Logarithmic Aggregation Operators of Picture Fuzzy Numbers for Multi-Attribute Decision Making Problems

The objective of this study was to create a logarithmic decision-making approach to deal with uncertainty in the form of a picture fuzzy set. Firstly, we define the logarithmic picture fuzzy number and define the basic operations. As a generalization of the sets, the picture fuzzy set provides a more profitable method to express the uncertainties in the data to deal with decision making problems. Picture fuzzy aggregation operators have a vital role in fuzzy decision-making problems. In this study, we propose a series of logarithmic aggregation operators: logarithmic picture fuzzy weighted averaging/geometric and logarithmic picture fuzzy ordered weighted averaging/geometric aggregation operators and characterized their desirable properties. Finally, a novel algorithm technique was developed to solve multi-attribute decision making (MADM) problems with picture fuzzy information. To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method, and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable.

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Evaluation of Investment Policy Based on Multi-Attribute Decision-Making Using Interval Valued T-Spherical Fuzzy Aggregation Operators

Symmetry ◽

10.3390/sym11030357 ◽

2019 ◽

Vol 11 (3) ◽

pp. 357 ◽

Cited By ~ 21

Author(s):

Kifayat Ullah ◽

Nasruddin Hassan ◽

Tahir Mahmood ◽

Naeem Jan ◽

Mazlan Hassan

Keyword(s):

Decision Making ◽

Fuzzy Set ◽

Intuitionistic Fuzzy Set ◽

Aggregation Operators ◽

Weighted Averaging ◽

Investment Policies ◽

Picture Fuzzy Set ◽

Fuzzy Aggregation ◽

Multi Attribute Decision Making ◽

Interval Valued

Expressing the measure of uncertainty, in terms of an interval instead of a crisp number, provides improved results in fuzzy mathematics. Several such concepts are established, including the interval-valued fuzzy set, the interval-valued intuitionistic fuzzy set, and the interval-valued picture fuzzy set. The goal of this article is to enhance the T-spherical fuzzy set (TSFS) by introducing the interval-valued TSFS (IVTSFS), which describes the uncertainty measure in terms of the membership, abstinence, non-membership, and the refusal degree. The novelty of the IVTSFS over the pre-existing fuzzy structures is analyzed. The basic operations are proposed for IVTSFSs and their properties are investigated. Two aggregation operators for IVTSFSs are developed, including weighted averaging and weighted geometric operators, and their validity is examined using the induction method. Several consequences of new operators, along with their comparative studies, are elaborated. A multi-attribute decision-making method in the context of IVTSFSs is developed, followed by a brief numerical example where the selection of the best policy, among a list of investment policies of a multinational company, is to be evaluated. The advantages of using the framework of IVTSFSs are described theoretically and numerically, hence showing the limitations of pre-existing aggregation operators.

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Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making

Symmetry ◽

10.3390/sym10110593 ◽

2018 ◽

Vol 10 (11) ◽

pp. 593 ◽

Cited By ~ 20

Author(s):

Hongran Zhang ◽

Runtong Zhang ◽

Huiqun Huang ◽

Jun Wang

Keyword(s):

Decision Making ◽

Fuzzy Set ◽

Intuitionistic Fuzzy Set ◽

Aggregation Operators ◽

Fuzzy Decision Making ◽

Good Ability ◽

Picture Fuzzy Set ◽

Fuzzy Aggregation ◽

Multi Attribute Decision Making ◽

Heronian Mean

As an extension of the intuitionistic fuzzy set (IFS), the recently proposed picture fuzzy set (PFS) is more suitable to describe decision-makers’ evaluation information in decision-making problems. Picture fuzzy aggregation operators are of high importance in multi-attribute decision-making (MADM) within a picture fuzzy decision-making environment. Hence, in this paper our main work is to introduce novel picture fuzzy aggregation operators. Firstly, we propose new picture fuzzy operational rules based on Dombi t-conorm and t-norm (DTT). Secondly, considering the existence of a broad and widespread correlation between attributes, we use Heronian mean (HM) information aggregation technology to fuse picture fuzzy numbers (PFNs) and propose new picture fuzzy aggregation operators. The proposed operators not only fuse individual attribute values, but also have a good ability to model the widespread correlation among attributes, making them more suitable for effectively solving increasingly complicated MADM problems. Hence, we introduce a new algorithm to handle MADM based on the proposed operators. Finally, we apply the newly developed method and algorithm in a supplier selection issue. The main novelties of this work are three-fold. Firstly, new operational laws for PFSs are proposed. Secondly, novel picture fuzzy aggregation operators are developed. Thirdly, a new approach for picture fuzzy MADM is proposed.

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Linguistic Spherical Fuzzy Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems

Mathematics ◽

10.3390/math7050413 ◽

2019 ◽

Vol 7 (5) ◽

pp. 413 ◽

Cited By ~ 19

Author(s):

Huanhuan Jin ◽

Shahzaib Ashraf ◽

Saleem Abdullah ◽

Muhammad Qiyas ◽

Mahwish Bano ◽

...

Keyword(s):

Decision Making ◽

Group Decision Making ◽

Fuzzy Set ◽

Group Decision ◽

Aggregation Operators ◽

Weighted Averaging ◽

Operational Laws ◽

Picture Fuzzy Set ◽

Fuzzy Aggregation ◽

Multi Attribute Decision Making

The key objective of the proposed work in this paper is to introduce a generalized form of linguistic picture fuzzy set, so-called linguistic spherical fuzzy set (LSFS), combining the notion of linguistic fuzzy set and spherical fuzzy set. In LSFS we deal with the vague and defective information in decision making. LSFS is characterized by linguistic positive, linguistic neutral and linguistic negative membership degree which satisfies the conditions that the square sum of its linguistic membership degrees is less than or equal to 1. In this paper, we investigate the basic operations of linguistic spherical fuzzy sets and discuss some related results. We extend operational laws of aggregation operators and propose linguistic spherical fuzzy weighted averaging and geometric operators based on spherical fuzzy numbers. Further, the proposed aggregation operators of linguistic spherical fuzzy number are applied to multi-attribute group decision-making problems. To implement the proposed models, we provide some numerical applications of group decision-making problems. In addition, compared with the previous model, we conclude that the proposed technique is more effective and reliable.

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Approach to Multi-Attribute Decision-Making Methods for Performance Evaluation Process Using Interval-Valued T-Spherical Fuzzy Hamacher Aggregation Information

Axioms ◽

10.3390/axioms10030145 ◽

2021 ◽

Vol 10 (3) ◽

pp. 145

Author(s):

Yun Jin ◽

Zareena Kousar ◽

Kifayat Ullah ◽

Tahir Mahmood ◽

Nimet Yapici Pehlivan ◽

...

Keyword(s):

Decision Making ◽

Fuzzy Set ◽

Evaluation Process ◽

Aggregation Operators ◽

Weighted Averaging ◽

Membership Degree ◽

Operational Laws ◽

Multi Attribute Decision Making ◽

Interval Valued ◽

Do So

Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work.

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Decision-Making Approach under Pythagorean Fuzzy Yager Weighted Operators

Mathematics ◽

10.3390/math8010070 ◽

2020 ◽

Vol 8 (1) ◽

pp. 70 ◽

Cited By ~ 16

Author(s):

Gulfam Shahzadi ◽

Muhammad Akram ◽

Ahmad N. Al-Kenani

Keyword(s):

Decision Making ◽

Fuzzy Sets ◽

Set Theory ◽

Fuzzy Set ◽

Fuzzy Set Theory ◽

Aggregation Operators ◽

Weighted Averaging ◽

Ordered Weighted Averaging ◽

Multi Attribute Decision Making ◽

Parametric Families

In fuzzy set theory, t-norms and t-conorms are fundamental binary operators. Yager proposed respective parametric families of both t-norms and t-conorms. In this paper, we apply these operators for the analysis of Pythagorean fuzzy sets. For this purpose, we introduce six families of aggregation operators named Pythagorean fuzzy Yager weighted averaging aggregation, Pythagorean fuzzy Yager ordered weighted averaging aggregation, Pythagorean fuzzy Yager hybrid weighted averaging aggregation, Pythagorean fuzzy Yager weighted geometric aggregation, Pythagorean fuzzy Yager ordered weighted geometric aggregation and Pythagorean fuzzy Yager hybrid weighted geometric aggregation. These tools inherit the operational advantages of the Yager parametric families. They enable us to study two multi-attribute decision-making problems. Ultimately we can choose the best option by comparison of the aggregate outputs through score values. We show this procedure with two practical fully developed examples.

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Protraction of Einstein operators for decision-making under q-rung orthopair fuzzy model

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201611 ◽

2020 ◽

pp. 1-20

Author(s):

Muhammad Akram ◽

Gulfam Shahzadi ◽

Sundas Shahzadi

Keyword(s):

Decision Making ◽

Fuzzy Set ◽

Intuitionistic Fuzzy Set ◽

Fuzzy Model ◽

Aggregation Operators ◽

Weighted Averaging ◽

Pythagorean Fuzzy Set ◽

Career Selection ◽

Ordered Weighted Averaging ◽

Multi Attribute Decision Making

An q-rung orthopair fuzzy set is a generalized structure that covers the modern extensions of fuzzy set, including intuitionistic fuzzy set and Pythagorean fuzzy set, with an adjustable parameter q that makes it flexible and adaptable to describe the inexact information in decision making. The condition of q-rung orthopair fuzzy set, i.e., sum of q th power of membership degree and nonmembership degree is bounded by one, makes it highly competent and adequate to get over the limitations of existing models. The basic purpose of this study is to establish some aggregation operators under the q-rung orthopair fuzzy environment with Einstein norm operations. Motivated by innovative features of Einstein operators and dominant behavior of q-rung orthopair fuzzy set, some new aggregation operators, namely, q-rung orthopair fuzzy Einstein weighted averaging, q-rung orthopair fuzzy Einstein ordered weighted averaging, generalized q-rung orthopair fuzzy Einstein weighted averaging and generalized q-rung orthopair fuzzy Einstein ordered weighted averaging operators are defined. Furthermore, some properties related to proposed operators are presented. Moreover, multi-attribute decision making problems related to career selection, agriculture land selection and residential place selection are presented under these operators to show the capability and proficiency of this new idea. The comparison analysis with existing theories shows the superiorities of proposed model.

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Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric Aggregation Operators

Journal of Applied Mathematics ◽

10.1155/2014/745617 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14 ◽

Cited By ~ 5

Author(s):

Xiaoqiang Zhou ◽

Qingguo Li

Keyword(s):

Decision Making ◽

Multiple Attribute Decision Making ◽

Aggregation Operators ◽

Fuzzy Decision Making ◽

Fuzzy Decision ◽

Fuzzy Information ◽

Operational Laws ◽

Multiple Attribute ◽

Fuzzy Aggregation ◽

Geometric Operator

We first define an accuracy function of hesitant fuzzy elements (HFEs) and develop a new method to compare two HFEs. Then, based on Einstein operators, we give some new operational laws on HFEs and some desirable properties of these operations. We also develop several new hesitant fuzzy aggregation operators, including the hesitant fuzzy Einstein weighted geometric (HFEWGε) operator and the hesitant fuzzy Einstein ordered weighted geometric (HFEWGε) operator, which are the extensions of the weighted geometric operator and the ordered weighted geometric (OWG) operator with hesitant fuzzy information, respectively. Furthermore, we establish the connections between the proposed and the existing hesitant fuzzy aggregation operators and discuss various properties of the proposed operators. Finally, we apply the HFEWGεoperator to solve the hesitant fuzzy decision making problems.

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Complex pythagorean fuzzy aggregation operators based on confidence levels and their applications

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022050 ◽

2021 ◽

Vol 19 (1) ◽

pp. 1078-1107

Author(s):

Tahir Mahmood ◽

◽

Zeeshan Ali ◽

Kifayat Ullah ◽

Qaisar Khan ◽

...

Keyword(s):

Decision Making ◽

Sensitivity Analysis ◽

Comparative Analysis ◽

Aggregation Operators ◽

Weighted Averaging ◽

Confidence Levels ◽

Ordered Weighted Averaging ◽

Operational Laws ◽

Fuzzy Aggregation ◽

Multi Attribute Decision Making

<abstract> <p>The most important influence of this assessment is to analyze some new operational laws based on confidential levels (CLs) for complex Pythagorean fuzzy (CPF) settings. Moreover, to demonstrate the closeness between finite numbers of alternatives, the conception of confidence CPF weighted averaging (CCPFWA), confidence CPF ordered weighted averaging (CCPFOWA), confidence CPF weighted geometric (CCPFWG), and confidence CPF ordered weighted geometric (CCPFOWG) operators are invented. Several significant features of the invented works are also diagnosed. Moreover, to investigate the beneficial optimal from a large number of alternatives, a multi-attribute decision-making (MADM) analysis is analyzed based on CPF data. A lot of examples are demonstrated based on invented works to evaluate the supremacy and ability of the initiated works. For massive convenience, the sensitivity analysis and merits of the identified works are also explored with the help of comparative analysis and they're graphical shown.</p> </abstract>

