Algorithms for minimizing maximum lateness with unit length tasks and resource constraints

AbstractThe problem we consider is that of scheduling n unit length tasks on identical processors in the presence of additional scarce resources. The objective is to minimize maximum lateness. It has been known for some time that the problem is NP-hard even for two processors and one resource type. In the present paper we show that the problem can be solved in O(n log n) time, even for an arbitrary number of resources if the instance of the problem has the saturation property (i.e., no resource unit is idle in an optimal schedule). For the more general problem without saturation, two heuristic algorithms and a branch and bound approach are proposed. The results of computational tests of the above methods are also reported

Efficient algorithms to solve scheduling problems with a variety of optimization criteria

La programmation par contraintes est une technique puissante pour résoudre, entre autres, des problèmes d'ordonnancement de grande envergure. L'ordonnancement vise à allouer dans le temps des tâches à des ressources. Lors de son exécution, une tâche consomme une ressource à un taux constant. Généralement, on cherche à optimiser une fonction objectif telle la durée totale d'un ordonnancement. Résoudre un problème d'ordonnancement signifie trouver quand chaque tâche doit débuter et quelle ressource doit l'exécuter. La plupart des problèmes d'ordonnancement sont NP-Difficiles. Conséquemment, il n'existe aucun algorithme connu capable de les résoudre en temps polynomial. Cependant, il existe des spécialisations aux problèmes d'ordonnancement qui ne sont pas NP-Complet. Ces problèmes peuvent être résolus en temps polynomial en utilisant des algorithmes qui leur sont propres. Notre objectif est d'explorer ces algorithmes d'ordonnancement dans plusieurs contextes variés. Les techniques de filtrage ont beaucoup évolué dans les dernières années en ordonnancement basé sur les contraintes. La proéminence des algorithmes de filtrage repose sur leur habilité à réduire l'arbre de recherche en excluant les valeurs des domaines qui ne participent pas à des solutions au problème. Nous proposons des améliorations et présentons des algorithmes de filtrage plus efficaces pour résoudre des problèmes classiques d'ordonnancement. De plus, nous présentons des adaptations de techniques de filtrage pour le cas où les tâches peuvent être retardées. Nous considérons aussi différentes propriétés de problèmes industriels et résolvons plus efficacement des problèmes où le critère d'optimisation n'est pas nécessairement le moment où la dernière tâche se termine. Par exemple, nous présentons des algorithmes à temps polynomial pour le cas où la quantité de ressources fluctue dans le temps, ou quand le coût d'exécuter une tâche au temps t dépend de t.Constraint programming is a powerful methodology to solve large scale and practical scheduling problems. Resource-constrained scheduling deals with temporal allocation of a variety of tasks to a set of resources, where the tasks consume a certain amount of resource during their execution. Ordinarily, a desired objective function such as the total length of a feasible schedule, called the makespan, is optimized in scheduling problems. Solving the scheduling problem is equivalent to finding out when each task starts and which resource executes it. In general, the scheduling problems are NP-Hard. Consequently, there exists no known algorithm that can solve the problem by executing a polynomial number of instructions. Nonetheless, there exist specializations for scheduling problems that are not NP-Complete. Such problems can be solved in polynomial time using dedicated algorithms. We tackle such algorithms for scheduling problems in a variety of contexts. Filtering techniques are being developed and improved over the past years in constraint-based scheduling. The prominency of filtering algorithms lies on their power to shrink the search tree by excluding values from the domains which do not yield a feasible solution. We propose improvements and present faster filtering algorithms for classical scheduling problems. Furthermore, we establish the adaptions of filtering techniques to the case that the tasks can be delayed. We also consider distinct properties of industrial scheduling problems and solve more efficiently the scheduling problems whose optimization criteria is not necessarily the makespan. For instance, we present polynomial time algorithms for the case that the amount of available resources fluctuates over time, or when the cost of executing a task at time t is dependent on t

Least space-time first scheduling algorithm : scheduling complex tasks with hard deadline on parallel machines

