Skip to main content
Top
Published in: Structural and Multidisciplinary Optimization 3/2015

01-03-2015 | RESEARCH PAPER

On the analytical derivation of the Pareto-optimal set with applications to structural design

Authors: M. Gobbi, F. Levi, G. Mastinu, G. Previati

Published in: Structural and Multidisciplinary Optimization | Issue 3/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The Fritz John conditions for Pareto-optimality have been set in matrix form and used for introducing a procedure for the analytical derivation of the Pareto-optimal set in the design variables domain. Subsequently, the derivation of the Pareto-optimal set in the objective functions domain can be obtained, if possible, by a proper analytical derivation. Both the objective and constraint functions are assumed to be available in analytical form and twice differentiable and convex (or pseudo-convex). The proposed procedure to find the Pareto-optimal set is relatively simple. The computation of the determinant of a matrix is required. A symbolic manipulator can be exploited. If there are two design variables and two objective functions, the Pareto-optimal set can be easily computed by applying a simple formula derived in the paper. If the number of design variables equals the number of objective functions, the Pareto-optimal set in the design variables domain can be found by computing the product of the constraint functions times the determinant of the Jacobian of the objective functions. A number of case studies have been proposed to test the effectiveness of the proposed procedure. The optimal structural design of, respectively, a pair of compressed spheres, a cantilever with rectangular cross section have been faced and solved. Additionally the test problem proposed by Fonseca and Fleming has been addressed and solved analytically. Optimization problems with low dimensionality (2 or 3 design variables and 2 objective functions, 2 or more constraints) have been easily solved. The proposed procedure can be useful in the actual engineering practice at the earliest design stage. In this case the designer can be made aware on the proper design variables setting to obtain the desired tradeoff among conflicting objective functions.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Footnotes
1
b and h may vary, respectively, within two well defined ranges
 
