An efficient algorithm for 3D rectangular box packing

Akyokuş, Selim (Dogus Author) -- Conference full title: 9th International Conference, ETAI 2009, Ohrid, September 26-29, Republic of Macedonia, 2009.Getting highest occupancy rate of capacity of a container is very important for the companies, which deals in shipping or has shipping as a part of their main activities. They have to fit 3D boxes in container with optimum or nearest to optimum placement in order to ship more products with a minimum cost. The problem of fitting the boxes which is different from or the same to each other into a big container in optimum level, is called 3-dimensional packing problem. In this problem, the main objective is to minimize used container volume or wasted container space. This provides the reduction of costs in shipments with the use minimum number of containers

Container Loading Problems: A State-of-the-Art Review

Container loading is a pivotal function for operating supply chains efficiently. Underperformance results in unnecessary costs (e.g. cost of additional containers to be shipped) and in an unsatisfactory customer service (e.g. violation of deadlines agreed to or set by clients). Thus, it is not surprising that container loading problems have been dealt with frequently in the operations research literature. It has been claimed though that the proposed approaches are of limited practical value since they do not pay enough attention to constraints encountered in practice.In this paper, a review of the state-of-the-art in the field of container loading will be given. We will identify factors which - from a practical point of view - need to be considered when dealing with container loading problems and we will analyze whether and how these factors are represented in methods for the solution of such problems. Modeling approaches, as well as exact and heuristic algorithms will be reviewed. This will allow for assessing the practical relevance of the research which has been carried out in the field. We will also mention several issues which have not been dealt with satisfactorily so far and give an outlook on future research opportunities

Um modelo matemático para o problema de carregamento de múltiplos contêineres

Orientador : Prof. Dr. Cassius Tadeu ScarpinDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 27/02/2015Inclui referências : f.73-81Resumo: Este trabalho apresenta um modelo de Programação Linear Inteira que visa carregar, de modo ortogonal e sem sobreposição, um subconjunto de caixas retangulares no interior de contêineres, de modo a minimizar o espaço não utilizado dos contêineres selecionados. Com base em propostas realizadas anteriormente na literatura, a formulação matemática descrita neste trabalho considera as restrições de limitação de peso do contêiner, orientação das caixas e estabilidade vertical da carga, além de utilizar uma técnica heurística para realizar o pré-processamento dos dados. Tanto conjuntos de teste gerados aleatoriamente quanto da literatura foram utilizados para avaliar o desempenho computacional da formulação matemática proposta, e um software de otimização foi empregado para a resolução dos modelos gerados. A análise dos resultados obtidos permite concluir que a proposta gera resultados satisfatórios, com padrões de carregamento que atendem as restrições abordadas neste trabalho, dentro de um limite de tempo estabelecido para a execução dos testes. Palavras-chave: Matemática Discreta e Combinatória. Programação Linear Inteira. Modelagem Matemática. Problemas de Corte e Empacotamento. Carregamento de Contêineres.Abstract: This work presents an Integer Linear Programming model that aims loading, orthogonally and without overlap, a subset of rectangular boxes inside containers, in order to minimize the idle space of the selected containers. Based on proposals previously made in the literature, the mathematical formulation described in this work regards the restrictions of weight limit of the container, box orientation and vertical stability of the load, and also uses a heuristic technique to preprocess the data. Both randomly generated sets of trials and ones from literature were used to evaluate the computational performance of the proposed mathematical formulation, and an optimization software was employed for the resolution of the generated models. The analysis of the obtained results allow the conclusion that the proposition generates satisfactory results, with loading patterns that meet the restrictions addressed in this work within a time limit set for the tests. Keywords: Discrete and Combinatorial Mathematics. Integer Linear Programming. Mathematical Modeling. Cutting and Packing Problems. Container Loading

Novel approaches to container loading: from heuristics to hybrid tabu search

A thesis submitted for the degree of Doctor of Philosophy of the University ofBedford shireThis work investigates new approaches to the container loading problem which address the issue of how to load three-dimensional, rectangular items (e.g. boxes) into the container in such a way that maximum utilisation is made of the container space. This problem occurs in several industry sectors where the loading approach places cargo effectively into aeroplanes, ships, trailers or trucks in order to save considerable cost. In carrying out this work, the investigation starts by developing a new heuristic approach to the two-dimensional bin packing problem, which has lower complexity than container loading in the aspects of constraints and geometry. A novel approach, including the heuristic strategies and handling method for remaining areas, is developed that can produce good results when testing with benchmark and real world data. Based on the research for two-dimensional bin packing, a novel heuristic approach is developed to deal with the container loading problem with some practical constraints. The heuristic approach to container loading also includes heuristic strategies and the handling of remaining spaces. The heuristic strategies construct effective loading arrangements where combinations of identical or different box types are loaded in blocks. The handling method for remaining spaces further improves the loading arrangements through the representation, partitioning and merging of remaining spaces. The heuristic approach obtains better volume utilisation and the highest stability compared with other published heuristic approaches. However, it does not achieve as high a volume utilisation as metaheuristic approaches, e.g. genetic algorithms and tabu search.To improve volume utilisation, a new hybrid heuristic approach to the container loading problem is further developed based on the tabu search technique which covers the encoding, evaluation criterion and configuration of neighbourhood and candidate solutions. The heuristic strategies as well as the handling method for remaining spaces developed in the heuristic approach are used in this new hybrid tabu search approach. It is shown that the hybrid approach has better volume utilisation than the published approaches under the condition that all loaded boxes with one hundred per cent support from below. In addition, the experimental results show that both the heuristic and hybrid tabu search approaches can also be applied to the multiple container loading problem

