WAMB is a television station that has 25 thirty-second advertising slots during each evening. It is early January and the station is selling advertising for Sunday, March 24. They could sell all of the slots right now for $4,000 each, but, because on this particular Sunday the station is televising the Oscar ceremonies, there will be an opportunity to sell slots during the week right before March 24 for a price of $10,000. For now, assume that a slot not sold in advance and not sold during the last week is worthless to WAMB. To help make this decision, the sales force has created the following probability distribution for last-minute sales:
|
Number of Slots, x
|
Probability Exactly x Slots Are Sold
|
|
8
|
0.00
|
|
9
|
0.05
|
|
10
|
0.10
|
|
11
|
0.15
|
|
12
|
0.20
|
|
13
|
0.10
|
|
14
|
0.10
|
|
15
|
0.10
|
|
16
|
0.10
|
|
17
|
0.05
|
|
18
|
0.05
|
|
19
|
0.00
|
a. How many slots should WAMB sell in advance?
b. In practice, there are companies willing to place standby advertising messages: if there is an empty slot available (i.e., this slot was not sold either in advance or during the last week), the standby message is placed into this slot. Since there is no guarantee that such a slot will be available, standby messages can be placed at a much lower cost. Now suppose that if a slot is not sold in advance and not sold during the last week, it will be used for a standby promotional message that costs advertisers $2,500. Now how many slots should WAMB sell in advance?
c. Suppose WAMB chooses a booking limit of 10 slots on advanced sales. In this case, what is the probability there will be slots left over for stand-by messages?
d. One problem with booking for March 24 in early January is that advertisers often withdraw their commitment to place the ad (typically this is a result of changes in promotional strategies; for example, a product may be found to be inferior or an ad may turn out to be ineffective). Because of such opportunistic behavior by advertisers, media companies often overbook advertising slots. WAMB estimates that in the past the number of withdrawn ads has a Poisson distribution with mean 9. Assume each withdrawn ad slot can still be sold at a standby price of $2,500 although the company misses an opportunity to sell these slots at $4,000 a piece. Any ad that was accepted by WAMB but cannot be accommodated (because there isn't a free slot) costs the company $10,000 in penalties. How many slots (at most) should be sold?
e. Over time, WAMB saw a steady increase in the number of withdrawn ads and decided to institute a penalty of $1,000 for withdrawals. (Actually, the company now requires a $1,000 deposit on any slot that is refunded only if WAMB is unable to provide a slot due to overbooking.) The expected number of withdrawn ads is expected to be cut in half (to only 4.5 slots). Now how many slots (at most) should be sold?