1. The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 board feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 board feet of redwood; each picnic table takes 6 labor hours and 35 board feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each.
a. How many benches and tables should Outdoor Furniture produce to obtain the largest possible profit? Use the graphical LP approach. (Submit both the QM and the graphical solution.)
2. MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs six technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e., all 160 hours of time) be maintained for each worker during next month's operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield $1,800 each.
a. Determine the most profitable number of each model of minicomputer to produce during the coming month.
b. What would the profit be if they decided to make at least 15 of each model?
3. Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $1.50, and each pound of grain costs $0.70. A pound of the dog food must contain at least 9 units of Vitamin 1, 10 units of Vitamin 2, and 6 units of Vitamin 3. A pound of beef contains 6 units of Vitamin 1, 12 units of Vitamin 2, and 4 units of Vitamin 3. A pound of grain contains 12 units of Vitamin 1, 9 units of Vitamin 2, and 10 units of Vitamin 3. Formulate this as a LP problem to minimize the cost of the dog food.
a. How many pounds of beef and grain should be included in each pound of dog food?
b. What is the cost and vitamin content of the final product?
c. What would the distribution of beef and grain be if the cost of beef increased to $3.50 per pound?
4. The Feed ‘N Ship Ranch fattens cattle for local farmers and ships them to meat markets in Kansas City and Omaha. The owners of the ranch seek to determine the amounts of cattle feed to buy so that minimum nutritional standards are satisfied, and at the same time total feed costs are minimized. The feed mix can be made up of the three grains that contain the following ingredients per pound of feed:
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FEED (OZ)
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INGREDIENT
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STOCK X
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STOCK Y
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STOCK Z
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A
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3
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2
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4
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B
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2
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3
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1
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C
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1
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0
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2
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D
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6
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8
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4
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The cost per pound of stocks X, Y, and Z are $3.50, $4.75, and $4.50, respectively. The minimum requirement per cow per month is 4 pounds of ingredient A, 5 pounds of ingredient B, 1 pound of ingredient C, and 8 pounds of ingredient D. The ranch faces one additional restriction: it can only obtain 500 pounds of stock Z per month from the feed supplier regardless of its need. Because there are usually 100 cows at the Feed ‘N Ship Ranch at any given time, this means that no more than 5 pounds of stock Z can be counted on for use in the feed of each cow per month.
a. Formulate this as an LP problem.
b. Solve using LP software.