This article deals with the resolution of over-constrained problems using constraint programming, which often imposes to add to the constraint network new side constraints. These side constraints control how the initial constraints of the model should be satisfied or violated, to obtain solutions that have a practical interest. They are specific to each application. In our experiments, we show the superiority of a framework where side constraints are encoded by global constraints on new domain variables, which are directly included into the model. The case-study is a cumulative scheduling problem with over-loads. The objective is to minimize the total amount of over-loads. We augment the Cumulative global constraint of the constraint programming solver Choco with sweep and task interval violation-based algorithms. We provide a theoretical and experimental comparison of the two main approaches for encoding over-constrained problems with side constraints.

This research report presents an extension of Cumulative of Choco constraint solver, which is useful to encode over-constrained cumulative problems. This new global constraint uses sweep and task interval violation-based algorithms.

This article shows the advantages of a variable-based framework for solving over-constrained problems with practicability constraints. The case-study is a cumulative scheduling problem with over-loads.

This paper describes new filtering methods for the 𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 constraint. The first method introduces bounds for the so called longest cumulative hole problem and shows how to use these bounds in the context of the non-overlapping constraint. The second method introduces balancing knapsack constraints which relate the total height of the tasks that end at a specific time-point with the total height of the tasks that start at the same time-point. Experiments on tight rectangle packing problems show that these methods drastically reduce both the time and the number of backtracks for finding all solutions as well as for finding the first solution. For example, we found without backtracking all solutions to 66 perfect square instances of order 23-25 and sizes ranging from 332 ×332 to 661 ×661.

This article first introduces an extension of the classical cumulative constraint: each task is no more a rectangle but rather a sequence of contiguous trapezoid sub-tasks with variable duration and heights. The resource function is no more constant but is a positive or negative piecewise linear function of time. Finally, a task is no more pre-assigned to one resource, but to a task corresponds a set of possible resource assignments. In this context, this article provides an O(p ·(logp + q)) for computing all the cumulated resource profiles where q is the number of resources and p is the total number of trapezoid sub-tasks of all the tasks.

This paper introduces a geometrical constraint kernel for handling the location in space and time of polymorphic k-dimensional objects subject to various geometrical and time constraints. The constraint kernel is generic in the sense that one of its parameters is a set of constraints on subsets of the objects. These constraints are handled globally by the kernel. We first illustrate how tomodel several placement problems with the constraint kernel. We then explain how new constraints can be introduced and plugged into the kernel. Based on these interfaces, we develop a generic k-dimensional lexicographic sweep algorithm for filtering the attributes of an object (i.e., its shape and the coordinates of its origin as well as its start, duration and end in time) according to all constraints where the object occurs. Experiments involving up to hundreds of thousands of objects and 1 million integer variables are provided in 2, 3 and 4 dimensions, both for simple shapes (i.e., rectangles, parallelepipeds) and for more complex shapes.

This paper considers a generalisation of the classical RCPSP problem: the resource consumption of each task is continuously varying over time and the duration and the start of each task may vary within real intervals. A first contribution is a general model for describing the resource consumption of a task over time. This model is justified when considering continuously divisible resources. The second contribution is the computation of the compulsory part or core time of such a task. The compulsory part gives the task's resource consumption common to all feasible schedules. Hence, it can be used in a global resolution process such as constraint programming or branch and bound approaches. The presented polynomial algorithms use only two particular schedules of that task.

The period constraint restricts the smallest period of a sequence of domain variables to be equal to a given domain variable. This paper first provides propositions for evaluating the feasible values of the smallest period of a sequence of domain variables. Then, it gives propositions for pruning the variables of the sequence in order to achieve one out of several possible smallest periods. The generalisation of the period constraint to the case where the equality between two domain variables is replaced by any condition is also considered.

La plupart des travaux sur l'ordonnancement avec contraintes de ressources (probleme RCPSP) sont lies a des problemes ou les tâches utilisent des quantites constantes de ressources durant toute leur execution. Cependant, dans de nombreux problemes pratiques, les consommations des tâches en ressources sont variables avec le temps. Ces variations peuvent etre discretes (besoin de main d'oeuvre) ou continues (electricite, petrole). En general, de tels profils de consommation sont approximes par des suites de rectangles lies par des contraintes de contiguites mais un nombre eleve de rectangles elementaires et de contraintes sont alors necessaires. Dans cette these, nous presentons dans un premier temps, une extension du probleme RCPSP classique : la consommation en ressource d'une tâche est une fonction du temps lineaire par morceaux et sa duree peut etre variable. Une tâche est representee par une suite de sous-tâches trapezoidales contigus. Nous presentons ensuite les limites des methodes de resolution exactes par separation et evaluation proposees dans la litterature et nous proposons une resolution de cette generalisation du RCPSP en utilisant le langage de Programmation Par Contraintes CHIP. Pour cela, une nouvelle contrainte globale cumulative traitant globalement ce modele de tâche a ete creee et ajoutee au langage CHIP. Elle se nomme cumulative_trapeze et a necessite l'ecriture d'algorithmes efficaces pour le calcul de la partie obligatoire et pour l'elagage des domaines des variables d'une tâche. Les principales difficultes rencontrees lors de sa realisation contrainte ont reside dans le traitement de suites d'objets plutot que de rectangles individuels et dans la gestion d'un profil cumule des consommations non entier. Enfin, les principales applications sont presentees : modele Producteur/Consommateur pour la gestion de stock et problemes de placements