# American Institute of Mathematical Sciences

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. [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. [3] D. Bijulal, J. 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. [4] E. K. Boukas, P. 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. [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. [6] M. W. Braun, D. E. Rivera, M. E. Flores, W. 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. [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. [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. [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. [10] J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, MA, 1961. [11] X. Fu, B. Yan and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005. [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. [13] A. Gharbi, J. 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. [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. [15] L. Huang, Linear Algebra in Systems and Control, Science Press, Beijing, 1984. [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. [17] P. H. Lin, D. S. Wong, S. S. Jang, S. 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. [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. [19] S. Sana, S. 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. [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. [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. [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. [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. [24] T. Tanthatemee and B. Phruksaphanrat, Fuzzy inventory control system for uncertain demand and supply, Proceedings of IMEC, 2 (2012), 1224-1229. [25] A. A. Teleizadeh, M. 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. [26] D. R. Towill, Dynamic analysis of an inventory and order based produciton control system, International Journal of Production Research, 20 (1982), 671-687. [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. [28] X. Wang, S. 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. [29] X. Wang, S. 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. [30] R. D. H. Warburton, Further insights into 'the stability of supply chains?, International Journal of Production Research, 42 (2004), 639-648. [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. [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.

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##### References:
 [1] S. C. Aggarwal, Purchase-inventory decision models for inflationary conditions, Interfaces, 11 (1981), 18-23. doi: 10.1287/inte.11.4.18. [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. [3] D. Bijulal, J. 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. [4] E. K. Boukas, P. 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. [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. [6] M. W. Braun, D. E. Rivera, M. E. Flores, W. 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. [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. [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. [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. [10] J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, MA, 1961. [11] X. Fu, B. Yan and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005. [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. [13] A. Gharbi, J. 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. [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. [15] L. Huang, Linear Algebra in Systems and Control, Science Press, Beijing, 1984. [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. [17] P. H. Lin, D. S. Wong, S. S. Jang, S. 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. [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. [19] S. Sana, S. 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. [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. [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. [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. [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. [24] T. Tanthatemee and B. Phruksaphanrat, Fuzzy inventory control system for uncertain demand and supply, Proceedings of IMEC, 2 (2012), 1224-1229. [25] A. A. Teleizadeh, M. 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. [26] D. R. Towill, Dynamic analysis of an inventory and order based produciton control system, International Journal of Production Research, 20 (1982), 671-687. [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. [28] X. Wang, S. 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. [29] X. Wang, S. 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. [30] R. D. H. Warburton, Further insights into 'the stability of supply chains?, International Journal of Production Research, 42 (2004), 639-648. [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. [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.
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 $t=90$
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 $b_2=2$
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 $t=50$.
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 $b_2=2$
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