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Symmetry
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The Fusion of Fuzzy Sets and Optimization Using Symmetry
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The Fusion of Fuzzy Sets and Optimization Using Symmetry

A Special Issue of Symmetry (ISSN 2073-8994) belonging to the section "B: Mathematics".

Deadline for manuscript submissions: 31 December 2026 | Viewed by 8014

Editor

Prof. Dr. Hsien-Chung Wu

E-Mail Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan
Interests: fuzzy sets; optimization; nonlinear analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical Programming is a discipline that helps decision-makers to make better decisions. The better decision often includes maximizing the profit, yield, or performance, or minimizing the cost, loss, or risk. In this case, the advanced analytical methods arising from mathematical analyses play an important role. Creating the suitable optimization problems that are heavily based on the data turns into a very important starting step. When the data in mathematical models involve the imprecision or fuzziness, the fuzzy sets theory becomes helpful in tackling the so-called fuzzy optimization problems.

On the other hand, studying the theory of fuzzy sets sometimes needs to utilize the technique of optimization. For example, the well-known Extension Principle that is frequently used in formulating and investigating the arithmetics of fuzzy numbers and fuzzification of crisp functions is expressed using the concept of supremum. Under suitable conditions, this kind of supremum can be realized as an optimization problem.

The relationship between maximum and minimum in optimization problems is termed “duality”. Studying this kind of symmetric concept will be helpful in solving the fuzzy optimization problems and fuzzy sets problems involving supremum. The topics of this Special Issue include, but are not limited to, the following:

  • Foundation of Fuzzy Sets (Fuzzy Arithmetic Operations, Extension Principle, Possibility Measures, etc.).
  • Fuzzy Logics (Many-Valued Logics, Type-2 Fuzzy Logics, Intuitionistic Fuzzy Logics, etc.).
  • Hybrid Systems (Fuzzy Control, Fuzzy Neural Networks, Genetic Fuzzy Systems, Fuzzy Intelligent Systems, Fuzzy Biomedical Systems, Fuzzy Chaotic Systems, Fuzzy Information Systems, etc.).
  • Nature of Computation (Ant Colony Optimization, Artificial Immune Systems, Genetic Algorithms, Particle Swarm Intelligence, Simulated Annealing, Tabu Search, etc.).
  • Numerical Methods of Fuzzy Optimization (Variants of Newton Method, Interior-point Method, Trust Region Method, etc.).
  • Operations Research and Management Sciences (Fuzzy Games Theory, Fuzzy Inventory Models, Fuzzy Queueing Theory, Fuzzy Scheduling Problems, Fuzzy Decision Making, Fuzzy Data Mining, Fuzzy Clustering, Stochastic Optimization, etc.).

Prof. Dr. Hsien-Chung Wu
Guest Editor

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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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

  • fuzzy games theory
  • fuzzy vector optimization
  • fuzzy goal programming
  • arithmetics of fuzzy intervals
  • fuzzification of crisp functions
  • extension principle

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Published Papers (7 papers)

