In order to be practical, solutions of engineering design optimization problems must be robust, i.e., competent and reliable in the face of uncertainties. While such uncertainties can emerge from a number of sources (imprecise variable values, errors in performance estimates, varying environmental conditions, etc.), this study focuses on problems where uncertainties emanate from the design variables. While approaches to identify robust optimal solutions of single and multi-objective optimization problems have been proposed in the past, we introduce a practical approach that is capable of solving robust optimization problems involving many objectives building on authors’ previous work. Two formulations of robustness have been considered in this paper, (a) feasibility robustness (FR), i.e., robustness against design failure and (b) feasibility and performance robustness (FPR), i.e., robustness against design failure and variation in performance. In order to solve such formulations, a decomposition based evolutionary algorithm (DBEA) relying on a generational model is proposed in this study. The algorithm is capable of identifying a set of uniformly distributed nondominated solutions with different sigma levels (feasibility and performance) simultaneously in a single run. Computational benefits offered by using polynomial chaos (PC) in conjunction with Latin hypercube sampling (LHS) for estimating expected mean and variance of the objective/constraint functions has also been studied in this paper. Last, the idea of redesign for robustness has been explored, wherein selective component(s) of an existing design are altered to improve its robustness. The performance of the strategies have been illustrated using two practical design optimization problems, namely, vehicle crash-worthiness optimization problem (VCOP) and a general aviation aircraft (GAA) product family design problem.

References

References
1.
Beyer
,
H.-G.
, and
Sendhoff
,
B.
,
2007
, “
Robust Optimization—A Comprehensive Survey
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
33–34
), pp.
3190
3218
.10.1016/j.cma.2007.03.003
2.
Rangavajhala
,
S.
, and
Mahadevan
,
S.
,
2013
, “
Design Optimization for Robustness in Multiple Performance Functions
,”
Struct. Multidiscip. Optim.
,
47
(
4
), pp.
523
538
.10.1007/s00158-012-0860-y
3.
Chen
,
W.
,
Allen
,
J. K.
,
Tsui
,
K.-L.
, and
Mistree
,
F.
,
1996
, “
A Procedure for Robust Design: Minimizing Variations Caused by Noise Factors and Control Factors
,”
ASME J. Mech. Des.
,
118
(4), pp.
478
485
.10.1115/1.2826915
4.
Sun
,
G.
,
Li
,
G.
,
Zhou
,
S.
,
Li
,
H.
,
Hou
,
S.
, and
Li
,
Q.
,
2011
, “
Crashworthiness Design of Vehicle by Using Multiobjective Robust Optimization
,”
Struct. Multidiscip. Optim.
,
44
(
1
), pp.
99
110
.10.1007/s00158-010-0601-z
5.
Sundaresan
,
S.
,
Ishii
,
K.
, and
Houser
,
D. R.
,
1995
, “
A Robust Optimization Procedure With Variations on Design Variables and Constraints
,”
Eng. Optim.
,
24
(
2
), pp.
101
117
.10.1080/03052159508941185
6.
Wang
,
L.
,
Grandhi
,
R. V.
, and
Hopkins
,
D. A.
,
1995
, “
Structural Reliability Optimization Using an Efficient Safety Index Calculation Procedure
,”
Int. J. Numer. Methods Eng.
,
38
(10), pp.
1721
1738
.10.1002/nme.1620381008
7.
Deb
,
K.
, and
Gupta
,
H.
,
2005
, “
Searching for Robust Pareto-Optimal Solutions in Multi-Objective Optimization
,”
Lect. Notes Comput. Sci.
,
3410
, pp.
150
164
.10.1007/b106458
8.
Jin
,
Y.
, and
Sendhoff
,
B.
,
2003
,
Trade-Off Between Performance and Robustness: An Evolutionary Multiobjective Approach
(Evolutionary Multi-Criterion Optimization Lecture Notes in Computer Science),
C. M.
Fonseca
,
P. J.
Fleming
,
E.
Zitzler
,
L.
Thiele
, and
K.
Deb
, eds., Vol.
