# 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
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##### References:
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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