Approximation algorithms for scheduling unrelated parallel machines

We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep _ij . The objective is to find a schedule that minimizes the makespan.Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints.In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.A preliminary version of this paper appeared in theProceedings of the 28th Annual IEEE Symposium on the Foundations of Computer Science (Computer Society Press of the IEEE, Washington, D.C., 1987) pp. 217–224.     

Approximation schemes for parallel machine scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to m parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which the unavailability is due to a priori assignment of some of the n jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that m is a constant, and that the jobs are non-resumable.For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even if w_i=p_i for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless P=NP.For the fixed job model, it has been shown that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless P=NP. In this paper, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.     

A novel parallel quantum genetic algorithm for stochastic job shop scheduling

In this paper, a Novel Parallel Quantum Genetic Algorithm (NPQGA) is proposed for the stochastic Job Shop Scheduling Problem with the objective of minimizing the expected value of makespan, where the processing times are subjected to independent normal distributions. Based on the parallel evolutionary idea and some concepts of quantum theory, we simulate a model of parallel quantum computation. In this frame, there are some demes (sub-populations) and some universes (groups of populations), which are structured in super star-shaped topologies. A new migration scheme based on penetration theory is developed to control migration rate and direction adaptively between demes, and a novel quantum crossover strategy is devised among universes. The quantum evolution is executed in every deme by applying some improvement operators (the coding mechanism aiming at job shop, the new quantum rotation angle and the catastrophe operator). Experiment results show NPQGA's effectiveness and applicability.     

Optimal policies in stochastic shop scheduling

In this paper a survey is presented of some of the recent results in stochastic open shop, flow shop and job shop scheduling. The distributions of the processing times of the jobs are known in advance, but the actual processing times are not known in advance. The jobs may have due dates. Optimal preemptive and nonpreemptive policies are determined for the minimization of various objective functions, such as the expected makespan, the expected flow time and the expected number of late jobs. The effect of various degrees of dependence between the processing times of any given job on the various machines is investigated. Under given conditions bounds are obtained for the expected makespan in the different models.Partially supported by the National Science Foundation (NSF), under grant ECS-8115344 with the Georgia Institute of Technology.     

Multi-resource allocation in stochastic project scheduling

We propose a resource allocation model for project scheduling. Our model accommodates multiple resources and decision-dependent activity durations inspired by microeconomic theory. First, we elaborate a deterministic problem formulation. In a second stage, we enhance this model to account for uncertain problem parameters. Assuming that the first and second moments of these parameters are known, the stochastic model minimises an approximation of the value-at-risk of the project makespan. As a salient feature, our approach employs a scenario-free formulation which is based on normal approximations of the activity path durations. We extend our model to situations in which the moments of the random parameters are ambiguous and describe an iterative solution procedure. Extensive numerical results are provided.     

Approximation of the supply scheduling problem

The supply scheduling problem consists in finding a minimum cost delivery plan from a set of providers to a manufacturing unit, subject to given bounds on the shipment sizes and subject to the demand at the manufacturing unit. We provide a fully polynomial time approximation scheme for this problem.     

Approximation algorithms for problems in scheduling with set-ups

In this paper, we present new approximation results for the offline problem of single machine scheduling with sequence-independent set-ups and item availability, where the jobs to be scheduled are independent (i.e., have no precedence constraints) and have a common release time.We present polynomial-time approximation algorithms for two versions of this problem. In the first version, the input includes a weight for each job, and the goal is to minimize the total weighted completion time. On any input, our algorithm produces a schedule whose total weighted completion time is within a factor 2 of optimal for that input.In the second version, the input includes a due date for each job, and the goal is to minimize the maximum lateness of any job. On any input, our algorithm produces a schedule with the following performance guarantee: the maximum lateness of a job is at most the maximum lateness of the optimal schedule on a machine that runs at half the speed of our machine.     

Assignment and scheduling in parallel matrix factorization

We consider the problem of factoring a dense n×n matrix on a network consisting of P MIMD processors, with no shared memory, when the network is smaller than the number of elements in the matrix (P<n²). The specific example analyzed is a computational network that arises in computing the LU, QR, or Cholesky factorizations. We prove that if the nodes of the network are evenly distributed among processors and if computations are scheduled by a round-robin or a least-recently-executed scheduling algorithm, then optimal order of speedup is achieved. However, such speedup is not necessarily achieved for other scheduling algorithms or if the computation for the nodes is inappropriately split across processors, and we give examples of these phenomena. Lower bounds on execution time for the algorithm are established for two important node-assignment strategies.     

Approximation of eigencharacteristics in nearly-completely decomposable stochastic systems

We analyse the accuracy with which the eigensystem of a nearly-completely decomposable stochastic matrix may be expressed as a function of the eigensystems of its principal submatrices and of the maximum degree of coupling between these submatrices.     

