1D vs 2D cutting optimization: choosing the right nesting approach

"Cutting optimization" covers two different problems that happen to share a name. Cutting bars, profiles or pipe to length is a one-dimensional problem: every piece and every stock length is a single number. Cutting plates or sheets is two-dimensional: every piece has a width and a height, and where you place it relative to the others is part of the answer, not just how much of it you need. The algorithms, and the trade-offs, are different for each.

1D: length is the only variable

Given a cut list of required lengths and a set of standard stock lengths, 1D optimization is a bin-packing problem: pack the required lengths into the fewest stock bars possible, accounting for kerf (the material lost to each cut). The main lever is stock length selection. Mixing available lengths well matters more than clever placement, because there's no placement to be clever about. The main output that matters on the shop floor is a clear per-bar cut sequence and the leftover length from each bar, so offcuts worth keeping don't get thrown out with the drop.

2D: placement is the problem

Plate and sheet cutting is a different kind of problem, because where each piece sits on the sheet determines how much usable material is left for the next piece. Two broad families of algorithm solve this, and they optimize for different things.

Free-rectangle packing (the MaxRects family is the common modern approach) tracks the free space left on a sheet as a set of rectangles and places each new piece into whichever free rectangle wastes the least space, in any position. This maximizes material yield: it's the mathematically best answer to "how little material can I get away with." The visible cost is that the resulting layout can look irregular, with pieces staggered at different heights and cuts that don't line up across the sheet.

Guillotine cutting adds a constraint that free packing doesn't have: every cut has to go straight from one edge of the remaining sheet to the opposite edge, exactly what a panel saw or a shear physically does, since neither can stop partway across a sheet. This produces layouts that are visually cleaner and match what simpler cutting equipment can execute. The trade-off is mathematical, not incidental: a guillotine-constrained layout can only match or lose material yield compared to unconstrained free packing. It can never win, because it's solving the same problem with a stricter rule. Every real project that chooses guillotine is trading some yield for producibility.

Which one matters for your shop

The honest answer is: it depends on what's cutting the plate.

• CNC plasma, laser or waterjet can cut arbitrary shapes and stop anywhere, so a free-packing layout is directly executable. There's no reason to accept a guillotine constraint you don't need, and free-rectangle packing gets you closer to the material yield you're paying for.
• Panel saws, shears, and any process limited to straight, full-width cuts can't execute a free-packed layout as drawn. Handed one anyway, an operator has to improvise a producible cutting order on the fly. At that point the "optimal" plan on paper isn't the plan being cut, and the real yield achieved is whatever the operator manages to figure out under time pressure.

Neither approach is "correct" in general. A layout that beats another on paper by two percent of material yield is not a win if it can't be cut on the equipment that has to execute it.

What a cutting plan needs beyond the layout

The layout itself is only useful if it comes with what the shop floor needs: kerf accounted for in every placement, a material yield percentage so the plan can be compared against alternatives, and leftover pieces clearly identified as usable surplus, worth registering for the next job, rather than scrap swept up with the drop.

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