Optimal Production Control of Hybrid Manufacturing/Remanufacturing Failure-Prone Systems under Diffusion-Type Demand

ABSTRACT

The problem of production control for a hybrid manufacturing/remanufacturing system under uncertainty is analyzed. Two sources of uncertainty are considered: machines are subject to random breakdowns and repairs, and demand level is modeled as a diffusion type stochastic process. Contrary to most of studies where the demand level is considered constant and fewer results where the demand is modeled as a Poisson process with few discrete levels and exponentially distributed switching time, the demand is modeled here as a diffusion type process. In particular Wiener and Ornstein-Uhlenbeck processes for cumulative demands are analyzed. We formulate the stochastic control problem and develop optimality conditions for it in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). We demonstrate that HJB equations are of the second order contrary to the case of constant demand rate (corresponding to the average demand in our case), where HJB equations are linear PDEs. We apply the Kushner-type finite difference scheme and the policy improvement procedure to solve HJB equations numerically and show that the optimal production policy is of hedging-point type for both demand models we have introduced, similarly to the known case of a constant demand. Obtained results allow to compute numerically the optimal production policy in hybrid manufacturing/ remanufacturing systems taking into account the demand variability, and also show that Kushner-type discrete scheme can be successfully applied for solving underlying second order HJB equations.

The problem of production control for a hybrid manufacturing/remanufacturing system under uncertainty is analyzed. Two sources of uncertainty are considered: machines are subject to random breakdowns and repairs, and demand level is modeled as a diffusion type stochastic process. Contrary to most of studies where the demand level is considered constant and fewer results where the demand is modeled as a Poisson process with few discrete levels and exponentially distributed switching time, the demand is modeled here as a diffusion type process. In particular Wiener and Ornstein-Uhlenbeck processes for cumulative demands are analyzed. We formulate the stochastic control problem and develop optimality conditions for it in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). We demonstrate that HJB equations are of the second order contrary to the case of constant demand rate (corresponding to the average demand in our case), where HJB equations are linear PDEs. We apply the Kushner-type finite difference scheme and the policy improvement procedure to solve HJB equations numerically and show that the optimal production policy is of hedging-point type for both demand models we have introduced, similarly to the known case of a constant demand. Obtained results allow to compute numerically the optimal production policy in hybrid manufacturing/ remanufacturing systems taking into account the demand variability, and also show that Kushner-type discrete scheme can be successfully applied for solving underlying second order HJB equations.

Cite this paper

S. Ouaret, V. Polotski, J. Kenné and A. Gharbi, "Optimal Production Control of Hybrid Manufacturing/Remanufacturing Failure-Prone Systems under Diffusion-Type Demand,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 550-559. doi: 10.4236/am.2013.43079.

S. Ouaret, V. Polotski, J. Kenné and A. Gharbi, "Optimal Production Control of Hybrid Manufacturing/Remanufacturing Failure-Prone Systems under Diffusion-Type Demand,"

References

[1] M. Fleischmann, “Quantitative Models for Reverse Logistics,” Springer Verlag, New York, 2001. doi:10.1007/978-3-642-56691-2

[2] M. P. De Brito, R. Dekker and S. D. P. Flapper, “Reverse Logistics: A Review of Case Studies,” ERIM Report Series Reference No. ERS-2003-012-LIS, 2003.

[3] G. P. Kiesmüller and C. W. Scherer, “Computational Issues in a Stochastic Finite Horizon One Product Recovery Inventory Model,” European Journal of Operational Research, Vol. 146, No. 3, 2003, pp. 553-579. doi:10.1016/S0377-2217(02)00249-7

[4] K. Inderfurth, “Optimal Policies in Hybrid Manufacturing/Remanufacturing System with Product Substitution,” International Journal of Production Economics, Vol. 90, No. 3, 2004, pp. 325-343. doi:10.1016/S0925-5273(02)00470-X

[5] M. E. Nikoofal and S. M. M. Husseini, “An Inventory Model with Dependent Returns and Disposal Cost,” International Journal of Industrial Engineering Computations, Vol. 1, No. 1, 2010, pp. 45-54.

[6] S. Oscar and F. Silva, “Suboptimal Production Planning Policies for Closed-Loop System with Uncertain Levels of Demand and Return,” The 18th IFAC World Congress, Milano, 28 August-2 September 2011.

[7] I. Dobos, “Optimal Production-Inventory Strategies for HMMS-Type Reverse Logistics System,” International Journal of Production Economics, Vol. 81-82, 2003, pp. 351-360. doi:10.1016/S0925-5273(02)00277-3

[8] W. H. Fleming, H. M. Soner and S. P. Sethi, “A Stochastic Production Planning Problem with Random Demand,” SIAM Journal on Control and Optimization, Vol. 25, No. 6, 1987, pp.1494-1502. doi:10.1137/0325082

[9] E. K. Boukas and A. Haurie, “Manufacturing Flow Control and Preventive Maintenance: A Stochastic Approach,” IEEE Transactions on Automatic Control, Vol. 33, No. 9, 1990, pp. 1024-1031. doi:10.1109/9.58530

[10] J. P. Kenné and E. K. Boukas, “Hierarchical Control of Production and Maintenance Rates in Manufacturing Systems,” Journal of Quality in Maintenance Engineering, Vol. 9, No. 1, 2003, pp. 66-82. doi:10.1108/13552510310466927

[11] J. P. Kenné, P. Dejax and A. Gharbi, “Production Planning of a Hybrid Manufacturing-Remanufacturing System under Uncertainty within a Closed-Loop Supply Chain,” International Journal of Production Economics, Vol. 135, No. 1, 2012, pp. 81-93. doi:10.1016/j.ijpe.2010.10.026

[12] H. Yan and Q. Zhang, “A Numerical Method in Optimal Production and Setup Scheduling of Stochastic Manufacturing Systems,” IEEE Transaction on Automatic Control, Vol. 42, No. 10, 1997, pp. 441-449. doi:10.1109/9.633837

[13] E. Khemlnitsky, E. Presman and S. P. Sethi, “Optimal Production Control of a Failure-Prone Machine,” Annals of Operations Research, Vol. 182, 2011, pp. 67-86. doi:10.1007/s10479-009-0668-3

[14] J. R. Perkins and R. Srikant, “Failure-Prone Production Systems with Uncertain Demand,” IEEE Transaction on Automatic Control, Vol. 46, No. 3, 2001, pp. 441-449. doi:10.1109/9.911420

[15] E. Presman and S. P. Sethi, “Inventory Models with Continuous and Poisson Demands and Discounted and Average Costs,” Production and Operations Management, Vol. 15, No. 2, 2006, pp. 279-293. doi:10.1111/j.1937-5956.2006.tb00245.x

[16] A. Bensoussan, R. Liu and S. P. Sethi, “Optimality of an (s, S) Policy with Compound Poisson and Diffusion Demands: A Q.V.I. Approach,” SIAM Journal of Control and Optimization, Vol. 44, No. 5, 2005, pp. 1650-1676. doi:10.1137/S0363012904443737

[17] H. J. Kushner and P. Dupuis, “Numerical Methods for Stochastic Control Problems in Continuous Time,” Springer Verlag, New York, 1992. doi:10.1007/978-1-4684-0441-8

[18] W. H. Fleming and H. M. Soner, “Controlled Markov Processes and Viscosity Soutions,” Springer Verlag, New York, 2005.

[19] G. Barles and E. R. Jakobsen, “On the Convergence Rate of Approximation Schemes for HJB Equations,” Mathematical Modeling and Numerical Analysis, Vol. 36, No. 1, 2002, pp. 33-54. doi:10.1051/m2an:2002002

[20] N. V. Krylov, “On the Rate of Convergence of Finite-Difference Approximations for Bellman’s Equations with Variable Coeffcients,” Probability Theory and Related Fields, Vol. 117, No. 1, 2000, pp. 1-16. doi:10.1007/s004400050264

[1] M. Fleischmann, “Quantitative Models for Reverse Logistics,” Springer Verlag, New York, 2001. doi:10.1007/978-3-642-56691-2

[2] M. P. De Brito, R. Dekker and S. D. P. Flapper, “Reverse Logistics: A Review of Case Studies,” ERIM Report Series Reference No. ERS-2003-012-LIS, 2003.

