Question

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?

Answer 1

a.  

6 copies are stocked.

At demand level 0, loss = 6$\times$50 cents - 6$\times$5 cents = 300 cents - 30 cents = 270 cents = $2.70

At demand level 1, loss = 6$\times$50 cents - $1 - 5$\times$5 cents = 300 cents -100 cents - 25 cents = 175 cents = $1.75

At demand level 2, loss = 6$\times$50 cents - 2$\times$$1 - 4$\times$5 cents = 300 cents -200 cents - 20 cents = 80 cents = $0.80

At demand level 3, profit = 3$\times$$1 + 3$\times$5 cents - 6$\times$50 cents = 300 cents +15 cents - 300 cents = 15 cents = $0.15

At demand level 4, profit = 4$\times$$1 + 2$\times$5 cents - 6$\times$50 cents = 400 cents + 10 cents - 300 cents = 110 cents = $1.10

At demand level 5, profit = 5$\times$$1 + 5 cents - 6$\times$50 cents = 500 cents + 5 cents - 300 cents = 215 cents = $2.15

At demand level 6, profit = 6$\times$$1 - 6$\times$50 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 = (-$2.70)$\times$0.1 + (-$1.75)$\times$0.2 + (-$0.80)$\times$0.3 + $0.15$\times$0.1 + $1.10$\times$0.1 + $2.15$\times$0.1 + $3$\times$0.1 = -$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 = -45$\times$0.1 cents + 50$\times$0.2 cents = 5.5 cents

When n=2 :-

At demand level 0, profit = 2$\times$5 cents - 2$\times$50 cents = -90 cents

At demand level 1, profit = $1 + 5 cents - 2$\times$50 cents = 5 cents

At demand level 2, profit = 2$\times$$1 - 2$\times$50 cents = 100 cents

Expected profit = -90$\times$0.1 cents + 5$\times$0.2 cents + 100$\times$0.3 cents = 38 cents

When n=3 :-

At demand level 0, profit = 3$\times$5 cents - 3$\times$50 cents = -135 cents

At demand level 1, profit = $1 + 2$\times$5 cents - 3$\times$50 cents = -40 cents

At demand level 2, profit = 2$\times$$1 + 5 cents - 3$\times$50 cents = 45 cents

At demand level 3, profit = 3$\times$$1 - 3$\times$50 cents = 150 cents

Expected profit (in cents) = -0.1$\times$135 cents - 0.2$\times$40 cents + 0.3$\times$45 cents + 0.1$\times$150 cents = 7 cents

When n=4 :-

At demand level 0, profit = 4$\times$5 cents - 4$\times$50 cents = -180 cents

At demand level 1, profit = $1 + 3$\times$5 cents - 4$\times$50 cents = -85 cents

At demand level 2, profit = 2$\times$$1 + 2$\times$5 cents - 4$\times$50 cents = 10 cents

At demand level 3, profit = 3$\times$$1 + 5 cents - 4$\times$50 cents = 105 cents

At demand level 4, profit = 4$\times$$1 - 4$\times$50 cents = 200 cents

Expected profit (in cents) = -0.1$\times$180 cents - 0.2$\times$85 cents + 0.3$\times$10 cents + 0.1$\times$105 cents + 0.1$\times$200 cents = -1.5 cents

When n=5 :-

At demand level 0, profit = 5$\times$5 cents - 5$\times$50 cents = -225 cents

At demand level 1, profit = $1 + 4$\times$5 cents - 5$\times$50 cents = -130 cents

At demand level 2, profit = 2$\times$$1 + 3$\times$5 cents - 5$\times$50 cents = -35 cents

At demand level 3, profit = 3$\times$$1 + 2$\times$5 cents - 5$\times$50 cents = 60 cents

At demand level 4, profit = 4$\times$$1 + 5 cents - 5$\times$50 cents = 155 cents

At demand level 5, profit = 5$\times$$1 - 5$\times$50 cents = 250 cents

Expected profit ( in cents) = -0.1$\times$225 cents - 0.2$\times$130 cents - 0.3$\times$35 cents + 0.1$\times$60 cents + 0.1$\times$155 cents + 0.1$\times$250 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.