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Linear Programming — Part 1: Terms, models and feasible regions

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Lesson

Linear Programming, Part 1

Linear programming finds the best plan when limited resources are shared between two activities.

In this lesson you will

Use the terms decision variable, objective function, constraint and optimisation
State advantages, limitations and application areas of linear programming
Formulate manufacturing, diet and transportation-type problems in two variables
Draw feasible regions and classify them as bounded, unbounded or empty

Part 1 at a glance.

Concept

The parts of a model

Decision variables, an objective function, constraints and non-negativity.

Every linear programming problem has these parts.
Term	Meaning	Dairy example
Decision variables	Quantities we choose	$x$ paneer batches, $y$ curd batches
Objective function	Linear quantity to maximise or minimise	$Z = 3000x + 4000y$ (profit)
Constraints	Linear limits on the variables	$2x + y \le 14$ , $x + 2y \le 13$
Non-negativity	Variables cannot be negative	$x \ge 0$ , $y \ge 0$
Optimisation	Finding the best feasible plan	Largest profit

Concept

Advantages and limitations

Best use of scarce resources
Systematic, clear decisions
But: linearity, exact data and a single objective are assumed

Know two of each for a 2-mark question.
Advantages	Limitations
Best use of scarce resources	Assumes everything is linear (proportional)
Clear, systematic decisions	Coefficients must be known exactly
Shows bottlenecks and spare capacity	Only one objective at a time
Easy to re-solve when data change	Answers may be fractional for whole items