Le volet applicatif de cette thèse porte sur l'agencement d'un logement destiné à une personne en situation de handicap. L'agencement désigne le choix de la position, de la forme et des dimensions des pièces, des portes et des couloirs. L'agencement est généralement élaboré par un architecte, dans le respect d'un nombre si élevé de contraintes qu'il lui est difficile de parvenir qu'il parvienne à toutes les satisfaire : il y a d'abord des contraintes architecturales évidentes : non recouvrement des pièces, largeur suffisante des couloirs, accessibilité à tout point du lieu à partir de tout autre point, nécessité de placer certaines pièces sur des arrivées ou évacuations … Il y a ensuite les contraintes imposées par le handicap : largeur accrue des couloirs (déplacement en fauteuil), nécessité d'assurer un effort quotidien minimum (lutte contre le vieillissement), limitation des escaliers (asthme sévère), éloignement d'une pièce des murs mitoyens (surdité) .... Et il y a finalement les souhaits exprimés par le futur occupant, par exemple minimiser certains trajets, maximiser l’éloignement entre deux pièces ou imposer l’orientation d’une pièce. D’un point de vue formel, notre travail a consisté à développer d'une part des modèles mathématiques et des méthodes algorithmiques capables de gérer ces contraintes et d'autre part des prototypes logiciels opérationnels. Les méthodes élaborées relèvent de deux approches : l'optimisation d'un agencement conçu par l'architecte et la synthèse d'un agencement sans suggestion initiale de l'architecte. La synthèse d'un plan a été abordée comme un problème de type « bin-packing » (réputé NP-difficile) avec des contraintes additionnelles : les objets à placer - les pièces - ont des tailles variables et ils sont soumis à des contraintes fonctionnelles. La méthode de résolution s'appuie sur un premier modèle mathématique, qui prend la forme d’un programme quadratique (linéarisé par la suite) en variables mixtes. Elle a été appliquée avec succès pour placer les pièces d'un logement, pour les dimensionner, pour déterminer les couloirs assurant une complète accessibilité au logement et pour prendre en compte certaines contraintes imposées par le handicap du futur occupant. Un deuxième modèle mathématique a été élaboré pour le placement des portes et une heuristique a enfin été développée pour affecter l'espace occupé par les couloirs non indispensables aux pièces avoisinantes. La totalité de cette démarche a été programmée dans un prototype logiciel pleinement opérationnel. Le deuxième ensemble de contributions concerne l'optimisation d'un agencement existant. Cette optimisation a été conçue comme un processus itératif enchaînant évaluation et modification (amélioration) d'un agencement. Il est décliné de quatre manières : une métaheuristique de type « recuit simulé » et trois méthodes de type « recherche locale », qui explorent l’espace des solutions en utilisant des voisinages spécialement définis. Cette approche a d'une part permis d’appréhender le caractère multicritère de cette problématique et a d'autre part exigé la mise en œuvre de nombreux algorithmes géométriques. Ces travaux sont implantés dans un deuxième prototype logiciel. Ce projet a nécessité la participation à de nombreuses manifestations au-delà du domaine de l’informatique, nationales et régionales, scientifiques et non-scientifiques, organisées par différents organismes politiques et associatifs travaillant sur la problématique du handicap et de l’accessibilité, afin de bien appréhender les attentes du monde scientifique et socioprofessionnel. Cette phase prospective a été concrétisée par la rédaction de nombreux rapports qui ont alimentés la bibliographie du mémoire de thèse / At an application level, this thesis deals with the layout of an accommodation intended for a disabled person. Determining the layout means choosing the position, shape and dimensions of rooms, doors and corridors. It is usually an architect's job but the complexity is such that it is very unlikely that he succeeds in optimally fulfilling all the constraints: first, there are architectural constraints: no room overlapping, sufficient width for the corridors, accessibility to and from any point, mandatory positioning of some rooms on some areas (e.g. water supply and outlet) … Then, there are constraints imposed by disabilities: enlarged corridors (wheelchairs), mandatory daily amount of efforts (fight against aging), reducing the number of steps (severe asthma), moving a room away from shared walls (deafness)... Finally, there are the wishes expressed by the future occupant, such as minimizing some journeys, maximizing the distance between two rooms or fixing a room's orientation. From a formal point of view, our work has consisted, firstly, in developing mathematical models and algorithmic methods to deal with all these constraints and, secondly, in realizing software prototypes applying these concepts. The tools we propose aim either at optimizing a layout previously designed by an architect or at synthesising a layout without any initial suggestions from the architect. Synthesis has been tackled as bin-packing-type problem (known to be NP-hard) but with additional constraints: the objects to be placed (the rooms) have variable sizes and they are submitted to functional constraints. The resolution is based on a first, initially quadratic and then linearized, mixed integer mathematical model. It has been successfully applied to position and dimension the rooms of an accommodation, to determine corridors allowing a full accessibility to all the rooms and to take into account a number of constraints coming from the disabilities of the future occupant. A second mathematical model has been formulated for the positioning of the doors and, finally, a heuristic method has been designed to assign the space used by useless corridors to adjacent rooms. The whole process has been embedded in a fully operational software. The second set of contributions is about the optimization of an existing layout. This task has been tackled through an iterative process, looping on evaluation and modification (improvement) of an accommodation. It has been implemented in four different ways: a metaheuristic (simulated annealing) and three local-search-type methods, which traverse the solution space by using specific definitions of the neighbourhood. This approach has firstly underlined the multicriteria feature of our problem and, secondly, has required the development of many computational geometry algorithms. All this work is integrated in another functional prototype software. To understand the expectations of the scientific, social and professional worlds, this project has implied to take part to various manifestations which were national or regional, in the computer science domain or in others, scientific or non-scientific, organised by various political or non-political organisations working in the field of disabilities and accessibility. This phase has resulted in many reports which have directly fed into the bibliography of this thesis.

Dans ce mémoire, nous nous intéressons à l’optimisation des mouvements improductifs de chargement/déchargement, appelés shiftings, dans les problèmes de transport. Dans le premier contexte,nous introduisons le problème de shifting dans le cas du voyageur de commerce. Notre objectif est de chercher un circuit hamiltonien qui optimise à la fois le coût distance et le coût shifting. Nous proposons une modélisation mathématique du problème, puis, nous adaptons la métaheuristique d’optimisation par colonies de fourmis sous sa forme séquentielle et parallèle pour le résoudre. Dans le deuxième contexte, nous abordons le problème d’optimisation des plans de chargement et d’arrimage des conteneurs dans des barges. Ce problème consiste à chercher l’emplacement le plus convenable de chaque conteneur dans les barges de façon à faciliter son déchargement dans la chronologie des ports à visiter. D'abord, nous introduisons une modélisation mathématique du problème dans le cas d’une seule barge ou différents ports du trajet ont des coûts shiftings non-uniformes. L’objectif est d’optimiser le coût total de shiftings, la stabilitélongitudinale de la barge et celle transversale. Ensuite, nous généralisons le problème au cas d’un système de convoi de barges. Nous proposons, d’abord, un modèle mathématique en nombres entiers, dans lequel, nous considérons l’aspect multi-objectif en optimisant le nombre de shiftings, la stabilité du convoi et le nombre de barges utilisées dans le convoi. Puis, nous adaptons la méthode nsga-II en se basant sur les heuristiques du problème de bin-packing.L'ensemble des résultats obtenus est évalué en utilisant des mesures de performances adaptées au problème. / This work outlines the optimization of unproductive loading/unloading movements, called shiftings, in transport problems. in the first context, we introduce the shifting in the case of the traveling salesman problem. our goal is to find a hamiltonian circuit that optimizes both distance and shifting costs. we propose a mathematical modeling of the problem, and then we adapt the ant colony optimization metaheuristic in its sequential and parallel form to solve it. in the second context, we address the 3d container stowage planning problem of barges. this problem consists in finding the most suitable location of each container in the barge in order to facilitate its retrieval in the chronology of ports to be visited. firstly, we introduce a mathematical modeling of the problem in the case of a single barge where different ports are of non-uniform operational costs. the main objective is to optimize the total shiftings fees, the longitudinal stability of the barge and the transverse one. then, we generalize our problem to the case of barge convoy systems. we first propose a suitable mathematical modeling, in which, we consider the multi-objective aspect by optimizing the total number of shiftings, the convoy stability and the number of the real-used barges in the convoy. in order to solve this new variant, we propose a novel adaptation of the multi-objective evolutionary algorithm nsga-ii (non-dominated sorting genetic algorithm-ii) based on a set of heuristics introduced by the bin-packing problem resolution methods. the numerical results are evaluated using performance measures adapted to theproblem.

