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

  • R. Timothy MarlerEmail author
  • Jasbir S. Arora
Research Paper

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.

Keywords

Multi-objective optimization Weighted sum Preferences 

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References

  1. Athan TW, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176CrossRefGoogle Scholar
  2. Chen W, Wiecek MM, Zhang J (1999) Quality utility—a compromise programming approach to robust design. J Mech Des 121:179–187CrossRefGoogle Scholar
  3. 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 14:63–69CrossRefGoogle Scholar
  4. Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8:631–657zbMATHCrossRefMathSciNetGoogle Scholar
  5. Eckenrode RT (1965) Weighting multiple criteria. Manage Sci 12:180–192CrossRefGoogle Scholar
  6. Gembicki FW (1974) Performance and sensitivity optimization: a vector index approach. PhD dissertation, Case Western Reserve University, Cleveland, OHGoogle Scholar
  7. 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 87–89Google Scholar
  8. Geoffrion AM (1968) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22:618–630zbMATHCrossRefMathSciNetGoogle Scholar
  9. Goicoechea A, Hansen DR, Duckstein L (1982) Multiobjective decision analysis with engineering and business applications. Wiley, New YorkGoogle Scholar
  10. Hobbs BF (1980) A comparison of weighting methods in power plant siting. Decis Sci 11:725–737CrossRefGoogle Scholar
  11. Holtzman JM, Halkin H (1966) Directional convexity and the maximum principle for discrete systems. SIAM J Control 4:263–275zbMATHCrossRefMathSciNetGoogle Scholar
  12. Huang C-H, Galuski J, Bloebaum CL (2007) Multi-objective Pareto concurrent subspace optimization for multidisciplinary design. AIAA J 45:1894–1906CrossRefGoogle Scholar
  13. 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, BerlinGoogle Scholar
  14. 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 731–737Google Scholar
  15. Koski J (1985) Defectiveness of weighting method in multicriterion optimization of structures. Commun Appl Numer Methods 1:333–337zbMATHCrossRefGoogle Scholar
  16. Koski J, Silvennoinen R (1987) Norm methods and partial weighting in multicriterion optimization of structures. Int J Numer Methods Eng 24:1101–1121zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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 YorkGoogle Scholar
  18. Marler RT (2009) A study of multi-objective optimization methods for engineering applications. VDM, SaarbruckenGoogle Scholar
  19. Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscipl Optim 26:369–395CrossRefMathSciNetGoogle Scholar
  20. Marler RT, Arora JS (2005) Transformation methods for multi-objective optimization. Eng Optim 37:551–569CrossRefMathSciNetGoogle Scholar
  21. Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Eng Optim 3:431–450zbMATHCrossRefGoogle Scholar
  22. Messac A, Sukam CP, Melachrinoudis E (2000a) Aggregate objective functions and Pareto frontiers: required relationships and practical implications. Optim Eng 1:171–188zbMATHCrossRefMathSciNetGoogle Scholar
  23. Messac A, Sundararaj GJ, Tappeta RV, Renaud JE (2000b) Ability of objective functions to generate points on nonconvex Pareto frontiers. AIAA J 38(6):1084–1091CrossRefGoogle Scholar
  24. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscipl Optim 25:86–98CrossRefMathSciNetGoogle Scholar
  25. Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic, BostonzbMATHGoogle Scholar
  26. Proos KA, Steven GP, Querin OM, Xie YM (2001) Multicriterion evolutionary structural optimization using the weighted and the global criterion methods. AIAA J 39:2006–2012CrossRefGoogle Scholar
  27. Rao JR, Roy N (1989) Fuzzy set theoretic approach of assigning weights to objectives in multicriteria decision making. Int J Syst Sci 20:1381–1386zbMATHCrossRefMathSciNetGoogle Scholar
  28. Saaty TL (1977) A scaling method for priorities in hierarchies, multiple objectives and fuzzy sets. J Math Psychol 15:234–281zbMATHCrossRefMathSciNetGoogle Scholar
  29. Saaty TL (2003) Decision-making with the AHP: why is the principal eigenvalue necessary. Eur J Oper Res 145:85–91zbMATHCrossRefMathSciNetGoogle Scholar
  30. Saaty TL, Hu G (1998) Ranking by eigenvector versus other methods in the analytic hierarchy process. Appl Math Lett 11:121–125zbMATHCrossRefMathSciNetGoogle Scholar
  31. 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 33:883–894zbMATHCrossRefMathSciNetGoogle Scholar
  32. Stadler W (1995) Caveats and boons of multicriteria optimization. Microcomput Civ Eng 10:291–299CrossRefGoogle Scholar
  33. 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, DCGoogle Scholar
  34. Steuer RE (1989) Multiple criteria optimization: theory, computation, and application. Krieger, MalabarzbMATHGoogle Scholar
  35. Tappeta RV, Renaud JE, Messac E, Sundararaj GJ (2000) Interactive physical programming: tradeoff analysis and decision making in multicriteria optimization. AIAA J 38:917–926CrossRefGoogle Scholar
  36. Voogd H (1983) Multicriteria evaluation for urban and regional planning. Pion, LondonGoogle Scholar
  37. Yoon KP, Hwang C-L (1995) Multiple attribute decision making, an introduction. Sage, LondonGoogle Scholar
  38. Zadeh LA (1963) Optimality and non-scalar-valued performance criteria. IEEE Trans Automat Contr AC-8:59–60CrossRefGoogle Scholar
  39. 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 46:2611–2622CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Center for Computer Aided Design, College of Engineering; 111 ERFThe University of IowaIowa CityUSA
  2. 2.Center for Computer Aided Design, College of Engineering; 238 ERFThe University of IowaIowa CityUSA

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