Linear programming problems:Formulation and solution

1. The Lakeside Boatworks is planning to manafacture three types of molded fiber glass recreational boats-a fishing (bass) boat, a ski boat, and a small speedboat. The estimated selling price and variable cost for each type of boat are summarized in the following table.

Boat Variable Cost Selling Price
Bass $ 12,500 $23,000
Ski $ 8,500 $18,000
Speed $ 13,700 $26,000

The company has incurred fixed costs of $2,800,000 to set up its manufacturing operation and begin production. Lakeside has also entered into agreement with several boat dealers in the region to provide a minimum of 70 bass boats, 50 ski boats, and 50 speed boats. Alternatively, the company is unsure of what actual demand will be, so it has decided to limit production to no more than 120 of any one boat. The company wants to determine the number of boats that it must sell to break even while minimizing its total variable cost.

A. Formulate a linear programming model for this problem.
B. Solve the model using the computer.

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2. The Mill Mountain Coffee Shop blends coffee on the premises for its customers. It sells three basic blends in 1-pound bags, Special Mountain Dark, and Mill Regular. It uses four different types of coffee to produce the blends - Brazilian, mocha, Columbian, and mild. The shop used the following blend recipe requirements.

Blend Mix Requirements Selling Price/lb($)
Special At least 40% Columbian, at least 30% mocha 6.50
Dark At least 60% Brazilian, no more than 10% mild 5.25
Regular No more than 60% mild, at least 30% Brazilian 3.75

The cost of Brazilian coffee is $2.00 per, the pound, the cost of mocha is $2.75 per pound, the cost of Columbian is $2.90 per pound, and the cost of mild is $1.70 per pound. The shop has 110 pounds of Brazilian coffee, 70 pounds of mocha, 80 pounds of Columbian, and 150 pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit.

A. Formulate a Linear programming model for this problem.
B. Solve this model using the computer.

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3. Joe Henderson runs a small metal part shop. The shop contains three machines- a drill press, a lathe, and a grinder. Joe has three operators, each certified to work on all three machines. However each operator performs better on some machines than on other. The shop has contracted to do a big job that requires all three machines. The times required by the various operators to perform the required operations on each machine are summarized as follows.

Operator Drill Press (min) Lathe (min) Grinder (min)
1 22 18 35
2 41 30 28
3 25 36 18

Joe Henderson wants to assign one operator to each machine so that the total operating time for all three operators in minimized.

A. Formulate a linear programming model for this problem
B. Solve the model using the computer
C. Joe's brother, Fred, has asked him to hire his wife, Kelly, who is a machine operator. Kelly can perform each of the three required machine operations in 20 minutes. Should Joe hire his sister-in-law?