A polynomial algorithm for some preemptive multiprocessor task scheduling problems

In this paper we consider a problem of preemptive scheduling of multiprocessor tasks on dedicated processors in order to minimize the sum of completion times. Using a standard notation, our problem can be denoted as P ∣ fix_j, pmtn ∣ ∑C_j. We give a polynomial-time algorithm to solve P ∣ fix_j, G = {P₄, dart}-free, pmtn ∣ ∑C_j problem. This result generalizes the following problems: P2 ∣ fix_j, pmtn ∣ ∑C_j, P ∣ ∣fix_j∣ ∈ {1, m}, pmtn ∣ ∑C_j and P4 ∣ fix_j = 2, pmtn ∣ ∑C_j.     

Optimal preemptive scheduling on a fixed number of identical parallel machines

In this paper, we consider the preemptive scheduling problem on a fixed number of identical parallel machines. We present a polynomial-time algorithm for finding a minimal length schedule for an order class which contains properly interval orders.     

Nonpreemptive open shop with restricted processing times

A polynomial time algorithm was given by Fiala for the nonpreemptivem-processor open shop problem whenever the sum of processing times for one processor is large enough with respect to the maximal processing time. Here a special case where all processing times are from a bounded cardinality set of nonnegative integers is studied. For such a situation we give anO(nm) algorithm while the algorithm of Fiala works inO(n ² m ³) wheren is the number of jobs.     

Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

This paper describes a complex scheduling problem taken from a hospital diagnostic testing center that schedules hundreds of patients in an open shop environment consisting of multiple facilities and multiple processors. This scheduling problem, known as the multiprocessor open shop (MPOS) problem, is strongly NP-hard with few published results. Realizing that in many MPOS environments processing times are stage-dependent, not both job and stage-dependent, this paper examines a new class of problems for the MPOS—proportionate ones. This paper exploits the structural nature of the proportionate MPOS and defines new terms. Despite the enormous complexity of the MPOS problem, this work demonstrates that polynomial time algorithms exist for two special cases. Since other applications of this problem exist in service and manufacturing environments, solving the proportionate MPOS problem is not only significant in the theory of optimization, but also in many real-world applications.     

On the optimality of static policy in stochastic open shop

Consider n jobs and two machines. Each job has to be processed on both machines. The order in which it is dome is immaterial. However, the decision maker has to decide in advance which jobs will be processes first on machine 1 (2). We assume that processing times on each machine are identically exponentially distributed random variables. We prove that assigning equal number of jobs to be first processed by machine 1 (2) stochastically minimizes the makespan.     

Minimizing makespan in a two-stage hybrid flow shop scheduling problem with open shop in one stage

This paper considers a scheduling problem in two-stage hybrid flow shop, where the first stage consists of two machines formed an open shop and the other stage has only one machine. The objective is to minimize the makespan, i.e., the maximum completion time of all jobs. We first show the problem is NP-hard in the strong sense, then we present two heuristics to solve the problem. Computational experiments show that the combined algorithm of the two heuristics performs well on randomly generated problem instances.     

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.     

A hybrid imperialist competitive algorithm for minimizing makespan in a multi-processor open shop

In this paper a new mixed-integer linear programming (MILP) model is proposed for the multi-processor open shop scheduling (MPOS) problems to minimize the makespan with considering independent setup time and sequence dependent removal time. A hybrid imperialist competitive algorithm (ICA) with genetic algorithm (GA) is presented to solve this problem. The parameters of the proposed algorithm are tuned by response surface methodology (RSM). The performance of the algorithm to solve small, medium and large sized instances of the problem is evaluated by introducing two performance metrics. The quality of obtained solutions is compared with that of the optimal solutions for small sized instances and with the lower bounds for medium sized instances. Also some computational results are presented for large sized instances.     

可中断的多任务平行机排序问题

Leung等(Preemptive multiprocessor order scheduling to minimize total weighted flowtime [J]. European Journal of Operational Research, 2008, 190: 40-51)研究了如下问题: 有 n 个订单, 其中每个订单 i 含有 n_i 个不同的工件. 所有的订单在零时刻已经到达, 并且工件的加工是可中断的. 每个订单 i有一个权重 \omega_i, 定义订单 i 的完工时间 C_i 为订单 i 最后一个完工工件的完工时间. 目标是找到一个可行排序使得加权总完工时间\sum\limits_{i=1}^n \omega_iC_i 最小. Leung等证明了这个问题是NP-难的, 给出了一个近似算法, 并且分析了该算法的最坏情况界. 但是定理2的证明存在一些错误. 证明了尽管定理2的证明过程存在错误, 但是其结论仍然正确. 另外, 对上述模型的一种特殊情形给出了更好的近似算法.     

