A fault map is worth most where the grade is worst

Two accounts of this account have ended at the same recommendation. A missing end is a fault the length of the piece observed that every expression in it with an exponential in it is an expression about ignorance rather than about cloth, and that a fault whose position is known needs no probability at all. A grade charges by the length and a cutter pays by the panel found the grading scheme uninformative about yield and suggested the same repair: record the positions and let the cutter do the arithmetic.

Neither computed what the recording is worth. A fault map costs nothing to make on a modern inspection frame, and what it records is the position of something a broken end leaves down the whole piece, and a recommendation that costs nothing is one nobody checks. This account checks it, and two of the three things it finds are not what the earlier accounts expected.

There are two things a map lets a cutter do, and only one is worth anything #

A marker is a plan of panels laid on a length of cloth. Given a map, a cutter has exactly two freedoms.

The marker can be slid. The panels stay in their rigid tiling and the whole plan moves along the piece, so a fault that fell in the middle of a panel may fall on a boundary instead. It costs one decision, wastes at most one panel’s length of cloth at the start, and it is the freedom most people would think of first.

The marker can be broken. A panel may start anywhere, so a spoilt stretch is skipped and the next panel begins after it, with the cloth between them scrapped. It costs whatever is skipped and it is what a cutting room actually does when it knows where the faults are.

Sliding the marker recovers two panels and breaking it recovers twelve. That is the first finding and it is worth stating as a ratio: the freedom everybody would reach for first is worth a sixth of the freedom that matters.

The reason is a counting argument rather than anything about cloth. A rigid tiling has one degree of freedom and the faults have many, so sliding it trades one fault’s position for another’s and gains only where two faults happened to be nearly a panel apart. Breaking the tiling gives the marker one degree of freedom per panel, which is what it takes to answer a field of faults.

The greedy sweep is the answer, not a heuristic #

Cutting around looks like an optimisation problem and it is one, and it is worth saying that it is a solved one, because a result that depended on a heuristic would be a result about the heuristic.

Sweep from one end of the bolt. At the current position, start a panel. If the panel would contain a fault, move to just past that fault and try again. Otherwise take the panel and move on by its length.

That is optimal. A panel started later than necessary cannot allow more panels after it, because every panel is the same length: any solution that starts its first panel later can be transformed, panel by panel, into the greedy one without ever losing a panel. So the number the sweep returns is the most panels the bolt can yield, and the comparison against the blind cut is a comparison against a true ceiling rather than against a good try.

That would stop being true the moment the panels were of different lengths, which is what a real marker has — and it is the honest limit of this calculation rather than a detail.

The gain has a maximum, and it is not where the cloth is worst #

Running the same bolt at every fault rate gives the shape of what a map is for.

A map is worth nothing on a clean bolt and nothing on a ruined one, which is the same shape a fault rate takes when it is multiplied by a piece size, and the maximum in between is at about 0.8 faults a metre for a nine-hundred-millimetre panel — a rate at which a blind cutter loses half the bolt and a mapped one loses a quarter.

Read as a share rather than as a count the picture is different and both readings matter. The proportional gain rises all the way: 6 per cent more panels at a tenth of a fault a metre, 43 per cent at 0.8, 69 at two, and at five faults a metre the map doubles a yield of nothing. The count peaks and the share does not, because at high rates the map is recovering a large fraction of a small number. A mill deciding whether to buy an inspection frame wants the count; a cutter deciding whether to bother with the map on a particular roll wants the share.

The previous account’s prediction is false #

The grading account ended with a prediction it called the sharpest thing to be said about the whole arrangement: a fault map should be worth most on the roll a grade says is best, because a roll whose faults are clustered leaves long clear runs, and a clear run is exactly what a mapped marker can exploit.

It is not true, and the arithmetic says why in one sentence.

Bunching leaves clear runs, and a clear run needs no map to find. A cutter working blind tiles the bolt from one end and every panel in a clear run comes out sound by accident. What the map buys is the difference between what accident supplies and what planning supplies, and a clustered bolt supplies most of it by accident.

So the prediction had the right mechanism and the wrong conclusion. Clustering does help; it helps the cutter who has no map at least as much as the one who has, and the gap — which is what a map is worth — closes rather than opening. A map is worth most where the blind cut does worst, which is a bolt whose faults are spread about one to a panel.

That is an uncomfortable result for the recommendation, because it says the map earns its keep on exactly the rolls a mill would most like to sell as good. It is also the useful form of the advice: ask for the map on the mediocre rolls, not on the awkward-looking ones.

The panel size decides the whole of it #

The one input that moves every number here is the one the mill does not have, which is the earlier account’s finding arriving from the other direction.

At 0.8 faults a metre, a three-hundred-millimetre panel goes from 80 per cent sound to 90 with a map, and a two-metre panel goes from 16 per cent to 36. The small panel gains a seventh and the large one gains two and a quarter times.

The reason is the same exponential the first of these essays established. A small panel is unlikely to contain a fault at all, so the blind cut is already near the ceiling and there is nothing for a map to recover; a large panel almost always contains one, so the blind cut is near the floor and the map’s freedom to place panels between faults is worth everything.

So the value of a fault map is a function of the panel, not of the cloth — and it rises exactly where the cloth was least usable to begin with. A furnishing cutter taking two-metre drops gains more from a map than a shirt cutter gains from a much better roll.

What the map costs, which is scrap #

Nothing here is free, and the cut-around line buys its panels with cloth.

Every time the sweep skips past a fault it abandons whatever lay between the last panel’s end and the fault — on average half a panel’s length per fault skipped. At 0.8 faults a metre on a fifty-metre bolt that is around forty faults, of which the sweep skips past most, so the scrap is a few metres of the fifty.

