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On the completeness and constructiveness of parametric characterizations to vector optimization problems

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Summary

Motivated by recent reviews of characterizations of optimal solutions to vector optimization problems and by applications to decision support systems, this paper presents a methodological approach to comparing such characterizations. After specifying attributes of constructiveness, alternative classes of characterizations are reviewed. Characterization theorems are quoted or presented in more detail in cases that supplement those given in recent reviews. One of alternative classes of characterizations — by aspiration levels and order-consistent achievement functions — is discussed in more detail. An impossibility theorem of complete and robustly computable characterization of efficient (as opposed to weakly or properly efficient) solutions to vector optimization problems is presented.

Zusammenfassung

Angeregt durch neuere Übersichten der Charakterisierung von optimalen Lösungen von Vektoroptimierungsproblemen und durch Anwendungen auf Entscheidungsunterstützungssysteme wird in diesem Beitrag ein methodischer Ansatz zum Vergleich solcher Charakterisierungen dargestellt. Nach der Spezifizierung von Attributen der Konstruktivität werden alternative Klassen von Charakterisierungen betrachtet. Charakterisierungstheoreme werden entweder zitiert oder, in Ergänzung neuerer Übersichten, dargestellt. Eine der alternativen Klassen der Charakterisierungen wird näher diskutiert. Ein Unmöglichkeitstheorem einer vollständigen und robust berechenbaren Charakterisierung von effizienten (im Gegensatz zu schwach oder streng effizienten) Lösungen der Vektoroptimierungsprobleme wird dargelegt.

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References

  1. Benson HP (1978) Existence of efficient solutions for vector-maximum problems. JOTA 26:569–580

    Article  Google Scholar 

  2. Bowman VJ Jr (1976) On the relationship of the Chebyshev norm and efficient frontier of multiple-criteria objectives. In: Thiriez H, Zionts S (eds) Multiple criteria decision making. Springer, Berlin Heidelberg New York (Lecture Notes in Economic and Mathematical Systems, vol 130)

    Google Scholar 

  3. Changkong V, Haimes YT (1978) The interactive surrogate worth trade-off (ISTW) for multiobjective decision making. In: Zionts S (ed) Multiple criteria problem solving. Springer, Berling Heidelberg New York (Lecture Notes in Economic and Mathematical Systems, vol 155)

    Google Scholar 

  4. Charnes A, Cooper W (1961) Management models and industrial applications of linear programming. John Wiley, New York

    Google Scholar 

  5. Charnes A, Cooper W (1975) Goal programming and multiple objective optimization. J Oper Res Soc 1:39–54

    Article  Google Scholar 

  6. Dinkelbach W (1971) Über einen Lösungsansatz zum Vectormaximumproblem. In: Beckman M (ed) Unternehmungs-forschung heute. Springer, Berlin Heidelberg New York (Lecture Notes in Operational Research and Mathematical Systems, vol 50, pp 1–30)

    Chapter  Google Scholar 

  7. Dinkelbach W (1982) Entscheidungsmodelle. Walter de Gruyter, Berlin New York

    Google Scholar 

  8. Dinkelbach W, Iserman H (1973) On decision making under multiple criteria and under incomplete information. In: Cochrane JL, Zeleny M (eds) Multiple criteria decision making. University of South Carolina Press, Columbia, South Carolina

    Google Scholar 

  9. Dyer JS (1972) Interactive goal programming. Manag Sci 19:62–70

    Article  Google Scholar 

  10. Ecker JG, Kouada IA (1975) Finding efficient points for linear multiple objective programs. Math Programming 8: 375–377

    Article  Google Scholar 

  11. Fandel G (1972) Optimale Entscheidung bei mehrfacher Zielsetzung. Springer, Berling Heidelberg New York (Lecture Notes in Economic and Mathematical Systems, vol 76)

    Google Scholar 

  12. Gal T (1982) On efficient sets in vector maximum problems — a brief survey. In: Hansen P (ed) Essays and surveys on multiple criteria decision making. Proceedings, Mons 1982. Springer, Berlin Heidelberg New York (Lecture Notes in Economic and Mathematical Systems, vol 209)

    Google Scholar 

  13. Gearhart WB (1983) Characterization of properly efficient solutions by generalized scalarization methods. JOTA 41: 618–630

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Grauer M, Lewandoswki A, Wierzbicki AP (1984) DIDAS: Theory, implementation and experiences. In: Grauer M, Wierzbicki AP (eds) Interactive decision analysis. Springer, Berlin Heidelberg New York Tokyo (Lecutre Notes in Economic and Mathematical Systems, vol 229)

    Chapter  Google Scholar 

  16. Haimes YY, Hall WA, Freedman HB (1975) Multiobjective optimization in water resources systems, the surrogate trade-off method. Elesevier Scientific, New York

    Google Scholar 

  17. Henig MI (1982) Proper efficiency with respect to cones. JOTA 36:387–407

    Article  Google Scholar 

  18. Ignizio JP (1983) Generalized goal programming. Comp Oper Res 10:277–291

    Article  Google Scholar 

  19. Jahn J (1984) Scalarization in vector optimization. Math Programming 29:203–218

    Article  Google Scholar 

  20. Jahn J (1985) Some characterizations of the optimal solutions of a vector optimization problem. OR Spektrum 7: 7–17

    Article  Google Scholar 

  21. Kalai E, Smorodinsky M (1975) Other solutions to Nash's bargaining problem. Econometrica 43: 513–518

    Article  Google Scholar 

  22. Kallio M, Lewandowski A, Orchard-Hays W (1980) An implementation of the reference point approach for multiobjective optimization. WP-80-35, International Institute for Applied Systems Analysis, Laxenburg, Austria

