Uncertain Dominance And Its Application

Some information and knowledge are usually represented by human language like"about 100km","roughly 80kg","low speed","middle age", and"big size". A lot ofsurveys showed that those imprecise quantities behave neither like randomness nor likefuzziness. In order to model those imprecise quantities, uncertainty theory was foundedas a branch of mathematics based on normality, monotonicity, self-duality, countablesubadditivity, and product measure axioms.A key problem in uncertainty theory is how to rank uncertain variables, and dif-ferent ranking criteria produce di?erent types of uncertain programming. Four rankingcriteria, which are expected value criterion, optimistic value criterion, pessimistic valuecriterion, and chance criterion, are used to build uncertain programming models. In thisdissertation, a new criterion is presented, which is called uncertain dominance. Andsome properties are investigated. Furthermore, a necessary and su?cient condition forlower order uncertain dominance and a necessary condition for higher order one areproved.Portfolio optimization is a classic problem in finance. In this dissertation, a newmodel is presented to deal with this problem. If the risk of a portfolio is less than therisk of the market index, it is a feasible solution, and the optimal solution is the one,which has the largest expected profit.In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects:The concept of uncertain dominance is proposed;A necessary and su?cient condition for lower order uncertain dominance and anecessary condition for higher order one are proved;Uncertain dominance is used to deal with portfolio optimization.