# Two-Stage Stochastic Programming for Precast Module Water Transportation: A Case Study in Hong Kong
## 3 Problem Statement
### Properties  
The mathematical model proposed in Section 3.2 is a complicated two-stage stochastic programming model, with complexity stemming primarily from two factors: multiple scenario representation and two-stage decision-making. The model must handle numerous future scenarios, each corresponding to different realizations of uncertainties, such as variations in the weekly demand for modules. As the number of scenarios increases, so does the complexity, since each scenario necessitates its own set of variables and constraints, leading to a larger, more intricate problem [53]. After the actual weekly demand is realizedin the second stage, the model is required to make decisions about renting barges temporarily. This leads to complex conditional decisions that depend on the outcomes of the first stage and the specific scenario encountered in the second stage [54]. Fortunately, it has an appealing property that can simplify the problem-solving process (see Proposition 1).  
For any given \(\mathbf{\alpha}\) and \(\mathbf{d}\), we have \(F(\mathbf{\alpha},\mathbf{d})=\sum_{t=1}^{T}\overline{F}_{t}(\mathbf{\alpha},\mathbf{d})\), where \(\overline{F}_{t}(\mathbf{\alpha},\mathbf{d}),t=1,\ldots,\)\(T\) is defined as follows.  
\[\overline{F}_{t}(\mathbf{\alpha},\mathbf{d})=\min\Bigl{\{}\sum_{k=1}^{K}p _{k}\overline{\beta}_{k}\Bigr{\}}, \tag{6}\] \[\text{s.t.}\ \sum_{k=1}^{K}N_{k}\bigl{(}\alpha_{k}+\overline{ \beta}_{k}\bigr{)}\geq d_{t}\] (7) \[\overline{\beta}_{k}\in Z_{+},k=1,\ldots,K. \tag{8}\]  
Given \(\mathbf{\alpha}\) and \(\mathbf{d}\), we define \(\beta^{*}_{t,k}\) as the optimal solution to \(F(\mathbf{\alpha},\mathbf{d})\) and \(\overline{\beta}^{*}_{t,k}\) as the optimal value of variable \(\overline{\beta}_{k}\) to \(\overline{F}_{t}(\mathbf{\alpha},\mathbf{d})\) for any \(t=1,\ldots,\ T\). To show Proposition 1, we need to show \(\min\Bigl{\{}\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\beta_{t,k}\Bigr{\}}=\sum_{t=1}^ {T}\min\Bigl{\{}\sum_{k=1}^{K}p_{k}\overline{\beta}_{k}\Bigr{\}}\), i.e., \(\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\delta^{*}_{t,k}=\sum_{t=1}^{T}\sum_{k=1}^{K} p_{k}\overline{\beta}^{*}_{t,k}\). \(\square\)  
First, according to the definition of \(\overline{F}_{t}(\mathbf{\alpha},\mathbf{d})\), \(\overline{\beta}^{*}_{t,k}\) satisfies Constraints (4) and (5) and is thus a feasible solution to \(F(\mathbf{\alpha},\mathbf{d})\). An optimal solution to a minimization problem is a feasible solution that yields the minimum objective value. Given that \(\overline{\beta}^{*}_{t,k}\) is a feasible solution and \(\beta^{*}_{t,k}\) is an optimal solution to \(F(\mathbf{\alpha},\mathbf{d})\), we have inequality (9) as follows:  
\[\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\beta^{*}_{t,k}\leq\sum_{t=1}^{T}\sum_{k=1}^ {K}p_{k}\overline{\beta}^{*}_{t,k}, \tag{9}\]  
Similarly, \(\beta^{*}_{t,k}\) satisfies Constraints (7) and (8) and is thus a feasible solution to \(\overline{F}_{t}(\mathbf{\alpha},\mathbf{d})\). Therefore, we have inequality (10) as follows:  
\[\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\beta^{*}_{t,k}\geq\sum_{t=1}^{T}\sum_{k=1}^ {K}p_{k}\overline{\beta}^{*}_{t,k}. \tag{10}\]  
Based on inequalities (9) and (10), we have shown that \(\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\beta^{*}_{t,k}=\sum_{t=1}^{T}\sum_{k=1}^{K} p_{k}\overline{\beta}^{*}_{t,k}\) and proven Proposition 1.  
This proposition indicates that the second-stage decision problem can be decomposed into \(T\) subproblems. Specifically, in any week \(t\), the decision-maker only needs to decide whether to temporarily rent ships, ensuring that the demand for modules is satisfied, to minimize the second-stage module transportation cost incurred in the current week \(t\). The sum of these \(T\) minimal costs is equivalent to the total second-stage module transportation cost calculated by considering the entire \(T\) periods.  
Given Proposition 1, the problem is significantly simplified and can be easily solved using commercial solvers, e.g., Gurobi. As solvers can efficiently calculate the optimal solution, no additional algorithms are developed.