Cette thèse a été réalisée dans le cadre du projet « Gestion et Optimisation de la Chaîne Documentaire », projet labellisé par le Pôle de compétitivité des Industries du Commerce. Le projet a pour but de concevoir et de développer un nouveau workflow et un outil d’aide à la décision. Ce système doit être capable de gérer et d’optimiser le flux complet dématérialisé de contrats reçus à COFIDIS.Nous présentons d’abord le framework retenu dans le cadre du projet pour modéliser et implémenter le workflow. En phase de conception BPMN a été choisi. Pour la partie développement, l’utilisation de BPEL a été préconisée pour implémenter et exécuter l'application finale (services web).Cependant la flexibilité offerte par BPMN peut conduire à des propriétés indésirables du processus telles que blocage et inaccessibilité. De plus, BPMN a été conçu pour fournir des modèles Orientés Process. Les données ou les ressources y sont donc peu représentées. En conséquence, l'analyse de performance sur un modèle BPMN est quasi inexistante.Afin de surmonter ces problèmes nous proposons d’insérer dans le framework deux nouvelles phases. Ces deux phases sont appliquées au modèle BPMN. La première est une phase de vérification et de validation et la deuxième une phase d'optimisation. Ces deux phases sont réalisées en transformant le modèle BPMN vers un langage formel. Notre choix dans ce travail a été d'utiliser les réseaux de Petri. Ce qui nous a permis de vérifier et de valider de bonnes propriétés du process. Quant à l’optimisation, nous avons défini une nouvelle variante du problème d’affectation (bin packing problem) et proposé une résolution à intégrer dans le processus d’aide à la décision / This thesis has been realized as a part of the project GOCD (French acronym for Management and optimization of document life cycle) and within the context of the French competitive cluster PICOM. The project aims to design and develop a new paperless workflow system and decision making tool to replace the current paper based system. The new workflow system must manage and optimize received credit demands at COFIDIS.The first part of this thesis presents and discusses a framework to model and implement workflow systems. The proposed framework allows more flexibility in workflow reengineering process and operational analysis for different business process. The proposed framework uses the most recent and promising language to model and execute workflow the Business Process Modeling Notation (BPMN) and Business Process Execution Language (BPEL).The flexibility offered by BPMN can also lead to undesirable properties for business process such as deadlocks and unreachablity. More, BPMN notation was designed to model business process, and little consideration was concentrated to represent data and resources. As a result, carrying out performance analysis on a BPMN model is also limited.To overcome these problems, we propose two additional phases in the reengineering process. They are applied to the target BPMN model. The first phase is verification and validation and the second one is optimization. These two phases are realized by transforming the BPMN model to a formal language, Petri nets. As for optimization, a new variant of bin packing problem has been defined. And we propose to integrate its resolution in a decision making tool

La fabrication additive est un domaine en plein essor. Cependant, les industriels sont aujourd’hui dans une phase d’interrogation sur l’utilisation de ce procédé dans le cadre d’une production de masse. La problématique posée dans le cadre de ces travaux de recherche est : Comment rendre viable, industriellement, le procédé de fusion sur lit de poudre ? Nos travaux abordent la conception et le pilotage d’ateliers intégrant la fabrication additive et le processus complet d’obtention de la pièce selon les trois niveaux de décision : stratégique, tactique et opérationnel. D’un point du vue stratégique, des décisions fortes d’investissement, de sélection de machines et de choix d’organisation sont à prendre avec des enjeux économiques importants. L’objectif est de définir une méthode d’optimisation multicritère pour la conception modulaire d’un système de production intégrant la fabrication additive en présence de données incertaines, optimale sur le long terme et sur le court terme. D’un point de vue tactique, toutes les pièces ne sont pas forcément des candidates pertinentes pour la fabrication additive. Dans ces travaux, nous avons développé un outil d’aide à la décision qui évalue la pertinence ou non de la fabrication additive pour l’obtention des pièces dans une approche globale des coûts. Au niveau opérationnel, nous proposons un outil basé sur la simulation de flux qui permet de passer des commandes aux ordres de