This paper focuses on multi-objective optimization under uncertainty for mechanical design, through a reliability-based formulation referring to the concept of probabilistic nondominance. To address this problem, the implementation of a co-evolutionary strategy is advocated, consisting of the concurrent evolution of two intertwined populations optimized according to coupled subproblems: the upper level optimizer handles the design variables, whereas the corresponding values of the probabilistic thresholds for the objectives (namely the reliable nondominated front) are retrieved at the lower stage. The proposed methodology is successfully applied to six analytical test cases, as well as to the sizing optimization of two truss structures, demonstrating an improved capacity to cover wider ranges of the reliable nondominated front in comparison with all-at-once strategies tackling all types of variables simultaneously.

References

References
1.
Coello Coello
,
C. A.
,
Van Veldhuizen
,
D. A.
, and
Lamont
,
G. B.
,
2002
,
Evolutionary Algorithms for Solving Multi-Objective Problems
,
Kluwer Academic/Plenum Publishers
,
New York
.
2.
Marler
,
R. T.
, and
Arora
,
J. S.
,
2004
, “
Survey of Multi-Objective Optimization Methods for Engineering
,”
Struct. Multidiscip. Optim.
,
26
, pp.
369
395
.10.1007/s00158-003-0368-6
3.
Breitkopf
,
P.
, and
Filomeno Coelho
,
R.
, eds.,
2010
,
Multidisciplinary Design Optimization in Computational Mechanics
,
ISTE/John Wiley & Sons
,
Chippenham, UK
.
4.
Beyer
,
H.-G.
, and
Sendhoff
,
B.
,
2007
, “
Robust Optimization—A Comprehensive Survey
,”
Comput. Methods Appl. Mech. Eng.
,
196
, pp.
3190
3218
.10.1016/j.cma.2007.03.003
5.
Der Kiureghian
,
A.
, and
Ditlevsen
,
O.
,
2009
, “
Aleatory or Epistemic? Does it Matter?
,”
Struct. Saf.
,
31
, pp.
105
112
.10.1016/j.strusafe.2008.06.020
6.
Achenie
,
L. E. K.
, and
Ostrovsky
,
G. M.
,
2005
, “
Multicriteria Optimization Under Parametric Uncertainty
,”
Applied Research in Uncertainty Modeling and Analysis
,
Springer
,
USA
.
7.
Barakat
,
S.
,
Bani-Hanib
,
K.
, and
Taha
,
M. Q.
,
2004
, “
Multi-Objective Reliability-Based Optimization of Prestressed Concrete Beams
,”
Struct. Saf.
,
26
, pp.
311
342
.10.1016/j.strusafe.2003.09.001
8.
Sinha
,
K.
,
2007
, “
Reliability-Based Multiobjective Optimization for Automotive Crashworthiness and Occupant Safety
,”
Struct. Multidiscip. Optim.
,
33
, pp.
255
268
.10.1007/s00158-006-0050-x
9.
Kumar
,
A.
,
Nair
,
P. B.
,
Keane
,
A. J.
, and
Shahpar
,
S.
,
2008
, “
Robust Design Using Bayesian Monte Carlo
,”
Int. J. Numer. Methods Eng.
,
73
, pp.
1497
1517
.10.1002/nme.2126
10.
Mourelatos
,
Z. P.
, and
Liang
,
J.
,
2006
, “
A Methodology for Trading-Off Performance and Robustness Under Uncertainty
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
856
863
.10.1115/1.2202883
11.
Deb
,
K.
, and
Gupta
,
H.
,
2006
, “
Introducing Robustness in Multi-Objective Optimization
,”
Evol. Comput.
,
14
(
4
), pp.
463
494
.10.1162/evco.2006.14.4.463
12.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evol. Comput.
,
6
(
2
), pp.
182
197
.10.1109/4235.996017
13.
Basseur
,
M.
, and
Zitzler
,
E.
,
2006
, “
Handling Uncertainty in Indicator-Based Multiobjective Optimization
,”
Int. J. Comput. Intell. Res.
,
2
(
3
), pp.
255
272
. Available at http://www.ripublication.com/ijcirv2/ijcirv2n3_3
14.
Köppen
,
M.
,
Vicente-Garcia
,
R.
, and
Nickolay
,
B.
,
2005
, “
Fuzzy-Pareto-Dominance and Its Application in Evolutionary Multi-Objective Optimization
,”
Evolutionary Multi-Criterion Optimization, 3rd International Conference, EMO 2005
,
Guanajuato, Mexico
,
March 9–11
, pp.
