Terminology and formulation

The essentials

A linear programming problem asks for the maximum or minimum value of a linear function, called the objective function, subject to a set of linear inequalities called constraints. The unknowns, such as the number of chairs and tables to make, are the decision variables. Because quantities cannot be negative, every LPP includes the non-negative constraints x ≥ 0, y ≥ 0. Finding the best (maximum or minimum) value is called optimisation.

Formulating an LPP means translating a situation into this form:

• Define the decision variables in words (for example, x = number of chairs made per day).
• Write the objective function using the profit or cost per unit (for example, maximise Z = 30x + 50y).
• Write one constraint for each limited resource or requirement.
• Add x ≥ 0, y ≥ 0.

Key points

• "At most", "not more than", "available" lead to ≤ constraints.
• "At least", "not less than", "minimum requirement" lead to ≥ constraints.
• Standard types: manufacturing problems (maximise profit), diet problems (minimise cost while meeting nutrient needs) and transportation problems (minimise shipping cost).
• The word "linear" means every variable appears only to the first power, with no products like xy.

Exam tips

• Make a small table of resources against products before writing constraints.
• Check the units: hours with hours, kg with kg.

Common mistakes

• Reversing the inequality sign in diet problems, where nutrient needs are minimums (≥).
• Forgetting the non-negative constraints.