## 3 Problem Description and Formulation
### Problem Description  
Due to the limited number of yard cranes, each must move back and forth between different bays rather than being dedicated to a specific unit. So, the state change of a bay and the division of operation rounds during the retrieval process of import containers are shown in Figure 1. As shown in Figure 1, we define an _operation round_ as the bay is in _busy_ state, whose period that begins when the yard crane moves into the destination bay and ends when the yard crane moves out the destination bay. After an operation round is completed, the bay state turns into _idle._ Figure 1 tells us that the whole retrieval process of a bay can be divided into three parts--completed operation rounds, the real-time operation round and future operation rounds. The completed operation rounds represent that the retrieval and relocation operations have been finished, which is irreversible. The operations within the real-time operation round are unrelated to the completed operation round but will impact future operations. Consider a bay with \(N\) containers piled on \(W\) stacks, and the height of each stack cannot exceed \(H\) tiers. In the initial state, all the \(N\) containers in the bay have been appointed their departure time windows according to the corresponding ETs' reservation arrival period. In turn, they can be classified into \(G\) groups based on their respective booking departure times and numbered \(1,\ \ldots,G\), which implied their retrieval priorities (the smaller group index, the higher retrieval priority). Suppose this instance requires \(K\) (\(K\geq 1\)) operation rounds to finish retrieving all the \(N\) containers. Let \(S^{k}\) denote the bay layout after the completion of \(k\)th operation round, especially indicating the initial layout when \(k=0\). Suppose we have completed the first \(k-1\) operational rounds, then we need to complete the \(k\)th operation round. We aim to find an optimized solution \(a^{k}\) for the \(k\)th operation round that minimizes the number of relocations during the \(k\)th operation round. Once the \(k\)th operation round finishes, the bay layout will be transformed from \(S^{k-1}\) to \(S^{k}\), which becomes the basis for \((k+1)\)th operation round. The change in the bay layout between two adjacent operation rounds is described by the following Equation (1). \[S^{k-1}\xrightarrow{a^{k}}S^{k},\ 1\leq k\leq K \tag{1}\]  
Figure 1: The retrieval operation process analysis of a bay. The \(a^{k}\) is a set of operation instructions. If \(a^{k}\) includes some relocation operation instructions, some containers will be moved from their stacks into other new stacks. For the restricted CRP, the relocated containers are only the blocking containers above the targets. The relocated containers may contain containers other than the blocking containers above the targets for the unrestricted CRP. However, these movements may result in the addition of the number of blocking containers for the moved-in stacks and impact the \((k+1)\)th and subsequent operation rounds. Our objective is to minimize the addition of the number of blocking containers in the real-time operation round. Moreover, during real-time operations, the retrieval sequence of target containers and the movement orders of blocking containers determine the changes in the bay layout, and different bay layouts will produce different increments in the number of blocking containers and other impacts on the subsequent operation process. In the real-time operation round, when multiple target containers need to be picked up, it makes sense to choose a reasonable target container pickup order that can produce fewer relocation operations, even though it may prolong the waiting time of some ETs. Therefore, the method proposed in this paper allows for adjusting the pickup order according to the bay layout, integrating the optimization of the pickup sequence and the reallocations of blocking containers to minimize the increment of the expected number of blocking containers (ENBC).