Three-Dimensional Knapsack Problem with Pre-Placed Boxes and Vertical Stability

A three-dimensional knapsack problem packs a subset of rectangular boxes inside a bin with fixed size such that the total value of packed boxes is maximized. Each box has its own value and size and can be freely rotated into any of the six positions while its edges are parallel to the bin\u27s edges. A Mixed Integer Linear Programming is developed for the 3D knapsack problem, while some practical constraints such as vertical stability are considered. However, the given model can be applied to two dimensional problems as well. The proposed solution methodology is based on the sequence triple. Simulated annealing technique is used to model the heuristic approach. Moreover, the situation where some boxes are pre-placed in the bin is investigated. These pre-placed boxes represent potential obstacles. Numerical experiments are conducted for bins with and without obstacles. The results show that the heuristic approach is successful and can handle different kinds of instances

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

A literature review on the Pallet Loading Problem Una revisión literaria del Problema de Carga del Pallet

Actualmente, las empresas enfrentan una competencia agresiva, por lo que implementar estrategias para alcanzar la competitividad es elemental. En este sentido, en Logística, el uso adecuado de los recursos es imprescindible. El impacto en la ganancia que tienen el almacenaje y el transporte, conlleva la implementación de acciones para contrarrestarlo. Un paletizado efectivo puede contribuir a reducir costos. El Problema de Carga del Pallet (PLP) procura la optimización del espacio del pallet para lograr cargar máxima de producto debidamente empacado. El uso práctico y beneficios del PLP han dado pie a su estudio en la búsqueda su solución. Este artículo presenta una revisión literaria de 30 estudios para mostrar las características principales y los métodos de solución propuestos para proveer la base teórica y las maneras como se ha tratado el PLP. Con el entendimiento de estas propuestas de solución, se busca tener el sustento para elaborar un modelo nuevo.Nowadays, businesses face a fierce competition. Hence, the search for strategies to achieve competitiveness is elemental. For that purpose, in Logistics, the proper use of resources is a must. Storing and transportation cause impact the overall profit, making it necessary to take actions to lower their effect. An efficient palletizing can contribute to reduce costs. The Pallet Loading Problem (PLP) focuses on finding space optimization to load the maximum quantity of packed product onto the pallet. The PLP’s practical use and benefits have made it subject of study throughout time. This article presents a literature review of 30 approaches to show the main characteristics and the solution methods researchers have proposed. The objective of this revision consists of providing the theoretical basis and the way the PLP has been treated. Thus, the understanding of these solution approaches can help in the development of a new proposed model

決定木・回帰木のための多変量判別ルール発見アルゴリズム

広島大学(Hiroshima University)博士(工学)Engineeringdoctora

Contributions to Methodology and Techniques of Decision Analysis (First Stage)

This collaborative volume reports on the results of a contracted study agreement between the System and Decision Analysis Program and its project on the Methodology of Decision Analysis at IIASA and a group of Polish institutes working in this area. The study includes research in four directions: mathematical programming techniques for decision support; applications of decision support systems new methodological developments in decision support; dissemination of results; and educational activities

Combinatorial approaches for the trunk packing problem

In this thesis we consider a three-dimensional packing problem arising in industry. The task is to pack a maximum number of rigid boxes with side length ratios of 4 : 2 : 1 into an irregularly shaped container. Motivated by the structure of manually constructed packings so far, we pursue a discrete approach. We discretize the shape of the container as well as the set of possible box placements. This discrete packing problem can be reduced to a maximum stable set problem. First we formulate the problem as an integer linear program, which admittedly can only be solved to optimality within reasonable runtime for very small instances. Therefore, we present several heuristics based, for example, on the linear programming relaxation or on local search. Other heuristics generate tight packings for the core of the container, thereby reducing the problem to a set of smaller subproblems. We compare all presented algorithms on real data sets. We achieve very good results for the majority of instances and for some instances we even surpass the manually constructed solutions.In dieser Arbeit behandeln wir ein dreidimensionales Packungsproblem aus der Industrie. Die Aufgabe besteht darin, möglichst viele starre Quader mit einem Seitenverhältnis von 4 : 2 : 1 in einen unregelmäßig geformten Container zu packen. Motiviert durch die Struktur der bisher manuell erstellten Packungen verfolgen wir einen diskreten Lösungsansatz. Dazu diskretisieren wir sowohl die Form des Containers als auch die Platzierungsmöglichkeiten der Quader. Dieses diskrete Packungsproblem lässt sich auf die Berechnung einer größtmöglichen unabhängigen Knotenmenge reduzieren. Wir formulieren das Problem zunächst als ganzzahliges lineares Programm, das allerdings nur für sehr kleine Instanzen mit angemessenem Rechenaufwand beweisbar optimal gelöst werden kann. Daher stellen wir verschiedene Heuristiken vor, die zum Beispiel auf einer Relaxierung des ganzzahligen linearen Programms oder lokaler Suche basieren. Andere Heuristiken generieren zunächst dichte Packungen für den Kern des Containers und reduzieren so das Problem auf eine Reihe kleinerer Teilprobleme. Wir vergleichen alle vorgestellten Algorithmen an Hand realer Datensätze. In der Mehrzahl der Fälle erreichen wir sehr gute Resultate, bei einigen Instanzen übertreffen wir sogar die manuell erstellten Packungen

Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

A New Quasi-Human Algorithm for Solving the Packing Problem of Unit Equilateral Triangles

The packing problem of unit equilateral triangles not only has the theoretical significance but also offers broad prospects in material processing and network resource optimization. Because this problem is nondeterministic polynomial (NP) hard and has the feature of continuity, it is necessary to limit the placements of unit equilateral triangles before optimizing and obtaining approximate solution (e.g., the unit equilateral triangles are not allowed to be rotated). This paper adopts a new quasi-human strategy to study the packing problem of unit equilateral triangles. Some new concepts are put forward such as side-clinging action, and an approximation algorithm for solving the addressed problem is designed. Time complexity analysis and the calculation results indicate that the proposed method is a polynomial time algorithm, which provides the possibility to solve the packing problem of arbitrary triangles