## 2 Literature Review
[34] proposed CDP that simultaneously incorporates the Drop-and-pull mode and the route balancing. It is worth to note that both [PERSON] et al. [33] and [PERSON] et al. [34] split the IFER and OFED requests into four basic requests, i.e., IF, ER, ED, and OF. Aside from container terminals, in reality, the empty containers generated by unpacking inbound full containers may have multiple destinations, including depots or shippers. Additionally, not every delivery request can be perfectly paired with a pickup request. Consequently, the repositioning of empty containers was formally introduced into the study of CDP by [PERSON] et al. [35]. They pioneered the extension of previous studies by incorporating four types of transport requests: IFER, OFED, IE, and OE, laying the foundation for subsequent research. Since the destination of the IE request and the origin of the OE request are not given in advance, the IE/OE requests are also called flexible requests. Additionally, they exclusively employed the Stay-with mode, formulated a multi-depot CDP as a MILP model based on a DAOV graph and developed a reactive TS algorithm. Nevertheless, they treated the determined movement between a request vertex and any other vertex as the activity on the request vertex, making the time calculation more complex and difficult to explicitly describe the allocation of empty containers. [PERSON] et al. [36] and [PERSON] [37] further extended [PERSON] et al. [35] to a CDP with multiple depots and multiple terminals. The latter designed a two-stage heuristic that outperformed the window partitioning method proposed by the former. [PERSON] and [PERSON] [38] explicitly modeled the repositioning of empty containers upon the framework presented by [PERSON] et al. [36], and designed an enhanced TS heuristic, demonstrating superior effectiveness and efficiency on both small and large instances compared to existing methods. [PERSON] et al. [39] investigated the scenario of one vehicle depot and multiple terminals where terminals are used to store empty containers, and proposed a sequential and an integrated approach, but their model only allowed the Drop-and-pull mode for IFER requests. [PERSON] et al. [40] investigated the CDP with multi-size containers utilizing a state-transition-based modeling method and improved reactive TS algorithm. However, they made the \"full-twin\" assumption, which states that if a truck is carrying two 20 ftfull containers, it can only begin transporting new requests after unloading both of them. [PERSON] et al. [10] formulated a bi-objective mathematical model based on the DAOV graph by considering empty containers constraints and designed a LNS algorithm to solve the problem. [PERSON] and [PERSON] [41] relaxed the \"full-twin\" assumption in [PERSON] et al. [40] by introducing a MILP model. [PERSON] et al. [42] extended [PERSON] and [PERSON] [41] by considering heterogeneous fleet and network flow constraints at depots, and solved this problem using a tailored generic algorithm (GA). However, the trucks were forced to start/finish at the terminal which reduced the difficulty of modeling. Due to trade imbalances, certain regions have become import-dominant or export-dominant areas where inbound empty and outbound empty containers may not exist simultaneously. [PERSON] et al. [43] then focused on the CDP in an export-dominant area, and restricted the three types of requests: IFER, OFED, and IE. Since the purchase cost of empty containers is relatively high, which means the supply of empty containers in container terminals or inland depots is not inexhaustible. Therefore, for the CDP, empty containers are also a necessary transportation resource, in addition to the two essential resources of trucks and drivers found in typical vehicle routing problems. [PERSON] et al. [43] first formulated the CDP with empty container constraints as a m-TSPTW and developed a reactive TS algorithm to solve it. Conversely, [PERSON] et al.