2632
,
Springer
,
Berlin, Germany
, pp.
237
251
10.1007/978-3-540-31880-4_11.
9.
Bennett
,
J. A.
, and
Lust
,
R. V.
,
1990
, “
Conservative Methods for Structural Optimization
,”
AIAA J.
,
28
(
8
), pp.
1491
1496
.10.2514/3.25243
10.
Taguchi
,
G.
,
1986
,
Introduction to Quality Engineering: Designing Quality into Products and Processes
,
Asian Productivity Organization
, University of Michigan.
11.
Wang
,
Z.
,
Huang
,
H.-Z.
, and
Liu
,
Y.
,
2010
, “
A Unified Framework for Integrated Optimization Under Uncertainty
,”
ASME J. Mech. Des.
,
132
, pp.
1
8
10.1115/1.4001526.
12.
Deb
,
K.
,
Gupta
,
S.
,
Daum
,
D.
,
Branke
,
J.
,
Mall
,
A. K.
, and
Padmanabhan
,
D.
,
2009
, “
Reliability-Based Optimization Using Evolutionary Algorithms
,”
IEEE Trans. Evol. Comput.
,
13
(
5
), pp.
154
174
10.1109/TEVC.2009.2014361.
13.
Jung
,
D. H.
, and
Lee
,
B. C.
,
2002
, “
Development of a Simple and Efficient Method for Robust Optimization
,”
Int. J. Numer. Methods Eng.
,
53
(9), pp.
2201
2215
.10.1002/nme.383
14.
Frangopol
,
D.
, and
Moses
,
F.
,
1994
,
Reliability-Based Structural Optimization
,
Chapman and Hall
,
London, UK
, pp.
492
570
.
15.
Al-Harthy
,
A. S.
, and
Frangopol
,
D. M.
,
1994
, “
Reliability-Based Design of Prestressed Concrete Beams
,”
J. Struct. Eng.
,
120
(11), pp.
3156
3177
.10.1061/(ASCE)0733-9445(1994)120:11(3156)
16.
Kleiber
,
M.
,
Siemaszko
,
A.
, and
Stocki
,
R.
,
1999
, “
Interactive Stability-Oriented Reliability-Based Design Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
168
(
1–4
), pp.
243
253
.10.1016/S0045-7825(98)00143-1
17.
Du
,
X.
, and
Chen
,
W.
,
2004
, “
Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design
,”
ASME J. Mech. Des.
,
126
(
2
), pp.
225
233
.10.1115/1.1649968
18.
Rangavajhala
,
S.
,
Mullur
,
A.
, and
Messac
,
A.
,
2007
, “
The Challenge of Equality Constraints in Robust Design Optimization: Examination and New Approach
,”
Struct. Multidiscip. Optim.
,
34
(
5
), pp.
381
401
.10.1007/s00158-007-0104-8
19.
Koch
,
P.
,
Yang
,
R.-J.
, and
Gu
,
L.
,
2004
, “
Design for Six Sigma Through Robust Optimization
,”
Struct. Multidiscip. Optim.
,
26
(
3–4
), pp.
235
248
.10.1007/s00158-003-0337-0
20.
Lei
,
G.
,
Zhu
,
J. G.
,
Guo
,
Y. G.
,
Hu
,
J. F.
,
Xu
,
W.
, and
Shao
,
K. R.
,
2013
, “
Robust Design Optimization of PM-SMC Motors for Six Sigma Quality Manufacturing
,”
IEEE Trans. Magn.
,
49
(
7
), pp.
3953
3956
.10.1109/TMAG.2013.2243123
21.
Lee
,
K.-H.
,
Eom
,
I.-S.
,
Park
,
G.-J.
, and
Lee
,
W.-I.
,
1996
, “
Robust Design for Unconstrained Optimization Problems Using the Taguchi Method
,”
AIAA J.
,
34
(5), pp.
1059
1063
.10.2514/3.13187
22.
Lee
,
K.-H.
, and
Park
,
G.-J.
,
2001
, “
Robust Optimization Considering Tolerances of Design Variables
,”
Comput. Struct.