399
412
.
15.
Limbourg
,
P.
,
2005
, “
Multi-Objective Optimization of Problems With Epistemic Uncertainty
,”
Evolutionary Multi-Criterion Optimization, 3rd International Conference, EMO 2005
,
Guanajuato, Mexico
,
March 9–11
, pp.
413
427
.
16.
Filomeno Coelho
,
R.
,
Lebon
,
J.
, and
Bouillard
,
Ph.
,
2011
, “
Hierarchical Stochastic Metamodels Based on Moving Least Squares and Polynomial Chaos Expansion—Application to the Multiobjective Reliability-Based Optimization of 3D Truss Structures
,”
Struct. Multidiscip. Optim.
,
43
(
5
), pp.
707
729
.10.1007/s00158-010-0608-5
17.
Filomeno Coelho
,
R.
, and
Bouillard
,
Ph.
,
2011
, “
Multi-Objective Reliability-Based Optimization With Stochastic Metamodels
,”
Evol. Comput.
,
19
(
4
), pp.
525
560
.10.1162/EVCO_a_00034
18.
Zou
,
T.
, and
Mahadevan
,
S.
,
2006
, “
Versatile Formulation for Multiobjective Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
128
(
6
), pp.
1217
1226
.10.1115/1.2218884
19.
Haldar
,
A.
, and
Mahadevan
,
S.
,
2000
,
Reliability Assessment Using Stochastic Finite Element Analysis
,
John Wiley & Sons
,
Chichester, UK
.
20.
Deb
,
K.
,
Padmanabhan
,
D.
,
Gupta
,
S.
, and
Mall
,
A. K.
,
2007
, “
Reliability-Based Multi-Objective Optimization Using Evolutionary Algorithms
,”
Evolutionary Multi-Criterion Optimization
,
Springer
,
Berlin/Heidelberg
.
21.
Deb
,
K.
,
Gupta
,
S.
,
Jaum
,
D.
,
Branke
,
J.
,
Mall
,
A. K.
, and
Padmanabhan
,
D.
,
2009
, “
Reliability-Based Optimization Using Evolutionary Algorithms
,”
IEEE Trans. Evol. Comput.
,
13
(
5
), pp.
1054
1074
.10.1109/TEVC.2009.2014361
22.
Levi
,
F.
,
Gobbi
,
M.
, and
Mastinu
,
G.
,
2005
, “
An Application of Multi-Objective Stochastic Optimisation to Structural Design
,”
Struct. Multidiscip. Optim.
,
29
, pp.
272
284
.10.1007/s00158-004-0456-2
23.
Caballero
,
R.
,
Cerdá
,
E.
,
Muñoz
,
M. M.
,
Rey
,
L.
, and
Stancu-Minasian
,
I. M.
,
2001
, “
Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming
,”
J. Optim. Theory Appl.
,
110
(
1
), pp.
53
74
.10.1023/A:1017591412366
24.
Wang
,
Z.
,
Huang
,
H.-Z.
, and
Liu
,
Y.
,
2010
, “
A Unified Framework for Integrated Optimization Under Uncertainty
,”
ASME J. Mech. Des.
,
132
(
5
), p.
051008
.10.1115/1.4001526
25.
Rangavajhala
,
S.
, and
Mahadevan
,
S.
,
2011
, “
Joint Probability Formulation for Multiobjective Optimization Under Uncertainty
,”
ASME J. Mech. Des.
,
133
, p.
051007
.10.1115/1.4003540
26.
Tichý
,
M.
,
1993
,
Applied Methods of Structural Reliability
,
Kluwer Academic Publishers
, The Netherlands.
27.
Teich
,
J.
,
2001
, “
Pareto-Front Exploration With Uncertain Objectives
,”
Evolutionary Multi-Criterion Optimization
, Vol.
1993
(Lecture Notes in Computer Science),
Springer
,
New York
, pp.
314
328
.
28.
Eiben
,
A. E.
,
Hinterding
,
R.
, and
Michalewicz
,
Z.
,
1999
, “
Parameter Control in Evolutionary Algorithms
,”
IEEE Trans. Evol. Comput.
,
3
(
2
), pp.
124
141
.10.1109/4235.771166
29.
Paredis
,
J.
,
1995
, “
Coevolutionary Computation
,”
Artif. Life
,
2
(
4
), pp.
355
375
.10.1162/artl.1995.2.4.355
30.
Michalewicz
,
Z.
, and
Nazhiyath
,
G.
