# Rescheduling with new orders and general maximum allowable time disruptions

- 316 Downloads
- 3 Citations

## Abstract

We study the rescheduling with new orders on a single machine under the general maximum allowable time disruptions. Under the restriction of general maximum allowable time disruptions, each original job has an upper bound for its time disruption (regarded as the maximum allowable time disruption of the job), or equivalently, in every feasible schedule, the difference of the completion time of each original job compared to that in the pre-schedule does not exceed its maximum allowable time disruption. We also consider a stronger restriction which additionally requires that, in a feasible schedule, the starting time of each original job is not allowed to be scheduled smaller than that in the pre-schedule. Scheduling objectives to be minimized are the maximum lateness and the total completion time, respectively, and the pre-schedules of original jobs are given by EDD-schedule and SPT-schedule, respectively. Then we have four problems for consideration. For the two problems for minimizing the maximum lateness, we present strong NP-hardness proof, provide a simple 2-approximation polynomial-time algorithm, and show that, unless \(\text {P}= \text {NP}\), the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 2. For the two problems for minimizing the total completion time, we present strong NP-hardness proof, provide a simple heuristic algorithm, and show that, unless \(\text {P}= \text {NP}\), the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 4/3. Moreover, by relaxing the maximum allowable time disruptions of the original jobs, we present a super-optimal dual-approximation polynomial-time algorithm. As a consequence, if the maximum allowable time disruption of each original job is at least its processing time, then the two problems for minimizing the total completion time are solvable in polynomial time. Finally, we show that, under the agreeability assumption (i.e., the nondecreasing order of the maximum allowable time disruptions of the original jobs coincides with their scheduling order in the pre-schedule), the four problems in consideration are solvable in polynomial time.

## Keywords

Rescheduling Time disruption Maximum lateness Total completion time## Mathematics Subject Classification

90B35## Notes

### Acknowledgments

We would like to thank the associate editor and two anonymous referees for their constructive comments and kind suggestions. This research was supported by NSFC (11271338), NSFC (11301528), NSFC (U1504103), and NSF-Henan (15IRTSTHN006).

