Engineering design often involves problems with multiple conflicting performance criteria, commonly referred to as multi-objective optimization problems (MOP). MOPs are known to be particularly challenging if the number of objectives is more than three. This has motivated recent attempts to solve MOPs with more than three objectives, which are now more specifically referred to as “many-objective” optimization problems (MaOPs). Evolutionary algorithms (EAs) used to solve such problems require numerous design evaluations prior to convergence. This is not practical for engineering applications involving computationally expensive evaluations such as computational fluid dynamics and finite element analysis. While the use of surrogates has been commonly studied for single-objective optimization, there is scarce literature on its use for MOPs/MaOPs. This paper attempts to bridge this research gap by introducing a surrogate-assisted optimization algorithm for solving MOP/MaOP within a limited computing budget. The algorithm relies on principles of decomposition and adaptation of reference vectors for effective search. The flexibility of function representation is offered through the use of multiple types of surrogate models. Furthermore, to efficiently deal with constrained MaOPs, marginally infeasible solutions are promoted during initial phases of the search. The performance of the proposed algorithm is benchmarked with the state-of-the-art approaches using a range of problems with up to ten objective problems. Thereafter, a case study involving vehicle design is presented to demonstrate the utility of the approach.

References

References
1.
Deb
,
K.
,
2001
,
Multi-Objective Optimization Using Evolutionary Algorithms
, Vol.
16
,
Wiley
, Chichester, UK.
2.
Asafuddoula
,
M.
,
Ray
,
T.
, and
Sarker
,
R.
,
2015
, “
A Decomposition-Based Evolutionary Algorithm for Many-Objective Optimization
,”
IEEE Trans. Evol. Comput.
,
19
(
3
), pp.
445
460
.
3.
Bhattacharjee
,
K. S.
,
Singh
,
H. K.
, and
Ray
,
T.
,
2016
, “
Multi-Objective Optimization With Multiple Spatially Distributed Surrogates
,”
ASME J. Mech. Des.
,
138
(
9
), p.
091401
.
4.
Bhattacharjee
,
K. S.
,
Singh
,
H. K.
, and
Ray
,
T.
,
2017
, “
A Novel Decomposition Based Evolutionary Algorithm for Engineering Design Optimization
,”
ASME J. Mech. Des.
,
139
(
4
), p.
041403
.
5.
Gal
,
T.
,
Stewart
,
T.
, and
Hanne
,
T.
,
2013
,
Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications
, Vol.
21
,
Springer Science & Business Media
, Berlin.
6.
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
.
7.
Jin
,
Y.
,
2005
, “
A Comprehensive Survey of Fitness Approximation in Evolutionary Computation
,”
Soft Comput.
,
9
(
1
), pp.
3
12
.
8.
Jin
,
Y.
,
2011
, “
Surrogate-Assisted Evolutionary Computation: Recent Advances and Future Challenges
,”
Swarm Evol. Comput.
,
1
(
2
), pp.
61
70
.
9.
Ishibuchi
,
H.
,
Tsukamoto
,
N.
, and
Nojima
,
Y.
,
2008
, “
Evolutionary Many-Objective Optimization: A Short Review
,”
IEEE Congress on Evolutionary Computation
(
CEC
), Hong Kong, China, June 1–6, pp.
2419
2426
.
10.
Trivedi
,
A.
,
Srinivasan
,
D.
,
Sanyal
,
K.
, and
Ghosh
,
A.
,
2017
, “
A Survey of Multi-Objective Evolutionary Algorithms Based on Decomposition
,”
IEEE Trans. Evol. Comput.
,
21
(
3
), pp.
440
462
.
11.
Ishibuchi
,
H.
,
Yu
,
S.
,
Hiroyuki
,
M.
, and
Yusuke
,
N.
,
2017
, “
Performance of Decomposition-Based Many-Objective Algorithms Strongly Depends on Pareto Front Shapes
,”
IEEE Trans. Evol. Comput.
,
21
(
2
), pp.
169
190
.
12.
Asafuddoula
,
M.
,
Singh
,
H.
, and
Ray
,
T.
,
2017
, “
An Enhanced Decomposition Based Evolutionary Algorithm With Adaptive Reference Vectors
,”
IEEE Trans. Cybern.
, in press.
13.
Chugh
,
T.
,
Jin
,
Y.
,
Meittinen
,
K.
,
Hakanen
,
J.
, and
Sindhya
,
K.
