Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2014 / Article
Special Issue

Recent Advances on Modeling, Control, and Optimization for Complex Engineering Systems

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 260415 | https://doi.org/10.1155/2014/260415

Jinwei Gu, Manzhan Gu, Xingsheng Gu, "Optimal Rules for Single Machine Scheduling with Stochastic Breakdowns", Mathematical Problems in Engineering, vol. 2014, Article ID 260415, 9 pages, 2014. https://doi.org/10.1155/2014/260415

Optimal Rules for Single Machine Scheduling with Stochastic Breakdowns

Academic Editor: Kang Li
Received02 Jun 2014
Accepted04 Sep 2014
Published29 Sep 2014

Abstract

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 the Longest Processing Time first (LPT) rule minimizes the expected makespan. For the large-scale problem, it is also showed that the Shortest Processing Time first (SPT) rule is optimal to minimize the expected total completion times of all jobs.

1. Introduction

Machine scheduling problems belong to the classic combinational optimization problems. These problems deal with the model where decision maker needs to arrange jobs to process on a limited number of machines or processors. Machine scheduling problems play an important role in manufacturing, parallel computing, or compiler optimization. These problems have been studied since the 1950s and a lot of results have been achieved until now. We refer to the books by Brucker [1] and Pinedo [2] for a general overview of literature in scheduling problems.

In the environment of classical scheduling problems, the machine is assumed to be workable continuously until the completion of all jobs. Nevertheless, some unexpected events during production (e.g., equipment damaged, error operation, and instrument breakdown) often occur in manufacturing environment. Therefore, it is very common that a machine breakdown happens during the processing of a job. Moreover, the information about the breakdowns may be uncertain. In the realistic situation, the decision maker has to consider how to utilize the available information to give a more effective scheduling plan in order to increase the output and reduce the cost. In this way, it is necessary and valuable to research the stochastic scheduling problems with random machine breakdowns.

According to the impact a machine breakdown exerts to the job being processed, the machine breakdowns could be categorized into two models: preemptive-resume model and preemptive-repeat. In the preemptive-resume model, if a breakdown happens during the processing of a job, the work done prior to the breakdown is not lost, and the job could be resumed when the machine becomes available again. In the preemptive-repeat model, the job has to be reprocessed in its entirety if the machine breakdown occurs before the job is completed.

The main purpose of this paper is to study the problem with machine breakdowns of preemptive-repeat model. There are many results on preemptive-resume model. Such as Glazebrook [3], Birge et al. [4], Mittenthal and Raghavachari [5], Cai and Zhou [6, 7], and Qi et al. [8]. Regarding the preemptive-repeat model, many authors have contributed remarkable achievements. Adiri et al. [9, 10] studied the problems with single breakdown; Cai et al. [1113] studied the problems in which the realizations of processing times for a job between breakdowns are the same. They referred to this model as the case of without resampling. Frostig [14] considered the resampling mechanism in which the repeated processing times of a job are i.i.d. between breakdowns. Khalil and Dimitrov [15] studied the flow time of a job under the preemptive-repeat and preemptive-resume models. Lee and Lin [16] considered the problem where the decision maker can decide whether to activate a maintenance or to leave the machine to run at a slower speed. Kasap et al. [17] studied the uptime distributions to ensure that the LPT rule minimizes the expected makespan. Tang and Zhao [18] designed an optimal algorithm for the problem with early and tardy penalties. Lee and Yu [19] gave algorithms to the problem with the objective to minimize the expected weighted completion times and expected maximum tardiness.

However, all the papers reviewed above (except Kasap et al. [17]) carry the implicit assumption that the breakdown process of the machine is dependent on the job it is processing. With this assumption, the problem with machine breakdowns could be converted to the traditional scheduling problem without breakdowns; see papers [11, 12, 18].

