A stochastic programming (SP) problem involves some or all of the parameters or variables being uncertain. Uncertainty is typically expressed as a probability distribution on the parameters. In reality, despite its precise description, uncertainty can manifest in various forms, ranging from a limited number of possible outcomes to precise joint probability distributions. In the water supply system, weather patterns (in the rainy season the rainfall is very high compared to the other seasons), water demand, and water availability are a few uncertain parameters. These uncertainties might not be sufficiently taken into account by conventional deterministic optimization techniques, resulting in less-than-ideal results. The water supply model will be enhanced in this study by SP ideas, resulting in a more stable and adaptable optimization strategy. In this research, we first analyze a 2-stage SP model by capturing more sample data and show the comparison of capturing more and less sample data. We will extend the 2-stage SP model to a 3-stage SP model by using the tree algorithm, and we will show the comparison between these two-stage and three-stage SP models.

This paper explores the use of stochastic optimization techniques to address the aircraft allocation problem under uncertain passenger demand. The proposed stochastic allocation model successfully meets the study’s objectives by demonstrating how uncertainty in passenger demand can be effectively incorporated into aircraft assignment decisions through a two-stage stochastic programming framework. Simulation results across multiple demand scenarios show that the model provides stable and adaptive allocations that minimize total cost while maintaining service quality, even under high variability. Incorporating the simple recourse approach enables post-decision flexibility, reducing penalties for unmet demand, and the use of Geometric Brownian Motion (GBM) offers a realistic representation of continuous demand fluctuations over time. These outcomes confirm the model’s practical value in bridging deterministic planning and real-time decision environments. While future research will focus on extending the model to a Markov Decision Process (MDP) framework and integrating real-time data streams, the current results establish a solid foundation by quantifying how uncertainty directly impacts fleet utilization, cost efficiency, and service reliability.

The Black-Scholes equation plays an important role in financial mathematics for the evaluation of European options. It is a fundamental PDE in financial mathematics, models the price dynamics of options and derivatives. While a closed-form of analytical solution exists for European options, numerical methods remain essential for validating computational approaches and extending solutions to more complex derivatives. This study explores and compares various numerical techniques for solving the Black-Scholes partial differential equation, including the finite difference method (explicit, implicit, and Crank-Nicolson schemes), and Monte Carlo simulation. Each method is implemented and tested against the analytical Black-Scholes formula to assess accuracy, convergence, and computational efficiency. The results demonstrate the strengths and limitations of each numerical approach, providing insights into their suitability for different option pricing scenarios. This comparative analysis highlights the importance of method selection in practical financial modeling applications.