fabrication et leur ordonnancement de manière à garantir l’efficience de l’atelier. Ces travaux de recherche sont développés en lien avec des acteurs du monde industriel : AddUp, MBDA et Dassault qui alimentent nos travaux et nous permettent de confronter nos outils à une réalité industrielle. / The additive manufacturing is a field on the rise. However, companies wonder about the use of additive manufacturing for mass production. The problem raised in the context of this thesis is: How to make the process of sintering laser melting industrially viable? Our work focuses on the design and on the management of workshops integrating the additive manufacturing and of the complete process to obtain part according to three levels of decision: strategic, tactic and operational. About the strategic level, strong decisions of investment, machines selection and organization choice are taken with important economic issues. The aim is to define a multicriteria optimization method for the modular design of a production system integrating the additive manufacturing in the presence of uncertain data, optimal in the long term and the short term. From a tactical point of view, not all parts are necessarily relevant candidates for additive manufacturing. In this work, we developed a decision support tool that evaluates the relevance or not of additive manufacturing to obtain parts in a global cost approach. At the operational level, we offer a tool based on flow simulation that allows orders to be placed to production orders and their scheduling in order to guarantee the efficiency of the workshop. This research work is developed in collaboration with companies: AddUp, MBDA and Dassault, who contribute to our work and enable us to compare our tools with an industrial reality.

The bin packing problem has been the corner stone of approximation algorithms and has been extensively studied starting from the early seventies. In the classical bin packing problem, we are given a list of real numbers in the range (0, 1], the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. In this thesis we study approximation algorithms for three generalizations of bin packing: geometric bin packing, vector bin packing and weighted bipartite edge coloring. In two-dimensional (2-D) geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. Geometric packing has vast applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. We consider the widely studied orthogonal packing case, where the items must be placed in the bin such that their sides are parallel to the sides of the bin. Here two variants are usually studied, (i) where the items cannot be rotated, and (ii) they can be rotated by 90 degrees. We give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for the versions with and without rotations. We have also shown the limitations of rounding based algorithms, ubiquitous in bin packing algorithms. We have shown that any algorithm that rounds at least one side of each large item to some number in a constant size collection values chosen independent of the problem instance, cannot achieve an asymptotic approximation ratio better than 3/2. In d-dimensional vector bin packing (VBP), each item is a d-dimensional vector that needs to be packed into unit vector bins. The problem is of great significance in resource constrained scheduling and also appears in recent virtual machine placement in cloud computing. Even in two dimensions, it has novel applications in layout design, logistics, loading and scheduling problems. We obtain a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for 2-D VBP. We also obtain a polynomial time algorithm with almost tight (absolute) approximation ratio of $1+\ln(1.5)$ for 2-D VBP. For $d$ dimensions, we give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(d/2) + 1.5 \approx \ln d+0.81$. We also consider vector bin packing under resource augmentation. We give a polynomial time algorithm that packs vectors into $(1+\epsilon)Opt$ bins when we allow augmentation in (d - 1) dimensions and $Opt$ is the minimum number of bins needed to pack the vectors into (1,1) bins. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite graph $G=(V,E)$ with weights $w: E \rightarrow [0,1]$. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. This problem is motivated by rearrangeability of 3-stage Clos networks which is very useful in various applications in interconnected networks and routing. We show a polynomial time approximation algorithm that returns a proper weighted coloring with at most $\lceil 2.2223m \rceil$ colors where $m$ is the minimum number of unit sized bins needed to pack the weight of all edges incident at any vertex. We also show that if all edge weights are $>1/4$ then $\lceil 2.2m \rceil$ colors are sufficient.