In this paper, the machine breakdowns are subject to preemptive-repeat model and are independent of job it is processing. The objective is to minimize the expected makespan or expected total completion times of all jobs. For this problem, Adiri et al. [9] firstly studied a special case of a single machine scheduling with only one breakdown, and the machine is assumed to be continuous workable after the breakdown. Subject to this restriction, Adiri et al. concluded that the LPT (SPT) rule minimizes the expected makespan when the distribution function of the uptime is convex (concave). This paper considers the general problem where the downtimes (repairing time) are i.i.d., and the uptimes are independently subject to a common uniform distribution. Under the assumptions above, it is proved that LPT rule is optimal to achieve the minimal expected makespan, and SPT rule minimizes the expected total completion times for large-scale problem, where the number of jobs is large enough.

The remainder of the paper is organized as follows. In Section 2, the model with stochastic preemptive-repeat breakdowns is formulated. Then, for a given processing order, we present a formulation of the expected completion time of a job. In Section 3, we show that the LPT rule minimizes the expected makespan. Section 4 demonstrates that SPT rule minimizes the expected total completion times for large-scale problems. Finally, some concluding remarks are made in Section 5.

2. Formulation of Problem

Suppose there are jobs available at time 0 and these jobs are to be processed on a single machine. Denote by a constant the time needed to process job if no breakdown occurs during its processing. Due to the breakdowns, the actual time needed to process job may be more than , and the time may vary in different processing orders. It is assumed that the machine could process one and only one job at a time, and once a job begins to be processed on the machine, it could not be preempted by other jobs (except by machine breakdowns) until its completion.

The machine is subject to stochastic breakdowns, and, after each maintenance, the machine will start anew. The breakdown process is characterized by a sequence of positive random vectors , where are, respectively, the durations of the th uptime and downtime of the machine. The uptimes and downtimes are defined to be independent of the jobs. If the machine breaks down during the processing of job , the work done on the job will be lost and the job has to be restarted. are defined to be i.i.d. and follow the uniform distribution with the distribution function with the support in . Hence, . are also i.i.d. with an arbitrary distribution with . The objective functions in this paper are the expected makespan and the expected total completion times. Our work focuses on the scheduling order of all jobs so as to minimize the objective functions.

Define a jobs processing order , and . Given a processing order , assume the machine begins to process only the th, thth jobs at time zero; then define that denotes the time to complete the jobs. So the makespan , and the completion time of th job is . Let be an indicator variable such that if event A occurs; otherwise . Based on the notations defined above, the expected completion time of job could be expressed as which implies Therefore, we obtain the expected makespan

3. LPT Minimizes Expected Makespan

In this section, we will prove the optimality of LPT to minimize the expected value of makespan. Define the processing order   , which is obtained by interchanging the processing order of the two jobs in . The following lemma shows that the processing of first two jobs is subject to LPT rule.

Lemma 1. Consider the processing order . If the uptimes are i.i.d and uniformly distributed and , then the expected makespan can be reduced by interchanging the first two jobs ; that is,

Proof. According to the definition of and by (3), it is obtained that By replacing (5) in (3), That is, By the same method, we could get the expression of . Since we obtain The conclusion follows by .

Corollary 2. Assume the uptimes are i.i.d and uniformly distributed. If , then

Proof. The proof is the same as that in Lemma 1.

Next we will give another expression for . Assume the machine begins to process jobs at time zero and . Let ; then we consider the following three cases.

Case  1 (). With the possibility , in this case, we have Case  2 (). With the possibility , we have Case  3 (). We have From the three cases above, we get Known from Corollary 2, we know . Hence That is, The following lemma shows that the processing of any two consecutive jobs is subject to LPT rule.

Lemma 3. Assume the uptimes are i.i.d and uniformly distributed. If , then

Proof. Let be the uptime where the job is completed, and assume the machine has been continuously processing jobs for time units when is finished at time ; that is, we have . Let , and let be the distribution function of . We have which implies that is uniformly distributed for any given . According to the definition of , we know . We now consider four possibilities depending on the value obtains.
Case  1 (). We have Case  2 (). We have Case  3 (). We have Case  4 (). We have Hence, Because By replacing (24) in (23), we obtain By (16) and , the conclusion in this lemma holds.

Based on Lemmas 1 and 3, the following theorem is immediate.

Theorem 4. Assume the uptimes are i.i.d and uniformly distributed with support in , where ; then the LPT rule is optimal to minimize the expected makespan.

