The daily demand for a local newspaper at a street vendor has the following probabilities:
Demand |
0 1 |
2 3 4 5 6 |
Probability |
.1 .2 |
.3 .1 .1 .1 .1 |
The news vendor must pay 50 cents for each copy. Each newspaper sells for $1 and unsold copies are returned to the distributor for a 5 cent refund.
a.[7 marks] When 6 copies are stocked, calculate the profit (or loss) for each possible level of demand.
b.[2 marks] When 6 copies are stocked, what is the expected profit?
c.[6 marks] How many copies should the news vendor stock to maximize expected profit?
a.
6 copies are stocked.
At demand level 0, loss = 650 cents - 65 cents = 300 cents - 30 cents = 270 cents = $2.70
At demand level 1, loss = 650 cents - $1 - 55 cents = 300 cents -100 cents - 25 cents = 175 cents = $1.75
At demand level 2, loss = 650 cents - 2$1 - 45 cents = 300 cents -200 cents - 20 cents = 80 cents = $0.80
At demand level 3, profit = 3$1 + 35 cents - 650 cents = 300 cents +15 cents - 300 cents = 15 cents = $0.15
At demand level 4, profit = 4$1 + 25 cents - 650 cents = 400 cents + 10 cents - 300 cents = 110 cents = $1.10
At demand level 5, profit = 5$1 + 5 cents - 650 cents = 500 cents + 5 cents - 300 cents = 215 cents = $2.15
At demand level 6, profit = 6$1 - 650 cents = 600 cents - 300 cents = 300 cents = $3
.
b.
Demand | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
Profit | -$2.70 | -$1.75 | -$0.80 | $0.15 | $1.10 | $2.15 | $3 |
Probability | 0.1 | 0.2 | 0.3 | 0.1 | 0.1 | 0.1 | 0.1 |
Expected profit = (-$2.70)0.1 + (-$1.75)0.2 + (-$0.80)0.3 + $0.150.1 + $1.100.1 + $2.150.1 + $30.1 = -$0.22
Thus the expected loss is 22 cents.
.
c.
Let the news vendor stock n copies, n=1,2,3,4,5,6
When n=1 :-
At demand level 0, profit = 5 cents - 50 cents = -45 cents
At demand level 1, profit = $1 - 50 cents = 50 cents
Expected profit = -450.1 cents + 500.2 cents = 5.5 cents
When n=2 :-
At demand level 0, profit = 25 cents - 250 cents = -90 cents
At demand level 1, profit = $1 + 5 cents - 250 cents = 5 cents
At demand level 2, profit = 2$1 - 250 cents = 100 cents
Expected profit = -900.1 cents + 50.2 cents + 1000.3 cents = 38 cents
When n=3 :-
At demand level 0, profit = 35 cents - 350 cents = -135 cents
At demand level 1, profit = $1 + 25 cents - 350 cents = -40 cents
At demand level 2, profit = 2$1 + 5 cents - 350 cents = 45 cents
At demand level 3, profit = 3$1 - 350 cents = 150 cents
Expected profit (in cents) = -0.1135 cents - 0.240 cents + 0.345 cents + 0.1150 cents = 7 cents
When n=4 :-
At demand level 0, profit = 45 cents - 450 cents = -180 cents
At demand level 1, profit = $1 + 35 cents - 450 cents = -85 cents
At demand level 2, profit = 2$1 + 25 cents - 450 cents = 10 cents
At demand level 3, profit = 3$1 + 5 cents - 450 cents = 105 cents
At demand level 4, profit = 4$1 - 450 cents = 200 cents
Expected profit (in cents) = -0.1180 cents - 0.285 cents + 0.310 cents + 0.1105 cents + 0.1200 cents = -1.5 cents
When n=5 :-
At demand level 0, profit = 55 cents - 550 cents = -225 cents
At demand level 1, profit = $1 + 45 cents - 550 cents = -130 cents
At demand level 2, profit = 2$1 + 35 cents - 550 cents = -35 cents
At demand level 3, profit = 3$1 + 25 cents - 550 cents = 60 cents
At demand level 4, profit = 4$1 + 5 cents - 550 cents = 155 cents
At demand level 5, profit = 5$1 - 550 cents = 250 cents
Expected profit ( in cents) = -0.1225 cents - 0.2130 cents - 0.335 cents + 0.160 cents + 0.1155 cents + 0.1250 cents = -12.5 cents
When n=6 , expected profit = -$0.22 = -22 cents
Thus it is seen that expected profit is maximum when n=2
The news vendor should stock 2 copies to maximize expected profit.
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