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What a formulation decides, what it assumes, and what it cannot see.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to read a model somebody else wrote: name its decision variables with units, say what its objective is a proxy for, turn each constraint back into a sentence, and identify what it cannot see by asking what the cheapest answer would do that a person would refuse. You will also be able to name which of proportionality, additivity and certainty a given model leans on hardest, and say where that assumption stops holding.
You can write a model: variables, objective, constraints, indices, integrality. This lesson is the other direction, and it is the skill that actually gets used — most models a person meets were written by somebody else, and the question is never can I solve this but what is this, and what does it assume.
Proportionality: doubling a decision doubles its contribution. What $c x$ claims, and what a volume discount breaks.
Additivity: contributions add with no interaction term. What $c_1x_1 + c_2x_2$ claims.
Divisibility: fractional amounts are meaningful. What dropping integrality claims.
Certainty: the parameters are known numbers rather than forecasts.
Proxy objective: the number being maximised, standing in for the goal you actually have.
Blind spot: anything in neither the objective nor a constraint. The solver spends it freely.
Range of validity: the region of decisions over which the model's assumptions hold well enough to act on.
Reading a model is a routine. Asked in this order, the four questions answer each other.
1. What is being chosen? Find the decision variables and say what one unit of each means, with its unit of measurement. Everything else refers to them.
2. What is being judged? The objective, and — separately — what real thing it stands in for. Cost is a proxy for run this well; distance is a proxy for drive less. Naming the gap between the proxy and the goal is most of what a reader can add.
3. What is not allowed? Each constraint, back into a sentence. If a constraint cannot be said in a sentence, either it is wrong or you have not understood it yet.
4. What can this model not see? The question people skip. Anything that is neither in the objective nor in a constraint is invisible, and the solver will spend it without limit because nothing tells it not to.
The three standing assumptions. Every linear model makes them, whether or not it says so:
| Assumption | What it claims | Where it fails |
|---|---|---|
| Proportionality | doubling the decision doubles the effect | bulk discounts, congestion, fatigue |
| Additivity | contributions add with no interaction | setup times, shared equipment |
| Certainty | the parameters are known numbers | every forecast there has ever been |
None of these is a defect. They are the price of a model that can be solved, and a reader's job is to say where they stop holding rather than to be scandalised that they were made.
Another way: picture
Two circles. One is everything the decision affects; the other is everything the model represents. The overlap is what the model can reason about; the part of the first circle outside the second is what it will spend freely. Reading a model is drawing that second circle accurately.
Another way: steps
Given someone else's model:
A solver will happily accept $15x \le 8$ where $15$ is minutes per item and $8$ is hours available. It is dimensionally nonsense and numerically fine: the model permits about half an item, reports optimal, and nobody is warned.
Writing the unit beside every parameter catches this in seconds and catches nothing else — which is the point. Most model errors produce a confident wrong answer rather than a failure, so the checks worth doing are the ones that look at something the solver does not.
The same applies to the objective. A cost in pounds and a penalty in hours cannot be added; if both matter, that is a multi-objective problem, and lesson 27 is where it is dealt with honestly rather than by inventing an exchange rate in passing.
Reading the objective as the goal. It is a proxy. The distance between the two is where every uncomfortable recommendation comes from.
Not asking what is missing. A constraint nobody wrote is a constraint the model will violate enthusiastically, and it will look like an efficiency gain when it does.
Blaming the data. When an answer looks wrong, the parameter is the first suspect and rarely the culprit. A missing constraint and a transposed index are both far more common.
Treating an optimal report as a guarantee. Optimal means optimal for the model. The model is a claim about the world that a person made, and the claim is the part worth checking.
The model says is not a reason, and it is the sentence this whole unit exists to prevent. A model is a written argument of the form: given these variables, this criterion and these rules, this decision is best. Every one of those three is a claim somebody made, and the conclusion inherits their weaknesses exactly. When the recommendation is uncomfortable, the productive response is not to distrust the arithmetic — the arithmetic is fine — but to find which of the three claims you disagree with, and to change it and re-solve. That is what makes optimization an argument you can have, rather than an answer you must accept.
$\min\ 4x_1 + 7x_2$ subject to $x_1 + x_2 \ge 100$, $x_1 \le 60$, $x_1, x_2 \ge 0$. Variables: units bought from supplier one and supplier two.
What is chosen.
Objective: total purchase cost, a proxy for buy well. It says nothing about lead time, reliability or the risk of depending on one supplier.
What is judged, and what it stands in for.
Constraints: buy at least 100 in total; supplier one can supply at most 60. What it cannot see: everything about quality and risk. So the answer will buy the cheap supplier's full 60 and make up the rest — correct for this model, and a single-supplier dependency nobody asked about.
And what it cannot see.
A model prices labour at $20$ an hour and lets hours run to any level the constraints allow.
Linear in hours.
In the world, hours past forty cost $30$, and past sixty they cost more than money. The straight line and the real cost agree over a range and diverge sharply outside it.
The approximation has a range.
So the reader's job is to check where the answer landed. Forty-two hours: the model is roughly right. Ninety hours: the model has been extrapolated past the region where anybody checked it, and the answer should not be used. The fix is a constraint capping hours, not a better cost figure.
Check where the answer landed, not only that it is optimal.
Minimise total delivery distance, subject to every parcel being delivered and each van's capacity. Variables: which parcels go on which van, and in what order.
Variables first.
Objective: distance, a proxy for cost and emissions. Constraints: coverage and capacity.
Not seen: time. Nothing in the model says a driver's shift ends, or that a delivery window exists, so the cheapest route is free to be a fourteen-hour day. Ask what the cheapest answer does that a person would refuse, and the missing constraint names itself.
Match each model statement to the assumption it makes.
| Proportionality | Additivity | Known parameters | |
|---|---|---|---|
| Revenue is $10x$ at every sales volume | |||
| Total cost is the sum of every route's cost | |||
| Demand is fixed at 200 units |
A model says revenue is $23x$: every unit sells for $23$, whatever the volume. Suppose instead that each extra unit pushes the price down by $1$, so $x$ units sell at $23 - x$ each. Write the revenue as an expression in $x$.
Answer:
A constraint says each item takes $29$ minutes and there are $5$ hours available. What should the right-hand side be, if the left-hand side is written in minutes?
Answer:
Put the steps of reading someone else's model into a sensible order.
Number the steps in order (write the number in the box):
A rostering model minimises total wage cost across $4$ staff, subject to cover requirements and maximum hours. It returns a roster nobody will work. What is most likely missing?
A model recommends closing $7$ depots. Which statement about that recommendation is accurate?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A constraint says each item takes $30$ minutes and there are $4$ hours available. What should the right-hand side be, if the left-hand side is written in minutes?
Answer:
You can read a model back into words, check its units, name its standing assumptions and say what it leaves out. That closes unit 1: you can now write a model and read one. Next: the property that decides whether any of the methods that follow can promise you anything — convexity.
10. Your turn: what is this model unable to see?, step 3