A MILP model for the teacher assignment problem considering teachers’ preferences

The Teacher Assignment Problem is part of the University Timetabling Problem and involves assigning teachers to courses, taking their preferences into consideration. This is a complex problem, usually solved by means of heuristic algorithms. In this paper a Mixed Integer Linear Programing model is developed to balance teachers’ teaching load (first optimization criterion), while maximizing teachers’ preferences for courses according to their category (second optimization criterion). The model is used to solve the teachers-courses assignment in the Department of Management at the School of Industrial Engineering of Barcelona, in the Universitat Politècnica de Catalunya. Results are discussed regarding the importance given to the optimization criteria. Moreover, to test the model's performance a computational experiment is carried out using randomly generated instances based on real patterns. Results show that the model is proven to be suitable for many situations (number of teachers-courses and weight of the criteria), being useful for departments with similar requests.Peer ReviewedPostprint (author's final draft

Decomposition, Reformulation, and Diving in University Course Timetabling

In many real-life optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective-/value-restricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality. Our study is performed on a university course timetabling problem used in the 2007 International Timetabling Competition, also known as the Udine Course Timetabling Problem. In the proposed heuristic, an objective-restricted neighbourhood generator produces assignments of periods to events, with decreasing numbers of violations of two period-related soft constraints. Those are relaxed into assignments of events to days, which define neighbourhoods that are easier to search with respect to all four soft constraints. Integer programming formulations for all subproblems are given and evaluated using ILOG CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table

On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported

Integer programming techniques for educational timetabling

Educational timetabling problems require the assignment of times and resources to events, while sets of required and desirable constraints must be considered. The XHSTT format was adopted in this work because it models the main features of educational timetabling and it is the most used format in recent studies in the field. This work presents new cuts and reformulations for the existing integer programming model for XHSTT. The proposed cuts improved hugely the linear relaxation of the formulation, leading to an average gap reduction of 32%. Applied to XHSTT-2014 instance set, the alternative formulation pro- vided four new best known lower bounds and, used in a matheuristic framework, improved eleven best known solutions. The computational experiments also show that the resulting integer programming mod- els from the proposed formulation are more effectively solved for most of the instances

Problema de atribuição de vigilâncias de exames a docentes

O presente trabalho foi realizado com o intuito de resolver o problema de alocação de vigilantes a exames do Instituto Superior de Engenharia do Porto, no departamento de Engenharia Mecânica. O modelo apresentado faz a atribuição das vigilâncias de uma forma hierárquica, utilizando vários critérios, desde a regência da unidade curricular até à simples vigilância. Devido ao facto de estar implementado informaticamente, apresenta reduzidos tempos na formulação e obtenção de uma solução, o que o torna uma boa ferramenta para a criação de cenários alternativos. Em suma, o modelo proposto neste trabalho apresenta soluções de melhor qualidade, em que a distribuição de afetações é proporcional entre os docentes, e o seu tempo de obtenção é muito reduzido em comparação com a alternativa atual.The present study was performed in order to solve the problem of allocating vigilantes to examinations in the Instituto Superior de Engenharia do Porto, in the department of Mechanical Engineering. The model presented makes the allocation of surveillances in a hierarchical manner, using various criteria, since the regency of course to the simple vigilance. Due to it being implemented on a computer, the model presents reduced time in formulating and obtaining a solution, which makes it a good tool for the creation of alternative scenarios. In sum, the proposed model presents better quality solutions, in which the distribution of the affectations is equitable among teachers, and require much less time to obtain compared to the current alternative

Coil batching to improve productivity and energy utilization in steel production

This paper investigates a practical batching decision problem that arises in the batch annealing operations in the cold rolling stage of steel production faced by most large iron and steel companies in the world. The problem is to select steel coils from a set of waiting coils to form batches to be annealed in available batch annealing furnaces and choose a median coil for each furnace. The objective is to maximize the total reward of the selected coils less the total coil'coil and coil'furnace mismatching cost. For a special case of the problem that arises frequently in practical settings where the coils are all similar and there is only one type of furnace available, we develop a polynomial-time dynamic programming algorithm to obtain an optimal solution. For the general case of the problem, which is strongly NP-hard, an exact branch-and-price-and-cut solution algorithm is developed using a column and row generation framework. A variable reduction strategy is also proposed to accelerate the algorithm. The algorithm is capable of solving medium-size instances to optimality within a reasonable computation time. In addition, a tabu search heuristic is proposed for solving larger instances. Three simple search neighborhoods, as well as a sophisticated variable-depth neighborhood, are developed. This heuristic can generate near-optimal solutions for large instances within a short computation time. Using both randomly generated and real-world production data sets, we show that our algorithms are superior to the typical rule-based planning approach used by many steel plants. A decision support system that embeds our algorithms was developed and implemented at Baosteel to replace their rule-based planning method. The use of the system brings significant benefits to Baosteel, including an annual net profit increase of at least 1.76 million U.S. dollars and a large reduction of standard coal consumption and carbon dioxide emissions

Optimisation d'un réseau de communication par fibre optique

Le présent travail concerne la modélisation d'un réseau de communication par fibre optique. Il s'agira de modéliser une topologie virtuelle prenant en compte les noeuds, les capacités de traitement aux noeuds, les exigences de qualité ainsi que les routes virtuelles ou successions de trajets sur lesquels sont embarqués les flux de données ou la matrice de trafic. Le modèle devra aussi tenir compte de divers paramètres (longueurs d'onde, modulations, type de routage, etc..). Nous utiliserons, après la modélisation en un programme linéaire mixte en nombres entiers (MILP), un ensemble de solveurs pour nous permettre d'apprécier les performances ainsi que les limites de la résolution sur ces différentes plates-formes au fur et à mesure que le problème croît. Un autre aspect du travail sera de faire le constat de la complexité de la programmation linéaire mixte en nombres entiers. Cela va d'ailleurs susciter le passage d'un MILP dit fort à un MILP dit faible. Un autre volet du travail sera la formalisation du MILP dit faible par une approche de décomposition structurée, prenant en compte la structure du modèle ainsi que des caractéristiques de certaines variables avec pour objectif ultime d'améliorer le processus d'optimisation

Μοντελοποίηση και επίλυση του προβλήματος κατάρτισης του ωρολόγιου προγράμματος του τμήματος Μ.Μ.Β. με μεθόδους ακέραιου προγραμματισμού σε διαδικτυακό γραφικό περιβάλλον

Η παρούσα διπλωματική εργασία αποτελεί επέκταση της δουλειάς που έγινε από τον Βασίλειο Κουτρουμπίνα για τη βελτιστοποίηση του ωρολογίου προγράμματος του Τμήματος Μηχανολόγων Μηχανικών Βιομηχανίας. Υιοθετήσαμε το αρχικό μοντέλο ακεραίου προγραμματισμού και το εξελίξαμε ώστε να μπορεί να έχει πρακτική εφαρμογή για την κάλυψη των αναγκών του τμήματος, δίνοντας τα επιθυμητά αποτελέσματα σε λογικούς χρόνους. Επιπλέον, δημιουργήθηκε ένα αρχικό γραφικό interface για την εισαγωγή και εξαγωγή δεδομένων, όπως επίσης και για την επικαιροποίηση των σχετικών βάσεων δεδομένων. Έτσι, καταλήξαμε σε ένα ουσιαστικά λειτουργικό λογισμικό που με τις απαραίτητες ανθρώπινες παρεμβάσεις μπορεί να καταστρώσει ταχύτατα το ωρολόγιο πρόγραμμα διδασκαλίας του Τμήματος Μηχανολόγων Μηχανικών Βιομηχανίας