Research Papers

Limitations of Pareto Front in Design Under Uncertainty and Their Reconciliation

[+] Author and Article Information
Vijitashwa Pandey

e-mail: pandey2@oakland.edu

Zissimos P. Mourelatos

e-mail: mourelat@oakland.edu
Department of Mechanical Engineering,
Oakland University,
2200 N Squirrel Road,
Rochester, MI 48309

Efstratios Nikolaidis

Department of Mechanical Industrial and Manufacturing Engineering,
University of Toledo,
2801 Bancroft Street,
Toledo, OH 43606
e-mail: enikolai@eng.utoledo.edu

The functions need not be monotonic; only surjections (onto) suffice provided the preference orders (e.g., less is better) over them are uniquely defined.

The attributes must be preferentially independent in order to use them in plotting a Pareto front. Preferential independence means that the preference order (e.g., less is better) over one attribute does not depend on the value of the other attributes. If this is not true, we cannot plot a Pareto front because the utopia point will change whenever the preference order changes. If we use functions of the attributes, they must also be preferentially independent.

Notice that G and H are the same if V1 = f1 and V2 = f2.

Note that for multi-attribute problems, x* is not a single vector but a collection of vectors represented by surfaces with respect to all attributes. Each vector in this collection has utility equal to that of x that maximizes U.

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 25, 2011; final manuscript received April 6, 2013; published online May 29, 2013. Assoc. Editor: Wei Chen.

J. Mech. Des 135(7), 071010 (May 29, 2013) (12 pages) Paper No: MD-11-1286; doi: 10.1115/1.4024224 History: Received June 25, 2011; Revised April 06, 2013

Engineering design reconciles design constraints with decision maker (DM) preferences. The task of eliciting and encoding decision maker preferences is, however, extremely difficult. A Pareto front representing the locus of the nondominated designs is, therefore, often generated to help a decision maker select the best design. In this paper, we show that this method has a shortcoming when there is uncertainty in both the decision problem variables and in the model of decision maker's preferences. In this case, the Pareto front is inconsistent with multi-attribute utility (MAU) theory, unless the decision maker trades off attributes or some functions of them linearly. This is a strong restriction. To account for this, we propose a methodology that enables a decision maker to select the best design on a modified pareto front (MPF) which is acquired using envelopes of a set of certainty equivalent (CE) surfaces. The proposed method does not require separability of the multi-attribute utility function into single-attribute utilities, nor does it require the decision maker to trade the attributes (or any function of them) linearly. We demonstrate our approach on a simple optimization problem and in design of a reduction gear.

Copyright © 2013 by ASME
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Fig. 1

Decision making with different combinations of the two types of uncertainties

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Fig. 2

Isopreference curves over two attributes

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Fig. 3

Pareto front over two attributes

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Fig. 4

Pareto front over individual value functions. Negative signs indicate a minimization problem

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Fig. 5

Process to select (a) the certainty equivalent curve or envelope for a given solution among all candidate utility functions and (b) the best solution from a given set of certainty equivalent envelopes for an attribute minimization decision problem

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Fig. 6

Inverting a utility function in a multi-attribute problem results in a CE curve

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Fig. 7

Performance of the solutions against all utility functions. For simplicity, only three curves in two dimensions are shown for each solution

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Fig. 8

Best valuations (left envelope) of the CE envelopes of Fig. 7

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Fig. 9

Modified Pareto front and possible selections by the DM

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Fig. 10

Contour plots of the three candidate utility functions

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Fig. 11

(a) Certainty equivalent curves for solution μ1 for the three utility functions and the corresponding conservative envelope. (b) Certainty equivalent curves for solution μ2 for the three utility functions and the corresponding conservative envelope. (c) Certainty equivalent curves for solution μ3 for the three utility functions and the corresponding conservative envelope.

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Fig. 12

Left envelope (black solid line) of conservative certainty equivalent curves from Figs. 11(a)11(c). The curves marked X_1, X_2 and X_3 indicate the conservative envelopes of the three designs, respectively.

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Fig. 13

Modified Pareto front and attribute realizations for the three solutions

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Fig. 14

Modified Pareto front and its relative position with respect to individual certainty equivalent curves

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Fig. 15

Modified Pareto front defined by the right envelope of certainty equivalent surfaces

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Fig. 16

A simple speed reducer (adapted from Ref. [4])




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