Abstract
We consider the planning of production over a prescribed period of time. More precisely, the problem is to minimize the cost integral (the time integral of the sum of the costs of production and storage) under the assumptions that the initial and final stocks are zero and that the production and the stock are nonnegative. Under this formulation, the problem can be considered as a Pontryagin-type problem with inequality constraints on the state variable and the control variable. We deduce from Pontryagin's maximum principle and Gamkrelidze's necessary conditions the existence and the uniqueness of an extremal trajectory.
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References
Arrow, K. J., Karlin, S., andScarf, H.,Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, California, 1958.
Barra, J. R., andBrodeau, M., Mimeographed Notes, Faculté des Sciences de Grenoble, 1964.
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley and Sons (Interscience Publishers), New York, 1962.
Leitmann, G.,An Introduction to Optimal Control, Chapter 4, McGraw-Hill Book Company, New York, 1966.
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Communicated by G. Leitmann
The author is indebted to Professors J. R. Barra, A. Blaquière, G. Leitmann, and H. Wan for their comments.
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Sprzeuzkouski, A.Y. A problem in optimal stock management. J Optim Theory Appl 1, 232–241 (1967). https://doi.org/10.1007/BF00926065
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DOI: https://doi.org/10.1007/BF00926065