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# Linear Programming (5 Problems)

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Question 1

Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs \$500 and requires 100 cubic feet of storage space, and each medium shelf costs \$300 and requires 90 cubic feet of storage space. The company has \$75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is \$300 and for each medium shelf is \$150. In order to maximize profit, how many big shelves (B) and how many medium shelves (M) should be purchased?

Question 2

The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are \$3.00 per case and profits for diet soft drink are \$2.00 per case. This problem is solved using a computer software package, and the optimal production quantities are 90 cases of regular and 75 cases of diet soft drink for a maximum profit of \$420. If the company decides to increase the amount of syrup it uses during production of these soft drinks to 990 lbs. will the current product mix change? How many cases of each type of soft drink should the company produce? How much will the profit increase?

Question 3

The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of salt. Profits for a bag of Lime chips are \$0.40, and for a bag of Vinegar chips \$0.50. What are the optimal production quantities of these products and what is the optimal profit?

Question 4

Consider the following integer linear programming problem

Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 30
4x1 + 2x2 28
x1 8
x1 ,x2 0 and integer
The solution to the Linear programming relaxation is: x1 = 5.714, x2= 2.571.
What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function.

Question 5

A large book publisher has five manuscripts that must be edited as soon as possible. Five editors are available for doing the work, however their working times on the various manuscripts will differ based on their backgrounds and interests. The publisher wants to use an assignment method to determine who does what manuscript. Estimates of editing times (in hours) for each manuscript by each editor are as follows:

Editor
Manuscript A B C D E

1 12 8 10 16 13
2 9 10 14 13 9
3 17 14 9 18 12
4 15 7 11 9 18
5 12 18 22 11 27

What is the total minimum editing time?

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(See attached file for full problem description)