## 4 Methodology
### Preliminaries
#### 4.1.1 The Evaluation Method of ENBC  
Let \(p(c)\) represent the retrieval priority value of container \(c\) and \(p_{min}(s)\) denote the minimum retrieval priority value of the container set \(s\) (if \(s\) is a stack number, it represent all the containers in stack \(s\)), the smaller value is the the higher priority. Specifically, \(p_{min}(s)=N+1\) if \(s\) is an empty set or stack. Let \(under(c)\) denote the containers set within the same stack with container \(c\) and below it.  
We define three placement state types of containers as follows. The placement state of the container \(c\) is to be known as: (1) _sequential-placement_ if \(p(c)<p_{min}(under(c))\) is satisfied; (2) _inverted-placement_ if \(p(c)>p_{min}(under(c))\) is satisfied; (3) _level-placement_ if \(p(c)=p_{min}(under(c))\) is _True_. A sequential-placement container means that all other containers placed below have lower retrieval priorities than it. Similarly, an inverted-placement container means that all other containers placed below have higher retrieval priorities than it, i.e., it has a bigger retrieval priority value than the other containers below it. A level-placement container has the same retrieval priority as the highest retrieval priority of the other containers placed below.  
Based on their predetermined precedence (e.g., appointment time window), the containers with higher precedence will be retrieved earlier than those with lower precedence during retrievals. Therefore, a sequential-placement container will be picked up earlier than other containers below it without relocation. Conversely, an inverted-placement container must be relocated at least once, as some containers below it will be picked up earlier. We assume the containers with the same retrieval priority have the same retrieval probability. So, a level-placement container causes a relocation when the same retrieval priority container below it is picked up earlier. The blocking containers in a bay layout include the inverted-placement and level-placement containers, and the ENBC is equal to the sum of the number of inverted-placement containers and the expected relocation number of the level-placement containers. The calculation method for the ENBC can be described as Equation (10), which has been investigated in [5]. In Equation (10), \(I(x)\) is an indicator function that equals one if \(x\) is _True_ and zero otherwise, and \(h(r)\) indicates the container number within stack \(r\).  
\[lb(r)=h(r)-\sum_{i=1}^{h(r)}\left(\frac{I\left(p(c_{i})=\min_{1\leq j\leq i} \left\{p(c_{j})\right\}\right)}{\sum\limits_{j=1}^{i}I(p(c_{i})=p(c_{j}))}\right) \tag{10}\]  
There is no increase in the number of blocking containers in a candidate stack when a blocking container becomes a sequential-placement container after being moved into the candidate stack during retrieval. We refer to the candidate stacks for the blocking container that satisfy the above condition as _sequential-stack_ (_SS_ for short). Contrarily, the ENBC of the candidate stack will increase when a blocking container becomes the inverted-placement or level-placement container after being moved into a candidate stack. The candidate stack that satisfies the condition is referred to as _inverted-stack_ (_IS_ for short) or _level-stack_ (_LS_ for short), respectively. The notation \(f(c,r)\) has been defined previously, which denotes the increment of ENBC of the stack \(r\) after relocating the container \(c\) into stack \(r\), and the formula is shown Equation (11).  
\[f(c,r)=\left\{\begin{array}{ll}0,&r\in SSs,\\ 1,&r\in ISs,\\ \frac{m-1}{m},&r\in LSs,\end{array}\right. \tag{11}\]  
where \(m=\sum\limits_{j=1}^{h(r)}I(p(c)=p(c_{j}))\) denotes the number of containers with the same priority as container \(c\) in the stack \(r\).