Modeling uncertainty using probabilistic-based possibility theory with applications to optimization.

摘要

It is shown that possibility distributions can be formulated within the context of probability theory and that membership values of fuzzy set theory can be interpreted as cumulative probabilities. The basic functions and operations of possibility theory are interpreted within this setting. The probabilistic information that can be derived from possibility distributions is examined. This leads to two functionals that provide estimates for the expected value of a random variable, the expected average of a single possibility distribution and the estimated expectation that requires two special possibility distributions to compute. Secondly, the space of fuzzy numbers is examined. It is shown that this space can be partitioned into a vector space and that the expected average functional motivates a norm on this space. It is shown that for most applications, Cauchy sequences converge in this space. Thirdly, applications of this theory to problems in optimization are examined. The concept of a fuzzy function is formulated. The minimum and minimizer of a fuzzy function are described. Objectives in fuzzy optimization are examined. Lastly, the ideas of the thesis are applied to the linear programming problem with uncertain coefficients.