Opportunity Losses Due to Uncertainty in Multiobjective Decision Problems

A multiobjective maximization problem is considered in which at least one objective function f_i(x, C) depends on a random parameter C. If a single-valued measure, such as weighting or an l_p distance, is used to determine the preferred solution among the nondominated solutions, then standard decision-theoretic methods can be used to determine the expected opportunity loss (EOL). By an example hydrologic problem, it is shown that EOL is highly dependent on the single-valued measure selected to solve the multiobjective problem. The expected multiobjective opportunity loss (EMOL) is developed as a vector-valued measure of the effect of uncertainty on the problem which is independent of the technique. Finding the decision point with minimum EOL or EMOL is a possible way of selecting the preferred point. Problems pertaining to a multiobjective formulation of the EOL concept are examined.