Question

The owner of a high end US apparel store, Feminine Fashions, is planning to open another faacility in her hometown of Izmir, Turkey. She can open a large store, a small store, or, to hedge her debts, she could open a medium-sized store. The market for high-end apparel in Izmir could be favorable or unfavorable. If the market is favorable, a large store will earn her a payoff of $200,000. If unfavorable, she will suffer a net loss of $180,000. If she opens a medium sized store and the market is unfavorable, her loss will be $100,000. By contrast, a favorable market for her medium-sized store will generate a payoff of $140,000. A small store with a favorable market will result in a payoff of $60,000 but a payoff of -$20,000 of the market is unfavorable. The probability of a favorable market is 0.60 and that of an unfavorable market is 0.4.

(a) What should she decide? Analyze the problem using a decision tree.

(b) Perform a sensitivity analysis for the P (unfavorable market) to provide a range of probability values where one decision alternative would be preferred over the other two.

(c) If she hires an expert in the apparel industry in Turkey to get information about whether or not the market will be favorable, how much should she pay for the information?

Answer 1

We have the following payoff matrix for setting up the stores of different sizes

We also know that the probability of favorable market = P(F)= 0.6

probability of unfavorable market = P(UF)=0.4

The expected payoff due to a decision E(V) is given by

E(V) = (payoff if the market is favorable)*(Probability that market is favorable)+ (payoff if the market is not favorable)*(Probability that market is unfavorable)

expected value for opening different store sizes are given below

small store

EV(S) = 60,000*0.6 - 20,000*0.4 = $28,000

medium store

EV(M) = 140,000*0.6 - 100,000*0.4 = $44,000

Large store

EV(L) = 200,000*0.6 - 180000*0.4 = $48,000

We draw the decision tree as below. The square node is the decision node and the circle is the chance node.

We can see that the expected value of the payoff is the highest if she goes for large size store and it is $48,000

b) Let p be the probability of unfavorable market. Then we know that 1-p is the probability of favorable market.

We can write the expected value of each of the decision as follows

small store

EV(S) = 60,000*(1-p) - 20,000*p = 60,000- 80000p

medium store

EV(M) = 140,000*(1-p) - 100,000*p = 140,000 - 240000p

​Large strore

EV(L) = 200,000*(1-p) - 180,000*p = 200,000 - 380000p

Find the probability at which EV(L)>=EV(S)

200000-380000p >=60000-80000p

rearranging we get

300000p <=140000

p<= 0.47

The large format has higher payoff than the small format for p <= 0.47

large format with medium size, the probability at which EV(L)>=EV(M)

200000-380000p >= 140,000 - 240000p

rearranging we get

140000p<= 60000

p<= 0.43

That means the large format sores have higher expected value than both medium and small stores for p [0,0.43]

Now we will see when medium stores have higher expected value compared to small stores

That is EV(M)>=EV(S)

140,000 - 240000p>=60000-80000p

rearranging we get

160000p<=80000

p<=0.5

That means the medium sizes stores have higher expected value than small stores for p<=0.5

Summarizing, the preferred store sizes for probability range os unfavorable market

small size : preferred over probability range of p= 0.5 to 1

medium size: preferred range p = 0.43 to 0.5

large format stores are preferred over other sizes for p = 0 to 0.43

c)

Consider the folowing pay off

If she knows that the market condition is going to be favorable then the best decision to take is to go for large store as the pay off is the $200000. If she gets to know that the market condition is unfavorable then the small size gives the lowest negative payoff of -$20,000. That means if she has this perfect information about the market condition then the expected pay off is

200000*0.6 - 20000*0.4 = $112,000

However if this information were not available her expected payoff is $48000 by going for the large format store

So the expected value of perfect information is

112,000 - 48,000 = $64,000

That means she should be ready to pay upto $64000 for this information