Blocks Relocation Problem Research Articles

A local-search based heuristic for the unrestricted block relocation problem

The unrestricted block relocation problem is an important optimization problem encountered at terminals, where containers are stored in stacks. It consists in determining the minimum number of container moves so as to empty the considered bay following a certain retrieval sequence. A container move can be either the retrieval of a container or the relocation of a certain container on top of a stack to another stack. The latter types of moves are necessary so as to provide access to containers which are currently not on top of a stack. They might also be useful to prepare future removals. In this paper, we propose the first local search type improvement heuristic for the block relocation problem. It relies on a clever definition of the state space which is explored by means of a dynamic programming algorithm so as to identify the locally optimal sequence of moves of a given container. Our results on large benchmark instances reveal unexpectedly high improvement potentials (up to 50%) compared to results obtained by state-of-the-art constructive heuristics.

The bounded beam search algorithm for the block relocation problem

In this paper we deal with the restricted Block Relocation Problem. We present a new lower bound and a heuristic approach for the problem. The proposed lower bound can be computed in polynomial time and it is provably better than some previously known lower bounds. We use it within a bounded beam search algorithm to solve the Block Relocation Problem and show that the considered heuristic approach outperforms the other existing algorithms on most of the instances in the literature. In order to test the approaches on real-size dimensions, new large instances of the Block Relocation Problem are also introduced.

An efficient ant colony optimization algorithm for the blocks relocation problem

In this paper we present an ant colony optimization (ACO) algorithm for the Blocks Relocation Problem (BRP). The method is applied to both versions of the problem most commonly considered in literature, i.e., the restricted (rBRP) and the unrestricted (uBRP) BRP with distinct due dates. In case of the uBRP a new heuristic is proposed and incorporated in a standard greedy algorithm. The performance of the basic greedy approach is enhanced by extending it to the ACO metaheuristic. In it, a novel approach for defining the pheromone matrix is proposed. More precisely, it only stores a small amount of information instead of the complete bay state. Further, we show that the proposed ACO method can easily be adapted for solving the BRP in which the objective function is related to the crane operation time. Our computational results show that the proposed approach manages to outperform existing methods for the BRP.

A GRASP approach for solving the Blocks Relocation Problem with Stowage Plan

In this paper, we present a method for finding high quality solutions for the Blocks Relocation Problem with Stowage Plan (BRLP). The interest for this problem comes from the fact that previous research has shown that a significant amount of savings can be achieved, if the process of specifying the loading sequence takes into account both the state of the bay and the stowage plan. In this work, we propose a two step greedy algorithm for the BRLP. In the first one, a heuristic is used to select the next container to be loaded into the vessel. In the second step, a new heuristic is used to relocate obstructing containers. The solutions acquired in this way are improved using a correction procedure. The idea of the correction procedure is to use a heuristic approach to recognize undesirable properties of a solution and remove them. Finally, the method is extended towards the GRASP metaheuristic. Our computational experiments show that the proposed approach manages to significantly outperform existing methods for the BRLP.

An exact algorithm for the Blocks Relocation Problem with new lower bounds

The Blocks Relocation Problem is an important problem in storage systems. An input instance for it consists of a set of blocks distributed in stacks where each block is identified by a retrieval number and each stack has a same maximum height limit. The objective is to retrieve all the blocks respecting their retrieval order and performing the minimum number of relocations. Only blocks at the top of a stack can be moved: either a block is retrieved, if it has the highest retrieval priority among the stacked blocks, or it is relocated to the top of another stack. Solving this problem is critical in storage systems because it saves operational time and resources. In this paper, we present two new lower bounds for the number of relocations of an optimal solution. We implemented an exact iterative deepening A* algorithm using these new proposed lower bounds and other well-known lower bounds from the literature. We performed several computational experiments to show that the new lower bounds improve the performance of the exact algorithm, solving to optimality more instances than when using other lower bounds when given the same amount of time.

