Abstract
We consider infinite‐horizon periodic‐review inventory models with unreliable suppliers where the demand, supply and cost parameters change with respect to a randomly changing environment. Although our analysis will be in the context of an inventory model, it is also appropriate for production systems with unreliable machines where planning is done on a periodic basis. It is assumed that the environmental process follows a Markov chain. The stock‐flow equations of the inventory system subject to environmental fluctuations is represented using a two‐dimensional stochastic process. We show that an environment‐dependentorder‐up‐to level (i.e., base‐stock) policy is optimal when the order cost is linearin order quantity. When there is also a fixed cost of ordering, we show that a two‐parameter environment‐dependent (s, S) policy is optimal under reasonable conditions. We also discuss computational issues and some extensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Archibald and E. Silver, (s, S) policies under continuous review and discrete compound Poisson demands, Management Science 24(1978)899-908.
C. Bell, Improved algorithms for inventory and replacement stock problems, SIAM Journal on Applied Mathematics 18(1970)558-566.
R. Bellman, I. Glicksberg and O. Gross, On the optimal inventory equation, Management Science 2(1955)83-104.
D.P. Bertsekas, Dynamic Programming, Deterministic and Stochastic Models, Prentice-Hall, Englewood Cliffs, NJ, 1987.
E. Çınlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ, 1975.
E. Çınlar and S. Özekici, Reliability of complex devices in random environments, Probability in the Engineering and Informational Sciences 1(1987)97-115.
M. Eisen and M. Tainiter, Stochastic variations in queueing processes, Operations Research 11 (1963)922-927.
R. Feldman, A continuous review (s, S) inventory system in a random environment, Journal of Applied Probability 15(1978)654-659.
D. Gupta, The (Q, r) inventory system with an unreliable supplier, Technical Report, School of Business, McMaster University, Hamilton, Ontario, 1993.
Ü. Gürler and M. Parlar, An inventory problem with two randomly available suppliers, Operations Research 45(1997)904-918.
D.L. Iglehart, Dynamic programming and stationary analysis of inventory problems, in: Multistage Inventory Models and Techniques, eds. H.E. Scarf, D.M. Gilford and M.W. Shelly, Stanford University Press, Stanford, CA, 1963.
D.L. Iglehart and S. Karlin, Optimal policy for dynamic inventory process with nonstationary stochastic demands, in: Studies in Applied Probability and Management Science, eds. K.J. Arrow, S. Karlin and H.E. Scarf, Stanford University Press, Stanford, CA, 1962.
B. Kalymon, Stochastic prices in a single item inventory purchasing model, Operations Research 19(1971)1434-1458.
S. Karlin, The application of renewal theory to the study of inventory policies, in: Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, CA, 1958.
S. Karlin, Dynamic inventory policy with varying stochastic demands, Management Science 6(1960)231-258.
P.J.M. Van Laarhoven and E.H.L. Aarts, Simulated Annealing: Theory and Applications, D. Reidel, Dordrecht, 1987.
H.L. Lee and C.A. Yano, Production control in multi-stage systems with variable yield losses, Operations Research 36(1988)269-278.
S. Nahmias, Production and Operations Analysis, Irwin, Homewood, IL, 2nd ed., 1993.
M.F. Neuts, The M/M/1 queue with randomly varying arrival and service rates, Opsearch 15 (1978)139-157.
S. Özekici, Optimal maintenance policies in random environments, European Journal of Operational Research 82(1995)283-294.
M. Parlar, Continuous-review inventory problem with random supply interruptions, European Journal of Operational Research 99(1997)366-385.
M. Parlar and D. Berkin, Future supply uncertainty in EOQ models, Naval Research Logistics 38(1991)107-121.
M. Parlar and D. Perry, Analysis of a (Q, r, T) inventory policy with deterministic and random yields when future supply is uncertain, European Journal of Operational Research 84(1995)431-443.
M. Parlar and Y. Wang, Diversification under yield randomness in inventory models, European Journal of Operational Research 66(1993)52-64.
M. Parlar, Y. Wang and Y. Gerchak, A periodic review inventory model with Markovian supply availability, International Journal of Production Economics 42(1995)131-136.
N.U. Prabhu and Y. Zhu, Markov-modulated queueing systems, Queueing Systems 5(1989)215-246.
H.E. Scarf, The optimality of (s, S) policies in the dynamic inventory problem, in: Mathematical Methods in the Social Sciences, eds. K.J. Arrow, S. Karlin and P. Suppes, Stanford University Press, Stanford, CA, 1960.
S.P. Sethi and F. Cheng, Optimality of (s, S) policies in inventory models with Markovian demand, Operations Research 45(1997)931-939.
W. Shih, Optimal inventory policies when stockouts result from defective products, International Journal of Production Research 18(1980)677-686.
E.A. Silver, Operations research in inventory management: A review and critique, Operations Research 29(1981)628-645.
E.A. Silver, Establishing the order quantity when the amount received is uncertain, INFOR 14(1976)32-39.
J.S. Song and P. Zipkin, Inventory control with information about supply conditions, Management Science 42(1996)1409-1419.
J.S. Song and P. Zipkin, Inventory control in fluctuating demand environment, Operations Research 41(1993)351-370.
A. Veinott and H. Wagner, Computing optimal (s, S) inventory policies, Management Science 11(1965)525-552.
Y.S. Zheng, A simple proof of optimality of (s, S) policies in infinite-horizon inventory systems, Journal of Applied Probability 28(1991)802-810.
Y.S. Zheng and A. Federgruen, Finding optimal (s, S) policies is about as simple as evaluating a single policy, Operations Research 39(1991)654-665.
Rights and permissions
About this article
Cite this article
Özekici, S., Parlar, M. Inventory models with unreliable suppliersin a random environment. Annals of Operations Research 91, 123–136 (1999). https://doi.org/10.1023/A:1018937420735
Published:
Issue Date:
DOI: https://doi.org/10.1023/A:1018937420735