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doi: 10.3934/jimo.2021011
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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

Received  February 2020 Revised  October 2020 Early access 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.

Keywords: Robust control, model misspecification, pollution accumulation, green production.
Mathematics Subject Classification: Primary: 93E20; Secondary: 49L05.
Citation: Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021011
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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

M. Funke and M. Paetz, Environmental policy under model uncertainty: A robust optimal control approach, Climatic Change, 107 (2011), 225-239.   Google Scholar

[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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[12]

L. P. Hansen and T. J. Sargent, Robust control of forward-looking models, Journal of monetary Economics, 50 (2003), 581-604.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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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. Google Scholar

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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.  Google Scholar

[17]

E. Keeler, M. Spence and R. Zeckhauser, The Optimal Control of Pollution, Taylor & Francis Oxford, 1971. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[21]

J. Miao and A. Rivera, Robust contracts in continuous time, Econometrica, 84 (2016), 1405-1440.  doi: 10.3982/ECTA13127.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

C. Roseta-Palma and A. Xepapadeas, Robust control in water management, Journal of Risk and Uncertainty, 29 (2004), 21-34.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

show all references

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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Funke and M. Paetz, Environmental policy under model uncertainty: A robust optimal control approach, Climatic Change, 107 (2011), 225-239.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

L. P. Hansen and T. J. Sargent, Robust control of forward-looking models, Journal of monetary Economics, 50 (2003), 581-604.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

E. Keeler, M. Spence and R. Zeckhauser, The Optimal Control of Pollution, Taylor & Francis Oxford, 1971. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[21]

J. Miao and A. Rivera, Robust contracts in continuous time, Econometrica, 84 (2016), 1405-1440.  doi: 10.3982/ECTA13127.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

C. Roseta-Palma and A. Xepapadeas, Robust control in water management, Journal of Risk and Uncertainty, 29 (2004), 21-34.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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 [17] and is adjusted.
4 This parameter mainly refers to [7] and is adjusted.
5 This parameter mainly refers to [29] and is adjusted.
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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.
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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$ 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
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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$ 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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