Fuzzy constraint levels for probabilistic performance measures

Performance risk measures are often used in environmental decision making applications as a means to enhance the understanding of a system subject to stochastic inputs and demands. These risk measures are usually probabilistic random variables describing some performance measure of a system. These risk measures are incorporated into the decision making (and optimisation) by specifying required levels of attainment, for example system reliability of .95 or 95%. Typically the specified constraint level is a crisp number even though there is imprecision or uncertainty about what level should be specified and the relative importance of the metrics themselves. Allowing a linguistic rating of the importance and incorporating fuzzy constraints better encapsulates the imprecision associated with the decision making. This paper examines the effect of imposing a fuzzy specification on a probabilistic random variable in an optimisation problem. The example chosen is a Markov decision process with probabilistic constraints. The practical example follows from formulating the operation of a reservoir as a MDP with constraints on reliability, resilience and vulnerability. The specified levels of performance are then given an imprecise specification i.e. a fuzzy representation.