Warm-upWarm-up Sketch the region bounded by the system of inequalities: 1) 2) Sketch the region bounded by the system of inequalities: 1) 2)

Objectives: 1.Write an objective function modeling a quantity to be maximized or minimized. 2.Use inequalities to model limitations in a situation 3.Use linear programming to solve the problem Linear Programming

1. Objective Function A linear objective function: Goal Goal : Determine the values x and y that will maximize (or minimize) the value z, subject to certain constraints. A linear objective function: Goal Goal : Determine the values x and y that will maximize (or minimize) the value z, subject to certain constraints.

2. Writing an Objective Function Example: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model is given: The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit ? Example: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model is given: The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit ? Model AModel B Assembling54 Painting23

3. Writing Constraints The constraints are the equations that will determine the feasible region for a solution. write the constraints for the previous problem. The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. The constraints are the equations that will determine the feasible region for a solution. write the constraints for the previous problem. The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. Model AModel B Assembling54 Painting23

4. Solving a Linear Programming Problem. The Linear Programming problem is: maximize : subject to: The Linear Programming problem is: maximize : subject to: 1: Graph the feasible region 2: Determine the corner points (vertices) 3: Find the value of the objective function at each corner (make a table) 4: The largest z value is the solution. 1: Graph the feasible region 2: Determine the corner points (vertices) 3: Find the value of the objective function at each corner (make a table) 4: The largest z value is the solution. corner (x,y)Objective function: z =

More practice p. 821 #10 maximize : subject to: p. 821 #10 maximize : subject to:

Warm-upWarm-up maximize : subject to: maximize : subject to:

5. Analyzing a Linear Programming problem Example: maximize : subject to: Example: maximize : subject to: Analyze : Graph the objective function for different values of z. Why is the max at a corner point? Let z = 10: Let z = 20: Let z = 22: What happens when z > 22? Analyze : Graph the objective function for different values of z. Why is the max at a corner point? Let z = 10: Let z = 20: Let z = 22: What happens when z > 22? Theorem: If a Linear Programming problem has a solution, it is located at a corner point. Why? Theorem: If a Linear Programming problem has a solution, it is located at a corner point. Why?