Problem:
1. In the investment example in this chapter, how would the solution be affected if the requirement that then entire $70,000 be invested were relaxed such that it is the maximum amount available for investment?
If the entire amount available for investment does not have to be invested and the amount available is increased by $10,000 (to 80,000), how much will the total optimal return increase? Will the entire $10,000 increase be invested in one alternative?
2. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows
Brand Cost/Gallon
Yodel $1.50
Shotz 0.90
Rainwater 0.50
The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of Shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
3. Grafton Metalworks Company produces metal alloys from six different ores it mines. The company has an order from a customer to produce an alloy that contains four metals according to the following specifications: at least 21% of metal A, no more than 12% of metal B, no more than 7% of metal C, and between 30% and 65% of metal D. The proportion of the four metals in each of the six ores and the level of impurities in each ore are provided in the following table:
Metal(%)
Ore A B C D Impurities (%) Cost/Ton
1 19 15 12 14 40 $27
2 43 10 25 7 15 25
3 17 0 0 53 30 32
4 20 12 0 18 50 22
5 0 24 10 31 35 20
6 12 18 16 25 29 24
When the metals are processed and refined, the impurities are removed.
The company wants to know the amount of each ore to use per ton of the alloy that will minimize the cost per ton of the alloy.
a. Formulate a linear programming model for this problem
b. Solve the model by using the computer
Question-4.
Brooks City has three consolidated high schools, each with a capacity of 1,200 students. The school board has partitioned the city into five busing districts- north, south, east, west, and central- each with different high school student populations. The three schools are located in the central, west, and south districts. Some students must be bused outside their districts, and the school board wants to minimize the total bus distance traveled by these students. The average distances from each district to the three schools and the total student population in each district are as follows:
Distance (miles)
Central West South Student
District School School School Population
North 8 11 14 700
South 12 9 -- 300
East 9 16 10 900
West 8 -- 9 600
Central -- 8 12 500
The school board wants to determine the number of students to bus from each district to each school to minimize the total busing miles traveled.
a. Formulate a linear programming model for this problem
b. Solve the model by using the computer.