# Two-Stage Stochastic Programming for Precast Module Water Transportation: A Case Study in Hong Kong
## 4 Case Study
### Comparative Experiments  
To demonstrate the effectiveness of our proposed two-stage stochastic programming model, three deterministic models were developed and tested on the same case. The three deterministic models [DM1], [DM2], and [DM3] were designed by determining booking barge types and their weekly round in advance based on (i) the minimum weekly demand for modules in the 20 weeks (i.e., 0 units/week in InnoCell case), (ii) the average weekly demand over 20 weeks (i.e., 21 units/week), and (iii) the maximum weekly demand for modules in the 20 weeks (i.e., 58 units/week) at the beginning and determining the optimal number and types of temporarily rented barges based on the actual module demand observed in each week. The actual demand values (\(d_{t}\)) used are consistent across both the two-stage model and three deterministic models. Note that in the model [DM3], the contractor books enough shipping capacity to transport the peak weekly module demand. Consequently, there will be no need for additional temporary rentals. The three deterministic models [DM1], [DM2], and [DM3] are presented as follows. Newly defined parameter:  
\(\dot{d}\)--the reference demand for deciding the number of trips to be finished by booked barges in deterministic models  
Mathematical model:  
\[[\text{M2}]\text{min}\Big{\{}T\!\sum_{k=1}^{K}c_{k}\alpha_{k}\Big{\}} \tag{11}\]  
s.t. \(\sum_{k=1}^{K}N_{k}\alpha_{k}\geq\dot{d}\)  
Constraints (2) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type of Ship** & \(N_{k}\) & \(\alpha_{k}^{*}\) & \(\alpha_{k}^{*}\)**from [DM1]** & \(\alpha_{k}^{*}\)**from [DM2]** & \(\alpha_{k}^{*}\)**from [DM3]** \\ \hline
28 TEU & 14 & 0 & 0 & 0 & 0 \\
36 TEU & 18 & 0 & 0 & 0 & 0 \\
48 TEU & 24 & 0 & 0 & 1 & 0 \\
60 TEU & 30 & 1 & 0 & 0 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The optimal type and number of barges booked in advance for a round trip are derived by solving the model [M1] and the three deterministic models. Figure 2: Optimal transportation plans solved by the two-stage stochastic programming model and two deterministic models in one scenario. Objective function (11) indicates that the contractor seeks to minimize the total cost of transporting modules based on reference module demand \(\hat{d}\). The contractor books trips in advance given to the value of \(\hat{d}\), as expressed by Constraint (12). Based on the definition of deterministic models, they are differentiated only by their \(\hat{d}\) values. We have \(\hat{d}=0\) in [DM1], \(\hat{d}=21\) in [DM2], and \(\hat{d}=58\) in [DM3]. By solving [M2], we can also obtain the optimal value of \(\alpha_{k}\), denoted as \(\alpha_{k}^{*}\). \[[\mathbf{M}3]\text{min}\Big{\{}\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k}\hat{p}_{t,k} \Big{\}} \tag{13}\]  
\[\text{s.t. }\sum_{k=1}^{K}N_{k}(\alpha_{k}^{*}+\beta_{t,k})\geq d_{t},t=1,\dots,T \tag{14}\]  
Constraints (5)  
The objective function (13) aims to minimize the cost of renting barges temporarily. Constraints (14) ensure that the capacity of booked and temporarily rented trips can satisfy the actual demand for modules. Likewise, we define the optimal value of \(\beta_{t,k}\) as \(\beta_{t,k}^{*}\). The total transportation costs solved by deterministic models are calculated by:  
\[C\text{T}\sum_{k=1}^{K}c_{k}\alpha_{k}^{*}+\sum_{t=1}^{T}\sum_{k=1}^{K}p_{k} \beta_{t,k}^{*} \tag{15}\]  
The optimal plans of these deterministic models about booked barges are presented in Table 5. [DM1] does not plan to finish any round trips using booked barges, as the reference demand is 0. [DM2] books one trip with a 48 TEU barge every week, while [DM3] schedules two trips using 60 TEU barges every week. This variation in booking strategies reflects different approaches to managing transport capacity based on reference demand levels in each model. Figure 1b-d illustrates the composition and value of transportation costs while Figure 3 compares the optimal transportation costs solved by the two-stage stochastic programming model and three deterministic models.