In this paper we address an exact algorithm for the block relocation problem with a stowage plan. This problem is an extension of the container (block) relocation problem that aims at minimizing the total number of relocations required for retrieving containers piled up in a container yard. The difference is that a stowage plan in a vessel is given in advance and it is taken into consideration when retrieving containers. More specifically, the retrieval order of containers should be feasible with respect to a given stowage plan. We construct a branch-and-bound algorithm with iterative deepening for unrestricted and restricted variants of this problem. To this end, we propose three lower bounds on the total number of relocations. We also extend dominance rules for the container relocation problem, in order to restrict the search space and improve the search efficiency. The effectiveness of the proposed algorithm is examined by numerical experiments for benchmark instances from the literature.

Introduction

Container transportation is a major means of maritime freight nowadays, and the number of containers handled in sea ports has been constantly increasing in recent decades. According to the statistics of the United Nations Conference on Trade and Development, the world container port throughput is 752 million TEUs (Twenty-foot Equivalent Units) in 2017, which exhibits a 34% growth from 560 million TEUs in 2010 (UNICTAD, 2018). To handle this many containers properly without causing serious delays, effective operations in sea ports are of crucial importance. Container terminals in sea ports provide temporal storage for connecting different transportation modes: Incoming containers via maritime and land transport are stored temporarily in a storage yard until they are retrieved for further maritime and land transport. Primarily due to space limitation, containers are usually piled up vertically in tiers in the yard. Since cranes can access only the topmost containers, relocations of containers within the yard are inevitable, unless their retrieval order is known in advance and, at the same time, containers are stacked in accordance with the order. Roughly speaking, three types of problems have been studied so far (Caserta, Schwarze, & Voß, ۲۰۱۱a) to reduce un-productive relocations and to achieve a better container handling performance. The stacking problem determines appropriate storage locations of incoming containers using incomplete and often stochastic information of their retrieval order. The pre-marshalling problem and the re-marshalling problem rearrange containers within the yard off-line so that no relocations are necessary when they are actually retrieved.