Uncertainties, inevitable in nature, can be classified as probability based and interval based uncertainties in terms of its representations. Corresponding optimization strategies have been proposed to deal with these two types of uncertainties. It is more likely that both types of uncertainty can occur in one single problem, and thus, it is trivial to treat all uncertainties the same. A novel formulation for reliability-based design optimization (RBDO) under mixed probability and interval uncertainties is proposed in this paper, in which the objective variation is concerned. Furthermore, it is proposed to efficiently solve the worst-case parameter resulted from the interval uncertainty by utilizing the Utopian solution presented in a single-looped robust optimization (RO) approach where the inner optimization can be solved by matrix operations. The remaining problem can be solved utilizing any existing RBDO method. This work applies the performance measure approach to search for the most probable failure point (MPFP) and sequential quadratic programing (SQP) to solve the entire problem. One engineering example is given to demonstrate the applicability of the proposed approach and to illustrate the necessity to consider the objective robustness under certain circumstances.

References

References
1.
Fine
,
T. L.
,
1973
,
Theories of Probability: An Examination of Foundations
,
Academic Press
,
New York, NY
.
2.
Haldar
,
A.
, and
Mahadevan
,
S.
,
2000
,
Probability, Reliability and Statistical Methods in Engineering Design
,
Wiley
,
New York
.
3.
Valdebenito
,
M. A.
, and
Schuëller
,
G. I.
,
2010
, “
A Survey on Approaches for Reliability-Based Optimization
,”
Struct. Multidiscip. Optim.
,
42
(
5
), pp.
645
663
.
4.
Zadeh
,
L. A.
,
1978
, “
Fuzzy Sets as a Basis for a Theory of Possibility
,”
Fuzzy Sets Syst.
,
1
(
1
), pp.
3
28
.
5.
Mourelatos
,
Z. P.
, and
Zhou
,
J.
,
2005
, “
Reliability Estimation and Design With Insufficient Data Based on Possibility Theory
,”
AIAA J.
,
43
(
8
), pp.
1696
1705
.
6.
Zadeh
,
L. A.
,
1965
, “
Fuzzy Sets
,”
Inf. Control
,
8
(
3
), pp.
338
353
.
7.
Möller
,
B.
, and
Beer
,
M.
,
2010
,
Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics
,
Springer
,
Berlin, Germany
.
8.
Oberkampf
,
W. L.
, and
Helton
,
J. C.
,
2005
, “
Evidence Theory for Engineering Applications
,”
Engineering Design Reliability
,
E.
Nikolaidis
,
D. M.
Ghiocel
, and
S.
Singhal
, eds.,
CRC Press
,
Boca Raton, FL
, pp.
10.1
10.30
.
9.
Moore
,
R. E.
,
1966
,
Interval Analysis
,
Prentice Hall
,
Englewood Cliffs, NJ
.
10.
Nakagiri
,
S.
, and
Suzuki
,
K.
,
1999
, “
Finite Element Interval Analysis of External Loads Identified by Displacement Input With Uncertainty
,”
Comput. Methods Appl. Mech. Eng.
,
168
(1–4), pp.
63
72
.
11.
Qiu
,
Z.
,
2005
, “
Convex Models and Interval Analysis Method to Predict the Effect of Uncertain-But-Bounded Parameters on the Buckling of Composite Structures
,”
Comput. Methods Appl. Mech. Eng.
,
194
(18–20), pp.
2175
2189
.
12.
McWilliam
,
S.
,
2001
, “
Anti-Optimisation of Uncertain Structures Using Interval Analysis
,”
Comput. Struct.
,
79
(
4
), pp.
421
430
.
13.
Impollonia
,
N.
, and
Muscolino
,
G.
,
2011
, “
Interval Analysis of Structures With Uncertain-But Bounded Axial Stiffness
,”
Comput. Methods Appl. Mech. Eng.
,
299
(21–22), pp.
1945
1962
.
14.
Du
,
X.
,
Sudjianto
,
A.
, and
Huang
,
B.
,
2005
, “
Reliability-Based Design With the Mixture of Random and Interval Variables
,”
ASME J. Mech. Des.
,
127
(
6
), pp.
1068
1076
.
15.
Bertsimas
,
D.
,
Brown
,
D. B.
, and
Caramanis
,
C.
,
2010
, “
Theory and Applications of Robust Optimization
,”
Math. Program., Ser. B
,
107
, pp.
464
501
.
16.
Lin
,
X.
,
Stacy
,
L. J.
, and
Floudas
,
C. A.
,
2003
, “
A New Robust Optimization Approach for Scheduling Under Uncertainty: I. Bounded Uncertainty
,”
Comput. Chem. Eng.
,
28
(
6–7
), pp.
1069
1085
.
17.
Li
,
M.
, and
Azarm
,
S.
,
2008
, “
Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation
,”
ASME J. Mech. Des.
,
130
(
8
), p.
081402
.
18.
Starks
,
S. A.
,
Kreinovich
,
V.
,
Longpre
,
L.
,
Ceberio
,
M.
,
Xiang
,
G.
,
Araiza
,
R.
,
Beck
,
J.
,
Kandathi
,
R.
,
Nayak
,
A.
, and
Torres
,
R.
,
2004
, “
Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations
,”
NSF Workshop on Reliable Engineering Computing
,
Savannah, GA
,
Sept. 15–17
, pp.
193
213
.
19.
Kreinovich
,
V.
,
Beck
,
J.
,
Ferregut
,
C.
,
Sanchez
,
A.
,
Keller
,
G. R.
,
Averill
,
M.
, and
Starks
,
S. A.
,
2014
, “
Monte Carlo-Type Techniques for Processing Interval Uncertainty, and Their Engineering Applications
,”
NSF Workshop on Reliable Engineering Computing
,
Savannah, GA
,
Sept. 15–17
, pp.
139
160
.
20.
Penmetsa
,
R. C.
, and
Grandhi
,
V.
,
2002
, “
Efficient Estimation of Reliability for Problems With Uncertain Intervals
,”
Comput. Struct.
,
80
(
12
), pp.
1103
1112
.
21.
Jiang
,
C.
,
Li
,
W. X.
,
Han
,
X.
,
Liu
,
L. X.
, and
Le
,
P. H.
,
2011
, “
Structural Reliability Analysis Based on Random Distributions With Interval Parameters
,”
Comput. Struct.
,
89
(
23
), pp.
2292
2302
.
22.
Zhou
,
J. H.
, and
Li
,
M.
,
2013
, “
Advanced Robust Optimization With Interval Uncertainty Using a Single-Looped Structure and Sequential Quadratic Programming
,”
ASME J. Mech. Des.
,
136
(
2
), p.
021008
.
23.
Gunawan
,
S.
,
2004
, “
Parameter Sensitivity Measures for Single Objective, Multi-Objective, and Feasibility Robust Design Optimization
,” Ph.D. dissertation, Department of Mechanical Engineering, University of Maryland, College Park, MD.
You do not currently have access to this content.