Sixteen commercial sofas for a hotel residents' lounge. On eight of them the front-left foot stood 2 mm above the floor. Every leg on every sofa was inside the tolerance the purchase order gave, because the fault was never in the leg.

The order, and the one line it wrote about legs

A hotel group was refitting the residents' lounge of a 120-room property and ordered sixteen commercial two-seat sofas from a factory in Foshan. The model was a steel base frame carrying four bolt-on steel legs, a 150 mm leg, with the leg centres 1,500 mm apart across and 700 mm front to back. The unit price was $320 and the order was $5,120. Thirty per cent, $1,536, was paid on order and seventy per cent, $3,584, was due against a pre-shipment inspection report.

The purchase order carried one line about the legs. Legs: steel, 150 mm, tolerance plus or minus 1.5 mm.

That was the whole of it, and it is worth noticing what the line does not say. It does not say from where the 150 mm is measured. From the leg's own mounting face down to the foot, it is the length of a part; from the underside of the frame's bottom rail down to the floor, it is the height at which the sofa stands. Both readings are taken with the same tape on the same sofa, and they can be 2 mm apart on a sofa whose legs are all the same part. Nothing on the order says which one the buyer is buying.

It also says nothing about the plate the leg bolts to, nothing about the four feet bearing on the floor at the same time, and nothing about the sixteen sofas matching each other when they were set in a row. This is a commercial sofa inspection case in which that one line, read the way it was written, was most of the dispute.

What the mock-up showed before anybody checked a number

The lounge seating is a continuous bench, so the client approved it as a run rather than as a sofa. Two runs of eight, back to back down the length of the lounge, thirteen metres of front line in each. The factory set all sixteen out in the finishing bay for that approval, on a taped floor line.

From the doorway the front of the bases did not read as one line. Eight of the sixteen sat with the front-left corner of the base about 2 mm proud of the floor, and the line rose and fell by that amount every few sofas.

Two millimetres over 1,500 mm of width is less than one tenth of a degree, and on a single sofa in a photograph it is not a thing anybody sees. In a run of eight it is a step in a thirteen-metre line, and the eye finds a break in a long straight edge at a fraction of the size it takes to notice a tilt in one object. The hotel's interior designer photographed it, and the photographs went to the buyer the same day.

Nothing had been measured. What had been established was that the run did not look like a run, and that no line on the purchase order said what a run ought to look like.

The inspection that measured the legs, and the legs were fine

The order was inspected at the Full Verification level, two man-days at $299, and because the mock-up had put the legs in question the dimensional work was carried out on all sixteen units rather than on a sample. Sampling was not the weak point in this order, and it is worth saying so, because it is the usual explanation offered afterwards. With 8 of 16 units affected, a random sample of 5 would have found the condition 98 times in 100. The odds of missing it completely are 1.3 in 100, since a lot of 16 falls in code letter C under ISO 2859-1 and general inspection level II gives a sample of 5. The weakness was in the criterion.

On all sixteen sofas the inspector measured every leg with a caliper, from the leg's own mounting face down to the foot, and read 149.9 to 150.1 mm. Sixty-four readings, every one inside 148.5 to 151.5. The legs were correct legs.

What the inspector also did, because the mock-up photographs had asked for it, was stand each sofa on a checked-flat reference surface and read the height of each of the four feet above it with a height gauge. On eight of the sixteen the front-left foot came out 2.0 mm above the surface and the other three were on it. On the other eight, all four were on it and the largest reading was 0.2 mm.

Both sets of numbers are in the report, and both are true. The report said the legs conformed, because they did, and it said that half the sofas did not stand on four feet, because that was also true. Only one of those two sentences had a criterion behind it.

Three measurements, and the two different diagonals

There were three measurements on that sofa, and they answer three different questions. That is why the report was able to say two apparently contradictory things without either of them being wrong.

The leg length, taken from the leg's own mounting face to the foot, answers the question of whether this leg is the right part. All sixty-four legs were 150.0 mm.

The plan diagonals, taken corner to corner between leg centres, answer the question of whether the frame is square. The calculated diagonal for a 1,500 by 700 mm leg pattern is 1,655.3 mm, and on every one of the sixteen sofas the two diagonals came out at 1,655.2 and 1,655.4. A difference of 0.2 mm. The frames were square.