,
79
(
1
), pp.
77
86
.10.1016/S0045-7949(00)00117-6
23.
Zaman
,
K.
, and
Mahadevan
,
S.
,
2013
, “
Robustness-Based Design Optimization of Multidisciplinary System Under Epistemic Uncertainty
,”
AIAA J.
,
51
(
5
), pp.
1021
1031
.10.2514/1.J051372
24.
Lee
,
S. H.
,
Chen
,
W.
, and
Kwak
,
B. M.
,
2009
, “
Robust Design With Arbitrary Distributions Using Gauss-Type Quadrature Formula
,”
Struct. Multidiscip. Optim.
,
39
(
3
), pp.
227
243
.10.1007/s00158-008-0328-2
25.
Chen
,
W.
,
Wiecek
,
M. M.
, and
Zhang
,
J.
,
2007
, “
Quality Utility—A Compromise Programming Approach to Robust Design
,”
ASME J. Mech. Des.
,
121
(
2
), pp.
179
187
10.1115/1.2829440.
26.
Du
,
X.
,
Sudjianto
,
A.
, and
Chen
,
W.
,
2004
, “
An Integrated Framework for Optimization Under Uncertainty Using Inverse Reliability Strategy
,”
ASME J. Mech. Des.
,
126
(
4
), pp.
562
570
.10.1115/1.1759358
27.
Chen
,
W.
,
Sahai
,
A.
,
Messac
,
A.
, and
Sundararaj
,
G. J.
,
2000
, “
Exploration of the Effectiveness of Physical Programming in Robust Design
,”
ASME J. Mech. Des.
,
122
(
2
), pp.
155
163
.10.1115/1.533565
28.
McAllister
,
C. D.
,
Simpson
,
T. W.
,
Lewis
,
K.
, and
Messac
,
A.
,
2004
, “
Robust Multiobjective Optimization Through Collaborative Optimization and Linear Physical Programming
,”
AIAA
Paper No. 2004-4549. 10.2514/6.2004-4549
29.
Yang
,
C.
, and
Du
,
X.
,
2014
, “
Robust Design for Multivariate Quality Characteristics Using Extreme Value Distribution
,”
ASME J. Mech. Des.
,
136
(
10
), p.
101405
.10.1115/1.4028016
30.
Li
,
M.
,
Azarm
,
S.
, and
Aute
,
V.
,
2005
, “
A Multi-Objective Genetic Algorithm for Robust Design Optimization
,” Proceedings of the
GECCO’05
Conference on Genetic and Evolutionary Computation
,
ACM
, Washington, DC, pp.
771
778
10.1145/1068009.1068140.
31.
Das
,
I.
,
2000
, “
Robustness Optimization for Constrained Nonlinear Programming Problems?
,”
Eng. Optim.
,
32
(
5
), pp.
585
618
.10.1080/03052150008941314
32.
Paenke
,
I.
,
Branke
,
J.
, and
Jin
,
Y.
,
2006
, “
Efficient Search for Robust Solutions by Means of Evolutionary Algorithm and Fitness Approximation
,”
IEEE Trans. Evol. Comput.
,
10
(
4
), pp.
405
420
.10.1109/TEVC.2005.859465
33.
Jin
,
Y.
, and
Branke
,
J.
,
2005
, “
Evolutionary Optimization in Uncertain Environments-A Survey
,”
IEEE Trans. Evol. Comput.
,
9
(
3
), pp.
303
317
.10.1109/TEVC.2005.846356
34.
Farmani
,
R.
,
Walters
,
G. A.
, and
Savic
,
D. A.
,
2005
, “
Trade-Off Between Total Cost and Reliability for Anytown Water Distribution Network
,”
J. Water Resour. Plann. Manage.
,
131
(
3
), pp.
161
171
.10.1061/(ASCE)0733-9496(2005)131:3(161)
35.
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
36.
Gupta
,
H.
, and
Deb
,
K.
,
2005
, “
Handling Constraints in Robust Multiobjective Optimization
,”
IEEE
Congress on Evolutionary Computation
, Edinburgh, Scotland, Sept. 5, pp.
