and the individual stock patterns.
- Apply machine-specific trims, clamping and sequencing before production.
Frequently asked questions
What is a linear cutting optimization problem?
It assigns requested one-dimensional cut lengths to repeated stock lengths without exceeding each stock length, while trying to minimize the number of stocks or waste.
How is kerf applied in PackMize?
Kerf is reserved between pieces and, by default, at the final cut separating an offcut. The terminal loss is clipped at the stock edge. End trim is entered separately.
Can I use the same tool for timber, pipe and metal profiles?
Yes when only one-dimensional length matters. Enter end trim in its own field; machine-specific clamps, defects and other constraints still require review.
Does PackMize prove the absolute minimum number of stock lengths?
For smaller instances PackMize V2 can prove the minimum using a mathematical lower bound and deterministic branch-and-bound search. Larger cases are labelled BEST SOLUTION FOUND unless the lower bound is reached.
Reproduce the calculation: the linear cutting methodology explains the kerf transformation and bounds. Run the controlled cases in the Benchmark Lab using the same engines as the calculator.