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Fuzzy Set Theory and Uncertainty Theory—Volume II

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

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 11201

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Prof. Dr. Jian Zhou

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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

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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
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Dr. Yuanyuan Liu

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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,

Due to the great success of our Special Issue "Fuzzy Set Theory and Uncertainty Theory", we decided to set up a second volume.

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.

Welcome to read the publications in "Fuzzy Set Theory and Uncertainty Theory" at https://www.mdpi.com/journal/symmetry/special_issues/Fuzzy_Set_Theory_Uncertainty_Theory.

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

Related Special Issue

Fuzzy Set Theory and Uncertainty Theory—Volume III in Symmetry

Published Papers (11 papers)

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Research

15 pages, 2188 KiB

Open AccessArticle

Generalized Orthopair Fuzzy Weighted Power Bonferroni Mean Operator and Its Application in Decision Making

by Bowen Hou and Yongming Chen

Symmetry 2023, 15(11), 2007; https://doi.org/10.3390/sym15112007 - 31 Oct 2023

Viewed by 624

Abstract

The generalized orthopair fuzzy set is more favored by decision-makers and extensively utilized in areas like supply chain management, risk investment, and pattern recognition because it offers a broader decision information boundary than the intuitionistic fuzzy set and Pythagorean fuzzy set. This enables it to express fuzzy information more comprehensively and accurately in multi-attribute decision-making problems. To this end, this paper combines the ability of the power average (PA) operator to eliminate the impact of extreme values and the advantage of the Bonferroni mean (

B M^{s, t}

) operator in reflecting the relationships between variables, then incorporates weight indicators for different attributes to define the generalized orthopair fuzzy weighted power Bonferroni mean operator. The effectiveness of this operator is demonstrated through aggregation laws for generalized orthopair fuzzy information. Subsequently, the desirable properties of this operator are discussed. Based on these findings, a novel generalized orthopair fuzzy multi-attribute decision-making method, with a correlation between attributes, is proposed. Lastly, an investment decision-making example illustrates the feasibility and superiority of this method. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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17 pages, 2075 KiB

Open AccessArticle

Generation of Polynomial Automorphisms Appropriate for the Generalization of Fuzzy Connectives

by Eleftherios Makariadis, Stefanos Makariadis, Avrilia Konguetsof and Basil Papadopoulos

Symmetry 2023, 15(11), 1992; https://doi.org/10.3390/sym15111992 - 28 Oct 2023

Viewed by 972

Abstract

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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26 pages, 1712 KiB

Open AccessArticle

An Extended TODIM Method and Applications for Multi-Attribute Group Decision-Making Based on Bonferroni Mean Operators under Probabilistic Linguistic Term Sets

by Juxiang Wang, Xiangyu Zhou, Si Li and Jianwei Hu

Symmetry 2023, 15(10), 1807; https://doi.org/10.3390/sym15101807 - 22 Sep 2023

Cited by 2 | Viewed by 999

Abstract

Due to the complexity and uncertainty of decision-making, probabilistic linguistic term sets (PLTSs) are currently important tools for qualitative evaluation of decision-makers. The asymmetry of evaluation information can easily lead to the loss of subjective preference information for decision-makers, and the existing operation of decision-maker evaluation information fusion operators is difficult to solve this problem. To solve such problems, this paper proposes some new operational methods for PLTSs based on Dombi T-conorm and T-norm. Considering the interrelationships between the input independent variables of PLTSs, the probabilistic linguistic weighted Dombi Bonferroni mean Power average (PLWDBMPA) operators are extended and the properties of these aggregation operators are proposed. Secondly, the PLWDBMPA operator is used to fuse the evaluation information of decision-makers, avoiding the loss of decision information as much as possible. This paper uses social media platforms and web crawler technology to obtain online comments from users on decision-making to obtain the public’s attitude towards decision events. TF-IDF and Word2Vec are used to calculate the weight of alternatives on each attribute. Under traditional group decision-making methods and integrating the wisdom of the public, a novel multi-attribute group decision-making method based on TODIM method is proposed. Finally, the case study of Turkey earthquake shelter selection proves this method is scientific and effective. Meanwhile, the superiority of this method was further verified through comparisons with the PL-TOPSIS, PLWA, SPOTIS and PROMETHEE method. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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22 pages, 2968 KiB

Open AccessArticle

A Bi-Objective Optimization Model for a Low-Carbon Supply Chain Network with Risk of Uncertain Disruptions

by Yingtong Wang, Xiaoyu Ji and Yutong Lang

Symmetry 2023, 15(9), 1707; https://doi.org/10.3390/sym15091707 - 06 Sep 2023

Cited by 1 | Viewed by 855

Abstract

Disruption risks exacerbate the complexity of low-carbon supply chain network design in an uncertain supply chain environment. Considering the low frequency and non-repeatability of these disruption events makes it impossible to collect data to obtain their probabilities. In this study, supply disruptions were regarded as uncertain events; supply chain uncertain disruption risk is defined and quantified based on the uncertainty theory, in which uncertain disruptions are characterized by the belief degree on account of expert estimation with duality, i.e., symmetry. Optimization models were constructed with the objective of minimizing expected carbon emissions and costs, which optimizes the selection of suppliers with uncertain disruptions, and the assignment of manufacturers and customers. The properties of the model were analyzed, and the models were solved separately using different methods according to different decision criteria. Finally, the validity of the proposed models and algorithm were verified using a real case study of a glass manufacturing company. The findings exhibit promising insights for designing a sustainable and resilient supply chain network in an uncertain environment. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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20 pages, 442 KiB