,
2016
, “
A Surrogate-Assisted Reference Vector Guided Evolutionary Algorithm for Computationally Expensive Many-Objective Optimization
,”
IEEE Trans. Evol. Comput.
,
22
(1), pp. 129–142.
14.
Chugh
,
T.
,
Sindhya
,
K.
,
Miettinen
,
K.
,
Hakanen
,
J.
, and
Jin
,
Y.
,
2016
, “
On Constraint Handling in Surrogate-Assisted Evolutionary Many-Objective Optimization
,”
Parallel Problem Solving From Nature
, Springer International Publishing AG, Cham, Switzerland, pp.
214
224
.
15.
Li
,
B.
,
Li
,
J.
,
Tang
,
K.
, and
Yao
,
X.
,
2015
, “
Many-Objective Evolutionary Algorithms: A Survey
,”
ACM Comput. Surv.
,
48
(
1
), p. 13.
16.
Zitzler
,
E.
, and
Künzli
,
S.
,
2004
, “
Indicator-Based Selection in Multiobjective Search
,”
Parallel Problem Solving From Nature
,
Springer
, Berlin, pp.
832
842
.
17.
Zhang
,
Q.
, and
Li
,
H.
,
2007
, “
MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition
,”
IEEE Trans. Evol. Comput.
,
11
(
6
), pp.
712
731
.
18.
Cheng
,
R.
,
Jin
,
Y.
,
Olhofer
,
M.
, and
Sendhoff
,
B.
,
2016
, “
A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization
,”
IEEE Trans. Evol. Comput.
,
20
(
5
), pp.
773
791
.
19.
Yuan
,
Y.
,
Xu
,
H.
,
Wang
,
B.
, and
Yao
,
X.
,
2016
, “
A New Dominance Relation-Based Evolutionary Algorithm for Many-Objective Optimization
,”
IEEE Trans. Evol. Comput.
,
20
(
1
), pp.
16
37
.
20.
Das
,
I.
, and
Dennis
,
J. E.
,
1998
, “
Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems
,”
SIAM J. Optim.
,
8
(
3
), pp.
631
657
.
21.
Miettinen
,
K.
,
1999
,
Nonlinear Multiobjective Optimization
, Vol.
12
, Kluwer Academic Publishers, Boston, MA.
22.
Voss
,
T.
,
Beume
,
N.
,
Rudolph
,
G.
, and
Igel
,
C.
,
2008
, “
Scalarization Versus Indicator-Based Selection in Multi-Objective CMA Evolution Strategies
,”
IEEE Congress Evolutionary Computation
(
CEC
), Hong Kong, China, June 1–6, pp.
3036
3043
.
23.
Bhattacharjee
,
K. S.
,
Singh
,
H. K.
,
Ray
,
T.
, and
Zhang
,
Q.
,
2017
, “
Decomposition Based Evolutionary Algorithm With a Dual Set of Reference Vectors
,”
IEEE Congress on Evolutionary Computation
(
CEC
), San Sebastian, Spain, June 5–8, pp.
105
112
.
24.
Jin
,
R.
,
Chen
,
W.
, and
Simpson
,
T. W.
,
2001
, “
Comparative Studies of Metamodelling Techniques Under Multiple Modelling Criteria
,”
Struct. Multidiscip. Optim.
,
23
(
1
), pp.
1
13
.
25.
Knowles
,
J.
, and
Nakayama
,
H.
,
2008
,
Meta-Modeling in Multiobjective Optimization
,
Springer, Berlin
, pp.
245
284
.
26.
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
.
27.
Goel
,
T.
,
Haftka
,
R. T.
,
Shyy
,
W.
, and
Queipo
,
N. V.
,
2007
, “
Ensemble of Surrogates
,”
Struct. Multidiscip. Optim.
,
33
(
3
), pp.
199
216
.
28.
Zerpa
,
L. E.
,
Queipo
,
N. V.
,
Pintos
,
S.
, and
Salager
,
J. L.
,
2005
, “
An Optimization Methodology of Alkaline-Surfactant-Polymer Flooding Processes Using Field Scale Numerical Simulation and Multiple Surrogates
,”
J. Pet. Sci. Eng.
,
47
(
3–4
), pp.
197
208
.
29.
Hamza
,
K.
, and
Saitou
,
K.
,
2012
, “
A Co-Evolutionary Approach for Design Optimization Via Ensembles of Surrogates With Application to Vehicle Crashworthiness
,”
ASME J. Mech. Des.
