The multi-item capacitated lot-sizing problem with setup times and shortage costs

International audienceWe address a multi-item capacitated lot-sizing problem with setup times and shortage costs that arises in real-world production planning problems. Demand cannot be backlogged, but can be totally or partially lost. The problem is NP-hard. A mixed integer mathematical formulation is presented. Our approach in this paper is to propose some classes of valid inequalities based on a generalization of Miller et al. [A.J. Miller, G.L. Nemhauser, M.W.P. Savelsbergh, On the polyhedral structure of a multi-item production planning model with setup times, Mathematical Programming 94 (2003) 375–405] and Marchand and Wolsey [H. Marchand, L.A. Wolsey, The 0–1 knapsack problem with a single continuous variable, Mathematical Programming 85 (1999) 15–33] results. We also describe fast combinatorial separation algorithms for these new inequalities. We use them in a branch-and-cut framework to solve the problem. Some experimental results showing the effectiveness of the approach are reported

Mixed integer formulations using natural variables for single machine scheduling around a common due date

34 pages, 10 figuresWhile almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods

Lagrangean based lower bounds for a multi-plant lot-sizing problem with capacity constraints

2pInternational audienceThe paper addresses a multi-item, multi-plant lot-sizing problem with capacity restrictions. A set of facilities (plants) is available for producing some items. For each period of a discrete planning horizon, a demand is de ned for each pair of item and plant. The problem consists in producing all the demands such that the total production, inventory, setup and transfer costs is minimized. Setup production times are considered as well as capacity constraints on the production. Moreover, transfers between plants are allowed, however, the total transferred quantity between each pair of plants is upper bounded as well as the total inventory at each plant for a given period. The problem considered is NP-hard. We quote the work of Sambivasan and Yahya that describes some Lagrangean-based heuristics to solve a relaxed version of the problem where no transfer and storage capacities are considered. In the present work, we propose a Lagrangean lower bound on the optimal cost value of the problem based on the decomposition of the problem into Facility Location and Multi-Commodity Flow problems

Dominance inequalities for scheduling around an unrestrictive common due date

The problem considered in this work consists in scheduling a set of tasks on a single machine, around an unrestrictive common due date to minimize the weighted sum of earliness and tardiness. This problem can be formulated as a compact mixed integer program (MIP). In this article, we focus on neighborhood-based dominance properties, where the neighborhood is associated to insert and swap operations. We derive from these properties a local search procedure providing a very good heuristic solution. The main contribution of this work stands in an exact solving context: we derive constraints eliminating the non locally optimal solutions with respect to the insert and swap operations. We propose linear inequalities translating these constraints to strengthen the MIP compact formulation. These inequalities, called dominance inequalities, are different from standard reinforcement inequalities. We provide a numerical analysis which shows that adding these inequalities significantly reduces the computation time required for solving the scheduling problem using a standard solver.Comment: 30 pages, 7 figures and 4 table

Scheduling Independent Tasks on Multi-cores with GPU Accelerators

Best PaperInternational audienceMore and more computers use hybrid architectures combin-ing multi-core processors and hardware accelerators like GPUs (Graphics Processing Units). We present in this paper a new method for scheduling efficiently parallel applications with $m$ CPUs and $k$ GPUs, where each task of the application can be processed either on a core (CPU) or on a GPU. The objective is to minimize the makespan. The corresponding scheduling problem is NP-hard, we propose an efficient approximation algorithm which achieves an approximation ratio of $\frac{4}{3} + \frac{1}{3k}$ . We first detail and analyze the method, based on a dual approximation scheme, that uses a dynamic programming scheme to balance evenly the load between the heterogeneous resources. Finally, we run some simulations based on realistic benchmarks and compare the solution obtained by a relaxed version of this method to the one provided by a classical greedy algorithm and to lower bounds on the value of the optimal makespan

