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On Efficient Sets in Vector Maximum Problems — A Brief Survey

  • Conference paper
Essays and Surveys on Multiple Criteria Decision Making

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 209))

Abstract

The notion of efficient (nondominated, noninferior. Pareto-optimal, functional efficient) set to a Vectormaximum Problem (VMP) has been analysed and developed in several directions during the last 30 years. Starting with the basic notion of efficiency given by Pareto [1896], formal descriptions of the efficient, properly efficient, locally (proper-) efficient, weak or strong efficient set have been developed beginning in the 50th of this century. Based on these notions various characteristics and properties of the efficient set have been studied, whereas the feasible set X and the functions zk(x), k = 1,.., K, constituting the vector-valued criterion z(x) of VMP have various properties (e.g., convexity of X, concavity of zk(x) for all k, differentiability etc.). Based on such properties, the structure of the efficient set and the existence of efficient solutions have been analysed. A part of the corresponding publications are rather of a pure theoretical character, others try to develope theories serving as the basis for working out methods for determining the efficient set or for interactive methods determining some compromise solutions. Duality theories for more or less general cases have been developed and various aspects of stability of VMP have been investigated. A brief survey is given.

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Authors and Affiliations

  1. Fachbereich Wirtschaftswissenschaft, Fernuniversität Hagen, D-5800, Hagen, West Germany

    Tomas Gal

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  1. Tomas Gal
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  1. Faculté de Sciences Economiques Appliquées, Faculté Universitaire Catholique de Mons, Chaussée de Binche, 151, B-7000, Mons, Belgium

    Pierre Hansen

  2. Institut d’Economie Scientifique et de Gestion, Lille, France

    Pierre Hansen

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© 1983 Springer-Verlag Berlin Heidelberg

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Gal, T. (1983). On Efficient Sets in Vector Maximum Problems — A Brief Survey. In: Hansen, P. (eds) Essays and Surveys on Multiple Criteria Decision Making. Lecture Notes in Economics and Mathematical Systems, vol 209. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46473-7_9

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  • DOI: https://doi.org/10.1007/978-3-642-46473-7_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11991-3

  • Online ISBN: 978-3-642-46473-7

  • eBook Packages: Springer Book Archive

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