Cut List Optimizer

Four 24 by 48 inch panels add up to exactly one 4 by 8 sheet. A calculator will confirm it. The first cut turns an eighth of an inch into dust, and the sheet holds three panels and a strip.

That one cut contains the whole subject. Getting parts out of sheet goods is not arithmetic about area. It is a packing problem with physical rules: a blade with thickness, a saw that travels in a straight line, veneer whose grain runs one way, and a sheet that is not quite the size on its label. Respect those rules and a layout can be cut at the bench. Ignore one and it is fiction with dimensions.

A problem with a long paper trail

Operations researchers call it the cutting stock problem. In 1961 P. C. Gilmore and R. E. Gomory defined it as filling an order at minimum cost for specified numbers of lengths cut from given stock lengths. The trouble, they wrote, is that the number of variables generally makes computation infeasible. Every way of dividing a piece of stock is a pattern, and lecture notes from Duke University list six worth considering even for a toy roll of paper 11 units wide. Gilmore and Gomory’s answer was to generate a promising new pattern only when needed.

Four years later the same authors moved to two dimensions, the plywood case, with bad news and good. The shortcut does not carry over in general, because no efficient method exists for the harder sub-problem a rectangle creates. But a wide class of industrial problems yields anyway: the ones in which the cutting is done in stages. Cut the sheet into strips, then the strips into parts. The oldest useful result in the field describes how a workshop already cuts.

Staged or not, the problem stays hard. Packing rectangles into the fewest bins is proven to be NP-hard, and with two or more part sizes the optimization problems of guillotine cutting are NP-hard too. No known method is guaranteed to find the best layout quickly as the parts list grows, so working software relies on heuristics: rules that are fast and usually good.

Why a saw insists on guillotine cuts

A guillotine cut is a straight line running from one edge of a rectangle to the opposite edge, like the blade of a paper guillotine. It turns one piece into two, and each of those can be cut again. Glass, steel plate, cardboard and furniture panels are all cut this way. A table saw makes exactly this cut, because the panel is pushed completely past the blade. So does a track saw, whose rail is a straight line laid across the work.

Free layouts drop the rule. Parts interlock so that no straight line crosses the sheet without slicing through one of them, and cutting that takes a tool that can stop and turn. Cabinet factories use both: one CNC router maker describes sending plain rectangles to the panel saw and nesting the rest on full sheets at the router. In Jukka Jylanki’s survey of rectangle packing, the guillotine variants were the faster algorithms and packed slightly worse than the best free-placement method. That small penalty buys a layout a saw can cut.

The blade, the grain and the real sheet

Start with the blade. The slot it leaves is the kerf, and that width leaves the sheet at every cut. Freud lists its full-kerf 10 inch plywood blade at .126 inch, the thin-kerf version at .098 inch and its 160 mm track saw blade at .087 inch. So 1/8 inch is an honest default, and the tool on this page starts there. It packs each part as its size plus one kerf into a sheet one kerf larger, so neighbors sit one blade width apart and the far edges cost nothing.

Return to the four panels. Enter one part 48 by 24 inches, quantity 4, set the kerf to 0, and the tool reports one sheet at 100 percent. Restore the 1/8 inch kerf and it reports two sheets at 50 percent. A thin-kerf blade changes nothing. A design change does: panels of 23 15/16 by 47 7/8 inches fit on one sheet again. Kerf is small, but it is charged at every line, and a layout with no slack cannot pay.

Grain comes next. PS 1, the US product standard for structural plywood, describes a panel of an odd number of layers, with the grain of adjacent layers at right angles and the outer layers generally parallel to the long dimension. APA calls that direction the strength axis. For a furniture maker the visible consequence matters more: turn one door 90 degrees and its figure runs sideways. Kreg’s layout guide says not to lay out similar parts in different directions, except on painted work, utility pieces and panels nobody will see.

That is the purpose of the switch that lets parts turn 90 degrees. Leave it on for MDF, where the only goal is to minimize waste. Turn it off and every part keeps its length along the sheet. The sample cabinet shows the cost: still two sheets, but the largest offcut shrinks from 48 by 27 1/8 inches to 26 3/4 by 22 1/2.

Then there is the sheet itself. PS 1 allows a panel’s length and width to be up to 1/16 inch under the stated size and never over it. Panels marked sized for spacing may be 1/8 inch under; APA explains that they are made slightly smaller than the nominal 48 by 96 inches on purpose, to allow for spacing during construction. Squareness is held to 1/64 inch per foot of the longest edge, or 1/8 inch on an 8 foot panel. A layout that needs all 48 inches is betting against the standard.

