Lecture 9
Linear Programming

Both the objective function and constraints are linear. 

Steps

  1. Plot the equality constraints
  2. Obtain the feasible region
  3. Determine the corner points of the feasible region
  4. Compute the objective function at those corner points. 
The corner point that yields the extremum of the objective function is the solution to the optimization problem.

Example 1
The corner points of the feasible region are: (1,1.5) and (4,0).

Compute the value of the objective function at those points.

Solution: x = 1, y = 1.5

Example 2

A farmer plants wheat and rye on his 10 acres of land. He has to plant at least 7 acres. He has $1,200 to spend. An acre of wheat costs $200 to plant while an acre of rye costs $100. The farmer has to get the planting done in 12 hours. It takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye, how many acres of each should the farmer plant in order to maximize profits? [Source]

Obtain the constraints and plot them. In the figure below, acres of wheat is on the horizontal axis.

Corner points of the feasible region:

Solution: To maximize profits, the farmer should plant 4 acres each of wheat and rye.
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