# Optimal Deployment of Container Weighing Equipment: Models and Properties
## 2 Problem Description  
Consider a maritime system (MS) that has \(P\) ports, the set of which is \(\mathcal{P}=\{1,2,\ldots,P\}\), each indexed by \(i\). Every week the MS needs to transfer containers between different ports. In most cases, containers are not shipped directly to their destinations, but are transshipped through intermediate ports before being delivered to their destinations. Assume that from the origin port \(i\) to the destination port \(j(i,j\in\mathcal{P},i\
eq j)\), the MS needs to transfer \(N_{ij}\) containers in total. Furthermore, supposing that there is an intermediate port \(k(k\in\mathcal{P}\setminus\{i,j\})\), we assume that the MS knows the number of containers to be transshipped at port \(k\) from the origin port \(i\) to the destination port \(j\), denoted by \(n^{k}_{ij}\), and the number of containers not to be transshipped, denoted by \(\overline{n}_{ij}\). Therefore, we have \(\sum_{k\in\mathcal{P}\setminus\{i,j\}}n^{k}_{ij}+\overline{n}_{ij}=N_{ij},i \
eq j\in\mathcal{P}\). Accurately weighing containers helps improve the overall safety and stability of the shipping system. Enhancements to security and safety measures can increase the efficiency of supply chains [35]. Shipping is widely regarded as one of the riskiest activities, involving a broad range of potential hazards [33]. The earlier a container is weighed, the more it contributes to the overall safety of the maritime system, so our ideal system is that all of the containers are weighed at their origin ports. However, due to limited resources (number of weighing machines and their capacity), practical implementation of container weighing requires optimization. We need to make decisions about the number of weighing machines owned by each port as well as the location where each container should be weighed. Assume that the MS has a total of \(Q\) machines to weigh containers. The MS needs to decide how many weighing machines, denoted by \(x_{i}\), to install in each port \(i\) (\(i\in\mathcal{P}\)). The maximum number of containers that can be weighed per weighing machine per week is denoted as \(M\). The MS needs to decide how many containers to be weighed at different ports. We assume that if a container is to be weighed, it is to be weighed only once. Specifically, it can be scheduled to be weighed at the origin port \(i\), the intermediate port \(k\), or the destination port \(j\) (\(i,j,k\in\mathcal{P},i\
eq j\
eq k\)). The objective of this problem is a little bit different from normal optimization problems whose optimization goals are measurable and concrete, such as distance and cost. Here, the optimization goal we are discussing is the safety of ships during navigation, which is abstract and difficult to measure. The difficulty in measuring safety lies in the uncertainty of whether a ship is truly safe or whether an accident will occur. Before the International Maritime Organization (IMO) emphasized the issue of container weighing, many cargo ships did not weigh their containers, resulting in many overweight containers. This situation is dangerous and unsafe, but even so, a ship carrying overweight containers may not necessarily have an accident during navigation. Historically, only a few cargo ships have had accidents due to the overweight issue. Therefore, the occurrence of accidents is a probabilistic issue, and it is impossible to definitively determine whether a ship will have an accident or not, unlike the measurable distance or cost. However, we can be certain that the safety level of a ship carrying overweight containers during navigation is lower than that of a ship without overweight containers (assuming other conditions are the same). To ensure the ship does not carry overweight containers, we can measure the weight of the containers and remove the overweight carriage as early as possible. The earlier the containers are weighed for this purpose, the shorter the distance that the ship will travel with overweight containers, and the safer the sailing process of the ship.