# American Institute of Mathematical Sciences

• Previous Article
Resource allocation flowshop scheduling with learning effect and slack due window assignment
• JIMO Home
• This Issue
• Next Article
Stability of a class of risk-averse multistage stochastic programs and their distributionally robust counterparts
doi: 10.3934/jimo.2021011

## Robust control in green production management

 1 Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China 2 The Research Center of Information Technology & Social and Economic Development, Hangzhou Dianzi University, Hangzhou 310018, China

* Corresponding author: Jian-Xin Guo, guojianxin@casisd.cn

Received  February 2020 Revised  October 2020 Published  December 2020

Fund Project: Support from National Key R&D Program of China, 2018YFC1509008; National Natural Science Foundation of China under grant No. 71801212 and 71701058.

This study proposes a robust control model for a production management problem related to dynamic pricing and green investment. Contaminants produced during the production process contribute to the accumulation of pollution stochastically. We derive optimal robust controls and identify conditions under which some concerns about model misspecification are discussed. We observe that optimal price and investment control decrease in the degree of robustness. We also examine the cost of robustness and the relevant importance of contributions in the overall value function. The theoretical results are applied to a calibrated model regarding production management. Finally, we compare robust choices with those in the benchmark stochastic model. Numerical simulations show that robust decision-making can indeed adjust investment decisions based on the level of uncertainty.

Citation: Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021011
##### References:

show all references

##### References:
$p^{*}(S)$ with different levels of robustness
$e^{*}(S)$ with different levels of model robustness
Price $p_t$ sensitivity analysis: the left image is $k_p = k_p(s, \theta)$, and the right image is $b_p = b_p(s, \theta)$
Abatement effort $e_t$ sensitivity analysis: the left image is $k_e = k_e(s, \theta)$, and the right image is $b_e = b_e(s, \theta)$
Demand $d_t$ sensitivity analysis: the left image is $k_d = k_d(s, \theta)$, and the right image is $b_d = b_d(s, \theta)$
Model parameters used in the simulation
 Parameter Description Value $T$ Time Duration 20 $S_0$ Pollution stock in the initial year 100 $r$ Compound Rate 5% $\alpha$ Potential market size 1001 $\beta$ Coefficient in the demand function associated with the sales price 11 $s$ Co-benefit of the abatement effort 0.12 $\tau$ Coefficient of environmental damage caused by accumulation of pollution 0.53 $c$ Cost coefficient associated with firm's pollution abatement effort 14 $\delta$ Pollution decay rate 0.13 $\sigma$ Volatility parameter in $S_t$ 105 $\theta$ Robust parameter depends 1 Parameters in the demand function mainly refer to the relevant literature [38, 5, 7] and are corrected. The special relative relationship must be kept reasonable. 2 This parameter mainly refers to [5], whose magnitude corresponds to $\alpha, \beta$. 3 This parameter mainly refers to [17] and is adjusted. 4 This parameter mainly refers to [7] and is adjusted. 5 This parameter mainly refers to [29] and is adjusted.
 Parameter Description Value $T$ Time Duration 20 $S_0$ Pollution stock in the initial year 100 $r$ Compound Rate 5% $\alpha$ Potential market size 1001 $\beta$ Coefficient in the demand function associated with the sales price 11 $s$ Co-benefit of the abatement effort 0.12 $\tau$ Coefficient of environmental damage caused by accumulation of pollution 0.53 $c$ Cost coefficient associated with firm's pollution abatement effort 14 $\delta$ Pollution decay rate 0.13 $\sigma$ Volatility parameter in $S_t$ 105 $\theta$ Robust parameter depends 1 Parameters in the demand function mainly refer to the relevant literature [38, 5, 7] and are corrected. The special relative relationship must be kept reasonable. 2 This parameter mainly refers to [5], whose magnitude corresponds to $\alpha, \beta$. 3 This parameter mainly refers to [17] and is adjusted. 4 This parameter mainly refers to [7] and is adjusted. 5 This parameter mainly refers to [29] and is adjusted.
Model parameters with different levels of robustness
 $l$ $m$ $n$ $\Delta_m$ $\Delta_n$ $\theta$=200 106304.82 -258.15 0.27 -138.15 0.11 $\theta$=500 81176.12 -153.22 0.16 -33.22 0.03 $\theta$=800 77749.76 -138.88 0.15 -18.88 0.02 $\theta=\infty$ 73244.95 -120.0 0.13 0 0
 $l$ $m$ $n$ $\Delta_m$ $\Delta_n$ $\theta$=200 106304.82 -258.15 0.27 -138.15 0.11 $\theta$=500 81176.12 -153.22 0.16 -33.22 0.03 $\theta$=800 77749.76 -138.88 0.15 -18.88 0.02 $\theta=\infty$ 73244.95 -120.0 0.13 0 0
Control processes with different levels of robustness
 $h_t$ $p_t$ $e_t$ $\theta$=200 -0.03$S$ + 12.90 -0.28$S$ + 185.34 -0.25$S$ + 125.43 $\theta$=500 -0.006$S$ + 3.06 -0.16$S$ + 130.38 -0.15$S$ + 75.46 $\theta$=800 -0.003$S$ + 1.73 -0.15$S$ + 122.87 -0.14$S$ + 68.63 $\theta=\infty$ 0 -0.13$S$ + 112.98 -0.12$S$ + 59.64
 $h_t$ $p_t$ $e_t$ $\theta$=200 -0.03$S$ + 12.90 -0.28$S$ + 185.34 -0.25$S$ + 125.43 $\theta$=500 -0.006$S$ + 3.06 -0.16$S$ + 130.38 -0.15$S$ + 75.46 $\theta$=800 -0.003$S$ + 1.73 -0.15$S$ + 122.87 -0.14$S$ + 68.63 $\theta=\infty$ 0 -0.13$S$ + 112.98 -0.12$S$ + 59.64
 [1] 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 [2] Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 [3] 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 [4] 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 [5] Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319 [6] David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial & Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121 [7] 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 [8] Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021002 [9] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [10] Yanjun He, Wei Zeng, Minghui Yu, Hongtao Zhou, Delie Ming. Incentives for production capacity improvement in construction supplier development. Journal of Industrial & Management Optimization, 2021, 17 (1) : 409-426. doi: 10.3934/jimo.2019118 [11] Shuai Huang, Zhi-Ping Fan, Xiaohuan Wang. Optimal financing and operational decisions of capital-constrained manufacturer under green credit and subsidy. Journal of Industrial & Management Optimization, 2021, 17 (1) : 261-277. doi: 10.3934/jimo.2019110 [12] Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 [13] 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 [14] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [15] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113 [16] Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 [17] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 [18] Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100 [19] Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65 [20] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409

2019 Impact Factor: 1.366