From a Described Problem to a Model

A description in words becomes a model by answering three questions in order: what is chosen, what is being optimised, and what limits the choice. Each answer has a syntactic form, and the order matters because the objective and the constraints are both written in terms of the variables. Most modelling faults are traceable to a variable that was never pinned down.

Definition

Modelling is the translation of a described decision problem into an optimization problem. Three questions settle it, and each maps to one part of the model.

What is chosen? The decision variables. Each needs a symbol, a meaning, and a unit: $x_{j}$ = tonnes of product $j$ produced per week, not $x_{j}$ = product $j$ . A variable whose unit is unstated cannot be checked for dimensional consistency later.

What is being optimised? The objective, written as a linear function of those variables, together with a direction. Every coefficient carries a unit too, and the product of a coefficient and its variable must have the unit of the objective.

What limits the choice? One constraint per restriction, each written in terms of the same variables. A restriction the description states is a constraint; a fact about the world that the variables already enforce is not.

The order is not stylistic. The objective and every constraint are functions of the variables, so they cannot be written until the variables have meanings. Attempting the objective first produces expressions whose symbols acquire their definitions retroactively, which is where units stop matching.

What does not belong in the model. Quantities the decision maker does not set are data, not variables. A number that is given, measured, or fixed by someone else appears as a coefficient. Promoting it to a variable makes it something the solution selects, which answers a different question than the one asked.

Assumptions and scope

A decision variable is something the decision maker sets directly. Quantities that are measured, given, or set by another party are data and appear as coefficients.
Every variable needs a unit as well as a meaning. Without units, dimensional consistency cannot be checked and a constraint mixing incompatible quantities looks the same as a correct one.
One restriction in the description corresponds to one constraint. A sentence that restates a restriction already encoded adds nothing, and a sentence that describes the world without limiting the decision is not a constraint at all.
The objective needs a direction as well as an expression. The same linear function maximised and minimised describes two different problems.
Nonnegativity is a modelling decision rather than a default. A quantity that cannot be negative needs the restriction stated; a variable that may genuinely take either sign must not carry it.

Worked material

Non-example

Four lines that are not constraints

A definition restated as a constraint. A modeller defines $T$ as total output and writes $T = x_{1} + x_{2}$ alongside the production limits. The equation is true, and it forbids nothing: $T$ was introduced to mean that sum, so no assignment of $x_{1}$ and $x_{2}$ can violate it. It is a definition, and treating it as a restriction suggests the model constrains more than it does.

A price that nobody chooses. The description mentions a raw material at £8 per cubic metre, so the model includes $p = 8$ . A variable fixed to its own value is not a decision. Either the price belongs in a cost coefficient or it belongs nowhere, and writing it as a variable invites a later edit in which the price becomes a decision.

A restriction the data already enforces. A workshop with 30 finishing hours, one hour per item, is given the constraint $x_{1} + x_{2} \leq 500$ "for safety". The finishing constraint already caps the total at 30. The safety line removes nothing from the feasible set.

A sentence about the world. "Demand for tables is seasonal." True, relevant to the business, and not a restriction on this week's plan unless it is turned into one: a maximum, a minimum, a relationship between the two products. As written there is nothing to encode, and inventing a number to make it encodable puts data into the model that the description never supplied.

What unites these is that each passes a reading test and fails a forbidding test. Checking that a proposed line rules something out is what separates the two.

Contrast

Decision or data

The same quantity can be a decision in one model and data in another, decided by who is doing the deciding. The question is never what kind of thing it is, but whether this decision maker sets it.

Quantity	Decision when	Data when
Selling price	The firm sets its own price	The price is quoted by the market
Staff hours	Overtime can be authorised	The roster is fixed for the period
Machine capacity	New capacity can be bought this period	The plant is what it is this week
Order quantity	The buyer is placing the order	The order has arrived and must be met
Delivery route	The planner chooses routes	A carrier has already committed

The consequence of getting it wrong runs in one direction more often than the other. Treating data as a decision gives the model freedom the decision maker does not have, and it returns a plan that cannot be carried out: an answer that assumes a price nobody will pay, or capacity that does not exist. Treating a decision as data gives away freedom that was available, and returns a plan that is feasible but worse than necessary.

Both are wrong, and the first is harder to notice, because a model with extra freedom still reports an optimum and still looks solved.

The test: can the decision maker walk out and change this number, within the period the model covers? A capacity that could be expanded next year is data in a weekly plan and a decision in an annual one, and the same description supports both models. Which one is being built has to be stated, because the algebra does not record it.