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Symmetry
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Fuzzy Set Theory and Uncertainty Theory
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Fuzzy Set Theory and Uncertainty Theory

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

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 42892

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
Co-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
Co-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
Prof. 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

Published Papers (30 papers)

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Research

22 pages, 1436 KiB  
Article
On Fuzzy Linear Fractional Programming Problems via α-Cut-Based Method with Application in Transportation Sector
by Abhishek Chauhan, Sumati Mahajan, Izhar Ahmad and Suliman Al-Homidan
Symmetry 2023, 15(2), 419; https://doi.org/10.3390/sym15020419 - 04 Feb 2023
Viewed by 1089
Abstract
The article provides an α-cut-based method that solves linear fractional programming problems with fuzzy variables and unrestricted parameters. The parameters and variables are considered as asymmetric triangular fuzzy numbers, which is a generalization of the symmetric case. The problem is solved by [...] Read more.
The article provides an α-cut-based method that solves linear fractional programming problems with fuzzy variables and unrestricted parameters. The parameters and variables are considered as asymmetric triangular fuzzy numbers, which is a generalization of the symmetric case. The problem is solved by using α-cut of fuzzy numbers wherein the α- and r-cut are applied to the objective function and constraints, respectively. This reduces the problem into an equivalent biobjective model which leads to the upper and lower bounds of the given problem. Afterwards, the membership functions corresponding to various values of r∈(0,1] are obtained using the optimal values of the biobjective model. The proposed method is illustrated by taking an example from the literature to highlight the fallacy of an existing approach. Finally, a fuzzy linear fractional transportation problem is modelled and solved using the aforementioned technique. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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11 pages, 575 KiB  
Article
An Optimal Location-Allocation Model for Equipment Supporting System Based on Uncertainty Theory
by Han Li, Wenshu Xie, Meilin Wen, Shuyu Li, Yi Yang and Linhan Guo
Symmetry 2023, 15(2), 338; https://doi.org/10.3390/sym15020338 - 25 Jan 2023
Viewed by 1100
Abstract
Scientific support depot location and reasonable spare parts transportation are the keys to improving the support level of complex systems. The current equipment support system has the problems of chaotic warehouse layout and low efficiency of spare parts. The reliability and completeness of [...] Read more.
Scientific support depot location and reasonable spare parts transportation are the keys to improving the support level of complex systems. The current equipment support system has the problems of chaotic warehouse layout and low efficiency of spare parts. The reliability and completeness of spare parts’ historical data are hard to believe. In order to deal with the cognitive uncertainty caused by the asymmetry of data, this paper adopts the uncertainty theory to optimize the depot location and transportation volume. Under the constraints of shortage rate, supply availability, average logistic delay time, and inventory limit, the uncertain chance-constrained model of equipment supporting depot is established. The optimization model is transformed into a deterministic model by using the inverse uncertainty distribution. The genetic algorithm is used to optimize the solution of this model. Finally, the practicability and operability of the model method are verified through the example analysis. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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22 pages, 6837 KiB  
Article
Uncertain Sensor–Weapon–Target Allocation Problem Based on Uncertainty Theory
by Guangjian Li, Guangjun He, Mingfa Zheng and Aoyu Zheng
Symmetry 2023, 15(1), 176; https://doi.org/10.3390/sym15010176 - 07 Jan 2023
Viewed by 933
Abstract
The sensor–weapon–target allocation (S-WTA) is a typical collaborative task allocation problem involved in network-centric warfare (NCW). The existing related studies have a limitation to the nature of cooperation and uncertainty in an air defense battle scenario, and most existing models have the assumption [...] Read more.
