# Optimal Deployment of Container Weighing Equipment: Models and Properties
## 1 Introduction
### Literature Review  
Research topics related to container shipping primarily focus on the design of container shipping routes and the repositioning of empty containers. In the past few years, there has been a growing discussion about container weighing. In terms of container shipping route design problems, [PERSON] and [PERSON] [7] developed a mathematical programming model to determine the optimal routing, container load, and service frequency for a chartered container ship. [PERSON] [8] conducted a detailed study of the routes taken by rented container ships or towed vessels in inland waterways. In addition, they found the best values for parameters that affect how efficiently these ships or tows are used, which impacts the commercial success of a container shipping business. [PERSON] et al. [9] conducted a study  
Figure 1: A busy port scene with containers. to optimize the shipping routes of barge container ships with the goal of maximizing the profit of a shipping company. This optimization determines the order in which the ships visit ports both upstream and downstream, as well as the number of loaded and empty containers transported between any two ports. [PERSON] et al. [10] developed a mixed-integer nonlinear programming model to enhance the efficiency of container transport from Shanghai Port to inland river ports on the Yangtze River by minimizing total operating costs and optimizing feeder departure times. [PERSON] et al. [11] proposed a hub-and-spoke network for container shipping on inland waterways, using the river's branching structure like a tree. [PERSON] et al. [12] examined the distribution of bulk cargo within a multimodal transportation network, considering inland waterway routes as well as containerization technology as transport modalities. In terms of empty containers' repositioning, [PERSON] et al. [13] focused on empty containers' repositioning for the first time when designing a container shipping route. They formulated the problem as a two-stage task and developed an heuristic based on the genetic algorithm. [PERSON] et al. [14] considered the problem of repositioning empty containers in multi-port, multi-vessel, and multi-voyage shipping systems characterized by dynamic, unpredictable, and uneven customer demand. [PERSON] and [PERSON] [15] suggested the problem of designing the network of liner shipping services that integrated both hub-and-spoke operations and multi-port calling strategies, along with the repositioning of empty containers. [PERSON] et al. [16] addressed the issue of cargo allocation while considering how empty containers must be repositioned and used the Dantzig-Wolfe principle to decompose an arc-flow problem formulation into path-flow problem formulations. [PERSON] et al. [17] devised a multi-route strategy that took into account weekly scheduling, cargo transshipment, and transport duration while aiming to minimize costs. In terms of container weighing, the importance of container weight measurement has been addressed in several safety regulations governing verified gross mass (VGM). Since about ten years ago, academics and stakeholders have tended to focus more on analyzing maritime transport risks and developing risk hierarchies [18; 19]. While maritime safety has seen notable improvements from 2008 to 2018, as evidenced by lower average losses [20] and better identification of accident sources [21], a study in 2018 found that more than 60% of shipping accidents were recurring incidents [22]. Additionally, the shipping and port industries are confronting elevated risks stemming from environmental changes [23]. This context underscores the need for the progressively stricter regulatory framework imposed by the International Maritime Organization (IMO). The IMO has adopted about 50 conventions and protocols, as well as more than 1000 codes and recommendations [24]. Research by [PERSON] and [PERSON] [25] has shown the positive impact of legislation on improving safety and environmental protection.