Math Problem Statement

Riya Khatri, contract negotiator for Nebula Airframe Company, is currently involved in bidding on a follow-up government contract. In gathering cost data from the first three units, which Nebula produced under a research and development contract, she found that the first unit took 3,000 labor hours, the second took 2,700 labor hours, and the third took 2,540 hours. Use Exhibit 6.5.

In a contract for three more units, how many labor hours should Khatri plan for?

Note: Round your answer to the nearest whole number. this is exhibit 6.5 "Cumulative Improvement Factor Unit 60% 65% 70% 75% 80% 85% 90% 95% 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2 1.600 1.650 1.700 1.750 1.800 1.850 1.900 1.950 3 2.045 2.155 2.268 2.384 2.502 2.623 2.746 2.872 4 2.405 2.578 2.758 2.946 3.142 3.345 3.556 3.774 5 2.710 2.946 3.195 3.459 3.738 4.031 4.339 4.662 6 2.977 3.274 3.593 3.934 4.299 4.688 5.101 5.538 7 3.216 3.572 3.960 4.380 4.834 5.322 5.845 6.404 8 3.432 3.847 4.303 4.802 5.346 5.936 6.574 7.261 9 3.630 4.102 4.626 5.204 5.839 6.533 7.290 8.111 10 3.813 4.341 4.931 5.589 6.315 7.116 7.994 8.955 12 4.144 4.780 5.501 6.315 7.227 8.244 9.374 10.62 14 4.438 5.177 6.026 6.994 8.092 9.331 10.72 12.27 16 4.704 5.541 6.514 7.635 8.920 10.38 12.04 13.91 18 4.946 5.879 6.972 8.245 9.716 11.41 13.33 15.52 20 5.171 6.195 7.407 8.828 10.48 12.40 14.61 17.13 22 5.379 6.492 7.819 9.388 11.23 13.38 15.86 18.72 24 5.574 6.773 8.213 9.928 11.95 14.33 17.10 20.31 25 5.668 6.909 8.404 10.19 12.31 14.80 17.71 21.10 30 6.097 7.540 9.305 11.45 14.02 17.09 20.73 25.00 35 6.478 8.109 10.13 12.72 15.64 19.29 23.67 28.86 40 6.821 8.631 10.90 13.72 17.19 21.43 26.54 32.68 45 7.134 9.114 11.62 14.77 18.68 23.50 29.37 36.47 50 7.422 9.565 12.31 15.78 20.12 25.51 32.14 40.22 60 7.941 10.39 13.57 17.67 22.87 29.41 37.57 47.65 70 8.401 11.13 14.74 19.43 25.47 33.17 42.87 54.99 80 8.814 11.82 15.82 21.09 27.96 36.80 48.05 62.25 90 9.191 12.45 16.83 22.67 30.35 40.32 53.14 69.45 100 9.539 13.03 17.79 24.18 32.65 43.75 58.14 76.59 120 10.16 14.11 19.57 27.02 37.05 50.39 67.93 90.71 140 10.72 15.08 21.20 29.67 41.22 56.78 77.46 104.7 160 11.21 15.97 22.72 32.17 45.20 62.95 86.80 118.5 180 11.67 16.79 24.14 34.54 49.03 68.95 95.96 132.1 200 12.09 17.55 25.48 36.80 52.72 74.79 105.0 145.7 250 13.01 19.28 28.56 42.05 61.47 88.83 126.9 179.2 300 13.81 20.81 31.34 46.94 69.66 102.2 148.2 212.2 350 14.51 22.18 33.89 51.48 77.43 115.1 169.0 244.8 400 15.14 23.44 36.26 55.75 84.85 127.6 189.3 277.0 450 15.72 24.60 38.48 59.80 91.97 139.7 209.2 309.0 500 16.26 25.68 40.58 63.68 98.85 151.5 228.8 340.6 600 17.21 27.67 44.47 70.97 112.0 174.2 267.1 403.3 700 18.06 29.45 48.04 77.77 124.4 196.1 304.5 465.3 800 18.82 31.09 51.36 84.18 136.3 217.3 341.0 526.5 900 19.51 32.60 54.46 90.26 147.7 237.9 376.9 587.2 1,000 20.15 31.01 57.40 96.07 158.7 257.9 412.2 647.4 1,200 21.30 36.59 62.85 107.0 179.7 296.6 481.2 766.6 1,400 22.32 38.92 67.85 117.2 199.6 333.9 548.4 884.2 1,600 23.23 41.04 72.49 126.8 218.6 369.9 614.2 1001 1,800 24.06 43.00 76.85 135.9 236.8 404.9 678.8 1116 2,000 24.83 44.84 80.96 144.7 254.4 438.9 742.3 1230 2,500 26.53 48.97 90.39 165.0 296.1 520.8 897.0 1513 3,000 27.99 52.62 98.90 183.7 335.2 598.9 1047 1791"