450
457
10.1109/CEC.2005.1554663.
37.
Chen
,
W.
,
Allen
,
J. K.
,
Mavris
,
D. N.
, and
Mistree
,
F.
,
1996
, “
A Concept Exploration Method for Determining Robust Top-Level Specifications
,”
Eng. Optim.
,
26
(
2
), pp.
137
158
.10.1080/03052159608941114
38.
Mattson
,
C. A.
, and
Messac
,
A.
,
2005
, “
Pareto Frontier Based Concept Selection Under Uncertainty, With Visualization
,”
Optim. Eng.
,
6
(
1
), pp.
85
115
.10.1023/B:OPTE.0000048538.35456.45
39.
Besharati
,
B.
,
Luo
,
L.
,
Azarm
,
S.
, and
Kannan
,
P.
,
2006
, “
Multi-Objective Single Product Robust Optimization: An Integrated Design and Marketing Approach
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
884
892
.10.1115/1.2202889
40.
Jin
,
Y.
,
Tang
,
K.
,
Yu
,
X.
,
Sendhoff
,
B.
, and
Yao
,
X.
,
2013
, “
A Framework for Finding Robust Optimal Solutions Over Time
,”
Memetic Comput.
,
5
(
1
), pp.
3
18
.10.1007/s12293-012-0090-2
41.
Lee
,
K.-H.
,
Eom
,
I.-S.
,
Park
,
G.-J.
, and
Lee
,
W.-I.
,
1994
, “
A Study on the Robust Design for Unconstrained Optimization Problems
,”
ASME J. Mech. Des.
,
18
, pp.
2825
2836
10.2514/3.13187.
42.
Balling
,
R. J.
,
Free
,
J. C.
, and
Parkinson
,
A. R.
,
1986
, “
Consideration of Worst-Case Manufacturing Tolerances in Design Optimization
,”
J. Mech. Trans. Autom.
,
108
(4), pp.
438
441
.10.1115/1.3258751
43.
Sundaresan
,
S.
,
Ishii
,
K.
, and
Houser
,
D. R.
,
1992
, “
Design Optimization for Robustness Using Performance Simulation Programs
,”
Eng. Optim.
,
20
(
3
), pp.
163
178
.10.1080/03052159208941278
44.
Zhu
,
J.
, and
Ting
,
K.-L.
,
2000
, “
Performance Distribution Analysis and Robust Design
,”
ASME J. Mech. Des.
,
123
(
1
), pp.
11
17
10.1115/1.1333095.
45.
Su
,
J.
, and
Renaud
,
J. E.
,
1997
, “
Automatic Differentiation in Robust Optimization
,”
AIAA J.
,
35
(
6
), pp.
1072
179
.10.2514/3.13628
46.
Phadke
,
M. S.
,
1989
,
Quality Engineering Using Robust Design
,
Prentice Hall
,
Englewood Cliffs, NJ
.
47.
Chao
,
L. P.
,
Gandhi
,
M. V.
, and
Thompson
,
B. S.
,
1993
, “
A Design-for-Manufacture Methodology for Incorporating Manufacturing Uncertainties in the Robust Design of Fibrous Laminated Composite Structures
,”
J. Compos. Mater.
,
27
(
2
), pp.
175
194
.10.1177/002199839302700204
48.
Poles
,
S.
, and
Lovison
,
A.
,
2009
, “
A Polynomial Chaos Approach to Robust Multiobjective Optimization
,”
Hybrid and Robust Approaches to Multiobjective Optimization, Dagstuhl Seminar Proceedings, Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik
,
K.
Deb
,
S.
Greco
,
K.
Miettinen
, and
E.
Zitzler
, eds., pp.
1862
4405
, Paper No. 09041.
49.
Tsutsui
,
S.
,
Ghosh
,
A.
, and
Fujimoto
,
Y.
,
1996
, “
A Robust Solution Searching Scheme in Genetic Search
,”
Parallel Problem Solving From Nature—PPSN IV
,
Springer-Verlag
, Germany, pp.
543
552
.
