Recent advances in simulation and computation capabilities have enabled designers to model increasingly complex engineering problems, taking into account many dimensions, or objectives, in the problem formulation. Increasing the dimensionality often results in a large trade space, where decision-makers (DM) must identify and negotiate conflicting objectives to select the best designs. Trade space exploration often involves the projection of nondominated solutions, that is, the Pareto front, onto two-objective trade spaces to help identify and negotiate tradeoffs between conflicting objectives. However, as the number of objectives increases, an exhaustive exploration of all of the two-dimensional (2D) Pareto fronts can be inefficient due to a combinatorial increase in objective pairs. Recently, an index was introduced to quantify the shape of a Pareto front without having to visualize the solution set. In this paper, a formal derivation of the Pareto Shape Index is presented and used to support multi-objective trade space exploration. Two approaches for trade space exploration are presented and their advantages are discussed, specifically: (1) using the Pareto shape index for weighting objectives and (2) using the Pareto shape index to rank objective pairs for visualization. By applying the two approaches to two multi-objective problems, the efficiency of using the Pareto shape index for weighting objectives to identify solutions is demonstrated. We also show that using the index to rank objective pairs provides DM with the flexibility to form preferences throughout the process without closely investigating all objective pairs. The limitations and future work are also discussed.

References

References
1.
Fleming
,
P. J.
,
Purshouse
,
R. C.
, and
Lygoe
,
R. J.
,
2005
, “
Many-Objective Optimization: An Engineering Design Perspective
,”
Evolutionary Multi-Criterion Optimization
,
Springer
,
Berlin
, pp.
14
32
.
2.
Ross
,
A. M.
, and
Hastings
,
D. E.
,
2005
, “
The Tradespace Exploration Paradigm
,”
INCOSE International Symposium
, Rochester, NY, July 10–14, pp.
1706
1718
.https://pdfs.semanticscholar.org/7f44/1744fb3fbc12ef26deb7df92b9167a584553.pdf
3.
Balling
,
R.
,
1999
, “
Design by Shopping: A New Paradigm?
,”
Third World Congress of Structural and Multidisciplinary Optimization (WCSMO)
, Buffalo, NY, May 17–21, pp.
295
297
.
4.
Woodruff
,
M. J.
,
Reed
,
P. M.
, and
Simpson
,
T. W.
,
2013
, “
Many Objective Visual Analytics: Rethinking the Design of Complex Engineered Systems
,”
Structural and Multidisciplinary Optimization
, Vol.
48
, Springer, Berlin, pp.
201
219
.
5.
Wong
,
P. C.
, and
Thomas
,
J.
,
2004
, “
Guest Editors' Introduction—Visual Analytics
,”
IEEE Comput. Graphics Appl.
,
24
(
5
), pp.
20
21
.
6.
Cook
,
K. A.
, and
Thomas
,
J. J.
,
2005
,
Illuminating the Path: The Research and Development Agenda for Visual Analytics
,
Pacific Northwest National Laboratory (PNNL)
,
Richland, WA
.
7.
Keim
,
D.
,
Kohlhammer
,
J.
,
Ellis
,
G.
, and
Mansmann
,
F.
,
2010
,
Mastering the Information Age Solving Problems With Visual Analytics
, Eurographics Association, Goslar, Germany.
8.
Chiu
,
P.-W.
, and
Bloebaum
,
C. L.
,
2008
, “
Hyper-Radial Visualization (HRV) With Weighted Preferences for Multi-Objective Decision Making
,”
AIAA
Paper No. 2008-5986.
9.
Agrawal
,
G.
,
Parashar
,
S.
, and
Bloebaum
,
C. L.
,
2006
, “
Intuitive Visualization of Hyperspace Pareto Frontier for Robustness in Multi-Attribute Decision-Making
,”
AIAA
Paper No. 2006-6962.
10.
Blasco
,
X.
,
Herrero
,
J. M.
,
Sanchis
,
J.
, and
Martínez
,
M.
,
2008
, “
A New Graphical Visualization of n-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization
,”
Inf. Sci.
,
178
(
20
), pp.
