Abstract

This paper presents a surrogate model-based computationally efficient optimization scheme for design problems with multiple, probabilistic objectives estimated through stochastic simulation. It examines the extension of the previously developed MODU-AIM (Multi-Objective Design under Uncertainty with Augmented Input Metamodels) algorithm, which performs well for bi-objective problem but encounters scalability difficulties for applications with more than two objectives. Computational efficiency is achieved by using a single surrogate model, adaptively refined within an iterative optimization setting, to simultaneously support the uncertainty quantification and the design optimization, and the MODU-AIM extension is established by replacing the originally used epsilon-constraint optimizer with a multi-objective evolutionary algorithm (MOEA). This requires various modifications to accommodate MOEA’s unique traits. For uncertainty quantification, a clustering-based importance sampling density selection is introduced to mitigate MOEA’s lack of direct control on Pareto solution density. To address the potentially large solution set of MOEAs, both the termination criterion of the iterative optimization scheme and the design of experiment (DoE) strategy for refinement of the surrogate model are modified, leveraging efficient performance comparison indicators. The importance of each objective in the different parts of the Pareto front is further integrated in the DoE to improve the adaptive selection of experiments.

References

References
1.
Fu
,
M. C.
,
Bayraksan
,
G.
,
Henderson
,
S. G.
,
Nelson
,
B. L.
,
Powell
,
W. B.
,
Ryzhov
,
I. O.
, and
Thengvall
,
B.
,
2014
, “
Simulation Optimization: A Panel on the State of the Art in Research and Practice
,”
Simulation Conference (WSC), 2014 Winter
,
Savannah, GA
,
Dec. 7–10
,
IEEE
, pp.
3696
3706
.
2.
Miettinen
,
K.
,
2012
,
Nonlinear Multiobjective Optimization
,
Springer Science, New York
.
3.
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
4.
Helton
,
J. C.
,
Johnson
,
J. D.
, and
Oberkampf
,
W. L.
,
2004
, “
An Exploration of Alternative Approaches to the Representation of Uncertainty in Model Predictions
,”
Reliab. Eng. Syst. Safety
,
85
(
1–3
), pp.
39
71
. 10.1016/j.ress.2004.03.025
5.
Zang
,
C.
,
Friswell
,
M.
, and
Mottershead
,
J.
,
2005
, “
A Review of Robust Optimal Design and its Application in Dynamics
,”
Comput. Struct.
,
83
(
4–5
), pp.
315
326
. 10.1016/j.compstruc.2004.10.007
6.
Beck
,
J. L.
, and
Taflanidis
,
A.
,
2013
, “
Prior and Posterior Robust Stochastic Predictions for Dynamical Systems Using Probability Logic
,”
Int. J. Uncertain. Quantif.
,
3
(
4
), pp.
271
288
. 10.1615/Int.J.UncertaintyQuantification.2012003641
7.
Papadimitriou
,
D. I.
, and
Papadimitriou
,
C.
,
2016
, “
Robust and Reliability-Based Structural Topology Optimization Using a Continuous Adjoint Method
,”
ASCE ASME J. Risk Uncertain. Eng. Syst. A: Civ. Eng.
,
2
(
3
), pp.
B4016002
. 10.1061/AJRUA6.0000869
8.
Pandita
,
P.
,
Bilionis
,
I.
, and
Panchal
,
J.
,
2016
, “
Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions
,”
ASME J. Mech. Des.
,
138
(
11
), p.
111412
. 10.1115/1.4034104
9.
Gidaris
,
I.
,
Taflanidis
,
A. A.
,
Lopez-Garcia
,
D.
, and
Mavroeidis
,
G. P.
,
2016
, “
Multi-Objective Risk-Informed Design of Floor Isolation Systems
,”
Earthquake Eng. Struct. Dynam.
,
45
(
8
), pp.
1293
1313
. 10.1002/eqe.2708
10.
Rosen
,
S. L.
,
Harmonosky
,
C. M.
, and
Traband
,
M. T.
,
2008
, “
Optimization of Systems With Multiple Performance Measures via Simulation: Survey and Recommendations
,”
Comput. Ind. Eng.
,
54
(
2
), pp.
327
339
. 10.1016/j.cie.2007.07.004
11.
Abdelaziz
,
F. B.
,
2012
, “
Solution Approaches for the Multiobjective Stochastic Programming
,”
Eur. J. Oper. Res.
,
216
(
1
), pp.
1
16
. 10.1016/j.ejor.2011.03.033
12.
Gutjahr
,
W. J.
, and
Pichler
,
A.
,
2016
, “
Stochastic Multi-Objective Optimization: A Survey on Non-Scalarizing Methods
,”
Ann. Oper. Res.
,
236
(
2
), pp.
475
499
. 10.1007/s10479-013-1369-5
13.
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
14.
Li
,
M.
,
Azarm
,
S.
, and
Aute
,
V.
,
2005
, “
A Multi-Objective Genetic Algorithm for Robust Design Optimization
,”
Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation
,
Washington, DC
,
June 25–29
,
ACM
, pp.
771
778
.
15.
Coelho
,
R. F.
,
2013
, “
Co-Evolutionary Optimization for Multi-Objective Design Under Uncertainty
,”
ASME J. Mech. Des.
,
135
(
2
), p.
021006
. 10.1115/1.4023184
16.
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
17.
Wang
,
G. G.
, and
Shan
,
S.
,
2007
, “
Review of Metamodeling Techniques in Support of Engineering Design Optimization
,”
ASME J. Mech. Des.
,
129
(
4
), pp.
370
380
. 10.1115/1.2429697
18.
Li
,
M.
,
Li
,
G.
, and
Azarm
,
S.
,
2008
, “
A Kriging Metamodel Assisted Multi-Objective Genetic Algorithm for Design Optimization
,”
ASME J. Mech. Des.
,
130
(
3
), p.
031401
. 10.1115/1.2829879
19.
Zhang
,
Q.
,
Liu
,
W.
,
Tsang
,
E.
, and
Virginas
,
B.
,
2010
, “
Expensive Multiobjective Optimization by MOEA/D With Gaussian Process Model
,”
IEEE Trans. Evol. Comput.
,
14
(
3
), pp.
456
474
. 10.1109/TEVC.2009.2033671
20.
Knowles
,
J.
,
2006
, “
ParEGO: A Hybrid Algorithm With On-Line Landscape Approximation for Expensive Multiobjective Optimization Problems
,”
IEEE Trans. Evol. Comput.
,
10
(
1
), pp.
50
66
. 10.1109/TEVC.2005.851274
21.
Dubourg
,
V.
,
Sudret
,
B.
, and
Bourinet
,
J.-M.
,
2011
, “
Reliability-Based Design Optimization Using Kriging Surrogates and Subset Simulation
,”
Struct. Multidiscipl. Optim.
,
44
(
5
), pp.
673
690
. 10.1007/s00158-011-0653-8
22.
Bichon
,
B. J.
,
Eldred
,
M. S.
,
Mahadevan
,
S.
, and
McFarland
,
J. M.
,
2013
, “
Efficient Global Surrogate Modeling for Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
135
(
1
), p.
011009
. 10.1115/1.4022999
23.
Janusevskis
,
J.
, and
Le Riche
,
R.
,
2013
, “
Simultaneous Kriging-Based Estimation and Optimization of Mean Response
,”
J. Glob. Optim.
,
55
(
2
), pp.
313
336
. 10.1007/s10898-011-9836-5
24.
Li
,
F.
,
Luo
,
Z.
,
Rong
,
J.
, and
Zhang
,
N.
,
2013
, “
Interval Multi-Objective Optimisation of Structures Using Adaptive Kriging Approximations
,”
Comput. Struct.
,
119
, pp.
68
84
. 10.1016/j.compstruc.2012.12.028
25.
Hu
,
W.
,
Li
,
M.
,
Azarm
,
S.
, and
Almansoori
,
A.
,
2011
, “
Multi-Objective Robust Optimization Under Interval Uncertainty Using Online Approximation and Constraint Cuts
,”
ASME J. Mech. Des.
,
133
(
6
), p.
061002
. 10.1115/1.4003918
26.
Zhou
,
Q.
,
Jiang
,
P.
,
Shao
,
X.
,
Zhou
,
H.
, and
Hu
,
J.
,
2017
, “
An On-Line Kriging Metamodel Assisted Robust Optimization Approach Under Interval Uncertainty
,”
Eng. Comput.
,
34
(
2
), pp.
420
446
. 10.1108/EC-01-2016-0020
27.
Leotardi
,
C.
,
Serani
,
A.
,
Iemma
,
U.
,
Campana
,
E. F.
, and
Diez
,
M.
,
2016
, “
A Variable-Accuracy Metamodel-Based Architecture for Global MDO Under Uncertainty
,”
Struct. Multidiscipl. Optim.
,
54
(
3
), pp.
573
593
. 10.1007/s00158-016-1423-4
28.
Pandita
,
P.
,
Bilionis
,
I.
,
Panchal
,
J.
,
Gautham
,
B.
,
Joshi
,
A.
, and
Zagade
,
P.
,
2018
, “
Stochastic Multiobjective Optimization on a Budget: Application to Multipass Wire Drawing With Quantified Uncertainties
,”
Int. J. Uncertain. Quan.
,
8
(
3
), pp.
233
249
. 10.1615/Int.J.UncertaintyQuantification.2018021315
29.
Tsoukalas
,
I.
, and
Makropoulos
,
C.
,
2015
, “
Multiobjective Optimisation on a Budget: Exploring Surrogate Modelling for Robust Multi-Reservoir Rules Generation Under Hydrological Uncertainty
,”
Environ. Model. Softw.
,
69
, pp.
