TRANSPORTES Y PESCA

The main representative methods are selected and briefly presented.

According to the three operation stages of stacking blocks, storing blocks and retrieving blocks, the literature concerning the main handling problem at each stage is divided into three parts.

The problem to be handled during the block stacking stage is the storage location assignment. The blocks relocation problem deals with the block retrieval operations during which relocations are carried out. The pre-marshaling happens during the time after the blocks are put away but before they are picked up. The aim is to prepare the blocks in such a sequence so that it is not necessary to do any relocations or very less relocations during retrieval operations. The three types of problems at the three of stages are formulated more or less with the same goal of minimizing the movement of blocks during stacking, pre-marshaling and/or retrieval operations.

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2.1 Storage location assignment

Formulas are proposed to estimate the expected number of relocations for retrieving an arbitrary container from a container stack [5, 6]. A dynamic programming model is formulated by Kim et al. [7] to determine the storage location for an arriving individual container to minimize the number of relocation movements expected during loading operations. In their problem formulation, the containers may be assigned three priorities depending on which weight group they belong to. The higher the weight of a container is, the higher the priority of the container has for retrieval. Yang and Kim [8]

address both static location problem with known arrival time and dynamic location problem with uncertain arrival time. The static location problem is solved using a genetic algorithm and the dynamic location problem is solved using heuristic rules.

Zhang et al. [9] propose a two-stage heuristic to handle the location assignment for arriving outbound containers even the proportion of the remaining containers on a weight group keeps changing during the container-receiving process. The priority sequence of stacking patterns for each weight group of containers is generated through a neighborhood searching heuristic in the first stage and the incumbent solution is improved by simulating more stack alternatives for each arriving container through a rollout-based heuristic in the second stage.

2.2 Blocks relocation problem

The blocks relocation problem presented by Kim and Hong [2] can be considered as the pioneer formulation of the basic BRP with the following assumptions:

 Only blocks from the top of stacks in the bay can be accessed.

 The relocation of blocks is carried out among stacks in the same bay.

 The relocated blocks can be put only on top of the other stacks. That means no rearrangement of blocks within a stack is allowed.

 No pre-sorting is carried out before retrieval. That means relocations occur only at the moment when blocks are retrieved.

 No block stacking occurs during retrieval operations.

 The pickup precedence of blocks to be retrieved is known in advance and does not change during retrieval operations.

 When a block is retrieved, it is removed from the container bay. Hence, the overall number of blocks in the bay decreases over time.

Obviously, the problem is to minimize the number of moves needed for retrieving some blocks or all of the blocks in the bay with a given retrieval sequence. Figure 2 shows an example of removing blocks with required relocation operations. The boxes simulate the blocks and the numbers inside tell the priorities of retrieving the blocks. Blocks with lower numbers should be removed earlier than the blocks with higher numbers. To remove four

blocks in the given sequence 1, 2, 3 and 4, it is necessary to conduct three relocations.

Figure 2: An example of retrieving a set of blocks.

Towards solving this kind of BRP, there are different optimization methods developed in the literature. Caserta et al. [4] utilize dynamic programming formulation in order to solve the BRP by means of the corridor method. In compare to the heuristic addressed by Kim und Hong [2], their method exhibited better results based on the computational experiments. A mathematical model firstly proposed by Caserta et al. [9] encompasses the complete set of features of the BRP. In the three-phase algorithm proposed by Lee and Lee [11], an extended objective combining both numbers of relocation and crane’s working time is formulated. In the heuristic approach proposed by Jovanovic and Voss [12], the properties of the block to be moved next are taken into account when the decision where to relocate a block is made. Kim et al. [13] propose a heuristic algorithm for BRP with not only minimizing relocation number but also the working time of the crane.

Galle et al. [14] propose the stochastic model for blocks relocation problem, in which the uncertainty of retrieval time is considered because of the unexpected arriving time of trucks. Zhao and Goodchild [15] examine the influence of completeness of retrieval sequence on container relocation operations. Furthermore, the assumption of no blocks to be stacked during retrieval operations is also questioned.

2.3 Pre-marshalling problem

Container pre-marshaling for container operations can be conducted before retrieval operations start when the blocks are stored in the storage for a certain time. The blocks are pre-sorted in the pickup sequence so that no extra relocations are needed during the retrieval process. Figure 3 shows an example of pre-marshaling of seven existing blocks. For the stock

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operations for reaching the retrieval sequence without any further relocation requirement.

Figure 3: An example of pre-marshaling all existing blocks.

When the two examples in Figure 2 and Figure 3 are compared, it is to notice that from the same initial situation, pre-marshaling needs more relocation operations to reach the final block configuration than doing relocation operations during retrieval. From the viewpoint of relocation reduction, it is not recommended to do pre-marshaling. However, to reduce the retrieval time which determines the loading time of a vessel or the waiting time of trucks, pre-marshaling is then the solution.

Container pre-marshaling problem is often modeled as a mathematical optimization problem with the optimization goal of minimizing the number of container movements during pre-marshaling [16]. Through 37 computational examples, their method is proved to be very effective for solving container re-marshaling problem to prepare the individual containers in a certain sequence or all container groups to be accessible. Lee and Chao [17] develop a heuristic which consists of a neighborhood search process, an integer programming model and three minor subroutines for container pre-marshaling. Izquierdo et al. [18] announce in their paper that they have developed a heuristic for a pre-marshaling problem with better solving quality than the previous ones in the literature.