Question

WVTU is a television station that has 20 thirty-second advertising slots during their regularly scheduled programming each evening. The station is now selling advertising for the first few days in November. They could sell all the slots immediately for $4,500 each, but because November 7 will be an election day, the station manager knows she may be able to sell slots at the last minute to political candidates in tight races for a price of $8,000 each. The demand for these last minute slots is estimated as follows:
8 9 10 11 12 13 14 15 16 17 18 19
Probability 0.03 0.05 0.1 0.15 0.2 0.15 0.1 0.05 0.05 0.05 0.05 0.02

Slots not sold in advance and not sold to political candidates at the last minute can be sold to local advertisers at a price of $2,000.

a. If the station manager sells all the advertising slots in advance, how much revenue will the station receive?
b. How many advertising slots should be sold in advance if the station manager wants to maximize expected revenue?
c. If the station manager sells in advance the number of slots identified in the previous question, what is the probability that the total revenue received will exceed the amount identified in part a where all slots are sold in advance?

Answer 1

Answer:

a. $90,000

b. 15 advertisement slots

c. 0.53

Step-by-step explanation:

Let the purchase price of the slots, c = 4,500

The salvage value of the slots, s = 2,000

The selling price of the slots, p = 8,000

a. If the store manager sells all the advertising slots in advance, the revenue received, 'R', is given as follows;

R = The number of slots, n × The purchase price for selling immediately, c

∴ R = 20 × $4,500 = $90,000

The revenue the station will receive by selling all the advertising slots in advance, R = $90,000

b. The Cost of Overage, $C_o$ = Purchase price - Salvage value

∴ The Cost of Overage, $C_o$ = 4,500 - 2,000 = 2,500

The Cost of Underage, $C_u$ = Selling price - Purchase price

∴ The Cost of Underage, $C_u$ = 8,000 - 4,500 = 3,500

P; 0.03, 0.05, 0.1, 0.15, 0.2, 0.15, 0.1, 0.05, 0.05, 0.05, 0.05, 0.02

With help of MS Excel, we get the following cumulative frequency

F(Q); 0.03, 0.08, 0.18, 0.33, 0.53, 0.68, 0.78, 0.83, 0.88, 0.93, 0.98, 1

The expected monetary value for the decision to sell the 8th slot at the last minute is given by the following relation;

$C_u$ × [1 - F(Q)] - $C_o$ × F(Q)

Where F(Q) = The cumulative frequency for the 8th stock

Therefore, we have;

3,500 × (1 - 0.03) - 2,500 × 0.03 = 3,300

For the 8th slot, the TV station is expected to make an extra $3,300

Using MS Excel, we get;

For the 9th slot, the profit over the purchase price = $3,020

Fot the 10th slot, the profit over the purchase price = $2,420

The profit over the purchase price for the 11th slot = $1,520

The profit over the purchase price for the 12th slot = $320

The profit over the purchase price for the 13th slot = $(-580)

The 14th = $(-1,180)

15th = $(-1,480)

16th = $(-1,780)

17th = $(-2,080)

18th = $(-2,380)

19th = $(-2,500)

Therefore, to maximize profit, only the 8th, 9th, 10th, 11th, and 12th, slots which are 5 slots should be sold late while 20 - 5 = 15 slots should be sold in advance

c. The probability that the total revenue will exceed the identified amount in part 'a' is the cumulative probability at the 12th slot which is given as follows;

P(D ≤ 12) = F(12) = 0.53.