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Research

28 pages, 622 KB  
Article
Fully Hesitant Fuzzy Bilevel Linear Programming and Its Application to Quantum Communication Resource Allocation
by Jintao Tan, Shengyue Deng, Lan Hu and Yong Zhang
Symmetry 2026, 18(6), 1055; https://doi.org/10.3390/sym18061055 - 18 Jun 2026
Viewed by 369
Abstract
The problem of bilevel decision-making under multi-expert uncertain information is addressed in this paper. Traditional fuzzy bilevel models are unable to accurately quantify expert consensus and capture evaluation hesitation. To overcome these limitations, a fully hesitant fuzzy bilevel linear programming model is proposed, [...] Read more.
The problem of bilevel decision-making under multi-expert uncertain information is addressed in this paper. Traditional fuzzy bilevel models are unable to accurately quantify expert consensus and capture evaluation hesitation. To overcome these limitations, a fully hesitant fuzzy bilevel linear programming model is proposed, in which all coefficients and decision variables are characterized by hesitant fuzzy numbers. By virtue of (α,k)-cuts, the original model is equivalently transformed into an interval-valued bilevel programming problem and further decomposed into best–best and worst–worst sub-models to derive the upper and lower bounds of optimal solutions. Under the Slater constraint qualification, Karush–Kuhn–Tucker (KKT) conditions are adopted to convert the two sub-models into single-level mathematical programs with complementarity constraints (MPCCs), thereby enabling efficient model solving. The proposed method is applied to the resource allocation problem in quantum communication networks. The numerical results demonstrate that the optimal solution interval converges to a unique core value as the membership-level α increases, while a larger consensus parameter k reduces the fuzzy support set without altering the core solution. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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19 pages, 314 KB  
Article
Solving Fuzzy Multiobjective Linear Programming Problems Using Embedding Theorem
by Hsien-Chung Wu
Symmetry 2026, 18(2), 313; https://doi.org/10.3390/sym18020313 - 9 Feb 2026
Cited by 1 | Viewed by 743
Abstract
Fuzzy multiobjective linear programming problems are solved in this paper by considering the concepts of the minimizer and ideal minimizer. A useful embedding theorem is provided to define the partial orderings among the space of fuzzy intervals in R, where the partial [...] Read more.
Fuzzy multiobjective linear programming problems are solved in this paper by considering the concepts of the minimizer and ideal minimizer. A useful embedding theorem is provided to define the partial orderings among the space of fuzzy intervals in R, where the partial orderings are generated by some convex cones. Using the partial orderings, we can propose the concept of minimal elements such that the concepts of the minimizer and ideal minimizer of fuzzy multiobjective linear programming problems can be proposed. The main issue is to separately derive the optimality conditions for the minimizer and ideal minimizer. Finally, we consider some practical problems by providing a specific convex cone such that the minimizer and ideal minimizer can be obtained by solving the conventional linear programming problems using the simplex method. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
22 pages, 2055 KB  
Article
Time-Dependent Route Optimization for Multimodal Hazardous Materials Transport Using Conditional Value-at-Risk Under Uncertainty
by Song Liu, Jingjing Li, Yazhi Lin, Dennis Z. Yu, Yong Peng, Yi Liu and Xianting Ma
Symmetry 2026, 18(2), 292; https://doi.org/10.3390/sym18020292 - 5 Feb 2026
Cited by 1 | Viewed by 729
Abstract
Transporting hazardous materials has low accident probabilities but potentially catastrophic consequences, making effective risk management essential in uncertain conditions such as population distribution, weather, traffic, and multimodal scheduling constraints. This study develops a Conditional Value-at-Risk (CVaR)-based optimization model for multimodal hazardous materials transportation [...] Read more.
Transporting hazardous materials has low accident probabilities but potentially catastrophic consequences, making effective risk management essential in uncertain conditions such as population distribution, weather, traffic, and multimodal scheduling constraints. This study develops a Conditional Value-at-Risk (CVaR)-based optimization model for multimodal hazardous materials transportation that incorporates transportation and transshipment risks, population exposure uncertainty, fixed departure schedules for rail and waterway transport, dual time-window constraints, and limits on the number of transshipments. The model also reflects the decision-maker’s risk aversion and time-varying travel times. To solve this NP-hard problem, an improved chaotic simulated annealing-ant colony optimization (CSAACO) algorithm is proposed. Numerical experiments show that CSAACO outperforms the standard ACO in terms of solution quality and stability. The results demonstrate that the model effectively captures tail risk in dynamic environments and that both the risk aversion coefficient μ and departure time significantly influence route selection. The proposed approach provides an efficient and practical decision-support tool for hazardous materials multimodal transportation planning under uncertainty. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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32 pages, 6406 KB  
Article
Incorporating Parameter Uncertainty into Copula Models: A Fuzzy Approach
by Irina Georgescu and Jani Kinnunen
Symmetry 2025, 17(11), 1892; https://doi.org/10.3390/sym17111892 - 6 Nov 2025
Cited by 2 | Viewed by 1645
Abstract
This paper proposes a fuzzy copula-based optimization framework for modeling dependence structures and financial risk under parameter uncertainty. The parameters of selected copula families are represented as trapezoidal fuzzy numbers, and their α-cut intervals capture both the support and core ranges of plausible [...] Read more.
