The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in the following Table. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $100 to rent a supercomputer for one month, $180 for two months, and $250 for three months. At the beginning of month 1, the IRS has no supercomputers. Determine the rental plan that meets the next 12 months’ requirements at minimum cost. Note: You may assume that fractional rentals are okay, so if your solution says to rent 140.6 computers for one month we can round this up or down (to 141 or 140) without having much effect on the total cost.

The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in the following Table. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $100 to rent a supercomputer for one month, $180 for two months, and $250 for three months. At the beginning of month 1, the IRS has no supercomputers. Determine the rental plan that meets the next 12 months’ requirements at minimum cost. Note: You may assume that fractional rentals are okay, so if your solution says to rent 140.6 computers for one month we can round this up or down (to 141 or 140) without having much effect on the total cost.

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 88P

See similar textbooks

Similar questions

You want to take out a 450,000 loan on a 20-year mortgage with end-of-month payments. The annual rate of interest is 3%. Twenty years from now, you will need to make a 50,000 ending balloon payment. Because you expect your income to increase, you want to structure the loan so at the beginning of each year, your monthly payments increase by 2%. a. Determine the amount of each years monthly payment. You should use a lookup table to look up each years monthly payment and to look up the year based on the month (e.g., month 13 is year 2, etc.). b. Suppose payment each month is to be the same, and there is no balloon payment. Show that the monthly payment you can calculate from your spreadsheet matches the value given by the Excel PMT function PMT(0.03/12,240, 450000,0,0).
Suppose you begin year 1 with 5000. At the beginning of each year, you put half of your money under a mattress and invest the other half in Whitewater stock. During each year, there is a 40% chance that the Whitewater stock will double, and there is a 60% chance that you will lose half of your investment. To illustrate, if the stock doubles during the first year, you will have 3750 under the mattress and 3750 invested in Whitewater during year 2. You want to estimate your annual return over a 30-year period. If you end with F dollars, your annual return is (F/5000)1/30 1. For example, if you end with 100,000, your annual return is 201/30 1 = 0.105, or 10.5%. Run 1000 replications of an appropriate simulation. Based on the results, you can be 95% certain that your annual return will be between which two values?
A European put option allows an investor to sell a share of stock at the exercise price on the exercise data. For example, if the exercise price is 48, and the stock price is 45 on the exercise date, the investor can sell the stock for 48 and then immediately buy it back (that is, cover his position) for 45, making 3 profit. But if the stock price on the exercise date is greater than the exercise price, the option is worthless at that date. So for a put, the investor is hoping that the price of the stock decreases. Using the same parameters as in Example 11.7, find a fair price for a European put option. (Note: As discussed in the text, an actual put option is usually for 100 shares.)
Suppose you currently have a portfolio of three stocks, A, B, and C. You own 500 shares of A, 300 of B, and 1000 of C. The current share prices are 42.76, 81.33, and, 58.22, respectively. You plan to hold this portfolio for at least a year. During the coming year, economists have predicted that the national economy will be awful, stable, or great with probabilities 0.2, 0.5, and 0.3. Given the state of the economy, the returns (one-year percentage changes) of the three stocks are independent and normally distributed. However, the means and standard deviations of these returns depend on the state of the economy, as indicated in the file P11_23.xlsx. a. Use @RISK to simulate the value of the portfolio and the portfolio return in the next year. How likely is it that you will have a negative return? How likely is it that you will have a return of at least 25%? b. Suppose you had a crystal ball where you could predict the state of the economy with certainty. The stock returns would still be uncertain, but you would know whether your means and standard deviations come from row 6, 7, or 8 of the P11_23.xlsx file. If you learn, with certainty, that the economy is going to be great in the next year, run the appropriate simulation to answer the same questions as in part a. Repeat this if you learn that the economy is going to be awful. How do these results compare with those in part a?
Based on Grossman and Hart (1983). A salesperson for Fuller Brush has three options: (1) quit, (2) put forth a low level of effort, or (3) put forth a high level of effort. Suppose for simplicity that each salesperson will sell 0, 5000, or 50,000 worth of brushes. The probability of each sales amount depends on the effort level as described in the file P07_71.xlsx. If a salesperson is paid w dollars, he or she regards this as a benefit of w1/2 units. In addition, low effort costs the salesperson 0 benefit units, whereas high effort costs 50 benefit units. If a salesperson were to quit Fuller and work elsewhere, he or she could earn a benefit of 20 units. Fuller wants all salespeople to put forth a high level of effort. The question is how to minimize the cost of encouraging them to do so. The company cannot observe the level of effort put forth by a salesperson, but it can observe the size of his or her sales. Thus, the wage paid to the salesperson is completely determined by the size of the sale. This means that Fuller must determine w0, the wage paid for sales of 0; w5000, the wage paid for sales of 5000; and w50,000, the wage paid for sales of 50,000. These wages must be set so that the salespeople value the expected benefit from high effort more than quitting and more than low effort. Determine how to minimize the expected cost of ensuring that all salespeople put forth high effort. (This problem is an example of agency theory.)
