Mathematics

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35 pages, 421 KB

Open AccessFeature PaperArticle

A Three-Dimensional Product-Based Circular Intuitionistic Fuzzy Potential Method for Transportation Problems

by Velichka Traneva and Stoyan Tranev

Mathematics 2026, 14(8), 1380; https://doi.org/10.3390/math14081380 - 20 Apr 2026

Transportation problems constitute a fundamental class of optimization models; however, real-world applications involve uncertainty, hesitation, and expert disagreement that cannot be adequately captured by deterministic or classical fuzzy approaches. This paper proposes a three-dimensional circular intuitionistic fuzzy potential method (3D–CIFMODI), which extends the [...] Read more.

Transportation problems constitute a fundamental class of optimization models; however, real-world applications involve uncertainty, hesitation, and expert disagreement that cannot be adequately captured by deterministic or classical fuzzy approaches. This paper proposes a three-dimensional circular intuitionistic fuzzy potential method (3D–CIFMODI), which extends the classical MODI framework to Circular Intuitionistic Fuzzy Triples (C-IFTs) through radius-aware operations and indexed matrix representations. Unlike existing circular intuitionistic fuzzy transportation methods, which are primarily feasibility-driven, the proposed approach introduces a dual-based optimality framework based on circular reduced costs, preserving the full structure of uncertainty without reducing it to crisp equivalents. The method retains polynomial-time computational complexity

O (m n (m + n))

, i.e.,

O (n^{3})

for square problems, with only a constant computational overhead due to circular operations. A numerical case study demonstrates the effectiveness and robustness of the proposed framework. Furthermore, a comparative analysis between classical intuitionistic fuzzy (IFS) and circular intuitionistic fuzzy (C-IFS) representations shows that incorporating the radius parameter significantly improves discrimination capability, solution stability, and interpretability. The results confirm that the proposed method provides a unified, interpretable, and computationally efficient framework for solving multi-layer transportation problems under circular intuitionistic fuzzy uncertainty. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

29 pages, 6216 KB

Open AccessFeature PaperArticle

Model Validation for Multivariate Functional Responses via Autoencoder-Based Dual-Layer Feature Extraction

by Dengyu Wu, Xiaodong Zhang, Daobo Sun, Haidong Lin, Jinhui Li and Baoqiang Zhang

Mathematics 2026, 14(4), 674; https://doi.org/10.3390/math14040674 - 13 Feb 2026

Viewed by 310

Abstract

Model validation for complex simulation models with multivariate functional responses poses significant challenges, as it involves the dual coupling of physical correlations among variables and field correlations in time-series data. A novel Autoencoder-based Dual-Layer Feature Extraction (AE-DLFE) method is proposed. The first layer [...] Read more.

Model validation for complex simulation models with multivariate functional responses poses significant challenges, as it involves the dual coupling of physical correlations among variables and field correlations in time-series data. A novel Autoencoder-based Dual-Layer Feature Extraction (AE-DLFE) method is proposed. The first layer uses joint principal component analysis to decouple physical correlations, while the second layer develops an Autoencoder-improved Feature Selective Validation (AE-FSV) method that adaptively extracts features of time-series data and measures feature discrepancies via deep representation learning. On this basis, a new validation metric named U-PCDM (Uncertainty Principal Component Difference Measure) is developed to quantify the discrepancies between simulation and experiment under uncertainty. Theoretical analysis confirms the boundedness and unique temporal permutation sensitivity of the proposed metric. Case study results demonstrate that the proposed AE-FSV enhances the evaluation accuracy of traditional FSV on transient data. Furthermore, compared to benchmark methods such as MD-pooling, the U-PCDM metric significantly improves computational efficiency—especially in high-dimensional scenarios—while maintaining consistent model rankings. This work effectively addresses the heterogeneous correlation coupling issue, offering a robust quantitative tool for model validation. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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21 pages, 3490 KB

Open AccessFeature PaperArticle

Inverse Operator over Index Matrices

by Veselina Bureva, Krassimir Atanassov, Vassia Atanassova and Ivo Umlenski

Mathematics 2026, 14(4), 615; https://doi.org/10.3390/math14040615 - 10 Feb 2026

Viewed by 432

Abstract

Index matrices are an extension of the ordinary matrices with explicitly assigned index sets on both their rows and columns, forming an advanced mathematical structure used for specialized data modeling and problem-solving. In the present paper, a new operator, called “Inverse” operator, is [...] Read more.

