Abstract
In this paper, we consider a periodic-review stochastic inventory model with an asymmetric or piecewise-quadratic holding cost function and nonnegative production levels. It is assumed that the cost of deviating from an ideal production level or existing capacity is symmetric quadratic. It is shown that the optimal order policy is similar to the (s, S) policies found in the literature, except that the order-up-to quantity is a nonlinear function of the entering inventory level. Dynamic programming is used to derive the optimal policy. We provide numerical examples and a sensitivity analysis on the problem parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bensoussan, A., Hurst, E. G., andNaslund, B.,Management Applications of Modern Control Theory, North-Holland, Amsterdam, Holland, 1974.
Sethi, S. P., andThompson, G. L.,Optimal Control Theory, Martinus Nijhoff, Boston, Massachusetts, 1981.
Baker, K. R., andPeterson, D. W.,An Analytic Framework for Evaluating Rolling Schedules, Management Science, Vol. 25, pp. 341–351, 1979.
Parlar, M., andVickson, R. G.,An Optimal Control Problem with Piecewise Quadratic Cost Functional Containing a Dead-Zone, Optimal Control Applications and Methods, Vol. 1, pp. 361–372, 1980.
Parlar, M.,A Decomposition Technique for an Optimal Control Problem with PQDZ Cost and Bounded Controls, IEEE Transactions on Automatic Control, Vol. AC-27, pp. 947–951, 1982.
Parlar, M.,A Continuous-Time Linear Tracking Problem with an Asymmetric Quadratic Objective Functional Containing a Cost-Free Interval, International Journal of Control, Vol. 41, pp. 1245–1252, 1985.
Parlar, M.,A Stochastic Production Planning Model with a Dynamic Chance Constraint, European Journal of Operational Research, Vol. 20, pp. 255–260, 1985.
Scarf, H.,The Optimality of (s, S) Policies for the Dynamic Inventory Problem, Mathematical Methods in the Social Sciences, Edited by K. J. Arrow, S. Karlin, and S. Suppes, Stanford University Press, Stanford, California, pp. 196–202, 1960.
Aneja, Y. P., andNoori, H.,The Optimality of (s, S) Policies for a Stochastic Inventory Problem with Proportional and Lump-Sum Penalty Cost, Management Science, Vol. 33, pp. 750–755, 1987.
Neave, E. H.,The Stochastic Cash Balance Problem with Fixed Costs for Increases and Decreases, Management Science, Vol. 16, pp. 472–490, 1970.
Rocklin, S. M., Kashper, A., andVarvaloucas, G.,Capacity Expansion/Contraction of a Facility with Demand Augmentation Dynamics, Operations Research, Vol. 32, pp. 133–147, 1984.
Bensoussan, A., Crouhy, M., andProth, J. M.,Mathematical Theory of Production Planning, North-Holland, Amsterdam, Holland, 1983.
Zabczyk, J. Stochastic Control of Discrete-Time Systems, Control Theory and Topics in Functional Analysis, International Atomic Energy Agency, Vienna, Austria, Vol. 3, pp. 187–274, 1976.
Bertsekas, D. P.,Dynamic Programming and Stochastic Control, Academic Press, New York, New York, 1976.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A5872. The authors wish to thank an anonymous referee for very helpful comments on an earlier version of this paper.
Rights and permissions
About this article
Cite this article
Parlar, M., Rempala, R. Stochastic inventory problem with piecewise quadratic holding cost function containing a cost-free interval. J Optim Theory Appl 75, 133–153 (1992). https://doi.org/10.1007/BF00939909
Issue Date:
DOI: https://doi.org/10.1007/BF00939909