4. SPT Minimizes Expected Total Completion Times

This section considers the single machine problem to minimize the expected value of total completion times, that is, . Assume there exist two constants such that for ; that is, the processing times of all jobs are uniformly bounded. For a given processing order , if machine begins to process the th, thth jobs at time zero, define that denotes the sum of the completion times of the jobs. So we have In this section, we focus on the large-scale scheduling problems; that is, the number of jobs is large enough. With the assumption, the SPT rule will be proved to be optimal.

Lemma 5. Assume the uptimes are i.i.d and uniformly distributed. If , then for large number .

Proof. Firstly we have By (3), we could get the expressions for and . Hence we obtain Known from (9), we have Therefore, we have Note , so we have and Since , we obtain as long as is large enough.

By Lemma 5, the corollary below follows immediately.

Corollary 6. Assume the uptimes are i.i.d and uniformly distributed. If , then as long as is large enough. Also, we have

Proof. The proof is the same as that in Lemma 5.

In order to prove the main conclusion in this section, we will give another expression for . Assume the machine begins to process jobs at time zero and . Let , and ( for ). We consider the following three cases.

Case  1 (). In this case, we have We obtain Case  2 (). In this case, we have Therefore, we get Case  3 (). We have And we obtain Based on the three cases above, we have for large number . The inequality holds by the conclusion in Corollary 6.

Lemma 7. Assume the uptimes are i.i.d and uniformly distributed. If (), then one has for large number .

Proof. We let be the uptime where the job is completed and let be the downtime after . Assume the machine has been continuously processing for time units when is finished at time ; that is, we have . Let , and let be the distribution function of . Here, we have Case  1 (). In this case, we have Therefore, we obtain Case  2 (). The case is similar to the case . So we get Case  3 (). The case is similar to the case . We have Case  4 (). The case is similar to the case . We obtain Based on the four cases above, we have By (35), we have We discuss the following cases..
Case  1 (). Because , we have That is, So we obtain Therefore, in this case we obtain That is, Case  2 (). In this case, we define , and . Next two cases are discussed for the value obtains.
Case  2.1 (). In this case, we have That is, Known from Case 1 above, we obtain Case  2.2 (). According to the definition of , we have , so Note that there exists a number such that , so we get for all . Therefore, for all . So we obtain Either or not, we always have for large number .
Based on Lemmas 5 and 7, the following theorem is immediate.

Theorem 8. Assume the uptimes are i.i.d. and uniformly distributed with support , where ; then the SPT rule is optimal to minimize the expected value of total completion times if the number of jobs is large enough.

5. Concluding Remarks

The stochastic scheduling problem on a single machine with random breakdowns has been investigated in this paper. We consider the situation where the uptimes are uniformly distributed and i.i.d; the downtimes are also assumed to be i.i.d and follow an arbitrary distribution. The machine breakdowns are defined to be independent of the job it is processing. Under the assumptions above, we prove that the LPT rule could achieve the minimal expected makespan; the SPT rule is optimal to minimize the expected value of total completion times for large scale problems. For the scheduling with stochastic breakdowns independent of job it is processing, the result obtained in this paper is the foundation in this area.

Some problems may be considered for the future research: (a) whether optimal rule exists when the uptimes are subject to other probability distributions; (b) problems with other objective functions are worth investigation; (c) the multimachine version will also be an interesting but difficult problem in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

We are grateful to the anonymous referees for their valuable comments and suggestions. This research is supported by the Young Scientists Fund of the National Natural Science Foundation of China (61304209 and 11201282), the Fund of scheme for training young teachers in Colleges and universities in Shanghai (ZZCD12006), the Ministry of Education of Humanities and Social Science Fund Project (10YJCZH032), and Innovation Program of Shanghai Municipal Education Commission (14YZ127).

