Mathematical Techniques Research Articles

Blood donation drive management: a discrete event simulation-based approach for enhanced operational efficiency

PurposeThis study aims to optimize blood donation drive efficiency by addressing operational bottlenecks and improving resource deployment, focusing on enhancing donor experience and reducing camp duration.Design/methodology/approachThe research uses a mixed-method approach, combining qualitative insights from blood banking officer interviews with quantitative data from 58 camp observations. “Simio” simulation software models various operational configurations, while mathematical techniques like queuing theory analyze key performance metrics. The process involves creating a baseline model, proposing optimizations and validating recommendations through real-world implementation.FindingsThe study achieved significant improvements: Reduced average donor time from 1.79 to 0.79 hours (56.4% improvement); Shortened camp operation time from 7.98 to 4.85 hours (39.2% improvement); Decreased waiting times at the medical check station from 45.23 to 0.60 minutes; Improved service rates across all stations, notably at registration (233.79% increase); and Streamlined processes through digitization and health check consolidation.Practical implicationsThe study provides actionable recommendations for blood bank managers, including digital pre-registration and optimized staff allocation, leading to substantial time and resource savings while enhancing donor experience.Social implicationsBy improving donation camp efficiency and experience, this research could increase donor retention rates and lead to a more stable blood supply, crucial for medical care.Originality/valueThis research presents a novel approach by combining discrete event simulation with real-world implementation and validation, offering an innovative solution to common bottlenecks in blood donation drives.

Optimization of Assembly Line Balancing Via Response Surface Methodology and Discrete Event Simulation

Objective: The objective of this study is to distribute the operations time between the operators according to the production mix for a new product type to be added to the relevant module in the existing production line, and to integrate the new product with high workload time into the existing production line without adding a new operator. Theoretical Framework: Assembly line balancing is an important performance enhancement method that is frequently used to improve the performance of manufacturing systems. Method: Response Surface Methodology (RSM) is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes. A discrete event simulation model can be defined as a model where the state variable changes only at these discrete points in time when events occur. Results and Discussion: As a result of the research, the production line can produce 386 products under the condition of not having two consecutive units of X202, with a product mix ratio of 85%-15%. Research Implications: The practicality of use and the consistency of the results obtained make the integration of response surface experimental design with simulation a method that can be applied in both academic and industrial settings. Originality/Value: This study contributes to the literature by developing to balance the assembly line under constraints, the simulation model of the line model is established and the optimization of the model parameters with the response surface has not been discussed in the literature.

SurfILE: An Open-Source Python Package for Surface Topography Analysis

Surface metrology deals with inspecting surfaces and profiles by using contact or non-contact profilometers. In this field, the characterization of the dimensional, morphological, and texture parameters of samples as well as the assessment of metrological characteristics of measuring instruments are key issues. Manufacturers of instruments provide commercial software tools to analyze topography data. There are also freely available tools, including open-source options, that provide a variety of algorithms and methods. The rapid growth of investigations aimed at better understanding the effects of the microscale phenomena requires the improved traceable calibration of samples, the development of new methodologies and measuring techniques, and the specification of new mathematical models and processing techniques. In this work, we present SurfILE, the launch of an open-source Python project that provides various procedures and algorithms for topography analysis. The open-source software presented in this article is intended to be modular, expandable, and customizable.

Determinan Pertumbuhan Laba pada Perusahaan Industri Makanan dan Minuman Periode 2018-2022

Financial reports can be used as an instrument to assess company profit growth. Profit growth can be measured using financial and non-financial indicators. The researcher re-examined the debt to equity ratio, current ratio, total asset turnover, institutional ownership and independent board of commissioners on profit growth in food and beverage subsector companies listed on the Indonesian stock exchange. Quantitative research is defined as a systematic investigation of phenomena by collecting data that can be measured by performing statistical, mathematical or computational techniques. The population in this study are companies in the group of manufacturing companies listed on the Indonesian Stock Exchange (BEI) during the observation year 2018 - 2022. The sample in this study used a purposive sampling method. This research uses multiple linear analysis methods, testing independent variables is carried out on more than one variable. Data testing was carried out using SPSS and was carried out in stages. This research aims to test and analyze the influence of the variables debt to equity ratio, current ratio, total asset turnover, institutional ownership and independent board of commissioners on profit growth. The population used in this research is food and beverage sub-sector companies listed on the Indonesia Stock Exchange (BEI) for the 2018 - 2022 period.

Synergistic Impact of Active Case Detection and Early Hospitalization for Controlling the Spread of Yellow Fever Outbreak in Nigeria: An Epidemiological Modeling and Optimal Control Analysis

Capturing the factors influencing yellow fever (YF) outbreaks is essential for effective public health interventions, especially in regions like Nigeria, where the disease poses significant health risks. This study explores the synergistic effects of active case detection (ACD) and early hospitalization on controlling YF transmission dynamics. We develop a dynamic model that integrates vaccination, active case detection, and hospitalization to enhance our understanding of disease spread and inform prevention strategies. Our methodology encompasses mechanistic dynamic modeling, optimal control analysis, parameter estimation, model fitting, and sensitivity analyses to study YF transmission dynamics, ensuring the robustness of control measures. We employ advanced mathematical techniques, including next-generation matrix methods, to accurately compute the reproduction number and assess outbreak transmissibility. Rigorous qualitative analysis of the model reveals two equilibria: disease-free and endemic, demonstrating global asymptotic stability and its impact on overall YF transmission dynamics, significantly affecting control and prevention mechanisms. Furthermore, through sensitivity analysis, we identify crucial parameters of the model that require urgent attention for more effective YF control. Moreover, our results highlight the critical roles of ACD and early hospitalization in reducing YF transmission. These insights provide a foundation for informed decision making and resource allocation in epidemic control efforts, ultimately contributing to the enhancement of public health strategies aimed at mitigating the impact of YF outbreaks.

Applications of Vedic Mathematics in Computer Science

The convergence of ancient wisdom and modern computation is an intriguing frontier explored in this research paper, which investigates the potential application of Vedic Mathematics in the domain of Computer Science. Vedic Mathematics, originating from ancient Indian texts known as the Vedas, offers a distinct approach to mathematical problem - solving that emphasizes mental agility, pattern recognition, and elegant algorithms. This paper delves into the historical context of Vedic Mathematics, tracing its origins and key principles, and highlights its relevance in contemporary Computer Science. The intersection of Vedic Mathematics and Computer Science is illuminated through the identification of shared principles, such as pattern recognition and optimization. The paper explores specific Vedic Mathematics techniques that have the potential to enhance computational efficiency, including the "Nikhilam" method for subtraction and the "Vertically and Crosswise" method for multiplication. Case studies are presented to demonstrate the practical application of Vedic Mathematics techniques in Computer Science, spanning areas such as data processing, algorithmic optimization, and error detection and correction. However, this integration is not without its challenges and ethical considerations. The paper addresses concerns about accuracy and reliability, particularly when integrating unconventional mathematical techniques into modern computational paradigms. Cultural sensitivity and proper acknowledgment of the origins of Vedic Mathematics are paramount, as is the need to ensure that the techniques are accessible to a diverse audience. The ethical implications of cross - cultural knowledge exchange are also explored, emphasizing the importance of collaboration and respect. Looking ahead, the paper outlines potential research directions and implications for education in Computer Science. The integration of Vedic Mathematics offers opportunities for developing novel algorithms, enhancing problem - solving skills, and fostering cross - cultural understanding. Collaborations between mathematicians and computer scientists are highlighted as a promising avenue for innovation in algorithmic design. In conclusion, the fusion of Vedic Mathematics and Computer Science represents a dynamic interplay of ancient traditions and modern advancements. As we navigate the complexities of integration, we embark on a journey that transcends disciplinary boundaries, fosters cultural appreciation, and holds the potential to reshape computational methodologies in the digital age.

Enhancing low-light images using Sakaguchi type function and Gegenbauer polynomial

Enhancing low-light images is crucial for various applications in computer vision, yet current approaches often fall short in balancing image quality and detail preservation. This study introduces a novel method designed to enhance low-light images by applying advanced mathematical techniques from geometric function theory. Specifically, we employ Sakaguchi-type class functions, subordinated with the Gegenbeur polynomial, to derive coefficient estimations. These estimations are then used in convolution kernels to produce enhanced image versions. The method was tested on the LOw-Light dataset (LOL), containing challenging low-light images with noise and artifacts. Our approach’s effectiveness is validated through quantitative metrics, including PSNR and SSIM, as well as visual comparisons. The results demonstrate significant improvements over existing state-of-the-art methods, offering better visibility and detail retention. This method holds promise for enhancing images in critical fields such as surveillance and medical imaging.

Impact of the Fitness Function on the Development of Mathematical Models by Symbolic Regression

Objective: This study aims to investigate the impact of the fitness function on the development of mathematical models using Symbolic Regression based on experimental data from the decolorization process of a synthetic effluent, employing a Taguchi matrix to evaluate the efficiency of this modeling process and its application to other processes. Theoretical Framework: The study is grounded in references on advanced oxidation processes, experimental design with an emphasis on the Taguchi method, and Symbolic Regression. Method: The research methodology involved modeling and simulation, with data collection conducted through photodegradation experiments on a synthetic effluent. Results and Discussion: The results demonstrated that the fitness function affects the predictive quality of the model obtained through Symbolic Regression. This impact is highlighted in the discussion section through validation experiments. Possible discrepancies and limitations of the study are also considered in this section. Research Implications: The practical and theoretical implications of this research are discussed, providing insights into how the findings can be applied or influence practices in the field of industrial process modeling and optimization. These implications are broadly applicable across various industrial sectors that involve transformation processes. Originality/Value: This study contributes to the literature with the application of new mathematical modeling techniques employed in conjunction with Experiment Design. The relevance and value of this research are evidenced by demonstrating that the use of Symbolic Regression is viable and may be superior to the Ordinary Least Squares Method.

Unravelling Yield and Yield-Related Traits in Soybean Using GGE Biplot and Path Analysis

Soybean (Glycine max) is a vital crop for food, animal feed, and industrial products. However, its yield performance is significantly affected by genotype-by-environment interaction (GEI), which complicates the selection of high-yielding, stable varieties. This study aimed to evaluate the yield performance and stability of 12 elite soybean varieties across five major production areas in Uganda using GGE biplot and path analysis. The varieties were planted in a randomized complete block design with three replications over two consecutive seasons. Results revealed significant differences in grain yield among the varieties, locations, and their interactions (p < 0.001). The highest-yielding varieties were Maksoy 5N (979 kg ha−1), Maksoy 4N (978 kg ha−1), Maksoy 3N (930 kg ha−1), and Signal (930 kg ha−1). GGE biplot analysis grouped the locations into two mega-environments, with the Maksoy varieties exhibiting greater yield stability compared to Seed Co. varieties. Path analysis showed that traits such as the number of lower internodes, central internode length, and filled pods had the highest positive direct effects on grain yield. This study provides insights into soybean breeding in tropical environments, highlighting traits that can be targeted to improve yield and stability. The findings offer a framework for breeding programs in Uganda and similar agro-ecological regions, promoting more resilient and productive soybean varieties. This study also illustrated the potential advantages of employing more complex mathematical techniques like path analysis to uncover yield and yield-related traits in soybean breeding programs.

Application of leak-off test data in formation fracture gradient correlation for Niger Delta Basin natural gas wells development

Offshore drilling of gas wells in the Niger Delta is becoming very expensive and requires highly skilled manpower. Therefore, it is essential to accurately update the fracture pressure data, maintain adequate measurement while drilling, and ensure acceptable prediction with conventional empirical correlations while drilling operations are ongoing to avoid non-productive time (NPT). To effectively drill an offshore well and deliver it safely while protecting personnel, the environment, and equipment, drillers must ensure that the wellbore pressure is sufficient to balance the pore pressure without exceeding the fracture pressure anywhere along the open section of the well. Calculating and predicting the fracture pressure is critical for reducing NPT in offshore drilling and ensuring the safety of workers and goods on-site. There are numerous relationships for predicting the fracture pressures. However, they are primarily limited to onshore and shallow water fields, and as wells are drilled deeper in the Offshore Niger Delta field, a correlation (equation) based on offshore leak-off test (LOT) data is required to correctly estimate the crack pressure. Thus, this study employs a mathematical modelling technique to create an improved offshore fracture gradient equation from leak-off test data tailored to the needs of the offshore Niger Delta. The generated correlation (equation) was statistically examined, yielding a reliable coefficient of determination. It was tested again with the field cases, and the results were similar to the LOT results.

Management Analysis of University Joint Supervision System Based on Multidimensional Data Fusion

A robust joint supervision approach is required for both academic instruction and boosting the performance of students. Conventional approaches frequently come across many difficulties when merging different kinds of information, such as files, supervisor comments, and academic achievement. For the purpose of increasing the effectiveness of university supervision, this paper assesses the joint supervision approach using multidimensional data fusion. To combine the different variables used in this study such as grades, feedback evaluations, and the load of supervision mathematical techniques are applied. Next, significant factors impacting the efficiency of university supervision are examined, and students and supervisors are divided into groups based on achievement and feedback assessment. Fusion of these data is performed and analyzed in terms of different measures, like precision, recall, accuracy, level of supervision satisfaction and level of student satisfaction. The outcomes of this paper provide important evidence of the link between pupil achievement and supervision strategies. The paper recommends a multi-data fusion for the university joint supervision scheme and established a link between supervision approaches and student achievement.

FACTORS INFLUENCING THE DETERMINATION OF AUDITORS' FEES IN ALGERIA (FROM THE PERSPECTIVE OF PROFESSIONALS) - AN EMPIRICAL STUDY USING DEMATEL ANALYSIS

Objective: This study aims to identify the factors influencing the determination of auditors' fees in Algeria from the perspective of the practitioners, by understanding the relative importance of these factors. Theoretical Framework: The theoretical aspect addressed the most important factors influencing the determination of auditors' fees in Algeria, which are: the duration of the audit process, the size of the audited institution, the reputation and fame of the audit office, the quality of the audit office, and the risks of the audit process. Method: In order to determine the interrelationship between the influencing factors that contribute to the determination of auditors' fees in Algeria, the "DEMATEL" method was used, which relies on the opinions of experts (audit experts). Mathematical techniques represented in matrices were employed to identify the most influential factor among the following: the duration of the audit process, the size of the audited institution, the reputation and fame of the audit office, the quality of the audit office, and the risks of the audit process. Results and Discussion: Through the cause-and-effect relationship, the study concluded that the "reputation and fame of the auditing office" is the most important perspective for evaluation with the highest impact value (5.942), while the "audit process risk factor" is the least important perspective with an estimated value of (4.392), in order to reach the appropriate input for determining the fees of auditors in Algeria. Research Implications: The study results highlight the importance of the factors influencing the determination of auditors' fees in Algeria. As it is necessary to understand the degree of influence of each criterion in determining these fees, and consequently to identify the necessary solutions to reach a unified model for fees in Algeria based on the influencing factors, instead of subjecting the fees of auditors in Algeria to the method of specifications. Originality/Value: This study contributes to the literature on the factors influencing the determination of auditors' fees in Algeria by providing empirical evidence from Career experts in Algeria and analyzing it using mathematical methods to identify the most significant influencing factors, enabling policymakers to formulate the necessary policies.

Dynamics and numerical analysis of a fractional-order toxoplasmosis model incorporating human and cat populations

Toxoplasmosis is a significant zoonotic disease that poses risks to public health and animal health, making the understanding of its transmission dynamics crucial. In this study, we present a novel fractional-order model that captures complex interactions among human, cat, and mouse populations, providing deeper insights into the disease spread and control. We utilize mathematical techniques to analyze the model properties, including the existence, uniqueness, positivity, and boundedness of solutions, along with stability analysis of the equilibrium states. The basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$R_{0}$\\end{document} is derived, revealing the threshold for potential outbreaks. Our findings indicate that key parameters significantly influence the dynamics of toxoplasmosis, with implications for targeted intervention strategies. We propose the QLM-FONP numerical scheme for efficient resolution of the model and provide a comprehensive convergence analysis, demonstrating the reliability of the numerical solutions. The results confirm the effectiveness of our approach, illustrating that the proposed model not only offers accurate predictions but also extends beyond previous efforts in the literature by incorporating fractional-order dynamics, which better reflect real-world transmission processes. Overall, this study enhances the understanding of toxoplasmosis transmission and informs future research and control efforts.

Understanding the relationship between preferential interactions of peptides in water-acetonitrile mixtures with protein-solvent contact surface area.

