### MATH 318, Chapter 4 Quiz

Name:
You may use a calculator on this quiz. You may not use a cell phone or computer. If you wish partial credit for incorrect answers, please show your work carefully and give justifications for your answers. If you find that you are spending a lot of time on one problem, leave it blank and move on to the next. There are questions on both sides of this quiz paper.
Alpha Airlines is trying to decide whether to launch a new "executive" shuttle service from Boston to Washington, D.C. or to add more "discount" seats to their current service. They have estimated the following quarterly profits for the two services; projected profits vary depending on whether demand for air service between these two cities is strong or weak. Profits listed are in thousands of dollars. (This problem is a modified version of one on page 134 of the text.)

 Strong Weak Executive 960 -490 Discount 670 320
1. (20 points) What are the states of nature being considered in the Alpha Airlines decision?

The two states of nature are strong and weak demand for air service.

2. (20 points) If the airline chooses between executive and discount service using a conservative approach, which will they choose and why?

They will choose to add discount seats to their current service.

The worst case for the executive service is a loss of 490.
The worst case for the discount service is a profit of 320.
Discount has the best worst case, so the conservative approach recommends this decision.

3. (20 points) What combination of decision alternatives and states of nature lead to the largest (maximum) regret for Alpha Airlines?

Deciding on the executive service and encountering the state of nature of a weak demand results in a loss of 490, when there was potential for a profit of 320 for that state of nature.  The regret here is 320 - (-490) = 810.

The regret for a discount service under strong demand is only 960-670 = 310.

The greatest regret occurs with the combination of the executive decision and the weak state of nature.

4. An independent research firm estimates that the probability of strong demand for flights between Boston and Washington, D.C. is 70% and the probability of weak demand is 30%.
a) (10 points) Compute the expected values of quarterly profits for the executive and discount services.

EV(executive) = (960)(.7) + (-490)(.3) = 525

EV(discount) = (670)(.7) + (320)(.3) = 565

b) (10 points) Which decision alternative has the highest expected value?

The discount service has the highest expected value.

c) (20 points) Consider the estimated quarterly profits for executive class service with weak demand. What would the estimated profits need to be in order to change your answer to part (b) of this question? (I.e. calculate sensitivity to changes in this value.)

EV(executive) = 565

(960)(.7) + x (.3) = 565

.3x = -107

x = -356.666...

The estimated profit for executive service with weak demand would have to be no more of a loss than 356 and 2/3 thousand dollars in order to change the answer to part (b).

Bonus (5 points) Compute the expected value of perfect information (EVPI) for the Alpha Airlines decision.

EVwoPI = 565 (from problem 4.)

EVwPI = (960)(.7) + (320)(.3) = 767

EVPI = | EVwPI - EVwoPI | = 767 - 565 = 203