The foot heights, read with a height gauge on a checked-flat reference surface, answer the question of whether it stands on four feet. And the way those four numbers are put together is a fourth measurement: the sum of two opposite feet against the sum of the other two. On any four points lying in one plane those two sums are equal, whatever the tilt of that plane, so the difference is a direct measure of how far the feet are from being coplanar. On the eight good sofas it came out between 0.0 and 0.2 mm. On the eight bad ones it came out at 2.0 mm, because the front-left foot was 2.0 mm up and the other three were not.

Both of those last two are called diagonals, and they answer different questions. This is where the case would have been lost if the inspector had stopped at one of them. A sofa can be perfectly square and still stand on three feet. A sofa can stand on all four feet and still be out of square, in which case the front rail runs out of true and the unit leans with every foot on the floor. The plan diagonal and the coplanarity residual have to be measured separately, and calling one of them a diagonals check is how a fault gets missed.

The arithmetic of what one tolerance line permits

Take the plus or minus 1.5 mm and apply it to the four legs of one sofa. If the front-left leg is at the bottom of its band at 148.5 mm and the other three are at the top at 151.5 mm, then on a level surface that front-left foot finishes 3.0 mm above the floor while the other three are on it. Every leg in that sofa is inside the tolerance the buyer wrote.

So the order, as written, permitted a 3.0 mm gap under one corner of a sofa — with no fault in the factory at all. The 2.0 mm that was actually found was inside what the order allowed.

And then there is the part that made the line worth nothing, which is not the tolerance at all. The fault was never in the leg. Every leg in the order was 150.0 mm and every one came from the same carton. The plate welded under the front-left of the bottom rail on the affected frames carried about 2.0 mm of extra weld build-up on its face, so the leg sat 2.0 mm higher off the floor than its three neighbours.

That is the difference between a tolerance on a part and a criterion on an assembly. The leg tolerance could not have caught this fault no matter how carefully it was applied, because the fault was 2 mm of extra weld metal on a plate the order never mentioned and the leg itself was perfect. A tolerance on a part tells you how good the part is. It says nothing about what the assembly does.

Why the front-left corner, on eight units in a row

The factory found it in an afternoon. The leg plates are welded under the bottom rail in two cells, and on the frames that came out of one of them the front-left plate carried about 2 mm of extra weld build-up on its face, left behind by a different weld preparation and a slightly different electrode angle on that fixture. A leg bolted to that plate stood 2 mm high, and its foot finished 2 mm above the floor while the other three finished on it.

The legs were never the problem and never would have been. Every leg in the order was the same part from the same carton of the same purchase order, cut to 150 mm, and every leg measured 150.0 or 150.1. A leg measured in isolation was the best leg in the world.

The other 8 sofas came out of the other welding cell, whose plates were clean, and they are correct. That is why the run of 16 held exactly half good units and half bad, and why the fitters could not tell them apart by looking at them.

Which returns the case to the one thing it is actually about. The bad frames and the good frames are the same part number, the same drawing, the same purchase order and the same tolerance line. Nothing in the manufacturing record distinguishes them except which cell they came out of, and the factory does not put a cell number on a carton. A buyer who wants that distinction has to put it there.

The single-unit tolerance and the group result

The tolerance the buyer wrote is a per-unit quantity. Plus or minus 1.5 mm on a leg describes one sofa standing on its own, and it says nothing whatever about how sixteen sofas relate to one another when they are butted into a bench.

Consider the same eight bad units in two different arrangements. Shipped in a mixed order to eight different hotel rooms, each tilted by less than one tenth of a degree, with a 2 mm gap under one foot on a carpeted lounge floor, they would have passed through years of service and nobody would ever have known. That is the point of a tolerance. It is a promise about a unit.

Set the same eight units in a row and the front of the bases steps 2 mm every few sofas along thirteen metres of front line. A fitter's 1.5 m straightedge reads it on the spot, and a hotel group that has signed off a lounge and then watched the photographs come back with a stepped line in it does not have a parts problem. It has a room problem.

So an order for furniture that goes in as a run needs two tolerances, not one, and the second one has to be written in the units the fitter will use on site: millimetres of step at a joint, and millimetres over a 1.5 m straightedge along the front line. A factory can only be held to a set tolerance that was written down, and it can only be inspected against a set tolerance that was measured. The unit tolerance and the set tolerance are different numbers — and the 8 units in this order would have been caught by either one.

