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May  2022, 18(3): 1795-1807. doi: 10.3934/jimo.2021044

## Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times

 Department of Industrial Engineering, Faculty of Engineering, Erciyes University, 38039, Turkey

* Corresponding author: Mehmet Duran Toksari

Received  September 2020 Revised  December 2020 Published  May 2022 Early access  March 2021

Fund Project: The first author is supported by Scientific Research Fund of Erciyes University grant: FBA-2014-5397

In recent years, significant on the past sequence dependent delivery times have been increasing for scheduling problems. An electronic component when waiting to process may be exposed to certain an electromagnetic field and is required to neutralize the effect of electromagnetism. In this case, it needs an extra time to eliminate adverse effect. In the scheduling literature, this extra time is called as past-sequence-dependent delivery times. In this paper we introduce single-machine scheduling problems with an exponential sum-of-actual-processing-time-based delivery times. By the exponential sum-of-actual-processing-time-based delivery times, we mean that the delivery times are defined by an exponential function of the sum of the actual processing times of the already processed jobs. On the other hand, the learning effect is reflected in decreasing processing times based on the job's position in schedule. In this paper, we also introduce both exponential past sequence dependent delivery times and learning effect where the job processing time is a function based on the sum of the logarithm of processing times of jobs already processed. We show that the single-machine scheduling problems to minimize makespan, total completion time, weighted total completion time and maximum tardiness with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times have polynomial time solutions.

Citation: Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1795-1807. doi: 10.3934/jimo.2021044
##### References:
 [1] A. B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Transactions on Engineering Management, 39 (1992), 176-188.  doi: 10.1109/17.141275. [2] T. C. E. Cheng, P. Lai, C. Wu and W. Lee, Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations, Information Sciences, 179 (2009), 3127-3135.  doi: 10.1016/j.ins.2009.05.002. [3] T. C. E. Cheng, W. Kuo and D. Yang, Scheduling with a position-weighted learning effect based on sum-of-logarithm-processing-times and job position, Information Sciences, 221 (2013), 490-500.  doi: 10.1016/j.ins.2012.09.001. [4] C. Miao and J. Zou, Scheduling problem with simple deterioration and past-sequence-dependent delivery times, Operations Research Transactions, 20 (2016), 61-68. [5] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A Survey, Annals of Discrete Mathematics, 5 (1979), 287-326.  doi: 10.1016/S0167-5060(08)70356-X. [6] C. Hsu, W. Kuo and D. Yang, Unrelated parallel machine scheduling with past-sequence-dependent setup time and learning effects, Applied Mathematical Modelling, 35 (2011), 1492-1496.  doi: 10.1016/j.apm.2010.09.026. [7] C. Koulamas and G. J. Kyparisis, Single-machine scheduling problems with past-sequence-dependent delivery times, International Journal of Production Economics, 126 (2010), 264-266. [8] W. Kuo and D. Yang, Single machine scheduling with past-sequence-dependent setup times and learning effects, Information Processing Letters, 102 (2007), 22-26.  doi: 10.1016/j.ipl.2006.11.002. [9] W. Lee, A note on single-machine scheduling with general learning effect and past-sequence-dependent setup time, Computers and Mathematics with Applications, 62 (2011), 2095-2100.  doi: 10.1016/j.camwa.2011.06.057. [10] M. Liu, F. Zheng, C. n Chu and Y. Xu, New results on single-machine scheduling with past-sequence-dependent delivery times, Theoretical Computer Science, 438 (2012), 55-61.  doi: 10.1016/j.tcs.2012.03.009. [11] M. Liu, F. Zheng, C. Chu and Y. Xu, Single-machine scheduling with past-sequence-dependent delivery times and release times, Information Processing Letters, 112 (2012), 835-838.  doi: 10.1016/j.ipl.2012.07.002. [12] M. Liu, S. Wang and C. Chu, Scheduling deteriorating jobs with past-sequence-dependent delivery times, International Journal of Production Economics, 144 (2013), 418-421.  doi: 10.1016/j.ijpe.2013.03.009. [13] M. Liu, Parallel-machine scheduling with past-sequence-dependent delivery times and learning effect, Applied Mathematical Modelling, 37 (2013), 9630-9633.  doi: 10.1016/j.apm.2013.05.025. [14] L. Mingze, Single-Machine Scheduling Problems with Non-Linear Past-Sequence-Dependent Setup Times and Delivery Times, A a, 1 (2017), 2. [15] L. Shen and Y. Wu, Single machine past-sequence-dependent delivery times scheduling with general position-dependent and time-dependent learning effects, Applied Mathematical Modelling, 37 (2013), 5444-5451.  doi: 10.1016/j.apm.2012.11.001. [16] J. Wang, L. Sun and L. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Applied Mathematical Modelling, 34 (2010), 2813-2819.  doi: 10.1016/j.apm.2009.12.015. [17] J. Wang, Single-machine scheduling with past-sequence-dependent setup times and time-dependent learning effect, Computers and Industrial Engineering, 55 (2008), 584-591. [18] J. Wang, D. Wang, L. Wang, L. Lin, N. Yin and W. Wang, Single machine scheduling with exponential time-dependent learning effect and past-sequence-dependent setup times, Computers and Mathematics with Applications, 57 (2009), 9-16.  doi: 10.1016/j.camwa.2008.09.025. [19] N. Yin, L. Kang, P. Ji and J. Wang, Single machine scheduling with sum-of-logarithm-processing-times based deterioration, Information Sciences, 274 (2014), 303-309.  doi: 10.1016/j.ins.2014.03.004. [20] W. Yu-Bin and W. Jian-Jun, Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times, Neural Computing and Applications, 27 (2016), 937-943. [21] X. Zhang, G. Yan, W. Huang and G. Tang, A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects, Information Sciences, 187 (2012), 298-304.  doi: 10.1016/j.ins.2011.11.001.

