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Problems in Scalarizing Multicriteria Approaches

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Multiple Criteria Decision Making in the New Millennium

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 507))

Abstract

A general approach to different types of scalarization in multicriteria optimization is discussed. Several special results which can be deduced from this approach are proved. These results refer to efficient, weakly efficient and properly efficient solutions of the vector optimization problem, where proper efficiency is defined in the sense of Geoffrion and turns out to be essential in approximating the efficient point set. The statements do not require convexity.

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References

  1. Benson, H.P. (1979): An improved definition of proper efficiency for vector maximization with respect to cones. J.Math.Anal.Appl. 71, 232–241

    Article  Google Scholar 

  2. Bernau, H. (1987): Interactive methods for vector optimization. In: Brosowski, B., Martensen, E. (eds.) Optimization in mathematical physics. Verlag Peter Lang, Frankfurt a.M.

    Google Scholar 

  3. Brosowski, B. (1987): A criterion for efficiency and some applications. In: Brosowski, B., Martensen, E. (eds.) Optimization in mathematical physics. Verlag Peter Lang, Frankfurt a.M.

    Google Scholar 

  4. Brosowski, B., Conci, A. (1983): On vector optimization and parametric programming. Segundas Jornadas Latini Americanos De Matematica Aplicada II, 483–495

    Google Scholar 

  5. Chankong, V., Haimes, Y.Y. (1983): Multiobjective decision making — theory and methodology. North-Holland, New York Amsterdam Oxford

    Google Scholar 

  6. Choo, E.U., Atkins, D.R. (1983): Proper efficiency in nonconvex multicriteria programming. Math. Oper. Res. 8, 467–470

    Article  Google Scholar 

  7. Gembicki, F. (1974): Performance and sensitivity optimization: a vector index approach. Ph.D. thesis, Case Western Research University

    Google Scholar 

  8. Geoffrion, A.M. (1968): Proper efficiency and the theory of vector maximization. J.Math.Anal.Appl. 22, 618–630

    Article  Google Scholar 

  9. Gerth, C., Weidner, P. (1990): Nonconvex separation theorems and some applications in vector optimization. J.Optimization Theory Appl. 67, 297–320

    Article  Google Scholar 

  10. Grauer, M., Lewandowski, A., Wierzbicki, A. (1984): D1DASS — theory, implementation and experiences. In: Grauer, M., Wierzbicki, A. (eds.) Interactive decision analysis. Springer, Berlin New York

    Google Scholar 

  11. Helbig, S. (1987): Parametric optimization with a bottleneck objective and vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent advances and historical development of vector optimization. Springer, Berlin Heidelberg New York

    Google Scholar 

  12. Henig, M.I. (1982): Proper efficiency with respect to cones. J. Optimization Theory Appl. 36, 387–407

    Article  Google Scholar 

  13. Hwang, C.L., Masud, A.S. (1979): Multiple objective decision making — methods and applications. Springer, Berlin Heidelberg New York

    Book  Google Scholar 

  14. Kaliszewski, I. (1987): Generating nested subsets of efficient solutions. In:: Jahn, J., Krabs, W. (eds.) Recent advances and historical development of vector optimization. Springer, Berlin Heidelberg New York

    Google Scholar 

  15. Korhonen, P., Laakso, J. (1985): A visual interactive method for solving the multiple criteria problem. Eur. J. Oper. Res.

    Google Scholar 

  16. Kuhn, H.W., Tucker, A.W. (1951): Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley Los Angeles

    Google Scholar 

  17. Michalowski, W. (1988): Use of the displaced worst compromise in interactive multiobjective programming. IEEE Trans. Syst. Man Cybern. 18, 472–477

    Article  Google Scholar 

  18. Michalowski, W., Szapiro, T. (1989): A procedure for worst outcomes displacement in multiple criteria decision making. Comput.Oper.Res. 16, 195–206

    Article  Google Scholar 

  19. Pascoletti, A., Serafini, P. (1984): Scalarizing vector optimization problems. J. Optimization Theory Appl. 42, 499–524

    Article  Google Scholar 

  20. Podinovskij, V.V., Nogin, V.D. (1982): Pareto-optimal’nye resenija mnogokriterial’nych zadac. Nauka, Moskva (Russian)

    Google Scholar 

  21. Roy, B. (1971): Problems and methods with multiple objective functions. Math. Program. 1, 239–265

    Article  Google Scholar 

  22. Steuer, R.E. (1986): Multiple criteria optimization: theory, computation, and application. Wiley, New York

    Google Scholar 

  23. Weidner, P. (1993): Advantages of hyperbola efficiency. In: Brosowski, B., Ester, J., Helbig, S., Nehse, R. (eds.) Multicriteria decision. Peter Lang, Frankfurt a.M.

    Google Scholar 

  24. Weidner, P. (1990): Comparison of six types of separating functionals. In: Sebastian, H.-J., Tammer, K. (eds.): System modelling and optimization. Springer, Berlin Heidelberg New York

    Google Scholar 

  25. Weidner, P. (1990): Complete efficiency and interdependencies between objective functions in vector optimization. Zeitschrift für Operations Research 34, 91–115

    Google Scholar 

  26. Weidner, P. (1990): Ein Trennungskonzept und seine Anwendung auf Vektoroptimierungsverfahren. Ph.D.(B)thesis, Martin-Luther-Universität Halle-Wittenberg (German)

    Google Scholar 

  27. Weidner, P. (1993): Separation of nonconvex sets. In: Guddat, J., Jongen, H.Th., Kummer, B., Nozicka, F. (eds.) Parametric optimization and related topics III. Peter Lang, Frankfurt a.M.

    Google Scholar 

  28. Weidner, P. (1994): The influence of proper efficiency on optimal solutions of scalarizing problems in multicriteria optimization. OR Spektrum 16, 255–260

    Article  Google Scholar 

  29. Wierzbicki, A.P. (1980): The use of reference objectives in multiobjective optimization. In: Fandel, G., Gal, T. (eds.) Multiple criteria decision making — theory and application. Springer, Berlin Heidelberg New York

    Google Scholar 

  30. Yu, P.L. (1974): Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives. J. Optimization Theory Appl. 14, 319–377

    Article  Google Scholar 

  31. Zionts, S., Wallenius, J. (1976): An interactive programming method for solving the multiple criteria problem. Manage. Sci. 22, B652–B663

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. FB Physik-, Mess- u. Feinwerktechnik, FH Hildesheim/Holzminden/Göttingen, Von-Ossietzky-Str.99, D-37085, Göttingen, Germany

    Petra Weidner

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  1. Petra Weidner
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Editor information

Editors and Affiliations

  1. Krannert School of Management, Purdue University, 1310 State Street, West Lafayette, IN, 47907, USA

    Murat Köksalan

  2. School of Management, State University of New York, Buffalo, NY, 14260-4000, USA

    Stanley Zionts

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© 2001 Springer-Verlag Berlin Heidelberg

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Weidner, P. (2001). Problems in Scalarizing Multicriteria Approaches. In: Köksalan, M., Zionts, S. (eds) Multiple Criteria Decision Making in the New Millennium. Lecture Notes in Economics and Mathematical Systems, vol 507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56680-6_18

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  • DOI: https://doi.org/10.1007/978-3-642-56680-6_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42377-5

  • Online ISBN: 978-3-642-56680-6

  • eBook Packages: Springer Book Archive

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