The highest margin product nobody makes

When a plan makes none of something, the room reaches for explanations in a fairly predictable order. The margin figure must be wrong. The model must be missing a customer commitment. Somebody has capped Show somewhere. Each of those is a reasonable thing to check and none of them is what happened here. Show's margin of 71 is the correct margin. There is no cap on Show anywhere in the problem. The solver is free to make as much of it as it can fit, and it chooses not to.

Hold on to how strange that feels, because the strangeness is the lesson. A result that merely disagrees with you gets argued with. A result that disagrees with you this hard either reveals a fault in the model or a fault in the question, and the skill being built tonight is telling those two apart quickly.

• Optimal plan. The feasible plan with the best objective value. Named in session 1, computed tonight.
• Continuous solution. The plan when quantities may be fractional. Ravensworth's is 30.6897 Standard, 64.8276 Dairy, 0 Show.
• Integer solution. The plan when quantities must be whole. Ravensworth's is 33 Standard, 63 Dairy, 0 Show.
• Ranking. Putting items in order by a per-unit figure. Answers which item is better per unit, and nothing else.
• Allocation. Deciding how a fixed pool of a resource is spent across competing uses. A different question with a different answer.
• Consumption rate. What one tonne of a product takes out of a constraint. Show takes 2.3 milling hours, Standard takes 1.1.
• Binding constraint. One the plan has pushed right up against, with nothing left over. Priced in session 3.
• Degeneracy. A problem where several different plans tie for best. Worth knowing as a concept, and not what is happening at Ravensworth: exactly one integer plan reaches 4,656.00.

A per-unit figure answers a per-unit question: if I could have one more tonne of exactly one product for free, which would I want. That question is well posed and margin a tonne answers it perfectly. Show wins, and would win again tomorrow.

An allocation asks something else entirely. It asks how a fixed pool gets divided among uses that all draw on it at different rates. The moment the pool is fixed, a product that earns more per tonne but consumes more of the pool per tonne is in a genuine contest, and the margin column has no information about the second half of that contest. Show consumes 2.3 milling hours a tonne. Standard consumes 1.1. Those two numbers belong in the decision and neither of them is a margin.

The general form is worth saying without the mill attached to it, because it travels: a per-unit ranking and a capacity allocation are different questions. Anywhere a scarce pool is shared, the ranking will look like an answer and will not be one. Revenue per customer ranks customers; it does not allocate a support team. Value per feature ranks features; it does not allocate an engineering quarter.

Two reasons, and both are structural rather than anybody being careless. The first is that the margin column is the one that gets reported. It is in the monthly pack, it has a currency symbol on it, and it is comparable across products without any conversion. The consumption rates live in operations, in different units, and rarely appear on the same page as the money.

The second is that the ranking is genuinely correct. This is not a case of a wrong number being believed. Show really does earn the most a tonne, the report really does say so, and anyone who checks will find the report was right. A correct answer to a question nobody asked out loud is far harder to catch than an incorrect one, because every check you run comes back clean.

They are not always different answers, and pretending they always are is its own error. If every product drew on the constrained resource at the same rate, then ordering by margin a tonne and ordering by what the hours buy would give the same sequence, and the ranking would be a perfectly good allocation rule. Plenty of businesses are close enough to that condition that the shortcut has worked for years.

That is exactly why the failure is so quiet when it arrives. The heuristic is not wrong in principle, it is wrong in this instance, and the instance changed when the product mix changed. At Ravensworth the rates are 1.1, 1.5 and 2.3 for the three products, which is a wide enough spread that the shortcut stops being safe. The useful habit is not to distrust rankings, it is to ask what each item consumes before treating a ranking as a plan.

Set the box above to one tonne of Show, then five, then ten. Nothing breaks. Each of those plans is perfectly legal, the mill can run every one of them, and the solver will happily hand you the best plan available under that requirement. What changes is the total, and it changes downward every time.

That is the honest shape of the finding, and it is worth stating more carefully than "do not make Show". The mill is not banned from making Show. It is being told what making Show costs in a week where milling hours are the thing in short supply. If a commercial reason to keep the Show customer exists, and there often is one, the figure in that box is the price of keeping them, and it is now a number rather than an argument.

The continuous answer is 30.6897 Standard and 64.8276 Dairy, earning 4,666.90. The integer answer is 33 Standard and 63 Dairy, earning 4,656.00. The second is lower, which is not a surprise: requiring whole tonnes removes options and removing options cannot help.

What is worth noticing tonight, and gets taken apart properly in the practitioner track, is that the integer plan makes more Standard than the continuous one, not less. Going from 30.6897 to 33 is not what anybody expects when they are asked to turn a fractional plan into a whole-tonne plan. Session 5 comes back to what that implies about handing a fractional plan to somebody and telling them to be sensible with it.

Grant everything above and there is still a live argument about what to do on Monday morning, and it is not a soft one.

One camp says the incentive is simply broken and should be rewritten to pay on what the plan is actually short of. On this view the commission scheme is a piece of machinery that was built from the wrong column, the mill has now measured what that costs, and continuing to pay people to push the operation off its best plan is indefensible once you can see it. Pay on hours well spent, not on margin a tonne, and the chain in the amber band stops working against itself.

The other camp says the objective is the thing that is wrong, not the incentive. The sales force is closest to the Show customer and knows things the margin row does not carry: that the Show account is the reference other buyers check, that dropping the line is a one-way door, that the relationship is worth more over three years than the week's hours are worth now. On this view the solver is answering honestly about a week, the incentive is encoding a longer horizon badly but is encoding it, and rewriting the incentive to match a one-week objective throws away the only part of the system that was looking past Friday.

Notice that both camps agree on every number on this page. The disagreement is entirely about what the objective should have been, and no amount of further computation settles it. It is worth knowing which one you hold before you walk into the meeting, and worth knowing which one the person opposite holds before you try to persuade them.