# Abstract:

A single-machine batch scheduling problem is investigated. Each job has a positive processing time and due-date. Setup times are assumed to be identical for all batches. All batch sizes cannot exceed a common upper bound. As in many practical situations, jobs have to be subject to flexible precedence constraints. The aim of this paper is to find an optimal batch sequence. The sequence is to minimize the maximal completion time and maximize the minimum value of desirability of the fuzzy precedence. However, there usually exists no batch sequence optimizing both objectives at a time. Therefore, we seek some non-dominated batch sequences after the definition of non-dominated batch sequence. Based on an iterative Procedure HL proposed by Cheng et al., an efficient algorithm is presented to find some non-dominated batch sequences.

# Keywords:

single-machine, batch scheduling, modified due-date, fuzzy precedence relation, non-dominated batch sequence

# Classification:

90B35, 90C29, 90C70, 68Q25

# References:

1. P. Brucker: Scheduling Algorithm. Springer-Verlag, Berlin 2006.   CrossRef
2. P. Brucker, A. Gladky, H. Hoogeveen, M. Y. Kovalyov, C. N. Potts, T. Tautenhahn and S. V. Velde: Scheduling a batching machine. J. Scheduling 1 (1998), 31-54.   CrossRef
3. T. C. E. Cheng and A. Janiak: Single machine batch scheduling with deadline and resource dependent processing times. Oper. Res. Lett. 17 (1995), 243-249.   CrossRef
4. T. C. E. Cheng, A. Janiak and M. Y. Kovalyov: Single machine batch scheduling with resource dependent setup and processing times. European J. Oper. Res. 135 (2001), 177-183.   CrossRef
5. E. G. Coffman Jr., M. Yannakakis, M. J. Magazine and C. Santos: Batch sizing and job sequencing on a single machine. Ann. Oper. Res. 26 (1990), 135-147.   CrossRef
6. G. Dobson, U. S. Karmarkar and J. L. Rummel: Batching to minimize flow times on one machine. Management Sci. 33 (1987), 784-799.   CrossRef
7. D. S. Hochbaum and D. Landy: Scheduling with batching: Minimizing the weighted number of tardy jobs. Oper. Res. Lett. 16 (1994), 79-86.   CrossRef
8. E. L. Lawler and J. M. Moore: A functional equation and its application to resource allocation and scheduling problems. Management Sci. 16 (1969), 77-84.   CrossRef
9. G. Mosheiov and D. Oron: A single machine batch scheduling problem with bounded batch size. European J. Oper. Res. 187 (2008), 1069-1079.   CrossRef
10. G. Mosheiov and D. Oron: Open-shop batch scheduling with identical jobs. European J. Oper. Res. 187 (2008), 1282-1292.   CrossRef
11. G. Mosheiov, D. Oron and Y. Ritov: Minimizing flow-time on a single machine with integer batch sizes. Oper. Res. Lett. 33 (2005), 497-501.   CrossRef
12. D. Naddef and C. Santos: One-pass batching algorithms for the one-machine problem. Discrete Appl. Math. 21 (1988), 133-145.   CrossRef
13. C. N. Pottsand and M. Y. Kovalyov: Scheduling with Batching: A review. European J. Oper. Res. 120 (2000), 228-249.   CrossRef
14. C. N. Pottsand and L. N. Van Wasenhove: Interacting scheduling with batching and lot sizing: a review of algorithm and complexity. J. Oper. Res. Soc. 43 (1992), 395-406.   CrossRef
15. C. Santos and M. Magazine: Batching in single operation manufacturing systems. Oper. Res. Lett. 4 (1985), 338-343.   CrossRef
16. D. F. Shallcross: A polynomial algorithm for a one machine batching problem. Oper. Res. Lett. 11 (1992), 213-218.   CrossRef
17. C. S. Sung and U. G. Joo: Batching to minimize weighted mean flow time on a single machine with batch size restrictions. Comput. and Industr. Engrg. 32 (1997), 333-340.   CrossRef