Linear Programming

1. Linear Programming

2. Linear Programming • Linear programming is used to find the best outcome (usually maximum profit or lowest cost) in a given mathematical model. • Examples of Linear Programming • Maximizing farm production profit • Maximizing vehicle production profit • Finding lowest cost of production

3. Setup of Linear Programming Models • Generally linear programming models can be set up using linear inequalities. • Begin by writing inequalities for the contraints of the problem (i.e. all restrictions faced) • Then graph each inequality on a coordinate axis

4. Setup cont. • After graphing and shading you will see a region made by the lines you drew. • The vertices of this region represent possible amounts of each product you want to make • Finally, set up a function that finds the profit or costs (use how much money you may make for each product to set this up).

5. Example 1. A farmer has 25 days to plant cotton and soybeans. The cotton can be planted at a rate of 9 acres per day, and the soybeans can be planted at a rate of 12 acres a day. The farmer has 275 acres available. If the profit for soybeans is $18 per acre and the profit for cotton is $25 per acre, how many acres of each crop should be planted to maximize profits?

6. Example 1 What are the unknown values? • Let c = the number of acres of corn • Let s = the number of acres of soybeans

7. Write the inequalities • The number of acres of corn must be greater than or equal to zero. • The number of acres of soybeans must be greater than or equal to zero. • The total number of acres planted must be less than or equal to 275 • The time available for planting must be less than or equal to 25 days

8. s c Graph each inequality • The purple region is the region bounded by all 4 curves. So, we want to find the vertices (corners) of the region.

9. Vertices • (0,275) • (225,0) • (0,0) • (75,200) • These are all the possible “ideal” amounts of corn and soybeans to plant.

10. Write an equation for finding profit • p(c,s) = 25c + 18s • Maximum profit = $25 times the number of acres of cotton planted + $18 times the number of acres of soybeans planted.

11. Calculating Max Profit • Now plug in each coordinate (c, s) into the profit equation to find the maximum profit.

12. Max Profit • So, 225 acres of corn and 0 acres of soybeans should be planted for a maximum profit of $5625.