United Research Associates (URA) had received a contract to produce two units of a new cruise missile guidance control. The first unit took 5,000 hours to complete and cost $40,000 in materials and equipment usage. The second took 4,750 hours and cost $34,000 in materials and equipment usage. Labor cost is charged at $20 per hour. The company expects “learning” to occur relative to labor and also the pricing of parts from suppliers.

The prime contractor has now approached URA and asked to submit a bid for the cost of producing another 20 guidance controls. What will the last unit cost to build? use ***Exhibit 4A.5 - Improvement Curves: Table of Unit Values***

To solve this problem, we will use the **learning curve model** to estimate how labor efficiency improves with experience and how costs will decrease for the subsequent units produced. The two primary factors are:

1. **Learning Curve Percentage**: This tells us the rate of improvement in labor as production increases.
2. **Improvement Curve Table of Unit Values (Exhibit 4A.5)**: This table provides multipliers based on the learning curve rate and the unit number, allowing us to estimate the time and cost for subsequent units.

### Step-by-Step Process

#### Step 1: Determine the Learning Rate

Given that the first unit took 5,000 hours and the second took 4,750 hours, we can calculate the learning curve percentage using the formula:

\[
\text{Learning Rate} = \frac{\text{Time for the 2nd unit}}{\text{Time for the 1st unit}} = \frac{4750}{5000}
\]

Let’s calculate that first:

\[
\text{Learning Rate} = 0.95 \text{ or 95%}
\]

This means that for each doubling of production, the labor time reduces to 95% of the previous level. This is known as a **95% learning curve**.

#### Step 2: Use the Improvement Curves Table

From **Exhibit 4A.5**, we need to find the unit value for the **21st unit** based on a 95% learning curve. The prime contractor is asking for the cost of the last unit if 22 total units are produced (including the first two that were already built). The **21st unit** will give us the estimate for labor hours of the last unit.

By checking the learning curve table for a **95% learning rate**:
- The unit value for the 21st unit is approximately **0.861**.

#### Step 3: Calculate Labor Time for the Last Unit

Now, use the unit value multiplier from the table to calculate the labor hours required for the 21st unit. Since the first unit took 5,000 hours, we can estimate the time for the 21st unit as follows:

\[
\text{Labor Time for 21st Unit} = 5000 \times 0.861 = 4305 \text{ hours}
\]

#### Step 4: Calculate Labor Cost for the Last Unit

Labor is charged at $20 per hour, so the labor cost for the 21st unit will be:

\[
\text{Labor Cost} = 4305 \times 20 = 86,100 \text{ USD}
\]

#### Step 5: Estimate Material and Equipment Cost

The material and equipment costs seem to be decreasing as production progresses, but we don't have a detailed trend for that. We can estimate that materials and equipment costs will follow a similar improvement trend. The cost for the first unit was $40,000, and for the second unit, it was $34,000. Assuming a steady decrease:

\[
\text{Material and Equipment Cost for 21st Unit} \approx 34,000 \times 0.95 \approx 32,300 \text{ USD}
\]

#### Step 6: Total Cost for the 21st Unit

To find the total cost for the 21st unit, sum the labor cost and material/equipment cost:

\[
\text{Total Cost} = 86,100 + 32,300 = 118,400 \text{ USD}
\]

Thus, the estimated cost to build the last (21st) unit is approximately **$118,400**.

### Do you have any questions or want further details?

Here are some related questions for deeper understanding:

1. What if the learning rate was different, say 90%? How would that change the labor costs?
2. How would the cost change if URA had to build more than 22 units?
3. Could other factors, such as economies of scale, affect material and equipment costs further?
4. What is the impact on total project cost if URA’s labor rate increased to $25 per hour?
5. How does the improvement curve apply to industries other than manufacturing?

**Tip:** The learning curve model is especially useful in manufacturing industries, where repetitive tasks become more efficient with experience, reducing time and cost over time.

United Research Associates (URA) had received a contract to produce two units of a new cruise missile guidance control. The first unit took 5,000 hours to complete and cost $40,000 in materials and equipment usage. The second took 4,750 hours and cost $34,000 in materials and equipment usage. Labor cost is charged at $20 per hour. The company expects “learning” to occur relative to labor and also the pricing of parts from suppliers. The prime contractor has now approached URA and asked to submit a bid for the cost of producing another 20 guidance controls. What will the last unit cost to build? Use Exhibit 4A.5 - Improvement Curves: Table of Unit Values.

Discover how to use the learning curve model to estimate labor and material costs for producing additional units of a cruise missile guidance control system. This step-by-step solution shows how to calculate labor hours, material costs, and total cost for the 21st unit, using a 95% learning curve and the Improvement Curves Table.

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