Soft Computing

, Volume 23, Issue 14, pp 5753–5783 | Cite as

Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making

  • Aliya FahmiEmail author
  • Saleem Abdullah
  • Fazli Amin
  • M. Sajjad Ali Khan
Methodologies and Application


In this paper, we define some Einstein operations on trapezoidal cubic fuzzy set and develop three arithmetic averaging operators, that is trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for aggregating trapezoidal cubic fuzzy information. The TrCFEHWA operator generalizes both the TrCFEWA and TrCFEOWA operators. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply on the TrCFEHWA operator to multiple attribute decision making with trapezoidal cubic fuzzy information. Finally, a numerical example is providing to demonstrate the submission of the established approach.


Trapezoidal cubic fuzzy set (TrCFS) Einstein t-norm Arithmetic averaging operator Multi-attribute decision making (MADM) 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of mathematicsHazara University MansehraDhodialPakistan
  2. 2.Department of mathematicsAbdul wali Khan University MardanMardanPakistan

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