You have made it to the final round of “Let’s Make aDeal.” You know there is $1 million behind either door 1,door 2, or door 3. It is equally likely that the prize is behindany of the three. The two doors without a prize have nothingbehind them. You randomly choose door 2, but before door2 is opened Monte reveals that there is no prize behinddoor 3. You now have the opportunity to switch and choosedoor 1. Should you switch? Assume that Monte plays asfollows: Monte knows where the prize is and will open anempty door, but he cannot open door 2. If the prize is reallybehind door 2, Monte is equally likely to open door 1 ordoor 3. If the prize is really behind door 1, Monte mustopen door 3. If the prize is really behind door 3, Montemust open door 1. What is your decision?

You have made it to the final round of “Let’s Make aDeal.” You know there is $1 million behind either door 1,door 2, or door 3. It is equally likely that the prize is behindany of the three. The two doors without a prize have nothingbehind them. You randomly choose door 2, but before door2 is opened Monte reveals that there is no prize behinddoor 3. You now have the opportunity to switch and choosedoor 1. Should you switch? Assume that Monte plays asfollows: Monte knows where the prize is and will open anempty door, but he cannot open door 2. If the prize is reallybehind door 2, Monte is equally likely to open door 1 ordoor 3. If the prize is really behind door 1, Monte mustopen door 3. If the prize is really behind door 3, Montemust open door 1. What is your decision?

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter10: Introduction To Simulation Modeling

Section10.4: Simulation With Built-in Excel Tools

Problem 12P: In August of the current year, a car dealer is trying to determine how many cars of the next model...

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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?
In August of the current year, a car dealer is trying to determine how many cars of the next model year to order. Each car ordered in August costs 20,000. The demand for the dealers next year models has the probability distribution shown in the file P10_12.xlsx. Each car sells for 25,000. If demand for next years cars exceeds the number of cars ordered in August, the dealer must reorder at a cost of 22,000 per car. Excess cars can be disposed of at 17,000 per car. Use simulation to determine how many cars to order in August. For your optimal order quantity, find a 95% confidence interval for the expected profit.
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.)
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?
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An investor has a certain amount of money available to invest now. Three alternative investmentsare available. The estimated profit in Kwacha of each investment under each economic conditionare indicated in the following payoff table:Event Investment SelectionA B CEconomy declines 500 -2000 -7000No charge 1000 2000 -1000Economy Expand 2000 5000 20,000Based on his own past experience, the investor assigns the following probabilities to eacheconomic condition:( )( )( )Economy declines 0.30No change 0.50Economy expands 0.20PPP===i. Determine the optimal action based on the maximax criterion ii. Determine the optimal action based on the maximin criterion iii. Compute the expected monetary value (EMV) for each investment iv. Compute the expected opportunity loss (EOL) for each investment v. Explain the meaning of the expected value of perfect information (EVPI) in thisproblem vi. Based on the results of (iii) and (iv), which investment would you choose?vii. Compute the coefficient of…
In the time interval between t and t 1 seconds beforethe departure of Braneast Airlines Flight 313, there is aprobability pt that the airline will receive a reservation for 19.4 Further Examples of Probabilistic Dynamic Programming Formulations 1035 the flight and a probability 1 pt that the airline will receiveno reservation. The flight can seat up to 100 passengers. Atdeparture time, if r reservations have been accepted by theairline, there is a probability q(y|r) that y passengers willshow up for the flight. Each passenger who boards the flightadds $500 to Braneast’s revenues, but each passenger whoshows up for the flight and cannot be seated receives $200in compensation. Formulate a dynamic programmingrecursion to enable the airline to maximize its expectedrevenue from Flight 313. Assume that no reservations arereceived more than 100,000 seconds before flight time.
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