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##### References:
 [1] A. B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Transactions on Engineering Management, 39 (1992), 176-188.  doi: 10.1109/17.141275. [2] T. C. E. Cheng, P. Lai, C. Wu and W. Lee, Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations, Information Sciences, 179 (2009), 3127-3135.  doi: 10.1016/j.ins.2009.05.002. [3] T. C. E. Cheng, W. Kuo and D. Yang, Scheduling with a position-weighted learning effect based on sum-of-logarithm-processing-times and job position, Information Sciences, 221 (2013), 490-500.  doi: 10.1016/j.ins.2012.09.001. [4] C. Miao and J. Zou, Scheduling problem with simple deterioration and past-sequence-dependent delivery times, Operations Research Transactions, 20 (2016), 61-68. [5] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A Survey, Annals of Discrete Mathematics, 5 (1979), 287-326.  doi: 10.1016/S0167-5060(08)70356-X. [6] C. Hsu, W. Kuo and D. Yang, Unrelated parallel machine scheduling with past-sequence-dependent setup time and learning effects, Applied Mathematical Modelling, 35 (2011), 1492-1496.  doi: 10.1016/j.apm.2010.09.026. [7] C. Koulamas and G. J. Kyparisis, Single-machine scheduling problems with past-sequence-dependent delivery times, International Journal of Production Economics, 126 (2010), 264-266. [8] W. Kuo and D. Yang, Single machine scheduling with past-sequence-dependent setup times and learning effects, Information Processing Letters, 102 (2007), 22-26.  doi: 10.1016/j.ipl.2006.11.002. [9] W. Lee, A note on single-machine scheduling with general learning effect and past-sequence-dependent setup time, Computers and Mathematics with Applications, 62 (2011), 2095-2100.  doi: 10.1016/j.camwa.2011.06.057. [10] M. Liu, F. Zheng, C. n Chu and Y. Xu, New results on single-machine scheduling with past-sequence-dependent delivery times, Theoretical Computer Science, 438 (2012), 55-61.  doi: 10.1016/j.tcs.2012.03.009. [11] M. Liu, F. Zheng, C. Chu and Y. Xu, Single-machine scheduling with past-sequence-dependent delivery times and release times, Information Processing Letters, 112 (2012), 835-838.  doi: 10.1016/j.ipl.2012.07.002. [12] M. Liu, S. Wang and C. Chu, Scheduling deteriorating jobs with past-sequence-dependent delivery times, International Journal of Production Economics, 144 (2013), 418-421.  doi: 10.1016/j.ijpe.2013.03.009. [13] M. Liu, Parallel-machine scheduling with past-sequence-dependent delivery times and learning effect, Applied Mathematical Modelling, 37 (2013), 9630-9633.  doi: 10.1016/j.apm.2013.05.025. [14] L. Mingze, Single-Machine Scheduling Problems with Non-Linear Past-Sequence-Dependent Setup Times and Delivery Times, A a, 1 (2017), 2. [15] L. Shen and Y. Wu, Single machine past-sequence-dependent delivery times scheduling with general position-dependent and time-dependent learning effects, Applied Mathematical Modelling, 37 (2013), 5444-5451.  doi: 10.1016/j.apm.2012.11.001. [16] J. Wang, L. Sun and L. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Applied Mathematical Modelling, 34 (2010), 2813-2819.  doi: 10.1016/j.apm.2009.12.015. [17] J. Wang, Single-machine scheduling with past-sequence-dependent setup times and time-dependent learning effect, Computers and Industrial Engineering, 55 (2008), 584-591. [18] J. Wang, D. Wang, L. Wang, L. Lin, N. Yin and W. Wang, Single machine scheduling with exponential time-dependent learning effect and past-sequence-dependent setup times, Computers and Mathematics with Applications, 57 (2009), 9-16.  