著色問題(graph coloring problem)的研究已行之有年，並衍生出廣泛的實際應用，但還缺乏一般化的著色問題模型。本論文建構一般化的著色問題模型，其目標函數包含顏色成本的固定支出和點著色變動成本。此著色模型為0/1整數線性規畫模型，其限制式含有選點問題(node packing problem)的限制式。我們利用圖中的極大團(maximal clique)所構成的強力限制式，取代原有的選點限制式，縮短求解時間。我們更進一步舉出一個特殊指派問題並將此著色模型應用於此指派問題上。本論文亦針對此指派問題發展了一個演算法來尋找極大團。計算結果顯示極大團限制式對於此著色問題模型的求解有極大的效益。 / The graph coloring problem (GCP) has been studied for a long time and it has a wide variety of applications. A straightforward formulation of graph coloring problem has not been formulated yet. In this paper, we formulate a general GCP model that concerns setup cost and variable cost of different colors. The resulting model is an integer program that involves the packing constraint. The packing constraint in the GCP model can be replaced by the maximal clique constraint in order to shorten the solution time. A special assignment problem is presented which essentially is a GCP model application. An algorithm of finding maximal cliques for this assignment problem is developed. The computational results show the efficiency of maximal clique constraints for the GCP problem.

Στην παρούσα μεταπτυχιακή εργασία περιγράφεται η διαδικασία συνδυασμού προβλημάτων Επιχειρησιακής Έρευνας με την μεθοδολογία εύρεσης συγκριτικής αποδοτικότητας (DEA). Αρχικά, παρουσιάζεται μια γενική περιγραφή της μεθόδου DEA και μια συνοπτική επισκόπηση της σχετικής βιβλιογραφίας. Παρουσιάζεται ο τρόπος συνδυασμού της μεθόδου DEA και δύο κλασσικών μοντέλων χωροθέτησης εγκαταστάσεων, του μοντέλου με περιορισμό και του αντίστοιχου μοντέλου χωρίς περιορισμό στην χωρητικότητα. Για την επίτευξη αυτού του στόχου γίνονται οι απαραίτητοι χειρισμοί στην μέθοδο DEA ούτως ώστε να μπορεί να υπολογίζεται η αποδοτικότητα για όλες τις μονάδες λήψης απόφασης ταυτόχρονα – μέθοδος ταυτόχρονης DEA (Simultaneous DEA), εφόσον το κλασσικό μοντέλο βρίσκει την αποδοτικότητα μιας μονάδας λύνοντας μια φορά το γραμμικό πρόβλημα με τους συντελεστές βαρύτητας αυτής της μονάδας. Η λύση του πολυκριτήριου προβλήματος αναδεικνύει την αλληλεπίδραση μεταξύ κόστους και αποδοτικότητας, για τη λήψη απόφασης ανάλογα με τις ανάγκες που μπορεί ενυπάρχουν σε ένα αντίστοιχο πραγματικό πρόβλημα. Στην συνέχεια αναπτύσσεται για πρώτη φορά στη διεθνή βιβλιογραφία μια μεθοδολογία για το συνδυασμό δύο άλλων βασικών προβλημάτων, της κάλυψης και της σύμπτυξης συνόλου, αντίστοιχα, με την μεθοδολογία DEA. Στόχος είναι να μορφοποιηθεί ένα μοντέλο γραμμικού προγραμματισμού έτσι ώστε εκτός από το μέτρο απόφασης του κόστους για την κάλυψη ή σύμπτυξη ενός συνόλου-στόχου, από διαθέσιμα υποσύνολα να ληφθεί υπόψη και η αποδοτικότητα του εκάστοτε υποσυνόλου, η οποία εν τέλει θα επηρεάσει και την συνολική αποδοτικότητα του συνόλου-στόχου. Γίνεται ο συνδυασμός των μεθοδολογιών και αναπτύσσονται μεθοδολογίες πολυκριτήριας ανάλυσης που μπορούν να χρησιμοποιηθούν για την λήψη αποφάσεων που αφορούν την αποδοτική και οικονομική κάλυψη ή σύμπτυξη ενός συνόλου. Για την πιστοποίηση και τη διαπίστωση της λειτουργικότητας των προτεινόμενων μεθοδολογιών αναπτύσσονται παραδείγματα προβλημάτων, τα οποία και επιλύονται επιτυχώς. / In the present thesis, the combination of Operation Research Problems with the Data Envelopment Analysis (DEA) is performed in order to make optimal and efficient decisions. Firstly, a general description of DEA and a breath literature review is presented. Then, we show and test location modeling formulations that utilize data envelopment analysis (DEA) efficiency measures to find optimal and efficient facility location/allocation patterns. In addition, to the authors’ best knowledge, the combinations of DEA with the Set Covering Problem as well as Set Packing Problem are formulated as multiobjective problems, for first time in the literature. The main aim of the proposed models is to make cost-effective and efficient decisions regarding the Set Covering and Packing Problem, respectively. Numerical examples are developed in order to validate and test the novel models. The numerical results of multiobjective analysis demonstrate that the proposed methods are able to successfully find optimal and efficient solutions for real set covering, packing and partitioning problems.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-10-20T16:09:47Z No. of bitstreams: 2 Dissertação - Lorrany Cristina da Silva - 2017.pdf: 8394886 bytes, checksum: 9cc1461b937a65a8c50964b3dea86623 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-10-23T10:05:52Z (GMT) No. of bitstreams: 2 Dissertação - Lorrany Cristina da Silva - 2017.pdf: 8394886 bytes, checksum: 9cc1461b937a65a8c50964b3dea86623 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-10-23T10:05:52Z (GMT). No. of bitstreams: 2 Dissertação - Lorrany Cristina da Silva - 2017.pdf: 8394886 bytes, checksum: 9cc1461b937a65a8c50964b3dea86623 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-06-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Three different integer linear programming models for the Vehicle Routing Problem with Two-dimensional Loading Constraints are developed in this work. The version of the problem studied considers that the unloading of the rectangular items can respect or not the sequence of the clients visited on the route, that is, we solve the sequential and unrestricted versions of the problem. The first model