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doi: 10.3934/jimo.2022013
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Stochastic production planning with regime switching

1. 

Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st district, Postal Code: 010374, Postal Office: 22, Bucharest, Romania

2. 

Department of Mathematical Methods and Models, University Politehnica of Bucharest, Splaiul Independenţei, 6th district, Postal Code: 060042, Postal Office: 313, Bucharest, Romania

3. 

Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada

*Corresponding author: Dragos-Patru Covei

Received  October 2021 Revised  December 2021 Early access February 2022

Fund Project: Traian A. Pirvu acknowledges that this research was supported by NSERC grant 5-36700

This paper considers a stochastic production planning problem with regime switching. There are two regimes corresponding to different economic cycles. A factory is planning its production so as to minimize production costs. We analyze this problem and the optimal production is characterized through an elliptic system of partial differential equations which can be numerically solved. We perform a sensitivity analysis for which we provide an intuition. A model risk analysis reveals the effect of adding regime switching to the modelling.

Citation: Dragos-Patru Covei, Elena Cristina Canepa, Traian A. Pirvu. Stochastic production planning with regime switching. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022013
References:
[1]

A. Aghighi, A. Goli, B. Malmir and E. B. Tirkolaee, The stochastic location-routing-inventory problem of perishable products with reneging and balking, J. Ambient Intell. Humaniz. Comput., (2021). doi: 10.1007/s12652-021-03524-y.

[2]

L. Arnold, Stochastic Differential Equations, New York: Wiley, 1974.

[3]

E. C. CanepaD.-P. Covei and T. A. Pirvu, A Stochastic production planning problem, Fixed Point Theory, 23 (2022), 179-198.  doi: 10.24193/fpt-ro.2022.1.11.

[4]

G. ChenJ. Zhou and W.-M. Ni, Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurc. Chaos, 10 (2000), 1565-1612.  doi: 10.1142/S0218127400001006.

[5]

D.-P. Covei, An elliptic partial differential equation modelling a production planning problem, J. Appl. Anal. Comput., 11 (2021), 903-910.  doi: 10.11948/20200112.

[6]

D.-P. Covei, Symmetric solutions for an elliptic partial differential equation that arises in stochastic production planning with production constraints, Appl. Math. Comput., 350 (2019), 190-197.  doi: 10.1016/j.amc.2019.01.015.

[7]

D.-P. Covei and T. A. Pirvu, An elliptic partial differential equation and its application, Appl. Math. Lett., 101 (2020), 106059, 7 pp. doi: 10.1016/j.aml.2019.106059.

[8]

D.-P. Covei and T. A. Pirvu, A stochastic control problem with regime switching, Carpathian J. Math., 37 (2021), 427-440.  doi: 10.37193/CJM.2021.03.06.

[9]

J. Dong, A. A. Malikopoulos, S. M. Djouadi and T. Kuruganti, Application of optimal production control theory for home energy management in a micro grid, 2016 American Control Conference (ACC), (2016), 5014–5019. doi: 10.1109/ACC.2016.7526148.

[10]

A. Gharbi and J. P. Kenne, Optimal production control problem in stochastic multiple-product multiple-machine manufacturing systems, IIE Transactions, 35 (2003), 941-952.  doi: 10.1080/07408170309342346.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

A. Goli, E. Babaee Tirkolaee and N. S. Aydin, Fuzzy integrated cell formation and production scheduling considering automated guided vehicles and Human factors, IEEE Transactions on Fuzzy Systems, 29 2021, 3686–3695. doi: 10.1109/TFUZZ.2021.3053838.

[13]

A. GoliH. Khademi ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numer. Algebra Control Optim., 9 (2019), 187-209.  doi: 10.3934/naco.2019014.

[14]

A. GoliH. Khademi-ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. 

[15]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, Int. J. ITS Res., 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2.

[16]

A. GoliH. Khademi-ZareR. Tavakkoli-MoghaddamA. SadeghiehM. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network: Computation in Neural Systems, 32 (2021), 1-35.  doi: 10.1080/0954898X.2020.1849841.

[17]

W. H. FlemingS. P. Sethi and H. M. Soner, An optimal stochastic production planning problem with randomly luctuating demand, SIAM J. Control Optim., 25 (1987), 1494-1502.  doi: 10.1137/0325082.

[18]

M. Katsoulakis and S. Koike, Viscosity solutions of monotone systems for Dirichlet problems, Differ. Integral Equ., 7 (1994), 367-382. 

[19]

N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. 

[20]

M. Kwak, T. A. Pirvu and H. Zhang, A multi period equilibrium pricing model, J. Appl. Math., 2014 (2014), Art. ID 408685, 14 pp. doi: 10.1155/2014/408685.

[21]

R. Lotfi, B. Kargar, A. Gharehbaghi and G.-W. Weber, Viable medical waste chain network design by considering risk and robustness, Environ. Sci. Pollut. Res. Int., (2021), 1–16. doi: 10.1007/s11356-021-16727-9.

[22]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G.-W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, Int. J. Energy. Res., 45 (2021), 17749-17766.  doi: 10.1002/er.6943.

[23]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G.-W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numer. Algebra Control Optim., 11 (2021), 221-253.  doi: 10.3934/naco.2020023.

[24]

A. Orpel, Connected sets of positive solutions of elliptic systems in exterior domains, Monatsh. Math., 191 (2020), 761-778.  doi: 10.1007/s00605-019-01343-0.

[25]

S. M. Pahlevan, S. M. S. Hosseini and A. Goli, Sustainable supply chain network design using products' life cycle in the aluminum industry, Environ Sci Pollut Res., (2021), 1–25. doi: 10.1007/s11356-020-12150-8.

[26]

T. A. Pirvu and H. Zhang, Utility indifference pricing: A time consistent approach, Appl. Math. Finance., 20 (2013), 304-326.  doi: 10.1080/1350486X.2012.700575.

[27]

T. A. Pirvu and H. Zhang, Investment-consumption with regime-switching discount rates, Math Social Sci., 71 (2014), 142-150.  doi: 10.1016/j.mathsocsci.2014.07.001.

[28]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Springer Berlin Heidelberg, Volume 309 of Lecture Notes in Mathematics, 1973.

[29]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science, Martinus Nijhoff, Boston, 1981 (481 pages).

[30]

G. L. Thompson and S. P. Sethi, Turnpike horizons for production planning, Management Sci., 26 (1980), 229–241. doi: 10.1287/mnsc.26.3.229.

[31]

D. D. Yao, Q. Zhang and X. Y. Zhou, A Regime-Switching Model for European Options Stochastic Processes, Optimization, and Control Theory, Applications in Financial Engineering, Queueing Networks, and Manufacturing.

show all references

References:
[1]

A. Aghighi, A. Goli, B. Malmir and E. B. Tirkolaee, The stochastic location-routing-inventory problem of perishable products with reneging and balking, J. Ambient Intell. Humaniz. Comput., (2021). doi: 10.1007/s12652-021-03524-y.

[2]

L. Arnold, Stochastic Differential Equations, New York: Wiley, 1974.

[3]

E. C. CanepaD.-P. Covei and T. A. Pirvu, A Stochastic production planning problem, Fixed Point Theory, 23 (2022), 179-198.  doi: 10.24193/fpt-ro.2022.1.11.

[4]

G. ChenJ. Zhou and W.-M. Ni, Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurc. Chaos, 10 (2000), 1565-1612.  doi: 10.1142/S0218127400001006.

[5]

D.-P. Covei, An elliptic partial differential equation modelling a production planning problem, J. Appl. Anal. Comput., 11 (2021), 903-910.  doi: 10.11948/20200112.

[6]

D.-P. Covei, Symmetric solutions for an elliptic partial differential equation that arises in stochastic production planning with production constraints, Appl. Math. Comput., 350 (2019), 190-197.  doi: 10.1016/j.amc.2019.01.015.

[7]

D.-P. Covei and T. A. Pirvu, An elliptic partial differential equation and its application, Appl. Math. Lett., 101 (2020), 106059, 7 pp. doi: 10.1016/j.aml.2019.106059.