[3] G. P. Kiesmüller and C. W. Scherer, “Computational Issues in a Stochastic Finite Horizon One Product Recovery Inventory Model,” European Journal of Operational Research, Vol. 146, No. 3, 2003, pp. 553-579. doi:10.1016/S0377-2217(02)00249-7

[4] K. Inderfurth, “Optimal Policies in Hybrid Manufacturing/Remanufacturing System with Product Substitution,” International Journal of Production Economics, Vol. 90, No. 3, 2004, pp. 325-343. doi:10.1016/S0925-5273(02)00470-X

[5] M. E. Nikoofal and S. M. M. Husseini, “An Inventory Model with Dependent Returns and Disposal Cost,” International Journal of Industrial Engineering Computations, Vol. 1, No. 1, 2010, pp. 45-54.

[6] S. Oscar and F. Silva, “Suboptimal Production Planning Policies for Closed-Loop System with Uncertain Levels of Demand and Return,” The 18th IFAC World Congress, Milano, 28 August-2 September 2011.

[7] I. Dobos, “Optimal Production-Inventory Strategies for HMMS-Type Reverse Logistics System,” International Journal of Production Economics, Vol. 81-82, 2003, pp. 351-360. doi:10.1016/S0925-5273(02)00277-3

[8] W. H. Fleming, H. M. Soner and S. P. Sethi, “A Stochastic Production Planning Problem with Random Demand,” SIAM Journal on Control and Optimization, Vol. 25, No. 6, 1987, pp.1494-1502. doi:10.1137/0325082

[9] E. K. Boukas and A. Haurie, “Manufacturing Flow Control and Preventive Maintenance: A Stochastic Approach,” IEEE Transactions on Automatic Control, Vol. 33, No. 9, 1990, pp. 1024-1031. doi:10.1109/9.58530

[10] J. P. Kenné and E. K. Boukas, “Hierarchical Control of Production and Maintenance Rates in Manufacturing Systems,” Journal of Quality in Maintenance Engineering, Vol. 9, No. 1, 2003, pp. 66-82. doi:10.1108/13552510310466927

[11] J. P. Kenné, P. Dejax and A. Gharbi, “Production Planning of a Hybrid Manufacturing-Remanufacturing System under Uncertainty within a Closed-Loop Supply Chain,” International Journal of Production Economics, Vol. 135, No. 1, 2012, pp. 81-93. doi:10.1016/j.ijpe.2010.10.026

[12] H. Yan and Q. Zhang, “A Numerical Method in Optimal Production and Setup Scheduling of Stochastic Manufacturing Systems,” IEEE Transaction on Automatic Control, Vol. 42, No. 10, 1997, pp. 441-449. doi:10.1109/9.633837

[13] E. Khemlnitsky, E. Presman and S. P. Sethi, “Optimal Production Control of a Failure-Prone Machine,” Annals of Operations Research, Vol. 182, 2011, pp. 67-86. doi:10.1007/s10479-009-0668-3

[14] J. R. Perkins and R. Srikant, “Failure-Prone Production Systems with Uncertain Demand,” IEEE Transaction on Automatic Control, Vol. 46, No. 3, 2001, pp. 441-449. doi:10.1109/9.911420

[15] E. Presman and S. P. Sethi, “Inventory Models with Continuous and Poisson Demands and Discounted and Average Costs,” Production and Operations Management, Vol. 15, No. 2, 2006, pp. 279-293. doi:10.1111/j.1937-5956.2006.tb00245.x

[16] A. Bensoussan, R. Liu and S. P. Sethi, “Optimality of an (s, S) Policy with Compound Poisson and Diffusion Demands: A Q.V.I. Approach,” SIAM Journal of Control and Optimization, Vol. 44, No. 5, 2005, pp. 1650-1676. doi:10.1137/S0363012904443737

[17] H. J. Kushner and P. Dupuis, “Numerical Methods for Stochastic Control Problems in Continuous Time,” Springer Verlag, New York, 1992. doi:10.1007/978-1-4684-0441-8

[18] W. H. Fleming and H. M. Soner, “Controlled Markov Processes and Viscosity Soutions,” Springer Verlag, New York, 2005.

[19] G. Barles and E. R. Jakobsen, “On the Convergence Rate of Approximation Schemes for HJB Equations,” Mathematical Modeling and Numerical Analysis, Vol. 36, No. 1, 2002, pp. 33-54. doi:10.1051/m2an:2002002

[20] N. V. Krylov, “On the Rate of Convergence of Finite-Difference Approximations for Bellman’s Equations with Variable Coeffcients,” Probability Theory and Related Fields, Vol. 117, No. 1, 2000, pp. 1-16. doi:10.1007/s004400050264