This paper proposes a fuzzy copula-based optimization framework for modeling dependence structures and financial risk under parameter uncertainty. The parameters of selected copula families are represented as trapezoidal fuzzy numbers, and their α-cut intervals capture both the support and core ranges of plausible dependence values. This fuzzification transforms the estimation of copula parameters into a fuzzy optimization problem, enhancing robustness against sampling variability. The methodology is empirically applied to gold and oil futures (1 January 2015–1 January 2025), comparing symmetric copulas, i.e., Gaussian and Frank and asymmetric copulas, i.e., Clayton, Gumbel and Student-t. The results prove that the fuzzy copula framework provides richer insights than classical point estimation by explicitly expressing uncertainty in dependence measures (Kendall’s τ, Spearman’s ρ) and risk indicators (Value-at-Risk, Conditional Value-at-Risk). Rolling-window analyses reveal that fuzzy VaR and fuzzy CVaR effectively capture temporal dependence shifts and tail severity, with fuzzy CVaR consistently producing more conservative risk estimates. This study highlights the potential of fuzzy optimization and fuzzy dependence modeling as powerful tools for quantifying uncertainty and managing extreme co-movements in financial markets. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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54 pages, 506 KB  
Article
Enhancing Complex Decision-Making Under Uncertainty: Theory and Applications of q-Rung Neutrosophic Fuzzy Sets
by Omniyyah Saad Alqurashi and Kholood Mohammad Alsager
Symmetry 2025, 17(8), 1224; https://doi.org/10.3390/sym17081224 - 3 Aug 2025
Cited by 3 | Viewed by 1190
Abstract
This thesis pioneers the development of q-Rung Neutrosophic Fuzzy Rough Sets (q-RNFRSs), establishing the first theoretical framework that integrates q-Rung Neutrosophic Sets with rough approximations to break through the conventional μq+ηq+νq≤1 constraint of existing [...] Read more.
This thesis pioneers the development of q-Rung Neutrosophic Fuzzy Rough Sets (q-RNFRSs), establishing the first theoretical framework that integrates q-Rung Neutrosophic Sets with rough approximations to break through the conventional μq+ηq+νq≤1 constraint of existing fuzzy–rough hybrids, achieving unprecedented capability in extreme uncertainty representation through our generalized model (Tq+Iq+Fq≤3). The work makes three fundamental contributions: (1) theoretical innovation through complete algebraic characterization of q-RNFRSs, including two distinct union/intersection operations and four novel classes of complement operators (with Theorem 1 verifying their involution properties via De Morgan’s Laws); (2) clinical breakthrough via a domain-independent medical decision algorithm featuring dynamic q-adaptation (q = 2–4) for criterion-specific uncertainty handling, demonstrating 90% diagnostic accuracy in validation trials—a 22% improvement over static models (p<0.001); and (3) practical impact through multi-dimensional uncertainty modeling (truth–indeterminacy–falsity), robust therapy prioritization under data incompleteness, and computationally efficient approximations for real-world clinical deployment. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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19 pages, 293 KB  
Article
On Some Characterization Theorems for New Classes of Multiple-Objective Control Models
by Tareq Saeed and Savin Treanţă
Symmetry 2025, 17(5), 705; https://doi.org/10.3390/sym17050705 - 5 May 2025
Viewed by 604
Abstract
This paper introduces a new class of multiple-objective control models driven by path-independent curvilinear integrals involving the partial derivatives of the control variable. We investigate its solution set by considering a dual problem. Various duality results are formulated and proved in order to [...] Read more.
This paper introduces a new class of multiple-objective control models driven by path-independent curvilinear integrals involving the partial derivatives of the control variable. We investigate its solution set by considering a dual problem. Various duality results are formulated and proved in order to study and investigate the relationships between the set of solutions for these two variational control problems. Specifically, first, we establish that the value of the cost functional associated with the primal model cannot be greater than the value of the cost functional associated with the dual model. Secondly, the following two results present a strong-type duality between the variational models considered. At the end, we illustrate the main findings of the current paper with a numerical example. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
33 pages, 981 KB  
Article
Parallel Smell Agent Optimization (SAO): Collaborative Subpopulations for Accelerated Convergence
by Glykeria Kyrou, Ioannis G. Tsoulos, Anna Maria Gianni and Vasileios Charilogis
Symmetry 2025, 17(4), 592; https://doi.org/10.3390/sym17040592 - 13 Apr 2025
Cited by 4 | Viewed by 1229
Abstract
In the dynamically evolving field of collective computational optimization, modern approaches increasingly incorporate bio-inspired techniques, such as Smell Agent Optimization (SAO), to address complex, high-dimensional problems inherent to contemporary scientific and industrial applications. While these methods are distinguished by their dynamic convergence and [...] Read more.
In the dynamically evolving field of collective computational optimization, modern approaches increasingly incorporate bio-inspired techniques, such as Smell Agent Optimization (SAO), to address complex, high-dimensional problems inherent to contemporary scientific and industrial applications. While these methods are distinguished by their dynamic convergence and heuristic ability to explore vast solution spaces, their growing computational complexity hinders their application in real-world, large-scale scenarios where simultaneous speed and precision are critical. To overcome this challenge, the present research advances a pioneering parallel implementation of SAO, which transcends simple workload distribution by integrating dynamic collaboration mechanisms and intelligent information dispersal among autonomous subpopulations. Concurrently, the method is enriched with innovative rules for exchanging optimal solutions between subpopulations. These rules not only prevent premature convergence to local minima but also establish a continuous flow of information that accelerates the global exploration of the solution space. Experimental validation of the proposed method demonstrated that, through optimized parameterization of the diffusion mechanisms, SAO’s efficiency can exceed 50%, achieving simultaneous reductions in both the number of objective function evaluations and total execution time. This outcome holds particular significance in high-dimensional problems, where balancing computational cost and accuracy is a decisive factor. These findings not only underscore the potential of parallel SAO to deliver sustainable solutions to real-world challenges but also open new horizons in the theory and practice of collective optimization. The implications extend to domains such as large-scale data analysis, autonomous systems, and adaptive resource management, where rapid and precise optimization is paramount. Full article
(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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