Using inequalities to plan
Linear programming finds the best way (the biggest profit or the lowest cost) to use limited resources. The limits are written as linear inequalities called constraints, and the quantity to be maximised or minimised is the objective function.
To draw an inequality such as x + y ≤ 10, draw the boundary line x + y = 10 and shade (or leave unshaded) the side that satisfies the inequality. The feasible region is the part satisfying all the inequalities at once. Its corners (vertices) are found by solving pairs of boundary equations.
A factory makes x chairs and y tables. It can make at most 10 items and at most 6 chairs. The profit is TSh 40 000 per chair and TSh 30 000 per table. Find the best plan.
- Vertices of the feasible region: (0, 0), (6, 0), (6, 4), (0, 10).
- Test each vertex.
Answer: The best plan is 6 chairs and 4 tables, a profit of TSh 360 000.
Choose a number of chairs x and tables y. The model checks the constraints and gives the profit.
Key terms
- Constraint
- A limit written as an inequality.
- Objective function
- The quantity to be maximised or minimised.
- Feasible region
- The region satisfying all constraints.
- Vertex
- A corner of the feasible region.
Summary
- Write the constraints as linear inequalities and draw the feasible region.
- Write the objective function.
- Test the vertices: the best value is at one of them.
- State the answer in the words of the problem.