Linear Programming Project Problem #1

Size: px

Start display at page:

Download "Linear Programming Project Problem #1"

Suzanna Reeves
5 years ago
Views:

1 Problem #1 1. Calvin Butterball is chief mathematician for Fly-By-Night Aircraft Corp. He is responsible for mathematical analysis of the manufacturing of the company s two model planes, the Sopwith Camel and the larger Sopwith Hippopotamus. Each department at Fly-By-Night has certain restrictions concerning the number of planes which can be manufactured per day. Production: No more than 7 hippopotami and no more than 11 camels can be manufactured per day. Shipping: No more than 12 total planes can be manufactured per day. Sales: The number of Hippopotami manufactured per day must be no more than twice the number of Camels. Labor: you must use more than 100 man-hours of labor per day. It takes 100 man-hours to manufacture each Camel and 200 man-hours to manufacture each Hippo. a) Select variables to represent the number of Camels and Hippopotami manufactured each day. Write constraints expressing the limitations above. Draw a feasible region graph. State the vertices. b) If Fly-By-Night makes a profit of $300 per Camel and $200 per Hippo, what is the objective function to maximize profits for this problem? c) State the optimal solution of how many Camels and Hippos it would take in order to make the maximum profit.

2 Problem #2 2. The Internal Canine Academy raises and trains Siberian sled dogs and dancing French poodles for sale. There are a number of restrictions that apply to the feeding, training, and raising of each dog that the ICA sells. Breeders can supply ICA with at most 20 poodles and 15 sled dogs each year. Each poodle eats 2 lbs of food a day and each sled dog will eat 6 lbs a day. ICA food supplies are restricted to at most 100 lbs of food each day. Poodles require 1000 hours of training per year, while sled dogs require 250 hours per year. The academy restricts training time to no more than 10,000 hours per year. a) Select variables to represent the poodles and sled dogs. Write constraints expressing the limitations above. Draw a feasible region graph. State the vertices. b) If the ICA makes a profit of $200 for each poodle and $80 for each sled dog, what would the objective function be to maximize the profits for this problem? c) State the optimal solution of how many of each type of dog it would take to get the maximum profit.

3 Problem #3 3. Suppose that you go into business raising Thoroughbreds and Quarter horses. Having studied linear programming, you decide to maximize the feasible profit you can make. Your supplier can get you at most 20 thoroughbreds and at most 15 quarter horses to raise each year. You must raise at least 12 horses, total, each year to make your business worthwhile. A thoroughbred eats 2 tons of food per year and a quarter horse eats 6 tons per year. You can handle no more than 96 tons of food per year. a) Assign your variables. Write your constraints. Graph the feasible region. Find and state your vertices. b) You can make a profit of $500 for each thoroughbred and $200 for each quarter horse. What is your objective function to maximize your profits c) State the optimal solution for this problem.

4 Problem #4 4. Sinkin Boat Company builds two styles of fiberglass boats, the Guppy and the Stickleback. Each stage in the manufacturing process requires different numbers of man-hours, as follows. Boat Fiberglass Casting Painting Assembly Guppy Stickleback There are several requirements in which Mr. Sinkin must observe in the manufacture of the boats The number of man-hours per day for the fiberglass casting is no more than 360. The number of man-hours per day for painting is at most 360. The number of man-hours per day for assembly is at most No more than 13 Guppies can be made per day. At least 10 boats, total, must be made each day. The number of Sticklebacks per day must be no more than 4 times the number of Guppies made per day. a) Define the variables for the problem. Write the constraints for the limitations stated above. Draw the feasible region. Find the vertices. b) The profit for each Guppy is $400 and for each Stickleback is $300, what would the objective function look like? c) State the optimal solution for this problem.

5 Problem #5 5. You have just been hired as the manger of Todaro s, a small business that makes frozen pizzas to sale to local markets. Todaro s makes 12-inch pizzas and 16-inch pizzas. Preparation and packaging takes 0.2 hours for each 12-inch Preparation and packaging takes 0.25 hours for each 16-inch. The staff at Todaro s can put at most 240 hours into preparation and packaging per week The staff must meet the company quota of 1000 boxes per week. Between refrigeration space and space in the storage room, Todaro s can stock only enough ingredients to make at most inch and inch pizzas. a) Select variables to represent the 12-inch and 16-inch pizzas Write the constraints expressing the limitations to the problem. Graph the feasible region State the coordinates of the vertices of the region b) If Todaro s makes a profit of $2 per box on a 12-inch pizza and $4 per box on a 16-inch write an objective function to maximize the profits for this problem c) State the optimal solution for the number of each type pizza Todaro s will have to make to maximize the profits.

6 Problem #6 6. One of the dolls that Dolls R Us manufactures is Talking Tommy. Another doll is Silent Sally since she is manufactured without a talking mechanism. The company must product at least 8 Talking Tommy dolls and 20 Silent Sally dolls. The company only has enough employees to produce no more than 48 dolls per week. The demand for Silent Sally is higher than for Talking Tommy. The company sells twice as many Silent Sallys as it does Talking Tommys. a. What are the constraints for this problem? Graph the feasible region State the vertices of the region b. If the profit for each Talking Tommy is $3 and for each Silent Sally it is $7.50, what is the objective function to maximize profits? c. What is the optimal solution to maximize profits

7 Problem #7 7. You are a producer of a popular talk show that runs for thirty minutes. You are negotiating for appearances by the reggae group Steel Beat and by the opera singer Maria Bunion. There are several conditions for their appearances. Both must appear on the show. As producer, you have a maximum of $9000 to spend on entertainment for Monday s show. Steal Beat charges $600 for each minute they play. Bunion s performance fee is $300 for each minute that she sings. Eight minutes is reserved for the show s opening, closing, and commercial segments. The remainder of the time is available for featured guests. Steel Beat will not appear on the show unless they are allowed to perform at least one number lasting 3 minutes Maria Bunion will not appear unless she is allowed to perform at least as long as Steel Beat. a. Write the constraints for the limitations listed above. Graph the feasible region State the coordinates of the vertices. b. What is the objective function to minimize costs? c. What is the optimal solution to minimize the cost?

8 Problem #8 8. Yamada Development Company will build a major shopping mall with 3 magnet (large) stores and 80 smaller shops. It must be decided how many public parking spaces and how many employee parking spaces the parking lot will have. o o o There is room available to build no more than 2500 parking spaces. The city requires a minimum of 2000 total parking spaces, and also requires that there be at least 4 times as many public spaces as employee spaces. Prospective businesses have been promised a minimum of 20 employee spaces for each magnet store and 2 employee spaces for each smaller shop a. Write the constraints for the problem above, graph the feasible region, and state the vertices of the region. b. Suppose the construction costs are $190 per public space and $215 per employee space. Write the objective function to minimize cost. c. What is the optimal solution to minimize cost?

2.1H Warm Up: Pet Sitters A Develop Understanding Task

5 2.1H Warm Up: Pet Sitters A Develop Understanding Task The Martinez twins, Carlos and Clarita, are trying to find a way to make money during summer vacation. When they overhear their aunt complaining