Evaluating cars is a common multi-attribute decision-making (MADM) problem because it requires considering multiple criteria at once. However, assessing these attributes often involves subjective judgments and incomplete data. Consequently, the decision-making process contains inherent uncertainties that necessitate advanced models to handle imprecise and ambiguous evaluations. This paper is aimed at developing novel complex hybrid hypersoft set structures: the possibility complex fuzzy hypersoft set (pCFHS), the possibility complex intuitionistic fuzzy hypersoft set (pCIFHS), and the possibility complex neutrosophic hypersoft set (pCNHS), by simultaneously integrating complex fuzzy hypersoft sets, complex intuitionistic fuzzy hypersoft sets, and complex neutrosophic hypersoft sets with possibility degree-based arrangements. By using the aggregations of three proposed frameworks, three algorithms are presented and validated through the MADM problem of car evaluation. This evaluation considers four main attributes along with eight corresponding sub-attribute values. Importantly, the study incorporates fuzzy values to reflect degrees of possibility. This possibility grade is meant to assess the biasing trends of decision-makers during the evaluation. In the end, the rankings of the three proposed algorithms are compared to identify the most reliable structure.
Data envelopment analysis (DEA) has been widely used to assess the technical performances of decision-making units with inputs and outputs. However, there are few research studies on performance assessment in service-oriented (without inputs) systems in the presence of contextual variables. In the theoretical part of the paper, we develop a without-input-DEA model with contextual variables for situations where we face variability and uncertainty in the data. In the application part, the work process in healthcare sector is considered as a service-oriented process with contextual variables, desirable and undesirable outputs. A two-step procedure consisting of the use of the proposed model along with a regression analysis is developed to evaluate performance in the healthcare sector and the impact of contextual variables on efficiency. The results revealed that 27% of the hospitals studied were classified as inefficient. We also found that hospitals are inefficient in terms of mortality rate and infectious waste. Regarding the Number of inpatients, on average, the performance of the hospitals was acceptable. The coefficient was 0.0046, indicating a direct impact on efficiency.
Given the nonlinearity, instability and uncertainty of wind speeds, effectively harnessing wind energy to ensure stable operation of power systems is critical. Based on uncertainty theory, this paper proposes a hybrid CEEMDAN-UAR-BiLSTM-Attention wind speed prediction model. The CEEMDAN algorithm decomposes the original wind speed time series into multiscale Intrinsic Mode Function (IMF) components, thereby reducing data complexity and mitigating noise interference. Subsequently, each component is modeled separately, the UAR model captures the linear trend of low-frequency components, while high-frequency components are fed into a bidirectional LSTM equipped with an attention mechanism. By leveraging the bidirectional structure of LSTM, the model reveals long-term dependencies while focusing on critical predictive information. During the prediction process, the combined model achieves reduced RMSE and MAE, attaining a coefficient of determination [Formula: see text] of 0.89. This provides robust support for wind power planning and operation, showcasing significant practical value and broad application prospects.
This paper extends the research and applications of the uncertain Gordon–Schaefer model. We examine two stability criteria for the equation solutions: stability in mean and stability in [Formula: see text]-th moment. Then we collect [Formula: see text] data points from the International Pacific Halibut Commission and analyze the catch data using the uncertain Gordon–Schaefer equation. Through the uncertain hypothesis test, we demonstrate that the uncertain Gordon–Schaefer equation provides an adequate fit. Finally, we establish why the stochastic Gordon–Schaefer equation fails to characterize the halibut catch data effectively.
Crude oil price time series data exhibit high complexity, strong correlation and long-term dependence. The accurate prediction of crude oil price trend has important significance for formulating macroeconomic policy, national energy policy and sustainable development policy. In this study, 228 monthly WTI crude oil price datasets from January 2001 to December 2019 were used. To estimate crude oil price time series, both the Uncertain Autoregressive (UAR) model and Long Short-Term Memory (LSTM) model are employed. As to get a better estimation effect, a hybrid model based on UAR and LSTM models is used for prediction. By comparing the Normalized Mean Squared Error (NMSE), Normalized Square Root Error (NRMSE), and Normalized Mean Absolute Error (NMAE) results of the three methods, the hybrid model has been proven to have the best prediction performance and the lowest prediction error. Specifically, the NMSE values of the UAR, LSTM, and UAR-LSTM models are 3.0419, 9.4696, and 1.1351, respectively, which further confirms the significant advantage of the hybrid model in terms of accuracy. The results of all applied models are consistent, which indicates the success of crude oil price prediction.
This study investigates the conditions for the existence and uniqueness of solutions to interval Fredholm integral equations (IFIEs). To achieve this objective, a metric is introduced on the space of continuous parameterized interval-valued functions (IV-functions) defined over a compact interval. Preliminary concepts, including self-mappings, contraction mappings on this function space and the fixed-point theorem are discussed to establish the theoretical foundation. The standard formulation of IFIEs is then presented. Utilizing the contraction mapping principle and the associated fixed-point theorem, sufficient conditions ensuring the existence and uniqueness of solutions to IFIEs are derived, constituting the primary contribution of this work. Finally, a systematic solution procedure for IFIEs is outlined and illustrated through numerical examples.
Susceptibility analysis is a useful technique that aids medical professionals in diagnosing and treating brain tumors as well as determining the severity of a patient’s suspected brain tumor. This approach has proven particularly beneficial in developing countries with limited resources and medical facilities. By utilizing set-based operations of an arithmetical model, specifically the fuzzy parameterized complex intuitionistic fuzzy hypersoft expert set (FPCIFHSES), this study seeks to develop a reliable multi-attribute decision support system for evaluating patients’ susceptibility to brain tumors. The FPCIFHSES is thought to be more dependable and comprehensive when managing information-based challenges due to its intricate components and fuzzy parameterization, which are intended to address the data’s periodic form and unclear characteristics (sub-characteristics), correspondingly. Based on the professional judgments of experts, the suggested FPCIFHSES-susceptibility framework approximates certain appropriate forms of brain tumors in terms of the most pertinent signs (characteristics) in units of complex intuitionistic fuzzy numbers (CIFNs). The scores for these kinds of cancers are calculated using a core matrix that connects them to fuzzy parameterized multi-argument-based pairs once the fuzzy parameterized values of the multi-argument-based tuples have been determined and the CIFNs have been converted into fuzzy values. The membership of score values in [Formula: see text] is used to assess a patient’s susceptibility.
Picture fuzzy graph is the generalization of fuzzy graph and intuitionistic fuzzy graph. [Formula: see text]s provide a more flexible and expressive way to represent uncertainty and vagueness in graph-based data, compared to traditional fuzzy graphs. Total domination is an important topic for its entire domination characteristics. In this paper, some domination parameters like complementary nil domination number, independent domination number and total dominating set, total domination number are introduced in a picture-fuzzy environment. We have introduced picture fuzzy vertex cardinality, picture fuzzy edge cardinality, cardinality of a [Formula: see text]. Studied their nature with domination parameters. A novel method to get a join of two [Formula: see text]s has been discussed. A new definition of complement of a [Formula: see text] is presented. Few findings regarding the complementary nil dominating set, complementary nil domination number in bipartite [Formula: see text], complete bipartite [Formula: see text] are developed. The concept of enclave is introduced in [Formula: see text] and proved some relations between enclave and complementary nil dominating set. Also, total domination number, and the independent domination number of complete, complement, and join of [Formula: see text]s have been presented. We presented some theorems associated with them. We proved that domination, independent domination and total domination number are equal for a complete [Formula: see text]. Finally, we demonstrate an application of the total domination concept to identify the most critical junction in a railway network.
Within the framework of uncertainty theory, this paper employs a Caputo–Hadamard uncertain fractional differential equation to model the dynamics of carbon emission allowance prices and investigates the pricing of carbon swaptions. Based on the constructed model, analytical expressions for the prices of carbon swaptions are derived. Actual trading data from the Shanghai Environment and Energy Exchange’s national carbon market for December 2024 to December 2025 are selected to estimate the model’s unknown parameters using the method of moments, and the model’s validity is verified through uncertainty hypothesis testing. The results indicate that the Caputo–Hadamard uncertain fractional differential equation effectively captures the dynamic characteristics of long-term carbon emission allowance prices. The proposed pricing methods demonstrate good applicability and can provide theoretical foundations and quantitative tools for risk management and contract design for carbon market participants under conditions of information asymmetry and market incompleteness.
As a type of differential equation driven by uncertain processes, uncertain differential equations are fundamental to modeling dynamic systems under uncertainty. While significant research attention has been directed toward this field, the theoretical foundations for their more complex forms, specifically multi-dimensional and higher-order uncertain differential equations, require further rigorous development. This paper addresses this gap by establishing comprehensive existence and uniqueness theorems for solutions to both multi-dimensional and higher-order uncertain differential equations under the condition of local Lipschitz continuity. We provide rigorous proofs for these theorems, thereby solidifying the mathematical underpinnings necessary for the application of these models.
