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Research Papers: Design Automation

Beyond Mean–Variance: The Mean–Gini Approach to Optimization Under Uncertainty

[+] Author and Article Information
Mengyu Wang

Department of Aerospace Engineering,
Iowa State University,
Ames, IA 50011-2271
e-mail: mengyuw@iastate.edu

Hanumanthrao Kannan

Department of Aerospace Engineering,
Iowa State University,
1620F Howe Hall,
Ames, IA 50011-2271
e-mail: hkannan@iastate.edu

Christina Bloebaum

Mem. ASME
Department of Aerospace Engineering,
Iowa State University,
1620F Howe Hall,
Ames, IA 50011-2271
e-mail: bloebaum@iastate.edu

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 6, 2017; final manuscript received October 30, 2017; published online December 21, 2017. Assoc. Editor: Harrison M. Kim.

J. Mech. Des 140(3), 031401 (Dec 21, 2017) (11 pages) Paper No: MD-17-1459; doi: 10.1115/1.4038566 History: Received July 06, 2017; Revised October 30, 2017

In probabilistic approaches to engineering design, including robust design, mean and variance are commonly used as the optimization objectives. This method, however, has significant limitations. For one, some mean–variance Pareto efficient designs may be stochastically dominated and should not be considered. Stochastic dominance is a mathematically rigorous concept commonly used in risk and decision analysis, based on the cumulative distribution function (CDFs), which establishes that one uncertain prospect is superior to another, while requiring minimal assumptions about the utility function of the outcome. This property makes it applicable to a wide range of engineering problems that ordinarily do not utilize techniques from normative decision analysis. In this work, we present a method to perform optimizations consistent with stochastic dominance: the Mean–Gini method. In macroeconomics, the Gini Index is the de facto metric for economic inequality, but statisticians have also proven a variant of it can be used to establish two conditions that are necessary and sufficient for both first and second-order stochastic dominance . These conditions can be used to reduce the Pareto frontier, eliminating stochastically dominated options. Remarkably, one of the conditions combines both mean and Gini, allowing for both expected outcome and uncertainty to be expressed in a single objective which, when maximized, produces a result that is not stochastically dominated given the Pareto front meets a convexity condition. We also find that, in a multi-objective optimization, the Mean–Gini optimization converges slightly faster than the mean–variance optimization.

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Figures

Grahic Jump Location
Fig. 2

First-order stochastic dominance demonstration

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Fig. 1

Different PDFs with same mean and variance

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Fig. 4

Pareto front from 100 generations of optimization a fit function of the Pareto front and the derivative of the fit function

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Fig. 5

Stochastic dominance test

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Fig. 6

(a) CDFs of designs from the dominated portion of the Pareto front. The corresponding designs on the Pareto front are emphasized on the right. (b) CDFs of SD designs on the Pareto Front. The corresponding designs on the Pareto front are emphasized on the right.

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Fig. 3

Design structure matrix of the satellite test problem

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Fig. 7

S-Indicator values across 100 generations from 15 pairs of optimizations

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Fig. 8

Pareto fronts at various generations

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Fig. 9

Pareto front from the multi-objective optimization along with final results from single-objective optimizations with the ϒ and weighted sums formulations

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