The sensor–weapon–target allocation (S-WTA) is a typical collaborative task allocation problem involved in network-centric warfare (NCW). The existing related studies have a limitation to the nature of cooperation and uncertainty in an air defense battle scenario, and most existing models have the assumption that they are determinate, i.e., the parameters in them are known certainly. For the actual battlefield environment, the asymmetric information in it could lead to the failure of the above assumption, and there are many uncertainties whose frequency can not be evaluated objectively. Based on uncertainty theory, this paper studied the S-WTA problem in an indeterminate battlefield environment. First, we analyze the uncertain factors existing in the actual battlefield environment and their influence on the S-WTA problem, and by considering the threat value of the target, the deviation parameters of the sensor tracking performance and weapon interception performance as uncertain variables, we then establish an uncertain S-WTA (USWTA) model, where the destruction value to targets is regarded as an objective function and four categories of typical constraints are set. Further, an equivalent transformation is presented to convert the unsolvable model into a determinate one by the expected value principle. To solve the proposed model efficiently, a permutation-based representation for the allocation scheme of the USWTA problem is introduced firstly, which can construct a feasible solution efficiently, and on this basis, a constructive heuristic algorithm based on maximum marginal return rule (MMRCH) is designed to construct a feasible solution with high quality. Additionally, a local search (LS) operation is proposed to explore for the better solution locally and further improve the quality of solution obtained by MMRCH. Finally, a set of instances are set to be solved by the designed algorithm, and the simulation experiment demonstrates the superiority of the designed algorithm and the feasibility of the proposed model. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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14 pages, 1487 KiB  
Article
One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model
by S. V. Gomathi and M. Jayalakshmi
Symmetry 2022, 14(10), 2056; https://doi.org/10.3390/sym14102056 - 02 Oct 2022
Cited by 2 | Viewed by 985
Abstract
This paper hinges upon the subject of an (n × n) assignment problem and the distinct symmetric fuzzy assignment problem byassigning n machines to n jobs. One’s orientation algorithm is developed for solving the assignment problems based on the position of [...] Read more.
This paper hinges upon the subject of an (n × n) assignment problem and the distinct symmetric fuzzy assignment problem byassigning n machines to n jobs. One’s orientation algorithm is developed for solving the assignment problems based on the position of one’s chosen in every row as well as every column to perform allocations and obtain the assignment cost at every (n − 1) reduced matrix. We also extended the two different symmetric concept to the problem to find the optimum solution based on symmetrical data and also used the ranking concept in our fuzzy assignment problem. In this proposed algorithm, the one’s position is associated with the successor of one in each iteration toobtain the optimal schedule and assignment cost. The comparative analysis is properly considered and discussed. The proposed technique is elaborated with the help of numerical computations and it gives clarity to the idea of this concept. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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20 pages, 910 KiB  
Article
Two-Stage Fuzzy Interactive Multi-Objective Approach under Interval Type-2 Fuzzy Environment with Application to the Remanufacture of Old Clothes
by Jian Zhou, Sisi Wu and Junjie Gao
Symmetry 2022, 14(9), 1785; https://doi.org/10.3390/sym14091785 - 27 Aug 2022
Cited by 3 | Viewed by 1004
Abstract
In this study, a two-stage approach is introduced to obtain a more interactive and flexible solution to deal with the multi-objective programming under interval type-2 fuzzy environment. In the first stage, the fuzzy multi-objective chance-constrained programming with regular symmetric triangular interval type-2 fuzzy [...] Read more.
In this study, a two-stage approach is introduced to obtain a more interactive and flexible solution to deal with the multi-objective programming under interval type-2 fuzzy environment. In the first stage, the fuzzy multi-objective chance-constrained programming with regular symmetric triangular interval type-2 fuzzy set parameters is proposed and transferred into its crisp equivalent form. In the second stage, we use the fuzzy interactive approach to address the crisp multi-objective programming obtained in the first stage by introducing the trade-off rate, which helps the decision maker react via updating their reference member values to obtain a satisfactory solution. Finally, taking a remanufacture of old clothes problem as an example, the comparison of experimental results obtained using a non-interactive method and interactive method shows that the proposed approach is conducive to obtaining satisfactory solutions effectively and efficiently, which broadens the application scope of the multi-objective programming with regular symmetric triangular interval type-2 fuzzy set parameters for sustainable manufacturing. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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18 pages, 470 KiB  
Article
Risk-Based Maintenance Optimization for a Subsea Production System with Epistemic Uncertainty
by Ying Liu, Liuying Ma, Luyang Sun, Xiao Zhang, Yunyun Yang, Qing Zhao and Zhigang Qu
Symmetry 2022, 14(8), 1672; https://doi.org/10.3390/sym14081672 - 11 Aug 2022
Cited by 1 | Viewed by 1253
Abstract
The lack of operation and maintenance data brings difficulties to traditional risk assessment based on probability methods. Therefore, experts are invited to evaluate the key performance indicators related to system risk. These evaluation results are usually described by ambiguous language, so they have [...] Read more.
