Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations
In this paper we present new concepts of efficiency for uncertain multi-objective optimization problems. We analyze the connection between the concept of minmax robust efficiency presented by Ehrgott et al. (Eur J Oper Res, <CitationRef CitationID="CR14">2014</CitationRef>, doi:<ExternalRef> <RefSource>10.1016/j.ejor.2014.03.013</RefSource> <RefTarget Address="10.1016/j.ejor.2014.03.013" TargetType="DOI"/> </ExternalRef>) and the upper set less order relation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\preceq _s^u$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mo>⪯</mo> <mi>s</mi> <mi>u</mi> </msubsup> </math> </EquationSource> </InlineEquation> introduced by Kuroiwa (<CitationRef CitationID="CR29">1998</CitationRef>, <CitationRef CitationID="CR30">1999</CitationRef>). From this connection we derive new concepts of efficiency for uncertain multi-objective optimization problems by replacing the set ordering with other set orderings. Those are namely the lower set less ordering (see Kuroiwa <CitationRef CitationID="CR29">1998</CitationRef>, <CitationRef CitationID="CR30">1999</CitationRef>), the set less ordering (see Nishnianidze in Soobshch Akad Nauk Gruzin SSR 114(3):489–491, <CitationRef CitationID="CR34">1984</CitationRef>; Young in Math Ann 104(1):260–290, <CitationRef CitationID="CR40">1931</CitationRef>, doi:<ExternalRef> <RefSource>10.1007/BF01457934</RefSource> <RefTarget Address="10.1007/BF01457934" TargetType="DOI"/> </ExternalRef>; Eichfelder and Jahn in Vector Optimization. Springer, Berlin, <CitationRef CitationID="CR15">2012</CitationRef>), the certainly less ordering (see Eichfelder and Jahn in Vector Optimization. Springer, Berlin, <CitationRef CitationID="CR15">2012</CitationRef>), and the alternative set less ordering (see Ide et al. in Fixed Point Theory Appl, <CitationRef CitationID="CR25">2014</CitationRef>, doi:<ExternalRef> <RefSource>10.1186/1687-1812-2014-83</RefSource> <RefTarget Address="10.1186/1687-1812-2014-83" TargetType="DOI"/> </ExternalRef>; Köbis <CitationRef CitationID="CR28">2014</CitationRef>). We analyze the resulting concepts of efficiency and present numerical results on the occurrence of the various concepts. We conclude the paper with a short comparison between the concepts, and an outlook to further work. Copyright Springer-Verlag Berlin Heidelberg 2014
