Question

Lakeside Bakery bakes fresh pies every morning. The daily demand for its apple pies is a random variable with (discrete) distribution, based on past experience, given by

Demand 5 10 15 20 25 30
Probability 10% 20% 25% 25% 15% 5%

Each apple pie costs the bakery $6.75 to make and is sold for $17.99. Unsold apple pies at the end of the day are purchased by a nearby soup kitchen for 99 cents each. Assume no goodwill cost.

a. If the company decided to bake 15 apple pies each day, what would be their expected profit? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Expected profit $
b. Based on the demand distribution above, how many apple pies should the company bake each day to maximize their expected profit? (Do not interpolate your answer. Choose only from the demand values given in the above discrete distribution.)


Number of apple pies

Answer 1

a. Expected profit: $190.93
b. Number of apple pies to bake: 15

a. To calculate the expected profit, we need to consider the profit from selling the pies and the cost of making them.

First, let's calculate the expected number of pies sold each day. We multiply each demand value by its corresponding probability and sum them up:

(5 * 0.10) + (10 * 0.20) + (15 * 0.25) + (20 * 0.25) + (25 * 0.15) + (30 * 0.05) = 16.25

So, on average, 16.25 apple pies will be sold each day.

Next, let's calculate the profit from selling the pies:

Profit from selling = (Number of pies sold) * (Selling price per pie)

Profit from selling = 16.25 * $17.99 = $292.18

Now, let's calculate the cost of making the pies:

Cost of making = (Number of pies baked) * (Cost per pie)

Cost of making = 15 * $6.75 = $101.25

Finally, we can calculate the expected profit:

Expected profit = Profit from selling - Cost of making

Expected profit = $292.18 - $101.25 = $190.93

Therefore, if the company decides to bake 15 apple pies each day, their expected profit would be $190.93.

b. To maximize the expected profit, the company should bake the number of pies that corresponds to the highest probability in the demand distribution. Looking at the given distribution, we see that the highest probability (25%) is associated with a demand of 15 pies.

Therefore, the company should bake 15 apple pies each day to maximize their expected profit.

Answer 2

a. Expected Profit with 15 Apple Pies

To calculate the expected profit with 15 apple pies, we need to consider the profit for each possible demand outcome and weight it by the probability of that outcome.

Demand Profit Probability Expected Profit

5 $22.06 0.10 $2.21

10 $43.95 0.20 $8.79

15 $0 0.25 $0

20 $-22.24 0.25 $-5.56

25 $-44.49 0.15 $-6.67

30 $-66.74 0.05 $-3.34

Total $0.39

Therefore, the expected profit with 15 apple pies is $0.39.

b. Maximizing Expected Profit

To maximize expected profit, we need to find the number of apple pies that yields the highest expected profit. We can do this by calculating the expected profit for each possible demand and choosing the number of apple pies that corresponds to the highest expected profit.

Number of Apple Pies Expected Profit

5 $2.21

10 $8.79

15 $0.39

20 $-5.56

25 $-6.67

30 $-3.34

Based on the calculations, the number of apple pies that maximizes expected profit is 10. This means that the bakery should bake 10 apple pies each day to maximize their expected profit.