doi: 10.1016/j.camwa.2008.09.025. [19] N. Yin, L. Kang, P. Ji and J. Wang, Single machine scheduling with sum-of-logarithm-processing-times based deterioration, Information Sciences, 274 (2014), 303-309.  doi: 10.1016/j.ins.2014.03.004. [20] W. Yu-Bin and W. Jian-Jun, Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times, Neural Computing and Applications, 27 (2016), 937-943. [21] X. Zhang, G. Yan, W. Huang and G. Tang, A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects, Information Sciences, 187 (2012), 298-304.  doi: 10.1016/j.ins.2011.11.001.
The gantt chart for Example 1
The gantt chart for Example 2
The Gantt chart for the practical application
The results for Example 1
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 0 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 4.87 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 9.63 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 30.06
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 0 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 4.87 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 9.63 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 30.06
The results for Example 2
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 78 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 97.11 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 104.09 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 93.15 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 111.18
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 78 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 97.11 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 104.09 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 93.15 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 111.18
Laptop components of processing times and due dates
 Job Part Name Processing Time Due Date 1 CPU/GPU 24 9 2 IC Chip 6 35 3 Oscillator 9 25 4 Copper coil 12 20 5 Capacitor 70 5 6 Card slot 15 10 7 Ports 25 8 8 Cooling fan 42 6 9 Sub-PCB 35 7
 Job Part Name Processing Time Due Date 1 CPU/GPU 24 9 2 IC Chip 6 35 3 Oscillator 9 25 4 Copper coil 12 20 5 Capacitor 70 5 6 Card slot 15 10 7 Ports 25 8 8 Cooling fan 42 6 9 Sub-PCB 35 7
The results for Example 2
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{3[2]}$ 5.39 $C_{3[2]}$ 11.39 $T_{3[2]}$ 0 $p_{4[3]}$ 5.37 $C_{4[3]}$ 16.76 $T_{4[3]}$ 0 $p_{6[4]}$ 5.49 $C_{6[4]}$ 22.25 $T_{6[4]}$ 12.25 $p_{1[5]}$ 7.52 $C_{1[5]}$ 29.77 $T_{1[5]}$ 20.77 $p_{7[6]}$ 6.84 $C_{7[6]}$ 36.61 $T_{7[6]}$ 28.61 $p_{9[7]}$ 8.60 $C_{9[7]}$ 45.20 $T_{9[7]}$ 38.20 $p_{8[8]}$ 9.36 $C_{8[8]}$ 54.56 $T_{8[8]}$ 48.56 $p_{9[9]}$ 14.33 $q_{psd}$ 84.68 $C_{9[9]}$ 68.89+84.68=153.57 $T_{9[9]}$ 63.89
 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{3[2]}$ 5.39 $C_{3[2]}$ 11.39 $T_{3[2]}$ 0 $p_{4[3]}$ 5.37 $C_{4[3]}$ 16.76 $T_{4[3]}$ 0 $p_{6[4]}$ 5.49 $C_{6[4]}$ 22.25 $T_{6[4]}$ 12.25 $p_{1[5]}$ 7.52 $C_{1[5]}$ 29.77 $T_{1[5]}$ 20.77 $p_{7[6]}$ 6.84 $C_{7[6]}$ 36.61 $T_{7[6]}$ 28.61 $p_{9[7]}$ 8.60 $C_{9[7]}$ 45.20 $T_{9[7]}$ 38.20 $p_{8[8]}$ 9.36 $C_{8[8]}$ 54.56 $T_{8[8]}$ 48.56 $p_{9[9]}$ 14.33 $q_{psd}$ 84.68 $C_{9[9]}$ 68.89+84.68=153.57 $T_{9[9]}$ 63.89
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