Decision Structure: The night crew is expensive because you pay double-time for night-shift hours ( 8 hours starting at 10pm ) whether it snows or not. If you called in the crew every winter night on which there was even a small chance of snow you'd be erring on the cautious side and busting your budget. But if you blow it and allow unplowed runways to greet the sunrise, you lose that day's landing fees (plus a lot of good will, the utility of which is hard to measure). To decide, you check the National Weather Service website at 9 pm to get a probability of snow (given as a percent chance in 10% increments), and then decide whether to call up the crew based on the probability and this decision struure: - The plow crew costs you $960 night. (That is 3 workers at 40hr,1.5 times the workday rate, aka night differential, for 8 hours, whether it snows or not) - A day of landing fees brings in $3,300 (you lose this if you don't plow on a snowy night). Recall from recitation and class that the expected utility of a decision in the face of two possible states of nature is the outcome utility for the first state of nature times the probability of that state, plus the outcome utility of the second state of nature times the probability of that state. And recall that the probability of a discreet event not occurring is one minus the probability of it happening. Here's your decision matrix: Now bow you assign the outcome utilities to the cells in the payoff matrix is somewhat subjective, and depends on what you consider to be your nominal operating condition--in the absence of a forecast would you have a crew every winter night? Or not? We'l assume that the crew is a variable, added cost, so when you don't have the erew you don't claim the savings as income. - Getting it "right" - having the crew and the snowstorm-brings you the income of the fees ( $3300) minas the crew's time ($960) for a net utility of $2340. - Getting it "right the other way," having no crew but also no snowstorm, is your best outcome, with no expenses and all of the income for a net of $3,300. - Getting it "wrong" - having no crew and a snowstorm-is your worst-case outcome, and you lose all of the landing fees and we'll say that your utility there is $0. - Getting it "wrong the other way" by having a crew but no snowstorm-gives you the landing fees ($3300) minus the ($960) for a net outcome of $2340, a diminished outcome from the best For each possible probability of snow you should choose the decision that has the largest net expected utility (keeping in mind that larger negative values are smaller utilities). Based on your decision analysis, answer these questions. (You can use the Crew Standby worksheet in the accompanying "Exercise. Two_Data" Excel spreadsheet for the calculations and to make the plot for Q3): (1) For a 20% chance of a snowstorm, fill out the probabilities, and outcome utilities (consequences) in the pay-off matrix above ( 4pts) (2) At 20% chance of snow, what decision (overnight crew or no crew) maximizes your expected utility? (1pt) (3) Now plot the expected utility of a crew and no crew for each 10 percent increment chance of snow from 10 to 90% (use the worksheet as shown in recitation) Insert your plot. ( 1 pt) To embed the plot, save it as an image by taking a screenshot (command shift +4 for Macs, snipping tool for PCs) or right clicking and selecting 'save as picture'. You can then click on the image box below and embed your plot. (4) What is the point of indifference (the probability at which the two decisions have the same expected utility)? %(1pt) Decision Structure: The night crew is expensive because you pay double-time for night-shift hours ( 8 hours starting at 10pm ) whether it snows or not. If you called in the crew every winter night on which there was even a small chance of snow you'd be erring on the cautious side and busting your budget. But if you blow it and allow unplowed runways to greet the sunrise, you lose that day's landing fees (plus a lot of good will, the utility of which is hard to measure). To decide, you check the National Weather Service website at 9 pm to get a probability of snow (given as a percent chance in 10% increments), and then decide whether to call up the crew based on the probability and this decision struure: - The plow crew costs you $960 night. (That is 3 workers at 40hr,1.5 times the workday rate, aka night differential, for 8 hours, whether it snows or not) - A day of landing fees brings in $3,300 (you lose this if you don't plow on a snowy night). Recall from recitation and class that the expected utility of a decision in the face of two possible states of nature is the outcome utility for the first state of nature times the probability of that state, plus the outcome utility of the second state of nature times the probability of that state. And recall that the probability of a discreet event not occurring is one minus the probability of it happening. Here's your decision matrix: Now bow you assign the outcome utilities to the cells in the payoff matrix is somewhat subjective, and depends on what you consider to be your nominal operating condition--in the absence of a forecast would you have a crew every winter night? Or not? We'l assume that the crew is a variable, added cost, so when you don't have the erew you don't claim the savings as income. - Getting it "right" - having the crew and the snowstorm-brings you the income of the fees ( $3300) minas the crew's time ($960) for a net utility of $2340. - Getting it "right the other way," having no crew but also no snowstorm, is your best outcome, with no expenses and all of the income for a net of $3,300. - Getting it "wrong" - having no crew and a snowstorm-is your worst-case outcome, and you lose all of the landing fees and we'll say that your utility there is $0. - Getting it "wrong the other way" by having a crew but no snowstorm-gives you the landing fees ($3300) minus the ($960) for a net outcome of $2340, a diminished outcome from the best For each possible probability of snow you should choose the decision that has the largest net expected utility (keeping in mind that larger negative values are smaller utilities). Based on your decision analysis, answer these questions. (You can use the Crew Standby worksheet in the accompanying "Exercise. Two_Data" Excel spreadsheet for the calculations and to make the plot for Q3): (1) For a 20% chance of a snowstorm, fill out the probabilities, and outcome utilities (consequences) in the pay-off matrix above ( 4pts) (2) At 20% chance of snow, what decision (overnight crew or no crew) maximizes your expected utility? (1pt) (3) Now plot the expected utility of a crew and no crew for each 10 percent increment chance of snow from 10 to 90% (use the worksheet as shown in recitation) Insert your plot. ( 1 pt) To embed the plot, save it as an image by taking a screenshot (command shift +4 for Macs, snipping tool for PCs) or right clicking and selecting 'save as picture'. You can then click on the image box below and embed your plot. (4) What is the point of indifference (the probability at which the two decisions have the same expected utility)? %(1pt)