Mathematical Sciences

Decision making has became increasingly complex as risky decisions are made in uncertain environments. To minimize risk, the problem of quantifying risk becomes very significant. In our review of the literature, we found a number of different risk measures. A closer study reveals that most of these risk measures define both risk and uncertainty in many different ways. In this dissertation, we model uncertainty and risk based on the Knightian definition of uncertainty and risk. We propose this alternative modeling paradigm and introduce our decision making methodology by solving an illustrative problem presented by a group of decision makers from Sandia Laboratories. Multicriteria decision problems for complex systems with interacting components require the algebra of operator representations and the separability of these representations to include the feedback due to the interacting uncertainties. However, the sum of these separable representations need not be separable. We propose a separable approximation for the sum and bound the errors due to the approximation