That scrap is cloth the blind cutter also lost, since it lay inside a panel they cut and threw away, so the comparison is fair as a count of sound panels and is not a comparison of waste. What changes is where the waste is: the blind cutter’s waste is full panels of cut cloth, the mapped cutter’s is short off-cuts. For a cutting room those are not the same thing — an off-cut may be usable for a facing or a pocket where a spoilt panel is not, and a cut edge frays whatever it is used for — so the map’s real gain is a little larger than the panel count says, by an amount that depends on what small pieces the garment needs.

What the map does to the specification #

The arithmetic has a consequence for the contract, and it is the one that would actually change anything.

A purchase order names a points figure, and the grading essay showed that number is nearly uninformative about yield. What a map offers is not a better number of the same kind; it is a different quantity altogether — the buyer’s own yield, computed for the buyer’s own panel, from the seller’s own inspection.

That changes what the two parties are arguing about. Under a points specification the mill sells a property of its loom shed and the buyer needs a property of their cutting room, and the gap between them is absorbed by a cutting allowance that neither can compute. Under a map the mill sells a list, the buyer computes a yield, and the allowance becomes a calculation with an exact answer for a stated marker.

It also changes which roll is which. Two rolls of one grade have different yields, and the yields are now readable before the cloth is bought — so the map turns a grade from a contract term into a summary statistic, which is what a grade should always have been. The number does not have to be better; it has to stop being the thing the contract is written on.

And there is a reading of the clustering result that belongs here rather than above. A mill whose faults cluster delivers better cloth at the same grade, gains least from publishing a map, and has the least incentive to publish one. A mill whose faults are spread delivers worse cloth at the same grade and gains most. So the incentive to publish a fault map runs opposite to the quality of the cloth, which is an unusual shape for a disclosure to have and is worth knowing before anybody legislates for one.

Why a cutting room does this by eye and gets most of it #

None of this is news to a cutting room, which has always cut around faults it could see, and the arithmetic says how much of the answer that habit reaches.

A cutter working from a marked roll — the mill’s own inspectors tie a string or stick a tab at each fault, which is the oldest form of fault map there is — is doing the greedy sweep by hand, one panel at a time, and the sweep is optimal. The habit is not an approximation to the calculation; it is the calculation, performed by somebody who never thought of it as one.

What the habit does not reach is the two-dimensional case and the mixed-panel case, which are exactly where a marker-making program earns its licence fee. The hand method gets the whole of the one-dimensional answer and none of the rest, and the rest is the part nobody can do by eye — which is a fair description of where computation has actually displaced judgement in a cutting room.

What was computed, and how #

A bolt’s faults are placed from a stated rate, either singly or in clusters, with the positions fixed by a seed so that every figure of one bolt reads the same bolt. The blind yield is the rigid tiling from one end with every panel containing a fault discarded; the phased yield is the best of sixty offsets of the same tiling; the mapped yield is the greedy sweep. All three count sound panels of the whole number the bolt’s length allows.

Five things are checked. Cutting around is never worse than phasing and phasing never worse than cutting blind, at every rate — a nesting that holds by construction and would fail if any of the three were counting differently. What a map buys has a maximum in the middle of the range and not at either end, which is the essay’s second finding stated so that a monotone result would fail it. Clustering leaves more panels clear even to a cutter who knows nothing. And therefore leaves a map less to recover — the earlier account’s prediction, tested and refused, stated in the direction the arithmetic actually goes. And a smaller panel loses less of the bolt blind, which is the first essay’s exponential checked in this arithmetic.

The fault rate, the cluster size, the panel length and the bolt’s dimensions are inputs. The faults are placed at random rather than measured off any frame.

Where the model stops #

The bolt is one-dimensional. Every fault here is a point along the length and every panel is the full width, which is the case for a curtain drop and is not the case for a garment. A real marker places panels in two dimensions, a fault has a position across the width as well as along it, and the greedy sweep is optimal for the first and is not optimal for the second.

The panels are all one length. That is what makes the sweep optimal, and a real marker mixes panel sizes deliberately so that a small piece can fill a space a large one cannot. A mixed marker is a harder problem and it can only do better than this, so the numbers here are a floor on what a map is worth rather than an estimate of it.

The faults are points. A fault with a length spoils every panel it touches, which the earlier account counts and this one does not; including it would raise every loss and would not change which of the three cuts wins.

And a fault is taken as spoiling a panel absolutely. A fault in a seam allowance costs nothing — and the selvedge is trimmed before any panel is cut, which is a band of the same exemption at every edge, and a cutter with a map places the marker to put faults exactly there — which is a third freedom, larger than either of the two counted here, and one that needs the garment rather than the cloth.

Still open: what the same arithmetic does in two dimensions #

The one-dimensional answer is exact and the two-dimensional one is the question a cutting room actually has, and the step between them is not small.

With faults at points in a rectangle and panels of a stated size to be placed without overlapping, the best marker is a packing problem, and packing problems of that shape are not solved by sweeping. What can be said in advance is that the two freedoms separate differently: across the width a bolt is only a few panels wide, so there is very little room to move, while along the length there is as much room as the bolt is long. So the second dimension should add much less than the first, and the one-dimensional answer may be most of the answer.

Whether it is, and by how much, is a computation the geometry here would support — it is simple and the sizes are small — and it has not been run.

Who worked it out #

Marker planning with known defects is ordinary practice in a cutting room and modern marker-making software does it automatically; the trade calls it defect mapping and the frames that record it have existed for decades. What is added here is the separation of the two freedoms and the finding that only one of them matters, the peak in what a map recovers, and the refusal of the clustering prediction the earlier account made.