Stochastically Minimizing Total Delay of Jobs Subject to Random Deadlines

This paper analyzes a scheduling system where a fixed number of nonpreemptive jobs is to be processed on multiple parallel processors with different processing speeds. Each processor has an exponential processing time distribution and the processors are ordered in ascending order of their mean processing times. Each job has its own deadline that is exponentially distributed with rate ß1, independent of the deadlines of other jobs and also independent of job processing times. A job departs the system as soon as either its processing completes or its deadline occurs. We show that there exists a simple threshold strategy that slochastically minimizes the total delay of all jobs. The policy depends on distributions of processing times and deadlines, but is independent of the rate of deadlines. When the rate of the deadline distribution is 0 (no deadlines), the total delay reduces to the flowtime (the sum of completion times of all jobs). If each job has its own probability of being correctly processed, then an extension of this policy stochastically maximizes the total number of correctly processed, nontardy jobs. We discuss possible generalizations and limitations of this result.

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Scheduling a proportionate flow shop of batching machines

Journal of Scheduling ◽

10.1007/s10951-020-00667-2 ◽

2020 ◽

Vol 23 (5) ◽

pp. 575-593

Author(s):

Christoph Hertrich ◽

Christian Weiß ◽

Heiner Ackermann ◽

Sandy Heydrich ◽

Sven O. Krumke

Keyword(s):

Polynomial Time ◽

Flow Shop ◽

Fixed Number ◽

Release Dates ◽

Proportionate Flow Shop ◽

Weighted Number ◽

Late Jobs ◽

Two Machines ◽

Completion Times ◽

Sum Of Completion Times

Abstract In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number $$m \ge 2$$ m ≥ 2 of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines, the makespan and the sum of completion times can be minimized in polynomial time. Furthermore, we show that the obtained algorithm can also be used to minimize the weighted total completion time, maximum lateness, total tardiness and (weighted) number of late jobs in polynomial time if all release dates are 0. Previously, polynomial time algorithms have only been known for two machines.

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A Note on Flow Shop Scheduling with the Effects of Learning and Deterioration

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.513-517.2149 ◽

2014 ◽

Vol 513-517 ◽

pp. 2149-2152

Author(s):

Yu Ping Niu ◽

Ji Bo Wang

Keyword(s):

Learning Effect ◽

Flow Shop ◽

Flow Shop Scheduling ◽

Scheduling Problems ◽

Shop Scheduling ◽

Processing Times ◽

Polynomially Solvable ◽

Machine Scheduling Problems ◽

Completion Times ◽

Sum Of Completion Times

In this note, we consider the machine scheduling problems with the effects of learning and deterioration. In this model, job processing times are defined by functions dependent on their starting times and positions in the sequence. The scheduling objectives are makespan, sum of completion times. It is shown that even with the introduction of learning effect and deterioration jobs to job processing times, several flow shop problems remain polynomially solvable.

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Minimizing sum of completion times for batch scheduling of jobs with deteriorating processing times

European Journal of Operational Research ◽

10.1016/j.ejor.2006.03.067 ◽

2008 ◽

Vol 187 (3) ◽

pp. 1090-1099 ◽

Cited By ~ 29

Author(s):

Joseph Y.-T. Leung ◽

C.T. Ng ◽

T.C. Edwin Cheng

Keyword(s):

Batch Scheduling ◽

Processing Times ◽

Completion Times ◽

Sum Of Completion Times

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Stochastic flowshop no-wait scheduling

Journal of Applied Probability ◽

10.2307/3213765 ◽

1985 ◽

Vol 22 (1) ◽

pp. 240-246 ◽

Cited By ~ 1

Author(s):

E. Frostig ◽

I. Adiri

Keyword(s):

Independent Random Variable ◽

Processing Time ◽

Random Variables ◽

Random Variable ◽

Schedule Length ◽

Special Cases ◽

No Wait ◽

Completion Times ◽

Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti, (i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

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Single-Machine Group Scheduling Problems with Deterioration to Minimize the Sum of Completion Times

Mathematical Problems in Engineering ◽

10.1155/2012/275239 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Yong He ◽

Li Sun

Keyword(s):

Single Machine ◽

Group Scheduling ◽

Scheduling Problems ◽

Minimization Problems ◽

Processing Times ◽

Machine Group ◽

Polynomially Solvable ◽

Decreasing Functions ◽

Completion Times ◽

Sum Of Completion Times

We consider two single-machine group scheduling problems with deteriorating group setup and job processing times. That is, the job processing times and group setup times are linearly increasing (or decreasing) functions of their starting times. Jobs in each group have the same deteriorating rate. The objective of scheduling problems is to minimize the sum of completion times. We show that the sum of completion times minimization problems remains polynomially solvable under the agreeable conditions.

