## Review Exercises for Finite MathematicsApplied to the Real World Topic: Linear Programming

Summary for This Topic
True/False Quiz
Index of Review Exercises
Everything for Finite Math
Everything for Calculus
Everything for Finite Math & Calculus
Utility: Pivot and Gauss-Jordan Tool
Utility: Simplex Method Tool

Question 1 Solve the following linear programming problems by graphing.

(a) Find the maximum value of p = 2x + 3y subject to

2x + y ≤ 15
x + 3y ≤ 20
x ≥ 0, y ≥ 0
 Answer: p = x = y =

(b) Find the minimum value of c = 4x + 5y subject to

x + 2y 10
2x + 3y 18
x ≥ 0, y ≥ 0
 Answer: c = x = y =

(c) Find the maximum value of p = 6x + y subject to

3x + y ≤ 15
x + y ≤ 10
2x ≤ y
x ≥ 0, y ≥ 0
 Answer: p = x = y =

(d) Find the minimum value of c = 2x + 3y subject to

x + 4y 12
2x + y 10
y ≤ 2x
x ≥ 0, y ≥ 0
 Answer: c = x = y =

Question 2   Lawsuits

Solve by graphing: The Sue All Law Firm handles two types of lawsuits: medical malpractice suits against unscrupulous heart surgeons for performing unnecessary surgery, and suits against hard-working math professors for failing students who do not deserve to pass. Math professor lawsuits each require 6 person-months of preparation and the hiring of 5 expert witnesses, whereas medical lawsuits each require 10 person-months of preparation and the hiring of 3 expert witnesses. The firm has a total of 30 person-months to work with and feels that it cannot afford to hire more than 15 expert witnesses. It makes an average profit of \$1 million per math professor sued and \$5 million per heart surgeon sued. How many of each type of lawsuit should it initiate in order to maximize its expected profits?

Question 3  Book Sales

Solve by graphing: O'Hagan Bookworm Booksellers buys books from two publishers. Duffin House offers a package of 5 mysteries and 5 romance novels for \$50, and Gorman Press offers a package of 5 mysteries and 10 romance novels for \$150. O'Hagan wants to buy at least 2,500 mysteries and 3,500 romance novels, and he has promised Gorman (who has influence on the Senate Textbook Committee) that at least 25% of the total number of packages he purchases will come from Gorman Press. How many packages should O'Hagan order from each publisher in order to minimize his cost and satisfy Gorman? What will the novels cost him?

Question 4  More Book Sales

Solve by graphing: In addition to the best selection of novels in South Park Mall, O'Hagan Booksellers also specializes in fantasy novels and art books. The manager at O'Hagan Booksellers, S. Shady, is considering a sales promotion of a new collection of fantasy novels and art books, and he plans to price the books so low as to actually take a loss: O'Hagan will lose \$3 on every fantasy novel and \$2 on every art book sold in the promotion. Since the store will only offer the art books to those who purchase two or more fantasy novels, the store will sell at least twice as many fantasy novels as art books, and also plans to sell at least 210 items in all. On the other hand, the store can spare up to 900 units of display space for the sale. S. Shady calculates that fantasy novels each require 3 units of display space, while art books require 2 units. Given these constraints, how many fantasy novels and art books should O'Hagan Booksellers order to lose the least amount of money in the sales promotion?

Question 5

Solve the following linear programming problems with the simplex method. (To assist you in your calculation, use the Pivot and Gauss-Jordan Tool. To have the computer do everything for you, try the Simplex Method Tool.)

(a) Find the maximum value of p = x + 2y + 3z subject to:

7x + z ≤ 6
x + 2y ≤ 20
3y + 4z ≤ 30
x ≥ 0, y ≥ 0, z ≥ 0
 Answer: p = x = y = z =

(b) Find the maximum value of p = 2x - 3y + 5z subject to:

2x + y ≤ 16
y + z ≤ 10
x + y + z ≤ 20
x ≥ 0, y ≥ 0, z ≥ 0
 Answer: p = x = y = z =

(c) Find the minimum value of c = x + 2y + 3z subject to:

x + y + z ≥ 500
x + 2y + 3z ≥ 700,
-y + 3z ≤ 0
x ≥ 0, y ≥ 0, z ≥ 0
 Answer: c = x = y = z =

(d) Find the minimum value of c = 50x + 150y + 100z subject to:

5x + 5y + 5z ≥ 2,500
5x + 10y + 15z ≥ 3,500
3x - y + 3z ≤ 0
x ≥ 0, y ≥ 0, z ≥ 0
 Answer: c = x = y = z =

Question 6  Politics

Senator Porkbarrel overdraws his accounts at the following banks: the Congressional Integrity Bank, Citizens' Trust, and "Checks R Us." There are no penalties for these withdrawals since the overdrafts are subsidized by the taxpayer. The Senate Ethics Committee tends to let slide irregular banking activities of this sort, provided they are not flagrant. At the moment (due to Congress' preoccupation with impeachment hearings) a total overdraft of up to \$10,000 will be overlooked. Porkbarrel's underlying sense of guilt makes him feel funny about writing overdrafts for banks whose names include expressions like "integrity" and "citizens' trust." The effect of this is that his bad check writing for the first two banks combined amounts to no more than one-quarter of the total. On the other hand, the financial officers at Integrity Bank are eager to please Senator Porkbarrel due to his influential position on the Senate Banking Committee, so they would like him to overdraw his account by as much as possible. Find the amount he should draw from each bank in order to avoid investigation by the Ethics Committee and overdraw his account at Integrity by as much as his sense of guilt will allow.

Question 7  Browser Wars

Your software company has launched the latest version of its web browser, "Java Cruise 4.0." As sales manager, you are planning to promote Java Cruise 4.0 by sending sales forces to software conventions running concurrently in Saint Louis and Detroit. You have 6 representatives available at each of your Little Rock, Ark. and Urbana, Ill. branches, and you would like to send at least 5 to the Saint Louis convention and at least 4 to the Detroit convention. The Saint Louis convention will last for three days, while the Detroit convention will last for two days. Air fares (per person) and hotel accommodation costs (per person) are shown in the following figure.

How many representatives should you send from each branch to each convention in order to minimize the total (air travel and accommodation) cost? What will the total cost amount to?