deals with the problem completely, that is, with all constraints inserted at once. The second and third models are based, respectively, on a three- and two-index formulation. Separation routines are considered to detect violated inequalities related with packing on the second and third models, while the third model also considers cuts on connectivity and capacity. Computational experiments were carried out over instances of the literature with the quantity of customers ranging from 15 to 36 and items from 15 to 114, besides to consider the cases in which the cost of traversing an edge is integer and real. The models with cuts on demand were better in relation to the first model, besides being competitive when comparing with the results fromthe literature. The first model solved 4 of the 80 instances, the three-index model solved 7 and, the two-index model solved 53. On the sequential version, the adopted model solved 33 instances for the case with integer costs (and 37 for the case with real costs). In comparing with a recent heuristic from the literature, the best model was capable of tying in 48 instances in the unrestricted version and 24 in the sequential version. / Neste trabalho desenvolvem-se três modelos de programação linear inteira para o Problema de Roteamento de Veículos com Restrições de Empacotamento Bidimensional. A versão do problema estudado considera que o descarregamento dos itens retangulares pode respeitar (ou não) a sequência de clientes visitados na rota, ou seja, resolve-se as versões sequencial e irrestrita do problema. O primeiro modelo trata do problema de forma completa, isto é, com todas as restrições inseridas de uma só vez. O segundo e o terceiro modelo são baseados, respectivamente, em uma formulação de três e dois índices. Rotinas de separação são consideradas para detectar desigualdades violadas de empacotamento no segundo e no terceiro modelo, enquanto o último modelo considera também cortes de conectividade e capacidade. Experimentos computacionais foram realizados em instâncias da literatura com número de clientes variando de 15 a 36 e itens de 15 até 114, além de considerar os casos em que o custo da aresta é inteiro ou real. Os modelos com cortes sob demanda foram melhores em relação ao primeiro modelo, além de serem competitivos quando comparado com a literatura. O modelo completo encontrou a solução ótima em 4 das 80 instâncias, o modelo de três índices 7 e o modelo de dois índices 53. Na versão sequencial, o modelo adotado resolveu 33 instâncias para o custo inteiro (e 37 para o custo real). Na comparação com uma heurística recente da literatura, o melhor modelo conseguiu empatar em 48 instâncias na versão irrestrita e em 24 na versão sequencial.

Les travaux de thèse présentés s’intéressent à l’optimisation des activités d’un bloc opératoire. Ces activités concernent les interventions chirurgicales à planifier et les interventions de maintenance préventive sur les équipements dans les salles d’opération. Une solution est la synchronisation de ces activités lors de la construction du planning opératoire au niveau opératoire. Nous dissocions deux stratégies de programmation opératoire : programmation ouverte et programmation avec allocation préalable des plages horaires aux chirurgiens. Pour chacune des stratégies, nous considérons deux cas : le cas où l’heure de début d’une intervention de maintenance dans la salle est fixée, ladite intervention précédant l’affection des interventions chirurgicales dans les salles. Le second cas étant celui où l’heure de début de maintenance varie dans un intervalle entre une heure de début minimum et une heure de début maximum, avec l’intervention de maintenance placée a posteriori.Nous faisons plusieurs propositions de méthodes (exactes et approchées), y compris une méthode hybride, qui repose sur le couplage entre une métaheuristique et une heuristique. Les résultats obtenus sur des instances générées en concertation avec le monde hospitalier sont intéressants. / The presented dissertation is about the optimization of hospital systems, more precisely the optimization of the activities of an operation theatre. These activities showcase the surgical procedures to be planned and the preventive maintenance interventions on the equipment in the operating rooms. One solution is the synchronization of these activities during the construction of the operational planning at the operational level.We dissociate two operating programming strategies: Open Scheduling or Open programming and Block Scheduling or Programming with prior allocation of times to surgeons. For each strategy two cases are considered: the first case is where the time of beginning of a maintenance intervention in the room is fixed - this intervention preceding the affection of the surgical interventions in the rooms. The second case is where the maintenance start time varies in the interval between a minimum start time and a maximum start time, with the maintenance intervention placed beforehand. We make several proposition’s methods (exact and approximate), including a hybrid method, which is based on the coupling between a metaheuristic and a heuristic. The results obtained on bodies generated in consultation with the hospital’s world are interesting.

