Skip to main content
SpringerLink
Log in

The weighted sum method for multi-objective optimization: new insights

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

As a common concept in multi-objective optimization, minimizing a weighted sum constitutes an independent method as well as a component of other methods. Consequently, insight into characteristics of the weighted sum method has far reaching implications. However, despite the many published applications for this method and the literature addressing its pitfalls with respect to depicting the Pareto optimal set, there is little comprehensive discussion concerning the conceptual significance of the weights and techniques for maximizing the effectiveness of the method with respect to a priori articulation of preferences. Thus, in this paper, we investigate the fundamental significance of the weights in terms of preferences, the Pareto optimal set, and objective-function values. We determine the factors that dictate which solution point results from a particular set of weights. Fundamental deficiencies are identified in terms of a priori articulation of preferences, and guidelines are provided to help avoid blind use of the method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Athan TW, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176

    Article  Google Scholar 

  • Chen W, Wiecek MM, Zhang J (1999) Quality utility—a compromise programming approach to robust design. J Mech Des 121:179–187

    Article  Google Scholar 

  • Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69

    Article  Google Scholar 

  • Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8:631–657

    Article  MATH  MathSciNet  Google Scholar 

  • Eckenrode RT (1965) Weighting multiple criteria. Manage Sci 12:180–192

    Article  Google Scholar 

  • Gembicki FW (1974) Performance and sensitivity optimization: a vector index approach. PhD dissertation, Case Western Reserve University, Cleveland, OH

  • Gennert MA, Yuille AL (1988) Determining the optimal weights in multiple objective function optimization. In: Second international conference on computer vision (held in Los Alamos, CA), Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp 87–89

  • Geoffrion AM (1968) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22:618–630

    Article  MATH  MathSciNet  Google Scholar 

  • Goicoechea A, Hansen DR, Duckstein L (1982) Multiobjective decision analysis with engineering and business applications. Wiley, New York

    Google Scholar 

  • Hobbs BF (1980) A comparison of weighting methods in power plant siting. Decis Sci 11:725–737

    Article  Google Scholar 

  • Holtzman JM, Halkin H (1966) Directional convexity and the maximum principle for discrete systems. SIAM J Control 4:263–275

    Article  MATH  MathSciNet  Google Scholar 

  • Huang C-H, Galuski J, Bloebaum CL (2007) Multi-objective Pareto concurrent subspace optimization for multidisciplinary design. AIAA J 45:1894–1906

    Article  Google Scholar 

  • Hwang C-L, Yoon K (1981) Multiple attribute decision making, methods and applications: a state-of-the-art survey. In: Beckmann M, Kunzi HP (eds) Lecture notes in economics and mathematical systems, no 186. Springer, Berlin

    Google Scholar 

  • Kassaimah SA, Mohamed AM, Kolkailah FA (1995) Bi-criteria optimum design of laminated plates under uniform load and shear. In: Proceedings of the 27th international SAMPLE technical conference (held in Albuquerque, NM), 27, pp 731–737

  • Koski J (1985) Defectiveness of weighting method in multicriterion optimization of structures. Commun Appl Numer Methods 1:333–337

    Article  MATH  Google Scholar 

  • Koski J, Silvennoinen R (1987) Norm methods and partial weighting in multicriterion optimization of structures. Int J Numer Methods Eng 24:1101–1121

    Article  MATH  MathSciNet  Google Scholar 

  • Lin JG (1975) Three methods for determining Pareto-optimal solutions of multiple-objective problems. In: Ho YC, Mitter SK (eds) Directions in large-scale systems. Plenum, New York

    Google Scholar 

  • Marler RT (2009) A study of multi-objective optimization methods for engineering applications. VDM, Saarbrucken

    Google Scholar 

  • Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscipl Optim 26:369–395

    Article  MathSciNet  Google Scholar 

  • Marler RT, Arora JS (2005) Transformation methods for multi-objective optimization. Eng Optim 37:551–569

    Article  MathSciNet  Google Scholar 

  • Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Eng Optim 3:431–450

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Messac A, Sundararaj GJ, Tappeta RV, Renaud JE (2000b) Ability of objective functions to generate points on nonconvex Pareto frontiers. AIAA J 38(6):1084–1091

    Article  Google Scholar 

  • Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscipl Optim 25:86–98

    Article  MathSciNet  Google Scholar 

  • Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic, Boston

    MATH  Google Scholar 

  • Proos KA, Steven GP, Querin OM, Xie YM (2001) Multicriterion evolutionary structural optimization using the weighted and the global criterion methods. AIAA J 39:2006–2012

    Article  Google Scholar 

  • Rao JR, Roy N (1989) Fuzzy set theoretic approach of assigning weights to objectives in multicriteria decision making. Int J Syst Sci 20:1381–1386

    Article  MATH  MathSciNet  Google Scholar 

  • Saaty TL (1977) A scaling method for priorities in hierarchies, multiple objectives and fuzzy sets. J Math Psychol 15:234–281

    Article  MATH  MathSciNet  Google Scholar 

  • Saaty TL (2003) Decision-making with the AHP: why is the principal eigenvalue necessary. Eur J Oper Res 145:85–91

    Article  MATH  MathSciNet  Google Scholar 

  • Saaty TL, Hu G (1998) Ranking by eigenvector versus other methods in the analytic hierarchy process. Appl Math Lett 11:121–125

    Article  MATH  MathSciNet  Google Scholar 

  • Saramago SFP, Steffen V Jr (1998) Optimization of the trajectory planning of robot manipulators taking into account the dynamics of the system. Mech Mach Theory 33:883–894

    Article  MATH  MathSciNet  Google Scholar 

  • Stadler W (1995) Caveats and boons of multicriteria optimization. Microcomput Civ Eng 10:291–299

    Article  Google Scholar 

  • Stadler W, Dauer JP (1992) Multicriteria optimization in engineering: a tutorial and survey. In: Kamat MP (ed) Structural optimization: status and promise. American Institute of Aeronautics and Astronautics, Washington, DC

    Google Scholar 

  • Steuer RE (1989) Multiple criteria optimization: theory, computation, and application. Krieger, Malabar

    MATH  Google Scholar 

  • Tappeta RV, Renaud JE, Messac E, Sundararaj GJ (2000) Interactive physical programming: tradeoff analysis and decision making in multicriteria optimization. AIAA J 38:917–926

    Article  Google Scholar 

  • Voogd H (1983) Multicriteria evaluation for urban and regional planning. Pion, London

    Google Scholar 

  • Yoon KP, Hwang C-L (1995) Multiple attribute decision making, an introduction. Sage, London

    Google Scholar 

  • Zadeh LA (1963) Optimality and non-scalar-valued performance criteria. IEEE Trans Automat Contr AC-8:59–60

    Article  Google Scholar 

  • Zhang K-S, Han Z-H, Li W-J, Song W-P (2008) Bilevel adaptive weighted sum method for multidisciplinary multi-objective optimization. AIAA J 46:2611–2622

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Computer Aided Design, College of Engineering; 111 ERF, The University of Iowa, Iowa City, IA, 52242, USA

    R. Timothy Marler

  2. Center for Computer Aided Design, College of Engineering; 238 ERF, The University of Iowa, Iowa City, IA, 52242, USA

    Jasbir S. Arora

Authors
  1. R. Timothy Marler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jasbir S. Arora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Timothy Marler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marler, R.T., Arora, J.S. The weighted sum method for multi-objective optimization: new insights. Struct Multidisc Optim 41, 853–862 (2010). https://doi.org/10.1007/s00158-009-0460-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-009-0460-7

Keywords

Search

Navigation