# American Institute of Mathematical Sciences

April  2015, 11(2): 685-700. doi: 10.3934/jimo.2015.11.685

## Two-machine scheduling with periodic availability constraints to minimize makespan

 1 Department of Mathematics, School of Science, East China University of Science and Technology, Shanghai 200237, China, China

Received  November 2013 Revised  May 2014 Published  September 2014

A two-machine scheduling problem where one machine has periodic availability constraints has been studied. The objective is to minimize makepan. For the nonresumable version, we give a better approximation algorithm with performance ratio of $4/3$. For the resumable version, we provide an offline $4/3$-approximation algorithm and an optimal online algorithm, respectively.
Citation: Ganggang Li, Xiwen Lu. Two-machine scheduling with periodic availability constraints to minimize makespan. Journal of Industrial and Management Optimization, 2015, 11 (2) : 685-700. doi: 10.3934/jimo.2015.11.685
##### References:
 [1] T. C. E. Cheng and G. Wang, An improved heuristic for two-machine flowshop scheduling with an availability constraint, Operation Research Letters, 26 (2000), 223-229. doi: 10.1016/S0167-6377(00)00033-X. [2] R. L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics, 17 (1969), 416-429. doi: 10.1137/0117039. [3] M. Ji, Y. He and T. C. E. Cheng, Single-machine scheduling with periodic maintenance to minimize makespan, Computer & Operations Research, 34 (2007), 1764-1770. doi: 10.1016/j.cor.2005.05.034. [4] C. J. Liao, D. L. Shyur and C. H. Lin, Makespan minimization for two parallel machines with an availability constraint, European journal of operational Research, 160 (2005), 445-456. doi: 10.1016/j.ejor.2003.08.034. [5] C. J. Liao and W. J. Chen, Single-machine scheduling with periodic maintenance and nonresumable jobs, Computers & operations Research, 30 (2003), 1335-1347. doi: 10.1016/S0305-0548(02)00074-6. [6] C. Y. Lee, Machine scheduling with an availability constraint, Journal of Global optimization, 9 (1996), 395-416. doi: 10.1007/BF00121681. [7] W. Luo and L. Chen, Approximation schemes for scheduling a maintenance and linear deteriorating jobs, Journal of Industrial and Management Optimization, 8 (2012), 271-283. doi: 10.3934/jimo.2012.8.271. [8] G. Wang and T. C. E. Cheng, Heuristics for two-machine no-wait flowshop scheduling with an availability constraint, Information Processing Letters, 80 (2001), 305-309. doi: 10.1016/S0020-0190(01)00181-8. [9] D. H. Xu, Z. M. Cheng, Y. Q. Yin and H. X. Li, Makespan minimization for two parallel machines scheduling with a periodic availability constraint, Computer & Operations Research, 36 (2009), 1809-1812. doi: 10.1016/j.cor.2008.05.001. [10] M. Y. Yue, A simple proof of the inequality $FFD(L)\leq \frac{11}{9}OPT(L)+1,\forall L$ for the FFD Bin-Packing algorithm, Acta Mathematics Application Sinica, 7 (1991), 321-331. doi: 10.1007/BF02009683.

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##### References:
 [1] T. C. E. Cheng and G. Wang, An improved heuristic for two-machine flowshop scheduling with an availability constraint, Operation Research Letters, 26 (2000), 223-229. doi: 10.1016/S0167-6377(00)00033-X. [2] R. L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics, 17 (1969), 416-429. doi: 10.1137/0117039. [3] M. Ji, Y. He and T. C. E. Cheng, Single-machine scheduling with periodic maintenance to minimize makespan, Computer & Operations Research, 34 (2007), 1764-1770. doi: 10.1016/j.cor.2005.05.034. [4] C. J. Liao, D. L. Shyur and C. H. Lin, Makespan minimization for two parallel machines with an availability constraint, European journal of operational Research, 160 (2005), 445-456. doi: 10.1016/j.ejor.2003.08.034. [5] C. J. Liao and W. J. Chen, Single-machine scheduling with periodic maintenance and nonresumable jobs, Computers & operations Research, 30 (2003), 1335-1347. doi: 10.1016/S0305-0548(02)00074-6. [6] C. Y. Lee, Machine scheduling with an availability constraint, Journal of Global optimization, 9 (1996), 395-416. doi: 10.1007/BF00121681. [7] W. Luo and L. Chen, Approximation schemes for scheduling a maintenance and linear deteriorating jobs, Journal of Industrial and Management Optimization, 8 (2012), 271-283. doi: 10.3934/jimo.2012.8.271. [8] G. Wang and T. C. E. Cheng, Heuristics for two-machine no-wait flowshop scheduling with an availability constraint, Information Processing Letters, 80 (2001), 305-309. doi: 10.1016/S0020-0190(01)00181-8. [9] D. H. Xu, Z. M. Cheng, Y. Q. Yin and H. X. Li, Makespan minimization for two parallel machines scheduling with a periodic availability constraint, Computer & Operations Research, 36 (2009), 1809-1812. doi: 10.1016/j.cor.2008.05.001. [10] M. Y. Yue, A simple proof of the inequality $FFD(L)\leq \frac{11}{9}OPT(L)+1,\forall L$ for the FFD Bin-Packing algorithm, Acta Mathematics Application Sinica, 7 (1991), 321-331. doi: 10.1007/BF02009683.
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