,
134
(
1
), p.
011001
.
30.
Breiman
,
L.
,
1996
, “
Bagging Predictors
,”
Mach. Learn.
,
24
(
2
), pp.
123
140
.
31.
Abney
,
S.
,
Schapire
,
R. E.
, and
Singer
,
Y.
,
1999
, “
Boosting Applied to Tagging and PP Attachment
,”
Joint SIGDAT Conference on Empirical Methods in Natural Language Processing and Very Large Corpora
, College Park, MD, June 21–22, pp. 38–45.http://www.vinartus.net/spa/98b.pdf
32.
Bhattacharjee
,
K. S.
,
Singh
,
H. K.
,
Ray
,
T.
, and
Branke
,
J.
,
2016
, “
Multiple Surrogate Assisted Multiobjective Optimization Using Improved Pre-Selection
,”
IEEE Congress on Evolutionary Computation
(
CEC
), Vancouver, BC, Canada, July 24–29, pp.
4328
4335
.
33.
Isaacs
,
A.
,
Ray
,
T.
, and
Smith
,
W.
,
2009
, “
Multi-Objective Design Optimisation Using Multiple Adaptive Spatially Distributed Surrogates
,”
Int. J. Prod. Dev.
,
9
(
1/2/3
), pp.
188
217
.
34.
Li
,
K.
,
Deb
,
K.
,
Zhang
,
Q.
, and
Kwong
,
S.
,
2015
, “
An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition
,”
IEEE Trans. Evol. Comput.
,
19
(
5
), pp.
694
716
.
35.
Singh
,
H. K.
,
Ray
,
T.
, and
Sarker
,
R.
,
2013
, “
Optimum Oil Production Planning Using Infeasibility Driven Evolutionary Algorithm
,”
Evol. Comput.
,
21
(
1
), pp.
65
82
.
36.
Singh
,
H. K.
,
Alam
,
K.
, and
Ray
,
T.
,
2016
, “
Use of Infeasible Solutions During Constrained Evolutionary Search: A Short Survey
,”
Australasian Conference on Artificial Life and Computational Intelligence
(
ACALCI
), Geelong, Australia, Jan. 31–Feb. 2, pp.
193
205
.
37.
Ray
,
T.
,
Singh
,
H. K.
,
Isaacs
,
A.
, and
Smith
,
W. F.
,
2009
, “
Infeasibility Driven Evolutionary Algorithm for Constrained Optimization
,”
Constraint-Handling in Evolutionary Optimization
, Springer, Berlin, pp.
145
165
.
38.
Deb
,
K.
,
Thiele
,
L.
,
Laumanns
,
M.
, and
Zitzler
,
E.
,
2005
, “
Scalable Test Problems for Evolutionary Multiobjective Optimization
,”
Evolutionary Multiobjective Optimization
, Springer, London, pp.
105
145
.
39.
Huband
,
S.
,
Hingston
,
P.
,
Barone
,
L.
, and
While
,
L.
,
2006
, “
A Review of Multi-Objective Test Problems and a Scalable Test Problem Toolkit
,”
IEEE Trans. Evol. Comput.
,
10
(
5
), pp.
477
506
.
40.
Barnum
,
G.
, and
Mattson
,
C.
,
2010
, “
A Computationally Assisted Methodology for Preference-Guided Conceptual Design
,”
ASME J. Mech. Des.
,
132
(
12
), p.
121003
.
41.
Ishibuchi
,
H.
,
Hitotsuyanagi
,
Y.
,
Tsukamoto
,
N.
, and
Nojima
,
Y.
,
2010
, “
Many-Objective Test Problems to Visually Examine the Behavior of Multiobjective Evolution in a Decision Space
,”
International Conference on Parallel Problem Solving From Nature
, Kraków, Poland, Sept. 11–15, pp.
91
100
.https://dl.acm.org/citation.cfm?id=1887266
42.
While
,
L.
,
Bradstreet
,
L.
, and
Barone
,
L.
,
2012
, “
A Fast Way of Calculating Exact Hypervolumes
,”
IEEE Trans. Evol. Comput.
,
16
(
1
), pp.
86
95
.
43.
Tian
,
Y.
,
Cheng
,
R.
,
Zhang
,
X.
, and
Jin
,
Y.
,
2017
, “
PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization
,”
IEEE Comput. Intell. Mag.
,
12
(4), pp. 73–87.
You do not currently have access to this content.