Section 3.5 Linear Programing In Two Variables

Optimization Example Soup Cans (Packaging) Maximize: Volume Minimize: Material Sales Profit Cost When one quantity is made as large as possible (maximized) while the other quantity is made as small as possible (minimized)

x - y ≥ -4 y≥ 0 x ≥ 0 y ≤ -x + 8 y ≤ x Graph the following system x + y ≤ 8 Shade the correct area

a. Vertices? {(0,0), (0,4), (2,6), (8,0)} b. Absolute maximum value? (2,6) c. Minimum value? (0,0) d. Will the maximum & minimum values always be at the vertices of the feasible region? Yes: extreme values 1. (cont.)

Def: The overlapping area is called the feasible region. Def: The boundries formed by the lines are called constraints. Def: The function that you want to maximize or minimize is known as the objective function.

x + y ≤ 8 2x + y ≤ 10 x ≥ 0 y ≥ 0 C = 3x + 2y y ≤ -x +8 y ≤ -2x + 10 Each grid mark = 1 unit 2. Find the maximum & minimum values for the following objective function under the given constraints.

Max & Min Vertices? (0,0) (0,8) (2,6) (5,0) C = 3x + 2y C = 3( ) + 2( ) Objective function C = 3( ) + 2( ) 00 C = 0 08 C = C = C = 15 Max? Min? Vertices

C = 3x + 2y Objective function? y = -(3/2)x + (1/2)C Slope = -(3/2) Intercept = (1/2)C What’s Going On? When C = 0 When C = 4 When C = 8 When C = 12 When C = 16 Max C occurs at the last point the obj. function intersects the feasibility region Max C occurs at the last point the obj. function intersects the feasibility region

Linear programming is an ideal computer application. Successive iterations yield the maximum or minimum value for an objective function.

Homework Practice worksheet 3-5 Page 158 Problems: 15,17,19, and 21