Feedback optimal control of dynamic stochastic two-machine flowshop with a finite buffer

Abstract: This paper examines the optimization of production involving a tandem two-machine system producing a single part type, with each machine being subject to random breakdowns and repairs. An analytical model is formulated with a view to solving an optimal stochastic production problem of the system with machines having up-downtime non-exponential distributions. The model developed is obtained by using a dynamic programming approach and a semi-Markov process. The control problem aims to find the production rates needed by the machines to meet the demand rate, through a minimization of the inventory/shortage cost. Using the Bellman principle, the optimality conditions obtained satisfy the Hamilton-Jacobi-Bellman equation, which depends on time and system states, and ultimately, leads to a feedback control. Consequently, the new model enables us to improve the coefficient of variation (CVup/down) to be less than one while it is equal to one in Markov model. Heuristics methods are used to involve the problem because of the difficulty of the analytical model using several states, and to show what control law should be used in each system state (i.e., including Kanban, feedback and CONWIP control). Numerical methods are used to solve the optimality conditions and to show how a machine should produce.

How to cite this paper

 Diep, T., Kenné, J & Dao, T. (2010). Feedback optimal control of dynamic stochastic two-machine flowshop with a finite buffer.International Journal of Industrial Engineering Computations , 1(2), 95-120.

Refrences

Abbad, M. and Filar, J.A. (1991). Perturbation theory for semi-Markov control problems. IEEE Conference on Decision and Control, 489-493.

Akella, R., Kumar, P.R. (1986). Optimal control of production rate in a failure prone manufacturing system. IEEE Transaction on Automation Control, 31 (2), 116-126.

Bai, S.X., Gershwin, S.B. (1996). Scheduling manufacturing systems with work-in-process inventoty control: reentrant systems. OR Spektrum, 18, 187-195.

Becker, G., Camarinopoulos, L. & Zioutas, G. (2000). A semi-Markov model allowing for inhomogenities with respect to process time. Reliability Engineering and System Safety, 70, 41-48.

Bielecki, T., Kumar, P.R. (1988). Optimality of zero-inventory policies for unreliable manufacturing systems. Operation Research, 36 (4), 532-541.

Bonvik, A.M. (1996). Performance analysis of manufacturing systems under hybrid control policies. In Electrical Engineering and Computer Science. 1996, Massachusetts Institute of Technology. 179p.

Boukas, E.K. (1988). Commande optimale stochastique appliqueé aux systems de production. In Génie Électrique. École Polytechnique de Montréal: Montréal.

Ciprut, P., Hongler, O.M. & Salama, Y. (2000). Fluctuations of the production output of transfer lines. Journal of Intelligent Manufacturing, 11, 183-189.

Dallery, Y., David, R. & Xie, X.L. (1989). Approximate Analysis of transfer lines with unreliable machines and finite buffers. IEEE Transaction on Automation Control, 34(9), 943-953.

D'Amico, G., Janssen, J. & Manca, R. (2005). Homogeneous semi-Markov reliability models for credit risk management. Decisions in Economics and Finance, 28, 79-93.

Davis, H.M.A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models (with discussion). Journal of the Royal Statistical Society S.B. Methodological, 46 (3), 353-388.

Engell, S. (2007). Feedback control for optimal process operation. Journal of Process Control, 17, 203-219.

Enginarlar, E., Li, J. & Meerkov S.M. (2005). Lean buffering in serial production lines with non-exponential machines. OP Spectrum, 27, 195-219.

Fleming, W.H., Soner, H.M. (2006) Controlled Markov processes and viscosity solutions. Spinger.

Fong, N.T., Zhou, X.Y. (2000). Optimal feedback controls in deterministic two-machine flowshops with finite buffers. IEEE Transaction on Automation Control, 45(6),1198-1203.

Gershwin, S.B. (2002). Manufacturing systems engineering. Massachusetts Institute of Technology.

Gershwin, S.B. (1989). Hierarchical flow control: A framework for scheduling and planning discrete events in manufacturing systems. Proceedings of the IEEE, 77, 195-209.

Glynn, P. (1989). A GSMP formalism for discrete event systems. IEEE Transaction on Automation Control, 77(1), 14-23.

Grabsky, F. (2003). The reliability of an object with semi-Markov failure rate. Applied Mathematics and Computation, 135,1-16.

Howard, R.A. (1971). Dynamic probabilitic systems. Vol. 1-2. New York: Wiley.

Janssen, J., Manca, R. (2007). Semi-Markov risk models for finance, insurance, and reliability. Springer.

Kenne, J.P., Gharbi, A. (2001). A simulation optimization approach in production planning of failure prone manufacturing systems. Journal of Intelligent Manufacturing, 12, 421-431.

Kim, J., Gershwin, S.B. (2008). Analysis of long flow lines with quality and operational failures. IIE Transactions, 40, 284-296.

Kim, J., Gershwin, S.B. (2005). Integrated quality and quantity modeling of production lines. OP Spectrum, 27, 287-314.

Kimemia, J., Gershwin, S.B. (1983) An algorithm for the computer control of a flexible manufacturing system. IIE Transaction, 15 (4), 353-362.

Kushner, H.J., Dupuis, P. (2001). Numerical method for stochastic control problems in continuous time. New York: Springer.

Lavoie, P., Kenne, J.P. & Gharbi, A. (2009). Optimization of production control policies in failure-prone homogenous transfer lines. IIE Transaction, 41, 209-222.

Lehoczky, J., Sethi, S.P., Soner, H.M. & Taksar. (1991). An asymptotic analysis of hierarchical control of manufacturing systems under uncertainty. Math Operation Research, 16, 596-608.

Li, J., Meerkov, S.M. (2005). On the coefficiens of variation of uptime and downtime in manufacturing equipment. Mathematical Problems in Engineering, 1, 1-6.

Perkins, J.R., Srikant, R. (1997). Scheduling multiple part-types in an unrelibale single-machine manufacturing system. IEEE Transaction on Automation Control, 42 (3), 364-376.

Presman, E., Sethi, R.P., Zhang, H. & Zhang, H. (2002). On optimality of stochastic N-Machine flowshop with long-run average cost. Stochastic Theory and Control, 280, 399-417.

Rishel, R. (1975). Dynamic programming and minimum principles for systems with jump Markov disturbances. SIAM Journal on Control, 13 (2), 338-371.

Ross, S.M. (2003). Introduction to probability models. 6th ed. Academic Press.

Sethi, S.P., Zhang, H. & Zhang, Q. (2005). Average-cost control of stochastic manufacturing systems. Stochastic Modeling and Applied Probability. New York: Springer.

Sethi, S.P., Zhang, Q. & Zhou, X.Y. (1997). Hierarchical production controls in a stochastic two-machine flowshop with a finite internal buffer. IEEE Transaction on Automation Control, 13 (1), 1-12.

Sethi, S.P., Zhang, Q. & Zhou, X.Y. (1994). Hierarchical controls in stochastic manufacturing systems with convex costs. Journal of Optimization Theory and Applications, 80(2), 299-317.

Shu, C., Perkins, J.R. (2001). Optimal PHP production of multiple part-types on a failure-prone machine with quadratic buffer costs. IEEE Transaction on Automation Control, 46(4), 541-549.

Tan, B., Gershwin, S.B. (2009). Analysis of a general Markovian two-stage continuous-flow production system with a finite buffer. International Journal of Production Economics, 120, 327-339.