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doi: 10.3934/dcdss.2020100

Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation

Kobamelo Mashaba 1, , Jianxing Li 2,, , Honglei Xu 1,, and  Xinhua Jiang 2,

1. 

School of Electrical Engineering, Computing, Mathematical Sciences, Curtin Univerity, Perth, 6845 WA, Australia
2. 

School of Information Science and Engineering, Fujian University of Technology, Fuzhou, Fujian 350118, China

* Corresponding author: Jianxing Li and Honglei Xu

Received  March 2018 Revised  September 2018 Published  September 2019

This paper studies new optimal control policies for solving complex decision-making problems encountered in industrial hybrid systems in a manufacturing setting where critical jobs exist in a busy structure. In such setting, different dynamical systems interlink each other and share common functions for smooth task execution. Entities arriving at shared resources compete for service. The interactions of industrial hybrid systems become more and more complex and need a suitable controller to achieve the best performance and to obtain the best possible service for each of the entities arriving at the system. To solve these challenges, we propose an optimal control policy to minimize the operational cost for the manufacturing system. Furthermore, we develop a hybrid model and a new smoothing algorithm for the cost balancing between the quality and the job tardiness by finding optimal service time of each job in the system.

Keywords: Hybrid systems, manufacturing process, optimal control, smoothing algorithms, system modeling.
Mathematics Subject Classification: Primary: 37C75, 93C10; Secondary: 34D20.
Citation: Kobamelo Mashaba, Jianxing Li, Honglei Xu, Xinhua Jiang. Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020100
References:
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M. Baker and J. Wurgler, Market timing and capital structure, The Journal of Finance, 57 (2002), 1-32. Google Scholar

[2]

P. I. Barton, C. K. Lee and M. Yunt, Optimization of hybrid systems, Computers and Chemical Engineering, 30 (2006), 1576-1589. doi: 10.1016/j.compchemeng.2006.05.024. Google Scholar

[3]

X. Cai, M. Lai, X. Li, Y. Li and X. Wu, Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels, European Journal of Operational Research, 233 (2014), 374-382. doi: 10.1016/j.ejor.2013.07.017. Google Scholar

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C. G. Cassandras, Q. Liu, K. Gokbayrak and D. L. Pepyne, Optimal control of a two-stage hybrid manufacturing system model, Decision and Control, Proceedings of the 38th IEEE Conference on, 1 (1999), 450-455. Google Scholar

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Y. C. Cho, C. G. Cassandras and D. L. Pepyne, Forward algorithms for optimal control of a class of hybrid systems, Decision and Control, 2000. Proceedings of the 39th IEEE Conference on, 1 (2000), 975-980. doi: 10.1109/CDC.2000.912900. Google Scholar

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A. K. Dhaiban, M. A. Baten and N. Aziz, An optimal inventory control in hybrid manufacturing/remanufacturing system with deteriorating and defective items, International Journal of Mathematics in Operational Research, 12 (2018), 66-90. doi: 10.1504/IJMOR.2018.088575. Google Scholar

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[10]

B. Liu, J. H. David and Z. Sun, Input-to-state-$KL$-stability and criteria for a class of hybrid dynamical systems, Applied Mathematics and Computation, 326 (2018), 124-140. doi: 10.1016/j.amc.2018.01.002. Google Scholar

[11]

M. Liu, X. Yang, J. Zhang and C. Chu, Scheduling a tempered glass manufacturing system: a three-stage hybrid flow shop model, International Journal of Production Research, 55 (2017), 6084-6107. doi: 10.1080/00207543.2017.1324222. Google Scholar

[12]

D. Mourtzis, M. Doukas and D. Bernidaki, Simulation in manufacturing: Review and challenges, Procedia CIRP, 25 (2014), 213-229. doi: 10.1016/j.procir.2014.10.032. Google Scholar

[13]

J.S. Pan, L. Kong, P.W. Tsai and V. Snasel, $\alpha$-fraction first strategy for hierarchical wireless sensor networks, Journal of Internet Technology, 19 (2018), 1717-1726. Google Scholar

[14]

D. L. Pepyne and C. G. Cassandras, Modeling, analysis, and optimal control of a class of hybrid systems, Discrete Event Dynamic Systems, 8 (1998), 175-201. doi: 10.1023/A:1008237701804. Google Scholar

[15]

D. L. Pepyne and C. G. Cassandras, Optimal control of hybrid systems in manufacturing, Proceedings of the IEEE, 88 (2000), 1108-1123. doi: 10.1109/5.871312. Google Scholar

[16]

A. J. Van Der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer London, 2000. doi: 10.1007/BFb0109998. Google Scholar

[17]

J. Wang, J. Zhao and X. Wang, Optimum policy in hybrid manufacturing/remanufacturing system, Computers and Industrial Engineering, 60 (2011), 411-419. doi: 10.1016/j.cie.2010.05.002. Google Scholar

[18]

