3-4 LINEAR PROGRAMMINGSolve problems by using linear programming.

Linear Programming

Method for finding a minimum or maximum value given constraints(limits).

The constraints come from a system of linear inequalities.

The graph of the system is call the feasible region.

Objective Function Models the quantity you are trying to

maximize or minimize. Often cost or profit Ex: the objective function is C = 2x + y Graphs of lines with various values for C are

parallel If there is a max or min for the objective

function, it occurs at one or more of the vertices of the feasible region. This is called the Vertex Principle of Linear Programming

Testing Vertices What point in the feasible region

maximizes P for the objective function P = 2x + y?

Graph the system to find the feasible region.

Vertices: (0,2.5) (3,1) (2,0) (0,0)

Evaluate P at each vertex. P has a maximum value of 7 at (3,1).

Using Linear Programming You have at most 20 hours to make T-shirts

&sweatshirts. You want to spend no more than $600 on supplies. You want to have at least 50 items to sell.

Continued How many of each to maximize profit? Create the constraints:

So the objective function is: P = 6x + 20y Graph constraints: Test vertices:

(50,0) (25,25) (75,15) (120,0)

P is maximized at (75,15) Sell 75 t-shirts and 15 sweatshirts.

Assignment

Odds p.160 #11-23