Abstract
The models discussed in the present paper are generalizations of the models introduced previously by A. Prékopa [6] and M. Ziermann [13]. In the mentioned papers the initial stock level of one basic commodity is determined provided that the delivery and demand process allow certain homogeneity (in time) assumptions if they are random. Here we are dealing with more than one basic commodity and drop the time homogeneity assumption. Only the delivery processes will be assumed to be random. They will be supposed to be stochastically independent. The first model discussed in this paper was already introduced in [9]. All these models are stochastic programming models and algorithms are used to determine the initial stock levels rather than simple formulas. We have to solve nonlinear programming problems where one of the constraints is probabilistic. The function and gradient values of the corresponding constraining function are determined by simulation. A numerical example is detailed.
Preview
Unable to display preview. Download preview PDF.
References
J.H. Ahrens and K. Dieter, “Computer methods for sampling from gamma, beta, Poisson and binomial distributions”, Computing 12 (1974) 223–246.
A.V. Fiacco and G.P. McCormick, Nonlinear programming sequential unconstrained minimization technique (Wiley, New York, 1968).
R. Hooke and T.A. Jeeves, “Direct search solution of numerical and statistical problems”, Journal of the Association for Computing Machinery 8 (1959) 215–229.
B.L. Miller and H.M. Wagner, “Chance constrained programming with joint constraints”, Operations Research 13 (1965) 930–945.
M.J.D. Powell, “An iterative method for finding the minimum of a function of several variables without calculating derivatives”, The Computer Journal 7 (1964), 155–162.
A. Prékopa, “Reliability equation for an inventory problem and its asymptotic solution”, in: A. Prékopa, ed., Colloquium on application of mathematics, to economics (Akadémia Kiadó, Budapest, 1965) pp. 317–327.
A. Prékopa, “On logarithmic concave measures with application to stochastic programming”, Acta Universitatis Szegedienis 32 (1971) 301–316.
A. Prékopa, “Contributions to the theory of stochastic programming”, Mathematical Programming 4 (1973) 202–221.
A. Prékopa, “Stochastic programming models for inventory control and water storage problems”, in: A. Prékopa, ed., Inventory control and water storage, Colloquia Mathematica Societatis János Bolyai, 7 (Bolyai János Mathematical Society. Budapest, and North-Holland, Amsterdam, 1973) pp. 229–246.
A. Prékopa, “Generalizations of the theorem of Smirnov with applications to a reliability type inventory problem”, Mathematische Operationsforschung und Statistik 4 (1973) 283–297.
H.H. Rosenbrock, “An automatic, method for finding the greatest or least value of a function”, The Computer Journal 3 (1960) 175–184.
S.S. Wilks, Mathematical statistics (wiley, New York, 1962).
M. Ziermann, “Anwendung des Smirnov’schen Sätzen auf Lagerhaltungsproblem”, Publications of the Mathematical Institute of the Hungarian Academy of Sciences, Series B, 8 (1965) 509–518.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1978 The Mathematical Programming Society
About this chapter
Cite this chapter
Prékopa, A., Kelle, P. (1978). Reliability type inventory models based on stochastic programming. In: Balinski, M.L., Lemarechal, C. (eds) Mathematical Programming in Use. Mathematical Programming Studies, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120825
Download citation
DOI: https://doi.org/10.1007/BFb0120825
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00795-8
Online ISBN: 978-3-642-00796-5
eBook Packages: Springer Book Archive