# Two-Stage Stochastic Programming for Precast Module Water Transportation: A Case Study in Hong Kong
## 2 Literature Review
[29] developed a bi-level programming model according to users' requirements to plan the transportation of prefabricated components, which was solved by an innovative heuristic algorithm based on the genetic algorithm and Partan Frank-Wolfe algorithm. These heuristic algorithms were prone to converging on local optima rather than achieving global optima. This limitation may result in suboptimal transportation plans, which could have negative impacts on the cost-effectiveness of transportation planning. To enhance solution quality, the utilization of professional mathematical optimization solvers, such as Gurobi and CPLEX, offers a promising alternative. [PERSON], [PERSON], [PERSON], and [PERSON] [24] obtained the optimal solutions to the quantity and types of prefabricated components to be transported in each round trip with the aid of Gurobi. The above research made plans about transporting prefabricated components, which belong to one-dimensional and two-dimensional PCs. MiC represents a higher level of prefabrication in off-site construction compared to PC, as it employs more integrated modules (three-dimensional prefabricated products) [3]. The high degree of integration results in larger modules requiring more precise transportation planning owing to their increased size and bulk. [PERSON] and [PERSON] [36] proposed a mixed-integer programming model by identifying relevant decision variables with the goal of reducing the total transportation and storage expenses. [PERSON], [PERSON], and [PERSON] [13] transformed a nonlinear model of transportation planning of prefabricated products into a linear integer programming model to embed it into CPLEX. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Study**} & **Transportation** & **Prefabrication** & **Whether Use** & \multirow{2}{*}{**Optimization Objective(s)**} & **Consideration of** \\  & **Mode** & **Level** & & **Optimization** & **Optimization** & **Uncertainty** \\ \hline [39] & Water & MiC & No (Qualitative approach) & N.A. & \(\times\) \\ [40] & Water & MiC & Yes (Exact method based on Solver) & Minimize the transportation cost & \(\times\) \\
**This** & **Water** & **MiC** & **Yes (Exact method based on Solver)** & **Minimize the transportation cost** & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. Road transportation of bulky MiC modules frequently faces challenges related to size constraints and safety concerns [3]. When modules exceed prescribed width limits, specialized traffic management measures, including police escorts, become mandatory. These requirements not only complicate operations and scheduling but simultaneously increase transportation costs as well. The size and load capacity of trucks inherently limit the number of modules that can be transported in a single trip. Additionally, the simultaneous presence of multiple trucks on the road can contribute to increased traffic congestion and, consequently, heightening traffic risks. This underscores the need for careful planning and scheduling of truck movements to ensure safe and efficient transportation operations. In contrast, water transportation is subjected to fewer limitations, given its higher cargo capacity and the typically greater distance of ports from residential areas. Therefore, it is imperative to place greater emphasis on water transportation planning for MiC projects. The application of workability assessment criteria on transportation planning, as proposed by [PERSON] et al. [39], can enhance the efficacy of water transportation planning for MiC projects from another perspective. [PERSON] et al. [40] developed an integer programming model to determine the allocation of prefabricated products in cargo holds, aiming to minimize the number of holds used and thereby reducing transportation costs. However, research on both road and water transportation in the MiC projects has rarely addressed uncertainty.