Both time constraints and logical correctness are essential to real-time systems and failure to specify and observe a time constraint may result in disaster. Two orthogonal issues arise in the design and analysis of real-time systems: one is the specification of the system, and the semantic model describing the properties of real-time programs; the other is the scheduling and allocation of resources that may be shared by real-time program modules. The problem of scheduling tasks with precedence and timing constraints onto a set of processors in a way that minimizes maximum tardiness is here considered. A new scheduling heuristic, Least Space Time First (LSTF), is proposed for this NP-Complete problem. Basic properties of LSTF are explored; for example, it is shown that (1) LSTF dominates Earliest-Deadline-First (EDF) for scheduling a set of tasks on a single processor (i.e., if a set of tasks are schedulable under EDF, they are also schedulable under LSTF); and (2) LSTF is more effective than EDF for scheduling a set of independent simple tasks on multiple processors. Within an idealized framework, theoretical bounds on maximum tardiness for scheduling algorithms in general, and tighter bounds for LSTF in particular, are proven for worst case behavior. Furthermore, simulation benchmarks are developed, comparing the performance of LSTF with other scheduling disciplines for average case behavior. Several techniques are introduced to integrate overhead (for example, scheduler and context switch) and more realistic assumptions (such as inter-processor communication cost) in various execution models. A workload generator and symbolic simulator have been implemented for comparing the performance of LSTF (and a variant -- LSTF+) with that of several standard scheduling algorithms. LSTF\u27s execution model, basic theories, and overhead considerations have been defined and developed. Based upon the evidence, it is proposed that LSTF is a good and practical scheduling algorithm for building predictable, analyzable, and reliable complex real-time systems. There remain some open issues to be explored, such as relaxing some current restrictions, discovering more properties and theorems of LSTF under different models, etc. We strongly believe that LSTF can be a practical scheduling algorithm in the near future

Project scheduling under undertainty – survey and research potentials.

The vast majority of the research efforts in project scheduling assume complete information about the scheduling problem to be solved and a static deterministic environment within which the pre-computed baseline schedule will be executed. However, in the real world, project activities are subject to considerable uncertainty, that is gradually resolved during project execution. In this survey we review the fundamental approaches for scheduling under uncertainty: reactive scheduling, stochastic project scheduling, stochastic GERT network scheduling, fuzzy project scheduling, robust (proactive) scheduling and sensitivity analysis. We discuss the potentials of these approaches for scheduling projects under uncertainty.Management; Project management; Robustness; Scheduling; Stability;

Algorithms for Scheduling Problems

This edited book presents new results in the area of algorithm development for different types of scheduling problems. In eleven chapters, algorithms for single machine problems, flow-shop and job-shop scheduling problems (including their hybrid (flexible) variants), the resource-constrained project scheduling problem, scheduling problems in complex manufacturing systems and supply chains, and workflow scheduling problems are given. The chapters address such subjects as insertion heuristics for energy-efficient scheduling, the re-scheduling of train traffic in real time, control algorithms for short-term scheduling in manufacturing systems, bi-objective optimization of tortilla production, scheduling problems with uncertain (interval) processing times, workflow scheduling for digital signal processor (DSP) clusters, and many more

Ordonnancement de camions dans une plateforme logistique : complexité, méthodes de résolution et incertitudes