3908
3924
.
11.
Kohonen
,
T.
,
2001
,
Self-Organizing Maps
,
Springer
, Berlin.
12.
Richardson
,
T.
,
Nekolny
,
B.
,
Holub
,
J.
, and
Winer
,
E. H.
,
2014
, “
Visualizing Design Spaces Using Two-Dimensional Contextual Self-Organizing Maps
,”
AIAA J.
,
52
(
4
), pp.
725
738
.
13.
Winer
,
E. H.
, and
Bloebaum
,
C. L.
,
2001
, “
Visual Design Steering for Optimization Solution Improvement
,”
Struct. Multidiscip. Optim.
,
22
(
3
), pp.
219
229
.
14.
Stump
,
G. M.
,
Yukish
,
M.
,
Simpson
,
T. W.
, and
Bennett
,
L.
,
2002
, “
Multidimensional Visualization and Its Application to a Design by Shopping Paradigm
,”
AIAA
Paper No. 2002-5622.
15.
Stump
,
G. M.
,
Simpson
,
T. W.
,
Donndelinger
,
J. A.
,
Lego
,
S.
, and
Yukish
,
M.
,
2009
, “
Visual Steering Commands for Trade Space Exploration: User-Guided Sampling With Example
,”
ASME J. Comput. Inf. Sci. Eng.
,
9
(
4
), p.
044501
.
16.
Kollat
,
J. B.
, and
Reed
,
P.
,
2007
, “
A Framework for Visually Interactive Decision-Making and Design Using Evolutionary Multi-Objective Optimization (VIDEO)
,”
Environ. Model. Software
,
22
(
12
), pp.
1691
1704
.
17.
Bang
,
H.
, and
Selva
,
D.
,
2016
, “
IFEED: Interactive Feature Extraction for Engineering Design
,”
ASME
Paper No. DETC2016-60077.
18.
Purshouse
,
R. C.
, and
Fleming
,
P. J.
,
2003
, “
Conflict, Harmony, and Independence: Relationships in Evolutionary Multi-Criterion Optimization
,”
Evolutionary Multi-Criterion Optimization
,
Springer
,
Berlin
, pp.
16
30
.
19.
Unal
,
M.
,
Warn
,
G. P.
, and
Simpson
,
T. W.
,
2016a
, “
Quantifying the Shape of a Pareto Front in Support of Many-Objective Trade Space Exploration
,”
ASME
Paper No. DETC2016-59716.
20.
Inselberg
,
A.
,
1997
, “
Multidimensional Detective
,”
IEEE
Symposium on Information Visualization
, Phoenix, AZ, Oct. 20–21, pp.
100
107
.
21.
Marler
,
R. T.
, and
Arora
,
J. S.
,
2004
, “
Survey of Multi-Objective Optimization Methods for Engineering
,”
Struct. Multidiscip. Optim.
,
26
(
6
), pp.
369
395
.
22.
Simpson
,
T. W.
,
Carlsen
,
D.
,
Malone
,
M.
, and
Kollat
,
J.
,
2011
, “
Trade Space Exploration: Assessing the Benefits of Putting Designers ‘Back-in-the-Loop’ During Engineering Optimization
,”
Human-in-the-Loop Simulations
,
Springer
,
London
, pp.
131
152
.
23.
Wang
,
J.
, and
Terpenny
,
J.
,
2003
, “
Interactive Evolutionary Solution Synthesis in Fuzzy Set-Based Preliminary Engineering Design
,”
J. Intell. Manuf.
,
14
(
2
), pp.
153
167
.
24.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast Elitist Multi-Objective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evolutionary Comput.
,
6
(
2
), pp.
182
197
.
25.
Pareto
,
V.
,
1971
,
Manual of Political Economy
,
A. M. Kelley
,
New York
.
26.
Clemen
,
R. T.
,
1996
,
Making Hard Decisions: An Introduction to Decision Analysis
,
2nd ed.
,
Duxbury Press
,
Belmont, CA
.
27.
Chambers
,
J. M.
,
Cleveland
,
W. S.
,
Kleiner
,
B.
