## 4 Methodology
### Moving Ahead for Sequential Stack (MSS)  
Given the blocking container \(c\) that is next to be relocated, its optimal reallocation stack \(s^{*}\) is determined according to the \(LL\) rule. If stack \(s^{*}\) is sequential-stack for blocking container \(c\), we try to find another inverted-placement container, which is the topmost item in its stack \(s\) (\(s\
eq s^{*}\)) and denotes as \(top(s)\). If stack \(s^{*}\) is sequential-stack for container \(top(s)\) and moving it into stack \(s^{*}\) before container \(c\) does not change the sequential-placement state of container \(c\). We define this method as a relocating policy, known as the Moving-ahead for Sequential Stack (MSS) rule. The MSS rule aims to move a inverted-placement container into a sequential-stack stack in advance and reduce the likelihood of the container \(top(s)\) becoming a blocking container again in the subsequent operation process. This rule needs to comply with the following conditions: (1) there are at least two empty slots in stack \(s^{*}\); (2) the topmost container \(top(s)\) in stack \(s\) must be in inverted-placement state, i.e., its retrieval priority satisfies the condition \(p(top(s))>p_{min}(s)\); and (3) the retrieval priority of container \(top(s)\) must satisfy the condition \(p(c)<p(top(s))<p_{min}(s^{*})\), i.e., the stack \(s^{*}\) is sequential-stack for containers \(c\) and \(top(s)\) and the retrieval priority of container \(top(s)\) is lower than container \(c\). Figure 3 shows us the processing of the rule \(MSS\).  
For a blocking container \(c\), if only inverted-stack stacks are available, directly moving container \(c\) into one of the stacks will add a new blocking container and, in turn, add a relocation during the subsequent retrieval processes. In this case, selecting a suitable stack \(s\) and moving its topmost container \(top(s)\) to another stack \(s^{\prime}\) will free up an sequential-placement for container \(c\). If stack \(s^{\prime}\) is a sequential-stack for container \(top(s)\), the total operation cost of the \"free up\" operations is the same as that of direct moving. As well as it may also increase the possibility of selecting sequential-placement for the subsequent relocations. We refer to the above method as a new relocation policy: freeing up for the sequential stack (FSS) rule. The FSS rule needs to comply with the following conditions: (1) the available reallocation positions for the blocking container \(c\) are inverted-placements; (2) if stack \(s\) is a candidate sequential-placement for the blocking container \(c\) by freeing up the topmost item \(c^{{}^{\prime}}\), while the retrieval priority value satisfies \(p(c^{{}^{\prime}})<p(c)<p(under(c^{{}^{\prime}}))\); and (3) the container \(c^{{}^{\prime}}\) will be moved into stack \(s^{{}^{\prime}}\), which must be the sequential-stack for container \(c^{{}^{\prime}}\).  
We now continue to analyze the instance depicted in Figure 3. In the final configuration, container \((1,2)\) is the current forced relocating item, and two available stacks, the second and fourth stack, are both inverted stacks. Let us move container \((4,1)\) from the fourth  
Figure 3: Processing of heuristic rule \(MSS\).  
to the second stack and obtain a sequential stack by consuming one practical relocation movement. Furthermore, this reshuffle is cost-effective since it may provide an additional sequential-placement in subsequent processes. Figure 4 illustrates the detailed processing of the FSS rule.