A new effective unified model for solving the Pre-marshalling and Block Relocation Problems

Container terminals are exchange hubs that interconnect many transportation modes and facilitate the flow of containers. Among other elements, terminals include a yard which serves as temporary storage space. In the yard, containers are piled up by cranes to form blocks of stacks. During the shipment process, containers are carried from the stacks to ships following a given sequence. Hence, if a high priority container is placed below low priority ones, such obstructing containers have to be moved (relocated) to other stacks. Given a set of stacks and a retrieval sequence, the aim in the Pre-marshalling Problem (pmp) is to sort the initial configuration according to the retrieval sequence using a minimum number of relocations, so that no new relocations are needed during the shipment. The objective in the Block Relocation Problem (brp) is to retrieve all the containers according to the retrieval sequence by using a minimum number of relocations. This paper presents a new unified integer programming model for solving the pmp, the brp, and the Restricted brp (r-brp) variant. The new formulations are compared with existing mathematical models for these problems, as well as with other exact methods that combines combinatorial lower bounds and the branch-and-bound (B&B) framework, by using a large set of instances available in the literature. The numerical experiments show that the proposed models are able to outperform the approaches based on mathematical programming. Nevertheless, the B&B algorithms achieve the best results both in terms of computation time and number of instances solved to optimality.

An exact algorithm for the unrestricted block relocation problem

The purpose of this study is to propose an exact algorithm for the unrestricted block relocation problem with distinct priorities. In this problem, a storage area is considered where blocks of the same size are stacked vertically in tiers. Because we can access only topmost blocks, relocations of blocks are required when other blocks are retrieved. The objective is to minimize the total number of such relocations necessary for retrieving all the blocks one by one according to a specified order. In the restricted version of this problem, only the topmost block above the target block is relocatable. On the other hand, no such restriction is imposed on the unrestricted problem, which is considered in this study. We also assume that each block is assigned a distinct retrieval priority and the retrieval order of blocks is unique. To improve the efficiency of a branch-and-bound algorithm for this problem, we propose several dominance properties to eliminate unnecessary nodes in the search tree. Furthermore, we propose a new lower bound of the total number of relocations. The effectiveness of the proposed exact algorithm is verified by numerical experiments for benchmark instances in the literature.

A new binary formulation of the restricted Container Relocation Problem based on a binary encoding of configurations

The Container Relocation Problem (CRP), also called Block Relocation Problem (BRP), 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. The restricted CRP enforces that only containers blocking the target container can be relocated. We improve upon and enhance an existing binary encoding and using it, formulate the restricted CRP as a binary integer programming problem in which we exploit structural properties of the optimal solution. This integer programming formulation reduces significantly the number of variables and constraints compared to existing formulations. Its efficiency is shown through computational results on small and medium sized instances taken from the literature.

On the quality of simple measures predicting block relocations in container yards

The blocks relocation problem aims to minimise the number of unproductive relocation moves where blocking containers have to be removed to other stacks when retrieving containers in a given sequence from a container yard. Sometimes there is not enough time to solve this complex optimisation problem and information on the crane performance, which mainly depends on the relocation effort, is required under great time pressure. In this case, simple measures are required, which are quickly computed and, nonetheless, accurately predict the amount of relocation moves. This paper is dedicated to comparing twelve different known and new measures. Our computational study shows that composite measures combining a lower bound argument with a simple rule-based upper bound lead to predictions of remarkable accuracy.

New insights on the block relocation problem

This article presents new methods for the block relocation problem (BRP). Although much of the existing work focuses on the restricted BRP, we tackle the unrestricted BRP, which yields more opportunities for optimisation. Our contributions include fast heuristics able to tackle very large instances within seconds, fast metaheuristics that provide very competitive performance on benchmark data sets, as well as a new lower bound that generalises existing ones. We embed it in a branch-and-bound algorithm, then assess the influence of various factors on the efficiency of branch-and-bound algorithms for the BRP.

ブロック積み替え問題に対するmemory-based dominanceの適用

ブロック積み替え問題に対するmemory-based dominanceの適用

Minimizing the Waiting Times of block retrieval operations in stacking facilities

This paper addresses the Blocks Relocation Problem with Waiting Times. Its objective is to retrieve a set of homogeneous blocks from a two-dimensional storage by minimizing the waiting times during their retrieval. An integer programming model and a heuristic algorithm are developed to solve this optimization problem. The mathematical model is able to solve small-size cases to optimality in reasonable computational times. Unfortunately, it requires large computational times when tackling medium and large-size scenarios. For its part, the heuristic algorithm overcomes the problems associated with the computational burden of the model by bringing forward the availability of blocks to retrieve from the storage. With this goal in mind, several look ahead strategies dedicated to perform the most promising predictive block relocation movements are proposed. The computational results disclose the proposed heuristic algorithm is able to report high-quality solutions through very short computational times, less than one second, in practical cases.

Container relocation problem with time windows for container departure

The blocks relocation problem is a classic combinatorial optimisation problem that occurs in daily operations for facilities that use block stacking systems. In the block stacking method, blocks can be stored on top of each other in order to utilise the limited surface of a storage area. When there is a predetermined pickup order among the blocks, this stacking method inevitably leads to the reshuffling moves for blocks stored above the target block and the minimisation of such unproductive reshuffling moves is of a primary concern to industry practitioners. A container terminal is a typical place where this problem arises, thus the problem being also referred to as the container relocation problem. In this study, we consider departure time windows for containers, which are usually revealed by the truck appointment system in port container terminals. We propose a stochastic dynamic programming model to calculate the minimum expected number of reshuffles for a stack of containers which all have departure time windows. The model is solved with a search-based algorithm in a tree search space, and an abstraction heuristic is proposed to improve the time performance. To overcome the computational limitation of exact methods, we develop a heuristic called the expected reshuffling index (ERI) and evaluate its performance.

A Faster Branch-and-Bound Algorithm for the Block Relocation Problem

The block relocation problem, which is also known as the container relocation problem, is an optimization problem to find an optimal sequence of operations for retrieving blocks (containers) from a container yard in a given order. The objective function to be minimized is the number of necessary relocations. In this study, we propose an efficient exact algorithm for two variants of the block relocation problem, those with distinct and duplicate retrieval priorities. In the former problem, the retrieval order of blocks is uniquely provided, whereas in the latter problem, it is given only among groups of blocks. The primary contribution of this study is a tighter lower bound of the number of relocations than existing ones. This enables us to construct a faster branch-and-bound algorithm. Its effectiveness is demonstrated by extensive numerical experiments.

On the abstraction method for the container relocation problem

The container relocation problem or the blocks relocation problem is a classic combinatorial optimisation problem that occurs in day-to-day operations for facilities that use block stacking systems. A typical place where this problem arises is a container terminal where containers can be stacked vertically in order to utilise the scarce resource of yard surface, thus at times resulting in the unproductive reshuffling moves for containers stacked above the target container for retrieval. Due to the problem class being NP-hard, a number of studies on this topic propose heuristic approaches to solve this problem. There are a few exact methods (search-based algorithms or mathematical programming) proposed for this problem but the feasible problem size of such methods is quite restricted, limiting their practical significance. In this paper, we propose a new insight into reducing the search space of this problem by the abstraction method. Our main contribution to the existing literature is two-fold: the reduction in the search space by the abstraction method and the bidirectional search using the pattern database. Our computational results confirm that our approach enables instances of a near-practical size to be solved optimally within a reasonable computation time.