,
1995
, “
Genocop III: A Co-Evolutionary Algorithm for Numerical Optimization Problems With Nonlinear Constraints
,”
2nd IEEE International Conference on Evolutionary Computation
,
Perth, Australia
,
November 29–December 1
, pp.
647
651
.
31.
Schaffer
,
J. D.
,
1985
, “
Multiple Objective Optimization With Vector Evaluated Genetic Algorithms
,”
Proceedings of the 1st International Conference on Genetic Algorithms
,
Pittsburgh, PA
,
July 24–26
, pp.
93
100
.
32.
Parmee
,
I. C.
, and
Watson
,
A. H.
,
1999
, “
Preliminary Airframe Design Using Co-Evolutionary Multiobjective Genetic Algorithms
,”
Genetic and Evolutionary Computation Conference—GECCO 1999
,
Orlando, Florida
,
July 13–17
, pp.
1657
1665
.
33.
Greiner
,
D.
, and
Hajela
,
P.
,
2012
, “
Truss Topology Optimization for Mass and Reliability Considerations–Co-Evolutionary Multiobjective Formulations
,”
Struct. Multidiscip. Optim.
,
45
, pp.
589
613
.10.1007/s00158-011-0709-9
34.
Nash
,
J.
,
1951
, “
Non-Cooperative Games
,”
Ann. Math.
,
54
(
2
), pp.
286
295
.10.2307/1969529
35.
Barbosa
,
H. J. C.
, and
Barreto
,
A. M. S.
,
2001
, “
An Interactive Genetic Algorithm With Co-Evolution of Weights for Multiobjective Problems
,”
Genetic and Evolutionary Computation Conference (GECCO-2001)
,
San Francisco, California
,
July 7–11
, pp.
203
210
.
36.
Cabrera
,
J.
,
Ortiz
,
A.
,
Estebanez
,
B.
,
Nadal
,
F.
, and
Simon
,
A.
,
2010
, “
A Coevolutionary Algorithm for Tyre Model Parameters Identification
,”
Struct. Multidiscip. Optim.
,
41
(
5
), pp.
749
763
.10.1007/s00158-009-0446-5
37.
Deb
,
K.
,
Gupta
,
S.
,
Dutta
,
J.
, and
Ranjan
,
B.
,
2011
, “
Solving Dual Problems Using a Coevolutionary Optimization Algorithm
,” Kanpur Genetic Algorithms Laboratory, Indian Institute of Technology, Kanpur, KanGAL Technical Report No. 2011009.
38.
Nema
,
S.
,
Goulermas
,
J.
,
Sparrow
,
G.
, and
Helman
,
P.
,
2011
, “
A Hybrid Cooperative Search Algorithm for Constrained Optimization
,”
Struct. Multidiscip. Optim.
,
43
(
1
), pp.
107
119
.10.1007/s00158-010-0543-5
39.
Zitzler
,
E.
,
Thiele
,
L.
,
Laumanns
,
M.
,
Fonseca
,
C. M.
, and
Grunert da Fonseca
,
V.
,
2003
, “
Performance Assessment of Multiobjective Optimizers: An Analysis and Review
,”
IEEE Trans. Evol. Comput.
,
7
(
2
), pp.
117
132
.10.1109/TEVC.2003.810758
40.
Haftka
,
R. T.
, and
Gürdal
,
Z.
,
1992
,
Elements of Structural Optimization
,
Kluwer Academic Publishers
,
Dordrecht
.
41.
Greco
,
M.
,
Gesualdo
,
F. A. R.
,
Venturini
,
W. S.
, and
Coda
,
H. B.
,
2006
, “
Nonlinear Positional Formulation for Space Truss Analysis
,”
Finite Elem. Anal. Des.
,
42
, pp.
1079
1086
.10.1016/j.finel.2006.04.007
42.
Knowles
,
J. D.
, and
Corne
,
D. W.
,
2000
, “
Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy
,”
Evol. Comput.
,
8
(
2
), pp.
149
172
.10.1162/106365600568167
43.
Gunawan
,
S.
, and
Papalambros
,
P. Y.
,
2006
, “
A Bayesian Approach to Reliability-Based Optimization With Incomplete Information
,”
ASME J. Mech. Des.
,
128
, pp.
909
918
.10.1115/1.2204969
44.
Mahadevan
,
S.
, and
Rebba
,
R.
,
2006
, “
Inclusion of Model Errors in Reliability-Based Optimization
,”
ASME J. Mech. Des.
,
128
, pp.
936
944
.10.1115/1.2204973
You do not currently have access to this content.