Skip to main content
SpringerLink
Log in

Considering dynamic Pareto frontiers in decision making

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

Considering how the resolution of conflicts changes over time is an aspect of multiobjective optimization that is not commonly explored. These considerations embody changes in both the preferences that dictate the selection of Pareto designs, and changes in the Pareto frontier itself over time, or s-Pareto frontier when a set of disparate design concepts are considered. As such, this paper explores the idea of dynamic s-Pareto frontiers and preferences. Specifically, this paper presents a dynamic multiobjective optimization problem formulation that provides a framework of identifying the s-Pareto frontier for a series of time steps. The application of the presented dynamic formulation is illustrated through a simple aircraft design example. Through this example it was observed that the identification of the dynamic s-Pareto frontier enabled the observation of the impact of design decisions on the offset of selected designs from the identified dynamic frontier. By measuring and minimizing the aircraft design offset, the selected aircraft design offset was improved by an average of roughly 60 % from the next best selected alternative identified using traditional selection methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Abbreviations

μ :

Vector of design objectives

x :

Vector of design variables/objects

y :

Vector of independent design objects

z :

Vector of dependent design objects

n [ ] :

indicates the number of [ ]

[ ] l :

indicates the lower limit of [ ]

[ ] u :

indicates the upper limit of [ ]

[ ]∗ :

indicates the optimal value of [ ]

References

  • Blackwell T, Branke J (2004) Multi-swarm optimization in dynamic environments. Lecture Notes in Computer Science, vol 3005. pp 489–500. doi:10.1007/978-3-540-24653-4_50

    Google Scholar 

  • Bowman MW (1999) Lockheed C-130 Hercules Crowood Press, Ramsbury, pp 4–54

    Google Scholar 

  • Cheng FY, Li D (1996) Multiobjective optimization of structures with and without control. J Guid Control Dyn 19(2):392–397

    Article  MATH  Google Scholar 

  • Curtis SK, Mattson CA, Hancock BJ, Lewis PK (2013) Divergent exploration in design with a dynamic multiobjective optimization formulation. Struct Multidiscip Optim 47(5):645–657

    Article  MathSciNet  MATH  Google Scholar 

  • Faulkenberg SL, Wiecek MM (2010) On the quality of discrete representations in multiple objective programming. Optim Eng 11:423–440. doi:10.1007/s11081-009-9099-x

    Article  MathSciNet  MATH  Google Scholar 

  • Gardenghi M, Wiecek MM (2011) Efficiency for multiobjective multidisciplinary optimization problems with quasiseparable subproblems. Optim Eng, 293–318. doi:10.1007/s11081-011-9136-4

  • Hatzakis I, Wallace D (2006) Dynamic multi-objective optimization with evolutionary algorithms: a forward-looking approach. In: Proceedings of the 8th annual conference on Genetic and evolutionary computation, GECCO ’06, New York, NY, USA, ACM, New York, pp 1201–1208. ISBN:1-59593-186-4. 10.1145/1143997.1144187

    Chapter  Google Scholar 

  • Heintz C (2002) Aircraft design made easy. EAA Exp. http://www.zenithair.com/images/kit-data/ht.html

  • Kasprzak EM, Lewis KE (2000) An approach to facilitate decision tradeoffs in Pareto solution sets. J Eng Valuat Cost Anal 3:173–187

    Google Scholar 

  • Khire RA, Messac A (2008) Selection-integrated optimization (SIO) methodology for optimal design of adaptive systems. J Mech Des 130(10):101401

    Article  Google Scholar 

  • Lewis PK (2012) An optimization-based method of traversing dynamic s-Pareto frontiers. PhD thesis, Brigham Young University

  • Lewis PK, Mattson CA (2012) A method for developing systems that traverse the Pareto frontiers of multiple system concepts through modularity. Struct Multidiscip Optim 45(4):467–478. doi:10.1007/s00158-010-055501

    Article  Google Scholar 

  • Lewis PK, Murray VR, Mattson CA (2011) A design optimization strategy for creating devices that traverse the Pareto frontier over time. Struct Multidiscip Optim 43(2):191–204. doi:10.1007/s00158-010-0555-1

