Using the monotone measure sum to enrich the measurement of the interaction of multiple decision criteria

In the field of multicriteria decision analysis, the most distinctive characteristic of the monotone measure, or called the nonadditive measure, the fuzzy measure, the capacity, is that it can adequately and flexibly describe the interactions between the decision criteria. Traditionally, the interaction described with the monotone measure can be measured by the various kinds of the probabilistic interaction indices, among which the Shapley interaction index is the most famous and suitable one for the multicriteria decision making. Inspired by the marginal interaction of the multiple decision criteria, which is a core notion of the probabilistic interaction indices, we define the extremely positive and negative interaction cases of the multiple decision criteria, and find interestingly that the monotone measure sum (see Definitions 6 and 7) can be taken as an index to measure the kind and degree of the interaction of the multiple decision criteria. Further, we propose a new type of interaction index, i.e., the sum interaction index, and investigate its properties with respect to the additive, subadditive and superadditive monotone measures. Some comparison analyses of the sum interaction index with the Shapley interaction index are also given by several illustrative examples.