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Scheduling two classes of exponential jobs on parallel processors: structural results and worst-case analysis

Advances in Applied Probability ◽

10.2307/1427684 ◽

1991 ◽

Vol 23 (4) ◽

pp. 925-944 ◽

Cited By ~ 6

Author(s):

Cheng-Shang Chang ◽

Randolph Nelson ◽

Michael Pinedo

Keyword(s):

Case Analysis ◽

Parallel Processors ◽

Worst Case Analysis ◽

Weighted Sum ◽

Scheduling Problems ◽

Worst Case ◽

Total Cost ◽

Class 1 ◽

Completion Times ◽

Sum Of Completion Times

In this paper, we consider scheduling problems with m machines in parallel and two classes of job. We assume that all jobs are present at time 0 and there are no further arrivals. The service times of class 1 (2) jobs are independent and exponentially distributed with mean . Each class 1 (2) job incurs a cost c1 (c2) per unit of time until it leaves the system. The objective is to minimize the expected total cost, that is the expected weighted sum of completion times. We show that the optimal policy among all preemptive policies is of threshold type. Based on these structural results, we also show that the ratio of the expected weighted sum of completion times under the cµ-rule to that under the optimal rule is less than 1·71.

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Scheduling two classes of exponential jobs on parallel processors: structural results and worst-case analysis

Advances in Applied Probability ◽

10.1017/s0001867800024022 ◽

1991 ◽

Vol 23 (04) ◽

pp. 925-944 ◽

Cited By ~ 2

Author(s):

Cheng-Shang Chang ◽

Randolph Nelson ◽

Michael Pinedo

Keyword(s):

Case Analysis ◽

Parallel Processors ◽

Worst Case Analysis ◽

Weighted Sum ◽

Scheduling Problems ◽

Worst Case ◽

Total Cost ◽

Class 1 ◽

Completion Times ◽

Sum Of Completion Times

In this paper, we consider scheduling problems with m machines in parallel and two classes of job. We assume that all jobs are present at time 0 and there are no further arrivals. The service times of class 1 (2) jobs are independent and exponentially distributed with mean . Each class 1 (2) job incurs a cost c 1 (c 2) per unit of time until it leaves the system. The objective is to minimize the expected total cost, that is the expected weighted sum of completion times. We show that the optimal policy among all preemptive policies is of threshold type. Based on these structural results, we also show that the ratio of the expected weighted sum of completion times under the cµ-rule to that under the optimal rule is less than 1·71.

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Optimal Rules for Single Machine Scheduling with Stochastic Breakdowns

Mathematical Problems in Engineering ◽

10.1155/2014/260415 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Jinwei Gu ◽

Manzhan Gu ◽

Xingsheng Gu

Keyword(s):

Single Machine ◽

Large Scale ◽

Processing Time ◽

Single Machine Scheduling ◽

Scale Problem ◽

Processing Times ◽

Spt Rule ◽

Lpt Rule ◽

Completion Times ◽

Stochastic Breakdowns

This paper studies the problem of scheduling a set of jobs on a single machine subject to stochastic breakdowns, where jobs have to be restarted if preemptions occur because of breakdowns. The breakdown process of the machine is independent of the jobs processed on the machine. The processing times required to complete the jobs are constants if no breakdown occurs. The machine uptimes are independently and identically distributed (i.i.d.) and are subject to a uniform distribution. It is proved that theLongest Processing Time first(LPT) rule minimizes the expected makespan. For the large-scale problem, it is also showed that theShortest Processing Time first(SPT) rule is optimal to minimize the expected total completion times of all jobs.

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Stochastic flowshop no-wait scheduling

Journal of Applied Probability ◽

10.1017/s0021900200029181 ◽

1985 ◽

Vol 22 (01) ◽

pp. 240-246

Author(s):

E. Frostig ◽

I. Adiri

Keyword(s):

Independent Random Variable ◽

Processing Time ◽

Random Variables ◽

Random Variable ◽

Schedule Length ◽

Special Cases ◽

No Wait ◽

Completion Times ◽

Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti . It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti , ( i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

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The Stochastic Optimality of SEPT in Parallel Machine Scheduling

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003326 ◽

1994 ◽

Vol 8 (2) ◽

pp. 179-188 ◽

Cited By ~ 2

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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