This thesis focuses on algorithms solving the on-line Bin-Covering problem, when the items are generated from a known, stationary distribution. We introduce the Prospect Algorithm. The main idea behind the Prospect Algorithm is to use information on the item distribution to estimate how easy it will be to fill a bin with small overfill as a function of the empty space left in it. This estimate is then used to determine where to place the items, so that all active bins either stay easily fillable, or are finished with small overfill. We test the performance of the algorithm by simulation, and discuss how it can be modified to cope with additional constraints and extended to solve the Bin-Packing problem as well. The Prospect Algorithm is then adapted to achieve perfect packing, yielding a new version, the Prospect+ Algorithm, that is a slight but consistent improvement. Next, a Markov Decision Process formulation is used to obtain an optimal Bin-Covering algorithm to compare with the Prospect Algorithm. Even though the optimal algorithm can only be applied to limited (small) cases, it gives useful insights that lead to another modification of the Prospect Algorithm. We also discuss two relaxations of the on-line constraint, and describe how algorithms that are based on solving the Subset-Sum problem are used to tackle these relaxed problems. Finally, several practical issues encountered when using the Prospect Algorithm in the real-world are analyzed, a computationally efficient way of doing the background calculations needed for the Prospect Algorithm is described, and the three versions of the Prospect Algorithm developed in this thesis are compared.

Thesis (PhD (Logistics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Packing problems are common in industry and there is a large body of literature on the subject. Two packing problems are considered in this dissertation: the strip packing problem and the bin packing problem. The aim in both problems is to pack a speci ed set of small items, the dimensions of which are all known prior to packing (hence giving rise to an o ine problem), into larger objects, called bins. The strip packing problem requires packing these items into a single bin, one dimension of which is unbounded (the bin is therefore referred to as a strip). In two dimensions the width of the strip is typically speci ed and the aim is to pack all the items into the strip, without overlapping, so that the resulting packing height is a minimum. The bin packing problem, on the other hand, is the problem of packing the items into a speci ed set of bins (all of whose dimensions are bounded) so that the wasted space remaining in the bins (which contain items) is a minimum. The bins may all have the same dimensions (in which case the problem is known as the single bin size bin packing problem), or may have di erent dimensions, in which case the problem is called the multiple bin size bin packing problem (MBSBPP). In two dimensions the wasted space is the sum total of areas of the bins (containing items) not covered by items. Many solution methodologies have been developed for above-mentioned problems, but the scope of the solution methodologies considered in this dissertation is restricted to heuristics. Packing heuristics follow a xed set of rules to pack items in such a manner as to nd good, feasible (but not necessarily optimal) solutions to the strip and bin packing problems within as short a time span as possible. Three types of heuristics are considered in this dissertation: (i) those that pack items into levels (the heights of which are determined by the heights of the tallest items in these levels) in such a manner that all items are packed along the bottom of the level, (ii) those that pack items into levels in such a manner that items may be packed anywhere between the horizontal boundaries that de ne the levels, and (iii) those heuristics that do not restrict the packing of items to levels. These three classes of heuristics are known as level algorithms, pseudolevel algorithms and plane algorithms, respectively. A computational approach is adopted in this dissertation in order to evaluate the performances of 218 new heuristics for the strip packing problem in relation to 34 known heuristics from the literature with respect to a set of 1 170 benchmark problem instances. It is found that the new level-packing heuristics do not yield signi cantly better solutions than the known heuristics, but several of the newly proposed pseudolevel heuristics do yield signi cantly better results than the best of the known pseudolevel heuristics in terms of both packing densities achieved and computation times expended. During the evaluation of the plane algorithms two classes of heuristics were identi ed for packing problems, namely sorting-dependent and sortingindependent algorithms. Two new sorting techniques are proposed for the sorting-independent algorithms and one of them yields the best-performing heuristic overall. A new heuristic