Open AccessArticle

Novel Approach to Multi-Criteria Decision-Making Based on the n,mPR-Fuzzy Weighted Power Average Operator

by Tareq Hamadneh, Hariwan Z. Ibrahim, Mayada Abualhomos, Maha Mohammed Saeed, Gharib Gharib, Maha Al Soudi and Abdallah Al-Husban

Symmetry 2023, 15(8), 1617; https://doi.org/10.3390/sym15081617 - 21 Aug 2023

Viewed by 969

Abstract

A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the n^th power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a n^th power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPA

_{m}^{n}

) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPA

_{m}^{n}

operator with the square-root fuzzy weighted power average (SR-FWPA), the n^th power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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24 pages, 1254 KiB

Open AccessArticle

An Evaluation Model for the Influence of KOLs in Short Video Advertising Based on Uncertainty Theory

by Meiling Jin, Yufu Ning, Fengming Liu, Fangyi Zhao, Yichang Gao and Dongmei Li

Symmetry 2023, 15(8), 1594; https://doi.org/10.3390/sym15081594 - 17 Aug 2023

Cited by 2 | Viewed by 1794

Abstract

In the era of rapid growth in the short video industry, it is very important to find more accurate suitable advertising promoters, namely Key Opinion Leaders, to promote the development of short video commerce. A mathematical method is needed to grade and evaluate KOL’s abilities. Only in this way can advertisers better determine the value of KOL and determine whether it is suitable for promoting its products. Moreover, in the hierarchical evaluation of KOL, there is not only structured and quantifiable information, but also a large amount of unstructured and linguistic non-quantifiable information. Therefore, this article regards unquantifiable information as an uncertain variable and uses a comprehensive evaluation method based on uncertainty theory to handle subjective uncertainty in the evaluation process. Among them, all uncertain variables are symmetric. The main contribution of this article is the provision of a new evaluation method for KOL grading. Firstly, a two-level evaluation index system for KOL was established. Secondly, the importance and annotation of the Index set are set as uncertain variables, and the KOL evaluation model is constructed. Finally, two KOLs on TikTok were selected for comparative analysis to determine the importance ranking and KOL scores of each level of indicator, verifying the effectiveness and practicality of this method. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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13 pages, 280 KiB

Open AccessArticle

Conditional Uncertainty Distribution of Two Uncertain Variables and Conditional Inverse Uncertainty Distribution

by Lihui Wang, Yufu Ning, Xiumei Chen, Shukun Chen and Hong Huang

Symmetry 2023, 15(8), 1592; https://doi.org/10.3390/sym15081592 - 16 Aug 2023

Viewed by 591

Abstract

It is noted that some uncertain variables are independent while others are not. In general, there is a symmetrical relationship between independence and dependence among uncertain variables. The utilization of conditional uncertain measures as well as conditional uncertainty distributions proves highly efficacious in resolving uncertainties pertaining to an event subsequent to the acquisition of knowledge about other events. In this paper, the theorem about the conditional uncertainty distribution of two uncertain variables is proposed. It is demonstrated that the theorem holds regardless of whether the two variables are independent or not. In addition, it is also found that uncertainty distribution possesses an inherent inverse function when it is a regular uncertainty distribution within the framework of Uncertainty Theory; therefore, this paper delves into investigating the conditional inverse uncertainty distribution, including specific cases of the conditional inverse uncertainty distributions. Meanwhile, illustrative examples are applied to clarify the findings. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

10 pages, 569 KiB

Open AccessArticle

Symmetric Difference Operators Derived from Overlap and Grouping Functions

by Bo Hu, Di He and Songsong Dai

Symmetry 2023, 15(8), 1569; https://doi.org/10.3390/sym15081569 - 11 Aug 2023

Cited by 1 | Viewed by 677

Abstract

This paper introduces the concept of symmetric difference operators in terms of overlap and grouping functions, for which the associativity property is not strongly required. These symmetric difference operators are weaker than symmetric difference operators in terms of positive and continuous t-norms and t-conorms. Therefore, in the sense of the characters of mathematics, these operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. We analyze several related important properties based on two models of symmetric differences. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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12 pages, 271 KiB

Open AccessArticle

Relative Error Linear Combination Forecasting Model Based on Uncertainty Theory

by Hongmei Shi, Lin Wei, Cui Wang, Shuai Wang and Yufu Ning

Symmetry 2023, 15(7), 1379; https://doi.org/10.3390/sym15071379 - 07 Jul 2023

Viewed by 678

Abstract

The traditional combination forecasting model has good forecasting effect, but it needs precise historical data. In fact, many random events are uncertain, and much of the data are imprecise; sometimes, historical data are lacking. We need to study combination forecasting problems by means of uncertainty theory. Uncertain least squares estimation is an important technique of uncertain statistics, an important way to deal with imprecise data, and one of the best methods to solve the unknown parameters of uncertain linear regression equations. On the basis of the traditional combination forecasting method and uncertain least squares estimation, this paper proposes two kinds of uncertain combination forecasting models, which are the unary uncertain linear combination forecasting model and the uncertain relative error combination forecasting model, respectively. We set up several piecewise linear regression models according to the data of different periods and, according to certain weights, These piecewise linear regression models are combined into a unary uncertain linear combination forecasting model with a better forecasting effect. The uncertain relative error combination forecasting model is a new forecasting model that combines the traditional relative error linear forecasting model and the uncertain least squares estimation. Compared with the traditional forecasting model, the model can better deal with the forecasting problem of imprecise data. We verify the feasibility of the uncertain combination forecasting model through a numerical example. According to the data analysis, compared with the existing model, the forecasting effect of the proposed model is better. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

21 pages, 3006 KiB

Open AccessArticle

Ranking Method of Intuitionistic Fuzzy Numbers and Multiple Attribute Decision Making Based on the Probabilistic Dominance Relationship

by Zhengwei Huang, Shizhou Weng, Yuejin Lv and Huayuan Liu

Symmetry 2023, 15(5), 1001; https://doi.org/10.3390/sym15051001 - 28 Apr 2023

Cited by 5 | Viewed by 1149

Abstract

The uncertainty of intuitionistic fuzzy numbers (IFNs) is further enhanced by the existence of the degree of hesitation (DH). The shortcomings of existing researches are mainly reflected in the following situations: when comparing IFNs, the comparison rules of IFNs are difficult to apply to the comparison of any two IFNs, or the relevant methods do not fully consider the uncertainty expressed by DH. Thus, the rationality of the decision results needs to be improved. On the other hand, multi-attribute decision making (DADM) based on IFNs is often not objective due to the need to determine the attribute weight. Moreover, the strict condition of attribute aggregation of classical dominance relation makes it a method that fails considering the practical application. Aiming at the comparison problem of IFNs, this paper takes probability conversion as the starting point and proposes an IFN comparison method based on the area method, which can better deal with the comparison problem of “either superior or inferior” IFNs. In addition, aiming at the MADM problem of an intuitionistic fuzzy information system, we propose an intuitionistic fuzzy probabilistic dominance relation model and construct the MADM method under the probabilistic dominance relation. The series properties of IFNs and probabilistic dominance relation were summarized and proved, which theoretically ensured the scientificity and rigor of the method. The results show that the comparison and ranking method of IFNs proposed in this paper can be applied to the comparison of any two IFNs, and the dominance degree of IFNs is presented in the form of probability, which is more flexible and practical than the classical method. The probabilistic dominance relation method based on IFNs avoids the problem of determining attribute weights subjectively or objectively, and the decision maker can reflect decision preference by adjusting decision parameters to better match the actual problem. The application of this model to a campus express site evaluation further verifies the feasibility of the proposed method and the rationality of the results. In addition, various extension problems of the model and method proposed in this paper are discussed, which pave the way for future related research. This paper constructs a complete decision-making framework through theoretical analysis and application from practical problems, which provides a reference for enriching and improving uncertain decision-making theory and the MADM method. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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14 pages, 334 KiB

Open AccessArticle

A Novel Fuzzy Covering Rough Set Model Based on Generalized Overlap Functions and Its Application in MCDM

by Jialin Su, Yane Wang and Jianhui Li

Symmetry 2023, 15(3), 647; https://doi.org/10.3390/sym15030647 - 04 Mar 2023

Cited by 2 | Viewed by 841

Abstract

As nonassociative fuzzy logic connectives, it is important to study fuzzy rough set models using overlap functions that replace the role of t-norms. Overlap functions and t-norms are logical operators with symmetry. Recently, intuitionistic fuzzy rough set and multi-granulation fuzzy rough set models have been proposed based on overlap functions. However, some results (that contain five propositions, two definitions, six examples and a proof) must be improved. In this work, we improved the existing results. Moreover, to extend the existing fuzzy rough sets, a new fuzzy covering rough set model was constructed by using the generalized overlap function, and it was applied to the diagnosis of medical diseases. First, we improve some existing results. Then, in order to overcome the limitations of the fuzzy covering rough set model based on overlap functions, a fuzzy

β

-covering rough set model based on generalized overlap functions was established. Third, some properties of the fuzzy

β

-covering rough set model based on generalized overlap functions are discussed. Finally, a multi-criteria decision-making (MCDM) method of the fuzzy

β

-covering rough set based on generalized overlap functions was proposed. Taking medical disease diagnosis as an example, the comparison with other methods shows that the proposed method is feasible and effective. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)

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