Explore: Applications of Linear Programming Solutions

Linear optimization has many real-world applications.
Linear programming word problems typically provide the information needed to write a system of inequalities so that you can graph and then optimize the solutions .
When writing a system of inequalities for a scenario, consider what quadrant(s) make the most sense for the solutions.

Example 5

Speedy Delivery has two types of delivery vehicles, trucks (x) and vans (y). The total daily mileage of the trucks and vans cannot exceed 1000 miles. If 3 trucks are on the road, then 2 less vans can be used to maintain a total mileage that is less than or equal to 1000 miles. Each day the vans will drive at least 100 miles and the box trucks will drive at least 100 miles. Write a system of inequalities to maximize deliveries using the equation $f (x, y) = 10 x + 15 y .$

Plan
Write and graph a system of inequalities.

Find the vertices.

Determine the optimization.

Implement

$x + y \leq 1000 3 x - 2 y \leq 1000 x \geq 100 y \geq 100$

Vertices: marked on the graph

$f (x, y) = 10 x + 15 y f (100, 100) = 2500 minimum f (100, 900) = 14500 maximum f (600, 400) = 12000 f (400, 100) = 5500$

Note

Remember that you can use a calculator to evaluate the objective function.

Because this is bounded, there will be a minimum and maximum.

Explain

The maximum value occurs at (100, 900). Ideally the box trucks would drive 100 miles and the vans would drive 900 miles in one day.