approach for the MBSBPP is also proposed, which may be combined with level and pseudolevel algorithms for the strip packing problem in order to nd solutions to the problem very rapidly. The best-performing plane-packing heuristic is modi ed to pack items into the largest bins rst, followed by an attempted repacking of the items in those bins into smaller bins with the aim of further minimising wasted space. It is found that the resulting plane-packing algorithm yields the best results in terms of time and packing density, but that the solution di erences between pseudolevel algorithms are not as marked for the MBSBPP as for the strip packing problem. / AFRIKAANSE OPSOMMING: Inpakkingsprobleme kom algemeen in die industrie voor en daar is 'n aansienlike volume literatuur oor hierdie onderwerp. Twee inpakkingsprobleme word in hierdie proefskrif oorweeg, naamlik die strook-inpakkingsprobleem en die houer-inpakkingsprobleem. In beide probleme is die doel om 'n gespesi seerde versameling klein voorwerpe, waarvan die dimensies almal voordat inpakking plaasvind, bekend is (en die probleem dus 'n sogenaamde a yn-probleem is), in een of meer groter houers te pak. In die strook-inpakkingsprobleem word hierdie voorwerpe in een houer, waarvan een dimensie onbegrens is, ingepak (hierdie houer word dus 'n strook genoem). In twee dimensies word die wydte van die strook gewoonlik gespesi seer en is die doel om al die voorwerpe sonder oorvleueling op s o 'n manier in die strook te pak dat die totale inpakkingshoogte geminineer word. In die houer-inpakkingsprobleem, daarenteen, is die doel om die voorwerpe op s o 'n manier in 'n gespesi seerde aantal houers (waarvan al die dimensies begrens is) te pak dat die vermorste of oorblywende ruimte in die houers (wat wel voorwerpe bevat) 'n minimum is. Die houers mag almal dieselfde dimensies h^e (in welke geval die probleem as die enkelgrootte houer-inpakkingsprobleem bekend staan), of mag verskillende dimensies h^e (in welke geval die probleem as die veelvuldige-grootte houer-inpakkingsprobleem bekend staan, afgekort as VGHIP). In twee dimensies word die vermorste ruimte geneem as die somtotaal van daardie deelareas van die houers (wat wel voorwerpe bevat) waar daar geen voorwerpe geplaas word nie. Verskeie oplossingsmetodologie e is al vir die bogenoemde twee inpakkingsprobleme ontwikkel, maar die bestek van die metodologie e wat in hierdie proefskrif oorweeg word, word beperk tot heuristieke. 'n Inpakkingsheuristiek volg 'n vaste stel re els waarvolgens voorwerpe in houers gepak word om so spoedig moontlik goeie, toelaatbare (maar nie noodwendig optimale) oplossings tot die strook-inpakkingsprobleem en die houer-inpakkingsprobleem te vind. Drie tipes inpakkingsheuristieke word in hierdie proefskrif oorweeg, naamlik (i) heuristieke wat voorwerpe langs die onderste randte van horisontale vlakke in die houers pak (die hoogtes van hierdie vlakke word bepaal deur die hoogtes van die hoogste item in elke vlak), (ii) heuristieke wat voorwerpe op enige plek binne horisontale stroke in die houers pak, en (iii) heuristieke waar inpakking nie volgens horisontale vlakke of stroke beperk word nie. Hierdie drie klasse heuristieke staan onderskeidelik as vlakalgoritmes, pseudo-vlakalgoritmes en platvlakalgoritmes bekend. 'n Berekeningsbenadering word in hierdie proefskrif gevolg deur die werkverrigting van die 218 nuwe heuristieke vir die strook-inpakkingsprobleem met die werkverrigting van 34 bekende heuristieke uit die literatuur te vergelyk, deur al die heuristieke op 1 170 toetsprobleme toe te pas. Daar word bevind dat die nuwe vlakalgoritmes nie 'n noemenswaardige verbetering in oplossingskwaliteit in vergeleke met soortgelyke bestaande algoritmes in die literatuur lewer nie, maar dat verskeie nuwe pseudo-vlakalgoritmes wel noemenswaardige verbeteringe in terme van beide inpakkingsdigthede en oplossingstye in vergeleke met die beste bestaande algoritmes in die literatuur lewer. Assessering van die platvlakalgoritmes het gelei tot die identi kasie van twee deelklasse van algoritmes, naamlik sorteringsafhanklike- en sorteringsonafhanklike algoritmes. Twee nuwe sorteringstegnieke word ook vir die deelklas van sorteringsonafhanklike algoritmes voorgestel, en een van hulle lewer die algeheel beste inpakkingsheursitiek. 'n Nuwe heuristiese benadering word ook vir die VGHIP ontwikkel. Hierdie benadering kan met vlak- of pseudo-vlakalgoritmes vir die strook-inpakkingsprobleem gekombineer word om baie vinnig oplossings vir die VGHIP te vind. Die beste platvlakheuristiek vir die strookinpakkingsprobleem word ook aangepas om voorwerpe eers in die grootste houers te pak, en daarna in kleiner houers te herpak met die doel om vermorste ruimte verder te minimeer. Daar word bevind dat die resulterende platvlakalgoritme die beste resultate in terme van beide inpakkingsdigtheid en oplossingstyd lewer, maar dat oplossingsverskille tussen die pseudovlakalgoritmes nie so opmerklik vir die VGHIP is as wat die geval met die strookinpakkingsprobleem was nie.

