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Research Papers: Design Automation

Quantifying the Shape of Pareto Fronts During Multi-Objective Trade Space Exploration

[+] Author and Article Information
Mehmet Unal

Civil and Environmental Engineering,
The Pennsylvania State University,
University Park, PA 16802
e-mail: mxu122@psu.edu

Gordon P. Warn

Civil and Environmental Engineering,
The Pennsylvania State University,
University Park, PA 16802
e-mail: gpw1@psu.edu

Timothy W. Simpson

Mechanical & Nuclear Engineering,
The Pennsylvania State University,
University Park, PA 16802
e-mail: tws8@psu.edu

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 17, 2017; final manuscript received August 30, 2017; published online December 13, 2017. Assoc. Editor: Gary Wang.

J. Mech. Des 140(2), 021402 (Dec 13, 2017) (13 pages) Paper No: MD-17-1138; doi: 10.1115/1.4038005 History: Received February 17, 2017; Revised August 30, 2017

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.

Copyright © 2018 by ASME
Topics: Shapes , Tradeoffs , Design
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Figures

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Fig. 1

Two-objective truss design problem and the Pareto front of the trade space

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Fig. 2

(a) A hypothetical four-objective problem trade space and (b) two-objective trade spaces for each objective pairs of the same hypothetical problem

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Fig. 3

(a) The parallel coordinate plot representation of a randomly produced hypothetical four objective optimization problem, (b) the parallel coordinate plot representation of the same four objective optimization problem with different ordering of objectives, (c) and (d) 2D trade spaces for the two objective pairs identified to have tradeoff from Figs. 3(a) and 3(b)

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Fig. 4

The Cartesian coordinates of the solution lines and their intersection for the derivation of the Pareto shape index

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Fig. 5

Pareto fronts with varying shapes, their associated parallel coordinate plots and the Pareto shape index values

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Fig. 6

A hypothetical Pareto front formed by different numbers and distributions of solutions. All objectives are assumed to be minimized.

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Fig. 7

Pareto fronts with multiple concave and convex subdomains and their regressions: (a) global linear form and (b) global convex form

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Fig. 8

Solution selection from the Pareto front

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Fig. 9

Solution selection from a specified area close to the Pareto front

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Fig. 10

Two-dimensional trade spaces for the subset of objective pairs with tradeoff for the modified Van Veldhuizen's benchmark problem

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Fig. 11

Parallel coordinate plot with the resulting solutions from the weighting objectives approach

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Fig. 12

Down-selection of solutions from 2D Pareto fronts for the Van Veldhuizen's problem

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Fig. 13

Down-selection of solutions from specified areas on two-dimensional Pareto fronts

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Fig. 14

Parallel coordinate plot with the resulting solutions selected from specified areas using the RA

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Fig. 15

The Parallel coordinate plots for the trade space of the GAA problem

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Fig. 16

Parallel coordinate plot with the resulting solutions selected from various techniques for the GAA problem

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