Probabilistic Design for Optimization and Robustness for Engineers

Answers to selected exercises

Chapter 2

2. 25.05

3. Probability (x₁ < X < x₂) = 0.095355

   Probability (X < x₁) = 0.566184

   Probability (X < x₂) = 0.661539

4. 23.42

5. 0.4512

6. μ = 42.50

   σ = 40.660

7. μ = 3.124

   σ = 1.809

8. β = 0.571

   θ = 52.527

9. 49.4

10. μ = 3.125

    σ = 3.407

11. 75.356

12. 0.2098

13. 315.28

14. 1.0

15. 0.0001515

Chapter 3

3. 0.598
4. 0.683
5. 4.16
6. 0.33
7. 
8. 

9. μ = 100

   σ = 4.47

Chapter 4

1. 2.43
2. 1.31
3. 0.27
4. 26%

5. μ = 100

   σ = 4.5

6. 1.2%
7.  

Chapter 5

1. 1. Nominal life: 22.05, standard deviation: 3.27
   2. Assuming normal distribution: below = 9.40%, above = 12.88%
2. 1. Lower power than spec: 29.8%

   2. Theoretical value: 2 400 000

      Actual value: 2 472 000

   3. Log mean parameter = 14.67

      Log STD parameter = 0.3096

   4. Below spec: 29.82%

      Above spec: 28.87%

3. 1. SD: 1003
   2. SD: 997
   3. Non-symmetrical distributions for input parameters have a small effect on the ability to predict the output standard deviation, usually less than 10% error on the predicted output distribution.
4. 1. 108
   2. 110.4
   3. SD is underestimated by 2.4.

Chapter 6

1. 1. 0.71
   2. 0.67
2. 1. 0.47
   2. The company should improve Y3 since it has the highest weightage for hitting the target and an increase in desirability would be the largest for this requirement. The new product desirability when Y3 hits the target is 0.89.
3. 1. 0.64

   2.  

      Requirements to be improved

          a    Strength of material ...