## 2 Literature Review
Recognizing these limitations, scholars are increasingly emphasizing the importance of broadening sales channels. Ref. [18] highlighted the significant advantages of shipping e-commerce in the face of COVID-19. Ref. [19] analyzed the SAPC in the e-commerce environment to maximize the marginal cost of liner companies. The result indicated that the e-commerce sales channel is predictable and outperforms traditional sales channels in terms of revenue. Furthermore, refs. [41; 42] devised a container-open booking platform. Building upon this platform, ref. [43] developed a multi-agent architecture to realize timely message interaction among autonomous agents, shippers, freight forwarders, and liner companies. To continuously increase liner companies' revenue, scholars have jointly optimized the SAPC with other issues. Refs. [44; 45] have researched the combination problem of route optimization and SAPC. Additionally, some scholars addressed the joint optimization problems of SAPC and empty container repositioning. Ref. [46] investigated SAPC with multi-type containers amid seasonal demand fluctuations, taking into account the empty container transportation. Their findings underscored the higher priority of refrigerated and 40' dry bulk containers over ordinary containers. Ref. [47] employed RM to tackle SAPC alongside the empty container repositioning issue for Asian ocean carriers, devising estimation methods for the expected cost of empty container repositioning. Ref. [8] established a bi-level linear programming model to address this jointly optimization problem. The upper model focused on the SAPC to maximize routes' profit, while the lower model minimized transportation costs associated with empty container repositioning. Ref. [46] explored the SAPC, considering empty container repositioning with the ship schedule adjustment, revealing that several factors, such as cargo transfer cost, potential profit of cargo handling, and profit of empty container repositioning, have significant effects on ship capacity decision making and port selection. Furthermore, ref. [1] combined the SAPC with empty container inventory management, considering market segmentation and no-shows scenarios. The brief summary of the literature is shown in Table 1.  
In the research on optimization methods, most of the literature relies on commercial software such as Cplex [1; 19], Gurobi [4], and WinQSB 2.0 [8]. These types of optimization software achieve high accuracy for small-scale problems; however, they struggle to deliver satisfactory solutions within a short time when addressing large-scale challenges. To overcome this limitation, some researchers have focused on developing optimization algorithms tailored for large-scale problem solving, such as the NSGA-II algorithm [30] and a pattern-search-based solution algorithm [48]. Others have proposed approximate algorithms to handle cabin allocation in stochastic scenarios, including sampling-based average approximation methods [5; 49] and approximate dynamic programming approaches [50; 51; 52]. In the existing literature on slot allocation, the research achievements of collaboratively optimizing empty container repositioning and slot allocation are scanty, requiring new models and empirical evidence. Much of the literature primarily focuses on the transportation demand for laden containers, neglecting the objective need for empty container repositioning. To the best of our knowledge, the current studies on empty container repositioning have overlooked the transformation between empty and laden containers. Many studies simplify the problem of empty container repositioning as a single-period problem [10; 12; 46; 53; 54; 55; 56; 57; 58; 59], disregarding its multi-period nature. Consequently, the collaborative optimization problem of multi-cycle empty container repositioning and slot allocation still has theoretical implications that need to be further explored. Moreover, most papers on