The plan that works on average

An optimiser answers the question it was given, exactly and without commentary. The question it was given at Ravensworth contained a single number for each product's demand, and that number was the average. There is nothing careless about that choice, it is the standard way to write the problem down, and the plan that comes back is genuinely the best plan for those inputs.

The trouble is what "those inputs" describe. A week in which Standard demand is exactly 33 tonnes and Dairy demand is exactly 63 tonnes is not a typical week. It is a week that essentially never happens. The plan is optimal for it anyway, because that is the week it was handed.

• Deterministic plan. One computed from single fixed numbers, with no spread anywhere in it. Ravensworth's 33 / 63 / 0 is one.
• Simulation over a plan. Draw uncertain demand many times, run the same fixed plan against every draw, and record what it earned. Ten thousand draws here.
• Seed. The starting point of the random draws. Change it and you get a different ten thousand weeks, which is how you check the answer is not an accident.
• Mean profit. The average of those ten thousand weeks. Ravensworth: 4,318 to 4,331.
• Median. The middle week. Half of them are worse.
• 5th percentile. The line one week in twenty falls below. Ravensworth: roughly 3,540 to 3,591.
• Shortfall. The gap between what a deterministic plan promises and what it delivers on average. Here 325 to 338.
• Sampling a posterior. A different technique with a similar name, belonging to the Bayesian course. Nothing tonight is walking a posterior; tonight draws inputs and re-runs a plan.

The mill commits to 33 tonnes of Standard before it knows what the week will ask for. In a week where Standard demand comes in at 28, five of those tonnes earn nothing this week. In a week where it comes in at 38, the mill sells all 33 and no more, because 33 is what it made. The upside is capped by the plan and the downside is not, so averaging across many weeks lands below the figure computed at the average.

That asymmetry is the whole mechanism and it is worth saying slowly, because it is not a modelling error and it does not go away with a better forecast. A fixed quantity committed before uncertain demand pays for every light week and is not compensated by the heavy ones.

About a quarter, in every one of the five runs. Precisely, it counts the weeks in which demand took every tonne the plan made, for both products at once. Three weeks in four, at least one product was made in a quantity the week did not want.

Say that figure out loud next to the plan and notice how differently the plan sounds. "The optimal plan, worth 4,656" and "the optimal plan, which works as intended about one week in four" are descriptions of the same object, and only one of them prompts the next question.

A single simulation run produces a number, and a number invites the reply that you got a bad draw. Five runs from five different starting points, ten thousand weeks each, produce five numbers that can be compared with each other. Here they land at 4,317.94, 4,327.93, 4,330.55, 4,328.35 and 4,323.93. The spread between the runs is a few pounds. The gap to the promise is over three hundred.

That contrast is the argument, and it is stronger than any single run could be. The measurement noise and the effect being measured are two orders of magnitude apart, so the finding survives the obvious objection without anybody having to defend a particular seed.

Replace the single figure with three and the conversation changes without anyone needing to understand a solver. How often does this plan do what it was designed to do. What does the middle week earn. What does one week in twenty look like. Those are answerable, they are answerable by simulation over a plan you already have, and none of them requires a new model.

The third one is usually the one that matters, because it is the only one that speaks to whether a bad week is survivable. A business with covenants, seasonal cash or a thin buffer is not indifferent between a plan that earns 4,440 in the middle and dips to 3,540 one week in twenty, and a steadier plan earning slightly less. The mean cannot tell those two apart, which is precisely why it is the number that gets reported.

In the box above, take Dairy down a few tonnes from 63 and leave Standard where it is. The deterministic margin falls immediately, by exactly what you would expect. Watch what happens to the 5th percentile and to how often the plan sells out, and try a few combinations before drawing a conclusion.

Whatever you find, notice the shape of the exercise: you are now trading a promised figure against a downside figure, by hand, because the problem as written contained no way to express that preference. That is not a failing of the mill or of the solver. It is a consequence of a problem statement with one number per product in it.

Nobody argues with the measurement. There is a live argument about what to plan for.

One camp plans for the mean and means it. Over a year the mill runs fifty-odd weeks, the good ones and the bad ones both turn up, and what accumulates is the average. Hedging every single week to protect a 5th percentile costs margin in all fifty-odd of them, most of which were never going to be bad weeks. On this view, protecting against the bad week is an expensive insurance premium paid continuously against an event that is survivable, and the discipline is to hold your nerve and let the averages arrive.

The other camp says the business does not experience the average, it experiences weeks, and weeks are where things break. A run of three bad ones in a quarter can breach a covenant, empty a cash buffer or cost a customer, and none of those are repaid by a good August. On this view the mean is the right number only for a business with enough balance sheet to be indifferent to sequence, which is most textbook firms and not many real ones, and the 5th percentile is the number that should sit next to the plan.

The honest position is that the answer depends on facts that are not in the model: how much buffer exists, whether bad weeks cluster, and what a bad week actually breaks. Notice that neither camp is arguing about the simulation. They are arguing about what the organisation can absorb, which is a question for the people in the room and not for the solver.