The influence of polar, water-miscible organic solvents (POS) on protein structure, stability, and functional activity is a subject of significant interest and complexity. This study examines the effects of acetonitrile (ACN), a semipolar, aprotic solvent, on the solvation properties of blocked Ace-Gly-X-Gly-Nme tripeptides (where Ace and Nme stands for acetyl and N-methyl amide groups respectively and X is any amino acid) through extensive molecular dynamics simulations. Individual simulations were conducted for each peptide, encompassing five different ACN concentrations within the range of χACN = 0.1-0.9. The preferential solvation parameter (Γ) calculated using the Kirkwood-Buff integral method was used for the assessment of peptide interactions with water/ACN. Additionally, weighted Voronoi tessellation was applied to obtain a three-way data set containing four time-averaged contact surface area types between peptide atoms and water/ACN atoms. A mathematical technique known as N-way Partial Least Squares (NPLS) was utilized to anticipate the preferential interactions between peptides and water/ACN from the contact surface areas. Furthermore, the temperature dependency of peptide-solvent interactions was investigated using a subset of 10 amino acids representing a range of hydrophobicities. MD simulations were conducted at five temperatures, spanning from 283 to 343K, with subsequent analysis of data focusing on both preferential solvation and peptide-solvent contact surface areas. The results demonstrate the efficacy of utilizing contact surface areas between the peptide and solvent constituents for successfully predicting preferential interactions in water/ACN mixtures across various ACN concentrations and temperatures.

Quantum mechanics and statistical physics: Novel frameworks for enhancing natural language processing

Abstract. This article explores the pioneering application of principles from quantum mechanics and statistical mechanics to the field of natural language processing (NLP). By drawing analogies between physical phenomena such as quantum entanglement, phase transitions, and statistical ensembles, and linguistic concepts like semantic relationships, language use dynamics, and lexical diversity, we offer a novel perspective on language analysis and processing. Quantum linguistic models, leveraging the intricacies of entanglement and quantum probability, provide a framework for understanding complex semantic networks and enhancing computational efficiency through quantum computing. Meanwhile, statistical mechanics inspires models for capturing lexical diversity and understanding the evolution of language patterns, akin to phase transitions in physical systems. This interdisciplinary approach not only deepens our understanding of linguistic phenomena but also introduces advanced mathematical and computational techniques to improve NLP tasks.

Optimum Planning of Carbon Capture and Storage Network Using Goal Programming

Carbon capture and storage (CCS) is a critical technology used for mitigating climate change by capturing carbon dioxide emissions from industrial sources and storing them underground to prevent their release into the atmosphere. Despite its potential, optimizing CCS systems for cost-effectiveness and efficiency improvement remains a significant challenge. In this paper, the optimization of CCS systems through the development and application of two mathematical optimization techniques is introduced. The first technique is based on using a superstructure optimization model, while the second technique relies on applying a goal programming optimization model. These models were solved using LINGO software version API 14.0.5099.166 to enhance the efficiency and cost-effectiveness of CCS systems. The first model, seeking to maximize the exchange of CO2 flowrate from sources to sinks, achieved a CO2 capture rate of 93.36% with an annual total cost of USD 1.175 billion. The second model introduced a novel mixed-integer non-linear programming (MINLP) approach for multi-objective optimization, targeting the minimization of total system cost, alternative storage, and unutilized storage while maximizing CO2 load exchange. The application of the second model, when prioritized to maximize CO2 flowrate exchange using the goal programming technique, resulted in a cost reduction of 36.46% and a CO2 capture rate of 75.87%. In contrast, when the second model prioritized minimizing the total annual cost, a 48% cost reduction was achieved, and the CO2 capture rate was decreased by 68.37%. A comparison of the two models’ results is presented. The results showed that the second model, with the priority of maximizing CO2 capture, provides the best economic–environmental objective balance, which offers notable cost reductions while keeping an efficient CO2 capture rate. This study highlights the potential of advanced mathematical modeling in increasing the feasibility of CCS as one of the very important strategies of mitigating climate change and reducing global warming.