Scheduling independent tasks on multi-cores with GPU accelerators

International audienceMore and more computers use hybrid architectures combining multi-core processors and hardware accelerators like GPUs (Graphics Process-ing Units). We present in this paper a new method for scheduling efficiently parallel applications with m CPUs and k GPUs, where each task of the appli-cation can be processed either on a core (CPU) or on a GPU. The objective is to minimize the maximum completion time (makespan). The corresponding scheduling problem is NP-hard, we propose an efficient approximation algo-rithm which achieves an approximation ratio of 4 3 + 1 3k . We first detail and analyze the method, based on a dual approximation scheme, that uses dynamic programming to balance evenly the load between the heterogeneous resources. Then, we present a faster approximation algorithm for a special case of the previous problem, where all the tasks are accelerated when affected to GPU, with a performance guarantee of 3 2 for any number of GPUs. We run some simulations based on realistic benchmarks and compare the solutions obtained by a relaxed version of the generic method to the one provided by a classical scheduling algorithm (HEFT). Finally, we present an implementation of the 4/3-approximation and its relaxed version on a classical linear algebra kernel into the scheduler of the xKaapi runtime system

A study of scheduling problems with preemptions on multi-core computers with GPU accelerators

International audienceFor many years, scheduling problems have been concerned either with parallel processor systems or with dedicated processors-job shop type systems. With a development of new computing architectures this partition is no longer so obvious. Multi-core (processor) computers equipped with GPU co-processors require new scheduling strategies. This paper is devoted to a characterization of this new type of scheduling problems. After a thorough introduction of the new model of a computing system, an extension of the classical notation of scheduling problems is proposed. A special attention is paid to preemptions, since this feature of the new architecture differs the most as compared with the classical model. In the paper, several scheduling algorithms, new ones and those refining classical approaches, are presented. Possible extensions of the model are also discussed

Décomposition d'un Problème de Lot-Sizing Multi-site en Problèmes de Localisation et de Multi-flots

2pNational audienceNous présentons une heuristique de résolution d'un problème difficile de Lot-Sizing à base de relaxation lagrangienne. Les modèles de Lot-Sizing concernent la planification de la production qui exploite les effets de regroupements de tâches en lots. Nous considérons ici, un ensemble de catégories de produits, un ensemble de sites, et un ensemble de périodes. Un site peut être simultanément producteur et demandeur et servir aussi de site de stockage ou encore de transporteur. Pour chaque période, chaque produit et chaque site, nous connaissons la demande. Il s'agit de définir les quantités à stocker et à transférer pour l'ensemble des sites et des demandes sur l'horizon de planification tout en minimisant les coûts de stockage, de transfert et de production et en respectant des contraintes de capacité. Les variables de décision sont les quantités à produire, à stocker, à transporter. Ces quantités sont soumises à des contraintes de capacité

Scheduling Independent Moldable Tasks on Multi-Cores with GPUs

The number of parallel systems using accelerators is growing up.The technology is now mature enough to allow sustainedpetaflop/s. However, reaching this performance scale requiresefficient scheduling algorithms to manage the heterogeneouscomputing resources.We present a new approach for scheduling independent tasks onmultiple CPUs and multiple GPUs. The tasks are assumed to beparallelizable on CPUs using the moldable model: the final numberof cores allotted to a task can be decided and set by thescheduler. More precisely, we design an algorithm aiming atminimizing the makespan---the maximum completion time of alltasks---for this scheduling problem. The proposed algorithmcombines a dual approximation scheme with a fast integer linearprogram (ILP). It determines both the partitioning of the tasks,ie whether a task should be mapped to CPUs or a GPU, and thenumber of CPUs allotted to a moldable task if mapped to the CPUs.A worst case analysis shows that the algorithm has anapproximation ratio of $\frac{3}{2} + \epsilon$. However, sincethe complexity of the ILP-based algorithm could benon-polynomial, we also present a proved polynomial-timealgorithm with an approximation ratio of $2+\epsilon$.We complement the theoretical analysis of our two novelalgorithms with an experimental study. In these experiments, wecompare our algorithms to a modified version of the classical\heft algorithm, adapted to handle moldable tasks. Theexperimental results show that our algorithm with the$\frac{3}{2} + \epsilon$ approximation ratio producessignificantly shorter schedules than the modified \heft for mostof the instances. In addition, the experiments provide evidencethat this ILP-based algorithm is also practically able to solvelarger problem instances in a reasonable amount of time