https://file.aiquickdraw.com/mathbot/file/1726378216119howdgd3j.jpg

Let's break down the problem shown in the image:

### Part (a):
The company bases its monthly orders on the **expected value** of the demand. 

- Unit demands and their respective probabilities are:
  - 300 units with a probability of 0.16
  - 400 units with a probability of 0.26
  - 500 units with a probability of 0.44
  - 600 units with a probability of 0.14

To find the **expected demand** (i.e., the average number of units the company should expect to sell), we compute the weighted average:

\[
\text{Expected Demand} = (300 \times 0.16) + (400 \times 0.26) + (500 \times 0.44) + (600 \times 0.14)
\]

Calculating this gives:

\[
\text{Expected Demand} = 48 + 104 + 220 + 84 = 456 \text{ units}
\]

So, the monthly order quantity should be **456 units**, which matches the result in the image.

---

### Part (b):
Here, the company places an order for 456 units based on the expected demand, but the actual demand turns out to be 405 units.

Each unit generates $79 in revenue, and each unit ordered costs $44.

Let's calculate the company's **gain or loss**:
- **Revenue**: If 405 units are demanded and sold, the total revenue is:
  \[
  \text{Revenue} = 405 \times 79 = 31,995 \text{ dollars}
  \]
  
- **Cost**: Since 456 units were ordered, the cost for those units is:
  \[
  \text{Cost} = 456 \times 44 = 20,064 \text{ dollars}
  \]

- **Net Gain**: The net gain is the revenue minus the cost:
  \[
  \text{Net Gain} = 31,995 - 20,064 = 11,931 \text{ dollars}
  \]

This differs from the image's displayed result of $16,179, so there might be an extra assumption (possibly involving leftover stock or penalties) not visible from the current view.

Let me know if you need further clarification or more details.

---

### Further Questions:
1. How is the cost of unsold inventory calculated, if applicable?
2. Could there be additional factors influencing the profit, such as storage costs or lost sales?
3. How does the expected value change if the probabilities shift?
4. What would happen if the actual demand exceeded the expected order quantity?
5. Could a probabilistic approach (like safety stock) better handle fluctuating demand?

**Tip:** When dealing with expected values in business scenarios, always factor in potential variations in demand and consider the costs of over- and under-stocking to maximize profits.

The demand for a product of Carolina Industries varies greatly from month to month. The probability distribution in the table shows the company's monthly demand. What should Carolina's monthly order quantity be based on the expected value of the demand? How much will the company gain or lose if it places an order based on the expected value but the actual demand is different?

Learn how to calculate the expected value of monthly demand for a product using probability distribution. This problem also includes revenue and cost calculations to determine the company's gain or loss when the actual demand differs from the expected value.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation