Induced continuous Choquet integral operators and their application to group decision making

• Two new induced continuous Choquet integral operators named induced continuous Choquet averaging (ICCA) operator and induced continuous Choquet geometric mean (ICCGM) operator are defined, which not only consider the importance of combinations, but also reflect the correlations among them.
• Based on the probabilistic generalized semivalue (Marichal, 2000), the probabilistic generalized semivalue ICCA (PGS-ICCA) operator and the probabilistic generalized semivalue ICCGM (PGS-ICCGM) operator are defined.
• Based on the generalized Shapley index, the generalized Shapley index ICCA (GSI-ICCA) operator and the generalized Shapley index ICCGM (GSI-ICCGM) operator are given. These operators not only globally consider the importance of combinations or their ordered positions, but also overall reflect the correlations among combinations or their ordered positions.
• Based on the consistency principle (Chiclana, Herrera-Viedma, Herrera, & Alonso, 2007) and the TOPSIS method (Negi, 1989), the model of getting the optimal fuzzy measure on the attribute set or experts set is given, respectively, where their weights are assigned to their Shapley values.
• Based on the given aggregation operators and the models of obtaining optimal fuzzy measures, we develop an approach to uncertain multi-attribute group decision making.

With respect to multi-attribute group decision making, in this study two induced continuous Choquet integral operators named as the induced continuous Choquet weighted averaging (ICCWA) operator and the induced continuous Choquet geometric mean (ICCGM) operator are defined, which reflect the interactive characteristics between elements. Meantime, some associated desirable properties are studied to provide assurance in applications. In order to globally reflect the interactions between elements, we further define the probabilistic generalized semivalue ICCWA (PGS-ICCWA) operator and the probabilistic generalized semivalue ICCGM (PGS-ICCGM) operator. If the information about the weights of experts and attributes is incompletely known, the models for the optimal fuzzy measures on experts set and on attribute set based on consistency principle and TOPSIS method are respectively established. Moreover, an approach to uncertain multi-attribute group decision making with incomplete weight information and interactive conditions is developed. Finally, a numerical example is provided to illustrate the practicality and feasibility of the developed procedure.