3 research outputs found
Calculating the interaction index of a fuzzy measure: a polynomial approach based on sampling
In this paper we address the problem of fuzzy measures index calculation. On
the basis of fuzzy sets, Murofushi and Soneda proposed an interaction index to
deal with the relations between two individuals. This index was later extended
in a common frame-work by Grabisch. Both indices are fundamental in the
literature of fuzzy measures. Nevertheless, the corresponding calculation still
presents a highly complex problem for which no approximation solution has been
proposed yet. Then, using a representation of the Shapley based on orders, here
we suggest an alternative calculation of the interaction index, both for the
simple case of pairs of individuals, and for the more complex situation in
which any set could be considered. This alternative representation facilitates
the handling of these indices. Moreover, we draw on this representation to
define two polynomial methods based on sampling to estimate the interaction
index, as well as a method to approximate the generalized version of it. We
provide some computational results to test the goodness of the proposed
algorithms.Comment: 17 page
The interval-valued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and 2-additive measure
Based on the operational laws on interval-valued intuitionistic fuzzy sets, the generalized Banzhaf interval-valued intuitionistic fuzzy geometric Choquet (GBIVIFGC) operator is proposed, which is also an interval-valued intuitionistic fuzzy value. It is worth pointing out that the GBIVIFGC operator can be seen as an extension of some geometric mean operators. Since the fuzzy measure is defined on the power set, it makes the problem exponentially complex. In order to overall reflect the interaction among elements and reduce the complexity of solving a fuzzy measure, we further introduce the GBIVIFGC operator w.r.t. 2-additive measures. Furthermore, if the information about weights of experts and attributes is incompletely known, the models of obtaining the optimal 2-additive measures on criteria set and expert set are given by using the introduced cross entropy measure and the Banzhaf index. Finally, an approach to pattern recognition and multi-criteria group decision making under interval-valued intuitionistic fuzzy environment is developed, respectively
Use of aggregation functions in decision making
A key component of many decision making processes is the aggregation step, whereby a set of numbers is summarised with a single representative value. This research showed that aggregation functions can provide a mathematical formalism to deal with issues like vagueness and uncertainty, which arise naturally in various decision contexts