Automobiles play a vital role in daily life, providing suitable and efficient transportation for work, school, and errands. They also support essential services like emergency response, goods delivery, and public transportation systems. This increased variety means that car manufacturers are competing intensely to attract customers and maximize their profits. However, making the right choice when buying a car can be challenging due to the wide range of factors to consider. This study introduces a new approach that uses Dombi operators combined with T-spherical fuzzy numbers (T-SFNs) to help improve the decision-making process. This method reduces the uncertainty and imprecision that often comes with decision-making, especially when selecting a car. The aim is to help customers make better, more informed choices and avoid financial difficulties. To achieve this, the study develops several innovative operators namely T-spherical fuzzy Dombi weighted averaging (T-SFDWA), T-spherical fuzzy Dombi ordered weighted averaging (T-SFDOWA), T-spherical fuzzy Dombi weighted geometric (T-SFDWG), T-spherical fuzzy Dombi ordered weighted geometric (T-SFDOWG). These methods offer flexibility, suppleness and can adapt to real-world problems where factors are constantly changing. By managing uncertainty and hesitation effectively, these approaches help decision-makers evaluate complex situations with multiple variables. A practical example, such as choosing a car, demonstrates how these approaches can evaluate important criteria like price, safety, and fuel efficiency. Ultimately, these methods ensure that consumers can make the best decision, even in uncertain and complex situations.

Neutrosophy is a special area of philosophy that explains the nature, genesis and scope of neutralities, like the interactions with diverse ideational hues. It showed the degree of indeterminacy as an independent component that was the extension of an intuitionistic set. In this paper, the interpretation of the linear equation of type $\mathcal{A}\mathcal{X} +\mathcal{B} =\mathcal{C}$ are discussed in a neutrosophic environment. It is observed that the equations $\mathcal{A}\mathcal{X} +\mathcal{B} =\mathcal{C}$, $\mathcal{A}\mathcal{X} =\mathcal{C} -\mathcal{B}$ and $\mathcal{A}\mathcal{X} -\mathcal{C} =-\mathcal{B}$ are same and their solution are also same in crisp sense. But, in the neutrosophic sense, the solutions to the above equations are different. Mathematical operations on intervals are considered for the purpose of solution and analysis. Further, an application of budgeting-financing is described with the help of neutrosophic fuzzy equation.

Accurate and robust image segmentation remains a fundamental challenge in computer vision, particularly in the presence of intensity inhomogeneity, noise, and weak object boundaries. To address these challenges, we propose a Robust Pythagorean Fuzzy Energy-Based Level Set (RPFELS) model, which integrates a novel fuzzy energy formulation with level set evolution to enhance segmentation precision and resilience against noise. The model introduces a Pythagorean fuzzy divergence term to refine energy optimization, ensuring adaptive boundary preservation and reducing sensitivity to intensity variations. Additionally, a bounded fuzzy energy constraint is incorporated to ensure numerical stability and prevent energy leakage during evolution. Extensive experiments on benchmark datasets, including medical and natural images, validate the effectiveness of RPFELS. The model consistently outperforms recent selective segmentation methods in terms of Dice Score, Jaccard Index, and Hausdorff Distance, achieving superior segmentation accuracy and reduced boundary errors. Furthermore, a detailed statistical significance analysis using paired t-tests confirms that the observed improvements are statistically significant (p-value $<$ 0.01), reinforcing the reliability of the proposed approach. Moreover, RPFELS exhibits higher computational efficiency, achieving faster convergence rates compared to existing methods. These findings highlight the robustness and versatility of the proposed approach in handling challenging segmentation scenarios, making it suitable for applications in medical imaging, remote sensing, and industrial defect detection. By ensuring bounded energy evolution and statistically validated performance gains, our model sets a new benchmark in selective segmentation.

In order to approximate several roots of nonlinear equations, we presented a novel family of two-step optimal iterative methods in this study. The method is fourth-order convergent, requiring just four function evaluations each iteration, and it is optimal in terms of Kung-Traub's conjecture. We use complex dynamical analysis, often known as basins of attraction, to study local convergence and dynamical behavior. Numerical experiments on nonlinear problems in biomedical engineering are carried out to determine the method's efficiency and robustness in comparison to other methods. In terms of convergence rate, computational complexity, and stability, numerical findings show that the novel approach outperforms the well-known existing algorithms, especially for functions with higher multiplicities of order.

In this paper, we derive a new conjugate gradient (CG) direction with random parameters which are obtained by minimizing the deviation between search direction matrix and self-scaled memoryless Broyden-Fletcher-Goldfard-Shanno (BFGS) update. We propose a new spectral three-term CG algorithm and establish the global convergence of new method for uniformly convex functions and general nonlinear functions, respectively. Numerical experiments show that our method has nice numerical performance on nonconvex functions and supply chain problems.

The forecasting of wheat commodity prices plays a crucial role in mitigating financial risks for stakeholders across the agricultural supply chain. In this study, the predictive performance of three models—Simple Moving Average (SMA), Extreme Gradient Boosting (XGBoost), and a hybrid SMA-XGBoost model—was evaluated to determine their efficacy in capturing both linear trends and complex nonlinear patterns inherent in wheat price data. A 10-lag structure was employed to integrate historical dependencies and seasonal fluctuations, thereby enhancing the accuracy of trend identification. The dataset was partitioned into training (75%) and testing (25%) subsets to facilitate an objective performance assessment. The XGBoost model, known for its capability in modelling nonlinear dependencies, demonstrated the highest forecasting precision, achieving a Mean Absolute Percentage Error (MAPE) of 1.64%. The hybrid SMA-XGBoost model, which leveraged the complementary strengths of both SMA and XGBoost, yielded a MAPE of 1.75%, outperforming the standalone SMA model, which exhibited a MAPE of 2.60%. While the hybrid model displayed slightly lower accuracy than XGBoost, it offered greater stability and robustness by effectively balancing trend extraction and nonlinear adaptability. These findings highlight the hybrid approach as a viable alternative to purely machine learning-based forecasting methods, particularly in scenarios requiring resilience to diverse market fluctuations. The proposed methodology provides a valuable tool for policymakers, agricultural producers, and market analysts seeking to enhance decision-making strategies and optimize risk management within the agricultural sector.

Graph structures (GSs) have appeared as a robust mathematical framework for modelling and resolving complex combinatorial problems across diverse realms. At the same time, the linear Diophantine fuzzy set (LDFS) is a noteworthy expansion of the conventional concepts of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and q-Rung orthopair fuzzy set (q-ROFS). The LDFS framework introduces a flexible parameterization strategy that independently relaxes membership and non-membership restraints through reference parameters, thereby attaining enhanced expressiveness in apprehending ambiguous real-world phenomena. In this paper, a novel concept of linear Diophantine fuzzy graph structure (LDFGS) is introduced as a generalization of intuitionistic fuzzy graph structure (IFGS) and linear Diophantine fuzzy graph (LDFG) to GSs. Several cardinal fundamental notions in LDFGSs, including $\breve{\rho}_i$-edge, $\breve{\rho}_i$-path, strength of $\breve{\rho}_i$-path, $\breve{\rho}_i$-strength of connectedness, $\breve{\rho}_i$-degree of a vertex, degree of a vertex, total $\breve{\rho}_i$-degree of a vertex, and the total degree of a vertex in an LDFGS are discussed. Additionally, $\breve{\rho}_i$-size of an LDFGS, the size of an LDFGS, and the order of an LDFGS are studied. Meanwhile, the ideas of the maximal product of two LDFGSs, strong LDFGS, degree, and $\breve{\rho}_i$-degree of the maximal product are introduced with several concrete illustrations. To empirically validate the efficacy and practical utility of the proposed LDFGS framework, this study presents a case study analyzing road crime patterns across heterogeneous urban regions in Sindh province, Pakistan.

Nanofluids, which are suspensions of nanoparticles in base fluids, have demonstrated considerable potential in enhancing thermal conductivity, energy storage, and lubrication properties, as well as improving the cooling efficiency of electronic devices. Despite their promising applications, the industrial utilization of nanofluids remains in the early stages, with further research needed to fully explore their capabilities. This study investigates a generalized nanofluid model, incorporating fractal-fractional derivative (FFD), to better understand the thermophysical behaviors in vertical channel flow. The nanofluid consists of polystyrene nanoparticles uniformly dispersed in kerosene oil. An exact solution to the model is obtained by employing the Laplace transform technique (LTT) in combination with the numerical Zakian’s algorithm. The FFD operator with an exponential kernel is applied to extend the classical nanofluid model. Discretization of the generalized model is achieved using the Crank-Nicolson method, and numerical simulations are performed to solve the resulting equations. The study reveals that, at a nanoparticle volume fraction of 4% (0.04), the heat transfer rate of the nanofluid is significantly higher than that of the base fluid. Furthermore, the enhanced heat transfer leads to improvements in various thermophysical properties, such as viscosity, thermal expansion, and heat capacity, which are crucial for industrial applications. The numerical results are presented graphically to highlight the dependence of the flow and thermal dispersion characteristics on key physical factors. These findings suggest that the use of fractal-fractional models can provide a more accurate representation of nanofluid behavior, particularly for high-precision applications in heat transfer and energy systems.