October  2019, 15(4): 1955-1964. doi: 10.3934/jimo.2018131

Scheduling with step-deteriorating jobs to minimize the makespan

1. 

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

2. 

School of Business Administration, University of New Brunswick, Fredericton, NB E3B 5A3, Canada

3. 

School of Management, Qufu Normal University, Rizhao 276826, China

* Corresponding author: Email: miaocuixia@126.com

Received  March 2018 Revised  April 2018 Published  October 2019 Early access  August 2018

In this paper, we consider the scheduling with step-deteriorating jobs. The objective is to minimize the makespan. We first propose a property of any optimal schedule after analyzing the NP-hardness for the parallel-machine scheduling with a common deteriorating date, and then present a fully polynomial time approximation scheme for the case where the number of machines $m$ is a constant and jobs have anti-agreeable parameters. Furthermore, we show that the single-machine scheduling is NP-hard in the strong sense if jobs have distinct release dates and distinct deteriorating dates.

Citation: Cuixia Miao, Yuzhong Zhang. Scheduling with step-deteriorating jobs to minimize the makespan. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1955-1964. doi: 10.3934/jimo.2018131
References:
[1]

T. C. E. Cheng and Q. Ding, Single machine scheduling with step-deteriorating processing times, European Journal of Operational Research, 134 (2001), 623-630.  doi: 10.1016/S0377-2217(00)00284-8.

[2]

L. R. Eduardo and V. Stefan, Modeling the Parallel Machine Scheduling Problem with Step Deteriorating Jobs, European Journal of Operational Research, 255 (2016), 21-33.  doi: 10.1016/j.ejor.2016.04.010.

[3]

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979.

[4]

R. L. GrahamE. L. LawlerJ. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics, 5 (1979), 287-326.  doi: 10.1016/S0167-5060(08)70356-X.

[5]

P. GuoW. M. Cheng and Y. Wang, A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs, Journal of Industrial and Magagement Optimization, 10 (2014), 1071-1090.  doi: 10.3934/jimo.2014.10.1071.

[6]

P. GuoW. M. Cheng and Y. Wang, Parallel machine scheduling with step-deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm, Engineering Optimization, 47 (2015), 1564-1585.  doi: 10.1080/0305215X.2014.982634.

[7]

A. A. K. Jeng and B. M. T. Lin, Minimizing the total completion time in single-machine scheduling with step-deteriorating jobs, Computers and Operations Research, 32 (2005), 521-536. 

[8]

M. Y. Kovalyov and W. Kubiak, A fully polynomial approximation scheme for minimizing makespan of deteriorating jobs, Journal of Heuristics, 3 (1998), 287-297. 

[9]

G. Mosheiov, Scheduling jobs with step-deterioration: Minimizing makespan on a single and multi-machine, Computer and Industrial Engineering, 28 (1995), 869-879.  doi: 10.1016/0360-8352(95)00006-M.

[10]

B. Mor and G. Mosheiov, Batch scheduling with step-deteriorating processing times to minimize flowtime, Naval Research Logistics, 59 (2012), 587-600.  doi: 10.1002/nav.21508.

[11]

P. S. Sundararaghavan and A. S. Kunnathur, Single machine scheduling with start time dependent processing times: Some solvable cases, European Journal of Operational Research, 78 (1994), 394-403.  doi: 10.1016/0377-2217(94)90048-5.

show all references

References:
[1]

T. C. E. Cheng and Q. Ding, Single machine scheduling with step-deteriorating processing times, European Journal of Operational Research, 134 (2001), 623-630.  doi: 10.1016/S0377-2217(00)00284-8.

[2]

L. R. Eduardo and V. Stefan, Modeling the Parallel Machine Scheduling Problem with Step Deteriorating Jobs, European Journal of Operational Research, 255 (2016), 21-33.  doi: 10.1016/j.ejor.2016.04.010.

[3]

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979.

[4]

R. L. GrahamE. L. LawlerJ. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics, 5 (1979), 287-326.  doi: 10.1016/S0167-5060(08)70356-X.

[5]

P. GuoW. M. Cheng and Y. Wang, A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs, Journal of Industrial and Magagement Optimization, 10 (2014), 1071-1090.  doi: 10.3934/jimo.2014.10.1071.

[6]

P. GuoW. M. Cheng and Y. Wang, Parallel machine scheduling with step-deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm, Engineering Optimization, 47 (2015), 1564-1585.  doi: 10.1080/0305215X.2014.982634.

[7]

A. A. K. Jeng and B. M. T. Lin, Minimizing the total completion time in single-machine scheduling with step-deteriorating jobs, Computers and Operations Research, 32 (2005), 521-536. 

[8]

M. Y. Kovalyov and W. Kubiak, A fully polynomial approximation scheme for minimizing makespan of deteriorating jobs, Journal of Heuristics, 3 (1998), 287-297. 

[9]

G. Mosheiov, Scheduling jobs with step-deterioration: Minimizing makespan on a single and multi-machine, Computer and Industrial Engineering, 28 (1995), 869-879.  doi: 10.1016/0360-8352(95)00006-M.

[10]

B. Mor and G. Mosheiov, Batch scheduling with step-deteriorating processing times to minimize flowtime, Naval Research Logistics, 59 (2012), 587-600.  doi: 10.1002/nav.21508.

[11]

P. S. Sundararaghavan and A. S. Kunnathur, Single machine scheduling with start time dependent processing times: Some solvable cases, European Journal of Operational Research, 78 (1994), 394-403.  doi: 10.1016/0377-2217(94)90048-5.

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