Models and Strategies for Variants of the Job Shop Scheduling Problem

Recently, a variety of constraint programming and Boolean satisfiability approaches to scheduling problems have been introduced. They have in common the use of relatively simple propagation mechanisms and an adaptive way to focus on the most constrained part of the problem. In some cases, these methods compare favorably to more classical constraint programming methods relying on propagation algorithms for global unary or cumulative resource constraints and dedicated search heuristics. In particular, we described an approach that combines restarting, with a generic adaptive heuristic and solution guided branching on a simple model based on a decomposition of disjunctive constraints. In this paper, we introduce an adaptation of this technique for an important subclass of job shop scheduling problems (JSPs), where the objective function involves minimization of earliness/tardiness costs. We further show that our technique can be improved by adding domain specific information for one variant of the JSP (involving time lag constraints). In particular we introduce a dedicated greedy heuristic, and an improved model for the case where the maximal time lag is 0 (also referred to as no-wait JSPs).Comment: Principles and Practice of Constraint Programming - CP 2011, Perugia : Italy (2011

Solving the Resource Constrained Project Scheduling Problem with Generalized Precedences by Lazy Clause Generation

The technical report presents a generic exact solution approach for minimizing the project duration of the resource-constrained project scheduling problem with generalized precedences (Rcpsp/max). The approach uses lazy clause generation, i.e., a hybrid of finite domain and Boolean satisfiability solving, in order to apply nogood learning and conflict-driven search on the solution generation. Our experiments show the benefit of lazy clause generation for finding an optimal solutions and proving its optimality in comparison to other state-of-the-art exact and non-exact methods. The method is highly robust: it matched or bettered the best known results on all of the 2340 instances we examined except 3, according to the currently available data on the PSPLib. Of the 631 open instances in this set it closed 573 and improved the bounds of 51 of the remaining 58 instances.Comment: 37 pages, 3 figures, 16 table

Why Cumulative Decomposition Is Not as Bad as It Sounds

International Conference on Principles and Practice of Constraint Programming (CP

Ordonnancement de tâches sous contraintes sur des métiers à tisser

Dans une usine de production de textile, il y a des métiers à tisser. Ces métiers à tisser peuvent être configurés de différentes façons. Des tâches doivent être exécutées sur ces métiers à tisser et le temps d’exécution d’une tâche est fonction du métier sur lequel elle est effectuée. De plus, chaque tâche est seulement compatible avec les métiers à tisser étant configurés de certaines façons. Un temps de mise en course peut permettre de configurer ou préparer un métier à tisser pour l’exécution d’une tâche. Le temps de mise en course est dépendant de la tâche qui précède et de celle qui suit. Nous souhaitons alors créer un horaire pour minimiser les temps de fabrication et les retards. Toutefois, certaines contraintes doivent être respectées. Lorsque des préparations surviennent sur des métiers différents en même temps, le nombre d’employés doit être suffisant. Un métier ne peut faire qu’une seule action à la fois. L’ordonnancement d’une seule machine est un problème NP-Difficile. Dans ce projet, il faut ordonnancer environ 800 tâches sur 90 machines dans un horizon de deux semaines, tout en respectant les contraintes de personnel. Des évènements stochastiques doivent être pris en compte pour obtenir un meilleur horaire. Le bris d’un fil n’étant pas un évènement rare, l’occurrence des bris est donnée sous la forme d’une loi de Poisson. Nous proposons alors une approche de résolution utilisant une heuristique de branchement basée sur le problème du commis voyageur. Cette approche permet d’obtenir de bonnes solutions pour le problème d’ordonnancement exploré. Les solutions trouvées sont 5 à 30% meilleures en termes de fonction objectif qu’une heuristique semblable à celle utilisée par l’équipe de planification de notre partenaire industriel. Nous présentons aussi un algorithme pour garantir la robustesse d’un horaire. Notre algorithme permet de générer des horaires plus réalistes et qui résistent bien aux évènements imprévus. La combinaison de ces deux pratiques mène à l’intégration et l’utilisation du produit final par notre partenaire industriel.In a textile factory, there are looms. Workers can configure the looms to weave different pieces of textiles. A loom can only weave a piece of textiles if the piece of textiles is compatible with its loom configuration. To change its configuration, a loom requires a setup. The setups are performed manually by workers. There are also sequence-dependent setups to prepare a loom for the upcoming piece of textiles. We wish to minimize the setups duration and the lateness. A solution must satisfy some constraints. The problem is subject to cumulative resources. The quantity of workers simultaneously configuring machines can’t exceed the total number of employees. A loom can only weave a piece of textiles at a time. Scheduling tasks on a single loom is an NP-Hard problem. In this project, we must schedule tasks an average of 800 tasks on 90 looms with a two-week horizon. Stochastic events might occur and must be accounted for. We must design an algorithm to create robust schedules under uncertainty. As a thread breaking during the weaving process isn’t a rare occurrence, a better schedule could greatly impact the performances of a company when applying the schedule to a real situation. We formulate that the number of breaks per task follows a Poisson distribution. First, we propose a branching heuristic based on the traveling salesperson problem in order to leverage computation times. The solutions found are 5 to 30% better according to their objective function than the ones of a greedy heuristic similar to what our industrial partner uses. We also present a filtering algorithm to guarantee robustness of solutions in respect to a confidence level. This algorithm improves robustness and creates more realist schedules. The algorithm is also efficient in computation time by achieving bound consistency in linear time. Combining both these techniques leads to the integration of our research in the decision system of our industrial partner

