By Paolo Serafini (eds.)

ISBN-10: 321181860X

ISBN-13: 9783211818602

ISBN-10: 3709128226

ISBN-13: 9783709128220

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**Extra info for Mathematics of Multi Objective Optimization**

**Sample text**

In dealing with uncertain outcomes, von Neumann and Morgenstern8 suggested to construct a real valued function u: R1 + Rl so that one alternative, represented by random variable X, is preferred to the other alternative, represented by random variable Y iff Eu(X) ) Eu(Y). Such a real valued function is known as a utility function for the preference over uncertain outcomes. 1 As noted earlier, much research has been devoted to studying the existence conditions of such a utility function. conditions must be extremely strict.

The basic idea of the nondominated set is to narrow the set of all available alternatives down to a set which contains the optimal choice by eliminating the inferior alternatives that are dominated by at least one alternative in the set. 5. (ii) (iii) N2 (o, s 2 ) • N3 (o, mv) if 0 is a class of normally distributed random variables. In the following figure we summarize relationships among the four dominance concepts. ___x_a Y___. 2 References 1. Fiahburn, P. , Utility Theory for Decision Making, John Wiley and Sons, New York, New York, 1970.

Some Relationships among Different Dominance Concepts In this section we investigate relationships among the four domi- nance concepts discussed in the previous sections. 1. 1 (i) X s 1 Y iff X u1 Y (ii) X s 2 Y iff X Proof. u2 Y See Radar and Russe~ 10 • 26 Hanoch and Levy, 9 and Bawa. 1 erence in the infinite dimensional space. alence. arge unless 0 is small and/or with some special structures. 2 .. _f.. x\t) • Pr[Y ~ tiX • t]. Then Ex[Fylx(X)] • FYIX(t)dFx(t) ~ ß is a necessary condition for X ß Y.

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