50.
Greiner
,
H.
,
1996
, “
Robust Optical Coating Design With Evolution Strategies
,”
Appl. Opt.
,
35
(
28
), pp.
5477
5483
.10.1364/AO.35.005477
51.
Wiesmann
,
D.
,
Hammel
,
U.
, and
Back
,
T.
,
1998
, “
Robust Design of Multilayer Optical Coatings by Means of Evolutionary Algorithms
,”
IEEE Trans. Evol. Comput.
,
2
(
4
), pp.
162
167
.10.1109/4235.738986
52.
Branke
,
J.
,
1998
, “
Creating Robust Solutions by Means of Evolutionary Algorithms
,”
Parallel Problem Solving From Nature—PPSN V
,
A. E.
Eiben
,
T.
Bck
,
M.
Schoenauer
, and
H.-P.
Schwefel
, eds., Vol.
1498
,
Springer
,
Berlin, Germany
, pp.
119
128
10.1007/BFb0056855.
53.
Forouraghi
,
B.
,
2000
, “
A Genetic Algorithm for Multiobjective Robust Design
,”
Appl. Intelligence
,
12
(
3
), pp.
151
161
.10.1023/A:1008356321921
54.
Tsutsui
,
S.
, and
Ghosh
,
A.
,
1997
, “
Genetic Algorithms With a Robust Solution Searching Scheme
,”
IEEE Trans. Evol. Comput.
,
1
(
3
), pp.
201
208
.10.1109/4235.661550
55.
Branke
,
J.
,
2001
, “
Reducing the Sampling Variance When Searching for Robust Solutions
,”
Proceedings of the Genetic and Evolutionary Computation Conference (GECCO)
,
L.
Spector
, ed.,
Morgan Kaufmann
, San Francisco, CA, pp.
235
242
.
56.
Rattray
,
M.
, and
Shapiro
,
J.
,
1997
, “
Noisy Fitness Evaluation in Genetic Algorithms and the Dynamics of Learning
,”
Found. Genet. Algorithms
,
4
, pp.
117
139
.
57.
Miller
,
B. L.
, and
Goldberg
,
D. E.
,
1996
, “
Genetic Algorithms, Selection Schemes, and the Varying Effects of Noise
,”
Evol. Comput.
,
4
(
2
), pp.
113
131
.10.1162/evco.1996.4.2.113
58.
Fitzpatrick
,
J. M.
, and
Grefenstette
,
J. J.
,
1988
, “
Genetic Algorithms in Noisy Environments
,”
Mach. Learn.
,
3
(
2–3
), pp.
101
120
10.1023/A:1022654003254.
59.
Kim
,
C.
, and
Choi
,
K. K.
,
2007
, “
Reliability-Based Design Optimization Using Response Surface Method With Prediction Interval Estimation
,”
ASME J. Mech. Des.
,
130
(
12
), pp.
1
12
10.1115/1.2988476.
60.
Chen
,
W.
,
Jin
,
R.
, and
Sudjianto
,
A.
,
2006
, “
Analytical Global Sensitivity Analysis and Uncertainty Propagation for Robust Design
,”
J. Qual. Technol.
,
38
(
4
), pp.
333
348
.
61.
Du
,
X.
, and
Chen
,
W.
,
2000
, “
Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design
,”
ASME J. Mech. Des.
,
122
(
4
), pp.
385
394
.10.1115/1.1290247
62.
McKay
,
M. D.
,
Beckmkan
,
R. J.
, and
Conover
,
W. J.
,
1979
, “
A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code
,”
Technometrics
,
21
(
2
), pp.
239
245
10.2307/1268522.
63.
Harbitz
,
A.
,
1986
, “
An Efficient Sampling Method for Probability of Failure Calculation
,”
Struct. Saf.
,
3
(
2
), pp.
109
115
.10.1016/0167-4730(86)90012-3
64.
Ditlevsen
,
O.
, and
Bjerager
,
P.
,
1989
, “
Plastic Reliability Analysis by Directional Simulation
,”
ASCE J. Eng. Mech.
,
115
(
6
), pp.