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.     

Preemptive scheduling and antichain polyhedra

We present a theoretical framework, which is based upon notions of ordered hypergraphs and antichain polyhedra, and which is dedicated to the combinatorial analysis of preemptive scheduling problems submitted to parallelization constraints.This framework allows us to characterize specific partially ordered structures which are such that induced preemptive scheduling problems may be solved through linear programming. To prove that, in the general case, optimal preemptive schedules may be searched inside some connected subset of the vertex set of an Antichain Polyhedron.     

A tabu search approach for proportionate multiprocessor open shop scheduling

In the multiprocessor open shop scheduling problem, jobs are to be processed on a set of processing centers—each having one or more parallel identical machines, while jobs do not have a pre-specified obligatory route. A special case is the proportionate multiprocessor open shop scheduling problem (PMOSP) in which the processing time on a given center is not job-dependent. Applications of the PMOSP are evident in health care systems, maintenance and repair shops, and quality auditing and final inspection operations in industry. In this paper, a tabu search (TS) approach is presented for solving the PMOSP with the objective of minimizing the makespan. The TS approach utilizes a neighborhood search function that is defined over a network representation of feasible solutions. A set of 100 benchmark problems from the literature is used to evaluate the performance of the developed approach. Experimentations show that the developed approach outperforms a previously developed genetic algorithm as it produces solutions with an average of less than 5 % deviation from a lower bound, and 40 % of its solutions are provably optimal.     

An optimal online algorithm for two-machine open shop preemptive scheduling with bounded processing times

This paper deals with a two-machine open shop scheduling problem. The objective is to minimize the makespan. Jobs arrive over time. We study preemption-resume model, i.e., the currently processed job may be preempted at any moment in necessary and be resumed some time later. Let p _{1, j} and p _{2, j} denote the processing time of a job J _j on the two machines M ₁ and M ₂, respectively. Bounded processing times mean that 1 ≤ p _{i, j}  ≤ α (i = 1, 2) for each job J _j , where α ≥ 1 is a constant number. We propose an optimal online algorithm with a competitive ratio ${\frac{5\alpha-1}{4\alpha}}$ .     

A linear time approximation algorithm for multiprocessor scheduling

Givenn jobs andm identical processors anO(n) approximation algorithm is presented which tries to determine a nonpreemptive schedule with minimum finish time. Ifr is the number of jobs placed onto the processor with maximum finish time, then the worst case ratio of the new algorithm's finish time to the optimal solution is shown to be less thanrm/(rm–m+1). Extensive empirical results show that the new algorithm is competitive with the LPT algorithm in terms of quality of solution and faster in terms of computing time.     

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

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, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

Open shop unit-time scheduling problems with symmetric objective functions

The paper addresses the open-shop scheduling problem with unit-time operations and nondecreasing symmetric objective function depending on job completion times. We construct two schedules, one being optimal for any symmetric convex function, the other one for any symmetric concave function. Both schedules are given by analytically defined formulas that determine in O(1) time for each operation the unit-length time slot for its processing.Received: June 2004, Revised: January 2005, AMS classification: 90B35, 68Q25     

A graph coloring approach to scheduling of multiprocessor tasks on dedicated machines with availability constraints

We address a generalization of the classical 1- and 2-processor unit execution time scheduling problem on dedicated machines. In our chromatic model of scheduling machines have non-simultaneous availability times and tasks have arbitrary release times and due dates. Also, the versatility of our approach makes it possible to generalize all known classical criteria of optimality. Under these stipulations we show that the problem of optimal scheduling of sparse tree-like instances can be solved in polynomial time. However, if we admit dense instances then the problem becomes NP-hard, even if there are only two machines.     

Open shop scheduling with some additional constraints

An open shop scheduling problem is presented; preemptions during processing of a job on a processorp is allowed but the job cannot be sent on another processorq before it is finished onp. A graph-theoretical model is described and a characterization is given for problems where schedules with such restricted preemptions useT time units whereT is the maximum of the processing times of the jobs and of the working times of the processors. The general case is shown to be NP-complete. We also consider the case where some constraints of simultaneity are present. Complexity of the problem is discussed and a solvable case is described.     

一种求解延迟工件数最小的混合流水车间调度问题的模拟退火算法

针对延迟工件数最小的混合流水车间调度问题,给出了一种改进的模拟退火求解算法. 该算法首先给出一个启发式算法来获得初始解,然后用模拟退火算法对初始解改进. 通过交换工件在第一阶段的排序来获得一个新的解,采用最先空闲设备分配规则和先到先被加工规则,对工件在剩余各级的工序进行调度. 实验仿真表明算法是可行有效的.