Managing an investment firm

1. An investment firm has $1 million to invest in stocks, bonds, certificates of deposit, and real estate. The firm wishes to determine the mix of investments that will maximize the cash value at the end of 6 years.
   Opportunities to invest in stocks and bonds will be available at the beginning of each of the next 6 years. Each dollar invested in stocks will return $1.20 (a profit of $0.20) 2 years later; the return can be immediately reinvested in any alternative. Each dollar invested in bonds will return $1.40 3 years later; the return can be reinvested immediately.
   Opportunities to invest in certificates of deposit will be available only once, at the beginning of the second year. Each dollar invested in certificates will return $1.80 four years later. Opportunities to invest in real estate will be available at the beginning of the fifth and sixth years. Each dollar invested will return $1.10 one year later.
   To minimize risk, the firm has decided to diversify its investments. The total amount invested in stocks cannot exceed 30% of total investments, and at least 25% of total investments must be in certificates of deposit.
   The firm’s management wishes to determine the optimal mix of investments in the various alternatives that will maximize the amount of cash at the end of the sixth year.
   a. Formulate a linear programming model for this problem.
   b. Solve the model by using the computer.
2. The manager of a department store in Seattle is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences.
   The television station representative indicates that a TV commercial, which costs $15,000, would reach 25,000 potential customers. The breakdown of the audience is as follows: Male Female
   Senior 5,000 5,000
   Young 5,000 10,000

The newspaper representative claims to be able to provide an audience of 10,000 potential customers at a cost of $4,000 per ad. The breakdown of the audience is as follows:

Male    Female

Senior 4,000 3,000
Young 2,000 1,000

The radio station representative says that the audience for one of the station’s commercials, which costs $6,000, is 15,000 customers. The breakdown of the audience is as follows:

Male    Female

Senior 1,500 1,500
Young 4,500 7,500
The store has the following advertising policy:
• Use at least twice as many radio commercials as newspaper ads.
• Reach at least 100,000 customers.
• Reach at least twice as many young people as senior citizens.
• Make sure that at least 30% of the audience is female.

Available space limits the number of newspaper ads to seven. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
c. Suppose a second radio station approaches the department store and indicates that its commercials, which cost $7,500, reach 18,000 customers with the following demographic breakdown:

Male    Female

Senior 2,400 3,600
Young 4,000 8,000

If the store considered this station along with the other media alternatives, how would this affect the solution?
Hint: we have 4 variables only