Uncertain differential equations have been expanded in many directions as an important modeling tool. As a class of differential equations including higher-order derivatives of uncertain processes, higher-order uncertain differential equations are widely used in the modeling of dynamic systems in uncertain environments. The equations usually involve unknown parameters to be estimated, and parameter estimation methods based on the difference format do not work when the time intervals between observations are not short enough. In this paper, residual-based moment estimation is applied to higher-order uncertain differential equations, some examples are given to illustrate the parameter estimation methods, and uncertain hypothesis test is used to verify whether the higher-order uncertain differential equations fit the observations.
This paper introduces the Caputo fractional differential equation for interval valued functions. A fractional differential equation incorporates memory sense through an iterated kernel for describing dynamical systems. Impreciseness exists in the process of quantification and analysis of the involved variables influencing such physical processes. In other words, memory and imprecision may coexist, necessitating the study of an imprecise fractional differential equation. Interval numbers and interval valued functions are mathematical tools to manifest uncertainty due to the variance of decision parameters between ranges. In this paper, an imprecise fractional differential equation is studied under Type 2 interval uncertainty, a generalization of interval uncertainty. This paper analyzes conditions for the existence of a unique solution of the Type 2 interval valued Caputo fractional differential equations. Riemann–Liouville fractional integral equations and metric spaces for Type 2 interval numbers and interval valued functions are employed for establishing results. Examples of linear and nonlinear Type 2 interval valued Caputo fractional differential equations are discussed, ensuring a smooth extension of the interval fractional differential equation to a wider domain. Economic and biological models are hinted at as possible applications of this proposed theory.
Precise forecasting of fluctuations in carbon allowance valuations is critical for shaping environmental policy and for bolstering the effectiveness of market-based regulatory mechanisms. Advanced statistical and machine-learning techniques afford regulators the capacity to fine-tune carbon taxation schemes, enhance the operational efficiency of emissions trading frameworks, and steer financial resources toward low-carbon development projects with greater assurance. This study examines the China Emissions Trading Scheme (CHNTS) one of China’s pioneering carbon markets established under the broader national decarbonization strategy and presents an innovative predictive model based on Gaussian Process Regression (GPR) whose hyperparameters are optimized through a Bayesian framework. By dynamically adjusting to latent market behaviors and unobserved structural shifts, this method adapts more responsively to evolving trading patterns. Our empirical investigation utilizes daily settlement data for China Emission Allowances spanning July 16, 2021, through April 9, 2025 a timeframe marked by key regulatory amendments, market maturation phases, and changing participant conduct as the scheme integrated into the wider national carbon pricing system. Model validation is performed on an out-of-sample window from June 28, 2024, to April 9, 2025, yielding notable performance metrics: a relative root-mean-square error (RRMSE) of 1.0771%, Root-Mean-Square Error (RMSE) of 1.0212, Mean Absolute Error (MAE) of 0.6773, and a Correlation Coefficient (CC) reaching 98.604%. To the best of our knowledge, this represents the first deployment of GPR in the context of China’s carbon trading exchanges. Beyond enriching theoretical understanding of price discovery in emergent emissions markets, the proposed approach provides a flexible analytical template that could readily be applied to analogous cap-and-trade systems worldwide.
This research adopts a cohesive analytical strategy, combining vector error-correction modeling (VECM) with directed acyclic graphs (DAGs), to rigorously assess evolving connections in monthly home prices across six key Guangdong cities. These cities include Dongguan, Zhongshan, Foshan, Guangzhou, Huizhou and Zhuhai. The analysis spans a significant period of nearly thirteen years (November 2011–July 2024). Supporting computational procedures determine the fundamental DAG structure: Initially, the PC algorithm identifies a preliminary set of potential causal relationships; subsequently, the LiNGAM (Linear Non-Gaussian Acyclic Model) approach leverages non-Gaussian data characteristics to eliminate directional ambiguity, establishing a conclusive causal sequence. Leveraging this DAG-based causal order, detailed innovation accounting — including impulse response functions — quantifies dynamics and shock effects within the price network. Findings reveal intricate transmission channels and diverse adjustment speeds within the provincial housing market following external disturbances. Empirical results show that policies designed to increase real estate values in Dongguan, Foshan, Huizhou and Zhuhai can substantially boost wider market recovery throughout Guangdong, attributable to their central network roles. For the remaining two cities exhibiting lower systemic influence, however, insights recommend localized revitalization approaches as more direct and efficient than provincial stimulus or measures targeting core areas, highlighting the necessity for policy differentiation grounded in spatial network functions.
