1. 3.4 Linear Programming p. 163

2. Optimization - Finding the minimum or maximum value of some quantity. Linear programming is a form of optimization where you optimize an objective function with a system of linear inequalities called constraints. The overlapped shaded region is called the feasible region.

3. Solving a linear programming problem 1.Graph the constraints. 2.Locate the ordered pairs of the vertices of the feasible region. 3.If the feasible region is bounded (or closed), it will have a minimum & a maximum. If the region is unbounded (or open), it will have only one (a minimum OR a maximum). 4. Plug the vertices into the linear equation (C=) to find the min. and/or max.

4. A note about: Unbounded Feasible Regions If the region is unbounded, but has a top on it, there will be a maximum only. If the region is unbounded, but has a bottom, there will be a minimum only.

5. Find the min. & max. values of C=-x+3y subject to the following constraints. x  2 x  5 y  0 y  -2x+12 Vertices of feasible region: (2,8) C= -2+3(8)= 22 (2,0) C= -2+3(0)= -2 (5,0) C= -5+3(0)= -5 (5,2) C= -5+3(2)= 1 Max. of 22 at (2,8) Min. of -5 at (5,0)

6. Ex: C=x+5y Find the max. & min. subject to the following constraints x0x0 y  2x+2 5  x+y Vertices? (0,2) C=0+5(2)=10 (1,4) C=1+5(4)=21 Maximum only! Max of 21 at (1,4)

7. Linear Programming You want to buy tickets for some concerts; rock and jazz. You want at least 5 rock tickets and at least 8 jazz tickets. No more than 20 total tickets. Rock tickets cost $3 and jazz tickets cost $5. You cannot spend more than $80. Find the number of each type of ticket you can buy at the lowest price total.