1. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Determination of MPFP of each kernel analysis in the original design space Figure Legend:

2. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Estimation of MPFP of each kernel reliability analysis in the original design space Figure Legend:

3. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Generated random distributions: (a) a heart-shaped distribution, (b) a like-shaped distribution, (c) a star-shaped distribution, and (d) a corona-shaped distribution. (Small dots represent the sampling points; the circle at the center represents the mean value of randomly distributed points; and dashed ellipses represent the ranges of 1σ, 2σ, and 3σ if treating the random points as normal distributions). Figure Legend:

4. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Iteration path of solving example 2 with corona-shaped distribution and Pf=30% using EoGRA with N=50,000 Figure Legend:

5. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Iteration path of solving example 3 with like-shaped distribution and Pf=30% using EoGRA with N=50,000 Figure Legend:

6. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 1 with Pf=1% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star- shaped, and (f) corona-shaped distributions Figure Legend:

7. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 1 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star- shaped, and (f) corona-shaped distributions Figure Legend:

8. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 2 with Pf=1% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star- shaped, and (f) corona-shaped distributions Figure Legend:

9. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 2 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star- shaped, and (f) corona-shaped distributions Figure Legend:

10. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 3 with Pf=1% and (a) normal and (b) uniform distributions Figure Legend:

11. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Using EoGRA with N=50,000 to solve example 3 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star- shaped, and (f) corona-shaped distributions Figure Legend:

12. Date of download: 6/21/2016 Copyright © ASME. All rights reserved. From: An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses J. Mech. Des. 2016;138(7):071403-071403-14. doi:10.1115/1.4033548 Results of photo enhancements: (a) original photo, (b) pixel distribution in CIELAB, (c) enhanced photo by treating image pixels as normally distributed (Pnorm = 0.2811%), and (d) enhanced photo by EoGRA (PEoGRA = 4.9485%) Figure Legend: