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Symmetry
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Fuzzy Set Theory and Uncertainty Theory—3rd Edition
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Fuzzy Set Theory and Uncertainty Theory—3rd Edition

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 6895

Special Issue Editors

Prof. Dr. Jian Zhou

E-Mail Website
Guest Editor
School of Management, Shanghai University, Shanghai 200044, China
Interests: logistics system design; quality innovation; uncertainty theory and its applications
Special Issues, Collections and Topics in MDPI journals
Dr. Ke Wang

E-Mail Website
Guest Editor
School of Management, Shanghai University, Shanghai 200444, China
Interests: uncertainty modeling and optimization; fuzzy programming; fuzzy stochastic optimization; system reliability risk analysis; fuzzy approaches for industrial and business applications
Special Issues, Collections and Topics in MDPI journals
Dr. Yuanyuan Liu

E-Mail Website
Guest Editor
School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, China
Interests: fuzzy sets theory; QFD; risk aversion
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fuzzy set theory was initiated by Prof. Zadeh in the early 1960s. It is a fundamental approach that can deal with problems relating to ambiguous, subjective, and imprecise judgments. Compared with probability theory, fuzzy set theory has a unique adaptation for the quantification in the linguistic facet of available data and preferences for individual or group decision making. Further, uncertainty theory, which was presented by Prof. Baoding Liu in the early twenty-one century, is a new branch of mathematics based on normality, monotonicity, self-duality, and countable subadditivity axioms. The outstanding advantages of uncertainty theory in the general properties of uncertain variables have gradually increased its acceptance and led to further studies conducted by worldwide researchers.

To date, both fuzzy sets theory and uncertainty theory have been studied widely and in depth both in theory and applications by worldwide researchers. The purpose of this Special Issue is to gather a collection of articles on the latest research and developments in this field of research. Specific topics of interest include but are not limited to:

  • Knowledge representation;
  • Information content measures;
  • Extensions and generalizations of fuzzy sets;
  • Multifold uncertainty and its arithmetic;
  • Aggregation operations;
  • Reasoning under uncertainty;
  • Preference modeling and multicriteria evaluation;
  • Fuzzy multiobjective and bi-level programming;
  • Uncertain multiobjective optimization.

The research direction of all manuscripts must be within the scope of the Symmetry journal.

Prof. Dr. Jian Zhou
Dr. Ke Wang
Dr. Yuanyuan Liu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • intelligent algorithms for solving fuzzy/uncertain optimization problems
  • properties of fuzzy optimal solutions
  • fuzzy linear/non-linear regression
  • fuzzy/uncertain clustering
  • fuzzy/uncertain risk management
  • fuzzy/uncertian reliability analysis
  • fuzzy/uncertain joint replenishment problem
  • fuzzy optimization in product design

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (5 papers)

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Research

40 pages, 12261 KiB  
Article
Integrating Reliability, Uncertainty, and Subjectivity in Design Knowledge Flow: A CMZ-BENR Augmented Framework for Kansei Engineering
by Haoyi Lin, Pohsun Wang, Jing Liu and Chiawei Chu
Symmetry 2025, 17(5), 758; https://doi.org/10.3390/sym17050758 - 14 May 2025
Viewed by 118
Abstract
As a knowledge-intensive activity, the Kansei engineering (KE) process encounters numerous challenges in the design knowledge flow, primarily due to issues related to information reliability, uncertainty, and subjectivity. Bridging this gap, this study introduces an advanced KE framework integrating a cloud model with [...] Read more.
As a knowledge-intensive activity, the Kansei engineering (KE) process encounters numerous challenges in the design knowledge flow, primarily due to issues related to information reliability, uncertainty, and subjectivity. Bridging this gap, this study introduces an advanced KE framework integrating a cloud model with Z-numbers (CMZ) and Bayesian elastic net regression (BENR). In stage-I of this KE, data mining techniques are employed to process online user reviews, coupled with a similarity analysis of affective word clusters to identify representative emotional descriptors. During stage-II, the CMZ algorithm refines K-means clustering outcomes for market-representative product forms, enabling precise feature characterization and experimental prototype development. Stage-III addresses linguistic uncertainties in affective modeling through CMZ-augmented semantic differential questionnaires, achieving a multi-granular representation of subjective evaluations. Subsequently, stage-IV employs BENR for automated hyperparameter optimization in design knowledge inference, eliminating manual intervention. The framework’s efficacy is empirically validated through a domestic cleaning robot case study, demonstrating superior performance in resolving multiple information processing challenges via comparative experiments. Results confirm that this KE framework significantly improves uncertainty management in design knowledge flow compared to conventional implementations. Furthermore, by leveraging the intrinsic symmetry of the normal cloud model with Z-numbers distributions and the balanced ℓ1/ℓ2 regularization of BENR, CMZ–BENR framework embodies the principle of structural harmony. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—3rd Edition)
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17 pages, 599 KiB  
Article
Centroid-Induced Ranking of Triangular Picture Fuzzy Numbers and Applications in Decision-Making
by Lorena Popa
Symmetry 2024, 16(11), 1492; https://doi.org/10.3390/sym16111492 - 7 Nov 2024
Viewed by 1325
Abstract
This paper proposes the concept of a centroid for picture fuzzy numbers and particularly for triangular picture fuzzy numbers. The concept allows the implementation of a ranking function for the triangular picture fuzzy numbers, which has the advantage of reuniting the symmetry and [...] Read more.
This paper proposes the concept of a centroid for picture fuzzy numbers and particularly for triangular picture fuzzy numbers. The concept allows the implementation of a ranking function for the triangular picture fuzzy numbers, which has the advantage of reuniting the symmetry and asymmetry of the information. Then, empirical applications are considered for the picture fuzzy numbers. Specifically, multiple TPFNs are considered. The ranked, A comparison study is conducted for said ranked TPFNs relative to other methodologies in the specialized literature, illustrating that these methods exhibit limitations in specific scenarios. An additional compelling example is provided: before elections, opinion surveys are extensively utilised to assess voter intentions about candidates. The survey findings can be analysed through PFNs and the ranking mechanism proposed in this study. Another contribution of this paper is the development an algorithm meant to solve decision making problems in an uncertain environment. This is applied in the practical context of comparing the performance of several standards in two successive evaluations. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—3rd Edition)
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18 pages, 411 KiB  
Article
Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares
by Han Wang, Yang Liu and Haiyan Shi
Symmetry 2024, 16(9), 1182; https://doi.org/10.3390/sym16091182 - 9 Sep 2024
Cited by 1 | Viewed by 1322
Abstract
In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only [...] Read more.
In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only takes into account the reduction in the deviations between predictions and observations, and fails to fully consider the inherent characteristics of the correlation uncertainty distributions under the uncertain regression framework. In light of this, this paper constructs a statistical invariant with symmetric uncertainty distribution based on the observations and the disturbance term. It also proposes the least squares estimation of unknown parameters and disturbance term in the uncertain regression model based on the least squares principle and, combined with the mathematical properties of the normal uncertainty distribution, gives a numerical algorithm for solving specific estimates. Finally, in order to verify the effectiveness of the least squares estimation method proposed in this paper, we also design two numerical examples and an empirical study of forecasting of electrical power output. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—3rd Edition)
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14 pages, 269 KiB  
Article
Further Results on Lusin’s Theorem for Uncertain Variables
by Deguo Yang, Zhaojun Zong and Feng Hu
Symmetry 2024, 16(8), 1034; https://doi.org/10.3390/sym16081034 - 13 Aug 2024
Viewed by 1225
Abstract
In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to [...] Read more.
In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin’s theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if {Λn} is a sequence of open sets and Λn↘∅, then limn→∞M{Λn}=0 are equivalent in compact metric spaces. It is shown that Lusin’s theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin’s theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin’s theorem for uncertain variables. As far as we know, our results are new in uncertainty theory. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—3rd Edition)
28 pages, 5214 KiB  
Article
An Online Review-Driven Picture Fuzzy Multi-Criteria Group Decision-Making Approach for Evaluating the Online Medical Service Quality of Doctors
by Kaiwen Shi and Juanjuan Peng
Symmetry 2024, 16(6), 639; https://doi.org/10.3390/sym16060639 - 21 May 2024
Viewed by 1965
Abstract
In order to further investigate the level of online medical services in China and improve the medical experience of patients, this study aims to establish an online review-driven picture fuzzy multi-criteria group decision-making (MCGDM) approach for the online medical service evaluation of doctors. [...] Read more.
In order to further investigate the level of online medical services in China and improve the medical experience of patients, this study aims to establish an online review-driven picture fuzzy multi-criteria group decision-making (MCGDM) approach for the online medical service evaluation of doctors. First, based on the Aczel–Alsina t-norm and t-conorm, the normal picture fuzzy Aczel–Alsina operations involving a variable parameter are defined to make the corresponding operations more flexible than other operations. Second, two picture fuzzy Aczel–Alsina aggregation operators are developed, and the corresponding properties are discussed as well. Third, combined with the online review information of China’s medical platform Haodaifu, the online review-driven evaluation attributes and their corresponding weights are obtained, which can make the evaluation model more objective. Fourth, an extended normal picture fuzzy complex proportional assessment (COPRAS) decision-making method for the service quality evaluation of online medical services is proposed. Finally, an empirical example is presented to verify the feasibility and validity of the proposed method. A sensitivity analysis and a comparison analysis are also conducted to demonstrate the effectiveness and flexibility of the proposed approach. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—3rd Edition)
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