CHAPTER 3 LINEAR SYSTEMS3.4 Linear Programming

LINEAR PROGRAMMING Linear Programming is a

method for finding a minimum or maximum value of some quantity, given a set of constraints Constraints are restrictions on the

variables of the objective function in a linear programming problem. The constraints form a system of

inequalities The graph of the system is the

feasible region, it contains all the points that satisfy all the constraints

The objective function is a model (equation) of the quantity that you want to make as large or as small as possible.

VERTEX PRINCIPLE OF LINEAR PROGRAMMING

If there is a maximum or minimum value of the linear objective function, it occurs at one or more vertices of the feasible region.

To find the maximum or minimum1. Graph the constraints to form the feasible

region2. Find the coordinates of each vertex

(Solve the system – for the two equations that intersect at that vertex – using the method of your choice)

3. Evaluate the objective function using each vertex to find the maximum or minimum

EXAMPLEGraph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

Constraints

Objective Function:

EXAMPLEGraph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

Constraints

Objective Function:

EXAMPLE – PAGE 159 You are screen printing T-shirts and sweatshirts to

sell at the Polk County Blues Festival and are working with the following constraints You have at most 20 hours to make shirts You want to spend no more than $600 on supplies You want to have at least 50 items to sell

How many T-shirts and how many sweatshirts should you make to maximize your profit?What is the maximum profit?

EXAMPLE

1. Organize the information in a table

2. Write the constraints and objective function

3. Solve and Answer the question

T-Shirts Sweatshirts

Total

ASSIGNMENT

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