Research Article
Extensions of Dombi Aggregation Operators for Decision
Making under m-Polar Fuzzy Information
Muhammad Akram ,
1
Naveed Yaqoob,
2
Ghous Ali,
1
and Wathek Chammam
3
1
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
2
Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad, Pakistan
3
Department of Mathematics, College of Science, Al Zulfi, Majmaah University, P.O. Box 66, Al-Majmaah 11952, Saudi Arabia
Correspondence should be addressed to Wathek Chammam; w.chammam@mu.edu.sa
Received 12 June 2020; Accepted 6 July 2020; Published 1 August 2020
Academic Editor: Tahir Mahmood
Copyright © 2020 Muhammad Akram et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
An m-polar fuzzy set is a powerful mathematical model to analyze multipolar, multiattribute, and multi-index data. e m-polar
fuzzy sets have appeared as a useful tool to portray uncertainty in multiattribute decision making. e purpose of this article is to
analyze the aggregation operators under the m-polar fuzzy environment with the help of Dombi norm operations. In this article,
we develop some averaging and geometric aggregation operators using Dombi t-norm and t-conorm to handle uncertainty in
m-polar fuzzy (mF, henceforth) information, which are mF Dombi weighted averaging (mFDWA) operator, mF Dombi ordered
weighted averaging (mFDOWA) operator, mF Dombi hybrid averaging (mFDHA) operator, mF Dombi weighted geometric
(mFDWG) operator, mF Dombi weighted ordered geometric operator, and mF Dombi hybrid geometric (mFDHG) operator. We
investigate properties, namely, idempotency, monotonicity, and boundedness, for the proposed operators. Moreover, we give an
algorithm to solve multicriteria decision-making issues which involve mF information with mFDWA and mFDWG operators. To
prove the validity and feasibility of the proposed model, we solve two numerical examples with our proposed models and give
comparison with mF-ELECTRE-I approach (Akram et al. 2019) and mF Hamacher aggregation operators (Waseem et al. 2019).
Finally, we check the effectiveness of the developed operators by a validity test.
1. Introduction
Multicriteria decision making (MCDM) is performing a vital
role in different areas, including social, physical, medical,
and environmental sciences. MCDM methods are not only
used to determine a suitable object but also used to rank the
objects in an appointed problem. To solve different un-
certain problems for decision making, Atanassov [1] pre-
sented the concept of intuitionistic fuzzy set (IFS) which
considers both membership and nonmembership parts, an
extension of fuzzy set [2] in which simple membership part
is characterized.
Aggregation operators (AOs) perform an important
role in order to combine data into a single form and solve
MCDM problems. For example, Yager [3] introduced
weighted AOs. Xu [4] proposed some new AOs under
IFSs. Xu and Yager [5] developed certain new geometric
AOs and solved some real-world MCDM problems. From
the inspection of an object, it can be easily seen that there
exist two properties of the object which are opposite to
each other. With this perspective, Zhang [6] presented
the idea of bipolar fuzzy set (BFS). BFSs provide gen-
eralized structure as compared to fuzzy sets [2] whose
memberships belong to [− 1, 0]×[0, 1]. Bipolarity plays
an important role in different research areas and pro-
vides more flexibility as compared to the fuzzy methods.
In the last decades, a lot of researchers, attracted by this
efficient concept, applied it to aggregate bipolar infor-
mation using different t-norms and their corresponding
conorms, including Hamacher and Dombi t-norms and
their corresponding conorms. For example, Wei et al. [7]
developed some bipolar fuzzy Hamacher weighted
Hindawi
Journal of Mathematics
Volume 2020, Article ID 4739567, 20 pages
https://doi.org/10.1155/2020/4739567