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Optimal feedback production for a single-echelon supply chain. (English) Zbl 1110.37067

Summary: 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.

MSC:

37N40 Dynamical systems in optimization and economics
49N35 Optimal feedback synthesis
93A30 Mathematical modelling of systems (MSC2010)
93B52 Feedback control
91B62 Economic growth models
34K35 Control problems for functional-differential equations
90B30 Production models
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