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

Strip packing problem with constraints in order and stability

Orientador: Flávio Keidi MiyazawaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Neste trabalho lidamos com o problema de Empacotamento em Faixa Bidimensional considerando o caso em que os itens devem ser dispostos de forma a manter o empacotamento estável e satisfazer uma ordem de descarregamento imposta. Consideramos o caso em que a orientação dos itens é fixa. Definimos uma metodologia para analisar a estabilidade do empacotamento observando as condições de equilíbrio estático para corpos rígidos. Desenvolvemos heurísticas e formulamos um programa linear inteiro para o problema de Empacotamento em Faixa sujeito a tais restrições. A resolução da formulação inteira ocorre através de uma estratégia do tipo branch-and-cut. As restrições de estabilidade foram inseridas como planos de corte de maneira a remover empacotamentos que não são estáveis. Em nossos experimentos computacionais, vemos que o modelo proposto é adequado para lidar com instâncias de pequeno até médio porte, dentro de um tempo computacional razoávelAbstract: This paper investigates the Two-Dimensional Strip Packing Problem considering the case in which the items should be arranged to form a stable packing and satisfy an order of unloading, so that after unloading, the packing is still stable. We consider the case where the items are oriented and rotations are not allowed. We present a methodology to analyze the stability of the packing observing the conditions for static equilibrium of rigid bodies. We present heuristics and formulate an integer linear programming model for the Strip Packing problem considering such constraints. To solve the integer formulation, we develop a branch-and-cut approach. For each integer solution obtained during the branch-and-cut algorithm, corresponding to a non-stable packing, we insert a cutting plane for which this integer solution is not satisfied. In our computational experiments, we see that the proposed model is suitable to deal with small and mid-sized instances. Some optimal solutions were obtained after few hours of CPU processingMestradoMestre em Ciência da Computaçã

Approximationsalgorithmen für Packungs- und Scheduling-Probleme

Algorithms for solving optimization problems play a major role in the industry. For example in the logistics industry, route plans have to be optimized according to various criteria. However, many natural optimization problems are hard to solve. That is, for many optimization problems no algorithms with running time polynomial in the size of the instance are known. Furthermore, it is a widely accepted assumption that many optimization problems do not allow algorithms that solve the problem optimally in polynomial time. One way of overcoming this dilemma is using approximation algorithms. These algorithms have a polynomial running time, but their solutions are in general not optimal but rather close to an optimum. The main subject of this thesis is approximation algorithms for packing and scheduling problems: For the three-dimensional orthogonal knapsack problem (OKP-3) without rotations we present algorithms with approximation ratios arbitrarily close to 9, 8 and 7. For OKP-3 with 90 degree rotations around the z-axis or around all axes, we present algorithms with approximation ratios arbitrarily close to 6 and 5, respectively. Both for the malleable and for the non-malleable case of the non-preemptive parallel job scheduling problem in which the number of available machines is polynomially bounded in the number of jobs, we present polynomial time approximation schemes. For the cases in which additionally the machines allotted to each job have to be contiguous, we show the existence of approximation algorithms with ratio arbitrarily close to 1.5

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Algorithms for cutting and packing problems

