Decision making under Z-information

Abstract

Rational decisions are based on information usually uncertain, imprecise and incomplete. The existing decision theories deal with three levels of generalization of decision making relevant information: numerical valuation, interval valuation and fuzzy number valuation. The classical decision theories, such as expected utility theory proposed by von Neumann and Morgenstern, and subjective expected utility theory proposed by Savage use the first level of generalization, i.e. numerical one. These approaches require that the objective probabilities or subjective probabilities and utility values be precisely known. But in real world in many cases it becomes impossible to determine the precise values of needed information. Interval analysis and classical fuzzy set theories have been applied in making decisions and many fruitful results have been achieved. But a problem is that in the mentioned above decision theories the reliability of the decision relevant information is not well taken into consideration. Prof. L. Zadeh introduced the concept of Z-numbers to describe the uncertain information which is more generalized notion closely related with confidence (reliability). Use of Z-information is more adequate and intuitively meaningful for formalizing information structure of a decision making problem. In this chapter we consider two approaches to decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. A fuzzy measure is calculated on a base of a given Z-information. The second approach is based on direct computation with Z-numbers. To illustrate a validity of suggested approaches to decision making with Z-information the numerical examples are used. © 2014 Springer-Verlag Berlin Heidelberg.