### Compliance with ethical standards

### Conflicts of interest

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

## References

- Akkan C (2015) Improving schedule stability in single-machine rescheduling for new operation insertion. Comput Oper Res 64:198–209CrossRefGoogle Scholar
- Aytug H, Lawley MA, McKay K, Mohan S, Uzsoy R (2005) Executing production schedules in the face of uncertainties: a review and some future direction. Eur J Oper Res 161:86–110CrossRefGoogle Scholar
- Church LK, Uzsoy R (1992) Analysis of periodic and event-driven rescheduling policies in dynamic shops. Int J Comput Integr Manuf 5:153–163CrossRefGoogle Scholar
- Gao Y, Yuan JJ (2015) Pareto minimizing total completion time and maximum cost with positional due indices. J Oper Res Soc China 3:381–387CrossRefGoogle Scholar
- Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San FranciscoGoogle Scholar
- Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326CrossRefGoogle Scholar
- Hall NG, Potts CN (2004) Rescheduling for new orders. Oper Res 52:440–453CrossRefGoogle Scholar
- Hall NG, Liu ZX, Potts CN (2007) Rescheduling for multiple new orders. INFORMS J Comput 19:633–645CrossRefGoogle Scholar
- Hoogeveen H, Lenté C, T’kindt V (2012) Rescheduling for new orders on a single machine with setup times. Eur J Oper Res 223:40–46CrossRefGoogle Scholar
- Jackson JR (1955) Scheduling a production line to minimize maximum tardiness. Research report 43, Management Science Research Project, University of California, Los Angeles, CAGoogle Scholar
- Jain AK, Elmaraghy HA (1997) Production scheduling/rescheduling in flexible manufacturing. Int J Prod Res 35:281–309CrossRefGoogle Scholar
- Katragjini K, Vallada E, Ruiz R (2013) Flow shop rescheduling under different types of disruption. Int J Prod Res 51:780–797CrossRefGoogle Scholar
- Koulamas C, Kyparisis GJ (2001) Single machine scheduling with release times, deadlines and tardiness objectives. Eur J Oper Res 133:447–453CrossRefGoogle Scholar
- Leus R, Herroelen W (2005) The complexity of machine scheduling for stability with a single disrupted job. Oper Res Lett 33:151–156CrossRefGoogle Scholar
- Liu L, Zhou H (2015) Single-machine rescheduling with deterioration and learning effects against the maximum sequence disruption. Int J Syst Sci 46:2640–2658CrossRefGoogle Scholar
- Moratori P, Petrovic S, Vázquez-Rodríguez JA (2012) Match-up approaches to a dynamic rescheduling problem. Int J Prod Res 50:261–276CrossRefGoogle Scholar
- Mu YD, Guo X (2009) On-line rescheduling to minimize makespan under a limit on the maximum sequence disruption. In: Proceeding of the 2009 international conference on services science, management and engineering, IEEE Computer Society, pp 479–482. doi: 10.1109/SSME.2009.44
- Mu YD, Guo X (2009) On-line rescheduling to minimize makespan under a limit on the maximum disruptions. In: Proceeding of the 2009 international conference on management of e-commerce and e-government. IEEE Computer Society, pp 141–144. doi: 10.1109/ICMeCG.2009.26
- Mu YD, Tian XZ (2010) Pareto optimizations of objective and disruptions for rescheduling problems. J Henan Univ (Nat Sci) 40:441–444Google Scholar
- Raman N, Talbot FB, Rachamadugu RV (1989) Due date based scheduling in a general flexible manufacturing system. J Oper Manag 8:115–132CrossRefGoogle Scholar
- Smith WE (1956) Various optimizers for single-stage production. Naval Res Logist Q 3:59–66CrossRefGoogle Scholar
- Teghem J, Tuyttens D (2014) A bi-objective approach to reschedule new jobs in a one machine model. Int Trans Oper Res 21:871–898CrossRefGoogle Scholar
- Unal AT, Uzsoy R, Kiran AS (1997) Rescheduling on a single machine with part-type dependent setup times and deadlines. Ann Oper Res 70:93–113CrossRefGoogle Scholar
- Vieira GE, Herrmann JW, Lin E (2003) Rescheduling manufacturing systems: a framework of strategies, policies and methods. J Sched 6:39–62CrossRefGoogle Scholar
- Wu SD, Storer RH, Chang P-C (1992) A rescheduling procedure for manufacturing systems under random disruptions. In: Fandel G, Gulledge T, Jone A (eds) New directions for operations research in manufacturing. Springer, Berlin, pp 292–308CrossRefGoogle Scholar
- Yuan JJ, Mu YD (2007) Rescheduling with release dates to minimize makespan under a limit on the maximum sequence disruption. Eur J Oper Res 182:936–944CrossRefGoogle Scholar
- Yuan JJ, Mu YD, Lu LF, Li WH (2007) Rescheduling with release dates to minimize total sequence disruption under a limit on the makespan. Asia-Pacific J Oper Res 24:789–796CrossRefGoogle Scholar
- Zhao QL (2012) Rescheduling research under pareto optimization, job-disruptions and weighted disruptions. Master Degree Thesis, Zhengzhou UniversityGoogle Scholar
- Zhao CL, Tang HY (2010) Rescheduling problems with deteriorating jobs under disruptions. Appl Math Model 34:238–243CrossRefGoogle Scholar
- Zhao QL, Yuan JJ (2013) Pareto optimization of rescheduling with release dates to minimize makespan and total sequence disruption. J Sched 16:253–260CrossRefGoogle Scholar
- Zhao QL, Yuan JJ (2007) Rescheduling to minimize the maximum lateness under the sequence disruption of original jobs. Asia-Pac J Oper Res (accepted)Google Scholar