Linear Programming

Graph the following system of inequalities

y

xy

xy

2

1

Graph the system

52

0

0

xy

y

x

Another example

yx

x

y

51

40

Graphing Inequalities

Graph the following system of inequalities:

More graphing. . .

Graph the following system of inequalities:

Linear Programming

A method used to find optimal solutions such as maximum or minimum profitsSteps:1. Assign variables2. Determine the constraints (inequalities)3. Find the feasible region (area of solution)4. Determine the vertices of feasible region5. Plug those values into the profit equation(also

called objective function)

Finding Max or Min

Given the following constraints, maximize the function

Example• Mr. Farmer wants to plant some corn and wheat

and he gets the following statistics from the US Bureau of Census

Crop Yield per acre Avg PriceCorn 113.5 bu $3.15/bu

Soybeans 34.9 bu $6.80 bu

Wheat 35.8 bu $4.45 bu

Cotton 540 lb $0.759/lb

Rice, rough 5621 lb $0.0865/lb

Example continued

• Mr. Farmer can have no more 120 acres of corn and wheat

• At least 20 and no more than 80 acres of corn• At least 30 acres of wheat

How many acres of each crop should Mr. Farmer plant to maximize the revenue from his harvest?

Working through the problem..

• Assign VariablesX=acres of corn and y=acres of wheat

• List the constraints

Graph it

Example continued

• List the vertices

• Determine Profit equation

• Which would yield the most?

Another Ex.• A snack bar cooks and sells hamburgers and hot dogs during

football games. To stay in business, it must sell at least 10 hamburgers but cannot cook more than 40. It must also sell at least 30 hot dogs but cannot cook more than 70. The snack bar cannot cook more than 90 items total. The profit on a hamburger is $0.33 and on a hot dog it is $0.21. How many of each item should it sell to make the maximum profit?

• Profit Equation: __________________________• Constraints: • Answer: _________________________

Another Example

• As a receptionist for a veterinarian, Sue scheduled appointments. She allots 20 minutes for a routine office visit and 40 minutes for surgery. The vet can not do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visits costs $55 and most surgeries costs $125, find a combination of office visits and surgeries that will maximize the income the veterinarian practice receives per day.

What do you know…

• Assign variablesx=number of office visitsy=number of surgeries• Constraints: 7 hours needs to be in terms of

minutes

Continued…• Graph and determine

coordinates of the vertices

• You should get (0,0) (0,6)(9,6)(21,0)

Continued….

• Determine the profit equation• $55v + $125s = P

• Test the points

• Highest profit would be when there are 6 surgeries and 9 visits