Tree search procedures for the blocks relocation problem with batch moves

This research investigates a blocks relocation problem with batch moves (BRP-BM) mainly arising from the operations at slab yards in iron and steel plants. Several recent types of bridge cranes used in hot rolling slab yards can handle up to three slabs in one move, resulting in a typical scenario of the BRP-BM. A lower bound of the number of moves for the BRP-BM is presented. A greedy algorithm and three tree search algorithms, including one based on compound moves, are designed to solve the BRP-BM. The lower bound and the algorithms are validated and extensively evaluated based on both existing instances and randomly generated instances. The results indicate that the tree search algorithm based on compound moves outperforms the other algorithms especially for large-sized instances. Roughly speaking, the relative number of moves decreases from 1 to 0.80 and to 0.75 if the batch size increases from 1 to 2 and then to 3.

An exact approach for the Blocks Relocation Problem

The Blocks Relocation Problem seeks to find the shortest sequence of movements to retrieve a set of homogeneous blocks placed in a two-dimensional storage according to a predefined order. In this paper we analyze an optimization model recently published in the literature. We illustrate that this model reports infeasible solutions and does not guarantee the optimality of the achieved solutions in some cases. In order to overcome this fact, we propose an alternative optimization model. However, its high computational consumption of temporal and space resources in large environments encourages us to also develop a branch and bound algorithm to solve realistic scenarios to optimality. This algorithm includes an intelligent strategy to explore the most promising nodes in the underlying tree. Additionally, its performance can be easily adapted to report high-quality solutions at the expense of sacrificing the optimality guarantee. In contrast to previous exact proposals, the computational experiments reveal the high efficiency of the branch and bound algorithm, reporting optimal solutions for the most widely extended benchmark suite from the related literature through short computational times.

An improved mathematical formulation for the blocks relocation problem

This paper deals with the blocks relocation problem (also called container relocation problem). First, it corrects the binary linear programming model BRP-II presented in Caserta et al. (2012). Second, it improves the initial model formulation by removing superfluous variables, tightening some constraints, introducing a new upper bound and applying a pre-processing step to fix several variables. Computational results show the efficiency of the improved model for small and medium sized instances.

A chain heuristic for the Blocks Relocation Problem

In the Blocks Relocation Problem (BRP) one is given a block retrieval sequence and is concerned with determining a relocation pattern minimizing the total number of moves required to enforce the given retrieval sequence. The importance of the BRP has been constantly growing in recent years, as a consequence of its close connection with the operations inside of a container terminal. Due to the complexity of the BRP, a large number of methods has been developed for finding near optimal solutions. These methods can be divided in two main categories greedy heuristics and more complex methods. The latter achieve results of higher quality, but at the cost of very long execution times. In many cases, this increased calculation time is not an option, and the fast heuristic methods need to be used. Greedy heuristic approaches, in general, apply the heuristic based only on the properties of the block that is being relocated and the current state of the bay. In this paper we propose a new heuristic approach in which when deciding where to relocate a block we also take into account the properties of the block that will be moved next. This idea is illustrated by improving the Min–Max heuristic for the BRP. We compare the new heuristic to several existing methods of this type, and show the effectiveness of our improvements. The tests have been conducted on a wide range of sizes of container bays, using standard test data sets.

A domain-specific knowledge-based heuristic for the Blocks Relocation Problem

The Blocks Relocation Problem consists in minimizing the number of movements performed by a gantry crane in order to retrieve a subset of containers placed into a bay of a container yard according to a predefined order. A study on the mathematical formulations proposed in the related literature reveals that they are not suitable for its solution due to their high computational burden. Moreover, in this paper we show that, in some cases, they do not guarantee the optimality of the obtained solutions. In this regard, several optimization methods based on the well-known A∗ search framework are introduced to tackle the problem from an exact point of view. Using our A∗ algorithm we have corrected the optimal objective function value of 17 solutions out of 45 instances considered by Caserta et al. (2012) [4]. In addition, this work presents a domain-specific knowledge-based heuristic algorithm to find high-quality solutions by means of short computational times. It is based on finding the most promising positions into the bay where to relocate those containers that are currently located on the next one to be retrieved, in such a way that, they do not require any additional relocation operation in the future. The computational tests indicate the higher effectiveness and efficiency of the suggested heuristic when solving real-world scenarios in comparison with the most competitive approaches from the literature.