Warm-Up 3.4

1) Solve the system.

x y 7

2x y 8

2) Graph the solution. y 2x 3

y x 5

x 0

y 1

3.4 Linear Programming3.4 Linear Programming3.4 Linear Programming3.4 Linear Programming•What is linear programming?•What is the difference between a bounded and an unbounded region?•What information do you need to find the maximum and the minimum?

Optimization

He ran a disk clean-up on his computer to optimize his computer’s performance.

The greenhouse was the optimum location for the new plants.

Optimization: Finding the maximum or minimum value of some quantity.

Linear Programming

The process of optimizing a linear objective function subject to a system of linear inequalities called constraints. The graph of the system of constraints is called the feasibility region.

Bounded Region

y

x

Unbounded Region

y

x

Optimal Solution of a Linear Programming

ProblemIf an objective function has a

maximum or a minimum value, then it must occur at a vertex of the feasible region. The objective function will have both a maximum and a minimum value if the feasible region is bounded.

Find the minimum value and the maximum value

ofC = 3x +4y

Subject to the following constraints.

8

0

0

yx

y

x

(0,8)

(8,0)(0,0)At (0,0): C= 3( ) + 4( ) =

At (8,0): C =3( ) + 4( ) =

At (0,8): C = 3( ) +4( ) =

Find the minimum value and the maximum value of

C = 5x +6y

1843

5

0

0

yx

yx

y

xSubject to the following constraints

At (0,5): C = 5( ) +6( ) =

At (2,3): C = 5( ) +6( ) =

At (6,0): C = 5( ) +6( )=

(0,5)

(2,3)

(6,0)

Example 1

*the inequalities contained in the problem are called the constraints

Let x = the # of afghans Let y = the # of sweaters

x y 100

10 x 60

y 20

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

*the solution to the set of constraints is called the feasible region

Example 1

*the function to be maximized or minimized is called the objective function

Write an objective function for the company’s total profit, P, from the sales of afghans and sweaters.

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

P =

10x

+ 12y

•What is linear programming?The process of optimizing a linear

objective function subject to a system of linear inequalities called constraints.

•What is the difference between a bounded and an unbounded region?

A bounded region it totally enclosed, an unbounded region is not.

•What information do you need to find the maximum and the minimum?

The point of intersection of two lines.

Homework

p.166, 9-11, 13-17 odd, 25-26

Warm-Up1) Graph the feasible region for the set of constraints below.

x 2y 8

2x y 10

x 0

y 0

2) Identify the vertices of the feasible region.