Concept explainers
Although the
The second argument determines the scale, in the sense that the product of it and the first argument equals the mean of the distribution. (The mean in this example is 30.) Also, the product of the second argument and the square root of the first argument is the standard deviation of the distribution. (In this example, it is
- a. Gamma distribution with parameters 2 and 85
- b. Gamma distribution with parameters 5 and 35
- c. Lognormal distribution with mean 170 and standard deviation 60
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Practical Management Science
- In Problem 11 from the previous section, we stated that the damage amount is normally distributed. Suppose instead that the damage amount is triangularly distributed with parameters 500, 1500, and 7000. That is, the damage in an accident can be as low as 500 or as high as 7000, the most likely value is 1500, and there is definite skewness to the right. (It turns out, as you can verify in @RISK, that the mean of this distribution is 3000, the same as in Problem 11.) Use @RISK to simulate the amount you pay for damage. Run 5000 iterations. Then answer the following questions. In each case, explain how the indicated event would occur. a. What is the probability that you pay a positive amount but less than 750? b. What is the probability that you pay more than 600? c. What is the probability that you pay exactly 1000 (the deductible)?arrow_forwardThe annual demand for Prizdol, a prescription drug manufactured and marketed by the NuFeel Company, is normally distributed with mean 50,000 and standard deviation 12,000. Assume that demand during each of the next 10 years is an independent random number from this distribution. NuFeel needs to determine how large a Prizdol plant to build to maximize its expected profit over the next 10 years. If the company builds a plant that can produce x units of Prizdol per year, it will cost 16 for each of these x units. NuFeel will produce only the amount demanded each year, and each unit of Prizdol produced will sell for 3.70. Each unit of Prizdol produced incurs a variable production cost of 0.20. It costs 0.40 per year to operate a unit of capacity. a. Among the capacity levels of 30,000, 35,000, 40,000, 45,000, 50,000, 55,000, and 60,000 units per year, which level maximizes expected profit? Use simulation to answer this question. b. Using the capacity from your answer to part a, NuFeel can be 95% certain that actual profit for the 10-year period will be between what two values?arrow_forwardContinuing the previous problem, assume, as in Problem 11, that the damage amount is normally distributed with mean 3000 and standard deviation 750. Run @RISK with 5000 iterations to simulate the amount you pay for damage. Compare your results with those in the previous problem. Does it appear to matter whether you assume a triangular distribution or a normal distribution for damage amounts? Why isnt this a totally fair comparison? (Hint: Use @RISKs Define Distributions tool to find the standard deviation for the triangular distribution.)arrow_forward
- You are considering a 10-year investment project. At present, the expected cash flow each year is 10,000. Suppose, however, that each years cash flow is normally distributed with mean equal to last years actual cash flow and standard deviation 1000. For example, suppose that the actual cash flow in year 1 is 12,000. Then year 2 cash flow is normal with mean 12,000 and standard deviation 1000. Also, at the end of year 1, your best guess is that each later years expected cash flow will be 12,000. a. Estimate the mean and standard deviation of the NPV of this project. Assume that cash flows are discounted at a rate of 10% per year. b. Now assume that the project has an abandonment option. At the end of each year you can abandon the project for the value given in the file P11_60.xlsx. For example, suppose that year 1 cash flow is 4000. Then at the end of year 1, you expect cash flow for each remaining year to be 4000. This has an NPV of less than 62,000, so you should abandon the project and collect 62,000 at the end of year 1. Estimate the mean and standard deviation of the project with the abandonment option. How much would you pay for the abandonment option? (Hint: You can abandon a project at most once. So in year 5, for example, you abandon only if the sum of future expected NPVs is less than the year 5 abandonment value and the project has not yet been abandoned. Also, once you abandon the project, the actual cash flows for future years are zero. So in this case the future cash flows after abandonment should be zero in your model.)arrow_forwardReferring to Example 11.1, if the average bid for each competitor stays the same, but their bids exhibit less variability, does Millers optimal bid increase or decrease? To study this question, assume that each competitors bid, expressed as a multiple of Millers cost to complete the project, follows each of the following distributions. a. Triangular with parameters 1.0, 1.3, and 2.4 b. Triangular with parameters 1.2, 1.3, and 2.2 c. Use @RISKs Define Distributions window to check that the distributions in parts a and b have the same mean as the original triangular distribution in the example, but smaller standard deviations. What is the common mean? Why is it not the same as the most likely value, 1.3?arrow_forwardSuppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are shown in the file P11_46.xlsx. The correlations between the annual returns on the four stocks are also shown in this file. a. What is the probability that your portfolios annual return will exceed 30%? b. What is the probability that your portfolio will lose money during the year?arrow_forward
