New Einstein Hybrid Aggregation Operators for Intuitionistic Fuzzy Sets and Applications in Multi-Criteria Decision-Making

New Einstein Hybrid Aggregation Operators for Intuitionistic Fuzzy Sets and Applications in Multi-Criteria Decision-Making

Bhagawati Prasad Joshi (Seemant Institute of Technology, India) and Abhay Kumar (Seemant Institute of Technology, India)
Copyright: © 2019 |Pages: 32
DOI: 10.4018/978-1-5225-5709-8.ch012

Abstract

The fusion of multidimensional intuitionistic fuzzy information plays an important part in decision making processes under an intuitionistic fuzzy environment. In this chapter, it is observed that existing intuitionistic fuzzy Einstein hybrid aggregation operators do not follow the idempotency and boundedness. This leads to sometimes illogical and even absurd results to the decision maker. Hence, some new intuitionistic fuzzy Einstein hybrid aggregation operators such as the new intuitionistic fuzzy Einstein hybrid weighted averaging (IFEHWA) and the new intuitionistic fuzzy Einstein hybrid weighted geometric (IFEHWG) were developed. The new IFEHWA and IFEHWG operators can weigh the arguments as well as their ordered positions the same as the intuitionistic fuzzy Einstein hybrid aggregation operators do. Further, it is validated that the defined operators are idempotent, bounded, monotonic and commutative. Then, based on the developed approach, a multi-criteria decision-making (MCDM) procedure is given. Finally, a numerical example is conducted to demonstrate the proposed method effectively.
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Introduction

Zadeh (1965) introduced the concept of fuzzy sets (FSs) as the generalization of traditional classical sets. Atanassov (1986) presented the concept of intuitionistic fuzzy sets (IFSs), which is defined by a membership degree, non-membership degree, and a hesitancy degree. It is observed that IFSs theory is more suitable and powerful tool to deal with uncertainty and vagueness in real applications than FSs, and successfully applied in decision-making, medical diagnosis, artificial intelligence, pattern recognition, etc. due to the handling property of IFSs with uncertainty. Joshi and Kumar (2012a, 2012b, 2013) successfully implemented the concept of IFSs in forecasting problems. The aggregation of intuitionistic fuzzy information for alternatives is a very important step in multi-criteria decision-making (MCDM) problems, so the aggregation operators play an important role during the fusion process of information and received much more attention to practitioners.

Atanassov (1986, 1989, 1994), De et. al. (2000), Xu (2005, 2007), Xu and Da (2002) and Xu and Yager (2006) did pioneering standard research on the intuitionistic fuzzy operational laws and the development of aggregation operators. Atanassov (1994) proposed addition and multiplication operation, and De, et. al. (2000) defined scalar multiplication operation and power operation over intuitionistic fuzzy numbers (IFNs). Based on these laws and operations, Xu (2007) introduced some intuitionistic fuzzy aggregation operators such as: intuitionistic fuzzy weighted averaging (IFWA) operator, intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and intuitionistic fuzzy hybrid averaging (IFHA) operator. Xu and Yager (2006) developed some intuitionistic fuzzy geometric aggregation operators such as: intuitionistic fuzzy weighted geometric averaging (IFWGA) operator, intuitionistic fuzzy ordered weighted geometric averaging (IFOWGA) operator and intuitionistic fuzzy hybrid geometric averaging (IFHGA) operator. After these interesting researches, aggregation operators have attracted much attention from researchers (Wei, 2009; Wei, 2010; Beliakov, James, Mordelova, Ruckschlossova, & Yager, 2010; Li, 2010; Tan & Chen, 2010; Zhao, Xu, & Liu, 2010; Ye, 2010; Beliakov, Bustince, Goswami, Mukherjee, & Pal, 2011; Xu & Xia, 2011; Xu, 2011; Xu & Yager, 2011; Wang & Liu, 2011; Zhou & Chen, 2011; Tan, 2011; Zhou & Chen, 2012; Xia, Xu, & Zhu, 2012; Zhou, & Chen, 2012; Wei & Zhao, 2012; Xu & Wang, 2012; Yu, 2012; Wang & Liu, 2012; Yu & Xu, 2013; Zhang, 2013; Zhao & Wei, 2013; He, Chen, Zhou, Liu, & Tao, 2014; Lion & Xu, 2014; Lin & Jiang, 2014; Joshi, 2016; Joshi & Kharayat, 2016; Joshi, 2017; Joshi, 2018)

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