The lack of operation and maintenance data brings difficulties to traditional risk assessment based on probability methods. Therefore, experts are invited to evaluate the key performance indicators related to system risk. These evaluation results are usually described by ambiguous language, so they have epistemic uncertainty. Uncertainty theory is a branch of mathematics used to model experts’ degrees of belief. The uncertain measure has duality, that is, some symmetry, which means that the sum of the uncertain measure of an event and the uncertain measure of its complementary set is equal to 1. Therefore, the risk occurrence time of each basic event evaluated by experts is modeled by the uncertain variable in this article. Then, the risk assessment method of systems with epistemic uncertainty is proposed based on an uncertain fault tree analysis. Furthermore, two risk-based maintenance optimization models for systems with epistemic uncertainty are established. In particular, the leakage risk assessment method and the two risk-based maintenance optimization models for a subsea production system are considered, and the optimization results are given. The optimization results can help practitioners to warn of the leakage risk and make scientific maintenance decisions based on expert knowledge, so as to extend the service life of subsea production systems. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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16 pages, 323 KiB  
Article
Law of Large Numbers, Central Limit Theorem, and Law of the Iterated Logarithm for Bernoulli Uncertain Sequence
by Ziyi Qu, Zhaojun Zong and Feng Hu
Symmetry 2022, 14(8), 1642; https://doi.org/10.3390/sym14081642 - 09 Aug 2022
Cited by 1 | Viewed by 1383
Abstract
In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers [...] Read more.
In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers (LLN), central limit theorem (CLT), and law of the iterated logarithm (LIL) play a critical role in probability theory. For uncertain variables, basic and important research is also to obtain the relevant limit theorems. However, up to now, there has been no research on these limit theorems for uncertain variables. The main results to emerge from this paper are a strong law of large numbers (SLLN), a weak law of large numbers (WLLN), a CLT, and an LIL for Bernoulli uncertain sequence. For studying these theorems, we first propose an assumption, which can be regarded as a generalization of the duality axiom for uncertain measure in the case that the uncertainty space can be finitely partitioned. Additionally, several new notions such as weakly dependent, Bernoulli uncertain sequence, and continuity from below or continuity from above of uncertain measure are introduced. As far as we know, this is the first study of the LLN, the CLT, and the LIL for uncertain variables. All the theorems proved in this paper can be applied to uncertain variables with symmetric or asymmetric distributions. In particular, the limit of uncertain variables is symmetric in (c) of the third theorem, and the asymptotic distribution of uncertain variables in the fifth theorem is symmetrical. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
22 pages, 1145 KiB  
Article
A Novel Inverse Credibility Distribution Approach for the Membership Functions of LR Fuzzy Intervals: A Case Study on a Completion Time Analysis
by Yujie Gu
Symmetry 2022, 14(8), 1554; https://doi.org/10.3390/sym14081554 - 28 Jul 2022
Viewed by 1003
Abstract
Fuzzy arithmetic is of great significance in dealing with vague information, especially the basic arithmetic operations (i.e., ⊕, ⊖, ⊗, ⊙). However, the classical and widely accepted accurate and approximate approaches, the interval arithmetic approach and standard approximation method, cannot output accurate or [...] Read more.
Fuzzy arithmetic is of great significance in dealing with vague information, especially the basic arithmetic operations (i.e., ⊕, ⊖, ⊗, ⊙). However, the classical and widely accepted accurate and approximate approaches, the interval arithmetic approach and standard approximation method, cannot output accurate or well-approximated expressions of the membership function, which may prevent decision makers from making the right decisions in real applications. To tackle this problem, this paper first discusses the relationships among the membership function, the credibility distribution, and the inverse credibility distribution and summarizes the relationships as several theorems. Then, by means of the theorems and the newly proposed operational law, this paper proposes an inverse credibility distribution approach that can output the accurate expression of the membership function for continuous and strictly monotone functions of regular LR fuzzy intervals. To better demonstrate the effectiveness of the raised approach, the commonly-used LR fuzzy interval, the symmetric trapezoidal fuzzy number, is employed, and several comparisons with the other two methods are made. The results show that the proposed approach can output an exact or well-approximated expression of the membership function, which the others cannot. In addition, some comparisons of the proposed approach with other methods are also made on a completion time analysis of a construction project to show the effectiveness of the proposed approach in real applications. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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19 pages, 363 KiB  
Article
Fuzzy Rough Programming Models: An Expected Value Perspective
by Guanshuang Jiang, Guang Wang, Haomin Zhang and Haoran Zheng
Symmetry 2022, 14(7), 1384; https://doi.org/10.3390/sym14071384 - 06 Jul 2022
Viewed by 943
Abstract
Usually, the quasi-normal fluctuations in practical applications are described via symmetric uncertainty variables, which is a common phenomenon in the manufacturing industry. However, it is relatively scarce in the literature to discuss two-fold uncertainty due to the its complexity. To deal with roughness [...] Read more.
