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Einstein Weighted Geometric Operator for Pythagorean Fuzzy Hypersoft with Its Application in Material Selection

Rana Muhammad Zulqarnain1, Imran Siddique2, Rifaqat Ali3, Fahd Jarad4,5,6,*, Aiyared Iampan7
1 Department of Mathematics, Zhejiang Normal University, Jinhua, China
2 Department of Mathematics, University of Management and Technology, Lahore, Pakistan
3 Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha, Saudi Arabia
4 Department of Mathematics, Cankaya University, Etimesgut, Ankara, Turkey
5 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
6 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
7 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao, Thailand
* Corresponding Author: Fahd Jarad. Email:
(This article belongs to this Special Issue: Decision making Modeling, Methods and Applications of Advanced Fuzzy Theory in Engineering and Science)

Computer Modeling in Engineering & Sciences 2023, 135(3), 2557-2583.

Received 06 April 2022; Accepted 05 August 2022; Issue published 23 November 2022


Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications. Pythagorean fuzzy hypersoft set (PFHSS) is the most influential and capable leeway of the hypersoft set (HSS) and Pythagorean fuzzy soft set (PFSS). It is also a general form of the intuitionistic fuzzy hypersoft set (IFHSS), which provides a better and more perfect assessment of the decision-making (DM) process. The fundamental objective of this work is to enrich the precision of decision-making. A novel mixed aggregation operator called Pythagorean fuzzy hypersoft Einstein weighted geometric (PFHSEWG) based on Einstein’s operational laws has been developed. Some necessary properties, such as idempotency, boundedness, and homogeneity, have been presented for the anticipated PFHSEWG operator. Multi-criteria decision-making (MCDM) plays an active role in dealing with the complications of manufacturing design for material selection. However, conventional methods of MCDM usually produce inconsistent results. Based on the proposed PFHSEWG operator, a robust MCDM procedure for material selection in manufacturing design is planned to address these inconveniences. The expected MCDM method for material selection (MS) of cryogenic storing vessels has been established in the real world. Significantly, the planned model for handling inaccurate data based on PFHSS is more operative and consistent.


Pythagorean fuzzy soft set; Pythagorean fuzzy hypersoft set; Pythagorean fuzzy hypersoft Einstein weighted geometric operator (PFHSEWG); MCDM

Cite This Article

Zulqarnain, R. M., Siddique, I., Ali, R., Jarad, F., Iampan, A. (2023). Einstein Weighted Geometric Operator for Pythagorean Fuzzy Hypersoft with Its Application in Material Selection. CMES-Computer Modeling in Engineering & Sciences, 135(3), 2557–2583.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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