Writing constraints as inequalities

Decision variables

• A decision variable is a quantity you get to choose in the problem.
• Define each one in words with a letter, e.g. let $x$ = number of tables made, $y$ = number of chairs made.
• Everything else in the problem — profit, hours used, material used — is built out of $x$ and $y$.
• Choosing clear variables first makes every later step (constraints, objective, graph) fall into place.

Turning a limit into an inequality

• A constraint is a limit the choice must respect — a fixed amount of time, material, money or space.
• Each constraint becomes one linear inequality:
  • Work out how much of the resource one unit of each variable uses.
  • Multiply by the variable and add the terms.
  • Compare with the total available using $\le$ or $\ge$.
• Example: if a table needs $2$ hours and a chair needs $1$ hour of assembly, and there are $10$ hours available:

$2x + y \le 10$

The non-negativity constraints

• You cannot make a negative number of a real object.
• So every linear programming problem carries the non-negativity constraints:

$x \ge 0, \qquad y \ge 0$

• Never forget these — they keep the feasible region in the first quadrant.

Common constraint types

• Resource limit — a shared resource is capped: $2x + y \le 10$ (an upper limit, $\le$).
• Minimum requirement — you must make at least a certain amount: $x + y \ge 4$ (a lower limit, $\ge$).
• Ratio / relationship — one variable compared to another: "at least twice as many chairs as tables" is $y \ge 2x$.

Reading "at least", "at most", "no more than"

• "at most" / "no more than" / "up to" / "maximum" → $\le$.
• "at least" / "no fewer than" / "minimum" → $\ge$.
• "exactly" → an equals sign (a line, not a region).
• Underline the key phrase in the question and write the symbol above it before forming the inequality.