Usually, the quasi-normal fluctuations in practical applications are described via symmetric uncertainty variables, which is a common phenomenon in the manufacturing industry. However, it is relatively scarce in the literature to discuss two-fold uncertainty due to the its complexity. To deal with roughness and ambiguity to accommodate inherent uncertainties, fuzzy rough programming approaches are put forward. In this paper, we pay attention to exploring two kinds of programming problems, namely fuzzy rough single-objective programming and fuzzy rough multi-objective programming, in which objective functions and/or constraints involve fuzzy rough variables (FRV). In accordance with the related existing research of FRVs, such as the chance measure and the expected value (EV) operator, this paper further discusses the EV model, convexity theory, and the crisp equivalent model of fuzzy rough programming. After that, combined with the latest published NIA-S fuzzy simulation technique, a new fuzzy rough simulation algorithm is developed to calculate the EVs of complicated functions for handling the presented fuzzy rough programming problems. In the end, the two types of numerical examples are provided for demonstration. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
13 pages, 449 KiB  
Article
Stability of Euler Methods for Fuzzy Differential Equation
by Cuilian You, Yan Cheng and Hongyan Ma
Symmetry 2022, 14(6), 1279; https://doi.org/10.3390/sym14061279 - 20 Jun 2022
Cited by 3 | Viewed by 1270
Abstract
The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit [...] Read more.
The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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14 pages, 1527 KiB  
Article
A Structural Credit Risk Model Driven by the Lévy Process under Knightian Uncertainty
by Hong Huang, Yufu Ning and Xiumei Chen
Symmetry 2022, 14(5), 1041; https://doi.org/10.3390/sym14051041 - 19 May 2022
Viewed by 1147
Abstract
The classic credit risk structured model assumes that risky asset values obey geometric Brownian motion. In reality, however, risky asset values are often not a continuous and symmetrical process, but rather they appear to jump and have asymmetric characteristics, such as higher peaks [...] Read more.
The classic credit risk structured model assumes that risky asset values obey geometric Brownian motion. In reality, however, risky asset values are often not a continuous and symmetrical process, but rather they appear to jump and have asymmetric characteristics, such as higher peaks and fat tails. On the other hand, there are real Knight uncertainty risks in financial markets that cannot be measured by a single probability measure. This work examined a structural credit risk model in the Lévy market under Knight uncertainty. Using the Lévy–Laplace exponent, we established dynamic pricing models and obtained intervals of prices for default probability, stock values, and bond values of enterprise, respectively. In particular, we also proved the explicit solutions for the three value processes above when the jump process is assumed to follow a log-normal distribution. Finally, the important impacts of Knightian uncertainty on the pricing of default probability and stock values of enterprise were studied through numerical analysis. The results showed that the default probability of enterprise, the stock values, and bond values were no longer a certain value, but an interval under Knightian uncertainty. In addition, the interval changed continuously with the increase in Knightian uncertainty. This result better reflected the impact of different market sentiments on the equilibrium value of assets, and expanded decision-making flexibility for investors. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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19 pages, 462 KiB  
Article
An Improved EDAS Method for the Multi-Attribute Decision Making Based on the Dynamic Expectation Level of Decision Makers
by Dan Peng, Jie Wang, Donghai Liu and Zaiming Liu
Symmetry 2022, 14(5), 979; https://doi.org/10.3390/sym14050979 - 10 May 2022
Cited by 5 | Viewed by 1236
Abstract
The improved evaluation based on the distance from average solution (EDAS) of the interval-valued intuitionistic trapezoidal fuzzy set is proposed. At first, we propose a new distance between interval-valued intuitionistic trapezoidal fuzzy numbers according to their interval endpoints and centroid point, and its [...] Read more.
The improved evaluation based on the distance from average solution (EDAS) of the interval-valued intuitionistic trapezoidal fuzzy set is proposed. At first, we propose a new distance between interval-valued intuitionistic trapezoidal fuzzy numbers according to their interval endpoints and centroid point, and its properties are also discussed. Furthermore, we apply the proposed distance measure to calculate the expectation level of the emergency plan, and the optimal dynamic expectation level of the emergency plan is obtained by solving the programming model. Then, we improve the EDAS method based on the dynamic expectation level of the decision makers and apply it to calculate the optimal emergency plan. Finally, a numerical example about flood disaster rescue is given to verify the feasibility and effectiveness of the proposed method, which is also compared with the existing methods. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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15 pages, 281 KiB  
Article
On the Aggregation of Comonotone or Countermonotone Fuzzy Relations
by Yuanyuan Liu and Fan Jia
Symmetry 2022, 14(5), 958; https://doi.org/10.3390/sym14050958 - 07 May 2022
Viewed by 915
Abstract
The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to [...] Read more.
