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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:

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##### 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
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