Multicriteria decision-making with imprecise importance weights

Our interest here is in multicriteria decision-making when we use a fuzzy measure to capture information about the importances and relationships between the criteria. We describe the use of an integral, such as the Choquet or Sugeno integral, to evaluate the overall satisfaction of each of the available alternatives. We discuss three measures particularly useful for these multicriteria decision problems, the additive, cardinality-based, and possibility measures. We note that the usefulness of these measures is a result of the fact that for each of these, the measure's values for any subset just depends on a small number of parameters. We then consider the situation in which we have some imprecision in these underlying parameters. We show how to represent this imprecision in the underlying parameters using a Dempster-Shafer belief structure. We then consider the evaluation of alternatives under this kind of imprecision using the Choquet, Sugeno, and median type aggregations. As a result of the imprecision in the parameters our overall evaluation for the alternatives, rather than being simple scalar values are imprecise, they are intervals. We discuss some methods for associating a scalar value with an interval. One notable method here is what we refer to as Golden Rule aggregation.