W. Xia and Z. Wu, An effective hybrid optimization approach for multi-objective flexible job-shop scheduling problem, Computers and Industrial Engineering, 48 (2005), 409-425. doi: 10.1016/j.cie.2005.01.018. Google Scholar

[19]

X. Xie, H. Xu, X. Cheng and Y. Yu, Improved results on exponential stability of discrete-time switched delay systems, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 199-208. doi: 10.3934/dcdsb.2017010. Google Scholar

[20]

J. Ye, H. Xu, E. Feng and Z. Xiu, et al., Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556–1569.Google Scholar

[21]

Y. Zhang, M. Wang, H. Xu and K. L. Teo, Global stabilization of switched control systems with time delay, Nonlinear Analysis: Hybrid Systems, 14 (2014), 86-98. doi: 10.1016/j.nahs.2014.05.004. Google Scholar

[22]

G. Zhou, K. C. Toh and J. Sun, Efficient algorithms for the smallest enclosing ball problem, Computational Optimization and Applications, 30 (2005), 147-160. doi: 10.1007/s10589-005-4565-7. Google Scholar

show all references

References:
[1]

M. Baker and J. Wurgler, Market timing and capital structure, The Journal of Finance, 57 (2002), 1-32. Google Scholar

[2]

P. I. Barton, C. K. Lee and M. Yunt, Optimization of hybrid systems, Computers and Chemical Engineering, 30 (2006), 1576-1589. doi: 10.1016/j.compchemeng.2006.05.024. Google Scholar

[3]

X. Cai, M. Lai, X. Li, Y. Li and X. Wu, Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels, European Journal of Operational Research, 233 (2014), 374-382. doi: 10.1016/j.ejor.2013.07.017. Google Scholar

[4]

C. G. Cassandras, Q. Liu, K. Gokbayrak and D. L. Pepyne, Optimal control of a two-stage hybrid manufacturing system model, Decision and Control, Proceedings of the 38th IEEE Conference on, 1 (1999), 450-455. Google Scholar

[5]

Y. C. Cho, C. G. Cassandras and D. L. Pepyne, Forward algorithms for optimal control of a class of hybrid systems, Decision and Control, 2000. Proceedings of the 39th IEEE Conference on, 1 (2000), 975-980. doi: 10.1109/CDC.2000.912900. Google Scholar

[6]

A. K. Dhaiban, M. A. Baten and N. Aziz, An optimal inventory control in hybrid manufacturing/remanufacturing system with deteriorating and defective items, International Journal of Mathematics in Operational Research, 12 (2018), 66-90. doi: 10.1504/IJMOR.2018.088575. Google Scholar

[7]

M. R. Garey, D. S. Johnson and R. Sethi, The complexity of flowshop and job-shop scheduling, Mathematics of Operations Research, 1 (1976), 117-129. doi: 10.1287/moor.1.2.117. Google Scholar

[8]

M. Gazarik and Y. Wardi, Optimal release times in a single server: An optimal control perspective, IEEE Trans. Automat. Control, 43 (1998), 998-1002. doi: 10.1109/9.701110. Google Scholar

[9]

S. Kowalewski, O. Stursberg, M. Fritz, H. Graf, I. Hoffmann, J. Preubig, M. Remelhe, S. Simon and H. Treseler, A case study in tool-aided analysis of discretely controlled continuous systems: The two tanks problem, Hybrid Systems V, 1567 (1997), 163-185. doi: 10.1007/3-540-49163-5_9. Google Scholar

[10]

B. Liu, J. H. David and Z. Sun, Input-to-state-$KL$-stability and criteria for a class of hybrid dynamical systems, Applied Mathematics and Computation, 326 (2018), 124-140. doi: 10.1016/j.amc.2018.01.002. Google Scholar

[11]

M. Liu, X. Yang, J. Zhang and C. Chu, Scheduling a tempered glass manufacturing system: a three-stage hybrid flow shop model, International Journal of Production Research, 55 (2017), 6084-6107. doi: 10.1080/00207543.2017.1324222. Google Scholar

[12]

D. Mourtzis, M. Doukas and D. Bernidaki, Simulation in manufacturing: Review and challenges, Procedia CIRP, 25 (2014), 213-229. doi: 10.1016/j.procir.2014.10.032. Google Scholar

[13]

J.S. Pan, L. Kong, P.W. Tsai and V. Snasel, $\alpha$-fraction first strategy for hierarchical wireless sensor networks, Journal of Internet Technology, 19 (2018), 1717-1726. Google Scholar

[14]

D. L. Pepyne and C. G. Cassandras, Modeling, analysis, and optimal control of a class of hybrid systems, Discrete Event Dynamic Systems, 8 (1998), 175-201. doi: 10.1023/A:1008237701804. Google Scholar

[15]

D. L. Pepyne and C. G. Cassandras, Optimal control of hybrid systems in manufacturing, Proceedings of the IEEE, 88 (2000), 1108-1123. doi: 10.1109/5.871312. Google Scholar

[16]

A. J. Van Der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer London, 2000. doi: 10.1007/BFb0109998. Google Scholar

