The Deviation Issue and Total Completion Time on a Single Machine Scale Subject to Decreasing and Growing Linear Deterioration are Theoretically Addressed

Mohammed Alsudani; Hanan Ali Chachan

doi:10.31185/wjps.569

Authors

Mohammed Alsudani Mathematics Department, College of Science, Mustansiriyah University, Bagdad, IRAQ.
Hanan Ali Chachan Mathematics Department, College of Science, Mustansiriyah University, Bagdad, IRAQ.

DOI:

https://doi.org/10.31185/wjps.569

Keywords:

single machine, linear deterioration, total completion time

Abstract

We addressed in this research the schedule of activities that are subject to the condition of linear degradation in declining and growing rate states, respectively, with the goal of guaranteeing that the primary process's duration would be fixed (ρ) and that its implementation would take place in a single machine. The objective was to find the optimal schedule for calculating the total completion time ∑_(ḭ=1)^ń▒C_ḭ , and the optimal schedule for the deviation problem (sum of squares of the difference between the tasks' total required time and the previously established due date time. where the objective is to minimize the squared deviation of job completion times from a due date), two theories that state as the Λ-shaped scheduling theory with increasing rate ƛ_ḭ of jobs and the v-shaped scheduling theory with decreasing rate đ_ḭ of jobs satisfy the conditions 0<đ_ḭ<1 ańd đ_ḭ<1/(ń-1), have been demonstrated in situations where the base is implemented at a constant basic processing time .

References

F. H. Ali and M. G. Ahmed, “Local Search Methods for Solving Total Completion Times, Range of Lateness and Maximum Tardiness Problem,” in Proceedings of the 6th International Engineering Conference “‘Sustainable Technology and Development’”, IEC 2020, IEEE, Feb. 2020, pp. 103–108. doi: 10.1109/IEC49899.2020.9122821.

H. A. Chachan, F. H. Ali, and M. H. Ibrahim, “Branch and Bound and Heuristic Methods for Solving Multi-Objective Function for Machine Scheduling Problems,” in 2020 6th International Engineering Conference “Sustainable Technology and Development" (IEC), IEEE, Feb. 2020, pp. 109–114. doi: 10.1109/IEC49899.2020.9122793.

M. H. Ibrahim, F. H. Ali, and H. A. Chachan, “Dominance Rules for Single Machine Scheduling to Minimize Weighted Multi Objective Functions,” in AIP Conference Proceedings, Feb. 2023, p. 040090. doi: 10.1063/5.0119047.

S. Browne and U. Yechiali, “Scheduling deteriorating jobs on a single processor,” Operations Research, vol. 38, no. 3, pp. 495–498, Jun. 1990, doi: 10.1287/opre.38.3.495.

G. Mosheiov, “V-shaped policies for scheduling deteriorating jobs,” Operations Research, vol. 39, no. 6, pp. 979–991, Dec. 1991, doi: 10.1287/opre.39.6.979.

B. Alidaee and N. K. Womer, “Scheduling with time dependent processing times: Review and extensions,” Journal of the Operational Research Society, vol. 50, no. 7, pp. 711–720, Jul. 1999, doi: 10.1057/palgrave.jors.2600740.

A. Bachman and A. Janiak, “Minimizing maximum lateness under linear deterioration,” European Journal of Operational Research, vol. 126, no. 3, pp. 557–566, Nov. 2000, doi: 10.1016/S0377-2217(99)00310-0.

C. T. Ng, T. C. E. Cheng, A. Bachman, and A. Janiak, “Three scheduling problems with deteriorating jobs to minimize the total completion time,” Information Processing Letters, vol. 81, no. 6, pp. 327–333, Mar. 2002, doi: 10.1016/S0020-0190(01)00244-7.

T. C. E. Cheng and Q. Ding, “Scheduling start time dependent tasks with deadlines and identical initial processing times on a single machine,” Computers and Operations Research, vol. 30, no. 1, pp. 51–62, Jan. 2003, doi: 10.1016/S0305-0548(01)00077-6.

Y.-H. Chung, H.-C. Liu, C.-C. Wu, and W.-C. Lee, “A deteriorating jobs problem with quadratic function of job lateness,” Computers & Industrial Engineering, vol. 57, no. 4, pp. 1182–1186, Nov. 2009, doi: 10.1016/j.cie.2009.05.012.

K. I. J. Ho, J. Y. T. Leung, and W. D. Wei, “Complexity of scheduling tasks with time-dependent execution times,” Information Processing Letters, vol. 48, no. 6, pp. 315–320, Dec. 1993, doi: 10.1016/0020-0190(93)90175-9.

A. Bachman, T. C. E. Cheng, A. Janiak, and C. T. Ng, “Scheduling start time dependent jobs to minimize the total weighted completion time,” Journal of the Operational Research Society, vol. 53, no. 6, pp. 688–693, Jun. 2002, doi: 10.1057/palgrave.jors.2601359.

J. B. Wang and Z. Q. Xia, “Scheduling jobs under decreasing linear deterioration,” Information Processing Letters, vol. 94, no. 2, pp. 63–69, Apr. 2005, doi: 10.1016/j.ipl.2004.12.018.

Y. Yin and D. Xu, “Some single-machine scheduling problems with general effects of learning and deterioration,” Computers and Mathematics with Applications, vol. 61, no. 1, pp. 100–108, 2011, doi: 10.1016/j.camwa.2010.10.036.

W.-C. Lee and P.-J. Lai, “Scheduling problems with general effects of deterioration and learning,” Information Sciences, vol. 181, no. 6, pp. 1164–1170, Mar. 2011, doi: 10.1016/j.ins.2010.11.026.

R. J. Freeman, “Letter to the Editor—A Generalized PERT,” Operations Research, vol. 8, no. 2, pp. 281–281, Apr. 1960, doi: 10.1287/opre.8.2.281.

A. Charnes and W. W. Cooper, “Chance-Constrained Programming,” Management Science, vol. 6, no. 1, pp. 73–79, Oct. 1959, doi: 10.1287/mnsc.6.1.73.

K. N. McKay, F. R. Safayeni, and J. A. Buzacott, “Job-Shop Scheduling Theory: What Is Relevant?,” Interfaces, vol. 18, no. 4, pp. 84–90, Aug. 1988, doi: 10.1287/inte.18.4.84.

B. Dodin, “Determining the optimal sequences and the distributional properties of their completion times in stochastic flow shops,” Computers and Operations Research, vol. 23, no. 9, pp. 829–843, Sep. 1996, doi: 10.1016/0305-0548(95)00083-6.

J. Mittenthal and M. Raghavachari, “Stochastic Single Machine Scheduling with Quadratic Early-Tardy Penalties,” Operations Research, vol. 41, no. 4, pp. 786–796, Aug. 1993, doi: 10.1287/opre.41.4.786.

H. A. Cheachan and M. T. Kadhim, “Exact Method for Solving Single Machine Scheduling Problem under Fuzzy Due Date to Minimize Multi-Objective Functions,” Journal of Physics: Conference Series, vol. 1879, no. 2, May 2021, doi: 10.1088/1742-6596/1879/2/022126.