Index matrices are an extension of the ordinary matrices with explicitly assigned index sets on both their rows and columns, forming an advanced mathematical structure used for specialized data modeling and problem-solving. In the present paper, a new operator, called “Inverse” operator, is defined over hierarchical index matrices with a description of its software implementation. This operator has no analogue in the classical theory of matrices. Its application allows various restructuring of datasets with multiple criteria, in order to outline data stratifications in the best possible way serving the particular decision maker’s needs. As an illustrative example for the new operator, a dataset of the recorded blood donors in Bulgaria is provided, with discussions of the different stratifications and perspectives in which the dataset can be rearranged using the “Inverse” operator. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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36 pages, 3446 KB

Open AccessFeature PaperArticle

Neurodegenerative Disease-Specific Relations Between Temporal and Kinetic Gait Features Identified Using InterCriteria Analysis

by Irena Jekova, Vessela Krasteva and Todor Stoyanov

Mathematics 2026, 14(2), 340; https://doi.org/10.3390/math14020340 - 19 Jan 2026

Viewed by 557

Abstract

Gait analysis is a non-invasive, cost-effective method for detecting subtle motor changes in neurodegenerative disorders. This study uses an exploratory approach to identify temporal–kinetic gait feature relationships specific to amyotrophic lateral sclerosis (ALS) and Huntington (HUNT) and Parkinson (PARK) disease versus healthy controls [...] Read more.

Gait analysis is a non-invasive, cost-effective method for detecting subtle motor changes in neurodegenerative disorders. This study uses an exploratory approach to identify temporal–kinetic gait feature relationships specific to amyotrophic lateral sclerosis (ALS) and Huntington (HUNT) and Parkinson (PARK) disease versus healthy controls (CONTROL) using recent advances in InterCriteria Analysis (ICrA). The novelty lies in the (i) comprehensive temporal–kinetic feature set, (ii) use of ICrA to characterize inter-feature coordination patterns at population and disease-group levels and (iii) interpretation in a neuromechanical context. Forty-one temporal/kinetic features were extracted from left/right leg ground reaction force and rate-of-force-development signals, considering laterality, gait phase (stance, swing, double support), magnitudes, waveform correlations, and inter-/intra-limb asymmetries. The analysis included 14,580 steps from 64 recordings in the Gait in Neurodegenerative Disease Database: 16 CONTROL (4054 steps), 13 ALS (2465), 20 HUNT (4730), 15 PARK (3331). Sensitivity analysis identified strict consonance thresholds (μ ≥ 0.75, ν ≤ 0.25), selecting <5% strongest inter-feature relations from 820 feature pairs: population level (16 positive, 14 negative), group-level (15–25 positive, 9–14 negative). ICrA identified group-specific consonances—present in one group but absent in others—highlighting disease-related alterations in gait coordination: ALS (15/11 positive/negative, disrupted bilateral stride coordination, prolonged stance/double-support, decoupled stride/cadence, desynchronized force-generation patterns—reflecting compensatory adaptations to muscle weakness and instability), HUNT (11/7, severe temporal–kinetic breakdown consistent with gait instability—loss of bilateral coordination, reduced swing time, slowed force development), PARK (1/2, subtle localized disruptions—prolonged stance and double-support intervals, reduced force during weight transfer, overall coordination remained largely preserved). Benchmarking vs. Pearson correlation showed strong linear agreement (R² = 0.847, p < 0.001), confirming that ICrA captures dominant dependencies while moderating the correlation via uncertainty. These results demonstrate that ICrA provides a quantitative, interpretable framework for characterizing gait coordination patterns and can guide principled feature selection in future predictive modeling. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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28 pages, 6981 KB

Open AccessArticle

Parameter Estimation and Forecasting Strategies for Cholera Dynamics: Insights from the 1991–1997 Peruvian Epidemic

by Hamed Karami, Gerardo Chowell, Oscar J. Mujica and Alexandra Smirnova

Mathematics 2025, 13(10), 1692; https://doi.org/10.3390/math13101692 - 21 May 2025

Cited by 1 | Viewed by 1493

Abstract

Environmental transmission is a critical driver of cholera dynamics and a key factor influencing model-based inference and forecasting. This study focuses on stable parameter estimation and forecasting of cholera outbreaks using a compartmental SIRB model informed by three formulations of the environmental transmission [...] Read more.

Environmental transmission is a critical driver of cholera dynamics and a key factor influencing model-based inference and forecasting. This study focuses on stable parameter estimation and forecasting of cholera outbreaks using a compartmental SIRB model informed by three formulations of the environmental transmission rate: (1) a pre-parameterized periodic function, (2) a temperature-driven function, and (3) a flexible, data-driven time-dependent function. We apply these methods to the 1991–1997 cholera epidemic in Peru, estimating key parameters; these include the case reporting rate and human-to-human transmission rate. We assess practical identifiability via parametric bootstrapping and compare the performance of each transmission formulation in fitting epidemic data and forecasting short-term incidence. Our results demonstrate that while the data-driven approach achieves superior in-sample fit, the temperature-dependent model offers better forecasting performance due to its ability to incorporate seasonal trends. The study highlights trade-offs between model flexibility and parameter identifiability and provides a framework for evaluating cholera transmission models under data limitations. These insights can inform public health strategies for outbreak preparedness and response. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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45 pages, 12946 KB

Open AccessArticle

Emphasizing Grey Systems Contribution to Decision-Making Field Under Uncertainty: A Global Bibliometric Exploration

by Andra Sandu, Paul Diaconu, Camelia Delcea and Adrian Domenteanu

Mathematics 2025, 13(8), 1278; https://doi.org/10.3390/math13081278 - 13 Apr 2025

Cited by 10 | Viewed by 1976

Abstract

Grey systems are applied in numerous domains, proving a high efficiency in predicting and investigating complex systems, where data is insufficient, unknown, or partially known. The systems have a strong contribution in the decision-making field under uncertainty, by identifying the connection between variables [...] Read more.