Chemical graph theory is an interdisciplinary field that integrates fuzzy-graph-theoretical and computational methods to model molecular structures as fuzzy graphs and to address associated mathematical problems. Topological indices assign numerical invariants to network structures; among them, the Sombor index, originally introduced in chemical graph theory, provides a useful measure for quantifying the structure of molecular graphs within a fuzzy-graph framework. In this paper, we investigate bounds for the Sombor index across several graph families and operations, including edge addition and deletion, broom graphs, fuzzy star graphs on n vertices, complete bipartite fuzzy graphs, and the star graph (K 1,t ). We also examine the relationship between the Sombor index and various properties of alkanes and octane isomers. Furthermore, we demonstrate a significant correlation between this index and multiple thermodynamic parameters, such as heat of vaporization, entropy, acentric factor, and enthalpy of vaporization, while observing a relatively weak correlation with the heat capacity of octane isomers. Finally, we apply the Sombor index in fuzzy graphs (SOF([Formula: see text] )) to identify and rank Indian states according to their crime rates. These results highlight the broad applicability of the Sombor index in both chemical graph theory and real-world network analysis.
Parisian option is a special type of barrier option. A general barrier option only requires a hitting time reaching the barrier level to activate (or terminate) the contract, which may be easily influenced by some short-term fluctuations in the market, when the price is close to the barrier. Thus, Parisian option is created to avoid exposure to such kind of risk. It requires the price should maintain below or above a certain level for a sustained period to trigger the contract. Considering the phenomenon of mean reversion in the market, this paper mainly investigates four different Parisian options’ pricing formulas based on the uncertain exponential Ornstein–Uhlenbeck model and designs the algorithms to calculate the price of the option. Besides, several numerical examples are given in this paper.
Fuzzy nonlinear equations are used to model real-world problems. Due to the nonlinearity, sometimes obtaining a fuzzy solution is challenging. This paper deals with two methods for solving such equations, the first is the fuzzy Newton–Raphson method (FNRM) and the other is the fuzzy Adomian decomposition method (FADM). Both methods are proposed and demonstrated. We observe that the FADM is better than the FNRM. FADM gives a fuzzy solution also, at the core the fuzzy solution matches with the crisp one. The FNRM initially gives a fuzzy solution, but as iterations increase, the solution does not remain fuzzy, and even the fuzzy solution does not match with the crisp one. Last, one numerical illustration is solved by both methods.
This paper explores premium principles in uncertain environment. First, we introduced several existing uncertain premium principles, including the uncertain net premium principle, the uncertain variance premium principle and the uncertain standard deviation premium principle. But all of them have a little shortcomings. So we proposed uncertain Dutch premium principle which is better than other principles. Some properties of uncertain Dutch premium principle are proved, such as risk loading and continuity. Finally, we give some examples to calculate the uncertain Dutch premium principle and compare it with others.
Due to the inexactness and uncertainty, most of the parameters involved in a supply chain system are imprecise. So, there arises a question — how can one handle such imprecise parameters to formulate a supply chain model under uncertainty? To find the answer to this question, this work focuses on the optimality of a classical supply chain model under interval uncertainty using interval parametric approach. In this chain, a manufacturer and a retailer are involved. Here, inventories of both manufacturer and retailer are considered with interval valued inventory components. The average costs of all members of the chain are obtained using parametric interval mathematics. Now, there arise another question — how can interval valued average costs of the chain under interval uncertainty be minimized? To do so, an Arithmetic Mean-Geometric Mean (AM–GM) inequality-based parametric geometric optimization approach has been introduced to find the optimality conditions for minimizing the manufacturer’s, retailer’s as well as their joint interval-valued average costs in parametric form. After that, using the Stackelberg game principle, interval valued minimum average costs for both the manufacturer and retailer are obtained in parametric form. Finally, theoretical findings are supported by numerical examples demonstrating the feasibility of the model and stability of the interval parametric approach. This work combines parametric interval analysis, inequality-based parametric optimization and game theory in supply chain cost modeling and contributes novel insights into uncertain decision-making.
In this paper, we propose a technique to solve a class of problems under uncertainty using the theory of fuzzy soft sets. First, we interpret and subsequently extract a mathematical form of such a decision making problem whose manipulation demands operations on fuzzy soft set and/or relation. The motivation of this research is to develop a mechanism which would follow the logic of the underlying set theory for a better understanding of the task of decision making under uncertainty. This will enable the user to make informed and effective choices while handling uncertainty due to imprecision. An algorithm is designed to facilitate the said task of decision making. Concrete examples are considered to show the consistency of the proposed task and a comparison is presented to establish effectiveness of the proposal. In the process, we use [Formula: see text]-norm of fuzzy soft sets to generate a resultant fuzzy set from a fuzzy soft set. On defuzzification of the resultant fuzzy set we obtained the desired result.
