Math Problem Statement
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
Solution
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:
- Learning Curve Percentage: This tells us the rate of improvement in labor as production increases.
- 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:
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:
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:
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:
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:
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:
- What if the learning rate was different, say 90%? How would that change the labor costs?
- How would the cost change if URA had to build more than 22 units?
- Could other factors, such as economies of scale, affect material and equipment costs further?
- What is the impact on total project cost if URA’s labor rate increased to $25 per hour?
- 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.
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Math Problem Analysis
Mathematical Concepts
Learning Curves
Cost Estimation
Multiplication
Percentages
Formulas
Learning Rate = Time for 2nd Unit / Time for 1st Unit
Labor Time for nth Unit = Time for 1st Unit * Improvement Factor
Total Cost = Labor Cost + Material and Equipment Cost
Theorems
Learning Curve Model
Suitable Grade Level
Grades 11-12 (High School) or University Level
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