Department of Mathematics, Dibrugarh University, Dibrugarh, Assam 786004, India
Uncertainty is an integral part of decision-making process which arises due to the lack of knowledge, data or information. Initially Fuzzy Set Theory (FST) was used to handle this type of uncertainty. Later, Intuitionistic Fuzzy Set (IFS) was developed to encounter uncertainty in a more specific manner. However, it is observed that due to the existence of different types of uncertainties, the Membership Function (MF) of IFS itself is uncertain and consequently, the concept of Interval-Valued Intuitionistic Fuzzy sets (IVIFS) came into the picture. But IVIFS is also not capable of handling uncertainty. To overcome the limitations of the existing IVIFS, Generalized Interval Valued Intuitionistic Fuzzy Sets (GIVIFS) have been defined and it has been observed that it has utmost applicability in real world situations as the parameter height characterises the degree of buoyancy of judgment of decision maker in a very specific compartment.
An arithmetic operation on GIVTIFNs is always a critical concern and the conventional way of performing arithmetic operations on GIVTIFNs has some shortcomings. This paper attempts to devise a novel technique to effectively resolve the drawbacks of conventional arithmetic operations on GIVTIFNs. Numerical examples are illustrated herewith and to justify the need of a new solution. Furthermore, an application of multi-criteria group decision-making problem was also carried out under this setting.
For the arithmetic operations on GIVTIFNs, the largest membership function is truncated at the minimum height first and the non-membership function is truncated at the maximum height. Accordingly, arithmetic operations on GIVTIFNs are defined. For this purpose, Decomposition theorems for GIVTrIFNs are discussed first.
The outputs are obtained as generalized interval-valued trapezoidal intuitionistic fuzzy numbers (GIVTrIFNs). The interesting part of the proposed approach is that it produces GIVTrIFNs. To check the validity and novelty of the approach, a multi criteria decision making was performed which obtained desirable results.
The arithmetic GIVTIFNs conventional approach produces invariant output in the form of GIVTIFNs for GIVTIFNs of different height. But for the same input GIVTIFNs, the present approach provided different GIVTrIFNs. It was observed that the proposed approach is efficient, simple, logical, technically sound and general enough for implementation. Researchers may apply this approach in any field where GIVTIFNs are involved.
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