Uma proposta de modelo matemático para o problema de carregamento de mútiplos contêineres heterogêneos com restrições adicionais

Orientador : Prof. Dr. Cassius Tadeu ScarpinDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 27/02/2015Inclui referências : fls. 95-103Área de concentração: Programação matemáticaResumo: Neste trabalho apresenta-se uma proposta de resolução para o problema de carregamento de múltiplos contêineres heterogêneos. Estes problemas consistem em empacotar caixas retangulares ortogonalmente e sem sobreposição dentro de contêineres, de modo a otimizar o valor total das caixas carregadas em um número limitado de contêineres ou maximizar a ocupação do espaço disponível. Tem-se como objetivo apresentar uma abordagem por meio de um modelo de programação linear inteira 0-1 capaz de considerar restrições práticas comumente encontradas em situações reais. Considerações de separação de itens, carregamento completo de grupos de caixas, estabilidade vertical e múltiplas orientações das caixas são descritas. Estas considerações, embora apareçam com grande frequência em situações reais, raramente são tratadas em trabalhos correlatos. Cenários foram modelados como problemas de programação linear por meio de um algoritmo em linguagem de programação e o solver CPLEX, com parâmetros default, foi utilizado para resolvê-los. Ao todo, trezentos e dezesseis problemas foram resolvidos com casos da literatura e dados gerados aleatoriamente. Os resultados obtidos mostram que, apesar do modelo proposto se limitar a resolver apenas problemas relativamente simples, o mesmo descreve de modo apropriado as considerações tratadas. Palavras-chave: Problema de carregamento de contêineres. Restrições práticas. Modelagem matemática.Abstract: In this work we present a proposal to solve the multiple heterogeneous container loading problem. These problems consist of packing orthogonally and without overlap rectangular boxes inside containers of available space, in order to optimize the total value of loaded boxes or to maximize the occupation. The objective of this work is to present an approach through a 0-1 integer linear programming model able to consider practical constraints usually found in real situations. Considerations of separations of items, complete shipment, vertical stability and multiple orientations of the boxes are described. Although these considerations appear frequently in real situations, some often are rarely treated in related work. Scenarios were modeled as linear programming problems by using an algorithm in a programming language and the CPLEX solver with default parameters was used to solve them. Three hundred sixteen problems were solved with instances of literature and data generated randomly. The results show that the model is able to solve only relatively simple problems, however, it describes appropriately the treated considerations. Key-words: Container loading problem. Practical constraints. Mathematical Modelling

A general purpose algorithm for three-dimensional packing

We present a fast and efficient heuristic algorithm for solving a large class of three-dimensional packing problems with the objective of maximizing the average volumetric utilization of containers that might be of different dimensions. The algorithm is a tree-search algorithm that implicitly explores the solution space. The algorithm relies on the fact that, in practice, (i) the number of different types of objects to pack is limited and known in advance, and (ii) the number of occurrences of those objects is sufficiently high to permit use of a pattern approach to solve, at least partially, the problem. Each node of the search tree is solved using an extension of a pallet-loading heuristic developed by Morabito and Morales to generate the patterns and of a container-loading heuristic developed by Pisinger to treat the objects not packed by the pattern approach. We report on our extensive computations on a new large library of instances derived from real world applications. © 2005 INFORMS.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

A general purpose algorithm for three-dimensional packing

The three--dimensional packing problem consists in finding the minimum number of containers needed to pack a set of different objects. We present a fast and efficient heuristic algorithm for solving a large class of three--dimensional packing problems with the objective of maximizing the average volumetric utilization of containers that might be of different dimensions. The algorithm is a tree search algorithm that implicitly explores the solution space. The algorithm relies on the fact that, in practice, (i) the number of different types of objects to pack is limited and known in advance and (ii) the number of occurrences of those objects are sufficiently high to permit to use a pattern approach to solve, at least partially, the problem. Each node of the search tree is solved using an extension of a pallet loading heuristic developed by Morabito and Morales to generate the patterns and of a container loading heuristic developed by Pisinger to treat the objects not packed by the pattern approach. We will report on our extensive computations on a new large library of instances deriving from real world applications