# Two-Stage Stochastic Programming for Precast Module Water Transportation: A Case Study in Hong Kong
## 3 Problem Statement  
### Problem Description  
Consider a MiC project where the modules are produced at a remote factory and subsequently transported to the construction site via barges and trucks. Water transportation, though only one segment of this water-road multimodal process, greatly influences overall delivery performance [49; 50]. On the contrary, the two inland road segments in multimodal transportation are of relatively shorter distances and the planning of module transportation via road has been extensively explored in the existing studies, e.g., [26; 36]. Therefore, the optimal transportation plan proposed in this paper focuses on the most critical part of multimodal transportation, i.e., the water transportation segment. We simplify the problem and only take barges into consideration. The contractor of the MiC project plans to complete all water transportation tasks within \(T\) weeks (indexed by \(t\)). To achieve this, the contractor decides to rent barges from a shipping company to transport the modules in a JIT manner. The shipping company operates a fleet comprising \(K\) types of barges (indexed by \(k\)), differing in their capacities for module transport, denoted as \(N_{k}\). The barges can be rented in two ways: the first option is to book in advance, and the second option is to rent temporarily. The cost of a round trip for booked barges of type \(k\) is \(c_{k}\), while the cost of a round trip for temporarily rented barges of type \(k\) is \(p_{k}\). Evidently, for all types of barges, the cost of temporary rental is higher than the cost of booking in advance (i.e., \(p_{k}>c_{k}\), for any \(k=1,\ \ldots,\ K\)). Once a contract has signed for the booked barges, it cannot be canceled. Even if the actual number of round trips completed by the booked barges is fewer than agreed upon in the contract, payment for the agreed-upon number of round trips is still required. In practice, the periodic demand for modules (e.g., on a weekly basis) is contingent on the progress of on-site assembly works, which is subject to various uncertainties, e.g., extreme weather [51] and malfunction of construction equipment [52]. Therefore, the demand for modules in week \(t\) should be defined as a stochastic parameter \(\overset{\sim}{d}_{t}\). For simplicity, the random demands over the entire planning horizon are defined as a vector \(\overset{\sim}{d}=(\overset{\sim}{d}_{1},\ \overset{\sim}{d}_{2},\ldots,\ \overset{\sim}{d}_{T})\). The objective of the contractor is to minimize the total transportation cost of modules through a two-stage decision-making. In the first stage, the contractor decides the number of trips to be finished by booked barges of type \(k\) per week (denoted as \(a_{k}\)). For simplicity, we define vector \(\boldsymbol{a}=(\alpha_{1},\alpha_{2},\ldots,\ \alpha_{k})\). In the second stage, the contractor decides the number of trips that need to be completed by temporarily rented barges of type \(k\) in week \(t\), denoted as \(\beta_{t,k}\), after observing the realization of random parameter \(\overset{\sim}{d}_{t}\) in week \(t\), denoted as \(d_{t}\). We define \(\boldsymbol{d}=(d_{1},\ \ d_{2},\ldots,\ d_{T})\) as the realization of \(\overset{\sim}{d}=(\overset{\sim}{d}_{1},\overset{\sim}{d}_{2},\ldots, \overset{\sim}{d}_{T})\). The two-stage stochastic module transportation problem investigated here can be briefly summarized as follows. Prior to the commencement of module transportation, the contractor anticipates the weekly demand for modules to determine the number of ships to book in advance. Then, each week, the contractor observes the actual demand for modules and decides whether additional barges need to be rented temporarily to cater this demand. Objectively, the minimization of the total costs of transporting the modules within the \(T\) weeks is desired. Before the formulation of the mathematical model, the assumptions of the investigated problem are summarized as follows.