Stochastic programming approaches to stochastic scheduling

Practical scheduling problems typically require decisions without full information about the outcomes of those decisions. Yields, resource availability, performance, demand, costs, and revenues may all vary. Incorporating these quantities into stochastic scheduling models often produces diffculties in analysis that may be addressed in a variety of ways. In this paper, we present results based on stochastic programming approaches to the hierarchy of decisions in typical stochastic scheduling situations. Our unifying framework allows us to treat all aspects of a decision in a similar framework. We show how views from different levels enable approximations that can overcome nonconvexities and duality gaps that appear in deterministic formulations. In particular, we show that the stochastic program structure leads to a vanishing Lagrangian duality gap in stochastic integer programs as the number of scenarios increases.This author's work was supported in part by the National Science Foundation under Grants ECS 88-15101, ECS 92-16819, and SES 92-11937.This author's work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-5489 and by the UK Engineering and Physical Sciences Research Council under Grants J90855 and K17897.     

Coordination mechanisms for parallel machine scheduling

We consider coordination mechanisms for the distributed scheduling of n jobs on m parallel machines, where each agent holding a job selects a machine to process his/her own job. Without a central authority to construct a schedule, each agent acts selfishly to minimize his/her own disutility, which is either the completion time of the job or the congestion time (defined as the load of the machine on which the job is scheduled). However, the overall system performance is measured by a central objective which is quite different from the agents’ objective. In the literature, makespan is often considered as the central objective. We, however, investigate problems with other central objectives that minimize the total congestion time, the total completion time, the maximum tardiness, the total tardiness, and the number of tardy jobs. The performance deterioration of the central objective by a lack of central coordination, referred to as the price of anarchy, is typically measured by the maximum ratio of the objective function value of a Nash equilibrium schedule versus that of an optimal, coordinated schedule. In this paper we give bounds for the price of anarchy for the above objectives. For problems with due date related objectives, the price of anarchy may not be defined since the optimal value may be zero. In this case, we consider the maximum difference between the objective function value of an equilibrium schedule and the optimal value. We refer to this metric as the absolute price of anarchy and analyze its lower and upper bounds.     

Deliberation scheduling using GSMDPs in stochastic asynchronous domains

We propose a new decision-theoretic approach for solving execution-time deliberation scheduling problems using recent advances in Generalized Semi-Markov Decision Processes (GSMDPs). In particular, we use GSMDPs to more accurately model domains in which planning and execution occur concurrently, plan improvement actions have uncertain effects and duration, and events (such as threats) occur asynchronously and stochastically. In this way, agents develop a continuous-time deliberation policy offline which can then be consulted to dynamically select deliberation-level and domain-level actions at plan execution-time. We demonstrate a significant improvement in expressibility over previous discrete-time approximate models in which mission phase duration was fixed, failure events were synchronized with phase transitions, and planning time was discretized into constant-sized planning quanta.     

Idle regulation in non-clairvoyant scheduling of parallel jobs

The optimization of parallel applications is difficult to achieve by classical optimization techniques because of their diversity and the variety of actual parallel and distributed platforms and/or environments. Adaptive algorithmic schemes, capable of dynamically changing the allocation of jobs during the execution to optimize global system behavior, are the best alternatives for solving this problem. In this paper, we focus on non-clairvoyant scheduling of parallel jobs with known resource requirements but unknown running times, with emphasis on the regulation of idle periods in the context of general list policies. We consider a new family of scheduling strategies based on two phases which successively combine sequential and parallel execution of jobs. We generalize known worst-case performance bounds by considering two extra parameters, in addition to the number of processors and maximum processor requirements considered in the literature, namely, job parallelization penalty and idle regulation factor. Furthermore, we prove that under certain conditions of idle regulation, the performance guarantee of parallel job scheduling in space-sharing mode can be improved.     

Minimizing earliness and tardiness costs in stochastic scheduling

We address the single-machine stochastic scheduling problem with an objective of minimizing total expected earliness and tardiness costs, assuming that processing times follow normal distributions and due dates are decisions. We develop a branch and bound algorithm to find optimal solutions to this problem and report the results of computational experiments. We also test some heuristic procedures and find that surprisingly good performance can be achieved by a list schedule followed by an adjacent pairwise interchange procedure.     

Batch scheduling on parallel units in acrylic-glass production

The cast-plate-method of manufacturing acrylic-glass gives raise to a batch scheduling problem on parallel processing units. This paper introduces the real world problem as well as an appropriate mathematical programming formulation. We describe a heuristical approach for gaining production sequences that employ the parallel processing units to capacity.     

A stochastic optimization approach for robot scheduling

In this paper we are concerned with a stochastic optimization approach for determining the optimal job-sequencing in a robot-handler production system, such that the best value of some performance indices which depend on control parameters are obtained. The idea is to reflect the possible control policies of the system in its Stochastic Petri Net model (SPN) and to select a suitable conflict resolution rule whenever the transitions representing the possible actions of the robot are enabled. This rule would depend on a vectorx ⁿ of control parameters, and the problem results in finding the values of those parameters which would be in some sense optimal for the system.The objective function is defined as a linear combination of several performance indices that are estimated simultaneously.We propose a combined simulation and optimization approach aimed at solving the conflict situations arising in the system due to simultaneous requests of the robot from jobs in different queues; then we establish a stochastic optimization approach for deriving control policies that govern the flow in the SPN model.The theoretical optimization criteria are presented along with a case study.