Inequalities and bounds for the scheduling of stochastic jobs on parallel machines

1985 ◽  
Vol 22 (3) ◽  
pp. 739-744 ◽  
Author(s):  
Michael Pinedo ◽  
Zvi Schechner
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Random Variable ◽  
Expected Time ◽  
Processing Times ◽  
Set Up ◽  
Expected Makespan

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj, a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.

Inequalities and bounds for the scheduling of stochastic jobs on parallel machines

1985 ◽  
Vol 22 (03) ◽  
pp. 739-744
Author(s):  
Michael Pinedo ◽  
Zvi Schechner
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Random Variable ◽  
Expected Time ◽  
Processing Times ◽  
Set Up ◽  
Expected Makespan

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj , a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.

STOCHASTIC BATCH SCHEDULING AND THE “SMALLEST VARIANCE FIRST” RULE

2007 ◽  
Vol 21 (4) ◽  
pp. 579-595 ◽  
Author(s):  
Michael Pinedo
Keyword(s):  
Completion Time ◽  
Processing Time ◽  
Random Variable ◽  
Combinatorial Problems ◽  
Set Partitioning ◽  
Batch Scheduling ◽  
Matching Problem ◽  
Batch Mode ◽  
Processing Times ◽  
Expected Makespan

Consider a single machine that can process multiple jobs in batch mode. We havenjobs and the processing time of jobjis a random variableXjwith distributionFj. Up tobjobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions, the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time but different variances. We show that for certain special classes of processing time distributions theSmallest Variance Firstrule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.

scholarly journals Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime

1986 ◽  
Vol 23 (03) ◽  
pp. 841-847 ◽  
Author(s):  
R. R. Weber ◽  
P. Varaiya ◽  
J. Walrand
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Preemptive Scheduling ◽  
Total Reward ◽  
Processing Times ◽  
Identical Machines ◽  
Scheduling Strategies

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

Scheduling jobs with exponential processing times on parallel machines

1988 ◽  
Vol 25 (04) ◽  
pp. 752-762 ◽  
Author(s):  
Tapani Lehtonen
Keyword(s):  
Binomial Distribution ◽  
Parallel Machines ◽  
Essential Feature ◽  
Stochastic Order ◽  
Partial Solution ◽  
Flow Time ◽  
Optimal Assignment ◽  
Processing Times ◽  
Expected Makespan

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.

scholarly journals Heuristic for Stochastic Online Flowshop Problem with Preemption Penalties

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Mohammad Bayat ◽  
Mehdi Heydari ◽  
Mohammad Mahdavi Mazdeh
Keyword(s):  
Standard Deviation ◽  
Processing Time ◽  
Heuristic Method ◽  
Heuristic Procedure ◽  
Release Date ◽  
Numerical Examples ◽  
Processing Times ◽  
Flowshop Problem ◽  
Expected Makespan ◽  
Actual Processing Time

The deterministic flowshop model is one of the most widely studied problems; whereas its stochastic equivalent has remained a challenge. Furthermore, the preemptive online stochastic flowshop problem has received much less attention, and most of the previous researches have considered a nonpreemptive version. Moreover, little attention has been devoted to the problems where a certain time penalty is incurred when preemption is allowed. This paper examines the preemptive stochastic online flowshop with the objective of minimizing the expected makespan. All the jobs arrive overtime, which means that the existence and the parameters of each job are unknown until its release date. The processing time of the jobs is stochastic and actual processing time is unknown until completion of the job. A heuristic procedure for this problem is presented, which is applicable whenever the job processing times are characterized by their means and standard deviation. The performance of the proposed heuristic method is explored using some numerical examples.

