To calculate the bid for 10 additional units based on the given information, we need to consider the learning curve effect, particularly for labor, and the cost of materials. Here’s how you can approach this problem:

Step 1: Determine the learning curve percentage

• The first unit took 5,000 hours, and the second unit took 3,750 hours.
• The learning curve can be calculated as the ratio of the time it took to produce the second unit to the time it took to produce the first unit:

$\text{Learning Curve (LC)} = \frac{\text{Time for 2nd Unit}}{\text{Time for 1st Unit}} = \frac{3,750}{5,000} = 0.75 \text{ or } 75\%$

This indicates a 75% learning curve.

Step 2: Calculate the total labor hours for 10 additional units

Given the 75% learning curve, the labor hours decrease as more units are produced. For each doubling of production, labor hours should reduce to 75% of the previous total.

Using the cumulative average time formula, the total labor hours for $n$ units after producing the first two units can be approximated using:

$T_n = T_1 \times n^b$

Where:

• $T_1 =$ labor hours for the first unit
• $n =$ cumulative number of units
• $b =$ log(learning curve)/log(2)

Given:

• $T_1 = 5000$ hours
• Learning curve = 0.75 → $b = \frac{\log(0.75)}{\log(2)} \approx -0.4150$

Step 3: Calculate the labor hours for the 10 additional units

To determine the labor hours for the next 10 units, you would calculate the cumulative hours required for 12 units and subtract the hours for the first 2 units.

1. Cumulative hours for 12 units:

$T_{12} = 5000 \times 12^{-0.4150} \approx 2211.35 \text{ hours per unit (average)}$

$\text{Total hours for 12 units} = 12 \times 2211.35 = 26,536.2 \text{ hours}$

2. Cumulative hours for the first 2 units:

$\text{Total hours for 2 units} = 5000 + 3750 = 8750 \text{ hours}$

3. Total hours for the next 10 units:

$\text{Total hours for 10 additional units} = 26,536.2 - 8750 = 17,786.2 \text{ hours}$

Step 4: Calculate the labor cost

The labor cost for the 10 additional units:

$\text{Labor Cost} = 17,786.2 \times 25 = 444,655 \text{ dollars}$

Step 5: Estimate material and other costs

Given that the material cost for the second unit was $212,500 (a 15% reduction from the first unit), and if we assume a similar learning effect:

$\text{Estimate for 3rd unit} = 212,500 \times 0.85 = 180,625 \text{ dollars per unit}$

Assuming this trend continues:

$\text{Total Material Cost} = 180,625 \times 10 = 1,806,250 \text{ dollars}$

Step 6: Calculate the total bid

Adding the labor cost and material costs together gives:

$\text{Total Bid} = 444,655 + 1,806,250 = 2,250,905 \text{ dollars}$

Therefore, the bid for the 10 additional units would be approximately $2,250,905.

Additional Considerations:

• This calculation assumes the learning curve continues similarly for both labor and material costs.
• If the learning curve flattens or the material costs don’t decrease as expected, the actual bid might need adjustments.

Would you like further details or explanations on any step? Here are some related questions that might help you understand this better:

1. How is a learning curve percentage determined in production?
2. What factors influence the material cost decrease in production?
3. How does labor cost impact overall project bidding?
4. What is the importance of accurate cost estimation in competitive bidding?
5. How would the bid change if the learning curve was different?

Tip: Always consider potential changes in the learning curve when scaling production to ensure accurate bids and forecasts.