How the report was written

The report was written against the drawing set and the purchase order rather than around them, and it was organised so that the buyer could see that the rest of the sofa was sound and the legs were the only open item.

The section that carried the most weight was the last one. It stated that the order contained no criterion for the four feet bearing simultaneously and no criterion for the sixteen units matching each other, that the 8 affected sofas were therefore recorded as an observation against the drawing rather than as a non-conformity against a clause that did not exist, and that the condition was static and had not been checked under load or after upholstery.

What the fault cost, and what it would have cost

The factory reworked the 8 affected sofas rather than ship them. The front-left leg plates were ground back flush and the legs re-bolted, $96 a sofa, $768 in total, and the factory absorbed it because the balance had not been released and because the follow-on orders were worth more to them than the plates were. Our own time on the order was the $598 for the full verification, plus one further man-day at $199 to re-check the 8 reworked sofas, $797 in all.

The alternative was not worth putting in a report. The client would have accepted 16 sofas into a dressed lounge, found the stepped front line after the furniture was in place, and had to take it back out. Eight replacements at $320 each and $256 of freight is $2,816, six weeks of lead time on a sofa the client had already designed the lounge around, and a residents' lounge standing closed while they waited. The original eight would not have been scrap, they would have been stock, but they would have been the wrong eight, and nobody could tell which was which without a height gauge.

Neither of those numbers is the cost of a 2 mm step. They are the cost of a line nobody measured, which is the general shape of this whole case.

And behind all of it sat $3,584 of unpaid balance, which was the reason the conversation was still a conversation.

Nine clauses, and every one of them is one line

Nothing here is exotic. Every clause below can be written by a buyer in fifteen minutes and understood by a factory immediately, and each one can be turned by an inspector into a pass or a fail with a height gauge, a tape and a flat surface.

• State the datum for the leg. Leg length 150 mm plus or minus 1.5 mm, measured from the leg's mounting face to the foot. If the height the sofa stands at is what is being bought, say that instead, measured from the underside of the frame's bottom rail down to the floor.
• Write the four feet, not just the four legs. All four feet shall bear on a level surface at the same time, with no foot more than 0.5 mm above it, checked with a height gauge on a reference surface flat to within 0.5 mm over 3 m.
• Write the coplanarity residual. The difference between the sum of the front-left and rear-right foot heights and the sum of the front-right and rear-left foot heights shall not exceed 1.0 mm. This is the clause that catches the case above, and it catches it whether the fault is in the leg, in the plate or in the frame.
• Write the squareness separately. The two plan diagonals of the leg pattern shall not differ by more than 2 mm, measured corner to corner. A square frame standing on three feet and an out-of-square frame standing on four are different faults and need different clauses.
• Write the leg plates. The welding of the leg plates shall leave no build-up on the plate face, and the plates shall be checked on a jig after welding and before assembly. On a frame with a fixed cell this is a two-minute check.
• Write the group tolerance, because the unit tolerance will not do it. All units for one area shall match in installed height to within 1.0 mm at the front rail, and the front line of an assembled run shall be within 2.0 mm over a 1.5 m straightedge. A row of furniture is bought as a row.
• Write the set. Units for one area shall be numbered as a set, packed in that order, and carry the set number on every carton together with the production cell or batch reference.
• Write the seat height across the lot. Seat height shall be within 1.0 mm of the mean for the order, measured at the front edge of the seat cushion, so that a bench reads as one bench.
• Write what the inspection measures. The four installed foot heights, the coplanarity residual, the plan diagonal and the leg lengths shall be recorded for every unit sampled, so the report carries the sofa and not only its legs.

Three questions for the next order

Does the order say from where the leg is measured, and from where the sofa's height is measured. A per-leg tolerance and a per-sofa tolerance are different numbers, and only one of them holds a row of furniture straight.

Does the order say how the four feet of one sofa must behave, and against what surface the sofa is standing when it is measured. Without both of those, the reading has no reference and the number is not comparable between one factory and the next.

Does the order say what the units going into one area must match, and how they are to be identified as a set on the cartons.

None of these needs a factory visit, a gauge or a sample size. They are three sentences on a purchase order, and the case above turned on their absence.