Abstract
This paper studies stochastic inventory problems with unbounded Markovian demands, ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are lower semicontinuous with polynomial growth. Finite-horizon problems, stationary and nonstationary discounted-cost infinite-horizon problems, and stationary long-run average-cost problems are addressed. Existence of optimal Markov or feedback policies is established. Furthermore, optimality of (s, S)-type policies is proved when, in addition, the ordering cost consists of fixed and proportional cost components and the surplus cost is convex.
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References
Karlin, S., Optimal Inventory Policy for the Arrow-Harris-Marschak Dynamic Model, Studies in Mathematical Theory of Inventory and Production, Edited by K. J. Arrow, S. Karlin, and H. Scarf, Stanford University Press, Stanford, California, pp. 135-154, 1958.
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.
Bensoussan, A., Crouhy, M., and Proth, J., Mathematical Theory of Production Planning, North-Holland, Amsterdam, Netherlands, 1983.
Holt, C., Modigliani, F., Muth, J. F., and Simon, H. A., Planning Production, Inventory, and Workforce, Prentice-Hall, Englewood Cliffs, New Jersey, 1960.
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, Vol. 45, pp. 931-939, 1997.
Beyer, D., and Sethi, S. P., Average-Cost Optimality in Inventory Models with Markovian Demands, Journal of Optimization Theory and Applications, Vol. 92, pp. 497-526, 1997.
Beyer, D., and Sethi, S. P., The Classical Average-Cost Inventory Models of Iglehart (1963) and Veinott and Wagner (1965) Revisited, Working Paper, Hewlwtt-Packard Laboratories, Palo Alto, California, 1998.
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.
Aubin, J. P., Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, Netherlands, 1979.
Ross, S. M., Stochastic Processes, Wiley Series in Probability and Statistics, Wiley, New York, New York, 1983.
Ross, S. M., Introduction to Probability Models, Academic Press, San Diego, California, 1989.
Chung, K. L., A Course in Probability Theory, Academic Press, New York, New York, 1974.
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.
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 of Control and Optimization, Vol. 31, pp. 282-344, 1993.
HernÁndez-Lerma, O., and Lasserre, J. B., Discrete-Time Markov Control Processes, Springer, New York, New York, 1996.
Beyer, D., Sethi, S. P., and Taksar, M., Production-Inventory Models with Average Cost, Book in Progress (to appear).
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Beyer, D., Sethi, S.P. & Taksar, M. Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth. Journal of Optimization Theory and Applications 98, 281–323 (1998). https://doi.org/10.1023/A:1022633400174
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DOI: https://doi.org/10.1023/A:1022633400174