Development of a Production Planning Method for the NIFOR Digester Screw Press

2013 ◽  
Vol 824 ◽  
pp. 499-504
Author(s):  
B.O. Abikoye ◽  
P.E. Amiolemhen
Keyword(s):  
Production Planning ◽  
Processing Time ◽  
Job Scheduling ◽  
Screw Press ◽  
Planning Method ◽  
Shop Floor ◽  
Planning And Scheduling ◽  
Processing Times ◽  
Set Up ◽  
Effective Use

In most production environments, the problem of efficiently scheduling production jobs on several machines is an important consideration when attempting to design a work plan that makes effective use of the available resources because job scheduling in manufacturing is at the crux of production planning because of its impact on revenues. This work examined the production planning and scheduling of jobs at the Nigerian Institute for Oil Palm Research (NIFOR) with the aim of developing an effective production planning method for the production of one of its products the Digester Screw Press (DSP). The material flow pattern and current manufacturing practice were observed and production data was collected on machine set up and run times. The data collected was used to estimate the actual average run times for all the machines and the total processing times for each of the three components that make up the Digester Screw Press. The Microsoft Project 2003 was used as a tool in producing an improved job schedule based on the shortest processing time rule. The technique has reduced the general processing time for the production of DSP from 157hours to 99hours. It provides low machine waiting time, high machine utilization and reduced job tardiness across the shop floor.

Scheduling jobs with exponential processing times on parallel machines

1988 ◽  
Vol 25 (4) ◽  
pp. 752-762 ◽  
Author(s):  
Tapani Lehtonen
Keyword(s):  
Binomial Distribution ◽  
Parallel Machines ◽  
Essential Feature ◽  
Stochastic Order ◽  
Partial Solution ◽  
Flow Time ◽  
Optimal Assignment ◽  
Processing Times ◽  
Expected Makespan

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.

Scheduling with Discretely Compressible Processing Times to Minimize Makespan

2014 ◽  
Vol 575 ◽  
pp. 926-930
Author(s):  
Shu Xia Zhang ◽  
Yu Zhong Zhang
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Processing Time ◽  
Parallel Machines ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Identical Parallel Machines ◽  
Processing Times ◽  
Release Times ◽  
Scheduling Model

In this paper, we address the scheduling model with discretely compressible processing times, where processing any job with a compressed processing time incurs a corresponding compression cost. We consider the following problem: scheduling with discretely compressible processing times to minimize makespan with the constraint of total compression cost on identical parallel machines. Jobs may have simultaneous release times. We design a pseudo-polynomial time algorithm by approach of dynamic programming and an FPTAS.

scholarly journals ANALYTICAL EVALUATION OF PROCESSING TIME OF A QUERY IN AN INFORMATION SYSTEM

2018 ◽  
pp. 69-73 ◽  
Author(s):  
M. M. Butaev ◽  
A. A. Tarasov
Keyword(s):  
Information Systems ◽  
Normal Distribution ◽  
Processing Time ◽  
Random Variables ◽  
Initial Approximation ◽  
Random Variable ◽  
Sequential Processing ◽  
Processing Times ◽  
Distribution Laws ◽  
Interval Distribution

The normal distribution of a random variable is usually used in studies of the probabilistic characteristics of information systems. However, the approximation by the normal distribution of distributions determined on a limited interval distorts the physical meaning of the model and the numerical results, and it can only be used as an initial approximation. The aim of the work is to improve the methods for calculating the probabilistic characteristics of information systems. The object of the study is an analytical method for calculating the processing time of the query in the system. The subject of the study are formulas for calculating the duration of sequential processing of the query by elements of the system with uniformly distributed random processing times. In deriving the formulas for calculating the probability characteristics of a sum of independent uniformly distributed random variables, the methods of the theory of probability and statistics are applied. It is proposed for random variables, determined only on the positive coordinate axis, to use finite-interval distribution laws, for example, beta distribution. Density formulas and probability functions for sums of two, three and four independent uniformly distributed random variables are derived.

The Stochastic Optimality of SEPT in Parallel Machine Scheduling

1994 ◽  
Vol 8 (2) ◽  
pp. 179-188 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Arie Hordijk ◽  
Rhonda Righter ◽  
Gideon Weiss
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Concave Function ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
Preemptive Scheduling ◽  
Processing Times ◽  
Schur Concave ◽  
Completion Times ◽  
Stochastic Optimality

We consider preemptive scheduling on parallel machines where processing times of jobs are i.i.d. but jobs may already have received distinct amounts of service. We show that when processing times are increasing in likelihood ratio, SEPT (shortest expected [remaining] processing time first) stochastically minimizes any increasing and Schur-concave function of the job completion times. The same result holds when processing times are exponential with possibly different means.

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