La problématique dite de crossdocking a été source de beaucoup d'attention ces dernières années dans la littérature. Un crossdock est une plateforme logistique favorisant, par une synchronisation efficace des camions entrants et sortants, une rotation rapide des produits, le volume de produits stockés devant être le plus faible possible. Le crossdocking soulève de nombreux problèmes logistiques, dont notamment celui de l'ordonnancement des camions entrants et sortants sur les quais de la plateforme. L'objectif classiquement considéré dans la littérature pour ce problème est la minimisation du makespan, critère très répandu en d'ordonnancement. Pour le crossdocking néanmoins, minimiser la date de départ du dernier camion ne garantie pas nécessairement une bonne synchronisation des camions et le makespan ne semble donc pas être l'objectif le plus pertinent. Pour répondre au besoin de synchronisation et favoriser les rotations rapides, notre travail propose alternativement de minimiser la somme des temps de séjour des palettes dans le stock. Nous étudions d'abord la version déterministe de ce problème d'ordonnancement. Sa complexité est détaillée selon différentes hypothèses pour identifier les éléments menant à sa NP-difficulté. Différentes méthodes de résolutions sont proposées. Une méthode classique de programmation linéaire en nombres entiers utilisant des variables de décision indexées par le temps. Une famille d'inégalités valides est également proposée et exploitée dans un algorithme avec ajout itératif de coupes. Des méthodes basées sur la programmation par contraintes sont enfin proposées. Une analyse comparative de ces différentes méthodes est proposée. Dans un deuxième temps, nous étudions une version non-déterministe de notre problème d'ordonnancement dans laquelle des incertitudes sur les dates d'arrivée des camions sont introduites sous la forme d'intervalles de temps équiprobables. Une méthode d'ordonnancement proactive-réactive utilisant le concept de groupes d'opérations permutables est proposée pour faire face aux incertitudes. Des groupes de camions permutables sont séquencés et affectés aux quais puis, durant l'exécution d'ordonnancement, en fonction de la réalisation des dates d'arrivée, un ordre est choisi dans chaque groupe à l'aide d'un algorithme réactif.Crossdocking has received a lot of attention in the literature in recent years. A crossdock is a logistic platform that promotes rapid product turnover through efficient synchronization of incoming and outgoing trucks, with the volume of products stored being kept as low as possible. Crossdocking raises many logistical problems, including the scheduling of incoming and outgoing trucks on the platform's docks. The classical objective considered in the literature for this problem is the minimization of the makespan, a very common criterion in scheduling. However, for crossdocking, minimizing the departure date of the last truck does not necessarily guarantee a good synchronization of the trucks and the makespan does not seem to be the most relevant objective. In order to meet the need for synchronization and to help fast rotations, our work proposes alternatively to minimize the sum of the pallets' sojourn times in the warehouse. We first study the deterministic version of this scheduling problem. Its complexity is detailed under different assumptions to identify the elements leading to its NP-hardness. Different solution methods are proposed. A classical integer linear programming method using time-indexed decision variables. A family of valid inequalities is also proposed and exploited in an algorithm with iterative addition of cuts. Finally, methods based on constraint programming are proposed. A comparative analysis of these different methods is proposed. In a second step, we study a non-deterministic version of our scheduling problem in which uncertainties on truck arrival dates are introduced in the form of equiprobable time intervals. A proactive-reactive scheduling method using the concept of permutable operation groups is proposed to cope with the uncertainties. Groups of permutable trucks are sequenced and assigned to the docks and then, during the scheduling run, based on the realization of arrival dates, an order is chosen in each group using a reactive algorithm

Datacenter Traffic Control: Understanding Techniques and Trade-offs

Datacenters provide cost-effective and flexible access to scalable compute and storage resources necessary for today's cloud computing needs. A typical datacenter is made up of thousands of servers connected with a large network and usually managed by one operator. To provide quality access to the variety of applications and services hosted on datacenters and maximize performance, it deems necessary to use datacenter networks effectively and efficiently. Datacenter traffic is often a mix of several classes with different priorities and requirements. This includes user-generated interactive traffic, traffic with deadlines, and long-running traffic. To this end, custom transport protocols and traffic management techniques have been developed to improve datacenter network performance. In this tutorial paper, we review the general architecture of datacenter networks, various topologies proposed for them, their traffic properties, general traffic control challenges in datacenters and general traffic control objectives. The purpose of this paper is to bring out the important characteristics of traffic control in datacenters and not to survey all existing solutions (as it is virtually impossible due to massive body of existing research). We hope to provide readers with a wide range of options and factors while considering a variety of traffic control mechanisms. We discuss various characteristics of datacenter traffic control including management schemes, transmission control, traffic shaping, prioritization, load balancing, multipathing, and traffic scheduling. Next, we point to several open challenges as well as new and interesting networking paradigms. At the end of this paper, we briefly review inter-datacenter networks that connect geographically dispersed datacenters which have been receiving increasing attention recently and pose interesting and novel research problems.Comment: Accepted for Publication in IEEE Communications Surveys and Tutorial

The Lazy Bureaucrat Scheduling Problem

We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine'') who performs the tasks. A typical worker's objective is to minimize the amount of work he does (he is ``lazy''), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse'' scheduling problems, which we denote ``Lazy Bureaucrat Problems,'' gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.Comment: 19 pages, 2 figures, Latex. To appear, Information and Computatio