Another view on the non-additivity index of capacity: a generalized index model and the unsupervised capacity identification with it

Non-additivity as a remarkable feature of non-additive measures is an important property of capacity. And the non-additivity index is a new concept presented in the current study on capacity, which gives an explicit treatment of non-additivity of a capacity and can reflect the type and degree of internal interaction among decision criteria. However, the influence and mechanism of the non-additivity index in capacity identification are still unclear for now, especially for the unsupervised learning identification methods. This paper is dedicated to an introduction of the generalized non-additivity index of capacity and an interesting unsupervised capacity identification with it, aiming to improve the quality of the modeling for the non-additivity index and reveal significant roles of the generalized index model in capacity identification with high efficiency. To start with, the definition of the generalized non-additivity index is presented, with the associated mathematical properties explored in detail. On this basis, a novel form of monotonicity constraints induced by this generalized non-additivity index is created to show the potential of improving the efficiency of capacity identification for practical purposes. This directly leads to our proposal of an interesting unsupervised capacity identification method, especially in the 2-additive case. By our technique, a desired monotone 2-additive capacity can be identified more efficiently by reducing the number of constraints from exponential to linear in the number of decision attributes, thus outperforming other sorts of methods in which a quadratic number of monotonicity constraints have to be considered. Finally, a real-world case of healthcare supplier selection is provided to attest to the practicality of our proposal.