Optimal feedback production for a single-echelon supply chain

K.H. Wong; Chi Kin Chan; H. W.J. Lee

doi:10.3934/dcdsb.2006.6.1431

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2006, Volume 6, Issue 6: 1431-1444. Doi: 10.3934/dcdsb.2006.6.1431

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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
3.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received: July 31, 2005

Revised: April 30, 2006

Published: October 2006

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Abstract

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.

Keywords:

Dynamical system in optimization and economics,
Optimal feedback synthesis,
Mathematical modeling,
Feedback control.

Mathematics Subject Classification: Primary: 37N40, 49N35, 93A30, 93B52.

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