October  2017, 13(4): 2033-2047. doi: 10.3934/jimo.2017030

Stability strategies of manufacturing-inventory systems with unknown time-varying demand

1. 

Department of Economics and Trade, Hunan University, Changsha, Hunan 410079, China

2. 

College of Business, Hunan Normal University, Changsha, Hunan 410081, China

3. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5

4. 

School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan 410004, China

Corresponding author: Mingyong Lai

Received  April 2016 Revised  February 2017 Published  April 2017

Fund Project: This work is partially supported by the National Natural Science Foundation of China (No.71501069, No.71420107027, No.71201013), the Hunan Provincial Natural Science Foundation of China (No.2015JJ3090, No.2017JJ3330), the China Postdoctoral Science Foundation Funded Project (No.2016M590742), the Humanities and Social Sciences Project of the Ministry of Education of China (No.14YJA860025), the Quantitative Economics Key Laboratory Program of Guangxi (No.2015ZD01)

For a manufacturing-inventory system, its stability and robustness are of particular important. In the literature, most manufacturing-inventory models are constructed based on deterministic demand assumption. However, demands for many real-world manufacturing-inventory systems are non-deterministic. To minimize the gap between theory and practice, we construct two models for the inventory control problem involving multi-machine and multi-product manufacturing-inventory systems with uncertain time-varying demand, where physical decay rate and shelf life are accounted for in the models. We then design state feedback control strategies to stabilize such systems. Based on the Lyapunov stability theory, sufficient conditions for robust stability, stabilization and control are derived in the form of linear matrix inequalities. Numerical examples are presented to show the potential applications of the proposed models.

Citation: Lizhao Yan, Fei Xu, Yongzeng Lai, Mingyong Lai. Stability strategies of manufacturing-inventory systems with unknown time-varying demand. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2033-2047. doi: 10.3934/jimo.2017030
References:
[1]

S. C. Aggarwal, Purchase-inventory decision models for inflationary conditions, Interfaces, 11 (1981), 18-23. doi: 10.1287/inte.11.4.18. Google Scholar

[2]

Z. T. Balkhi and L. Benkherouf, A production lot size inventory model for deteriorating items and arbitrary production and demand rates, European Journal of Operational Research, 92 (1996), 302-309. doi: 10.1016/0377-2217(95)00148-4. Google Scholar

[3]

D. BijulalJ. Venkateswaran and N. Hemachandra, Service levels, system cost and stability of production-inventory control systems, International Journal of Production Research, 49 (2011), 7085-7105. doi: 10.1080/00207543.2010.538744. Google Scholar

[4]

E. K. BoukasP. Shi and R. K. Agarwal, An application of robust control technique to manufacturing systems with uncertain processing time, Optimal Control Applications and Methods, 21 (2000), 257-268. doi: 10.1002/oca.677. Google Scholar

[5]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. Google Scholar

[6]

M. W. BraunD. E. RiveraM. E. FloresW. M. Carlyle and K. G. Kempf, A model predictive control framework for robust management of multi-product, multi-echelon demand networks, Annual Reviews in Control, 27 (2003), 229-245. Google Scholar

[7]

S. M. Disney and D. R. Towill, A discrete transfer function model to determine the dynamic stability of a vendor managed inventory supply chain, International Journal of Production Research, 40 (2002), 179-204. doi: 10.1080/00207540110072975. Google Scholar

[8]

G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Springer, New York, NY, 2000. doi: 10.1007/978-1-4757-3290-0. Google Scholar

[9]

Y. Feng and H. Yan, Optimal production control in a discrete manufacturing system with unreliable machines and random demand, IEEE Transactions on Automatic Control, 45 (2000), 2280-2296. doi: 10.1109/9.895564. Google Scholar

[10]

J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, MA, 1961. Google Scholar

[11]

X. Fu, B. Yan and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005.Google Scholar

[12]

A. Gharbi and J. P. Kenne, Optimal production control problem in stochastic multipleproduct multiple-machine manufacturing systems, IIE Transactions, 35 (2003), 941-952. doi: 10.1080/07408170309342346. Google Scholar

[13]

A. GharbiJ. P. Kenne and A. Hajji, Operational level-based policies in production rate control of unreliable manufacturing systems with set-ups, International Journal of Production Research, 44 (2006), 545-567. doi: 10.1080/00207540500270364. Google Scholar

[14]

R. W. Gruvvstrom and T. T. Huynh, Multi-level, multi-stage capacity constrained production-inventory systems in discrete-time with non-zero lead times using MRP theory, International Journal of Production Economics, 101 (2006), 53-62. Google Scholar