[17]

J. Wang, J. Zhao and X. Wang, Optimum policy in hybrid manufacturing/remanufacturing system, Computers and Industrial Engineering, 60 (2011), 411-419. doi: 10.1016/j.cie.2010.05.002. Google Scholar

[18]

W. Xia and Z. Wu, An effective hybrid optimization approach for multi-objective flexible job-shop scheduling problem, Computers and Industrial Engineering, 48 (2005), 409-425. doi: 10.1016/j.cie.2005.01.018. Google Scholar

[19]

X. Xie, H. Xu, X. Cheng and Y. Yu, Improved results on exponential stability of discrete-time switched delay systems, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 199-208. doi: 10.3934/dcdsb.2017010. Google Scholar

[20]

J. Ye, H. Xu, E. Feng and Z. Xiu, et al., Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556–1569.Google Scholar

[21]

Y. Zhang, M. Wang, H. Xu and K. L. Teo, Global stabilization of switched control systems with time delay, Nonlinear Analysis: Hybrid Systems, 14 (2014), 86-98. doi: 10.1016/j.nahs.2014.05.004. Google Scholar

[22]

G. Zhou, K. C. Toh and J. Sun, Efficient algorithms for the smallest enclosing ball problem, Computational Optimization and Applications, 30 (2005), 147-160. doi: 10.1007/s10589-005-4565-7. Google Scholar

Table 1.  Job arrival $ 0.4\le r_i \le 1.5 $
$ F_1 \, | \, \text{no-wait, seq-dependent}\, | \, x_{\max} $ $ F_1 \, | \, \text{wait, seq-dependent}\, | \, x_{\max} $
Job Arrival time Control input Completion time Processing cost Control input Completion time Processing cost
0.4 0.3162 0.5000 2.4500 0.4404 0.5940 1.4872
0.5 0.4472 0.7000 1.5900 0.4404 0.7950 1.7664
0.7 0.4472 0.9000 1.9100 0.4404 0.9959 2.1262
0.9 0.5300 1.1809 2.1777 0.5132 1.2593 2.4211
1.3 0.4472 1.5000 3.3500 0.447214 1.5000 3.3500
1.5 0.4423 1.6956 3.9998 0.4423 1.6956 3.9998
Total cost: 15.4775 Total cost: 15.1507
Table Options

Download as excel

$ F_1 \, | \, \text{no-wait, seq-dependent}\, | \, x_{\max} $ $ F_1 \, | \, \text{wait, seq-dependent}\, | \, x_{\max} $
Job Arrival time Control input Completion time Processing cost Control input Completion time Processing cost
0.4 0.3162 0.5000 2.4500 0.4404 0.5940 1.4872
0.5 0.4472 0.7000 1.5900 0.4404 0.7950 1.7664
0.7 0.4472 0.9000 1.9100 0.4404 0.9959 2.1262
0.9 0.5300 1.1809 2.1777 0.5132 1.2593 2.4211
1.3 0.4472 1.5000 3.3500 0.447214 1.5000 3.3500
1.5 0.4423 1.6956 3.9998 0.4423 1.6956 3.9998
Total cost: 15.4775 Total cost: 15.1507
Table 2.  Job arrival $ 1.1\le r_i \le 2.3 $
$ F_1 \, | \, \text{no-wait, seq-dependent}\, | \, x_{\max} $ $ F_1 \, | \, \text{wait, seq-dependent}\, | \, x_{\max} $
Job Arrival time Control input Completion time Processing cost Control input Completion time Processing cost
1.1 0.4963 1.3463 2.7057 0.4963 1.3463 2.7057
1.4 0.4472 1.6000 3.6600 0.4472 1.6000 3.6600
1.6 0.3162 1.7000 5.0900 0.4310 1.7857 4.3733
1.7 0.4204 1.8767 4.7668 0.4121 1.9530 5.1098
2.2 0.3162 2.3000 7.4900 0.3652 2.3393 7.2480
2.3 0.3690 2.4361 7.5503 0.3633 2.4713 7.9814
Total cost: 31.2628 Total cost: 31.0782
Table Options

Download as excel

$ F_1 \, | \, \text{no-wait, seq-dependent}\, | \, x_{\max} $ $ F_1 \, | \, \text{wait, seq-dependent}\, | \, x_{\max} $
Job Arrival time Control input Completion time Processing cost Control input Completion time Processing cost
1.1 0.4963 1.3463 2.7057 0.4963 1.3463 2.7057
1.4 0.4472 1.6000 3.6600 0.4472 1.6000 3.6600
1.6 0.3162 1.7000 5.0900 0.4310 1.7857 4.3733
1.7 0.4204 1.8767 4.7668 0.4121 1.9530 5.1098
2.2 0.3162 2.3000 7.4900 0.3652 2.3393 7.2480
2.3 0.3690 2.4361 7.5503 0.3633 2.4713 7.9814
Total cost: 31.2628 Total cost: 31.0782
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