Linear Linear

Programming

Programming1.3 M31.3 M3

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

Warm-UpGraph the system.

x 0

y 0

y 3x 2

y x 4

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

Graph the system.

x 0

y 0

y 3x 2

y x 4

What

is

What

is

Linear

Linear

Progra

mm

ing

Progra

mm

ing ??

1.1.Way to maximize or

Way to maximize or minimize a linear

minimize a linear objective function

objective function2.2.Has constraint

Has constraint inequalitiesinequalities3.3.Solutions (intersections)

Solutions (intersections)

of the constraints are

of the constraints are

possible solutions to the

possible solutions to the

objective function

objective function (equation)(equation)

Genera

l Exa

mple

Genera

l Exa

mple

Concession Stand that Sells

Concession Stand that Sells

Hot Dogs & Hamburgers

Hot Dogs & Hamburgers

You have a certain amount

You have a certain amount

of money to buy them.

of money to buy them.

Only so many hot dogs will

Only so many hot dogs will

fit on the grill. (same

fit on the grill. (same

with hamburgers)

with hamburgers)You want to know how

You want to know how much of each to buy to

much of each to buy to

give you the

give you the Maximum

Maximum

Profit.Profit.

You H

ave

3

You H

ave

3

Unkn

ow

ns

Unkn

ow

ns

Hot Dogs (x)

Hot Dogs (x)Hamburgers (y)

Hamburgers (y)Maximum Profit (z)

Maximum Profit (z)2 unknowns are graphed

2 unknowns are graphed

on a coordinate plane.

on a coordinate plane.

3 unknowns will create a

3 unknowns will create a

3 dimensional graph.

3 dimensional graph. (x, y, z)(x, y, z)

x

z

y

To S

olv

eTo

Solv

e

1.1.Graph the Inequalities

Graph the Inequalities

2.2.Identify the solutions

Identify the solutions

(intersections)

(intersections)3.3.Substitute the solutions

Substitute the solutions

into the objective

into the objective function (equation)

function (equation)4.4.Look for the answer to

Look for the answer to

the problem.

the problem. (maximum or minimum

(maximum or minimum

value)value)

003 15

4 16

xyx yx y

Find the minimum value and the maximum value of

the objective function C = 3x + 2y subject to the following constraints.

Unders

tandin

g

Unders

tandin

g

wit

h P

lay-

doh

wit

h P

lay-

doh

1.1.Place you play-doh on top

Place you play-doh on top

of your graph.

of your graph.2.2.Trim the edges of your play-

Trim the edges of your play-

doh to be constraint

doh to be constraint

equations.equations.3.3.The linear programming

The linear programming

concept builds a 3D model

concept builds a 3D model

that has its top base sliced

that has its top base sliced

at an angle. The

at an angle. The highest highest

vertexvertex is at the

is at the maximum

maximum

value value and the lowest

and the lowest

vertex is at the minimum

vertex is at the minimum

value. value. 4.4.Identify the maximum

Identify the maximum

vertex and slice the top

vertex and slice the top

base at an angle towards

base at an angle towards

your minimum vertex.

your minimum vertex.