[15]

L. Huang, Linear Algebra in Systems and Control, Science Press, Beijing, 1984.Google Scholar

[16]

S. Khanra and K. S. Chaudhuri, A note on an order-level inventory model for a deteriorating item with time dependent quadratic demand, Computers and Operations Research, 30 (2003), 1901-1916. doi: 10.1016/S0305-0548(02)00113-2. Google Scholar

[17]

P. H. LinD. S. WongS. S. JangS. S. Shieh and J. Z. Chu, Controller design and reduction of bullwhip for a model supply chain system using z-transform analysis, Journal of Process Control, 14 (2004), 487-499. doi: 10.1016/j.jprocont.2003.09.005. Google Scholar

[18]

M. Ortega and L. Lin, Control theory applications to the production-inventory problem: A review, International Journal of Production Research, 42 (2003), 2303-2322. doi: 10.1080/00207540410001666260. Google Scholar

[19]

S. SanaS. K. Goyal and K. S. Chaudhuri, A production-inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[20]

S. Sana, A production-inventory model in an imperfect production process, European Journal of Operational Research, 200 (2010), 451-464. doi: 10.1016/j.ejor.2009.01.041. Google Scholar

[21]

P. S. Simeonov and D. D. Bainov, Stability of the solutions of singularly perturbed systems with impulsive effect, Journal of Mathematical Analysis and Applications, 136 (1988), 575-588. doi: 10.1016/0022-247X(88)90106-0. Google Scholar

[22]

C. K. Sivashankari and S. Panayappan, Productive inventory model for two-level production with deteriorative items and shortages, The International Journal of Advanced Manufacturing Technology, 76 (2015), 2003-2014. Google Scholar

[23]

D. P. Song and Y. X. Sun, Optimal service control of a serial production line with unreliable workstations and random demand, Automatica, 34 (1998), 1047-1060. doi: 10.1016/S0005-1098(98)00050-8. Google Scholar

[24]

T. Tanthatemee and B. Phruksaphanrat, Fuzzy inventory control system for uncertain demand and supply, Proceedings of IMEC, 2 (2012), 1224-1229. Google Scholar

[25]

A. A. TeleizadehM. Noori-daryan and L. E. Cardenas-Barron, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. Google Scholar

[26]

D. R. Towill, Dynamic analysis of an inventory and order based produciton control system, International Journal of Production Research, 20 (1982), 671-687. Google Scholar

[27]

J. Venkateswaran and Y. J. Son, Effect of information update frequency on the stability of production-inventory control systems, International Journal of Production Economics, 106 (2007), 171-190. doi: 10.1016/j.ijpe.2006.06.001. Google Scholar

[28]

X. WangS. M. Disney and J. Wang, Stability analysis of constrained inventory systems with transportation delay, European Journal of Operational Research, 223 (2012), 86-95. doi: 10.1016/j.ejor.2012.06.014. Google Scholar

[29]

X. WangS. M. Disney and J. Wang, Exploring the oscillatory dynamics of a forbidden returns inventory system, International Journal of Production Economics, 147 (2014), 3-12. doi: 10.1016/j.ijpe.2012.08.013. Google Scholar

[30]

R. D. H. Warburton, Further insights into 'the stability of supply chains?, International Journal of Production Research, 42 (2004), 639-648. Google Scholar

[31]

H. P. Wiendahl and J. W. Breithaupt, Automatic production control applying control theory, International Journal of Production Economics, 63 (2000), 33-46. doi: 10.1016/S0925-5273(98)00253-9. Google Scholar

[32]

Y. W. Zhou, A production-inventory model for a finite time-horizon with linear trend in demand and shortages, Systems Engineering-Theory and Practice, 5 (1995), 43-49. Google Scholar

show all references

References:
[1]

S. C. Aggarwal, Purchase-inventory decision models for inflationary conditions, Interfaces, 11 (1981), 18-23. doi: 10.1287/inte.11.4.18. Google Scholar

[2]

Z. T. Balkhi and L. Benkherouf, A production lot size inventory model for deteriorating items and arbitrary production and demand rates, European Journal of Operational Research, 92 (1996), 302-309. doi: 10.1016/0377-2217(95)00148-4. Google Scholar

[3]

D. BijulalJ. Venkateswaran and N. Hemachandra, Service levels, system cost and stability of production-inventory control systems, International Journal of Production Research, 49 (2011), 7085-7105. doi: 10.1080/00207543.2010.538744. Google Scholar