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A method to solve strategy based decision making problems with logarithmic T-spherical fuzzy aggregation framework

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-211003 ◽

2021 ◽

pp. 1-19

Author(s):

Shouzhen Zeng ◽

Amina Azam ◽

Kifayat Ullah ◽

Zeeshan Ali ◽

Awais Asif

Keyword(s):

Decision Making ◽

Fuzzy Set ◽

Fuzzy Numbers ◽

Aggregation Operators ◽

Weighted Averaging ◽

Membership Degree ◽

Human Judgment ◽

Induction Method ◽

Fuzzy Aggregation ◽

The Impact

T-Spherical fuzzy set (TSFS) is an improved extension in fuzzy set (FS) theory that takes into account four angles of the human judgment under uncertainty about a phenomenon that is membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). The purpose of this manuscript is to introduce and investigate logarithmic aggregation operators (LAOs) in the layout of TSFSs after observing the shortcomings of the previously existing AOs. First, we introduce the notions of logarithmic operations for T-spherical fuzzy numbers (TSFNs) and investigate some of their characteristics. The study is extended to develop T-spherical fuzzy (TSF) logarithmic AOs using the TSF logarithmic operations. The main theory includes the logarithmic TSF weighted averaging (LTSFWA) operator, and logarithmic TSF weighted geometric (LTSFWG) operator along with the conception of ordered weighted and hybrid AOs. An investigation about the validity of the logarithmic TSF AOs is established by using the induction method and examples are solved to examine the practicality of newly developed operators. Additionally, an algorithm for solving the problem of best production choice is developed using TSF information and logarithmic TSF AOs. An illustrative example is solved based on the proposed algorithm where the impact of the associated parameters is examined. We also did a comparative analysis to examine the advantages of the logarithmic TSF AOs.

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New Fuzzy Aggregation Operators Based on the Finite Choquet Integral — Application in the MADM Problem

International Journal of Information Technology & Decision Making ◽

10.1142/s0219622016500127 ◽

2016 ◽

Vol 15 (03) ◽

pp. 517-551 ◽

Cited By ~ 9

Author(s):

Gia Sirbiladze

Keyword(s):

Decision Making ◽

Choquet Integral ◽

Aggregation Operators ◽

Fuzzy Measure ◽

Weighted Averaging ◽

Positive Real ◽

Information Measures ◽

Fuzzy Aggregation ◽

Multi Attribute Decision Making ◽

States Of Nature

In this paper, new generalizations of the probabilistic averaging operator — Associated Fuzzy Probabilistic Averaging (As-PA and As-FPA) and Immediate Probabilistic Fuzzy Ordered Weighted Averaging (As-IP-OWA and As-IP-FOWA) operators are presented in the environment of fuzzy uncertainty. An uncertainty is presented by associated probabilities of a fuzzy measure. Expert’s evaluations as arguments of the aggregation operators are described by a variable, values of which are compatibility levels on the states of nature defined in positive real or triangular fuzzy numbers (TFNs). Two propositions on the As-FPA operator are proved: (1) The As-FPA operator for the fuzzy measure — capacity of order two coincides with the finite Choquet Averaging (CA) Operator; (2) the As-FPA operator coincides with the FPA operator when a probability measure is used in the role of a fuzzy measure. Analogous propositions for the As-IP-FOWA operator are proved. Some propositions on the connection of the As-FPA and As-IP-FOWA operators are also proved. Information measures — Orness and Divergence for the constructed operators are defined. Some propositions on the connections of these parameters with the corresponding parameters of the finite CA Operator are proved. Two illustrative examples on the applicability of the As-FPA and As-IP-FOWA operators are presented: (1) Several variants of the As-FPA and As-IP-FOWA operators are used for comparison of decision-making results for the problems regarding the fiscal policy of a country; (2) The As-FPA operator is used in the Multi-attribute decision-making (MADM) problem of choosing the best version of the students’ project.

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