Orientador: Flávio Keidi MiyazawaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Problemas de Corte e Empacotamento são, em sua maioria, NP-difíceis e não existem algoritmos exatos de tempo polinomial para tais se for considerado P ¿ NP. Aplicações práticas envolvendo estes problemas incluem a alocação de recursos para computadores; o corte de chapas de ferro, de madeira, de vidro, de alumínio, peças em couro, etc.; a estocagem de objetos; e, o carregamento de objetos dentro de contêineres ou caminhões-baú. Nesta tese investigamos problemas de Corte e Empacotamento NP-difíceis, nas suas versões bi- e tridimensionais, considerando diversas restrições práticas impostas a tais, a saber: que permitem a rotação ortogonal dos itens; cujos cortes sejam feitos por uma guilhotina; cujos cortes sejam feitos por uma guilhotina respeitando um número máximo de estágios de corte; cujos cortes sejam não-guilhotinados; cujos itens tenham demanda (não) unitária; cujos recipientes tenham tamanhos diferentes; cujos itens sejam representados por polígonos convexos e não-convexos (formas irregulares); cujo empacotamento respeite critérios de estabilidade para corpos rígidos; cujo empacotamento satisfaça uma dada ordem de descarregamento; e, cujos empacotamentos intermediários e final tenham seu centro de gravidade dentro de uma região considerada "segura". Para estes problemas foram propostos algoritmos baseados em programação dinâmica; modelos de programação inteira; técnicas do tipo branch-and-cut; heurísticas, incluindo as baseadas na técnica de geração de colunas; e, meta-heurísticas como o GRASP. Resultados teóricos também foram obtidos. Provamos uma questão em aberto levantada na literatura sobre cortes não-guilhotinados restritos a um conjunto de pontos. Uma extensiva série de testes computacionais considerando instâncias reais e várias outras geradas de forma aleatória foram realizados com os algoritmos desenvolvidos. Os resultados computacionais, sendo alguns deles comparados com a literatura, comprovam a validade dos algoritmos propostos e a sua aplicabilidade prática para resolver os problemas investigadosAbstract: Several versions of Cutting and Packing problems are considered NP-hard and, if we consider that P ¿ NP, we do not have any exact polynomial algorithm for solve them. Practical applications arises for such problems and include: resources allocation for computers; cut of steel, wood, glass, aluminum, etc.; packing of objects; and, loading objects into containers and trucks. In this thesis we investigate Cutting and Packing problems that are NP-hard considering theirs two- and three-dimensional versions, and subject to several practical constraints, that are: that allows the items to be orthogonally rotated; whose cuts are guillotine type; whose cuts are guillotine type and performed in at most k stages; whose cuts are non-guillotine type; whose items have varying and unit demand; whose bins are of variable sizes; whose items are represented by convex and non-convex polygons (irregular shapes); whose packing must satisfy the conditions for static equilibrium of rigid bodies; whose packing must satisfy an order to unloading; and, whose intermediaries and resultant packing have theirs center of gravity inside a safety region; Such cutting and packing problems were solved by dynamic programming algorithms; integer linear programming models; branch-and-cut algorithms; several heuristics, including those ones based on column generation approaches, and metaheuristics like GRASP. Theoretical results were also provided, so a recent open question arised by literature about non-guillotine patterns restricted to a set of points was demonstrated. We performed an extensive series of computational experiments for algorithms developed considering several instances presented in literature and others generated at random. These results, some of them compared with the literature, validate the approaches proposed and suggest their applicability to deal with practical situations involving the problems here investigatedDoutoradoDoutor em Ciência da Computaçã

Algorithmes d'approximation pour des programmes linéaires et les problèmes de Packing avec des contraintes géometriques

In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is a model from convex optimization. This problem includes a large class of linear programs. The algorithms generate approximately feasible solutions within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and O(M epsilon{-2} ln (M epsilon^{-1}))$iterations, respectively, where in each iteration a block problem which depends on the specific application has to be solved. Both algorithms, applied to linear programs, can result in column generation algorithms. In Chapter 3, we implement an algorithm for the so-called max-min-resource sharing problem. This is a certain convex optimization problem which, similar to the problem in Chapter 1, includes a large class of linear programs. The implementation, which is included in the appendix, is done in C++. We use the implementation in the context of an AFPTAS for Strip Packing in order to evaluate dynamic optimization of a parameter in the algorithm, namely the step length used for interpolation. We compare our choice to the static step length proposed in the analysis of the algorithm and conclude that dynamic optimization of the step length significantly reduces the number of iterations. In Chapter 4, we study two closely related scheduling problems, namely non-preemptive scheduling with fixed jobs and scheduling with non-availability for sequential jobs on m identical machines under the makespan objective, where m is constant. For the first problem, which does not admit an FPTAS unless P=NP, we obtain a new PTAS. For the second problem, we show that a suitable restriction (namely the permanent availability of one machine) is necessary to obtain a bounded approximation ratio. For this restriction, which does not admit an FPTAS unless P=NP, we present a PTAS; we also discuss the complexity of various special cases. In total, the results are basically best possible. In Chapter 5, we continue the studies from Chapter 4 where now the number m of machines is part of the input, which makes the problem algorithmically harder. Scheduling with fixed jobs does not admit an approximation ratio better than 3/2, unless P=NP; here we obtain an approximation ratio of 3/2+epsilon for any epsilon>0. For scheduling with non-availability, we require a constant percentage of the machines to be permanently available. This restriction also does not admit an approximation ratio better than 3/2 unless P=NP; we also obtain an approximation ratio of$3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible. Finally, in Chapter 6, we conclude with some remarks and open research problems