[4]

E. K. BoukasP. Shi and R. K. Agarwal, An application of robust control technique to manufacturing systems with uncertain processing time, Optimal Control Applications and Methods, 21 (2000), 257-268. doi: 10.1002/oca.677. Google Scholar

[5]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. Google Scholar

[6]

M. W. BraunD. E. RiveraM. E. FloresW. M. Carlyle and K. G. Kempf, A model predictive control framework for robust management of multi-product, multi-echelon demand networks, Annual Reviews in Control, 27 (2003), 229-245. Google Scholar

[7]

S. M. Disney and D. R. Towill, A discrete transfer function model to determine the dynamic stability of a vendor managed inventory supply chain, International Journal of Production Research, 40 (2002), 179-204. doi: 10.1080/00207540110072975. Google Scholar

[8]

G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Springer, New York, NY, 2000. doi: 10.1007/978-1-4757-3290-0. Google Scholar

[9]

Y. Feng and H. Yan, Optimal production control in a discrete manufacturing system with unreliable machines and random demand, IEEE Transactions on Automatic Control, 45 (2000), 2280-2296. doi: 10.1109/9.895564. Google Scholar

[10]

J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, MA, 1961. Google Scholar

[11]

X. Fu, B. Yan and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005.Google Scholar

[12]

A. Gharbi and J. P. Kenne, Optimal production control problem in stochastic multipleproduct multiple-machine manufacturing systems, IIE Transactions, 35 (2003), 941-952. doi: 10.1080/07408170309342346. Google Scholar

[13]

A. GharbiJ. P. Kenne and A. Hajji, Operational level-based policies in production rate control of unreliable manufacturing systems with set-ups, International Journal of Production Research, 44 (2006), 545-567. doi: 10.1080/00207540500270364. Google Scholar

[14]

R. W. Gruvvstrom and T. T. Huynh, Multi-level, multi-stage capacity constrained production-inventory systems in discrete-time with non-zero lead times using MRP theory, International Journal of Production Economics, 101 (2006), 53-62. Google Scholar

[15]

L. Huang, Linear Algebra in Systems and Control, Science Press, Beijing, 1984.Google Scholar

[16]

S. Khanra and K. S. Chaudhuri, A note on an order-level inventory model for a deteriorating item with time dependent quadratic demand, Computers and Operations Research, 30 (2003), 1901-1916. doi: 10.1016/S0305-0548(02)00113-2. Google Scholar

[17]

P. H. LinD. S. WongS. S. JangS. S. Shieh and J. Z. Chu, Controller design and reduction of bullwhip for a model supply chain system using z-transform analysis, Journal of Process Control, 14 (2004), 487-499. doi: 10.1016/j.jprocont.2003.09.005. Google Scholar

[18]

M. Ortega and L. Lin, Control theory applications to the production-inventory problem: A review, International Journal of Production Research, 42 (2003), 2303-2322. doi: 10.1080/00207540410001666260. Google Scholar

[19]

S. SanaS. K. Goyal and K. S. Chaudhuri, A production-inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[20]

S. Sana, A production-inventory model in an imperfect production process, European Journal of Operational Research, 200 (2010), 451-464. doi: 10.1016/j.ejor.2009.01.041. Google Scholar

[21]

P. S. Simeonov and D. D. Bainov, Stability of the solutions of singularly perturbed systems with impulsive effect, Journal of Mathematical Analysis and Applications, 136 (1988), 575-588. doi: 10.1016/0022-247X(88)90106-0. Google Scholar

[22]

C. K. Sivashankari and S. Panayappan, Productive inventory model for two-level production with deteriorative items and shortages, The International Journal of Advanced Manufacturing Technology, 76 (2015), 2003-2014. Google Scholar

[23]

D. P. Song and Y. X. Sun, Optimal service control of a serial production line with unreliable workstations and random demand, Automatica, 34 (1998), 1047-1060. doi: 10.1016/S0005-1098(98)00050-8. Google Scholar

[24]

T. Tanthatemee and B. Phruksaphanrat, Fuzzy inventory control system for uncertain demand and supply, Proceedings of IMEC, 2 (2012), 1224-1229. Google Scholar

[25]

A. A. TeleizadehM. Noori-daryan and L. E. Cardenas-Barron, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. Google Scholar

[26]

D. R. Towill, Dynamic analysis of an inventory and order based produciton control system, International Journal of Production Research, 20 (1982), 671-687. Google Scholar

[27]

