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Research Papers: Design Automation

A Projection-Based Approach for Constructing Piecewise Linear Pareto Front Approximations

[+] Author and Article Information
Hemant Kumar Singh

School of Engineering and IT,
The University of New South Wales,
Canberra, ACT 2600, Australia

Kalyan Shankar Bhattacharjee

School of Engineering and IT,
The University of New South Wales,
Canberra, ACT 2600, Australia

Tapabrata Ray

School of Engineering and IT,
The University of New South Wales,
Canberra, ACT 2600, Australia

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 22, 2015; final manuscript received June 7, 2016; published online July 21, 2016. Assoc. Editor: Kazuhiro Saitou.

J. Mech. Des 138(9), 091404 (Jul 21, 2016) (12 pages) Paper No: MD-15-1843; doi: 10.1115/1.4033991 History: Received December 22, 2015; Revised June 07, 2016

Abstract

Real-life design problems often require simultaneous optimization of multiple conflicting criteria resulting in a set of best trade-off solutions. This best trade-off set of solutions is referred to as Pareto optimal front (POF) in the outcome space. Obtaining the complete POF becomes impractical for problems where evaluation of each solution is computationally expensive. Such problems are commonly encountered in several fields, such as engineering, management, and scheduling. A practical approach in such cases is to construct suitable POF approximations, which can aid visualization, decision-making, and interactive optimization. In this paper, we propose a method to generate piecewise linear Pareto front approximation from a given set of N Pareto optimal outcomes. The approximations are represented using geometrical linear objects known as polytopes, which are formed by triangulating the given M-objective outcomes in a reduced $(M−1)$-objective space. The proposed approach is hence referred to as projection-based Pareto interpolation (PROP). The performance of PROP is demonstrated on a number of benchmark problems and practical applications with linear and nonlinear fronts to illustrate its strengths and limitations. While being novel and theoretically interesting, PROP also improves on the computational complexity required in generating such approximations when compared with existing Pareto interpolation (PAINT) algorithm.

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Figures

Fig. 1

Self-dominance and point-polytope dominance

Fig. 2

Final polytopes for HE problem [15]

Fig. 3

PAINT using projected points: (a) DTLZ2 data projected on f1f2 plane, (b) initial polytopes using Delaunay triangulation of (a), and (c) final polytopes starting from (b)

Fig. 4

Transformation of projected points

Fig. 5

PAINT using transformed projection: (a) DTLZ2 data projected and transformed on f1f2 plane, (b) initial polytopes using Delaunay triangulation of (a), and (c) final polytopes starting from (b)

Fig. 6

Results using PROP on DTLZ1 problem: (a) data, (b) projection, (c) initial polytopes, and (d) final polytopes

Fig. 7

Results using PROP on DTLZ2 problem: (a) data, (b) projection, (c) initial polytopes, and (d) final polytopes

Fig. 8

Results using PROP on (a) WFG2, (b) DTLZ7, and (c) hourglass

Fig. 9

Results using PROP on degenerate problems: (a) Viennet and (b) DTLZ5

Fig. 10

Results using PROP on practical application problems: (a) forest management planning, (b) liquid-rocket injector design, and (c) wastewater treatment

Fig. 11

Results using PROP on HE network synthesis problem: (a) and (b) using f1 as projection axis, (c) and (d) using f2 as projection axis, and (e) and (f) using f3 as projection axis. (a), (c), and (e) Initial polytopes and (b), (d), and (f) final polytopes.

Fig. 12

Delaunay triangulation for the dataset of HE problem using f2 as projection axis. (a) Projections on f1f3 plane. (b) Triangulation resulting from transformed space as viewed in (scaled) original space. The point P2 lies below (has lower f2 value) the polytope formed by P1–P3–P4. (c) Initial polytopes in (scaled) original space. P2 dominates three polytopes.

Fig. 13

Results using PROP on DTLZ2 problem: (a) projection axis f1 and (b) projection axis f2

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