Solving resource-constrained shceuling problems with exact methods

Scheduling problems mainly consist in finding an assignment of execution times (a schedule) to a set of activities of a project that optimizes an objective function. There are many constraints imposed over the activities that any schedule must satisfy. The most usual constraints establish precedence relations between activities, or limit the amount of some resources that the activities can consume. There are many scheduling problems in the literature that have been and are currently still being studied. A paradigmatic example is the Resource-Constraint Project Scheduling Problem (RCPSP). It consists in finding a start time for each one of the activities of a project, respecting pre-defined precedence relations between activities and without exceeding the capacity of a set of resources that the activities consume. The goal is to find a schedule with the minimum makespan (total execution time of the project). The RCPSP has many generalizations, one of which is the Multimode Resource-Constrained Project Scheduling Problem (MRCPSP). In this variation, each activity has several available execution modes that differ in the duration of the activity or the demand of resources. A solution for the MRCPSP determines the start times of the activities and also an execution mode for each one. These problems are NP-hard, and are known in the literature to be especially hard, with moderately small instances of 50 activities that are still open. There are many approaches to solving RCPSP and MRCPSP in the literature. They are often tackled with metaheuristics due to their high complexity, but there are also some exact approaches, including Mixed Integer Linear Programming (MILP), Branch-and-Bound algorithms or Boolean Satisfiability (SAT), which have shown to be competitive and in many cases even better than metaheuristics. One of the exact methods that is growing in use in the field of constrained optimization is SAT Modulo Theories (SMT). This thesis is the continuation of previous works carried out in the Logic and Programming (L ∧ P) group of Universitat de Girona, which used SMT to tackle RCPSP and MRCPSP. Excluding these, there have not been any other attempts to use SMT to solve the MRCPSP. SMT solvers (like other generic methods such as SAT or MILP) do not know which is the problem they are dealing with. It is the work of the modeler to provide a representation of the problem (i.e. an encoding) in the language that the solver admits. The main goal of this thesis is to use SMT to solve the Multimode Resource-Constraint Project Scheduling Problem. We focus on two already existing encodings for the MRCPSP, namely the time encoding and the task encoding. We use some existing preprocessing methods that contribute to the formulation of time and task, and present new preprocessings. Most of them are based on the idea of incompatibility between two activities, i.e., the impossibility that two activities run at the same time instant. These incompatibilities let us discharge some con- figurations of the solutions prior to encode the problem. Consequently, the use of preprocessings helps to reduce the size of the encodings in terms of variables and clauses. Another contribution of this work is the study of the time and task encodings and the differences that they present. We refine these encodings to provide more compact versions. Moreover, two new versions of these encodings are presented, which mainly differ in the codification of the constraints over the use of resources. One of them is based on Linear Integer Arithmetic expressions, and the other one in Pseudo-Boolean constraints and Integer Difference Logic. Another contribution of this work is the presentation of an ad-hoc optimization algorithm based on a linear search that mainly consists in three steps. First of all it simplifies the problem to efficiently ensure or discharge the feasibility of the instance, then it finds a first non-optimal solution by using a quick heuristic method, and finally it optimizes the problem making use of the knowledge acquired with the preprocessings to boost the search. We also present an initial work on a more intrusive approach consisting in modifying the internal heuristic of the SMT solver for the decision of literals. This work involves the study of a state-of-the-art implementation of an SMT solver, and its modification to include a framework to specify heuristics related with the encoding of the problem. We give some initial results on custom heuristics for the time and task encodings of the MRCPSP. Finally, we test our system with the benchmark sets of instances for the MRCPSP available in the literature, and compare our performance with a state-of-the-art exact solver for the MRCPSP. The results show that we are able to solve the major part of the benchmark sets. Moreover, we show to be competitive with the state-of-the-art solver of Vílim et. al. for the MRCPSP, being our system slower in solving the easiest benchmark instances, but outperforming the solver of Vílim et. al. in solving the hardest instance

Solving hard industrial combinatorial problems with SAT

The topic of this thesis is the development of SAT-based techniques and tools for solving industrial combinatorial problems. First, it describes the architecture of state-of-the-art SAT and SMT Solvers based on the classical DPLL procedure. These systems can be used as black boxes for solving combinatorial problems. However, sometimes we can increase their efficiency with slight modifications of the basic algorithm. Therefore, the study and development of techniques for adjusting SAT Solvers to specific combinatorial problems is the first goal of this thesis. Namely, SAT Solvers can only deal with propositional logic. For solving general combinatorial problems, two different approaches are possible: - Reducing the complex constraints into propositional clauses. - Enriching the SAT Solver language. The first approach corresponds to encoding the constraint into SAT. The second one corresponds to using propagators, the basis for SMT Solvers. Regarding the first approach, in this document we improve the encoding of two of the most important combinatorial constraints: cardinality constraints and pseudo-Boolean constraints. After that, we present a new mixed approach, called lazy decomposition, which combines the advantages of encodings and propagators. The other part of the thesis uses these theoretical improvements in industrial combinatorial problems. We give a method for efficiently scheduling some professional sport leagues with SAT. The results are promising and show that a SAT approach is valid for these problems. However, the chaotical behavior of CDCL-based SAT Solvers due to VSIDS heuristics makes it difficult to obtain a similar solution for two similar problems. This may be inconvenient in real-world problems, since a user expects similar solutions when it makes slight modifications to the problem specification. In order to overcome this limitation, we have studied and solved the close solution problem, i.e., the problem of quickly finding a close solution when a similar problem is considered