Four Lines, Not Fifteen: Sizing a Can Plant on the Right Calendar

Twelve million people at six cans a week is 3.744 billion cans a year, which over the 525,600 minutes a calendar year offers is 7,123 cans a minute, or 3.56 production lines, or 4.19 once 85 percent uptime is allowed. Size the same demand on a 40-hour week and you get exactly 15 lines, because the two calendars differ by 219/52 = 4.2115, a systematic factor that no numerator guess can cancel. Sweeping all four inputs over 200,000 draws moves the count between 1.4 and 7.7 with a median of 3.4, so the conclusion is sturdier than any of the guesses inside it.

How big a plant does it take to make the aluminium cans for a state of twelve million people? Nobody asking this wants a number to three decimal places. They want to watch you decide what to guess, admit that you are guessing, and then not fumble the division.

The division is where it goes wrong. Most attempts get the annual volume roughly right and then miss the answer by a factor of four, because they size a factory on a working week.

A factory is a rate, so the answer is a quotient

Before any number, get the shape. A plant is not a quantity of cans, it is a quantity of cans per minute, and a line is a quantity of cans per minute too. So the answer is a ratio of two rates, and it is dimensionless:

N = \frac{P \cdot c}{M \cdot r \cdot u}

(1)

with $P$ the population, $c$ the cans one person gets through in a year, $M$ the minutes a year offers, $r$ the cans a line does per minute and $u$ the fraction of the time it is actually running. The units cancel the way they should:

\frac{[\text{people}] \cdot [\text{cans}/\text{person}/\text{year}]}{[\text{min}/\text{year}] \cdot [\text{cans}/\text{min}/\text{line}]} = [\text{lines}].

(2)

That check is not pedantry. Half of the wrong answers to this question are wrong because something was divided that should have been multiplied, and (2) catches every one of them before a calculator gets involved.

What is asserted here and what is not

Nothing below claims a true number of cans. Three quantities are declared guesses: six cans a person a week, two thousand cans a minute on a line, and 85 percent uptime. What is asserted is the arithmetic, the calendar ratio, and the range the answer occupies when all the guesses are allowed to move.

The numerator, and the one guess that carries it

Six cans a week per person, averaged over everybody including the people who drink none. That is $6 \times 52 = 312$ cans a year, so

12{,}000{,}000 \times 312 = 3{,}744{,}000{,}000 \ \text{cans a year}.

(3)

Twelve million is a rounding, and it is worth saying so out loud rather than hoping nobody checks: Ohio counted 11.8 million people at the 2020 census, so the working figure is 1.7 percent high in exchange for arithmetic you can do while talking.

The per-person guess is the only input with no anchor, so give it a second route. Six standard 355 millilitre cans a week is

\frac{6 \times 52 \times 355}{1000} = 110.76 \ \text{litres a year} \approx 0.30 \ \text{litres a day},

(4)

which is a third of a litre of canned drink a day per person, counting infants. Anybody can hold that up against their own fridge and decide whether it is too high or too low. This is the discipline that separates an estimate from a number: the guess has to be checked against something the guesser can feel.

The calendar is the whole problem

Now the denominator, where the trap is. A canning line does not go home at five. It runs nights, weekends, and the whole of August, stopping only for maintenance and changeovers. The minutes available are therefore

M = 60 \times 24 \times 365 = 525{,}600,

(5)

and not the working year that a person instinctively reaches for,

52 \times 40 \times 60 = 124{,}800.

(6)

The ratio between them is exact and pleasant:

\frac{525{,}600}{124{,}800} = \frac{219}{52} = 4.2115,

(7)

which factors into $168/40 = 4.2$ for the hours in a week against the hours in a working week, times $365/364$ for the day that 52 weeks leaves out. The error is not noise and it does not partially cancel. It is a systematic factor of 4.2 sitting in the denominator, and it multiplies the answer by 4.2 whatever the numerator turns out to be.

Fig. 1 — Four office years fit inside one calendar year, with a fifth of one to spare. Every wrong answer to this question is somewhere inside that gap.

From a rate to a count of lines

Divide (3) by (5) and the plant has to average

\frac{3{,}744{,}000{,}000}{525{,}600} = 7{,}123 \ \text{cans a minute}.

(8)

A modern high speed line runs somewhere around 2,000 cans a minute, so that is 3.56 lines. Rounding up to four would be the lazy finish, and it happens to land on the right answer for the wrong reason. The honest step is uptime. Nothing runs 100 percent of the calendar: maintenance, changeovers between can sizes and breakdowns cost you something, and 85 percent is a reasonable figure for a well run line. So

N = \frac{3.5616}{0.85} = 4.19 \ \longrightarrow \ 4 \ \text{lines}.

(9)

Four lines is one plant. Fifteen lines, which is what the office calendar returns, would be three or four plants and a completely different capital plan. Notice that the office-hours branch lands on exactly 15: $3{,}744{,}000{,}000 / 124{,}800 = 30{,}000$ cans a minute, which is 15 lines of 2,000 with nothing left over. Arithmetic that comes out round is no evidence of being right.

Fig. 2 — The chain, with the wrong turn drawn beside it. Only the third step differs between the two answers.

Every guess is linear, which is the good news

Equation (1) is a single product and quotient, so $N$ has elasticity exactly $\pm 1$ in every input. Double the per-person guess and the answer doubles. Halve the line rate and it doubles. Nothing compounds, and no error can hide behind another error except by being exactly reciprocal to it.

That is what makes the conclusion sturdier than any of its inputs. Sweeping the population from 11 to 12.5 million, consumption from 200 to 400 cans a person a year, line rate from 1,500 to 3,000 a minute and uptime from 80 to 95 percent, over 200,000 draws, the line count runs from 1.4 to 7.7 with a median of 3.4. Not one combination in that whole box asks for more than eight lines, so "one plant" survives every plausible set of guesses. Run the same sweep on the office calendar and the median is 14.4, and the ratio between the two sweeps is 4.2115 at every single draw, which is the check that the trap is structural rather than statistical.

What this calculation is not

It is not a plant design. A real facility is not sized to average demand, because demand for canned drink is seasonal and inventory is expensive, so the summer peak rather than the annual mean sets the capacity. The 85 percent uptime allowance quietly carries roughly the slack that an average-based calculation ignores, which is convenient and is not the same thing as having modelled it.

It also ignores geography. Cans are mostly air, so freight is expensive relative to the value of the good, which pushes canning plants close to the filling plants they serve rather than into one optimal spot. Four lines might well be two sites of two. And it ignores the market: no producer serves a whole state, so the question "what would it take to supply Ohio" is a capacity question, not a business plan.

What survives all of that is the structure. The answer is a rate divided by a rate, the numerator errors are linear and forgiving, and the single input that can move the answer by a factor of four is the one nobody thinks to question, which is how many minutes there are in a year.

Sources and further reading

The general method, including the habit of a second independent route — Fermi problem
The units check in (2) — Dimensional analysis
The uptime factor $u$ , and how it is measured in practice — Overall equipment effectiveness
The population figure quoted against the rounding — Ohio
The 355 millilitre can behind (4) — Drink can

Every step above was recomputed in exact integer and rational arithmetic, so no floating point enters the chain: $6 \times 52 = 312$ , $60 \times 24 \times 365 = 525{,}600$ , $52 \times 40 \times 60 = 124{,}800$ and the office branch landing on 15 exactly. An earlier draft of this problem carried 3.6 billion cans instead of 3.744 billion, which the sensitivity sweep caught: the two give 4.03 and 4.19 lines, so the conclusion was never at risk, but the chain did not close and a chain that does not close is a chain a reader can catch you on.