    Article  MathSciNet  MATH  Google Scholar 

  • Messac A (2000) From dubious construction of objective functions to the application of physical programming. AIAA J 38(1):155–163

    Article  Google Scholar 

  • Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. In: Optimization and engineering, vol 3. Kluwer Academic, Norwell, pp 431–450

    Google Scholar 

  • Messac A, Puemi-Sukam C, Melachrinoudis E (2000) Aggregate objective functions and Pareto frontiers: required relationships and practical implications. Optim Eng 1:171–188

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen KM (1999) Nonlinear multiobjective optimization. International series in operations research & management science. Kluwer Academic, Norwell, pp 5–36

    MATH  Google Scholar 

  • Nigam N, Kroo I (2008) Control and design of multiple unmanned air vehicles for a persistent surveillance task. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria, British Columbia, Sep. 10–12, pp 1–14. Number AIAA-2008-5913

    Google Scholar 

  • Olewnik A, Brauen T, Ferguson S, Lewis K (2004) A framework for flexible systems and its implementation in multiattribute decision making. J Mech Des 126(3):412–419

    Article  Google Scholar 

  • Siddiqi A, de Weck OL (2008) Modeling methods and conceptual design principles for reconfigurable systems. J Mech Des 130:101102

    Article  Google Scholar 

  • Simpson TW (1998) Product platform design and optimization: status and promise. Artif Intell Eng Des Anal Manuf 18(1):3–20

    Google Scholar 

  • Smith PC (2001) Lockheed C-130 Hercules: the World’s favourite military transport. Crowood, Wiltshire, pp 12–140

    Google Scholar 

  • Trautmann H, Mehnen J (2009) Preference-based Pareto optimization in certain and noisy environments. Eng Optim 41(1):23–38. doi:10.1080/03052150802347926

    Article  MathSciNet  Google Scholar 

  • Yang TG, Beiter KA, Ishii K (2004) Product platform development: an approach for products in the conceptual stages of design. In: 2004 ASME international mechanical engineering congress and RDD expo, pp 97–108. Number IMECE2004-62171

    Google Scholar 

  • Yun Y, Yoon M, Nakayama H (2009) Multi-objective optimization based on meta-modeling by using support vector regression. Optim Eng 10:167–181. doi:10.1007/s11081-008-9063-1

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Funding for this research was provided by the National Science Foundation Grant 0954580.

Author information

Authors and Affiliations

  1. Dept. of Mechanical Engineering, Brigham Young University, Provo, UT, 84602, USA

    Patrick K. Lewis, Morgan W. P. Tackett & Christopher A. Mattson

Authors
  1. Patrick K. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morgan W. P. Tackett
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christopher A. Mattson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher A. Mattson.

Appendix A: Dynamic s-Pareto formulation inputs

Appendix A: Dynamic s-Pareto formulation inputs

For the aircraft example given in this paper there are three concepts. The variable values for these different concepts are presented in Tables 6, 7 and 8, and were selected using data provided in Heintz (2002) for various aircraft and wing configurations. The units and descriptions of these variables are given in Table 4. The rows of the tables at each time-step are the values of the diagonal matrix (\(w^{(k^{(t)})}\)) and design object limits (\(x^{(k^{(t)})}_{l}\), \(x^{(k^{(t)})}_{u}\)) for the corresponding concept. For example, the rows corresponding to χ i,i , x l , and x u for Concept 1 at t=1 represent the \(\chi^{(1^{(1)})}_{i,i}\) values of Eq. (5), \(x^{(1^{(1)})}_{l}\) values of Eq. (3), and \(x^{(1^{(1)})}_{u}\) values of Eq. (3), respectively.

Table 6 Values of the design object identifiers and limits for aircraft Concept 1 (k=1)
Full size table
Table 7 Values of the design object identifiers and limits for aircraft Concept 2 (k=2)
Full size table
Table 8 Values of the design object identifiers and limits for aircraft Concept 3 (k=3)
Full size table

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lewis, P.K., Tackett, M.W.P. & Mattson, C.A. Considering dynamic Pareto frontiers in decision making. Optim Eng 15, 837–854 (2014). https://doi.org/10.1007/s11081-013-9238-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-013-9238-2

Keywords

Search

Navigation