, and
Tukey
,
P. A.
,
1983
,
Graphical Methods for Data Analysis
,
Chapman and Hall
,
New York
.
28.
Unal
,
M.
,
Warn
,
G. P.
, and
Simpson
,
T. W.
,
2015
, “
Introduction of a Tradeoff Index for Efficient Trade Space Exploration
,”
ASME
Paper No. DETC2015-46895.
29.
Winer
,
E. H.
, and
Bloebaum
,
C. L.
,
2002
, “
Development of Visual Design Steering as an Aid in Large-Scale Multidisciplinary Design Optimization—Part I: Method Development
,”
Struct. Multidiscip. Optim.
,
23
(
6
), pp.
412
424
.
30.
Eddy
,
J.
, and
Lewis
,
K. E.
,
2002
, “
Visualization of Multidimensional Design and Optimization Using Cloud Visualization
,”
ASME
Paper No. DETC2002/DAC-34130.
31.
Kendall
,
M. G.
,
1938
, “
A New Measure of Rank Correlation
,”
Biometrika
,
30
(
1/2
), pp.
81
93
.
32.
Unal
,
M.
,
Warn
,
G. P.
, and
Simpson
,
T. W.
,
2016b
, “
Quantifying Tradeoffs to Reduce the Dimensionality of Complex Design Optimization Problems and Expedite Trade Space Exploration
,”
Struct. Multidiscip. Optim. J.
,
54
(
2
), pp.
233
248
.
33.
Li
,
Y.
,
Fadel
,
G. M.
,
Wiecek
,
M.
, and
Blouin
,
V. Y.
,
2003
, “
Minimum Effort Approximation of the Pareto Space of Convex Bi-Criteria Problems
,”
Optim. Eng.
,
4
(
3
), pp.
231
261
.
34.
Engau
,
A.
, and
Wiecek
,
M. M.
,
2007
, “
2D Decision-Making for Multicriteria Design Optimization
,”
Struct. Multidiscip. Optim.
,
34
(
4
), pp.
301
315
.
35.
Charnes
,
A.
, and
Cooper
,
W. W.
,
1977
, “
Goal Programming and Multiple Objective Optimization—Part 1
,”
Eur. J. Oper. Res.
,
1
(
1
), pp.
39
54
.
36.
Rao
,
J. R.
, and
Roy
,
N.
,
1989
, “
Fuzzy Set Theoretic Approach of Assigning Weights to Objectives in Multicriteria Decision Making
,”
Int. J. Syst. Sci.
,
20
(
8
), pp.
1381
1386
.
37.
Shah
,
R.
,
Reed
,
P. M.
, and
Simpson
,
T. W.
,
2011
, “
Many-Objective Evolutionary Optimization and Visual Analytics for Product Family Design
,”
Multi-Objective Evolutionary Optimisation for Product Design and Manufacturing
,
L.
Wang
,
A.
Ng
, and
K.
Deb
, eds.,
Springer
,
London
, pp.
137
159
.
38.
Van Veldhuizen
,
D.
,
1999
, “
Multiobjective Evolutionary Algorithm: Classifications, Analyses and New Innovations
,”
Ph.D. dissertation
, Air Force Institute of Technology, Wright-Patterson AFB, OH.https://dl.acm.org/citation.cfm?id=929368
39.
Zitzler
,
E.
,
Thiele
,
L.
,
Laumanns
,
M.
,
Fonseca
,
C. M.
, and
Da Fonseca
,
V. G.
,
2003
, “
Performance Assessment of Multiobjective Optimizers: An Analysis and Review
,”
IEEE Trans. Evol. Comput.
,
7
(
2
), pp.
117
132
.
40.
Hadka
,
D.
, and
Reed
,
P.
,
2013
, “
Borg: An Auto-Adaptive Many-Objective Evolutionary Computing Framework
,”
Evol. Comput.
,
21
(
2
), pp.
231
259
.
41.
Reed
,
P. M.
,
2017
, “
Decision Analytics for Complex Systems
,” Cornell University, Ithaca, NY, accessed Oct. 3, 2017, https://reed.cee.cornell.edu/index.php/Main_Page
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