Markov‐modulated processes have been used in stochastic inventory models with setup costs for modeling demand under the influence of uncertain environmental factors, such as fluctuating economic and market conditions. The analyses of these models have been carried out in the literature only under the assumption that unsatisfied demand is fully backlogged. The lost sales situation occurs in many retail establishments such as department stores and supermarkets. We use the analysis of the Markovian demand model with backlogging to analyze the lost sales case; in particular, we establish the optimality of an (s, S)‐type policy under fairly general conditions.
BensoussanA.CrouhyM.andProthJ. (1983), Mathematical Theory of Production Planning, North‐Holland, Amsterdam.
2.
BertsekasD. (1976), Dynamic Programming and Stochastic Control, Academic Press, New York.
3.
BeyerD.andSethiS. P. (1997), “Average Cost Optimality in Inventory Models with Markovian Demand”, Journal of Optimization Theory and Applications, 92, 3, 497–526.
4.
BeyerD.andSethiS. P. (1999), “Average Cost Optimality of (s, S) ‐type policies in Inventory Models with Markovian Demand and Lost Sales” forthcoming.
5.
BeyerD.andSethiS. P., andTaksarM. (1998), “Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth”, Journal of Optimization Theory and Application, 98, 2, 281–323.
6.
ChengF. (1996), “Inventoty Models with Markovian Demands,” Doctoral Dissertation, University of Toronto.
7.
ChengF.andSethiS. P. (1999), “A Periodic Review Inventory with Demands Influenced by Promotions”, Management Science, forthcoming.
8.
FellerW. (1971), An Introduction to Probability Theory and Its Application, Vol. II, 2nd Edition, Wiley, New York.
9.
KarlinS., andScarfH. (1958), “Inventory Models of the Arrow‐Harris‐Marschak Type with Time Lag” in Studies in Mathematical Theory of Inventory and Production, ArrowK. J.KarlinS., and ScarfH. (eds.), Stanford University Press, Stanford, CA, 155–178.
10.
KarlinS., andFabensA. (1959), “The (s, S) Inventory Model under Markovian Demand Process” in Marhematical Methods in the Social Sciences, Stanford University Press, Stanford, CA.
11.
OzekiciS.andParlarM. (1998), “Inventory Models with Unreliable Suppliers in Random Environments”, Annals of Operations Research, forthcoming.
12.
ScarfH. (1960), “The Optimality of (S, s) Policies in the Dynamic Inventory” in Mathematical Methods in Social Sciences, ArrowJ.KarlinS., and SuppesP. (eds.), Stanford University Press, Stanford, CA.
13.
SethiS. P.andChengF. (1993), “Optimality of (s, S) Policies in Inventory Models with Markovian Demand ProcessesC2MIT Working Paper, University of Toronto.
14.
SethiS. P.andChengF. (1997), “Optimality of (s, S) Policies in Inventory Models with Markovian Demand Processes”, Operations Research, 45, 6, 931–939.
15.
ShreveS. E. (1976), “Abbreviated proof [in the lost sales case]”, in Dynamic Programming and Stochastic Control, D. P. Bertsekas, Academic Press, New York, 105–106.
16.
SongJ.‐S.andZipkinP. (1993), “Inventory Control in a Fluctuating Demand Environment”, Operations Research, 41, 2, 351–370.
17.
VeinottA.Jr. (1966), “On the Optimality of (s, S) Inventory Policies: New Conditions and a New Proof”, SIAM J. Appl. Math., 14, 1067–1083.
