Problem 4 (12 points) The Welsh Rabbit is a retailer of European cheese and rela

Question

Problem 4 (12 points) The Welsh Rabbit is a retailer of European cheese and related products. In one of their local stores they plan to offer a gift basket of cheese, crackers, and tea for the holiday season. They plan to place one order for the gift baskets, and any leftover inventory will be discounted at the end of the season. Expected demand for the gift baskets at this location is 4.5 baskets, Poisson-distributed. The gift basket sells for $55, the purchase cost to The Welsh Rabbit is $32, and leftover baskets will be sold at 40% of the original price (60% off). Suppose they purchase only 3 gift baskets. What is the probability that some demand will not be satisfied? a. b. Suppose they purchase 10 gift baskets. What is the expected left over inventory? Suppose they purchase 4 baskets. How many baskets can they expect to sell? How many baskets should The Welsh Rabbit purchase to maximize the expected profit? c. d. What is the expected profit for that order quantity?

Explanation / Answer

a)

x be the baskets sold and it follows poisonn distribution. it's mean s 4.5

purchases 3 gift basets, some demand will not be satisfied means x is greater than 3

P (x> 3) = 1 - P (x <3) = 1 - (p(x=0)+p(x=1)+p(x=2)+p(x=3))

p(x) = e^(-mu) mu^x/ factorial(x)

p(x>3) = 1-(0.0111+0.05+0.1125+0.1687) = 0.6577

b)

purchased 10 baskets. There is some inventory means "x" the sold baskets is 9 or less.

expected sold = x * p(x)

for all x less than 10, sum (x *p (x) ) = 4.3188

expected inventory = 10- 4.32 = 4.68 or approximately 5

c)

purchased 4

expected to sell = E(4) = 0.1898*4 = 3.04

d)

This is the matrix of profit for sold and not sold items

Find for each x what is expected to sell (E(x) = x * p(x))

x - E(x) is the number of not sold items.

Multiply E(x) by 23 to get profit from sold items

Multiply x- E(x) by -10 to get profit from notsold items.

Add both to get net profit

As you can see for 6 items the net profit is at the highest value. It is 92.21

x P(x) = e^-mu mu^x/ x! 0 0.0111 1 0.0500 2 0.1125 3 0.1687 4 0.1898 5 0.1708 6 0.1281 7 0.0824 8 0.0463 9 0.0232 10 0.0104

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