Question

A fruit vendor purchases apples to sell at the fruit stand in a hospital cafeteria. The apples are purchased at the beginning of every week for $4.00 per pound and sold at $6 per pound. Due to strict safety regulations, any apples that are left over at the end of the week cannot be sold the next week, and are sold to a local zoo for $.75 per pound. The demand for apples (in pounds) is as follows:

Demand (lbs)	Frequency
20	.05
21	.10
22	.05
23	.15
24	.40
25	.25

1. What is the average demand for apples? Assuming that the fruit vendor purchases 23 pounds of apples every week what is the average sales in pounds?
2. What is the cost of under-estimating demand? What is the cost of over-estimating demand?
3. What is the amount of apples the vendor should purchase to maximize profit?

Answer 1

Demand (lbs)	Frequency	Demand*Freq	Squared Dev
20	0.05	1	12.25
21	0.1	2.1	6.25
22	0.05	1.1	2.25
23	0.15	3.45	0.25
24	0.4	9.6	0.25
25	0.25	6.25	2.25
	Average	23.5	4.85

The sum of Demand * Frequency gives the average demand of the apples, μ = 23.5 lbs

Squared Deviation = (Average-Individual Demand)^2

The square root of the sum of deviations gives the standard deviation.

From the above table, the standard deviation, σ = 4.85 lbs

The cost price of apples, c = $4/lb

The selling price of apples, p = $6/lb

Unsold apples sold to zoo (Salvage value), s = $0.75/lb

The underage cost, Cu = p-c = 6-4 = $2/lb

The overage cost, Co = c-s = 4-0.75 = $3.25/lb

Critical Ratio = Cu/(Cu+Co) = 2/(2+3.25) = 0.38

z value for 0.38 from the standard normal table = -0.30

Quantity to order to maximize profit = μ +zσ = 23.5+(-0.3)*4.85 = 22.05 rounded to 22 lb

If fruit vendor purchases 23 pounds of apples each week,

Q = 23

z = (Q-μ)/σ = (23-23.5)/4.85 = -0.10

Expected lost sales = σ*L(z) = 4.85*0.45 = 2.18

Expected Sales = μ-Expected lost sales = 23.5-2.18 = 21.32 rounded to 21

a. Average Demand of apples = 23.5

Assuming that the fruit vendor purchases 23 pounds of apples every week,  the average sales in pounds is 21 lb

b. Cost of underestimating demand = The underage cost, Cu = $2/lb

Cost of overestimating demand = The overage cost, Co = $3.25/lb

c.The vendor should purchase 22 lbs of apple to maximize profit.