# American Institute of Mathematical Sciences

November  2006, 6(6): 1431-1444. doi: 10.3934/dcdsb.2006.6.1431

## Optimal feedback production for a single-echelon supply chain

 1 Department of Computational and Applied Mathematics, University of The Witwatersrand, Johannesburg 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China 3 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received  August 2005 Revised  May 2006 Published  August 2006

The dynamics of a supply chain have been modelled by several authors, yet no attempt has ever been made for finding the vendor's optimal production policy when facing such dynamics. In this paper, we model the dynamics of a supply chain as an infinite-horizon time-delayed optimal control problem. By approximating the time interval $[ 0,\infty )$ by $0,T_f$, we obtain an approximated problem $P(T_f)$ which can be easily solved by the control parametrization method. Moreover, we can show that the objective function of the approximated problem converges to that of the original problem as $T_f \to \infty$. Lastly, we also extend our method to solving a stochastic problem where the demand is a stochastic process with white noise input. Several examples for both the deterministic and the stochastic problems are solved to illustrate the efficiency of our method. In these examples, some important results relating the production rate to the demand are developed.
Citation: K.H. Wong, Chi Kin Chan, H. W.J. Lee. Optimal feedback production for a single-echelon supply chain. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1431-1444. doi: 10.3934/dcdsb.2006.6.1431
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