Linear Programming

A Summary

What??

Linear Programming is an algebraic strategy used to find optimal solutions.– Uses linear inequalities called constraints.– The solution of the set of constraints is called

the feasible region.• How do we identify the feasible region?

– The function to be maximized is called the objective function.

Warm Up

1) Solve the following system of equations:

82

7

yx

yx

)2,5(

2

75

5

153

y

y

x

x

2) Graph the solution:

1

0

5

32

y

x

xy

xy

Connections

On a piece of graph paper, graph all the numbers x and y whose sum is less than or equal to 8.What mathematical statement can represent these points?

8: yxAnswer

Suppose we add the constraints:

2

52

y

xHow does this change your graph?

Connections Continued

Given the graph we just found, solve the following:

If D = x – y, find the least and greatest possible values for D within the region.

1) What are our critical (test) points?

)2,5)(3,5)(6,2)(2,2(

2) Test these points to find the largest and smallest values for D.

325:)2,5(

235:)3,5(

462:)6,2(

022:)2,2(

D

Smallest: (2,6) Largest: (5,2)

D is our OBJECTIVE FUNCTION!

Hint: Find the points of intersection for the lines!

Practice

Choose one of the following problems to solve.

For each problem, clearly identify the following:– The linear equalities that

produce the constraints.

– Graph the feasible region.

– Identify the test points.

– State the objective function.

– Find the solution.

Manufacturing

A ski manufacturer makes two types of skis and has a fabricating department and a finishing department.

A pair of downhill skis requires 6 hours to fabricate and 1 hour to finish.

A pair of cross-country skis requires 4 hours to fabricate and 1 hour to finish. The fabricating department has 108 hours of labor available per day.

The finishing department has 24 hours of labor available per day.

The company makes a profit of $40 on each pair of downhill skis and a profit of $30 on each pair of cross-country skis.

Find the maximum profit.

Transportation

Trenton, Michigan, a small community, is trying to establish a public transportation system of large and small vans.

It can spend no more that $100,000 for both sizes of vehicles and no more than $500 per month for maintenance.

The community can purchase a small van for $10,000 and maintain it for $100 per month.

The large vans cost $20,000 each and can be maintained for $75 per month.

Each large van carries a maximum of 15 passengers and each small van carries a maximum of 7 passengers.

We need to maximize the number of passengers.

Business A tourist agency can sell up to 1200 travel packages for a football

game. The package includes airfare, weekend accommodations, and the

choice of two types of flights: a nonstop flight or a two-stop flight. The nonstop flight can carry up to 150 passengers. The two-stop flight can carry up to 100 passengers. The agency can locate no more than 10 planes for the travel

packages. Each package with a non-stop flight sells for $1200 and each

package with a two-stop flight sells for $900. Assume that each plane will carry the maximum number of

passengers. Find the maximum revenue.

Health A school dietician wants to prepare a meal of meat and

vegetables that has the lowest possible fat and that meets the FDA recommended daily allowance of iron and protein.

The minimums are 20 mg of iron and 45 grams of protein. Each 3 oz serving of meat contains 45 grams of protein, 10 mg

of iron, and 4 grams of fat. Each 1 cup serving of vegetables contains 9 grams of protein, 6

mg of iron, and 2 grams of fat. Let x be the number of 3 oz servings of meat and let y be the

number of 1 cup servings of vegetables Find the minimum number of grams of fat for the given

constraints.

Agriculture

A farmer has 90 acres available for planting millet and alfalfa.

See costs $4 per acre for millet and $6 per acre for alfalfa. Labor costs are $20 per acre for millet and $10 per acre

for alfalfa. The expected income is $110 per acre for millet and $150

per acre for alfalfa. The farmer intends to spend no more than $480 for seed

and $1400 for labor. Find the maximum income.

Review Identify x and y. Write a system of inequalities

based on the given constraints. Graph the inequalities to find

the feasible region. Find the vertices to use as test

points. Write the objective function to

be maximized or minimized. Substitute the test points to

find the minimum or maximum value.