Os problemas de corte e empacotamento são muito comuns nas indústrias e na logística. Dado um conjunto de N itens com diferentes pesos e um conjunto de M contentores com capacidade C, o problema de empacotamento unidimensional consiste em determinar o menor número de contentores a serem utilizados para alocar todos os itens respeitando a restrição de capacidade dos contentores. Nesse estudo pretende-se resolver o problema com instâncias benchmark da literatura, por meio de sessenta heurísticas diferentes, que são comparadas a quatro limitantes inferiores propostos na literatura com o intuito de avaliar a qualidade da solução heurística. Quatro limitantes inferiores e dez heurísticas construtivas diferentes foram programados em C++ num mesmo ambiente computacional, permitindo sua comparação tanto em termos de qualidade das soluções, quanto em termos dos tempos de processamento. Uma heurística simples de troca de itens entre contentores chamada Diferença-de-Quadrados foi proposta para melhorar as soluções iniciais do problema. A metaheurística simulated annealing foi acionada para melhorar a solução inicial quando o limitante inferior não foi atingido. Os parâmetros dos simulated annealing foram determinados com os dados das instâncias de forma diferente da utilizada na literatura. As combinações entre as dez soluções iniciais, a heurística Diferença-de-Quadrados e o simulated annealing geraram um conjunto de sessenta heurísticas diferentes. Os resultados mostraram que o algoritmo proposto é eficiente para resolver o problema com tempos de processamento adequados a tomada de decisão. / Cutting and packing problems are very common in industries and logistics. Given a set of N items with different weights and a set of M bins with full capacity C, the one-dimensional bin packing problem consists of determining the smallest number of bins capable to allocate all items respecting the capacity constraint of the bins. that impose that the sum of the weights of the items allocated to the bin is less than or equal to their capacity. In this study we intend to solve the problem with benchmark instances of the literature, by means of sixty different heuristics, which are compared to four lower bounds proposed in the literature in order to evaluate the quality of the heuristic solution. Four lower bounds and ten different constructive heuristics were programmed in C++ in the same computational environment, allowing their comparison both in terms of the quality of the solutions and in terms of processing times. A simple heuristic of item exchange between bins called Difference-of-Squares was proposed to improve the initial solutions of the problem. The simulated annealing metaheuristic was triggered to improve the initial solution when the lower bounds was not reached. The parameters of the simulated annealing were determined with the data of the instances differently from that used in the literature. The combinations of the ten initial solutions, the Difference-of-Squares heuristic and the simulated annealing generated a set of sixty different heuristics. The results showed that the proposed algorithm is efficient to solve the problem with adequate processing times for decision making.

Neste trabalho desenvolvemos o estudo da álgebra linear, secções cônicas e aplicações. Apresentamos os conceitos mais importantes da álgebra linear, estudando os espaços vetorias, subespaços vetoriais, matriz de mudança de base, transformações lineares e produto interno. O principal resultado do trabalho é o teorema espectral que fornece ferramentas para se estudar as secções cônicas não elementares, ou seja, aquelas nas quais uma parábola, elipse ou hipérbole são apresentadas com seus eixos não paralelos aos eixos coordenados do plano cartesiano. Uma vez de posse deste teorema é mostrado um processo prático no qual transformamos uma equação ax2 +bxy +cy2 +dx +ey + g = 0 na equação k1 (x\')2 + k2 (y\')2 + (dx1 + ey1) x\' + (dx2 + ey2) y\' + g = 0 sem o termo misto xy, onde após a eliminação deste, podemos deduzir a equação da cônica identificando assim esta curva. Apresentamos exemplos de cônicas com eixos paralelos e não paralelos aos coordenados do plano cartesiano e utilizamos o software geogebra para visualização. Também discutimos algumas aplicações das cônicas como trajetória de corpos celestes (planeta Terra e um cometa), princípio de reflexão da parábola mostrando o porquê das antenas e dos captadores de ondas sonoras serem parabólicos. Demonstramos um teorema que denominei de identificador de uma curva cônica pois com ele é possível classificar a cônica sem realizar o processo prático, apenas para isso identificamos através da equação ax2 +bxy + cy2 +dx + ey +g = 0, quais os valores de a;b e c e feito isto calculamos o discriminante b2 - 4ac, analisamos os sinais e a nulidade, ou seja, se é maior que zero, menor