An Exact Solution for a Single Machine Scheduling Under Uncertainty

Rebaz Abdulla Sharif; Ayad M. Ramadan

doi:10.34312/jjom.v5i1.16156

An Exact Solution for a Single Machine Scheduling Under Uncertainty

Rebaz Abdulla Sharif, Ayad M. Ramadan

Abstract

Here we have n jobs on one machine where the processing times are triangular fuzzy numbers. The jobs are available to process without interruption. The purpose is to find a best sequence of the jobs that minimizes total fuzzy completion times and maximum fuzzy tardiness. In this paper a new definition is presented called D-strongly positive fuzzy number, then an exact solution of the problem through this definition is found. This definition opens new ideas about converting scheduling problems into fuzzy cases.

Keywords

Scheduling; Fuzzy Processing Times; Fuzzy Tardiness; Efficient Solutions

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References

L. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, jun 1965, doi: 10.1016/S0019-9958(65)90241-X. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S001999586590241X

A. Kaufmann and M. M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science. New York: Elsevier Science Inc., 1988.

H. Prade, “Using fuzzy set theory in a scheduling problem: A case study,” Fuzzy Sets and Systems, vol. 2, no. 2, pp. 153–165, apr 1979, doi: 10.1016/0165-0114(79)90022-8. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/0165011479900228

M. Hapke, A. Jaszkiewicz, and R. Slowinski, “Fuzzy project scheduling system for software development,” Fuzzy Sets and Systems, vol. 67, no. 1, pp. 101–117, oct 1994, doi: 10.1016/0165-0114(94)90211-9. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/0165011494902119

S. Han, H. Ishii, and S. Fujii, “One machine scheduling problem with fuzzy duedates,” European Journal of Operational Research, vol. 79, no. 1, pp. 1–12, nov 1994, doi: 10.1016/0377-2217(94)90391-3. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/0377221794903913

H. Ishii, M. Tada, and T. Masuda, “Two scheduling problems with fuzzy due-dates,” Fuzzy Sets and Systems, vol. 46, no. 3, pp. 339–347, mar 1992, doi: 10.1016/0165-0114(92)90372-B. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/016501149290372B

S. Lam and X. Cai, “Earliness and tardiness scheduling with a fuzzy due date and job dependent weights,” Journal of the Chinese Institute of Industrial Engineers, vol. 17, no. 5, pp. 477–487, sep 2000, doi: 10.1080/10170669.2000.10432868. [Online]. Available: http://www.tandfonline.com/doi/abs/10.1080/10170669.2000.10432868

H. Ishibuchi, T. Murata, and Kyu Hung Lee, “Formulation of fuzzy flowshop scheduling problems with fuzzy processing time,” in Proceedings of IEEE 5th International Fuzzy Systems, vol. 1. IEEE, 1996. ISBN 0-7803-3645-3 pp. 199–205, doi: 10.1109/FUZZY.1996.551742. [Online]. Available: http://ieeexplore.ieee.org/document/551742/

Y. Tsujimura, M. Gen, and E. Kubota, “Solving Job-shop Scheduling Problem with Fuzzy Processing Time Using Genetic Algorithm,” Journal of Japan Society for Fuzzy Theory and Systems, vol. 7, no. 5, pp. 1073–1083, 1995, doi: 10.3156/jfuzzy.7.5 1073. [Online]. Available: https://www.jstage.jst.go.jp/article/jfuzzy/7/5/7KJ00002088534/article

S. Chanas and A. Kasperski, “On two single machine scheduling problems with fuzzy processing times and fuzzy due dates,” European Journal of Operational Research, vol. 147, no. 2, pp. 281–296, jun 2003, doi: 10.1016/S0377-2217(02)00561-1. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0377221702005611

H.-C. Wu, “Solving the fuzzy earliness and tardiness in scheduling problems by using genetic algorithms,” Expert Systems with Applications, vol. 37, no. 7, pp. 4860–4866, Jul 2010, doi: 10.1016/j.eswa.2009.12.029. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0957417409010756

S. Steiner and T. Radzik, “Computing all efficient solutions of the objective minimum spanning tree problem,” Computers & Operations Research, vol. 35, no. 1, pp. 198–211, jan 2008, doi: 10.1016/j.cor.2006.02.023. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S030505480600061X

L. N. Van Wassenhove and L. F. Gelders, “Solving a bicriterion scheduling problem,” European Journal of Operational Research, vol. 4, no. 1, pp. 42–48, jan 1980, doi: 10.1016/0377-2217(80)90038-7.[Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/0377221780900387

H. Khalifa, “On Single Machine Scheduling Problem with Distinct due Dates under Fuzzy Environment,” International Journal of Supply and Operations Management, vol. 7, no. 3, pp. 272–278, 2020, doi: 10.22034/IJSOM.2020.3.5.

A. Ramadan, “On Pareto Set for a Bi-criterion Scheduling Problem Under Fuzziness,” Iraqi Journal of Statistical Sciences, vol. 18, no. 33, pp. 64–71, jun 2021, doi: 10.33899/iqjoss.2021.168375. [Online]. Available: https://stats.mosuljournals.com/article_168375.html

DOI: https://doi.org/10.34312/jjom.v5i1.16156