# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 1 Introduction  
In the context of globalization, the volume of global trade has increased significantly, and more than 80% of it is transported by sea. More and more manufactured goods are transported in standardized containers because of their advantages in cargo security, standardization, and ease of interchange between different modes of transport. In 2022, container trade exceeded 850 million units and is expected to grow rapidly in the coming years. Container terminals (CTs), serving as important hubs that connect maritime trade with other modes of transportation, have become key infrastructures for global trade. In recent years, terminal congestion has become a common problem at major CTs around the world, leading to significant increases in container schedule delays and turnaround times. In 2021, the average global container schedule delay doubled compared to 2020, and the average time at the terminal reached 0.8 days [1]. Delays and waiting affect shipowners' business plans and timely cargo delivery, reducing shipowner satisfaction. Research indicates that increased shipowner satisfaction leads to greater loyalty and a stronger intent for repeat purchases [2]. Therefore, as one of the indicators of terminal service quality and competitiveness, the improvement of shipowner satisfaction is of great significance to CTs, and the key is to improve operational efficiency and reduce the waiting time and delay of vessels. Berths and quay cranes (QCs) are bottleneck resources in the seaside area of CTs. The berthing plan is a key input to yard allocation and equipment deployment planning, and effective utilization of berths is critical to improving shipowner satisfaction and increasing vessel capacity at the terminal [3]. Therefore, one of the most important issues facing CTs is the berth allocation problem (BAP). This problem determines the preferred berthing time and position for vessels arriving within a planning horizon based on information such as vessel length, expected arrival time, and processing time. In BAP, the processing time of a vessel is usually estimated in advance based on the task volume and is kept fixed. Equally important to the BAP is the quay crane assignment problem (QCAP), which involves the assignment of QCs to vessels at berth. For technical and contractual reasons, the number of cranes servicing a vessel at the same time is usually limited to a range, from a minimum to a maximum range. Research is increasingly focusing on jointly considering BAP and QCAP, as a vessel's processing time depends on the number of assigned QCs, influenced by its berthing time and position. The joint problem is referred to as the berth and quay crane allocation problem (BACAP), where each vessel is assigned a specific number of QCs, along with specified berthing times and positions. A more realistic extended version is the berth and specific quay crane allocation problem (BACASP), in which each vessel is assigned a specific set of QCs, thus ensuring that the QCs are able to service different vessels without crossing each other [4]. There are two versions of BACAP and BACASP [5]. In the time-invariant version, a vessel keeps its QC assignment constant during processing. In the time-variant version, the number of QCs is variable, and the QCs assigned to berthed vessels may change after a vessel departs or a new vessel is berthed. This situation may result in many crane movements, which can be difficult for human operators to process. Suppose the duration of these movements is considered non-negligible. In that case, it leads to unplanned additional waiting time, leading to unexpected changes in berthing schedules and delays in scheduled vessel departures, thus increasing operational costs and reducing shipowner satisfaction. In contrast, the time-invariant version results in fewer QC movements and provides a more stable berth plan, which is easier for terminal managers to implement. However, it may lead to lower QC resource utilization.