1. [10 pts.] Alpine Attic is the charity sponsored by local Episcopal churches in Denver, Colorado. Literally thousands of items, including televisions and stereos, are donated each year, most in need of repair. When Alpine Attic receives either a television or a stereo, it determines whether it can be sold “as is” or should be scrapped for parts. Those not sold “as is” are sent directly to JKL Electronics, whose owner, John K. Lucas, is a deacon at St. Paul’s Episcopal Church. In addition to his primary business, each month John donates 45 hours of an electrician’s time and 30 hours of a technician’s time to rebuild and test televisions and stereos for Alpine Attic. In addition to a tax write-off, he feels rewarded by helping out his church.
The recycled televisions and stereos typically sell for $50 and $30 each at the Alpine Attic Thrift Store. Each recycled television averages 90 minutes of an electrician’s time to rebuild and 30 minutes of a technician’s time to test, whereas each stereo averages 30 minutes of an electrician’s time to rebuild and 60 minutes of a technician’s time to test. What is the best use of the electrician’s and technician’s time to help Alpine Attic realize its optimal profit each month?
2. [15 pts.] Intronix uses copy editors, computer graphics specialists, and Java programmers to produce ads for magazines and the Internet. The average new ad for magazines typically requires 180 hours of a copy editor’s time and 135 hours of a computer graphics specialist’s time, whereas ads produced for the Internet require 35 hours of copy editor time, 195 hours of computer graphics time, and 60 hours of a Java programmer’s time.
Lassie Foods, a dog food manufacturer, has hired Intronix to produce adds in the next four weeks. Although currently it considers magazine ads 3 times more valuable than Internet ads, it still wishes to have at least 2 of each produced within the next four weeks. Intronix has assigned up to 3 copy editors, 4 computer graphics specialists, and 1 Java programmer, each committed to work up to 70 hours per week on the project. How many of each type of ad should be produced to maximize the overall value to Lassie Foods? (Note: make sure that you consider 4 weeks in work time limits.)
3. [15 pts.] Lawn Master produces 19-inch and 21-inch lawn mower, which it sells to membership warehouses and discount stores nationwide. Each lawn mower is powered by a Briggs and Stratton 3.5-horsepower engine. The 19-inch model is a “side-bagger” and requires 40 minutes (2/3 hour) to assemble, test, and package. The 21-inch model is a “rear-bagger” with a variable speed assembly and requires one hour to perform the same operations.
Each week Lawn Master can receive up to 200 Briggs and Stratton engines and has production facilities to manufacture up to 100 variable-speed assemblies. There are four production lines, each working eight hours a day, five days a week, for assembly, testing, and packaging. Each 19-inch model nets Lawn Master a $50 profit, whereas each 21-inch model nets a $60 profit.
a. Formulate and solve a linear programming problem for Lawn Master to determine an optimal weekly production schedule of 19-inch and 21-inch lawn mowers. What is the optimal weekly profit?
b. Determine and interpret the range of feasibility for (i) engines; (ii) variable speed assemblies; (iii) production hours.
c. What is a shadow price of engines?
d. What would be the profit if Lawn Master received 210 engines?
4. [20 pts.] Bay City Movers is a local company that specializes in intercity moves. In the business plan submitted to its backers, Bay City has committed itself to a total trucking capacity of at least 42 tons.
The company is in the process of replacing its entire fleet of trucks with 1-ton pickup trucks and 2 ½-ton moving van-type tracks. The 1-ton pickup trucks will be manned by one worker, whereas the large vans will utilize a total of four workers for larger moves.
Bay City Movers currently employs 63 workers and has facilities for at most 50 trucks. Pickup trucks cost the company 24,000 each; the moving vans cost 60,000 each. The company wishes to make a minimum investment of capital that will provide a trucking capacity of at least 42 tons while not requiring any additional workers or trucking facilities.
a. Use a linear programming model to determine the optimal number of pickup trucks and moving vans Bay City Movers should purchase.