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Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

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Abstract

This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the \( {m} \)-dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies.

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Abbreviations

J :

Objective function vector

x :

Design vector

p :

Vector of fixed parameters

g :

Inequality constraint vector

h :

Equality constraint vector

m :

Number of objectives

α i :

Weighting factor

\(\ifmmode\expandafter\bar\else\expandafter\=\fi{J}_{i} \) :

Normalized objective function

J Utopia :

Utopia point

J Nadir :

Nadir point

J i*:

ith anchor point

P j :

Position vector of the jth expected solution on the hyperplane

References

  • 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 Pareto optimal points in multicriteria optimization problems. SIAM J Optim 8:631–657

    Article  MathSciNet  MATH  Google Scholar 

  • Fonseca C, Fleming P (1995) An overview of evolutionary algorithms in multiobjective optimization. Evol Comput 3:1–18

    Google Scholar 

  • Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison Wesley, Reading

    MATH  Google Scholar 

  • Kim IY, de Weck OL (2005) Adaptive weighted sum method for bi-objective optimization: Pareto front generation. Struct Multidiscipl Optim 29:149–158

    Google Scholar 

  • 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 (1988) Multicriteria truss optimization. In: Stadler W (ed) Multicriteria optimization in engineering and in the sciences. Plenum, New York

    Google Scholar 

  • Lin J (1976) Multiple objective problems: Pareto-optimal solutions by method of proper equality constraints. IEEE Trans Automat Contr 21:641–650

    MATH  Google Scholar 

  • Marglin S (1967) Public investment criteria. MIT, Cambridge, MA

    Google Scholar 

  • Mattson CA, Messac A (2003) Concept selection using s-Pareto frontiers. AIAA J 41:1190–1204

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Messac A, Mattson CA (2004) Normal constraint method with guarantee of even representation of complete Pareto frontier. AIAA J 42:2101–2111

    Article  Google Scholar 

  • Stadler W (1979) A survey of multicriteria optimization, or the vector maximum problem. Jota 29:1–52

    Article  MATH  MathSciNet  Google Scholar 

  • Stadler W (1984) Applications of multicriteria optimization in engineering and the sciences (A Survey). In: Zeleny M (ed) Multiple criteria decision making—past decade and future trends. JAI, Greenwich

    Google Scholar 

  • Suppapitnarm A et al (1999) Design by multiobjective optimization using simulated annealing. International conference on engineering design ICED 99, Munich, Germany

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

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Engineering Systems Division, 33-410, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

    O. L. de Weck

  2. Department of Mechanical Engineering, Queen's University, Kingston, Ontario, K7L 3N6, Canada

    I. Y. Kim

Authors
  1. I. Y. Kim
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  2. O. L. de Weck
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Correspondence to O. L. de Weck.

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Kim, I.Y., de Weck, O.L. Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation. Struct Multidisc Optim 31, 105–116 (2006). https://doi.org/10.1007/s00158-005-0557-6

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  • DOI: https://doi.org/10.1007/s00158-005-0557-6

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