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

Jun Ye

doi:10.1186/s40064-016-3143-z

Abstract

An interval neutrosophic set (INS) is a subclass of a neutrosophic set and a generalization of an interval-valued intuitionistic fuzzy set, and then the characteristics of INS are independently described by the interval numbers of its truth-membership, indeterminacy-membership, and falsity-membership degrees. However, the exponential parameters (weights) of all the existing exponential operational laws of INSs and the corresponding exponential aggregation operators are crisp values in interval neutrosophic decision making problems. As a supplement, this paper firstly introduces new exponential operational laws of INSs, where the bases are crisp values or interval numbers and the exponents are interval neutrosophic numbers (INNs), which are basic elements in INSs. Then, we propose an interval neutrosophic weighted exponential aggregation (INWEA) operator and a dual interval neutrosophic weighted exponential aggregation (DINWEA) operator based on these exponential operational laws and introduce comparative methods based on cosine measure functions for INNs and dual INNs. Further, we develop decision-making methods based on the INWEA and DINWEA operators. Finally, a practical example on the selecting problem of global suppliers is provided to illustrate the applicability and rationality of the proposed methods.

Highlights

Neutrosophic sets and single-valued neutrosophic sets (SVNSs) were introduced for the first time by Smarandache (1998)
We develop multiple attribute decision making (MADM) methods by using the interval neutrosophic weighted exponential aggregation (INWEA) and dual interval neutrosophic weighted exponential aggregation (DINWEA) operators, where the data in the decision matrix are given by using crisp values or interval numbers as the evaluation values of attributes and the attribute weights are provided by interval neutrosophic numbers (INNs)
Based on the INWEA and DINWEA operators, we can deal with some decision making problems, where the weights of attributes are expressed by INNs and the attribute values are represented by crisp values or interval numbers

Summary

Background

Neutrosophic sets and single-valued neutrosophic sets (SVNSs) were introduced for the first time by Smarandache (1998). Definition 4 Leta = [TaL, TaU ], [IaL, IaU ], [FaL, FaU ] andb = [TbL, TbU ], [IbL, IbU ], [FbL, FbU ] be two INNs, the comparative method based on the cosine measure function C(a) can be defined as follows: 1. Theorem 1 Let a = [TaL, TaU ], [IaL, IaU ], [FaL, FaU ] and b = [TbL, TbU ], [IbL, IbU ], [FbL, FbU ] be two INNs and μ ∈ [0, 1] be a real number, there are the following commutative laws: 1. Definition 6 Let a = [TaL, TaU ], [IaL, IaU ], [FaL, FaU ] be an INN and μ = [μL, μU] be an interval number, the exponential operational law of the INN a is defined as μL. By the similar proof of the INWEA operator in Theorem 6, it is obvious that the DINWEA operator in Theorem 7 holds for all n and its aggregated value is a DINN

Decision making methods based on the INWEA and DINWEA operators

Practical example and comparative analysis

Conclusion