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

Time Relaxed Round Robin Tournament and the NBA Scheduling Problem

This dissertation study was inspired by the National Basketball Association regular reason scheduling problem. NBA uses the time-relaxed round robin tournament format, which has drawn less research attention compared to the other scheduling formats. Besides NBA, the National Hockey League and many amateur leagues use the time-relaxed round robin tournament as well. This dissertation study is the first ever to examine the properties of general time-relaxed round robin tournaments. Single round, double round and multiple round time-relaxed round robin tournaments are defined. The integer programming and constraint programming models for those tournaments scheduling are developed and presented. Because of the complexity of this problem, several decomposition methods are presented as well. Traveling distance is an important factor in the tournament scheduling. Traveling tournament problem defined in the time constrained conditions has been well studied. This dissertation defines the novel problem of time-relaxed traveling tournament problem. Three algorithms has been developed and compared to address this problem. In addition, this dissertation study presents all major constraints for the NBA regular season scheduling. These constraints are grouped into three categories: structural, external and fairness. Both integer programming and constraint programming are used to model these constraints and the computation studies are presente

Mathematical Modeling and Optimization Approaches for Scheduling the Regular-Season Games of the National Hockey League

RÉSUMÉ : La Ligue nationale de hockey (LNH) est une association sportive professionnelle de hockey sur glace regroupant des équipes du Canada et des États-Unis. Chaque année, la LNH dois compter sur un calendrier de haute qualité concernant des questions économiques et d'équité pour les 1230 matchs de sa saison régulière. Dans cette thèse, nous proposons le premier modèle de programmation linéaire en nombres entiers (PLNE) pour le problème de la planification de ces matchs. Basé sur la littérature scientifique en planification des horaires sportifs, et aussi sur un raisonnement pratique, nous identifions et soulignons des exigences essentielles et des préférences qui doivent être satisfaites par des calendriers de haute qualité pour la LNH. La construction de tels calendriers, tout comme la planification des horaires sportifs en général, s'avère une tâche très difficile qui doit prendre en compte des intérêts concurrents et, dans plusieurs cas, subjectifs. En particulier, les expérimentations numériques que nous décrivons dans cette étude fournissent des évidences solides suggérant qu'une approche basée sur la PLNE est actuellement incapable de résoudre des instances de taille réaliste pour le problème. Pour surmonter cet inconvénient, nous proposons ensuite un algorithme de recherche adaptative à voisinage large (ALNS) qui intègre à la fois des nouvelles stratégies et des heuristiques spécialisées provenant de la littérature scientifique. Afin de tester cette approche, nous générons plusieurs instances du problème. Toutes les instances sont basées sur les calendriers officiels de la LNH et, en particulier, utilisent les dates de matchs à domicile de chaque équipe comme des dates de disponibilité de son aréna. Dans les situations les plus difficiles, la disponibilité des arénas est rare ou est à son minimum. Dans tous les cas, en ce qui concerne les indicateurs de qualité soulevés, l'algorithme ALNS a été capable de générer des calendriers clairement meilleur que leur correspondants adoptés par la LNH. Les résultats obtenus suggèrent que notre approche pourrait certainement permettre aux gestionnaires de la LNH de trouver des calendriers de meilleur qualité par rapport à une variété de nouvelles préférences.----------ABSTRACT : The National Hockey League (NHL) is a major professional ice hockey league composed of 30 teams located throughout the United States and Canada. Every year, the NHL must rely on a high-quality schedule regarding both economic and fairness issues for the 1230 games of its regular season. In this thesis, we propose the first integer linear programming (IP) model for the problem of scheduling those games. Based both on the pertinent sports scheduling literature and on practical reasoning, we identify and point out essential requirements and preferences that should be satisfied by good NHL schedules. Finding such schedules, as many other sports scheduling problems, is a very difficult task that involves several stakeholders with many conflicting, and often subjective, interests. In fact, computational experiments that we describe in this study, provide compelling evidence that an IP approach is currently unable to solve instances of realistic size for the problem. To overcome such drawback, we propose then an Adaptive Large Neighborhood Search (ALNS) algorithm that integrates both novel strategies and specialized heuristics from the scientific literature. To test the approach, we generate instances based on past NHL schedules and on a given number of arena-available dates that are suitable for the home games of each team. In the most challenging instances, availability of arenas is scarce or at its minimum. In all cases, regarding the identified concerns, the ALNS algorithm was able to generate much better schedules than those implemented by the NHL. Results obtained suggest that our approach could certainly identify unnecessary weakness in NHL schedules, makes the NHL managers aware of better schedules with respect to different requirements, and even lead them to consider other desired features they might not have previously taken into account

Formulations and Reformulations in Integer Programming

Creating good integer programming formulations had, as a basic axiom, the rule “Find formulations with tighter linear relaxations”. This rule, while useful when using unsophisticated branch-and-bound codes,is insufficient when using state-of-the-art codes that understand and embed many of the obvious formulation improvements. As these optimization codes become more sophisticated it is important to have finer control over their operation. Modelers need to be even more creative in reformulating their integer programs in order to improve on the automatic reformulations of the optimization codes

Formulations and Reformulations in Integer Programming

Abstract. Creating good integer programming formulations had, as a basic axiom, the rule “Find formulations with tighter linear relaxations”. This rule, while useful when using unsophisticated branch-and-bound codes,is insufficient when using state-of-the-art codes that understand and embed many of the obvious formulation improvements. As these optimization codes become more sophisticated it is important to have finer control over their operation. Modelers need to be even more creative in reformulating their integer programs in order to improve on the automatic reformulations of the optimization codes.