• Previous Article
    Earth pressure field modeling for tunnel face stability evaluation of EPB shield machines based on optimization solution
  • DCDS-S Home
  • This Issue
  • Next Article
    Time-delay optimal control of a fed-batch production involving multiple feeds
June  2020, 13(6): 1711-1719. doi: 10.3934/dcdss.2020100

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

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.

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, 2020, 13 (6) : 1711-1719. 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. BartonC. 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. CaiM. LaiX. LiY. 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. CassandrasQ. LiuK. 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. ChoC. 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. DhaibanM. 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. GareyD. 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. KowalewskiO. StursbergM. FritzH. GrafI. HoffmannJ. PreubigM. RemelheS. 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. LiuJ. 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. LiuX. YangJ. 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. MourtzisM. 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. PanL. KongP.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. WangJ. 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. XieH. XuX. 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. ZhangM. WangH. 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. ZhouK. 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. BartonC. 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. CaiM. LaiX. LiY. 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. CassandrasQ. LiuK. 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. ChoC. 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. DhaibanM. 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. GareyD. 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. KowalewskiO. StursbergM. FritzH. GrafI. HoffmannJ. PreubigM. RemelheS. 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. LiuJ. 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. LiuX. YangJ. 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. MourtzisM. 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. PanL. KongP.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. WangJ. 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. XieH. XuX. 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. ZhangM. WangH. 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. ZhouK. 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
$ 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
$ 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]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[3]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[4]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[5]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[6]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[7]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[8]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[9]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179

[10]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[11]

Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144

[12]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[13]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322

[14]

Min Ji, Xinna Ye, Fangyao Qian, T.C.E. Cheng, Yiwei Jiang. Parallel-machine scheduling in shared manufacturing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020174

[15]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[16]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

[17]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[18]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[19]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051

[20]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (59)
  • HTML views (283)
  • Cited by (0)

[Back to Top]