The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min/max and commonly used increasing/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
20 pages, 1896 KiB  
Article
Novel Aczel–Alsina Operators for Pythagorean Fuzzy Sets with Application in Multi-Attribute Decision Making
by Abrar Hussain, Kifayat Ullah, Mohammed Nasser Alshahrani, Miin-Shen Yang and Dragan Pamucar
Symmetry 2022, 14(5), 940; https://doi.org/10.3390/sym14050940 - 05 May 2022
Cited by 50 | Viewed by 2047
Abstract
Multi-attribute decision-making (MADM) is usually used to aggregate fuzzy data successfully. Choosing the best option regarding data is not generally symmetric on the grounds that it does not provide complete information. Since Aczel-Alsina aggregation operators (AOs) have great impact due to their parameter [...] Read more.
Multi-attribute decision-making (MADM) is usually used to aggregate fuzzy data successfully. Choosing the best option regarding data is not generally symmetric on the grounds that it does not provide complete information. Since Aczel-Alsina aggregation operators (AOs) have great impact due to their parameter variableness, they have been well applied in MADM under fuzzy construction. Recently, the Aczel-Alsina AOs on intuitionistic fuzzy sets (IFSs), interval-valued IFSs and T-spherical fuzzy sets have been proposed in the literature. In this article, we develop new types of Pythagorean fuzzy AOs by using Aczel-Alsina t-norm and Aczel-Alsina t-conorm. Thus, we give these new operations Aczel-Alsina sum and Aczel-Alsina product on Pythagorean fuzzy sets based on Aczel-Alsina t-norm and Aczel-Alsina t-conorm. We also develop new types of Pythagorean fuzzy AOs including Pythagorean fuzzy Aczel-Alsina weighted averaging and Pythagorean fuzzy Aczel-Alsina weighted geometric operators. We elaborate some characteristics of these proposed Aczel-Alsina AOs on Pythagorean fuzzy sets, such as idempotency, monotonicity, and boundedness. By utilizing the proposed works, we solve an example of MADM in the information of the multinational company under the evaluation of impacts in MADM. We also illustrate the comparisons of the proposed works with previously existing AOs in different fuzzy environments. These comparison results demonstrate the effectiveness of the proposed Aczel-Alsina AOs on Pythagorean fuzzy sets. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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17 pages, 318 KiB  
Article
Entropy and Semi-Entropies of Regular Symmetrical Triangular Interval Type-2 Fuzzy Variables
by Meixi Zhang and Zhiyi Wang
Symmetry 2022, 14(5), 930; https://doi.org/10.3390/sym14050930 - 02 May 2022
Cited by 2 | Viewed by 1103
Abstract
Fuzzy entropy has a wide range of applications in uncertainty problems. Due to the dual-complexity of its characteristics and calculation, the study on type-2 fuzzy entropy is rare, let alone the semi-ones. Given this, the paper takes the lead in proposing the credibility-based [...] Read more.
Fuzzy entropy has a wide range of applications in uncertainty problems. Due to the dual-complexity of its characteristics and calculation, the study on type-2 fuzzy entropy is rare, let alone the semi-ones. Given this, the paper takes the lead in proposing the credibility-based type-2 entropy and semi-entropies delivered around a specific symmetric type-2 fuzzy variable. After presenting the relevant theorems and definitions, we give the corresponding examples of linear entropy and semi-entropies to illustrate and verify the favorable property of our study. This series of formulas on type-2 entropy proposed has a strong advantage in reducing computational complexity. It can be commonly applied to measure fuzziness and solve return-oriented and cost-oriented problems in the business field. A sequence of measures on type-2 fuzzy entropy developed in this paper delivers fresh insights into this field. It also provides a new reasonable tool for the decision-making on cost and investment control in companies. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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17 pages, 1591 KiB  
Article
Spare Parts Transportation Optimization Considering Supportability Based on Uncertainty Theory
by Yi Yang, Jiaying Gu, Siyu Huang, Meilin Wen, Yong Qin, Wei Liu and Linhan Guo
Symmetry 2022, 14(5), 891; https://doi.org/10.3390/sym14050891 - 27 Apr 2022
Cited by 1 | Viewed by 1677
Abstract
Ensuring a consistent, continuous, and efficient spare parts supply is a critical issue that must be addressed in the equipment support system. In order to effectively improve the coverage level and handle the common asymmetry information present in practical applications, the spare parts [...] Read more.
Ensuring a consistent, continuous, and efficient spare parts supply is a critical issue that must be addressed in the equipment support system. In order to effectively improve the coverage level and handle the common asymmetry information present in practical applications, the spare parts transport vehicle routing and scheduling model was further optimized. We integrated supportability requirements and uncertainty theory into the model to better describe the actual uncertain demand of each site. We selected three critical supportability indicators as constraints, redefined them with uncertain variables, and then completed the chance-constrained model on this basis. Once the confidence level is specified, the uncertain constraints can be transformed into deterministic constraints, and finally, the equivalent deterministic model can be solved easily. In addition, a feasible solution can be found through a genetic algorithm, and a numerical example is provided to validate the model’s rationality. The proposed method successfully seeks the balance between the total cost and supportability. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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27 pages, 565 KiB  
Article
Research on Comparision between Deterministic Method and Uncertain Method for Solving Uncertain Multiobjective Programming
by Mingfa Zheng, Haitao Zhong, Aoyu Zheng, Lin Zhou and Guoqiang Yuan
Symmetry 2022, 14(3), 470; https://doi.org/10.3390/sym14030470 - 25 Feb 2022
Viewed by 1082
Abstract
Since there are often few or no samples and asymmetry information in the problems, uncertainty theory is introduced to study uncertain multi-objective programming (UMP), which cannot be solved by probability theory. Generally speaking, there are two types of methods for solving the UMP [...] Read more.
Since there are often few or no samples and asymmetry information in the problems, uncertainty theory is introduced to study uncertain multi-objective programming (UMP), which cannot be solved by probability theory. Generally speaking, there are two types of methods for solving the UMP problem: in deterministic method, using the numerical characteristics of an uncertain variable, the UMP problem is transformed into a deterministic multiobjective programming, and then solved by the weighting method and ideal point method; in the uncertain method, the UMP problem is transformed into an uncertain single-objective programming, and then is solved by the evaluation criteria of the uncertain variables. The theoretical analysis and the data results for numerical examples solved by the AC algorithm designed in the paper show that the two types of methods are obviously different. Further, using this comparison, the essential difference between the two methods is whether the uncertainty relation between objective functions sholud be considered. Therefore, when the uncertainty relation is closely related, the uncertain method is more appropriate; otherwise, the deterministic method should be chosen. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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24 pages, 2056 KiB  
Article
Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations
by Ahmed Abdel Aziz Elsayed, Bassem Saassouh, Nazihah Ahmad and Ghassan Malkawi
Symmetry 2022, 14(3), 446; https://doi.org/10.3390/sym14030446 - 23 Feb 2022
Cited by 7 | Viewed by 1449
Abstract
Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). [...] Read more.
Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing analytical methods for solving FFSME are based on Vec-operator and Kronecker product. Nevertheless, these methods are only applicable for nonnegative fuzzy numbers, which limits the applications of the existing methods. Thus, this paper proposes a new numerical method for solving arbitrary Trapezoidal FFSME (TrFFSME), which includes near-zero trapezoidal fuzzy numbers to overcome this limitation. The TrFFSME is converted to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication operations. Then the non-linear system is solved using a newly developed two-stage algorithm. In the first stage algorithm, initial values are determined. Subsequently, the second stage algorithm obtains all possible finite fuzzy solutions. A numerical example is solved to illustrate the proposed method. Besides, this proposed method can solve other forms of fuzzy matrix equations and produces finite fuzzy and non-fuzzy solutions compared to the existing methods. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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20 pages, 901 KiB  
Article
Variance and Semi-Variances of Regular Interval Type-2 Fuzzy Variables
by Wenjing Tang and Yitao Chen
Symmetry 2022, 14(2), 278; https://doi.org/10.3390/sym14020278 - 29 Jan 2022
Cited by 2 | Viewed by 1664
Abstract
In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric [...] Read more.
In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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13 pages, 278 KiB  
Article
Existence and Uniqueness Theorem for Uncertain Wave Equation
by Rong Gao
Symmetry 2022, 14(2), 191; https://doi.org/10.3390/sym14020191 - 19 Jan 2022
Cited by 1 | Viewed by 1280
Abstract
In the real world, the indeterminate phenomenon and determinate phenomenon are symmetric; however, the indeterminate phenomenon absolutely exists. Hence, the indeterminate dynamic phenomenon is studied in this paper by using uncertainty theory, where the indeterminate dynamic phenomenon is associated with the belief degree [...] Read more.