J&J has given you $12 million to spend on advertising Huggys diapers during the next 12 months. Atthe beginning of January, Huggys has a 30% market7.9 Conclusion 401share. During any month, 10% of the people whopurchase Huggys defect to brand X, and a fraction0.2a1y2 of customers who usually buy brand X switchto Huggys, where a is the amount spent on advertisingin millions of dollars. For example, if you spend $4million during a month, 40% of brand X’s customersswitch to Huggys. Your goal is to maximize J&J’saverage market share during the next 12 months,where the average is computed from each month’sending share. Determine an appropriate advertisingpolicy. (Hint: Make sure you enter a nonzero trialvalue for each month’s advertising expense or Solvermight give you an error message.)
We are going to invest $1,000 for a period of 6 months.Two potential investments are available: T-bills and gold. Ifthe $1,000 is invested in T-bills, we are certain to end the6-month period with $1,296. If we invest in gold, there is a34chance that we will end the 6-month period with $400 anda 14chance that we will end the 6-month period with $10,000.If we end up with x dollars, our utility function is given byu(x) x1/2. Should we invest in gold or T-bills?
2. Provident Capital Corp. specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client contacted Provident with P2,000,000 available to invest. Provident’s investment advisor recommends a portfolio consisting of two investment funds: the Dynamic fund and the Diversified fund. The Dynamic fund has a projected annual return of 10%, and the Diversified fund has a projected annual return of 8%. The investment advisor requires that at most P1,400,000 of the client’s funds should be invested in the Dynamic fund. Provident’s services include a risk rating for each investment alternative. The Dynamic fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per P40,000 invested. The Diversified fund has a risk rating of 4 per P40,000 invested. For example, if P400,000 is invested in each of the two investment funds, Provident’s risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, Provident developed…
Suppose an investor has the opportunity to buy the followingcontract, a stock call option, on March 1. Thecontract allows him to buy 100 shares of ABC stock atthe end of March, April, or May at a guaranteed priceof $50 per share. He can exercise this option at mostonce. For example, if he purchases the stock at the endof March, he can’t purchase more in April or May atthe guaranteed price. The current price of the stock is$50. Each month, assume that the stock price eithergoes up by a dollar (with probability 0.55) or goesdown by a dollar (with probability 0.45). If the investorbuys the contract, he is hoping that the stock price willgo up. The reasoning is that if he buys the contract, theprice goes up to $51, and he buys the stock (that is, heexercises his option) for $50, he can then sell the stockfor $51 and make a profit of $1 per share. On the otherhand, if the stock price goes down, he doesn’t haveto exercise his option; he can just throw the contractaway.a. Use a decision tree…
Ariel holds a $10,000 portfolio that consists of four stocks. Her investment in each stock, as well as each stock’s beta, is listed in the following table: Stock Investment Beta Standard Deviation Andalusian Limited (AL) $3,500 0.80 15.00% Kulatsu Motors Co. (KMC) $2,000 1.50 12.00% Western Gas & Electric Co. (WGC) $1,500 1.20 16.00% Makissi Corp. (MC) $3,000 0.50 22.50% Suppose all stocks in Ariel’s portfolio were equally weighted. Which of these stocks would contribute the least market risk to the portfolio? Makissi Corp. Andalusian Limited Western Gas & Electric Co. Kulatsu Motors Co. Suppose all stocks in the portfolio were equally weighted. Which of these stocks would have the least amount of standalone risk? Western Gas & Electric Co. Makissi Corp. Andalusian Limited Kulatsu Motors Co. If the risk-free rate is 4% and the market risk premium is 6%, what is Ariel’s…
Consider a capital budgeting problem with seven projects represented by binary (0 or 1) variables X1, X2, X3, X4, X5, X6, X7. Write a constraint modeling the situation in which only 2 of the projects from 1, 2, 3, and 4 must be selected. Write a constraint modeling the situation in which at least 2 of the project from 1, 3, 4, and 7 must be selected. Write a constraint modeling the situation project 3 or 6 must be selected, but not both. Write a constraint modeling the situation in which at most 4 projects from the 7 can be selected.
2) A buyer for a large sporting goods store chain must place orders for professional footballs with the footballmanufacturer six months prior to the time the footballs will be sold in the stores. The buyer must decidein November how many footballs to order for sale during the upcoming late summer and fall months.Assume that each football costs the chain $45. Furthermore, assume that each can be sold for a retail priceof $90. If the footballs are still on the shelves after next Christmas, they can be discounted and sold for$35 each. The probability distribution of consumer demand for these footballs (in hundreds) during theupcoming season has been assessed by the market research specialists and is presented below. Finally,assume that the sporting goods store chain must purchase the footballs in lots of 100 units. Demand Probability 4 0.30 5 0.50 6 0.20 a) Construct a decision tree…