The Stochastic Container Relocation Problem

Abstract

The container relocation problem (CRP) is concerned with finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers, while respecting a given order of retrieval. However, the assumption of knowing the full retrieval order of containers is particularly unrealistic in real operations. This paper studies the stochastic CRP, which relaxes this assumption. A new multistage stochastic model, called the batch model, is introduced, motivated, and compared with an existing model (the online model). The two main contributions are an optimal algorithm called Pruning-Best-First-Search (PBFS) and a randomized approximate algorithm called PBFS-Approximate with a bounded average error. Both algorithms, applicable in the batch and online models, are based on a new family of lower bounds for which we show some theoretical properties. Moreover, we introduce two new heuristics outperforming the best existing heuristics. Algorithms, bounds, and heuristics are tested in an extensive computational section. Finally, based on strong computational evidence, we conjecture the optimality of the “leveling” heuristic in a special “no information” case, where, at any retrieval stage, any of the remaining containers is equally likely to be retrieved next. The online appendix is available at https://doi.org/10.1287/trsc.2018.0828 .

Optimizing pre-processing and relocation moves in the Stochastic Container Relocation Problem

“…A container is relocated to the stack with the smallest number of reshuffles. In Galle, Manshadi, Borjian Boroujeni, Barnhart, and Jaillet (2018) , another heuristic is introduced, which is called the Expected Minmax (EM) heuristic. In the EM heuristic, containers are relocated to stacks in which they are relocated as late as possible.…”

“…In the EM heuristic, containers are relocated to stacks in which they are relocated as late as possible. Besides the EM heuristic, in Galle et al (2018) an optimal formulation for the SCRP using decision trees is presented. For small instances, the optimal solution can be computed which is close to the solution of the EM heuristic.…”

“…Whereas, in the batch model , at the beginning of each interval the exact order of all containers in that single interval is revealed. The batch model is more suitable for larger terminals in which the waiting time of a truck is often larger than the time length of the interval ( Galle et al, 2018 ). As our problem setting is more applicable to less busy ports, we choose to use the online model.…”

“…Finding a good lower bound for the SCRPPP is more complicated than the upper bound of the previous section. There are multiple lower bounds known for the (S)CRP, (see, e.g., Galle et al, 2018 andScholl, Boywitz, &Boysen, 2018 ). One lower bound for the CRP is to count how many containers have a container with a lower time frame underneath them.…”

“…The lower bound for the SCRP from ( Galle et al, 2018 ) is similar but also takes into account the fact that two containers in a stack might have the same time frame. In this lower bound, the probability that a container is blocking another container is calculated.…”