Robust control in green production management

Jian-Xin Guo; Xing-Long Qu

doi:10.3934/jimo.2021011

Article Contents

2022, Volume 18, Issue 2: 1115-1132. Doi: 10.3934/jimo.2021011

This issue Previous Article Spectral norm and nuclear norm of a third order tensor Next Article A new adaptive method to nonlinear semi-infinite programming

Robust control in green production management

Jian-Xin Guo^1, , and
Xing-Long Qu^2,

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

Received: January 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: March 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 49L05.

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Figure 1. $ p^{*}(S) $ with different levels of robustness

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Figure 2. $ e^{*}(S) $ with different levels of model robustness

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Figure 3. 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) $

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Figure 4. 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) $

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Figure 5. 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) $

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Table 1. 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 [16] and is adjusted. ⁴ This parameter mainly refers to [7] and is adjusted. ⁵ This parameter mainly refers to [28] and is adjusted.

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Table 2. 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

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Table 3. 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

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References

[1]	S. Athanassoglou and A. Xepapadeas, Pollution control with uncertain stock dynamics: When, and how, to be precautious, Journal of Environmental Economics and Management, 63 (2012), 304-320.
[2]	H. Barman, M. Pervin, S. K. Roy and G. W. Weber, Back-ordered inventory model with inflation in a cloudy-fuzzy environment, Journal of Industrial and Management Optimization, 13 (2020), Sourced from Microsoft Academic - https://academic.microsoft.com/paper/3009545728.
[3]	W. A. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control and hot spots in spatiotemporal economic systems, Dyn. Games Appl., 4 (2014), 257-289. doi: 10.1007/s13235-014-0109-z.
[4]	S. H. Chung, R. D. Weaver and T. L. Friesz, Strategic response to pollution taxes in supply chain networks: Dynamic, spatial, and organizational dimensions, European J. Oper. Res., 231 (2013), 314-327. doi: 10.1016/j.ejor.2013.05.036.
[5]	C. Dong, B. Shen, P.-S. Chow, L. Yang and C. T. Ng, Sustainability investment under cap-and-trade regulation, Ann. Oper. Res., 240 (2016), 509-531. doi: 10.1007/s10479-013-1514-1.
[6]	F. El Ouardighi, H. Benchekroun and D. Grass, Controlling pollution and environmental absorption capacity, Ann. Oper. Res., 220 (2014), 111-133. doi: 10.1007/s10479-011-0982-4.
[7]	F. El Ouardighi, J. E. Sim and B. Kim, Pollution accumulation and abatement policy in a supply chain, European J. Oper. Res., 248 (2016), 982-996. doi: 10.1016/j.ejor.2015.08.009.
[8]	M. Funke and M. Paetz, Environmental policy under model uncertainty: A robust optimal control approach, Climatic Change, 107 (2011), 225-239.
[9]	I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econom., 18 (1989), 141-151. doi: 10.1016/0304-4068(89)90018-9.
[10]	H. Golpira and E. B. Tirkolaee, Stable maintenance tasks scheduling: A bi-objective robust optimization model, Computers & Industrial Engineering, 137 (2019), Sourced from Microsoft Academic - https://academic.microsoft.com/paper/2964639183.
[11]	L. P. Hansen and T. J. Sargent, Robustness and ambiguity in continuous time, J. Econom. Theory, 146 (2011), 1195-1223. doi: 10.1016/j.jet.2011.01.004.
[12]	L. P. Hansen and T. J. Sargent, Robust control of forward-looking models, Journal of monetary Economics, 50 (2003), 581-604.
[13]	L. P. Hansen, T. J. Sargent, G. Turmuhambetova and N. Williams, Robust control and model misspecification, J. Econom. Theory, 128 (2006), 45-90. doi: 10.1016/j.jet.2004.12.006.
