A new type of fuzzy integrals for decision making based on bivariate symmetric means

Abstract

We propose a new generalization of the discrete Choquet integral based on an arbitrary bivariate symmetric averaging function (mean). So far only the means with a natural multivariate extension were used for this purpose. In this paper, we use a general method based on a pruned binary tree to extend symmetric means with no obvious multivariate form, such as the logarithmic, identric, Heronian, Lagrangean, and Cauchy means. The generalized Choquet integral is built by computing the extensions of the bivariate means of the ordered inputs, and includes some existing extensions as special cases. Our construction is illustrated with multiple examples.

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

“…The nonadditivity with respect to disjoint subsets extends the additivity of the traditional probability measure and enables the capacity to adequately and flexibly represent the dependency or interaction phenomena among multiple decision criteria . Combined with nonlinear integrals, such as Choquet integral, Sugeno integral and so on, capacity‐based decision making and analysis methods have been applied in numerous studies and applications …”

“…[6][7][8][9] Combined with nonlinear integrals, such as Choquet integral, Sugeno integral and so on, 10 capacity-based decision making and analysis methods have been applied in numerous studies and applications. 6,7,[11][12][13][14][15][16] The main task of a capacity-based decision model is to provide the final ranking order of all alternatives in terms of their overall evaluations, or the dominance relationship on all alternatives, based on the decision maker's initial preference information on the correlated decision criteria and on some of the alternatives. Usually, the decision maker can provide some explicit preferences on the importance and interaction of decision criteria on a comparison or interval scale, like the following: criterion i is more important than criterion j, the interaction between criteria i and j is greater than that between criteria k and l, the interaction between i and j belongs to the interval a b[ , ], and so on.…”

“…We transform the new model into MGLP according to model (15), and we have the optimal objective function is 0.0474, but all the negative deviations are all zero. In this situation, it is better to add some small perturbations into the model, for example, we can change all righthand sides of preference constraints to 0.001.…”

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

“…The nonadditivity enables the capacity to represent the interaction phenomenon of the decision criteria adequately and flexibly 5 . There are various methods in multicriteria decision making (MCDM) based on the Choquet and Sugeno integrals, [5][6][7][8][9][10][11][12][13][14][15][16] which can accommodate redundancies of the decision criteria. The understanding and proper mathematical modeling of the interaction phenomena is thus an important issue in MCDM.…”