A bagel company bakes a specialty bagel that it sells by the dozen every day.These specialty bagels can only be baked early in the morning before the store opens for business.The company estimates that the daily demand (in dozens)for its specialty bagel is distributed as follows: Specialty bagels are sold by the dozen only at a cost of $9.00 per dozen.The cost to make one bagel is $0.50.Leftover specialty bagels are sold by the dozen the next day for a 50% discount.The optimal number of specialty bagels that should be baked tomorrow (in dozens)is

Question

A bagel company bakes a specialty bagel that it sells by the dozen every day.These specialty bagels can only be baked early in the morning before the store opens for business.The company estimates that the daily demand (in dozens)for its specialty bagel is distributed as follows:  Specialty bagels are sold by the dozen only at a cost of $9.00 per dozen.The cost to make one bagel is $0.50.Leftover specialty bagels are sold by the dozen the next day for a 50% discount.The optimal number of specialty bagels that should be baked tomorrow (in dozens)is

Quizplus · Accepted Answer

The company wants to maximize its profit by baking the optimal number of specialty bagels. Let's calculate the expected profit for different quantities of bagels baked:

- If 1 dozen bagels are baked, the probability of selling all of them is only 0.2, and the expected profit is: 
    - Probability of selling 1 dozen = 0.2
    - Revenue from selling 1 dozen = $9.00
    - Cost of making 1 dozen = $6.00 (12 bagels * $0.50 per bagel)
    - Expected profit from selling 1 dozen = ($9.00 - $6.00) * 0.2 = $0.60
- If 2 dozen bagels are baked, the probability of selling all of them is still low at 0.3, but the probability of selling some of the leftovers the next day is now 0.2. The expected profit is: 
    - Probability of selling 2 dozen = 0.3
    - Probability of selling 1 dozen of leftovers = 0.2
    - Revenue from selling 2 dozen = $18.00
    - Revenue from selling 1 dozen of leftovers = $4.50 (50% discount)
    - Cost of making 2 dozen = $12.00
    - Expected profit from selling 2 dozen = ($18.00 + $4.50 - $12.00) * 0.3 = $2.55
- If 3 dozen bagels are baked, the probability of selling all of them is higher at 0.4, but the probability of selling leftovers the next day is still only 0.1. The expected profit is: 
    - Probability of selling 3 dozen = 0.4
    - Probability of selling 0.5 dozen of leftovers = 0.1
    - Revenue from selling 3 dozen = $27.00
    - Revenue from selling 0.5 dozen of leftovers = $2.25 (50% discount)
    - Cost of making 3 dozen = $18.00
    - Expected profit from selling 3 dozen = ($27.00 + $2.25 - $18.00) * 0.4 = $4.90
- If 4 dozen bagels are baked, the probability of selling all of them is highest at 0.5, and the probability of selling leftovers the next day is also at 0.1. The expected profit is: 
    - Probability of selling 4 dozen = 0.5
    - Probability of selling 0.5 dozen of leftovers = 0.1
    - Revenue from selling 4 dozen = $36.00
    - Revenue from selling 0.5 dozen of leftovers = $2.25 (50% discount)
    - Cost of making 4 dozen = $24.00
    - Expected profit from selling 4 dozen = ($36.00 + $2.25 - $24.00) * 0.5 = $7.13
- If 5 dozen bagels are baked, the probability of selling all of them is now decreasing at 0.4, and the probability of selling leftovers the next day is even lower at 0.05. The expected profit is: 
    - Probability of selling 5 dozen = 0.4
    - Probability of selling 0.5 dozen of leftovers = 0.05
    - Revenue from selling 5 dozen = $45.00
    - Revenue from selling 0.5 dozen of leftovers = $2.25 (50% discount)
    - Cost of making 5 dozen = $30.00
    - Expected profit from selling 5 dozen = ($45.00 + $2.25 - $30.00) * 0.4 = $6.90

Therefore, the optimal number of bagels to bake is 4 dozen, which will give an expected profit of $7.13. Any more bagels would result in diminishing returns, as the probability of selling them all or as leftovers decreases.

A Bagel Company Bakes a Specialty Bagel That It Sells