When the hard part isn't the software, it's the maths
Everyone in your plant knows the sheets could be nested better. An experienced planner does it by eye and gets close. But "close" is a cost you pay on every order, permanently.
The reason it stays that way is rarely complacency. It is that the problem is genuinely hard — and that most software companies turn off the road at exactly this point.
The optimal solution exists. It can rarely be proven.
This is what separates it from most tasks in manufacturing software. With a bill of materials the question is how to model it cleanly. With cutting, the question is whether it can be answered at all in acceptable time.
The one-dimensional cutting stock problem — bars, profiles and tubes cut to length — has been well studied since the early 1960s. The classical approach uses column generation: rather than enumerating every conceivable cutting pattern, which becomes impossible with only a handful of lengths, it generates only the patterns actually worth considering.
Two dimensions are less forgiving. Rectangular parts on sheets are the easier case. Irregular contours — sheet metal parts, facade cuts, anything not rectangular — require geometry: for every pair of parts you must determine how closely two contours may sit without overlapping. That core computation is expensive, and it recurs for every combination of parts and every rotation angle.
Both problems are NP-hard. That does not mean nothing can be solved exactly: for smaller and structurally favourable instances, exact methods do find the optimal solution and prove it. The difficulty is order size. For many practically relevant instances, exact methods are too slow to prove optimality within the available time budget. Industrial practice therefore works predominantly with heuristics and hybrid methods.
That is not an academic point. It is why nobody can credibly promise you the proven optimum on every order — and why the question of how long the calculation may run matters more than the question of the optimum.
The right comparison is not the perfect solution, it is today's
If the best solution cannot be proven in time, the question shifts. It is no longer how close you get to the theoretical optimum, but how much better you are than what happens today — within the time your process allows.
Well-tuned heuristics can deliver very good solutions in practice, often far faster than an exact method could prove optimality. How good they are in a given case depends on the problem class, the constraints and the structure of your parts; there is no general guarantee. For your decision a different gap matters anyway: not the one to the best possible solution, but the one between your nesting today and what a tuned method achieves on your parts. That is the gap you are paying for.
The time budget is a design constraint
A method that runs overnight is a different thing from one that must answer during order entry. Both are legitimate, but they are different systems. If your planner is sitting at a screen waiting, the permissible compute time may be one second. If the optimisation runs at night across the whole order book, it is hours — and a great deal more can be done with hours.
That is the first question we ask in a project like this: when does the answer have to be there? The answer determines the method, not the other way round.
Anyone who quotes you a percentage before seeing your parts, your materials and your machines is guessing. The achievable improvement depends entirely on how good your starting point already is.
The algorithm is not always the biggest lever
This is the part demonstrations leave out, because it does not show well. In many plants the larger gain sits not in a better packing method but in the constraints around it.
A single order rarely fills a sheet cleanly. Nesting several orders of the same material and thickness together creates the freedom the optimisation needs in the first place. This can be one of the biggest single levers — and it is not a question of mathematics but of planning, because it assumes you are allowed to batch orders, and that touches delivery dates
Last week's offcut is this week's raw material, but only if somebody knows it exists, where it is and what size it is. Without stock control over remnants, much of the calculated gain goes back into the skip. Managed remnant stock looks unspectacular on paper and can in practice achieve more than a better packing method under unchanged conditions.
Kerf consumes material on every cut and is routinely estimated too coarsely in manual planning. Grain or rolling direction restricts permissible rotations, which worsens the solution but is non-negotiable. Clamping zones, gripper areas and lead-ins are surface the machine occupies and the method has to know about.
On classic panel saws, cuts are typically run right through from edge to edge. That makes many geometrically possible arrangements unusable in production. Laser, plasma and waterjet do not have that restriction; depending on the machine and the material they have others, such as common cut lines, pierce points or thermal distortion where parts sit close together. A method that doesn't know your machine's rules produces nests that look better on screen and don't work on the floor.
An optimisation that isn't connected to your machine and your stock produces a picture. A picture saves no material.
Three examples from fourteen years
Two-dimensional bin packing against two weights at once
Packing cartons is the familiar task. The complication was that carriers charge on the higher of two figures: actual weight and volumetric weight. A packing that improves one regularly worsens the other. What had to be optimised was therefore not space but freight cost — a different objective function, and one standard software does not provide for.
Skeletonisation for centre-line extraction
Deriving the centre line from an arbitrary contour and generating a toolpath from it. This is image processing and geometry in one, and it is what allows a machine to follow a shape for which nobody has drawn a path.
Dimension estimation from a single photograph
Determining real-world dimensions from an image, accurately enough to cost a facade from it. No site survey, no model — a photograph and the mathematics behind it.
What these three have in common is that none could be solved with a better interface. They were computationally hard, and the usual vendors routed around them.
Six questions that have to survive the demonstration
This list is not tailored to us. Put it to anyone selling you optimisation, ourselves included.
Do you nest across orders, or only within one order?
If only within, the biggest lever is not connected.
How are remnants managed?
Stock, dimensions, storage location — or not at all.
What compute time is assumed, and at which point in the process?
One second during order entry and a night across the order book are two different systems.
Edge-to-edge cuts, kerf, permissible rotations, clamping and gripper zones, common cut lines.
How is the improvement measured?
Yield per sheet is the obvious metric and not always the right one. Where material is expensive it counts; where machine time is expensive, cut length counts; where freight dominates, something else does.
Does the result go to the machine, or to the screen?
The answer decides whether you have bought an optimisation or a visualisation.
Do the arithmetic yourself before believing anyone: compare the material consumed today for a real set of orders with the material that optimised cutting plans actually require for those same orders. The difference in euros is the starting point for the business case. Do not calculate with percentage points of yield — five percentage points more yield is not five percent less material. That measurement belongs at the start of a project like this, not at the end.
Thirty minutes, one real cutting plan.
Bring an order you cut last week, with your parts and your material. We will run it against your cutting plan. The result is either a number you can work with, or the information that there is little to gain in your case.