Section 5.6Linear Programming

Objective Functions in Linear Programming

We will look at the important application of

systems of linear inequalities. Such systems arise

in linear programming, a method for solving

problems in which a particular quantity that

must be maximized or minimized is limited by

other factors. Linear programming is one of the

most widely used tools in management science.

It helps businesses allocate resources to manufacture

products in a way that will maximize profits.

An objective function is an algebraic expression in

two or more variables describing a quantity that

must be maximized or minimized.

Example

An adjunct college professor makes $12 an hour tutoring

at a local tutoring center, and $24 an hour teaching at the

local communicty college. Write an objective function that

describes total weekly earnings if hours worked tutoring is

x and hours worked teaching is y.

Example

A company manufactures kayaks and canoes. If

the company's profits are $200 on the kayaks and

$150 on canoes, write the objective equation for

the profit, z, made on x kayaks and y canoes that

can be produced in one month.

Constraints in Linear Programming

Example

The truck that takes the canoes and kayaks from the factory

can carry only a limited number with cargo space of 750

cubic feet. If each kayak takes up 5 cubic feet and each canoe

takes up 15 cubic feet of space, write an equation that describes

this situation where x is the number of kayaks and y is the

number of canoes. This is a constraint for the manufacturing

company.

Example

The truck that carries the kayaks and canoes can

only carry a maximum of 4000 lbs. The kayaks and

canoes both weigh the same amount, 40 lbs. If x is the

number of kayaks and y is the number of canoes, write

an equation that describes this situation. This is another

constraint that the manufacturing company must

consider.

Solving Problems with Linear Programming

ExampleStep 1:

A: Write down the Objective equation that exists

for the kayak/canoe manufacturing company.

B: Write the two constraint inequalities that you

found in the previous two examples.

C: Graph the two constraints on the same graph and

note the intersection of the regions.

Continued on the next screen.

x

y

ExampleStep 2:

A: From step 1 you should have the graph shown below.

B: Now locate the vertices of this region by finding the point

of intersection of the two lines, and the x and y intercepts.

Put these points in the chart below and plug those points

into the objective equation to find which vertex gives you

the maximum profit.

Corner (x,y)

Objective: 200x+150y=z

Example A plane carrying relief food and water to a tidal wave ravaged

community can carry a maximum of 50,000 lbs, and is limited in

space to carrying no more than 6000 cubic feet. Each container of

water weighs 60 lbs and takes up 1 cubic foot and each food container

weighs 50 lbs and takes up 10 cubic feet of space. The relief

organization wants the plane to assist as many people as possible and

it is known that the water containers can take care of 4 people and

the food containers can feed 10 people. Draw the region of constraint

and make recommendations on how many

containers of water and food should be

taken on the plane. How many people

will get food? water?Corner (x,y) Objective:

Example

An adjunct professor makes $12/hour tutoring at a local tutoring center

and $24/hr teaching at the local community college. Let x be the number of

hours tutoring and y the number of hours teaching. You have already

written the objective equation in a previous problem. There are however,

more constraints on her time. In order to take care of her children, she

can only work 20 hours a week, and the college requires that she

teach at least 5 hours a week for them, but no more than 8 hours.

How many hours should she work tutoring and

teaching. What will be her maximum income

each week?

Corner (x,y) Objective:

(a)

(b)

(c)

(d)

A farmer grows peaches (p) and apples(a). He knows that to

prevent a certain pest infection, the number of peach trees

cannot exceed the number of apple trees. Also because of

space requirements for each tree, the number of peach trees

plus twice the number of apple trees cannot exceed 100 trees.

He wants to produce the maximum number of bushels of fruit

from his orchard and he knows that each peach tree produces

80 bushels and apple trees each produce 100 bushels. What is the

constraint for space requirements.

2 100

80 100

80 100

p a

p a

p a z

a p z

(a)

(b)

(c)

(d)

A farmer grows apples and peaches. He knows that to

prevent a certain pest infection, the number of peach trees

cannot exceed the number of apple trees. Also because of

space requirements for each tree, the number of peach trees

plus twice the number of apple trees cannot exceed 100 trees.

He wants to produce the maximum number of bushels of fruit

from his orchard and he knows that each peach tree produces

80 bushels and apple trees each produce 100 bushels. What is the

objective function for this example.

2 100

80 100

80 100

p a

p a

p a z

a p z