Grey systems are applied in numerous domains, proving a high efficiency in predicting and investigating complex systems, where data is insufficient, unknown, or partially known. The systems have a strong contribution in the decision-making field under uncertainty, by identifying the connection between variables and optimizing the process of choosing the strategies. With time, the methods offered by the grey systems theory have faced a continuous adoption process in various research fields associated with decision-making. In this context, this paper aims to provide an in-depth bibliometric exploration, focusing on a filtered dataset, gathered from Clarivate Analytics’ Web of Science Core Collection database (WoS) for the purpose of better highlighting the adoption process faced by grey systems theory in the decision-making field under uncertainty. Based on the extracted dataset, the value registered for the annual growth rate is 17.1%, proving that the scientific community’s focus in this field is significant, and it has maintained academics’ interest for a long time. Also, the results of the bibliometric analysis showed that the Journal of Grey System was the most relevant source, while Sifeng Liu provided the greatest contribution to the field based on the number of published papers. Nanjing University of Aeronautics and Astronautics is ranked first in the top of most relevant affiliation based on the number of published papers, while China—the homeland of grey systems theory—assumes the leading contributor country place. The review of the top 10 most cited papers revealed the advantages of using grey systems theory in decision-making field under uncertainty. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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24 pages, 464 KB

Open AccessArticle

Probabilistic Linguistic Multiple Attribute Group Decision-Making Based on a Choquet Operator and Its Application in Supplier Selection

by Weijia Kang, Xin Liang and Yan Peng

Mathematics 2025, 13(5), 740; https://doi.org/10.3390/math13050740 - 25 Feb 2025

Cited by 1 | Viewed by 1186

Abstract

As an enhanced version of traditional linguistic term sets, Probabilistic Linguistic Term Sets (PLTS) incorporate probabilistic information, thereby offering a more robust approach to Multiple Attribute Group Decision-Making (MAGDM) and significantly improving its efficacy. This paper proposes two novel information aggregation operators for [...] Read more.

As an enhanced version of traditional linguistic term sets, Probabilistic Linguistic Term Sets (PLTS) incorporate probabilistic information, thereby offering a more robust approach to Multiple Attribute Group Decision-Making (MAGDM) and significantly improving its efficacy. This paper proposes two novel information aggregation operators for PLTS to address MAGDM problems in the PLTS context. Firstly, we introduce Choquet integral-based generalized arithmetic and geometric operators, which are designed to fuse decision information expressed by different PLTSs, thereby more comprehensively considering the interrelationships among various attributes. Subsequently, we further define measures of group consistency and inconsistency for individual decision information in MAGDM, which are used to determine the information weights of decision-makers. Finally, the group decision information is aggregated using the proposed PLTS aggregation operators. The effectiveness as well as the applicability of the developed method are illustrated through numerical examples and comparative analysis. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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24 pages, 420 KB

Open AccessEditor’s ChoiceArticle

A Group Consensus Measure That Takes into Account the Relative Importance of the Decision-Makers

by József Dombi, Jenő Fáró and Tamás Jónás

Mathematics 2025, 13(3), 526; https://doi.org/10.3390/math13030526 - 5 Feb 2025

Cited by 1 | Viewed by 2072

Abstract

In group decision making, the knowledge, skills, and experience of the decision-makers may not be at the same level. Hence, the need arises to take into account not only the opinion, but also the relative importance of the opinion of each decision-maker. These [...] Read more.

In group decision making, the knowledge, skills, and experience of the decision-makers may not be at the same level. Hence, the need arises to take into account not only the opinion, but also the relative importance of the opinion of each decision-maker. These relative importance values can be treated as weights. In a group decision making situation, it is not only the weighted aggregate output that matters, but also the weighted measure of the group consensus. Noting that weighted group consensus measures have not yet been intensely studied, in this study, based on well-known requirements for non-weighted consensus measures, we define six reasonable requirements for the weighted case. Then, we propose a function family and prove that it satisfies the above requirements for a weighted consensus measure. Hence, the proposed measure can be used in group decision making situations where the decision-makers have various weight values that reflect the relative importance of their opinions. The proposed weighted consensus measure is based on the fuzziness degree of the decumulative distribution function of the input scores, taking into account the weights. Hence, it may be viewed as a weighted adaptation of the so-called fuzziness measure-based consensus measure. The novel weighted consensus measure is determined by a fuzzy entropy function; i.e., this function may be regarded as a generator of the consensus measure. This property of the proposed weighted consensus measure family makes it very versatile and flexible. The nice properties of the proposed weighted consensus measure family are demonstrated by means of concrete numerical examples. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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