On Rearrangement of Items Stored in Stacks

There are $n \ge 2$ stacks, each filled with $d$ items, and one empty stack. Every stack has capacity $d > 0$. A robot arm, in one stack operation (step), may pop one item from the top of a non-empty stack and subsequently push it onto a stack not at capacity. In a {\em labeled} problem, all $nd$ items are distinguishable and are initially randomly scattered in the $n$ stacks. The items must be rearranged using pop-and-pushs so that in the end, the $k^{\rm th}$ stack holds items $(k-1)d +1, \ldots, kd$, in that order, from the top to the bottom for all $1 \le k \le n$. In an {\em unlabeled} problem, the $nd$ items are of $n$ types of $d$ each. The goal is to rearrange items so that items of type $k$ are located in the $k^{\rm th}$ stack for all $1 \le k \le n$. In carrying out the rearrangement, a natural question is to find the least number of required pop-and-pushes. Our main contributions are: (1) an algorithm for restoring the order of $n^2$ items stored in an $n \times n$ table using only $2n$ column and row permutations, and its generalization, and (2) an algorithm with a guaranteed upper bound of $O(nd)$ steps for solving both versions of the stack rearrangement problem when $d \le \lceil cn \rceil$ for arbitrary fixed positive number $c$. In terms of the required number of steps, the labeled and unlabeled version have lower bounds $\Omega(nd + nd{\frac{\log d}{\log n}})$ and $\Omega(nd)$, respectively

Approaches for solving the container stacking problem with route distance minimization and stack rearrangement considerations

We consider an optimization problem of sequencing the operations of cranes that are used for internal movement of containers in maritime ports. Some features of this problem have been studied in the literature as the stacker crane problem (SCP). However, the scope of most literature (including SCP) is restricted to minimizing the route or distance traveled by cranes and the resulting movement-related costs. In practice, cargo containers are generally stacked or piled up in multiple separate columns, heaps or stacks at ports. So, the cranes need to often rearrange or shuffle such container stacks, in order to pick up any required container. If substantial re-stacking is involved, cranes expend considerable effort in container stack rearrangement operations. The problem of minimizing the total efforts/time of the crane must therefore account for both - the stack rearrangement costs and also the movement-related (route distance) costs. The consolidated problem differs from standard route distance minimization situations if stack rearrangement activities are considered. We formally define the consolidated problem, identify its characteristic features and hence devise suitable models for it. We formulate several alternative MIP approaches to solve the problem. We compare the performance of our MIP formulations and analyze their suitability for various possible situations