Probabilistic bipartition interaction index of multiple decision criteria associated with the nonadditivity of fuzzy measures

Abstract

The probabilistic simultaneous interaction index has been widely adopted to measure the interaction among the decision criteria. However, this type of indices sometimes fails to reflect the kind of interaction associated with the nonadditivity of a fuzzy measure (capacity). For example, any simultaneous interaction index of the universal set of criteria w.r.t. a strictly superadditive capacity is not always positive. The main reason is that the simultaneous interaction index generalizes the notion of value by replacing the marginal contribution of a single criterion with the marginal simultaneous interaction of criteria subset. In this paper, we reform the generalization process and replace the marginal contribution with the marginal bipartition interaction, which can better reflect the kind of interaction associated with the nonadditivity, for example, superadditivity, subadditivity, strict‐superadditivity, or strict‐subadditivity. We construct a family of probabilistic bipartition interaction indices of subsets of criteria and study its properties. We discuss the issue of capacity identification based on the bipartition interaction index and demonstrate that the new type of interaction index can be adopted as a feasible alternative to describing the interaction phenomenon among the decision criteria.

Nonadditive robust ordinal regression with nonadditivity index and multiple goal linear programming

“…However, the Möbius representation and Shapley interaction index, as well as the probabilistic simultaneous interaction indices, do not naturally reflect the kind of interaction associated with the nonadditivity . The main reason is that the probabilistic interaction indices are designed to describe the simultaneous interaction among the decision criteria, which mathematically stems from the simultaneous marginal interaction embedded in the structure of every probabilistic interaction index, see Definition .…”

“…The above-mentioned nonadditivity property of capacity, such as superadditivity, subadditivity, strict-superadditivity, and strict-subadditivity, enables capacities to explicitly represent a variety of interaction phenomena among the decision criteria. 6,29,30 Generally speaking, an additive capacity means that decision criterion are all independent from each other. A strict superadditive (resp.…”

“…associated with the nonadditivity. 17,29 The main reason is that the probabilistic interaction indices are designed to describe the simultaneous interaction among the decision criteria, which mathematically stems from the simultaneous marginal interaction embedded in the structure of every probabilistic interaction index, see Definition 3. The common structure also leads to different ranges of these indices for subsets of different cardinalities and unintuitive interaction properties of a dummy criterion of the simultaneous interaction indices.…”

Fitting Sugeno integral for learning fuzzy measures using PAVA isotone regression

“…Another index characterising mutual dependence among the inputs is the nonadditivity index from Wu and Beliakov . There are also other indices, like the bipartition interaction index and the nonmodularity index which can also be used instead of the Shapley interaction index.…”

Multicriteria Correlation Preference Information (MCCPI)-Based Ordinary Capacity Identification Method

“…Capacity [1] (or fuzzy measure [2]) flexibly describes the importance and interaction among multiple decision criteria by means of evaluating the measures of all coalitions of the decision criteria [3][4][5]. However, this evaluation process is actually double-edged, also bringing an exponential complexity, a major obstacle for the application of capacity-based multiple criteria analysis, such as multicriteria decision making [6][7][8]. There are two main directions as regards overcoming this major obstacle.…”