In the real world, the indeterminate phenomenon and determinate phenomenon are symmetric; however, the indeterminate phenomenon absolutely exists. Hence, the indeterminate dynamic phenomenon is studied in this paper by using uncertainty theory, where the indeterminate dynamic phenomenon is associated with the belief degree and called the uncertain dynamic phenomenon. Based on uncertainty theory, the uncertain wave equation derived by the Liu process is constructed to model the propagation of various types of wave with uncertain disturbance in nature, where the Liu process is Lipschitz-continuous and has stationary and independent increments. First important of all, only the equation has solution which can be used to clearly depict the wave propagation influenced by uncertain disturbance. Therefore, the aims of this paper is to propose and prove a theorem of existence and uniqueness with Lipschitz and linear growth conditions. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
12 pages, 754 KiB  
Article
Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes
by Xiumei Chen, Yufu Ning, Lihui Wang, Shuai Wang and Hong Huang
Symmetry 2022, 14(1), 14; https://doi.org/10.3390/sym14010014 - 23 Dec 2021
Cited by 1 | Viewed by 1845
Abstract
In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse [...] Read more.
In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
12 pages, 974 KiB  
Article
An Uncertain APP Model with Allowed Stockout and Service Level Constraint for Vegetables
by Yufu Ning, Na Pang, Shuai Wang and Xiumei Chen
Symmetry 2021, 13(12), 2332; https://doi.org/10.3390/sym13122332 - 05 Dec 2021
Cited by 2 | Viewed by 1117
Abstract
Volatile markets and uncertain deterioration rate make it extremely difficult for manufacturers to make the quantity of saleable vegetables just meet the fluctuating demands, which will lead to inevitable out of stock or over production. Aggregate production planning (APP) is to find the [...] Read more.
Volatile markets and uncertain deterioration rate make it extremely difficult for manufacturers to make the quantity of saleable vegetables just meet the fluctuating demands, which will lead to inevitable out of stock or over production. Aggregate production planning (APP) is to find the optimal yield of vegetables, shortage and overstock symmetry, are not conducive to the final benefit.The essence of aggregate production planning is to deal with the symmetrical relation between shortage and overproduction. In order to reduce the adverse effects caused by shortage, we regard the service level as an important constraint to meet the customer demand and ensure the market share. So an uncertain aggregate production planning model for vegetables under condition of allowed stockout and considering service level constraint is constructed, whose objective is to find the optimal output while minimizing the expected total cost. Moreover, two methods are proposed in different cases to solve the model. A crisp equivalent form can be transformed when uncertain variables obey linear uncertain distributions and for general case, a hybrid intelligent algorithm integrating the 99-method and genetic algorithm is employed. Finally, two numerical examples are carried out to illustrate the effectiveness of the proposed model. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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11 pages, 261 KiB  
Article
Risk-Neutral Pricing Method of Options Based on Uncertainty Theory
by Hong Huang and Yufu Ning
Symmetry 2021, 13(12), 2285; https://doi.org/10.3390/sym13122285 - 01 Dec 2021
Viewed by 1634
Abstract
In order to rationally deal with the belief degree, Liu proposed uncertainty theory and refined into a branch of mathematics based on normality, self-duality, sub-additivity and product axioms. Subsequently, Liu defined the uncertainty process to describe the evolution of uncertainty phenomena over time. [...] Read more.
In order to rationally deal with the belief degree, Liu proposed uncertainty theory and refined into a branch of mathematics based on normality, self-duality, sub-additivity and product axioms. Subsequently, Liu defined the uncertainty process to describe the evolution of uncertainty phenomena over time. This paper proposes a risk-neutral option pricing method under the assumption that the stock price is driven by Liu process, which is a special kind of uncertain process with a stationary independent increment. Based on uncertainty theory, the stock price’s distribution and inverse distribution function under the risk-neutral measure are first derived. Then these two proposed functions are applied to price the European and American options, and verify the parity relationship of European call and put options. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
22 pages, 329 KiB  
Article
On Some Laws of Large Numbers for Uncertain Random Variables
by Piotr Nowak and Olgierd Hryniewicz
Symmetry 2021, 13(12), 2258; https://doi.org/10.3390/sym13122258 - 27 Nov 2021
Cited by 4 | Viewed by 1540
Abstract
Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one [...] Read more.
Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
23 pages, 1224 KiB  
Article
Arithmetic Operations and Expected Values of Regular Interval Type-2 Fuzzy Variables
by Hui Li and Junyang Cai
Symmetry 2021, 13(11), 2196; https://doi.org/10.3390/sym13112196 - 17 Nov 2021
Cited by 3 | Viewed by 1447
Abstract
High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership [...] Read more.