- 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.)arrow_forwardThe method for rating teams in Example 7.8 is based on actual and predicted point spreads. This method can be biased if some teams run up the score in a few games. An alternative possibility is to base the ratings only on wins and losses. For each game, you observe whether the home team wins. Then from the proposed ratings, you predict whether the home team will win. (You predict the home team will win if the home team advantage plus the home teams rating is greater than the visitor teams rating.) You want the ratings such that the number of predictions that match the actual outcomes is maximized. Try modeling this. Do you run into difficulties? (Remember that Solver doesnt like IF functions.) EXAMPLE 7.8 RATING NFL TEAMS9 We obtained the results of the 256 regular-season NFL games from the 2015 season (the 2016 season was still ongoing as we wrote this) and entered the data into a spreadsheet, shown at the bottom of Figure 7.38. See the file NFL Ratings Finished.xlsx. (Some of these results are hidden in Figure 7.38 to conserve space.) The teams are indexed 1 to 32, as shown at the top of the sheet. For example, team 1 is Arizona, team 2 is Atlanta, and so on. The first game entered (row 6) is team 19 New England versus team 25 Pittsburgh, played at New England. New England won the game by a score of 28 to 21, and the point spread (home team score minus visitor team score) is calculated in column J. A positive point spread in column J means that the home team won; a negative point spread indicates that the visiting team won. The goal is to determine a set of ratings for the 32 NFL teams that most accurately predicts the actual outcomes of the games played.arrow_forwardAn automobile manufacturer is considering whether to introduce a new model called the Racer. The profitability of the Racer depends on the following factors: The fixed cost of developing the Racer is triangularly distributed with parameters 3, 4, and 5, all in billions. Year 1 sales are normally distributed with mean 200,000 and standard deviation 50,000. Year 2 sales are normally distributed with mean equal to actual year 1 sales and standard deviation 50,000. Year 3 sales are normally distributed with mean equal to actual year 2 sales and standard deviation 50,000. The selling price in year 1 is 25,000. The year 2 selling price will be 1.05[year 1 price + 50 (% diff1)] where % diff1 is the number of percentage points by which actual year 1 sales differ from expected year 1 sales. The 1.05 factor accounts for inflation. For example, if the year 1 sales figure is 180,000, which is 10 percentage points below the expected year 1 sales, then the year 2 price will be 1.05[25,000 + 50( 10)] = 25,725. Similarly, the year 3 price will be 1.05[year 2 price + 50(% diff2)] where % diff2 is the percentage by which actual year 2 sales differ from expected year 2 sales. The variable cost in year 1 is triangularly distributed with parameters 10,000, 12,000, and 15,000, and it is assumed to increase by 5% each year. Your goal is to estimate the NPV of the new car during its first three years. Assume that the company is able to produce exactly as many cars as it can sell. Also, assume that cash flows are discounted at 10%. Simulate 1000 trials to estimate the mean and standard deviation of the NPV for the first three years of sales. Also, determine an interval such that you are 95% certain that the NPV of the Racer during its first three years of operation will be within this interval.arrow_forward
- 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?arrow_forwardSimulation can be used to illustrate a number of results from statistics that are difficult to understand with nonsimulation arguments. One is the famous central limit theorem, which says that if you sample enough values from any population distribution and then average these values, the resulting average will be approximately normally distributed. Confirm this by using @ RISK with the following population distributions (run a separate simulation for each): (a) discrete with possible values 1 and 2 and probabilities 0.2 and 0.8; (b) exponential with mean 1 (use the RISKEXPON function with the single argument 1); (c) triangular with minimum, most likely, and maximum values equal to 1,9, and 10. (Note that each of these distributions is very skewed.) Run each simulation with 10 values in each average, and run 1000 iterations to simulate 1000 averages. Create a histogram of the averages to see whether it is indeed bell-shaped. Then repeat, using 30 values in each average. Are the histograms based on 10 values qualitatively different from those, based on 30?arrow_forwardA 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.)arrow_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,