Abstract
A two-stage flexible flow shop scheduling is a manufacturing infrastructure designed to process a set of jobs, in which a single machine is available at the first stage and m parallel machines are available at the second stage. At the second stage, each task can be processed by multiple parallel machines. The objective is to minimize the maximum job completion time, i.e., the makespan. Sun et al. (J Softw 25:298–313, 2014) presented an \(O(n\log n)\)-time 3-approximation algorithm for \(F2(1, Pm)~|~size_i~|~C_{\max }\) under some special conditions. Zhang et al. (J Comb Optim 39:1–14, 2020) presented a 2.5-approximation algorithm for \(F2(1, P2)~|~line_i~|~C_{\max }\) and a 2.67-approximation algorithm for \(F2(1, P3)~|~line_i~|~C_{\max }\), which both run in linear time. In this paper, we achieved following improved results: for \(F2(1, P2)~|~line_i~|~C_{\max }\), we present an \(O(n\log n)\)-time 2.25-approximation algorithm, for \(F2(1, P3)~|~line_i~|~C_{\max }\), we present an \(O(n\log n)\)-time 7/3-approximation algorithm, for \(F2(1, Pm)~|~size_i~|~C_{\max }\) with the assumption \( \mathop {\min }_{1 \le i \le n} \left\{ {{p_{1i}}} \right\} \ge \mathop {\max }_{1 \le i \le n} \left\{ {{p_{2i}}} \right\} \), we present a linear time optimal algorithm.
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References
Alisantoso D, Khoo LP, Jiang PY (2003) An immune algorithm approach to the scheduling of a flexible PCB flow shop. Int J Adv Manuf Technol 22:819–827
Almeder C, Hartl RF (2013) A metaheuristic optimization approach for a real-world stochastic flexible flow shop problem with limited buffer. Int J Prod Econ 145:88–95
Arthanari TS, Ramamurthy KG (1971) An extension of two machines sequencing problem. Opsearch 8:10–22
Choi BC, Lee K (2013) Two-stage proportionate flexible flow shop to minimize the makespan. J Comb Optim 25:123–134
Graham RL, Lawler EL, Lenstra JK, Kan R (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326
Gupta J, Krüger K, Lauff V, Werner F, Sotskov Y (2002) Heuristics for hybrid flow shops with controllable processing times and assignable due dates. Comput Oper Res 29:1417–1439
He LM, Sun SJ, Luo RZ (2008) Two-stage flexible flow shop scheduling problems with a batch processor on second stage. Chin J Eng Math 25:829–842
Lee CY, Vairaktarakis GL (1994) Minimizing makespan in hybrid flowshops. Oper Res Lett 16:149–158
Lin HT, Liao CJ (2003) A case study in a two-stage hybrid flow shop with setup time and dedicated machines. Int J Prod Econ 86:133–143
Moseley B, Dasgupta A, Kumar R, Sarlós T (2011) On scheduling in map-reduce and flow-shops. In: Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures(SPAA ’11), pp 289–298
Salvador MS (1973) A solution to a special class of flow shop scheduling problems. In: Symposium of theory of scheduling and its applications, pp 83–91
Sun JH, Deng QX, Meng YK (2014) Two-stage workload scheduling problem on GPU architectures: formulation and approximation algorithm. J Softw 25:298–313
Zhang MH, Lan Y, Han X (2020) Approximation algorithms for two-stage flexible flow shop scheduling. J Comb Optim 39:1–14
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This research is supported by the Fundamental Research Funds for the Central Universities (Grant No. 20720190068)
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Peng, A., Liu, L. & Lin, W. Improved approximation algorithms for two-stage flexible flow shop scheduling. J Comb Optim 41, 28–42 (2021). https://doi.org/10.1007/s10878-020-00657-2
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DOI: https://doi.org/10.1007/s10878-020-00657-2