    Google Scholar 

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

  24. Koopmans TC (1951) Analysis of production as an efficient combination of activities. In: Koopmans TC (ed) Activity analysis of production and allocation. Yale University Press, New Haven

    Google Scholar 

  25. Kuhn HW, Tucker AW (1951) Nonlinear programming. In: Neyman J (ed) Proceedings of the 2-nd Berkeley Symposium on Mathematical Statistics and Probability

  26. Lewandowski A, Grauer M (1982) The reference point optimization approach — methods of efficient implementation. WP-82-019, International Institute for Applied Systems Analysis, Laxenburg, Austria

    Google Scholar 

  27. Lewandowski A, Toth F, Wierzbicki A (1985) A prototype selection committee decision support system — implementation, tutorial example and user's manual. Mimeograph. International Institute for Applied Systems Analysis, Laxenburg, Austria

    Google Scholar 

  28. Luce RD, Raiffa H (1957) Games and decisions. John Wiley, New York

    Google Scholar 

  29. Masud AS, Hwang CL (1981) Interactive sequential goal programming. J Oper Res Soc 32:391–400

    Google Scholar 

  30. Michalevich VS (1965) Sequential optimization algorithms and their applications (in Russian). Cybernetics (Kybernetika)

  31. Nakayama H (1985) On the components in interactive multiobjective programming methods. In: Grauer M, Thompson M, Wierzbicki AP (eds) Plural rationality and interactive decision processes. Springer, Berlin Heidelberg New York Tokyo (Lecture Notes in Economic and Mathematical Systems, vol 248)

    Google Scholar 

  32. Nash JF (1950) The bargaining problem. Econometrica 18:155–162

    Article  Google Scholar 

  33. Sakawa M (1983) Interactive fuzzy decision making for multiobjective nonlinear programming problems. In: Gauer M, Wierzbicki AP (eds) Interactive decision analysis. Springer, Berlin Heidelberg New York Tokyo (Lecture Notes in Economic and Mathematical Systems, vol 229)

    Google Scholar 

  34. Salukvadze ME (1979) Vector-valued optimization problems in control theory. Academic Press, New York

    Google Scholar 

  35. Sawaragi Y, Nakayama H, Tanino T (1985) Theory of multiobjective optimization. Academic Press, New York

    Google Scholar 

  36. Steuer RE, Choo EV (1983) An interactive weighted Chebyshev procedure for multiple objective programming. Mathe Programming 26:326–344

    Article  Google Scholar 

  37. Wierzbicki AP (1975) Penalty methods in solving optimization problems with vector performance criteria. Working Paper of the Institute of Automatic Control, Technical University of Warsaw (presented at the VI-th IFAC World Congress, Cambridge, Mass.)

  38. Wierzbicki AP (1977) Basic properties of scalarizing functionals for multiobjective optimization. Mathematische Operationsforschung und Statistik, s. Optimization, 8:55–60

    Article  Google Scholar 

  39. Wierzbicki AP (1978) On the use of penaly functions in multiobjective optimization. In: Oettli W, Steffens F et al (eds) Proceedings of the III-rd Symposium on Operations Research, Universität Mannheim. Athenäum, Wiesbaden

    Google Scholar 

  40. Wierzbicki AP (1980) Multiobjective trajectory optimization and model semiregularization. WP-80-181, International Institute for Applied Systems Analysis, Laxenburg, Austria

    Google Scholar 

  41. Wierzbicki AP (1982) A mathematical basis for satisficing decision making. Math Modelling 3:391–405

    Article  Google Scholar 

  42. Wierzbicki AP (1983) Negotiation and mediation in conflicts I: The role of mathematical approaches and methods. In: Chestnut H et al (eds) Supplemental ways for improving international stability. Pergamon Press, Oxford

    Google Scholar 

  43. Wierzbicki AP (1983) Negotiation and mediation in conflicts II: Plural rationality and interactive decision processes. In: Grauer M, Thompson M, Wierzbicki AP (eds) Plural rationality and interactive decision processes. Springer, Berlin Heidelberg New York Tokyo (Lecture Notes in Economic and Mathematical Systems, vol 248)

    Google Scholar 

  44. Yu PL, Leitmann G (1974) Compromise solutions, domination structures and Salukvadze's solution. JOTA 13:362–378

    Article  Google Scholar 

  45. Zeleny M (1973) Compromise programming. In: Cochrane JL, Zeleny M (eds) Multiple criteria decision making. University of South Carolina Press, Columbia, South Carolina

    Google Scholar 

  46. Zeleny M (1982) Multiple criteria decision making. McGraw-Hill, New York

    Google Scholar 

  47. Zionts S, Wallenius I (1976) An interactive programming method for solving the multiple criteria problem. Manag Sci 22:652–663

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Automatic Control, Technical University of Warsaw, Nowowiejska 15/19, PL-00-665, Warsaw, Poland

    A. P. Wierzbicki

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Additional information

This paper was written during a visit at the Fernuniversität Hagen, D-5800 Hagen, FRG, and is based on materials obtained through joint research with the Systems and Decision Sciences Program, International Institute for Applied Systems Analysis, Laxenburg, Austria

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Wierzbicki, A.P. On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8, 73–87 (1986). https://doi.org/10.1007/BF01719738

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