References

  1. P. Brucker, Scheduling Algorithms, Springer, 4th edition, 2004. View at: Publisher Site | MathSciNet
  2. M. Pinedo, Scheduling: Theory, Algorithms, and Systems, Springer, Heidelberg, Germany, 3rd edition, 2002. View at: MathSciNet
  3. K. D. Glazebrook, “Evaluating the effects of machine breakdowns in stochastic scheduling problems,” Naval Research Logistics, vol. 34, no. 3, pp. 319–335, 1987. View at: Google Scholar | MathSciNet
  4. J. Birge, J. B. G. Frenk, J. Mittenthal, and A. H. G. Rinnooy Kan, “Single-machine scheduling subject to stochastic breakdowns,” Naval Research Logistics, vol. 37, no. 5, pp. 661–677, 1990. View at: Google Scholar | MathSciNet
  5. J. Mittenthal and M. Raghavachari, “Stochastic single machine scheduling with quadratic early-tardy penalties,” Operations Research, vol. 41, no. 4, pp. 786–796, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. X. Cai and X. Zhou, “Asymmetric earliness and tardiness scheduling with exponential processing times on an unreliable machine,” Annals of Operations Research, vol. 98, pp. 313–331, 2000. View at: Publisher Site | Google Scholar | MathSciNet
  7. X. Cai and S. Zhou, “Stochastic scheduling on parallel machines subject to random breakdowns to minimize expected costs for earliness and tardy jobs,” Operations Research, vol. 47, no. 3, pp. 422–437, 1999. View at: Publisher Site | Google Scholar | MathSciNet
  8. X. D. Qi, G. Yin, and J. R. Birge, “Single-machine scheduling with random machine breakdowns and randomly compressible processing times,” Stochastic Analysis and Applications, vol. 18, no. 4, pp. 635–653, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. I. Adiri, J. Bruno, E. Frostig, and A. H. G. Rinnooy Kan, “Single machine flow-time scheduling with a single breakdown,” Acta Informatica, vol. 26, no. 7, pp. 679–696, 1989. View at: Publisher Site | Google Scholar | MathSciNet
  10. I. Adiri, E. Frostig, and A. H. G. Rinnooy Kan, “Scheduling on a single machine with a single breakdown to minimize stochastically the number of tardy jobs,” Naval Research Logistics, vol. 38, no. 2, pp. 261–271, 1991. View at: Google Scholar | MathSciNet
  11. X. Cai, X. Sun, and X. Zhou, “Stochastic scheduling with preemptive-repeat machine breakdowns to minimize the expected weighted flow time,” Probability in the Engineering and Informational Sciences, vol. 17, no. 4, pp. 467–485, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. X. Cai, X. Sun, and X. Zhou, “Stochastic scheduling subject to machine breakdowns: the preemptive-repeat model with discounted reward and other criteria,” Naval Research Logistics, vol. 51, no. 6, pp. 800–817, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  13. X. Cai, X. Wu, and X. Zhou, “Dynamically optimal policies for stochastic scheduling subject to preemptive-repeat machine breakdowns,” IEEE Transactions on Automation Science and Engineering, vol. 2, no. 2, pp. 158–172, 2005. View at: Publisher Site | Google Scholar
  14. E. Frostig, “A note on stochastic scheduling on a single machine subject to breakdown—the preemptive repeat model,” Probability in the Engineering and Informational Sciences, vol. 5, no. 3, pp. 349–354, 1991. View at: Publisher Site | Google Scholar | MathSciNet
  15. Z. Khalil and B. Dimitrov, “The service time properties of an unreliable server characterize the exponential distribution,” Advances in Applied Probability, vol. 26, no. 1, pp. 172–182, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. C. Y. Lee and C. S. Lin, “Single-machine scheduling with maintenance and repair rate-modifying activities,” European Journal of Operational Research, vol. 135, no. 3, pp. 493–513, 2001. View at: Publisher Site | Google Scholar | MathSciNet
  17. N. Kasap, H. Aytug, and A. Paul, “Minimizing makespan on a single machine subject to random breakdowns,” Operations Research Letters, vol. 34, no. 1, pp. 29–36, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. H. Tang and C. Zhao, “Stochastic single machine scheduling subject to machines breakdowns with quadratic early-tardy penalties for the preemptive-repeat model,” Journal of Applied Mathematics and Computing, vol. 25, no. 1-2, pp. 183–199, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  19. C. Y. Lee and G. Yu, “Single machine scheduling under potential disruption,” Operations Research Letters, vol. 35, no. 4, pp. 541–548, 2007. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2014 Jinwei Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

927 Views | 443 Downloads | 0 Citations
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.