Question

A bakery would like you to recommend how many loaves of its famous marble rye bread to bake at the beginning of the day. Each loaf costs the bakery $2.00 and can be sold for $7.00. Leftover loaves at the end of each day are donated to charity. Research has shown that the probabilities for demands of 25, 50, and 75 loaves are 35%, 20%, and 45%, respectively. Make a recommendation for the bakery to bake 25, 50, or 75 loaves each morning.

Find the expected monetary value when baking 25 loaves.

Find the expected monetary value when baking 50 loaves.

Find the expected monetary value when baking 75 loaves.

Answer 1

Let Event A: Bakery sells 25 loaves, P(A) = 0.35
Let Event B: Bakery sells 50 loaves, P(B) = 0.20
Let Event C: Bakery sells 75 loaves, P(C) = 0.45
Cost of 1 loaf = $2
Selling price of 1 loaf = $7
Profit earned on 1 loaf = $5
Loss on each loaf that is donated to charity = $2

Expected monetary value when baking 25 loaves:
Profit earned on selling 25 loaves = 25 x (7-2) = 125
Expected monetary value when baking 25 loaves = $125

Expected monetary value when baking 50 loaves:
= P(A).(Profit earned on selling 25 loaves) + (1-P(A)).(Profit earned on selling 50 loaves)
*(1-P(A)) = P(B) + P(C) , even if the demand is 75 for the day, all 50 loves would still sell.
P(A) = 0.35
Profit earned on selling 25 loaves = Profit on selling 25 loaves - Loss on giving away leftover 25 loaves
Profit earned on selling 25 loaves = (25 x (7-2)) - (25 x 2) = 75
1-P(A) = 1 - 0.65
Profit earned on selling 50 loaves = 50 x (7-2) = 250
Expected monetary value when baking 50 loaves = (0.35 x 75) + (0.65 x 250) = $188.75

Expected monetary value when baking 75 loaves:
= P(A).(Profit earned on selling 25 loaves) + P(B).(Profit earned on selling 50 loaves) + P(C).(Profit earned on selling 75 loaves)
P(A) = 0.35
Profit earned on selling 25 loaves = Profit on selling 25 loaves - Loss on giving away leftover 50 loaves
Profit earned on selling 25 loaves = (25 x (7-2)) - (50 x 2) = 25
P(B) = 0.20
Profit earned on selling 50 loaves = Profit on selling 50 loaves - Loss on giving away leftover 25 loaves
Profit earned on selling 50 loaves = (50 x (7-2)) - (25 x 2) = 200
P(C) = 0.45
Profit earned on selling 75 loaves = 75 x (7-2) = 375
Expected monetary value when baking 75 loaves = (0.35 x 25) + (0.20 x 200) + (0.45 x 375) = $217.5

Since the expected monetary value is highest when baking 75 loaves, bakery should bake 75 lovaes each morning.