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]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[2]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[3]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[4]

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

[5]

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

[6]

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

[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[8]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[9]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[10]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[11]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[12]

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

[13]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

[14]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[15]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[16]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[17]

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

[18]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[19]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[20]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (109)
  • HTML views (395)
  • Cited by (1)

Other articles
by authors

[Back to Top]