1347
1362
.10.1061/(ASCE)0733-9399(1989)115:6(1347)
65.
Mourelatos
,
Z. P.
, and
Liang
,
J.
,
2005
, “
A Methodology for Trading-Off Performance and Robustness Under Uncertainty
,”
ASME J. Mech. Des.
,
128
, pp.
856
863
10.1115/1.2202883.
66.
Gunawan
,
S.
, and
Azarm
,
S.
,
2004
, “
On a Combined Multi-Objective and Feasibility Robustness Method for Design Optimization
,” 10th
AIAA/ISSMO
Multidisciplinary Analysis and Optimization Conference
, pp.
1
10
.10.2514/6.2004-4357
67.
Gunawan
,
S.
, and
Azarm
,
S.
,
2005
, “
A Feasibility Robust Optimization Method Using Sensitivity Region Concept
,”
ASME J. Mech. Des.
,
127
(
5
), pp.
858
865
.10.1115/1.1903000
68.
Li
,
M.
,
Azarm
,
S.
, and
Boyars
,
A.
,
2006
, “
A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
874
883
.10.1115/1.2202884
69.
Du
,
X.
, and
Chen
,
W.
,
2001
, “
A Most Probable Point-Based Method for Efficient Uncertainty Analysis
,”
J. Des. Manuf. Autom.
,
4
(
1
), pp.
47
66
10.1.1.26.3453.
70.
Hasofer
,
A. M.
, and
Lind
,
N. C.
,
1974
, “
Exact and Invariant Second-Moment Code Format
,”
ASCE J. Eng. Mech. Div.
,
100
(
1
), pp.
111
121
.
71.
Hohenbichler
,
M.
,
Gollwitzer
,
S.
,
Kruse
,
W.
, and
Rackwitz
,
R.
,
1987
, “
New Light on First- and Second-Order Reliability Methods
,”
Struct. Saf.
,
4
(
4
), pp.
267
284
.10.1016/0167-4730(87)90002-6
72.
Du
,
X.
, and
Sudjianto
,
A.
,
2004
, “
First-Order Saddlepoint Approximation for Reliability Analysis
,”
AIAA J.
,
42
(6), pp.
1199
1207
.10.2514/1.3877
73.
Hamel
,
J. M.
, and
Azarm
,
S.
,
2011
, “
Reducible Uncertain Interval Design by Kriging Metamodel Assisted Multi-Objective Optimization
,”
ASME J. Mech. Des.
,
133
(
1
), p.
011002
.10.1115/1.4002974
74.
Siddiqui
,
S.
,
Azarm
,
S.
, and
Gabriel
,
S.
,
2011
, “
A Modified Benders Decomposition Method for Efficient Robust Optimization Under Interval Uncertainty
,”
Struct. Multidiscip. Optim.
,
44
(
2
), pp.
259
275
.10.1007/s00158-011-0631-1
75.
Chen
,
W.
, and
Lewis
,
K.
,
1999
, “
Robust Design Approach for Achieving Flexibility in Multidisciplinary Design
,”
AIAA J.
,
37
(
8
), pp.
982
989
.10.2514/2.805
76.
Du
,
X.
,
Wang
,
Y.
, and
Chen
,
W.
,
2000
, “
Methods for Robust Multidisciplinary Design
,” 41st
AIAA/ASME/ASCE/AHS/ASC
Structures
,
Structural Dynamics, and Materials Conference and Exhibit
,
Atlanta, GA
, pp.
1
10
10.2514/6.2000-1785.
77.
Ishibuchi
,
H.
,
Tsukamoto
,
N.
, and
Nojima
,
Y.
,
2008
, “
Evolutionary Many-Objective Optimization: A Short Review
,”
IEEE Congress on Evolutionary Computation
, pp.
2419
2426
.
78.
Koppen
,
M.
, and
Yoshida
,
K.
,
2007
,
Substitute Distance Assignment in NSGA-II for Handling Many-Objective Optimization Problems (EMO LNCS)
,
S.
Obayashi
, ed., Vol.
4403
, Springer Berlin Heidelberg, Matsushima, Japan, pp.
727
741
.
79.
Asafuddoula
,
M.
,
Ray
,
T.
, and
Sarker
,
R.
,
2013
, “
A Decomposition Based Evolutionary Algorithm for Many Objective Optimization With Systematic Sampling and Adaptive Epsilon Control
,”
Evolutionary Multi-Criterion Optimization Lecture Notes in Computer Science
,
R.
Purshouse
, ed., Vol.
7811
,
Springer
,
Berlin, Germany
, pp.
413
427
.10.1007/978-3-642-37140-0_32
80.
Jain
,
H.
, and
Deb
,
K.
,
2013
, “
An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Non-Dominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach
,”
IEEE Trans. Evol. Comput.
,
18
(
4
), pp.
602
622
10.1109/TEVC.2013.2281534.
81.
Asafuddoula
,
M.
,
Singh
,
H.
, and
Ray
,
T.
, “
Six-Sigma Robust Design Optimization Using a Many-Objective Decomposition Based Evolutionary Algorithm
,”
IEEE Trans. Evol. Comput.
(published online)10.1109/TEVC.2014.2343791.
82.
Singh
,
H. K.
,
Isaacs
,
A.
, and
Ray
,
T.
,
2011
, “
A Pareto Corner Search Evolutionary Algorithm and Dimensionality Reduction in Many-Objective Optimization Problems
,”
IEEE Trans. Evol. Comput.
,
15
(
4
), pp.
539
556
.10.1109/TEVC.2010.2093579
83.
Das.
,
I.
, and
Dennis
,
J. E.
,
1998
, “
Normal-Boundary Intersection: A New Method for Generating Pareto Optimal Points in Multicriteria Optimization Problems
,”
SIAM J. Optim.
,
8
(
3
), pp.
631
657
.10.1137/S1052623496307510
84.
Huband
,
S.
,
Hingston
,
P.
,
Barone
,
L.
, and
While
,
L.
,
2006
, “
A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit
,”
IEEE Trans. Evol. Comput.
,
10
(
5
), pp.
477
506
.10.1109/TEVC.2005.861417
85.
Bader
,
J.
,
Deb
,
K.
, and
Zitzler
,
E.
,
2010
, “
Faster Hypervolume-Based Search Using Monte Carlo Sampling
,”
Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems Lecture Notes in Economics and Mathematical Systems
,
M.
Ehrgott
,
B.
Naujoks
,
T. J.
Stewart
, and
J.
Wallenius
, eds., Vol.
634
,
Springer
,
Berlin, Germany
, pp.
313
326
10.1007/978-3-642-04045-0_27.
86.
Sun
,
G.
,
Li
,
G.
,
Zhou
,
S.
,
Li
,
H.
,
Hou
,
S.
, and
Li
,
Q.
,
2011
, “
Crashworthiness Design of Vehicle by Using Multiobjective Robust Optimization
,”
Struct. Multidiscip. Optim.
,
44
(
1
), pp.
99
110
.10.1007/s00158-010-0601-z
87.
Simpson
,
T. W.
,
Chen
,
W.
,
Allen
,
J. K.
, and
Mistree
,
F.
,
1996
, “
Conceptual Design of a Family of Products Through the Use of the Robust Concept Exploration Method
,”
AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
, Vol.
2
, pp.
1535
1545
.
88.
Hadka
,
D.
,
Reed
,
P. M.
, and
Simpson
,
T. W.
,
2012
, “
Diagnostic Assessment of the Borg MOEA for Many-Objective Product Family Design Problems
,”
IEEE
World Congress on Computational Intelligence
, pp.
10
15
10.1109/CEC.2012.6256466.
89.
Youn
,
B. D.
,
Choi
,
K. K.
, and
Yi
,
K.
,
2005
, “
Performance Moment Integration (PMI) Method for Quality Assessment in Reliability-Based Robust Design Optimization
,”
Mech. Based Des. Struct. Mach.
,
33
(
2
), pp.
185
213
.10.1081/SME-200067066
You do not currently have access to this content.