[14]	N. Jaakkola and F. van der Ploeg, Non-cooperative and cooperative climate policies with anticipated breakthrough technology, Journal of Environmental Economics and Management, 97 (2019), 42-66.
[15]	K. Jiang, R. Merrill, D. You, P. Pan and Z. Li, Optimal control for transboundary pollution under ecological compensation: A stochastic differential game approach, Journal of Cleaner Production, 241 (2019), 118391.
[16]	S. Jørgensen and G. Zaccour, Incentive equilibrium strategies and welfare allocation in a dynamic game of pollution control, Automatica J. IFAC, 37 (2001), 29-36. doi: 10.1016/S0005-1098(00)00119-9.
[17]	E. Keeler, M. Spence and R. Zeckhauser, The Optimal Control of Pollution, Taylor & Francis Oxford, 1971.
[18]	K. Kogan, F. El Ouardighi and A Herbon, Production with learning and forgetting in a competitive environment, International Journal of Production Economics, 189 (2017), 52-62.
[19]	N. Masoudi, M. Santugini and G. Zaccour, A dynamic game of emissions pollution with uncertainty and learning, Environmental and Resource Economics, 64 (2016), 349-372.
[20]	N. Masoudi and G. Zaccour, Emissions control policies under uncertainty and rational learning in a linear-state dynamic model, Automatica J. IFAC, 50 (2014), 719-726. doi: 10.1016/j.automatica.2013.11.040.
[21]	J. Miao and A. Rivera, Robust contracts in continuous time, Econometrica, 84 (2016), 1405-1440. doi: 10.3982/ECTA13127.
[22]	M. Pervin, S. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Ann. Oper. Res., 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5.
[23]	M. Pervin, S. K. Roy and G. W. Weber, An integrated vendor-buyer model with quadratic demand under inspection policy and preservation technology, Hacettepe Journal of Mathematics and Statistics, 49 (2020), 1168-1189.
[24]	M. Pervin, S. K. Roy and G. W.Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numer. Algebra Control Optim., 8 (2018), 169-191. doi: 10.3934/naco.2018010.
[25]	M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price-and stock-sensitive demand, J. Ind. Manag. Optim., 16 (2020), 1585-1612. doi: 10.3934/jimo.2019019.
[26]	M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, J. Ind. Manag. Optim., 15 (2019), 1345-1373. doi: 10.3934/jimo.2018098.
[27]	M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numer. Algebra Control Optim., 7 (2017), 21-50. doi: 10.3934/naco.2017002.
[28]	C. Roseta-Palma and A. Xepapadeas, Robust control in water management, Journal of Risk and Uncertainty, 29 (2004), 21-34.
[29]	S. K. Roy, M. Pervin and G. W. Weber, Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model, Numer. Algebra Control Optim., 10 (2020), 45-74. doi: 10.3934/naco.2019032.
[30]	S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, J. Ind. Manag. Optim., 16 (2020), 553-578. doi: 10.3934/jimo.2018167.
[31]	A. K. Sangaiah, E. B. Tirkolaee, A. Goli and S. Dehnavi-Arani, Robust optimization and mixed-integer linear programming model for LNG supply chain planning problem, Soft Computing, 24 (2020), 7885-7905.
[32]	E. B. Tirkolaee, S. Hadian, G.-W. Weber and I. Mahdavi, A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.
[33]	E. B. Tirkolaee, I. Mahdavi, M. M. S. Esfahani and G.-W. Weber, A robust green location-allocation-inventory problem to design an urban waste management system under uncertainty, Waste Management, 102 (2020), 340-350.
[34]	G. Vardas and A. Xepapadeas, Model uncertainty, ambiguity and the precautionary principle: implications for biodiversity management, Environmental and Resource Economics, 45 (2010), 379-404.
[35]	A. Yenipazarli, To collaborate or not to collaborate: Prompting upstream eco-efficient innovation in a supply chain, European J. Oper. Res., 260 (2017), 571-587. doi: 10.1016/j.ejor.2016.12.035.
[36]	J. Yu and M. L. Mallory, An optimal hybrid emission control system in a multiple compliance period model, Resource and Energy Economics, 39 (2015), 16-28.
[37]	Q. Zhang, W. Tang and J. Zhang, Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production, 112 (2016), 3267-3284.