Dans cette thèse, nous nous intéressons à trois problèmes issus de l'informatique théorique, à savoir le placement de formes rectangulaires dans un conteneur (OPP), la coloration dite "forte" d'arêtes des graphes et les codes identifiants dans les graphes. L'OPP consiste à décider si un ensemble d'items rectangulaires peut être placé sans chevauchement dans un conteneur rectangulaire et sans dépassement des bords de celui-ci. Une contrainte supplémentaire est prise en compte, à savoir l'interdiction de rotation des items. Le problème est NP-difficile même dans le cas où le conteneur et les formes sont des carrés. Nous présentons un algorithme de résolution efficace basé sur une caractérisation du problème par des graphes d'intervalles, proposée par Fekete et Schepers. L'algorithme est exact et utilise les MPQ-arbres - structures de données qui encodent ces graphes de manière compacte tout en capturant leurs propriétés remarquables. Nous montrons les résultats expérimentaux de notre approche en les comparant aux performances d'autres algorithmes existants. L'étude de la coloration forte d'arêtes et des codes identifiants porte sur les aspects structurels et de calculabilité de ces deux problèmes. Dans le cas de la coloration forte d'arêtes nous nous intéressons plus particulièrement aux familles des graphes planaires et des graphes subcubiques. Nous montrons des bornes optimales pour l'indice chromatique fort des graphes subcubiques en fonction du degré moyen maximum et montrons que tout graphe planaire subcubique sans cycles induits de longueur 4 et 5 est coloriable avec neuf couleurs. Enfin nous confirmons la difficulté du problème de décision associé, en prouvant qu'il est NP-complet dans des sous-classes restreintes des graphes planaires subcubiques.La troisième partie de la thèse est consacrée aux codes identifiants. Nous proposons une caractérisation des graphes identifiables dont la cardinalité du code identifiant minimum ID est n-1, où n est l'ordre du graphe. Nous étudions la classe des graphes adjoints et nous prouvons des bornes inférieures et supérieures serrées pour le paramètre ID dans cette classe. Finalement, nous montrons qu'il existe un algorithme linéaire de calcul de ID dans la classe des graphes adjoints L(G) où G a une largeur arborescente bornée par une constante. En revanche nous nous apercevons que le problème est NP-complet dans des sous-classes très restreintes des graphes parfaits. / In this thesis we study three theoretical computer science problems, namely the orthogonal packing problem (OPP for short), strong edge-colouring and identifying codes.OPP consists in testing whether a set of rectangular items can be packed in a rectangular container without overlapping and without exceeding the borders of this container. An additional constraint is that the rotation of the items is not allowed. The problem is NP-hard even when the problem is reduced to packing squares in a square. We propose an exact algorithm for solving OPP efficiently using the characterization of the problem by interval graphs proposed by Fekete and Schepers. For this purpose we use some compact representation of interval graphs - MPQ-trees. We show experimental results of our approach by comparing them to the results of other algorithms known in the literature. we observe promising gains.The study of strong edge-colouring and identifying codes is focused on the structural and computational aspects of these combinatorial problems. In the case of strong edge-colouring we are interested in the families of planar graphs and subcubic graphs. We show optimal upper bounds for the strong chromatic index of subcubic graphs as a function of the maximum average degree. We also show that every planar subcubic graph without induced cycles of length 4 and 5 can be strong edge-coloured with at most nine colours. Finally, we confirm the difficulty of the problem by showing that it remains NP-complete even in some restricted classes of planar subcubic graphs.For the subject of identifying codes we propose a characterization of non-trivial graphs having maximum identifying code number ID, that is n-1, where n is the number of vertices. We study the case of line graphs and prove lower and upper bounds for ID parameter in this class. At last we investigate the complexity of the corresponding decision problem and show the existence of a linear algorithm for computing ID of the line graph L(G) where G has the size of the tree-width bounded by a constant. On the other hand, we show that the identifying code problem is NP-complete in various subclasses of planar graphs.