Literature
go back to reference Aghezzaf B, Hachimi M (1999) Second-order optimality conditions in multi-objective optimization problems. J Optim Theory Appl 102(1):37–50CrossRefMATHMathSciNet Aghezzaf B, Hachimi M (1999) Second-order optimality conditions in multi-objective optimization problems. J Optim Theory Appl 102(1):37–50CrossRefMATHMathSciNet
go back to reference Anton H, Rorres C (2005) Elementary linear algebra, 9th edn. Wiley. (ISBN 978-0-471-66959-3) Anton H, Rorres C (2005) Elementary linear algebra, 9th edn. Wiley. (ISBN 978-0-471-66959-3)
go back to reference Arora JS (2004) Introduction to optimal design. Elsevier Academic Press, UK Arora JS (2004) Introduction to optimal design. Elsevier Academic Press, UK
go back to reference Askar SS, Tiwari A (2009) Finding exact solutions for multi-objective optimization problems using a symbolic algorithm. In: Proc IEEE Congress Evol Comput, pp 24–30 Askar SS, Tiwari A (2009) Finding exact solutions for multi-objective optimization problems using a symbolic algorithm. In: Proc IEEE Congress Evol Comput, pp 24–30
go back to reference Benoist J (1998) Connectedness of the efficient solutions for vector maximization problems. J Optim Theor Appl 96(3) Benoist J (1998) Connectedness of the efficient solutions for vector maximization problems. J Optim Theor Appl 96(3)
go back to reference Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer Verlag, Berlin Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer Verlag, Berlin
go back to reference Bjorck A (1996) Numerical methods for least squares problems. Society for industrial and applied mathematics. Philadelphia Bjorck A (1996) Numerical methods for least squares problems. Society for industrial and applied mathematics. Philadelphia
go back to reference Chawla N, Deng X (2005) Microstructure and mechanical behavior of porous sintered steels. Mater Sci Eng A 390:98–112CrossRef Chawla N, Deng X (2005) Microstructure and mechanical behavior of porous sintered steels. Mater Sci Eng A 390:98–112CrossRef
go back to reference Ehrgott M, Klamroth K (1997) Connectedness of the efficient solutions in multiple criteria combinatorial optimization. Eur J Oper Res 97(1) Ehrgott M, Klamroth K (1997) Connectedness of the efficient solutions in multiple criteria combinatorial optimization. Eur J Oper Res 97(1)
go back to reference Erfani T, Utyuzhnikov SV, Kolo B (2013) A modified directed search domain algorithm for multi-objective engineering and design optimization. Struct Multidiscip Optim:1–13 Erfani T, Utyuzhnikov SV, Kolo B (2013) A modified directed search domain algorithm for multi-objective engineering and design optimization. Struct Multidiscip Optim:1–13
go back to reference Fonseca CM, Fleming PJ (1998) Multi-objective optimization and multiple constraint handling with evolutionary algorithms Part II: application example. IEEE Trans Syst Man Cybern Fonseca CM, Fleming PJ (1998) Multi-objective optimization and multiple constraint handling with evolutionary algorithms Part II: application example. IEEE Trans Syst Man Cybern
go back to reference Gobbi M, Levi F, Mastinu G (2006) Multi-objective stochastic optimisation of the suspension system of road vehicles. J Sound Vib 298(4–5):1005–1072 Gobbi M, Levi F, Mastinu G (2006) Multi-objective stochastic optimisation of the suspension system of road vehicles. J Sound Vib 298(4–5):1005–1072
go back to reference Gobbi M, Levi F, Mastinu G (2005) An application of multi-objective stochastic optimisation to structural design. Struct Multidiscip Optim 29(4):272–284CrossRef Gobbi M, Levi F, Mastinu G (2005) An application of multi-objective stochastic optimisation to structural design. Struct Multidiscip Optim 29(4):272–284CrossRef
go back to reference Gobbi M, Mastinu G (2001) On the optimal design of composite material tubular helical springs. Meccanica 36(5):525–553CrossRefMATH Gobbi M, Mastinu G (2001) On the optimal design of composite material tubular helical springs. Meccanica 36(5):525–553CrossRefMATH
go back to reference Haimes Y, Lasdon LS, Wismer DA (1971) On a Bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1(3):296–297CrossRefMATHMathSciNet Haimes Y, Lasdon LS, Wismer DA (1971) On a Bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1(3):296–297CrossRefMATHMathSciNet
go back to reference Kim DS, et al. (2001) Counterexample and optimality conditions in differentiable multi-objective programming. J Optim Theory Appl 109(1):187–192CrossRefMATHMathSciNet Kim DS, et al. (2001) Counterexample and optimality conditions in differentiable multi-objective programming. J Optim Theory Appl 109(1):187–192CrossRefMATHMathSciNet
go back to reference Levi F, Gobbi M (2006) An application of analytical multi-objective optimization to truss structures., (AIAA 2006-6975). In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference. Portsmouth Levi F, Gobbi M (2006) An application of analytical multi-objective optimization to truss structures., (AIAA 2006-6975). In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference. Portsmouth
go back to reference Marusciac I (1982) On the Fritz John type optimality criterion in multi-objective goal programming. Revue d’Analyse Numerique et de Theorie de l’Approximation 11 No. 1–2 Marusciac I (1982) On the Fritz John type optimality criterion in multi-objective goal programming. Revue d’Analyse Numerique et de Theorie de l’Approximation 11 No. 1–2
go back to reference Mastinu G, Gobbi M, Miano C (2006) Optimal design of complex mechanical systems: with applications to vehicle engineering. Springer-Verlag. (ISBN-13:978-3540343547) Mastinu G, Gobbi M, Miano C (2006) Optimal design of complex mechanical systems: with applications to vehicle engineering. Springer-Verlag. (ISBN-13:978-3540343547)
go back to reference Meyer CD (2001) Matrix analysis and applied linear algebra. Society for industrial and applied mathematics (SIAM). Aurora Meyer CD (2001) Matrix analysis and applied linear algebra. Society for industrial and applied mathematics (SIAM). Aurora
go back to reference Miettinen K (1999) Nonlinear multi-objective optimization. Kluwer Academic Publishers, Boston Miettinen K (1999) Nonlinear multi-objective optimization. Kluwer Academic Publishers, Boston
go back to reference Papalambros PY, Wilde DJ (2000) Principles of optimal design. Modeling and computation. Cambridge University Press, CambridgeCrossRefMATH Papalambros PY, Wilde DJ (2000) Principles of optimal design. Modeling and computation. Cambridge University Press, CambridgeCrossRefMATH
go back to reference Pedersen P, Pedersen NL (2009) Analytical optimal designs for long and short statically determinate beam structures. Struct Multidiscip Optim 39(4):343–357CrossRefMathSciNet Pedersen P, Pedersen NL (2009) Analytical optimal designs for long and short statically determinate beam structures. Struct Multidiscip Optim 39(4):343–357CrossRefMathSciNet
go back to reference Siddiqui S, Azarm S, Gabriel SA (2012) On improving normal boundary intersection method for generation of Pareto frontier. Struct Multidiscip Optim 46(6):839–852CrossRefMATHMathSciNet Siddiqui S, Azarm S, Gabriel SA (2012) On improving normal boundary intersection method for generation of Pareto frontier. Struct Multidiscip Optim 46(6):839–852CrossRefMATHMathSciNet
go back to reference Simon CP, Blume LE (1994) Mathematics for economists. W. W. Norton and Company Simon CP, Blume LE (1994) Mathematics for economists. W. W. Norton and Company
go back to reference Takayama A (1993) Analytical methods in economics. University of Michigan Press Takayama A (1993) Analytical methods in economics. University of Michigan Press
go back to reference Yang XQ, Jeyakumar V (1997) First and second-order optimality conditions for convex composite multi-objective optimization. J Optim Theory Appl 95(1):209–224CrossRefMATHMathSciNet Yang XQ, Jeyakumar V (1997) First and second-order optimality conditions for convex composite multi-objective optimization. J Optim Theory Appl 95(1):209–224CrossRefMATHMathSciNet
go back to reference Young WC, Budynas RG (2002) Roark’s formulas for stress and strain. Mc Graw Hill, New York Young WC, Budynas RG (2002) Roark’s formulas for stress and strain. Mc Graw Hill, New York
go back to reference Zhang WH, Yang HC (2002) Efficient gradient calculation of the Pareto optimal curve in multicriteria optimisation. Struct Multidiscip Optim 23:311–319CrossRefMATH Zhang WH, Yang HC (2002) Efficient gradient calculation of the Pareto optimal curve in multicriteria optimisation. Struct Multidiscip Optim 23:311–319CrossRefMATH
Metadata
Title
On the analytical derivation of the Pareto-optimal set with applications to structural design
Authors
M. Gobbi
F. Levi
G. Mastinu
G. Previati
Publication date
01-03-2015
Publisher
Springer Berlin Heidelberg
Published in
Structural and Multidisciplinary Optimization / Issue 3/2015
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI
https://doi.org/10.1007/s00158-014-1152-5

Other articles of this Issue 3/2015

Structural and Multidisciplinary Optimization 3/2015 Go to the issue

Premium Partners