Formulating a Linear Programming Problem

A linear programming problem (LPP) is a problem of finding the maximum or minimum value of a linear function, subject to a collection of linear constraints in the same variables, together with the non-negativity restrictions of the previous section. Formulating an LPP from a word problem means turning its everyday business statement into this precise mathematical shape, in four steps.

Step 1 — identify the decision variables. These are the unknown quantities the business actually controls and wants to decide upon — typically how many units of one product to make and how many units of a second product to make. Call them $x$ and $y$.

Step 2 — write the objective function. This is the linear expression $Z = ax + by$ that the problem wants to maximise (e.g. total profit or total output) or minimise (e.g. total cost or total time), built from the per-unit contribution of each decision variable.

Step 3 — write the constraints. Every limited resource the problem mentions — machine hours, labour hours, raw material, a minimum requirement — becomes one linear inequality in $x$ and $y$: an at most statement becomes a $\le$ constraint, and an at least statement becomes a $\ge$ constraint.

Step 4 — add the non-negativity restrictions, $x \ge 0$ and $y \ge 0$, since a negative quantity of tables, chairs, or any physical product has no meaning.