American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball
  • DCDS-S Home
  • This Issue
  • Next Article
    Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation
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:
[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

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
[1]

David L. Russell. Modeling and control of hybrid beam systems with rotating tip component. Evolution Equations & Control Theory, 2014, 3 (2) : 305-329. doi: 10.3934/eect.2014.3.305
[2]

Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633
[3]

Jean-Michel Coron, Matthias Kawski, Zhiqiang Wang. Analysis of a conservation law modeling a highly re-entrant manufacturing system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1337-1359. doi: 10.3934/dcdsb.2010.14.1337
[4]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585
[5]

Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007
[6]

Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
[7]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082
[8]

Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25
[9]

Yanan Mao, Caixia Gao, Ruidong Yan, Aruna Bai. Modeling and identification of hybrid dynamic system in microbial continuous fermentation. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 359-368. doi: 10.3934/naco.2015.5.359
[10]

Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355
[11]

Sébastien Court, Karl Kunisch, Laurent Pfeiffer. Hybrid optimal control problems for a class of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1031-1060. doi: 10.3934/dcdss.2018060
[12]

M. Alipour, M. A. Vali, A. H. Borzabadi. A hybrid parametrization approach for a class of nonlinear optimal control problems. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 493-506. doi: 10.3934/naco.2019037
[13]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100
[14]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[15]

Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1
[16]

David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
[17]

Divya Thakur, Belinda Marchand. Hybrid optimal control for HIV multi-drug therapies: A finite set control transcription approach. Mathematical Biosciences & Engineering, 2012, 9 (4) : 899-914. doi: 10.3934/mbe.2012.9.899
[18]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071
[19]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2020020
[20]

Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (25)
  • HTML views (232)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences