Abstract
This paper is concerned with long-run average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, and convex surplus cost. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dynamic programming equation and the corresponding verification theorem are established. Finally, the existence of an optimal state-dependent (s, S) policy is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
KARLIN, S., and FABENS, A., The (s, S) Inventory Model under Markovian Demand Process, Mathematical Methods in the Social Sciences, Edited by K. Arrow, S. Karlin, and P. Suppes, Stanford University Press, Stanford, California, pp. 159–175, 1960.
SONG, J. S., and ZIPKIN, P., Inventory Control in a Fluctuating Demand Environment, Operations Research, Vol. 41, pp. 351–370, 1993.
SETHI, S., and CHENG, F., Optimality of (s, S) Policies in Inventory Models with Markovian Demand, Operations Research (to appear).
OZEKICI, S., and PARLAR, M., Periodic Review Inventory Models in Random Environment, Working Paper, Bogazici University, Istanbul, Turkey, 1995.
ARAPOSTATHIS, A., BORKAR, V. S., FERNANDEZ-GAUCHERAND, E., GHOSH, M. K., and MARCUS, S. I., Discrete-Time Controlled Markov Processes with Average Cost Criterion: A Survey, SIAM Journal on Control and Optimization, Vol. 31, pp. 282–344, 1993.
IGLEHART, D., Dynamic Programming and Stationary Analysis of Inventory Problems, Multistage Inventory Models and Techniques, Edited by H. Scarf, D. Gilford, and M. Shelly, Stanford University Press, Stanford, California, pp. 1–31, 1963.
VEINOTT, A., and WAGNER, H., Computing Optimal (s, S) Policies, Management Science, Vol. 11, pp. 525–552, 1965.
KARLIN, S., Steady-State Solutions, Studies in the Mathematical Theory of Inventory and Production, Edited by K. Arrow, S. Karlin, and H. Scarf, Stanford University Press, Stanford, California, pp. 223–269, 1958.
KARLIN, S., The Application of Renewal Theory to the Study of Inventory Policies, Studies in the Mathematical Theory of Inventory and Production, Edited by K. Arrow, S. Karlin, and H. Scarf, Stanford University Press, Stanford, California, pp. 270–297, 1958.
VEINOTT, A., On the Optimality of (s, S) Inventory Policies: New Conditions and a New Proof, SIAM Journal on Applied Mathematics, Vol. 14, pp. 1067–1083, 1966.
ZHENG, Y. S., A Simple Proof for Optimality of (s, S) Policies in Infinite-Horizon Inventory Systems, Journal of Applied Probability, Vol. 28, pp. 802–810, 1991.
STIDHAM, S., JR., Cost Models for Stochastic Clearing Systems, Operations Research, Vol. 25, pp. 100–127, 1977.
ZHENG, Y. S., and FEDERGRUEN, A., Finding Optimal (s, S) Policies Is about as Simple as Evaluating a Single Policy, Operations Research, Vol. 39, pp. 654–665, 1991.
FEDERGRUEN, A., and ZIPKIN, P., An Efficient Algorithm for Computing Optimal (s, S) Policies, Operations Research, Vol. 32, pp. 1268–1285, 1984.
HU, J. Q., NANANUKUL, S., and GONG, W. B., A New Approach to (s, S) Inventory Systems, Journal of Applied Probability, Vol. 30, pp. 898–912, 1993.
FU, M., Sample Path Derivatives for (s, S) Inventory Systems, Operations Research, Vol. 42, pp. 351–364, 1994.
PORTEUS, E., Numerical Comparison of Inventory Policies for Periodic Review Systems, Operations Research, Vol. 33, pp. 134–152, 1985.
SAHIN, I., Regeneration Inventory Systems: Operating Characteristics and Optimization, Bilkent University Lecture Series, Springer Verlag, New York, New York, 1990.
BEYER, D., and SETHI, S. P., The Classical Average Cost Inventory Models of Iglehart (1963) and Veinott and Wagner (1965) Revisited, Working Paper, University of Toronto, Toronto, Ontario, Canada, 1996.
SETHI, S. P., SUO, W., TAKSAR, M. I., and ZHANG, Q., Optimal Production Planning in a Stochastic Manufacturing System with Long-Run Average Cost, Journal of Optimization Theory and Applications, Vol. 92, pp. 161–168, 1997.
SCARF, H., The Optimality of (s, S) Policies in the Dynamic Inventory Problem, Mathematical Methods in the Social Sciences, Edited by K. Arrow, S. Karlin, and P. Suppes, Stanford University Press, Stanford, California, pp. 196–202, 1960.
SETHI, S., and CHENG, F., Optimality of (s, S) Policies in Inventory Models with Markovian Demand Processes, C2MIT Working Paper, University of Toronto, Toronto, Ontario, Canada, 1993.
FELLER, W., An Introduction to Probability Theory and Its Application, Vol. 2, 2nd Edition, Wiley, New York, New York, 1971.
YOSIDA, K., Functional Analysis, 6th Edition, Springer Verlag, New York, New York, 1980.
SZNADJER, R., and FILAR, J. A., Some Comments on a Theorem of Hardy and Littlewood, Journal of Optimization Theory and Applications, Vol. 75, pp. 201–208, 1992.
GIKHMAN, I., and SKOROKHOD, A., Stochastic Differential Equations, Springer Verlag, Berlin, Germany, 1972.
BEYER, D., SETHI, S. P., and TAKSAR, M., Inventory Models with Markovian Demand and Cost Functions of Polynomial Growth, Working Paper, University of Toronto, Toronto, Ontario, Canada, 1996.
BEYER, D., SETHI, S. P., and TAKSAR, M., Inventory Models with Average Cost, book, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beyer, D., Sethi, S.P. Average Cost Optimality in Inventory Models with Markovian Demands. Journal of Optimization Theory and Applications 92, 497–526 (1997). https://doi.org/10.1023/A:1022651322174
Issue Date:
DOI: https://doi.org/10.1023/A:1022651322174