During the 2019 Masters Golf tournament Patrick Reed failed to repeat as champion. His scores relative to par in each round, on each hole are reported in the table. Hole Round 1 Round 2 Round 3 Round 4 -1 -1 -1 1 4 1 1. 1 6. 1. 8. 0. -1 9. 10 -1 1. 11 1. 1. 12 13 -2 -1 14 1 15 -1 -1 16 1. 17 -1 18
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- The 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.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.)A company that supplies gasoline to a city has recorded the weeklyusage (tons/week) for the past 3 years. The file BA3653GasolineRecord.xlsxcontains this record. (c) In general, the company delivers the demanded gasoline to customersthat store it in their own facilities. However, the company would like toobtain its own storage tanks to better handle variability. The companyhas identified several solutions: a 500 ton, a 1000 ton, a 2000 ton and a3000 ton system, with each tank being proportionately more expensiveas its size grows. Given your predictions and the expected variability,what size of tank would you choose?(d) Reaching the city is sometimes hazardous due to weather conditions.Even with these challenges, the company estimates that the most thatshipments to the city may be delayed is 1 week. Given this information,which tank choice (of those in part c) do you think would be appropriate? week demand (tons) 1 1174.5 2 1316.2 3 1197 4 1127.3 5 1193.1 6 1260.7 7…