Thickness is nominal too. PS 1 sorts panels into performance categories: a sanded panel in the 3/4 category may measure from 0.734 to 0.766 inch, and the 23/32 category runs from 0.703 to 0.734. That never changes a layout, but it changes the parts list. The sample cabinet assumes 3/4 inch: a 34 1/2 inch bottom plus two sides equals the 36 inch back. Built from 0.703 inch stock, the box is 35.906 inches wide and the back overhangs by about 3/32.

PS 1 speaks only for structural plywood, not for cabinet-grade birch or MDF, and one supplier lists its standard 3/4 inch MDF board at 49 by 97 inches. Measure the sheet you bought and type that in.

From full sheet to parts

Breaking down a full sheet has a named hazard. OSHA describes a kickback as a saw seizing the stock and hurling it back at the operator. It happens more often when ripping, and the agency tells operators to support every piece and stand to the side of the blade. DeWalt’s safety sheet gives the mechanism: large panels sag under their own weight, the kerf closes and the blade binds. Its remedy is support on both sides of the cut. One magazine simply lays the sheet on 2 inch foam insulation.

The order of cuts follows the logic of the layout. Woodsmith’s table saw method is to break the sheet into manageable rough-sized pieces first, starting with a rip into approximate halves, then cut to finished size. TSO’s track saw method begins by accepting that no sheet is square: cut a fresh straight edge along one long side, rip from it, then trim the factory end square before crosscutting. Either way, cuts that cross the entire sheet come first, and a guillotine layout always has them. On the sample cabinet’s first sheet, two crosscuts leave three manageable pieces.

Decide about factory edges before the layout. Woodsmith’s verdict is that they are usually straight enough to guide a cut but not clean enough to be a finished edge, so they always get trimmed. A tight layout treats every line as a finish cut and sets parts hard against the sheet’s edges. If you plan to trim 1/4 inch from one long edge and one end, enter a sheet of 47 3/4 by 95 3/4 inches. The sample cabinet still fits on two.

What forty attempts can and cannot promise

Load the sample cabinet and check it by hand. It has two sides at 30 by 23 1/4 inches, a top and bottom at 34 1/2 by 23 1/4, and two shelves at 34 1/2 by 22 1/2. Add a back at 36 by 30, two doors at 29 1/2 by 17 3/4 and a toe kick at 36 by 4. The ten parts total 6,823 square inches, or 47.4 square feet. One 48 by 96 sheet is 4,608 square inches, or 32 square feet, so no layout can use fewer than two. With a 1/8 inch kerf the tool fits everything on two: 6,823 of 9,216 square inches, 74.0 percent, with 16.6 square feet left over.

The method is the guillotine algorithm from Jylanki’s survey. It keeps a list of free rectangles, places a part in the corner of one, splits the L-shaped remainder in two along one straight line, and merges neighbors that form a larger rectangle. Three choices stay open: the order of the parts, which free rectangle to fill, and which way to split. Jylanki observes that changing one small rule can produce very different packings, so he tests every combination. The tool does the same on a small scale: five orderings, two fit rules and four split rules make 40 layouts.

It keeps the layout with the fewest sheets and breaks ties by the largest reusable offcut. On the sample cabinet, 14 of the 40 attempts need a third sheet, which is the argument for trying many. It is also the limit of the promise. A heuristic finds a good layout quickly. It does not prove that no better one exists. Here the proof comes from outside, because the area count already ruled out one sheet. When the area says three and the tool says four, nothing on the screen tells you whether the fourth is real.

Other things are outside its view. It cannot see a knot you want to cut around or match the figure of one door to the next. It assumes rectangular parts and one stock size. It gives positions, not a cutting sequence, and OSHA’s advice to avoid crosscutting long boards on a table saw is yours to remember. And as Jylanki notes, the algorithm never places a part across one of its own split lines, so it can miss space that is really there.

That leaves the offcuts. The waste figure counts every square inch outside a part, but waste is not one thing. More than half of the sample cabinet’s 16.6 square feet is one rectangle, 48 by 27 1/8 inches, the full width of the sheet. The rest is strips. That is why the tie-break favors one large offcut, and why Kreg advises aligning parts of similar size: the cutoffs come out more usable. Keep rectangles big enough to become a part you actually make. Let the slivers go.

A cutting diagram is a claim about one saw and one sheet. Feed it the real kerf, the real size and the truth about grain, and it holds at the bench. The searching is the computer’s job. The measuring is yours.