Metaphorical meaning dynamics: Identifying patterns in the metaphorical evolution of English words using mathematical modeling techniques

Abstract Conceptual metaphor theory has been criticized due to its emphasis on concepts instead of words and its top-down direction of analysis. In response to these criticisms, this paper employs a new strategy, utilizing established mathematical modeling methods to allow a systematic, quantitative analysis of the entire dataset produced by the Mapping Metaphor project at the University of Glasgow. This dataset consists of 9609 words performing 18916 metaphorical mappings between 414 domains. The data is represented as a network consisting of 414 nodes, the domains, connected by shared words. Words are represented by groups of directed mappings between all domains in which they occur. This is made possible by the use of a directed hypergraph representation, a tool commonly used in discrete mathematics and various areas of computer science but not previously applied to the metaphorical meanings of words. Examining the dataset as a whole, rather than focusing on individual words or metaphors, allows global patterns of behavior to emerge from the data without pre-filtering or selection by the authors. Outcomes of the analysis relating to the distributions of source and target domains within the network, the growth mechanisms at work in the spread of metaphorical meanings and how these relate to existing concepts in CMT are discussed.

Modeling China’s Per Capita Disposable Income by Uncertain Statistics

Uncertain statistics is a set of mathematical techniques for collecting, analyzing and interpreting data by uncertainty theory. There are mainly three modeling methods in uncertain statistics: uncertain time series analysis, uncertain regression analysis, and uncertain differential equation. This paper applies these tools to modeling China’s per capita disposable income, and employs uncertain hypothesis test to determine whether the estimated uncertain statistical models fit China’s per capita disposable income. In addition, this paper shows that it is necessary to use uncertain statistics instead of probability statistics to model China’s per capita disposable income by investigating the corresponding residuals.

Green spectrophotometric approaches applied to tertiary mixture for management of common cold and COVID-19 symptoms

Recently, cold and cough dosage forms have gained significant attention due to their use in the supportive protocols for managing COVID-19 symptoms. In this study, a pharmaceutical formulation containing Paracetamol (PAR), Guaifenesin (GUA), and Phenylephrine hydrochloride (PHE) was investigated for spectral resolution and quantification using advanced spectrophotometric methods. The spectra of these components were significantly overlapped and present in their combined tablet in a challenging ratio of 250:100:5 for PAR, GUA, and PHE, respectively. The established approaches were employed for the simultaneous determination of these drugs in their pharmaceutical formulation without interference from matrix excipients. The study involved various manipulation steps, allowing each component in the combination to be analyzed by more than one approach. Integrating these methods with smart mathematical techniques, the issue of spectral data overlap was resolved without the need for preliminary separation steps. The developed methods are dual wavelength, first derivative, derivative ratio, ratio difference, constant center coupled with spectrum subtraction, and constant multiplication paired with spectrum subtraction. The proposed methods were linear over the concentration range of 3.0–35.0 μg/mL for GUA and 3.0–30.0 μg/mL for PHE. While the PAR ranges for the first derivative and constant multiplication methods were 2.5–35.0 μg/mL and 2.5–25.0 μg/mL, respectively. Excellent linearity of the suggested methods was demonstrated by the high correlation coefficients (R2), ≥ 0.9998 for all the tested compounds. These methodologies were validated according to ICH guidelines. Validation results demonstrated excellent accuracy, with recovery percentages ranging from 98 to 102 %, and precision, with RSD values less than 2 %. The obtained results were statistically compared with the official ones using F-test, Student’s t-test, and one-way ANOVA, revealing no significant differences. The proposed methods are accurate, green, smart, fast, and cost-effective. Their compliance with Green Analytical Chemistry principles was evaluated and compared to a published method using various tools to enable a more holistic evaluation from different perspectives. The promising results revealed that the investigated methods are superior green alternatives for routine analysis of the cited drugs in laboratories with limited resources and without access to expensive instruments.

Numerical Methods and Computation in Financial Mathematics: Solving PDEs, Monte Carlo Simulations, and Machine Learning Applications

Abstract. This paper delves into three fundamental numerical methods and computational techniques in financial mathematics: Finite Difference Methods (FDM), Monte Carlo Simulations (MCS), and Machine Learning (ML) applications. Finite Difference Methods are widely utilized for solving partial differential equations (PDEs) in option pricing, with various schemes offering different stability and convergence properties. Monte Carlo Simulations provide a powerful approach for pricing complex derivatives and risk management, addressing the challenges of high-dimensionality and computational complexity. Machine Learning has revolutionized predictive modeling in finance, enabling sophisticated analysis of large datasets to uncover hidden patterns and enhance trading strategies. Through a detailed examination of these methods, including specific examples and data, this paper highlights their theoretical foundations, practical implementations, and the advancements they bring to computational finance. By bridging theoretical approaches with practical applications, we aim to offer insights into the future directions and challenges in financial mathematics.