# Two-Stage Stochastic Programming for Precast Module Water Transportation: A Case Study in Hong Kong
## 2 Literature Review
Uncertainty significantly impacts the transportation of MiC projects and should be factored into when making plans to reduce unnecessary costs [46]. Previous research often ignored the uncertainties inherent in MiC projects, especially the uncertain weekly demand for modules. However, some researchers attempted to combine these models with other technologies, such as redundancy [13; 30; 31] and BIM 4D [24], to optimize transportation planning under uncertain conditions. Through setting redundancy like resilience [30] and buffer time [13], they typically assumed worst-case scenarios to avoid difficulties from uncertainty, which may lead to resource waste and inflated transportation costs. Moreover, leveraging BIM 4D for real-time tracking and optimization has limited applicability for water transportation, because shipping routes and vessels usually require advanced reservations. Given the significant impact of uncertain weekly demands for modules on transportation planning, it is crucial for contractors to arrange barges in accordance with fluctuating demands [10]. Robust optimization and Monte Carlo simulation are two commonly used methods for handling uncertainty. Robust optimization addresses uncertainty by introducing redundancies to mitigate its impact, but it often focuses on the worst-case scenario. On the other hand, Monte Carlo simulation, while relatively easy to set up, requires a large number of simulations to ensure the accuracy of the results. Additionally, it does not provide an exact optimal solution but rather a statistical estimate. Inevitably, there is a pressing need for an effective method to proactively plan water transportation that considers uncertainties while aiming to reduce costs. Additionally, although the optimization of transportation costs for prefabricated products has been a focal point in prior research, the specific influence of barge rentals on transportation costs remains unclear. Two-stage stochastic optimization is a framework used to make decisions under uncertainty, where the decision-making process is divided into two distinct stages. [PERSON] et al. [47] applied two-stage stochastic optimization to assess the impact of uncertain factors on the efficient operation of energy microgrids. Similarly, a two-stage stochastic programming model was proposed by [48] for sustainable utility systems, focusing on multi-objective optimization. The two-stage stochastic optimization model holds the potential to solve the uncertain demand in MiC projects. In summary, existing research on the transportation planning of prefabricated products in off-site construction has laid a foundational basis for this study. Mathematical optimization is a powerful tool for transportation planning and possesses the potential to integrate with other approaches and technologies to address uncertainties in MiC projects. Nevertheless, making transportation plans in MiC projects faces two primary challenges. Firstly, most existing optimization models were tailored for road transportation, with limited research focusing on water transportation. Furthermore, they usually overlooked the uncertainty in the demand for modules which may hinder their applicability in real-world projects. Secondly, existing transportation planning studies failed to identify influential factors in the total costs. This may lead contractors to doubt the reliability of the optimization results and prevent them from fully understanding the underlying cost drivers. To bridge these gaps, this research develops a two-stage stochastic programming model that incorporates the uncertain weekly demand for modules when developing water transportation plans. In addition, a sensitivity analysis was conducted to assist MiC contractors in understanding the influence of barge rentals, thereby supporting informed decision-making under uncertainty.