Ronald Indradjaja - California State Polytechnic ...



Ronald Indradjaja

Hwk #4(3)

Parisay’s comments are in red.

Pg. 765 #5 (completed with previous solutions and utility function)

During the summer, Olympic swimmer Adam Johnson swims every day. On sunny summer days, he goes to an outdoor pool, where he may swim for no charge. On rainy days, he must go to a domed pool. At the beginning of the summer, he has the option of purchasing $15 season pass to the domed pool, which allows him use for the entire summer. He doesn’t buy the season pass he must pay $1 each time he goes there. Past meteorological records indicate that there is a 60% chance that the summer will be sunny (in which case there is an average of 6 rainy days during the summer) and a 40% chance the summer will be rainy (an average of 30 rainy days during the summer).

Before the summer begins, Adam has the option of purchasing a long-range weather forecast for $1. The forecast predicts a sunny summer 80% of the time and a rainy summer 20% of the time. If the forecast predicts a sunny summer, there is a 70% chance that the summer will actually be sunny. If the forecast predicts a rainy summer, there is an 80% chance that the summer will actually be rainy. Assuming that Adam’s goal is to minimize his total expected cost for the summer, what should he do?

Find EVPI and EVSI. Complete your report to a manager and explain your findings. Consider a risk-averse personality of your choice and solve by using WinQSB. Write a report to a manager.

Table 1: WinQSB-Decision Tree Analysis Input Data

[pic]

Table 2: WinQSB-Decision Tree Analysis Output Data

[pic]

Graph 1: WinQSB Decision Tree Graph

[pic]

EVSI (Expected Value of Sample Information) Calculation:

Forecast Data Value: -$13.56

Meteorological Data Value: -(-$15.00)

$1.44

Only invest in a long-range weather forecast if it costs ≤ $1.44.

EVPI (Expected Value of Perfect Information) Calculation:

Class’s style:

Looking at Meterological Data

If perfect information indicates sunny then the best decision will be not to buy pass and pay $6 with probability of 0.6. If perfect information indicates rainy then the best decision will be to buy pass and pay $15 with probability of 0.4.

(-$6) * (0.6) + (-$15) * (0.4) = -$9.60

Book’s style by Mr. David Chavez (is corrected):

[pic]

(Perfect Info) – (No Info)

(-$9.60) – (-$15) = $5.40

Only purchase perfect info if it costs ≤ $5.40

Report to Manager:

Report 1:

This was the previous report. Both reports are correct.

The best decision is to pay $1 for forecasting and the minimum total expected cost is $14.56. There is 80% probability that the forecast will predict a sunny summer (with 6 days of rain). In this case, do not purchase the pass and the expected cost will be $13.2.

On the other hand there is 20% probability that the forecast will predict a rainy summer (with 30 days of rain). In this case, purchase the pass and the cost will be $15. Please note that the expected value means that if the process is repeated for many times the average cost will be as the expected value. However in one time effort the total cost will be $7, $31, or $16.

Report 2:

Considering the given information, the probabilities of the respective costs, and the objective function to minimize the total cost, Adam Johnson will have an expected cost of $14.56. However, this is not a fixed cost, due to the probabilities that play under the circumstances. Adam Johnson can expect to pay $14.56 in the long run, however, the cost can range from $6 to $31.

The best decision is to PURCHASE a long-range weather forecast for at a cost of $1. This will provide an expected cost of $14.56. Continuing further, if the Adam Johnson does decide to purchase the weather forecast, then the forecast can predict one of two conclusions, which will provide the following expected outcomes:

If forecast predicts that the summer will be sunny (80%), then the best decision is not to buy pass with an expected cost of $14.2. If forecast predicts that the summer will not be sunny (20%), then the best decision is to buy pass with a cost of $15 (notice there is no expected cost here).

In another scenario, if Adam decides NOT to purchase weather forecast it best to buy the pass for a cost of $15.

The Estimated Value of Sample Information is $1.44, which is the maximum we are willing to pay for a weather forecast. In the problem, purchasing a weather forecast will only cost Adam only $1. That is why the final decision is to purchase the weather forecast.

The Estimated Value of Perfect Information is $5.40, which is the maximum we are willing to pay for perfect information about the future.

Risk-Averse Personality of Choice:

Notice that all $ should be at terminal points before using utility. Therefore maximum is -6 and minimum is -31 as correctly indicated in the following graph.

Graph 2: Utility Function

[pic]

Table 3: WinQSB-Decision Tree Analysis Input Data for Risk Averse Person

[pic]

Table 4: WinQSB-Decision Tree Analysis Output Data for Risk Averse Person

[pic]

Graph 3: WinQSB Decision Tree for Risk Averse Person

[pic]

Utility Translation:

The expected utility cost that Adam Johnson will incur in the summer = 0.95

Utility of 0.95 = $15 on the Risk Averse Chart, shown in Graph 2: Utility Function.

Note: if we had used risk-neutral utility function the expected utility would have been 0.6576.

Report to Manager:

Considering the given information, the probabilities of the respective costs, and the objective function to minimize the total cost, Adam Johnson will have a cost of $15. (Notice in this case it is a fix cost and not an expected cost because of the nature of the situation. There is no probability involved for the related decision which is to buy the pass.)

The best decision is to NOT PURCHASE a long-range weather forecast and to PURCHASE a SWIM PASS (Note: this is the complete opposite of the original solution, which was to purchase the weather forecast and not purchase the swim pass).

-----------------------

Page 764 # 5 EVWPI

EVWPI =

-$9.60

Sunny

Rainy

Buy pass

.60

.40

No Buy pass

$-15

$-6

$-6

$-15

$-6

Buy pass

No Buy pass

$-15

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download