[8]

D.-P. Covei and T. A. Pirvu, A stochastic control problem with regime switching, Carpathian J. Math., 37 (2021), 427-440.  doi: 10.37193/CJM.2021.03.06.

[9]

J. Dong, A. A. Malikopoulos, S. M. Djouadi and T. Kuruganti, Application of optimal production control theory for home energy management in a micro grid, 2016 American Control Conference (ACC), (2016), 5014–5019. doi: 10.1109/ACC.2016.7526148.

[10]

A. Gharbi and J. P. Kenne, Optimal production control problem in stochastic multiple-product multiple-machine manufacturing systems, IIE Transactions, 35 (2003), 941-952.  doi: 10.1080/07408170309342346.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

A. Goli, E. Babaee Tirkolaee and N. S. Aydin, Fuzzy integrated cell formation and production scheduling considering automated guided vehicles and Human factors, IEEE Transactions on Fuzzy Systems, 29 2021, 3686–3695. doi: 10.1109/TFUZZ.2021.3053838.

[13]

A. GoliH. Khademi ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numer. Algebra Control Optim., 9 (2019), 187-209.  doi: 10.3934/naco.2019014.

[14]

A. GoliH. Khademi-ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. 

[15]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, Int. J. ITS Res., 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2.

[16]

A. GoliH. Khademi-ZareR. Tavakkoli-MoghaddamA. SadeghiehM. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network: Computation in Neural Systems, 32 (2021), 1-35.  doi: 10.1080/0954898X.2020.1849841.

[17]

W. H. FlemingS. P. Sethi and H. M. Soner, An optimal stochastic production planning problem with randomly luctuating demand, SIAM J. Control Optim., 25 (1987), 1494-1502.  doi: 10.1137/0325082.

[18]

M. Katsoulakis and S. Koike, Viscosity solutions of monotone systems for Dirichlet problems, Differ. Integral Equ., 7 (1994), 367-382. 

[19]

N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. 

[20]

M. Kwak, T. A. Pirvu and H. Zhang, A multi period equilibrium pricing model, J. Appl. Math., 2014 (2014), Art. ID 408685, 14 pp. doi: 10.1155/2014/408685.

[21]

R. Lotfi, B. Kargar, A. Gharehbaghi and G.-W. Weber, Viable medical waste chain network design by considering risk and robustness, Environ. Sci. Pollut. Res. Int., (2021), 1–16. doi: 10.1007/s11356-021-16727-9.

[22]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G.-W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, Int. J. Energy. Res., 45 (2021), 17749-17766.  doi: 10.1002/er.6943.

[23]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G.-W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numer. Algebra Control Optim., 11 (2021), 221-253.  doi: 10.3934/naco.2020023.

[24]

A. Orpel, Connected sets of positive solutions of elliptic systems in exterior domains, Monatsh. Math., 191 (2020), 761-778.  doi: 10.1007/s00605-019-01343-0.

[25]

S. M. Pahlevan, S. M. S. Hosseini and A. Goli, Sustainable supply chain network design using products' life cycle in the aluminum industry, Environ Sci Pollut Res., (2021), 1–25. doi: 10.1007/s11356-020-12150-8.

[26]

T. A. Pirvu and H. Zhang, Utility indifference pricing: A time consistent approach, Appl. Math. Finance., 20 (2013), 304-326.  doi: 10.1080/1350486X.2012.700575.

[27]

T. A. Pirvu and H. Zhang, Investment-consumption with regime-switching discount rates, Math Social Sci., 71 (2014), 142-150.  doi: 10.1016/j.mathsocsci.2014.07.001.

[28]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Springer Berlin Heidelberg, Volume 309 of Lecture Notes in Mathematics, 1973.

[29]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science, Martinus Nijhoff, Boston, 1981 (481 pages).

[30]

G. L. Thompson and S. P. Sethi, Turnpike horizons for production planning, Management Sci., 26 (1980), 229–241. doi: 10.1287/mnsc.26.3.229.

[31]

D. D. Yao, Q. Zhang and X. Y. Zhou, A Regime-Switching Model for European Options Stochastic Processes, Optimization, and Control Theory, Applications in Financial Engineering, Queueing Networks, and Manufacturing.

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