J. Venkateswaran and Y. J. Son, Effect of information update frequency on the stability of production-inventory control systems, International Journal of Production Economics, 106 (2007), 171-190. doi: 10.1016/j.ijpe.2006.06.001. Google Scholar

[28]

X. WangS. M. Disney and J. Wang, Stability analysis of constrained inventory systems with transportation delay, European Journal of Operational Research, 223 (2012), 86-95. doi: 10.1016/j.ejor.2012.06.014. Google Scholar

[29]

X. WangS. M. Disney and J. Wang, Exploring the oscillatory dynamics of a forbidden returns inventory system, International Journal of Production Economics, 147 (2014), 3-12. doi: 10.1016/j.ijpe.2012.08.013. Google Scholar

[30]

R. D. H. Warburton, Further insights into 'the stability of supply chains?, International Journal of Production Research, 42 (2004), 639-648. Google Scholar

[31]

H. P. Wiendahl and J. W. Breithaupt, Automatic production control applying control theory, International Journal of Production Economics, 63 (2000), 33-46. doi: 10.1016/S0925-5273(98)00253-9. Google Scholar

[32]

Y. W. Zhou, A production-inventory model for a finite time-horizon with linear trend in demand and shortages, Systems Engineering-Theory and Practice, 5 (1995), 43-49. Google Scholar

Figure 1.  Time history of system (4.1) with $u(t)=[u_1(t), u_2(t)]^T= [1.6, 1.2]^T $ and $\omega(t)=[\omega_1(t), \omega_2(t)]^T= [\omega_{01} \sin(a_1 t +b_1), \omega_{02} \sin(a_2 t +b_2)]^T $, where $\omega_{01}=0.15$, $\omega_{02}=0.1$, $a_1=16$, $b_1=10$, $a_2=20$ and $b_2=2$. Feedback control is applied to the system when $t=90$
Figure 2.  Time history of $z(t)$ and $\omega (t)$ for system (4.1) with feedback control applied to the system. Here, $u(t)=[u_1(t), u_2(t)]^T= [1.6, 1.2]^T $ and $\omega(t)=[\omega_1(t), \omega_2(t)]^T= [\omega_{01} \sin(a_1 t +b_1), \omega_{02} \sin(a_2 t +b_2)]^T $, where $\omega_{01}=0.15$, $\omega_{02}=0.1$, $a_1=16$, $b_1=10$, $a_2=20$ and $b_2=2$
Figure 3.  Time history of system (4.2) with $u(t)=[u_1(t), u_2(t)]^T= [1.5, 0.9]^T $ and $\omega(t)=[\omega_1(t), \omega_2(t)]^T= [\omega_{01} \sin(a_1 t +b_1), \omega_{02} \sin(a_2 t +b_2)]^T $, where $w_{01}=0.055$, $w_{02}=0.04$, $a_1=16$, $b_1=10$, $a_2=20$ and $b_2=2$. Feedback control is applied to the system when $t=50$.
Figure 4.  Time history of $z(t)$ and $\omega (t)$ for system (4.2) with feedback control applied to the system. Here, $u(t)=[u_1(t), u_2(t)]^T= [1.5, 0.9]^T $ and $\omega(t)=[\omega_1(t), \omega_2(t)]^T= [\omega_{01} \sin(a_1 t +b_1), \omega_{02} \sin(a_2 t +b_2)]^T $, where $w_{01}=0.055$, $w_{02}=0.04$, $a_1=16$, $b_1=10$, $a_2=20$ and $b_2=2$
[1]

Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653

[2]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667

[3]

Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193

[4]

Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

[5]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693

[6]

Po-Chung Yang, Hui-Ming Wee, Shen-Lian Chung, Yong-Yan Huang. Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand. Journal of Industrial & Management Optimization, 2013, 9 (4) : 769-787. doi: 10.3934/jimo.2013.9.769

[7]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

[8]

Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481

[9]

Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253

[10]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263

[11]

Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527

[12]

Tingwen Huang, Guanrong Chen, Juergen Kurths. Synchronization of chaotic systems with time-varying coupling delays. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1071-1082. doi: 10.3934/dcdsb.2011.16.1071

[13]

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

[14]

Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020

[15]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827

[16]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407

[17]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024

[18]

Mohammad-Sahadet Hossain. Projection-based model reduction for time-varying descriptor systems: New results. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 73-90. doi: 10.3934/naco.2016.6.73

[19]

Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535

[20]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (19)
  • HTML views (225)
  • Cited by (0)

Other articles
by authors

[Back to Top]