High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership degrees of 1. Firstly, by defining the medium of an RSTIT2-FV, its membership function, credibility distribution, and inverse distribution are analytically and explicitly expressed. Secondly, an operational law for fuzzy arithmetic operations regarding mutually independent RSTIT2-FVs is proposed, which can simplify the calculations and directly output the inverse credibility of the functions. Afterwards, the operational law is applied to define the expected value operator of the IT2-FV and prove the linearity of the operator. Finally, some comparative examples are provided to verify the efficiency of the proposed operational law. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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12 pages, 265 KiB  
Article
Sine Entropy of Uncertain Random Variables
by Gang Shi, Rujun Zhuang and Yuhong Sheng
Symmetry 2021, 13(11), 2023; https://doi.org/10.3390/sym13112023 - 26 Oct 2021
Cited by 3 | Viewed by 1534
Abstract
Entropy is usually used to measure the uncertainty of uncertain random variables. It has been defined by logarithmic entropy with chance theory. However, this logarithmic entropy sometimes fails to measure the uncertainty of some uncertain random variables. In order to solve this problem, [...] Read more.
Entropy is usually used to measure the uncertainty of uncertain random variables. It has been defined by logarithmic entropy with chance theory. However, this logarithmic entropy sometimes fails to measure the uncertainty of some uncertain random variables. In order to solve this problem, this paper proposes two types of entropy for uncertain random variables: sine entropy and partial sine entropy, and studies some of their properties. Some important properties of sine entropy and partial sine entropy, such as translation invariance and positive linearity, are obtained. In addition, the calculation formulas of sine entropy and partial sine entropy of uncertain random variables are given. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
11 pages, 267 KiB  
Article
Delayed Renewal Process with Uncertain Random Inter-Arrival Times
by Xiaoli Wang, Gang Shi and Yuhong Sheng
Symmetry 2021, 13(10), 1943; https://doi.org/10.3390/sym13101943 - 15 Oct 2021
Cited by 1 | Viewed by 1268
Abstract
An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. [...] Read more.
An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
13 pages, 3151 KiB  
Article
Research on Airport Scheduling of FGAP Multi-Objective Programming Model Based on Uncertainty Theory
by Xijian Hu, Jiaqi Teng, Wei Wu, Yan Li and Yuhong Sheng
Symmetry 2021, 13(10), 1915; https://doi.org/10.3390/sym13101915 - 12 Oct 2021
Cited by 6 | Viewed by 1424
Abstract
Based on the current background of airport management and flight-gate scheduling in China, this paper takes Shanghai Pudong International Airport’s flight number of the rising and landing aircraft in a certain day as the research object, and it establishes an uncertain FGAP (Flight-Gate [...] Read more.
Based on the current background of airport management and flight-gate scheduling in China, this paper takes Shanghai Pudong International Airport’s flight number of the rising and landing aircraft in a certain day as the research object, and it establishes an uncertain FGAP (Flight-Gate Assignment Problem) multi-objective programming model under the framework of uncertainty theory. Using genetic algorithm to solve the model, the specific flight-gate assignment scheduling plan is given. The research results show that the model in this paper can effectively alleviate the problems, such as unbalanced flight-gate allocation and excessive operating pressure of a single gate, in the conventional model, and make the allocation and scheduling more reasonable and efficient. Finally, we give the future application of uncertainty theory in finance and management, as well as the prospect of combining it with symmetry in physics. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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10 pages, 380 KiB  
Article
The Uncertain Network DEA Model for Two-Stage System with Additive Relationship
by Bao Jiang, Chen Yang and Jian Li
Symmetry 2021, 13(10), 1893; https://doi.org/10.3390/sym13101893 - 07 Oct 2021
Cited by 2 | Viewed by 1630
Abstract
When decision making units (DMUs) have internal structures with imprecise inputs and outputs, uncertain network data envelopment analysis (UNDEA) is appropriate to deal with the efficiency evaluation of these DMUs. However, a deep insight into clarifying the power’s differences between the internal structures [...] Read more.
When decision making units (DMUs) have internal structures with imprecise inputs and outputs, uncertain network data envelopment analysis (UNDEA) is appropriate to deal with the efficiency evaluation of these DMUs. However, a deep insight into clarifying the power’s differences between the internal structures of DMUs is a deficiency in the current UNDEA model. To address this issue, in this paper, we propose a new UNDEA model by differentiating the power asymmetry of each sub-stage with assumption of a two-stage system and demonstrate an additive relationship between stage 1 and stage 2 of each DMU. Moreover, the equivalent form and its proof of the new model are also presented for accurate calculation. Finally, a numerical example reflecting three different additive relationships between two sub-stages of DMUs is given to illustrate the results of evaluation. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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21 pages, 679 KiB  
Article
The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables
by Guang Wang, Yixuan Shen, Yujiao Jiang and Jiahao Chen
Symmetry 2021, 13(8), 1428; https://doi.org/10.3390/sym13081428 - 04 Aug 2021
Cited by 2 | Viewed by 1466
Abstract
As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical [...] Read more.
As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory)
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