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
Your Session has timed out. Please sign back in to continue.


Howard, R., 1988, “Decision Analysis: Practice and Promise,” Manage. Sci., 34(6), pp. 679–695. [CrossRef]
Saaty, T. L., 1990, Multicriteria Decision Making: The Analytic Hierarchy Process: Planning, Priority Setting Resource Allocation, 2nd ed., RWS Publications, Pittsburgh, PA.
Savage, L. G., 1954, The Foundations of Statistics, John Wiley & Sons, New York.
von Neumann, J., and Morgenstern, O., 1947, Theory of Games and Economic Behavior, Princeton, Princeton, NJ.
Thurston, D. L., 1991, “A Formal Method for Subjective Design Evaluation With Multiple Attributes,” Res. Eng. Des., 3(2), pp. 105–122. [CrossRef]
Hazelrigg, G. A., 1998, “A Framework for Decision-Based Engineering Design,” ASME J. Mech. Des., 120(4), pp. 653–658. [CrossRef]
Gurnani, A., and Lewis, K., 2005, “Robust Multiattribute Decision Making Under Risk and Uncertainty in Engineering Design,” Eng. Optimiz., 37(8), pp. 813–830. [CrossRef]
Scott, M. J., and Antonsson, E. K., 1998 “Aggregation Functions for Engineering Design Trade-Offs,” Fuzzy Sets Syst., 99(3), pp. 253–264. [CrossRef]
Lewis, K., Chen, W., and Schmidt, L. C., eds., 2006, Decision Making in Engineering Design, ASME Press, New York.
Hula, A., Jalali, K., Hamza, K., Skerlos, S. J., and Saitou, K., 2003, “Multi-Criteria Decision-Making for Optimization of Product Disassembly Under Multiple Situations,” Environ. Sci. Technol., 37(23), pp. 5303–5313. [CrossRef] [PubMed]
Jolly, M., Winward, P., Pandey, V., and Thurston, D. L., 2006, “Mass Customization Optimization Using Social Network Based Product Architecture Representation,” Proceedings of ASME International Design Engineering Technical Conferences, Philadelphia, September 10–13.
Deb, K., and Jain, S., 2003, “Multi-Speed Gearbox Design Using Multi-Objective Evolutionary Algorithms,” ASME J. Mech. Des., 125(3), pp. 609–619. [CrossRef]
MacDonald, E. F., Gonzalez, R., and Papalambros, P. Y., 2009, “Preference Inconsistency in Multidisciplinary Design Decision Making,” ASME J. Mech. Des., 131(3), p. 031009. [CrossRef]
Train, K., 2003, Discrete Choice Analysis With Simulation, Cambridge University Press, London, UK, p. 152.
Louviere, J., Street, D., Carson, R., Ainslie, A., Deshazo, J. R., Cameron, T., Hensher, D., Kohn, R., and Marley, T., 2002, “Dissecting the Random Component of Utility,” Mark. Lett., 13(3), pp. 177–193. [CrossRef]
Busemeyer, J. R., and Townsend, J. T., 1993, “Decision Field Theory: A Dynamic Cognition Approach to Decision Making,” Psychol. Rev., 100(3), pp. 432–459. [CrossRef] [PubMed]
Srinivas, N., and Deb, K., 1995, “Multi-Objective Function Optimization Using Nondominated Sorting Genetic Algorithms,” Evol. Comput., 2(3), pp. 221–248. [CrossRef]
Keeney, R. L., and Raiffa, H., 1994, Decisions With Multiple Objectives, Cambridge University Press, Cambridge, United Kingdom.
Clemen, R. T., 1997, Making Hard Decisions – An Introduction to Decision Analysis, 2nd ed., Duxbury Press , CA.
Wassenaar, H. J., Chen, W., Cheng, J., and Sudjianto, A., 2005, “Enhancing Discrete Choice Demand Modeling for Decision-Based Design,” ASME J. Mech. Des., 127(4), pp. 514–524. [CrossRef]
Mark, T. L., and Swait, J., 2004. “Using Stated Preference and Revealed Preference Modeling to Evaluate Prescribing Decisions,” Health Econ., 13(6), pp. 563–573. [CrossRef] [PubMed]
Fitzsimmons, P. J., 2007, “Converse Jensen Inequality,” http://math.ucsd.edu/∼pfitz/downloads/conjen.pdf, accessed May 15, 2011.
Farhang-Mehr, A., and Azarm, S., 2003, “An Information-Theoretic Entropy Metric for Assessing Multi-Objective Optimization Solution Set Quality,” ASME J. Mech. Des., 125(4), pp. 655–663. [CrossRef]
Morris, P. A., 1977, “Combining Expert Judgments: A Bayesian Approach,” Manage. Sci., 23(7), pp. 679–693. [CrossRef]
Schervish, M. J., 1986, “Comments on Some Axioms for Combining Expert Judgments,” Manage. Sci., 32(3), pp. 306–312. [CrossRef]
Deb, K., Pratap, A., Agrawal, S., and Meyarivan, T., 2002, “A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II,” IEEE Transactions on Evolutionary Computation, 6(2), pp. 182–197. [CrossRef]


Grahic Jump Location
Fig. 1

Decision making with different combinations of the two types of uncertainties

Grahic Jump Location
Fig. 2

Isopreference curves over two attributes

Grahic Jump Location
Fig. 3

Pareto front over two attributes

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
Fig. 9

Modified Pareto front and possible selections by the DM

Grahic Jump Location
Fig. 10

Contour plots of the three candidate utility functions

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 13

Modified Pareto front and attribute realizations for the three solutions

Grahic Jump Location
Fig. 14

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

Grahic Jump Location
Fig. 15

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

Grahic Jump Location
Fig. 16

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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In