que zero ou igual a zero, assim é possível classificar a cônica. / The paper develops the study of linear algebra, conic sections and applications. I present the most important concepts of linear algebra, studying vector spaces, vector subspaces, base change matrix, linear transformations, internal product. The main result of the work is the spectral theorem, which provides tools to study the non-elementary conic sections, that is, those in which a parabola, ellipse or hyperbola are presented with their axes not parallel to the cartesian planes coordinate axes. Using this theorem we show a practical process in which we transform an equation ax2 +bxy + cy2 +dx +ey +g = 0 into the equation k1 (x\')2 +k2 (y\')2 + (dx1 +ey1) x\' (dx2 + ey2) y\' +g = 0 without the mixed term xy, where after its elimination we can deduce the conic equation thus identifying the curve we are looking for. I present examples of conic with parallel and non-parallel axes to the coordinates of the Cartesian plane and use the geogebra software for visualization. I discuss some applications of the conic as a trajectory of celestial bodies (planet Earth and a comet), principle of reflection of parabola showing why the antennas and sound wave pickups are parabolics. I demonstrate a theorem that I named the identifier of a conic curve, with it it is possible to classify the conic without realizing the practical process only for this. I identify through the equation ax2 +bxy + cy2 +dx + ey + g = 0, what are the values of a;b, and c and, with this done, I compute the discriminant b2 - 4ac and analyze the signs and the nullity, that is, if it is greater than zero, less than zero or equal to zero, therefore is possible to classify the conic.

O problema de empacotamento de itens irregulares com balanceamento da carga é encontrado no carregamento de aviões, caminhões e navios. O objetivo é empacotar itens irregulares utilizando o menor número de recipientes possível de forma que os recipientes estejam balanceados, que os itens não se sobreponham e estejam inteiramente contidos no recipiente. Neste trabalho, propomos três heurísticas bases com três variações cada para o problema com recipientes retangulares e irregulares. As heurísticas utilizam abordagens diferentes para representar os itens e para fazer o balanceamento. Uma das heurísticas utiliza malha para representação dos itens e faz o balanceamento dividindo o recipiente em quadrantes e revezando a alocação dos itens entre eles de forma que o balanceamento é feito de forma indireta. Tal heurística resolve o problema tanto para recipientes retangulares quanto irregulares. A segunda heurística utiliza a representação dos itens por polígonos e impossibilita a sobreposição de itens utilizando a técnica do nofit polygon. A heurística constrói a solução item por item, sem posições fixas e a cada item alocado, os itens são deslocados em direção ao centro de gravidade desejado do recipiente. Esta heurística resolve apenas problemas com recipientes retangulares. A última heurística é uma adaptação da heurística anterior para a resolução do problema com recipientes irregulares, de forma que o problema é resolvido em duas fases. Cada heurística base possui três variações cada, totalizando nove heurísticas. As heurísticas foram comparadas com outro trabalho da literatura e conseguiram melhorar os resultados para nove das dezenove instâncias testadas. / The irregular bin packing problem with load balancing is found in the loading of airplanes, trucks and ships. The aim is to use as few bins as possible to pack all the items so that all bins are balanced, items do not overlap and are fully contained in the bin. In this work, we propose three base heuristics with three variations each for the problem with rectangular and irregular bin. The three heuristics use different approaches to represent the items and to balance the bin. One of the heuristics uses a grid to represent the items and does the balancing by dividing the container into quadrants and alternating the allocation of items between them so that the balancing is done indirectly. Such heuristic solves the problem for both rectangular and irregular bins. The second heuristic uses the representation of items by polygons and uses the nofit polygon technique. The heuristic constructs the solution item by item, with no fixed positions and with each item allocated, the items are shifted towards the desired center of gravity of the bin. This heuristic only solves problems with rectangular bins. The last heuristic is an adaptation of the previous one to solve the problem with irregular bins, so that the problem is solved in two phases. Each base heuristic has three variations, totaling nine heuristics. The heuristics were compared with other work in the literature and managed to improve the results for nine of the nineteen instances tested.