Solution

To solve this problem, we'll use the concept of learning curves, which are often used to estimate the time needed for future units based on past performance. The cumulative improvement factor, found in Exhibit 6.5, helps us determine the total labor hours for a given number of units.

Given:

  • The first unit took 3,000 labor hours.
  • The second unit took 2,700 labor hours.
  • The third unit took 2,540 labor hours.

Step 1: Calculate the Learning Rate

The learning rate (rr) is calculated using the formula:

r=(Time for the second unitTime for the first unit)r = \left(\frac{\text{Time for the second unit}}{\text{Time for the first unit}}\right)

Using the given dataTo solve this problem, we'll use the concept of learning curves, which are often used to estimate the time needed for future units based on past performance. The cumulative improvement factor, found in Exhibit 6.5, helps us determine the total labor hours for a given number of units.

Given:

  • The first unit took 3,000 labor hours.
  • The second unit took 2,700 labor hours.
  • The third unit took 2,540 labor hours.

Step 1: Calculate the Learning Rate

The learning rate (rr) is calculated using the formula:

r=(Time for the second unitTime for the first unit)r = \left(\frac{\text{Time for the second unit}}{\text{Time for the first unit}}\right)

Using the given

r=27003000=0.9r = \frac{2700}{3000} = 0.9

This implies a 90% learning rate.

Step 2: Calculate the Cumulative Improvement Factor

For a 90% learning rate, use the cumulative improvement factors from Exhibit 6.5.

Given the cumulative improvement factors for a 90% learning rate:

  • For 1 unit: 1.000
  • For 2 units: 1.900
  • For 3 units: 2.746
  • For 6 units (3 + 3 more units): 5.101

Step 3: Estimate the Total Labor Hours for 6 Units

The total labor hours for the first three units are:

Total hours for first 3 units=3000+2700+2540=8240 hours\text{Total hours for first 3 units} = 3000 + 2700 + 2540 = 8240 \text{ hours}

The cumulative improvement factor for the first three units is 2.746.

The total labor hours for 6 units:

Total hours for 6 units=Total hours for first 3 units×Cumulative improvement factor for 6 unitsCumulative improvement factor for 3 units\text{Total hours for 6 units} = \text{Total hours for first 3 units} \times \frac{\text{Cumulative improvement factor for 6 units}}{\text{Cumulative improvement factor for 3 units}}

Total hours for 6 units=8240×5.1012.74615297 hours\text{Total hours for 6 units} = 8240 \times \frac{5.101}{2.746} \approx 15297 \text{ hours}

Step 4: Calculate the Labor Hours for the Next 3 Units

To find the labor hours for the next 3 units:

Labor hours for the next 3 units=Total hours for 6 unitsTotal hours for first 3 units\text{Labor hours for the next 3 units} = \text{Total hours for 6 units} - \text{Total hours for first 3 units}

Labor hours for the next 3 units=152978240=7057 hours\text{Labor hours for the next 3 units} = 15297 - 8240 = 7057 \text{ hours}

Final Answer

The labor hours Khatri should plan for the next three units are 7,057 hours (rounded to the nearest whole number).


Do you want more details or have any questions? Here are five related questions:

  1. How does the learning rate impact the time estimation for additional units?
  2. What are some practical implications of using a learning curve in contract negotiations?
  3. How can variations in the learning rate affect project cost projections?
  4. What are some assumptions behind the use of cumulative improvement factors?
  5. How can labor hours savings from learning curves be optimized in manufacturing?

Tip: Understanding the learning curve concept can help in making more accurate labor cost estimations and improve pricing strategies for future contracts.

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Math Problem Analysis

Mathematical Concepts

Learning Curves
Cumulative Improvement Factors

Formulas

Learning rate formula: r = (Time for the second unit) / (Time for the first unit)

Theorems

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Suitable Grade Level

Graduate Level