396
413
. 10.1016/j.envsoft.2014.09.023
30.
Poles
,
S.
, and
Lovison
,
A.
,
2009
, “
A Polynomial Chaos Approach to Robust Multiobjective Optimization
,”
Dagstuhl Seminar Proceedings, Schloss Dagstuhl-Leibniz-Zentrum für Informatik
,
Wadern, Germany
.
31.
Taflanidis
,
A. A.
, and
Beck
,
J. L.
,
2008
, “
An Efficient Framework for Optimal Robust Stochastic System Design Using Stochastic Simulation
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
1
), pp.
88
101
. 10.1016/j.cma.2008.03.029
32.
Zhang
,
J.
,
Taflanidis
,
A.
, and
Medina
,
J.
,
2017
, “
Sequential Approximate Optimization for Design Under Uncertainty Problems Utilizing Kriging Metamodeling in Augmented Input Space
,”
Comput. Methods Appl. Mech. Eng.
,
315
, pp.
369
395
. 10.1016/j.cma.2016.10.042
33.
Li
,
G.
,
Hwai-yong Tan
,
M.
, and
Hui Ng
,
S.
,
2019
, “
Metamodel-Based Optimization of Stochastic Computer Models for Engineering Design Under Uncertain Objective Function
,”
IISE Trans
,
51
(
5
), pp.
517
530
.
34.
Sacks
,
J.
,
Welch
,
W. J.
,
Mitchell
,
T. J.
, and
Wynn
,
H. P.
,
1989
, “
Design and Analysis of Computer Experiments
,”
Stat. Sci.
,
4
(
4
), pp.
409
435
. 10.1214/ss/1177012413
35.
Zhang
,
J.
, and
Taflanidis
,
A. A.
,
2019
, “
Multi-Objective Optimization for Design Under Uncertainty Problems Through Surrogate Modeling in Augmented Input Space
,”
Struct. Multidiscipl. Optim.
,
59
(
2
), pp.
357
372
. 10.1007/s00158-018-2069-1
36.
Haimes
,
Y. Y.
,
Ladson
,
L.
, and
Wismer
,
D. A.
,
1971
, “
Bicriterion Formulation of Problems of Integrated System Identification and System Optimization
,”
IEEE Trans. Syst. Man Cybern.
,
SMC-1
(
3
), pp.
296
297
.
37.
Coello
,
C. A. C.
,
Lamont
,
G. B.
, and
Van Veldhuizen
,
D. A.
,
2007
,
Evolutionary Algorithms for Solving Multi-Objective Problems
,
Springer
,
New York
.
38.
Emmerich
,
M. T.
, and
Deutz
,
A. H.
,
2018
, “
A Tutorial on Multiobjective Optimization: Fundamentals and Evolutionary Methods
,”
Nat Comput.
,
17
(
3
), pp.
585
609
. 10.1007/s11047-018-9685-y
39.
Gidaris
,
I.
,
Taflanidis
,
A. A.
, and
Mavroeidis
,
G. P.
,
2017
, “
Multiobjective Design of Supplemental Seismic Protective Devices Utilizing Lifecycle Performance Criteria
,”
J. Struct. Eng.
,
144
(
3
), pp.
04017225
. 10.1061/(ASCE)ST.1943-541X.0001969
40.
Zitzler
,
E.
,
Deb
,
K.
, and
Thiele
,
L.
,
2000
, “
Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
,”
Evol. Comput.
,
8
(
2
), pp.
173
195
. 10.1162/106365600568202
41.
Branke
,
J.
,
Deb
,
K.
,
Dierolf
,
H.
, and
Osswald
,
M.
,
2004
, “
Finding Knees in Multi-Objective Optimization
,”
International Conference on Parallel Problem Solving From Nature
,
Springer
,
New York
, pp.
722
731
.
42.
Robert
,
C.
, and
Casella
,
G.
,
2013
,
Monte Carlo Statistical Methods
,
Springer Science, New York
.
43.
Apley
,
D. W.
,
Liu
,
J.
, and
Chen
,
W.
,
2006
, “
Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
945
958
. 10.1115/1.2204974
44.
Yen
,
G. G.
, and
He
,
Z.
,
2014
, “
Performance Metric Ensemble for Multiobjective Evolutionary Algorithms
,”
IEEE Trans. Evol. Comput.
,
18
(
1
), pp.
131
144
. 10.1109/TEVC.2013.2240687
45.
Deb
,
K.
, and
Jain
,
H.
,
2014
, “
An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints
,”
IEEE Trans. Evol. Comput.
,
18
(
4
), pp.
577
601
. 10.1109/TEVC.2013.2281535
46.
Knowles
,
J.
,
Corne
,
D.
, and
Reynolds
,
A.
,
2009
, “
Noisy Multiobjective Optimization on a Budget of 250 Evaluations
,”
International Conference on Evolutionary Multi-Criterion Optimization
,
Springer
,
New York
, pp.
36
50
.
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