Robust and stochastic, in outline

It is easy to hear "stochastic optimisation" as the fix for what session 7 found, and it is not. The mill still commits one production plan on Monday and then lives through one week. Writing the distribution into the problem changes which single plan you get and it cannot change the fact that a single plan faces many futures. The shortfall against a figure computed at the average does not vanish; the plan is simply chosen with the shortfall in view instead of in ignorance of it.

What actually changes is the objective you are optimising. At the moment the objective is profit in one imagined average week. It can become expected profit across the whole distribution, or profit in the worst week of some set, or profit subject to a promise about how often the plan works. Those are three different numbers and they pick three different plans.

• Scenario. One complete set of values for the uncertain inputs. A draw from session 7 is a scenario.
• Expected value formulation. Maximise the average objective across many scenarios, all of them in the problem at once. Commits you to a probability for every scenario.
• Two-stage problem. Some decisions are made before the uncertainty resolves and some after. Ravensworth's production is a first-stage decision; nothing here is second stage.
• Uncertainty set. The collection of futures a robust formulation promises to survive. Commits you to a boundary: inside it you are protected, outside it you are not.
• Robust formulation. Maximise the worst objective over the uncertainty set. Commits you to the set, and pays for the protection in average performance.
• Price of robustness. The average performance given up to buy that worst-case floor. Always positive unless the set is trivial.
• Chance constraint. A requirement that a condition hold in at least a stated fraction of futures. Commits you to naming the fraction.
• Conservatism. How wide the set or how high the fraction. The dial that turns a formulation from useless to paralysing, and it is set by a person.

This is the honest limit and it deserves saying before anyone gets attached to the idea. Put five hundred scenarios in the problem, weight them, maximise the average, and what comes back is one vector of tonnages. It is a better-informed vector than the one computed at the mean, because it knows that light weeks exist and that tonnes made in a light week earn nothing. It is still a single commitment facing a distribution.

There is a second thing worth being clear about. Maximising an average is a statement that you are indifferent between a steady plan and a volatile one with the same mean. If that is not true of the business, the expected value formulation is optimising the wrong thing, and it will do so very competently.

The case where a two-stage version genuinely changes the picture is when some decision can wait. If the mill could hold part of its protein unallocated and blend late, that later decision is second stage, it is allowed to see the week, and the value of having it is exactly the sort of thing a stochastic formulation is good at pricing. Ravensworth as written has no second stage, so the plan is one commitment and nothing else.

"Feasible in 95 percent of weeks" sounds like a guarantee and is a statement about a model of the weeks. It is only as good as the distribution it was computed against, and it says nothing whatever about the other 5 percent. A plan that fails gently in one week in twenty and a plan that fails catastrophically in one week in twenty satisfy the same chance constraint.

That is why the fraction is rarely the interesting question. The interesting question is what happens on the wrong side of it, and answering that means going back to a simulation and reading the tail rather than the threshold. A chance constraint is a good way to write a service promise into a model and a poor way to think about risk on its own.

Nothing above has been solved for Ravensworth, and the reason is that the inputs each formulation needs do not exist for the mill. The list is short and worth writing out, because it is the same list every time somebody proposes one of these.

• A demand distribution somebody will defend. Not just a mean and a spread but the shape, fitted to a stated number of weeks of order history, with a view on whether it is stable across the season.
• Whether the products move together. Two independent demands and two demands that rise and fall with the same customer give very different tails, and the difference is larger than most people expect.
• A scenario set and its probabilities, for the expected value version. Sampling from the fitted distribution is the usual route, and then the number of scenarios is a convergence question of exactly the kind session 7 measured.
• An uncertainty set, for the robust version. A boundary in demand space, with somebody prepared to say what is inside it and what is deliberately outside.
• A fraction, for the chance-constrained version, named by whoever owns the consequence rather than by whoever writes the model.
• What a bad week actually breaks. Until that is written down, there is no basis for choosing how much mean to spend, and the conservatism dial gets set by temperament.

Every item on that list is a measurement or a decision, and none of them is a solver feature. That is the honest state of the problem, and it is a more useful thing to hand a client than a plan computed from assumed inputs.

Press the button and the page walks every whole-tonne plan in a range, checks it against the three capacity rows, simulates the feasible ones, and reports the best few by average and the best few by 5th percentile. They are different plans, which is the trade made visible with no formulation at all.

It is worth being exact about what that is. It is enumeration, and it works here only because Ravensworth has two active products and whole tonnes, so the grid has a few hundred entries. Add a third active product and it has tens of thousands; add a week-by-week horizon and it stops being possible. A stochastic or robust formulation is what you reach for when you can no longer afford to look at every plan, and its job is to get the same answer without the enumeration.

It is also not a robust solution in the technical sense. Nothing here maximises a worst case over a declared set. It samples one distribution, at a stated seed and draw count, and reads a percentile off it, which is a different object and a weaker promise.

In the results, several different plans often share exactly the same 5th percentile. That is not a rounding artefact and it is worth a minute. If the draw sitting at that percentile is one where demand came in below the plan on both products, then everything the mill made was surplus to that week, what got sold was the demand itself, and the profit for that draw does not depend on the plan at all.

The consequence is useful and slightly deflating: over part of the range, raising production does not move the bad week. The tail is being set by the demand draw rather than by the decision, and no amount of extra production buys anything there. Knowing which part of a distribution your decision can actually reach is worth more than the percentile itself.

A box is the easiest set to write and the easiest to defend line by line, because each product gets its own honest range. The trouble is what it implies jointly. It says that every product can sit at its worst value in the same week, and if there are three products with independent demand, the corner it protects against is a week nobody has ever seen.

The ellipse and the budget set are both attempts to fix that by limiting how many things may go wrong at once. Both are more realistic and both introduce a new dial that somebody has to set. There is no version of this where the conservatism is derived rather than chosen, and pretending otherwise is the main way robust plans get oversold.

The unresolved argument is not about any of the mathematics above. It is whether a business should plan for the worst plausible week or the average one, and it is decided by facts about the business rather than by the model.

One camp plans for the average and holds its nerve. A mill runs fifty-odd weeks a year; good and bad both turn up, and what accumulates is the average. Every pound of mean given up to protect a 5th percentile is given up in all fifty-odd weeks, most of which were never going to be bad. On this view robustness is an insurance premium paid continuously against an event that is survivable, and the discipline is to stop buying it.

The other camp says the business does not experience the average, it experiences weeks, and weeks are where things break. Three bad ones in a quarter can breach a covenant, empty a cash buffer or lose a customer, and a strong August repays none of that. On this view the mean is the right number only for a firm with enough balance sheet to be indifferent to sequence, and the floor is what should be optimised.

Both positions are held by people who run real operations and neither is a misunderstanding of the other. What settles it in a particular case is a small number of facts nobody usually writes down: how much buffer exists, whether bad weeks cluster, and what precisely a bad week breaks. Notice that the argument is about the business and not the formulation, and that a robust model does not resolve it, it just moves the same judgement into the width of a set.