Exponential operations of interval-valued intuitionistic fuzzy numbers

Abstract

The operational laws of intuitionistic fuzzy numbers are useful in dealing with the problems under intuitionistic fuzzy circumstances. As a supplementation, the exponential operational law of intuitionistic fuzzy numbers has been defined. Consider that we may utilize interval-valued intuitionistic fuzzy numbers (IVIFNs) to express uncertain weight information comprehensively in the complex multi-attribute decision making problems, in this paper, we discuss two exponential operational laws of IVIFNs, namely, the exponential operational law of IVIFNs with crisp parameter and the exponential operational law of IVIFNs with interval-valued parameter. Several properties and aggregation methods based on the exponential operational laws are discussed. Furthermore, we propose a new method to compare the novel dual interval-valued intuitionistic fuzzy numbers. Additionally, we give two information aggregation methods over these two exponential operational laws for dealing with some special problems in multi-attribute decision making, and finally, we apply the methods to a practical problem involving the choice of the optimal powered roof support for coal extraction with a high recovery rate.

Exponential operations and aggregation operators of interval neutrosophic sets and their decision making methods

“…To extend the existing exponential operations of IVIFSs (Gou et al 2015b), this paper presented the exponential operational laws of INSs and INNs as a useful supplement of the existing operational laws of INSs and INNs, where the bases are crisp values or interval numbers and the exponents are INSs and INNs. Then, we proposed the INWEA and DINWEA operators with crisp parameters and interval-valued parameters and the comparative methods based on cosine measure functions for INNs and DINNs.…”

“…Recently, Gou et al (2015a) defined exponential operational laws of IFSs, where the bases are crisp values and the exponents are IFSs, and presented an intuitionistic fuzzy exponential aggregation operator with crisp parameters and its application in MADM problems. Further, Gou et al (2015b) defined exponential operational laws of IVIFSs, where the bases are crisp values or interval numbers and the exponents are IVIFSs, and presented the corresponding exponential aggregation operators and their applications in MADM problems. In the existing decision making problems with single-valued neutrosophic information and interval neutrosophic information, the exponential values (weights) of all the existing exponential operational laws of SVNSs and INSs and the corresponding aggregation operators are positive real numbers within the unit interval [0, 1].…”

“…Hence, we lack some important operational laws and aggregation operators with crisp parameters or interval-valued parameters by using SVNSs or INSs as exponents to handle the decision making problems. Motivated by the exponential operational laws of IFSs and IVIFSs and the corresponding exponential aggregation methods (Gou et al 2015a, b), it is necessary that we should extend the existing exponential operations of IFSs and IVIFSs to the exponential operational laws of SVNSs and INSs with crisp parameters and interval-valued parameters and propose the corresponding exponential aggregation operators as an important supplement of the existing simplified neutrosophic aggregation techniques. Since INSs reduce to SVNSs when upper and lower ends in interval numbers are equal, this paper only defines the exponential operational laws of INSs with crisp parameters and interval-valued parameters, where the bases are crisp values and interval numbers and the exponents are INSs, and proposes an interval neutrosophic weighted exponential aggregation (INWEA) operator, a dual interval neutrosophic weighted exponential aggregation (DINWEA) operator, and comparative methods for interval neutrosophic numbers (INNs) and dual interval neutrosophic numbers (DINNs).…”

“…“Exponential operational laws of INSs” section proposes exponential operational laws of INSs with crisp parameters and interval-valued parameters as the extension of the existing exponential operations of IVIFSs (Gou et al 2015b). “Exponential aggregation operators of INNs” section presents the exponential aggregation operators of INSs based on these exponential operational laws and comparative methods for INNs and DINNs.…”

An overview of interval-valued intuitionistic fuzzy information aggregations and applications

“…Similarly, based on the Einstein operations, developed the dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator, and then applied them to develop some approaches for MADM with IVIFNs, an illustrative example for evaluating the computer network security was given to verify the developed approach and to demonstrate its practicality and effectiveness. Gou et al (2015) also defined two exponential operational laws of IVIFNs as well as discussed the interval-valued intuitionistic fuzzy weighted exponential aggregation (IVIFWEA) operator and the dual intervalvalued intuitionistic fuzzy weighted exponential aggregation (DIVIFWEA) operator. Meanwhile, they applied these two operators to handle a practical MADM problem involving the choice of the optimal powered roof support for coal extraction with a high recovery rate.…”