Abstract
In many production/inventory systems, not only is the production/inventory capacity finite, but the systems are also subject to random production yields that are influenced by factors such as breakdowns, repairs, maintenance, learning, and the introduction of new technologies. In this paper, we consider a single-item, single-location, periodic-review model with finite capacity and Markov modulated demand and supply processes. When demand and supply processes are driven by two independent, discrete-time, finite-state, time-homogeneous Markov chains, we show that a modified, state-dependent, inflated base-stock policy is optimal for both the finite and infinite horizon planning problems. We also show that the finite-horizon solution converges to the infinite-horizon solution.
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Aviv, Y. and A. Federgruen. (1997). “Stochastic Inventory Models with Limited Production Capacity and Varying Parameters.” Probability in the Engineering and Informational Science 11, 107–135.
Bertsekas, D.P. and S.E. Shreve. (1978). Stochastic Optimal Control. New York: Academic Press.
Beyer, D. and S.P. Sethi. (1997). “Average Cost Optimality in Inventory Models with Markovian Demands.” Journal of Optimization Theory and Application 92, 497–526.
Chao, X. and G. Gallego. (1999). “Studies of Stochastic Production Systems.” Not published technical note.
Chen, F. andJ.-S. Song. (2001). “Optimal Policies for Multi-Echelon Inventory Problems with Markov-Modulated Demand.” Operations Research 49(2), 226–234.
Ciarallo, F.W., R. Akella, and T.E. Morton. (1994). “A Periodic Review Production Planning Models with Uncertain Capacity and Uncertain Demand-Optimality of Extended Myopic Policies.” Management Science 40, 320–334.
DeCroix, G. and T. Arreola-Risa. (1998). “Optimal Production and Inventory Policy for Multiple Products under Resource Constraints.” Management Science 44, 950–961.
Ehrhardt, R. and L. Taube. (1987). “An Inventory Model with Random Replacement Quantities.” International Journal of Production Research 25, 1795–1804.
Federgruen, A. and Y. S. Zheng. (1993). “Optimal Control Policies for Stochastic Inventory Systems with Endogenous Supply.” Probability in the Engineering and Informational Science 7, 257–272.
Federgruen, A. and P. Zipkin. (1986a). “An Inventory Model with Limited Production Capacity and Uncertain Demands. Part 1. The Average-Cost Criterion.” Mathematics of Operations Research 11, 193–207.
Federgruen, A. and P. Zipkin. (1986b). “An Inventory Model with Limited Production Capacity and Uncertain Demands. Part 2. The Discounted-Cost Criterion.” Mathematics of Operations Research 11, 208–215.
Gallego, G. “Coordinating the Production of Components and Assemblies when Component Yields and Demands Are Random.” Not published technical note.
Gallego, G. and A. Wolf. (2001). “Single Item Capacity Constrained Inventory Policies.” European Journal of Operational Research 33, 861–874.
Gerchak, Y., R.G. Vickson, and M. Palar. (1988). “Periodic Review Production Models with Variable Yield and Uncertain Demand.” IIE Transactions 20, 144–150.
Glasserman, P. and S. Tayur. (1994). “The Stability of a Capacitated, Multi-Echelon Production-Inventory System under a Base-Stock Policy.” Operations Research 42, 913–925.
Glasserman, P. and S. Tayur. (1995). “Sensitivity Analysis for Base-Stock Levels in Multi-Echelon Production-Inventory System.” Management Science 42, 263–281.
Glasserman, P. and S. Tayur. (1996). “A Simple Approximation for Multi-Stage Capacitated Production-Inventory System.” Naval Research of Logistics 43, 41–58.
Henig, M. and Y. Gerchak. (1990). “The Structure of Periodic Review Policies in the Presence of Random Yield.” Operations Research 38, 634–643.
Heyman, D.P. and M.J. Sobel. (1984). Stochastic Models in Operations Research, Vol. 2. New York: McGraw-Hill.
Hwang, J. and M.R. Singh. (1998). “Optimal Production Policies for Multi-Stage Systems with Setup Costs and Uncertain Capacities.” Management Science 44, 1279–1294.
Iglehart, D.L. (1963). “Optimality of (s, S) Policies in the Infinite Horizon Dynamic Inventory Problem.” Management Science 9, 259–267.
Iglehart, D. and S. Karlin. (1962). “Optimal Policy for Dynamic Inventory Process with Nonstationary Stochastic Demands.” In K.J. Arrow, S. Karlin, and H. Scarf (eds.), Studies in Applied Probability and Management Science, Chapter 8. Stanford, CA: Stanford University Press.
Kapuscinski, R. and S. Tayur. (1998). “A Capacitated Production-Inventory Model with Periodic Demand. Part 1. The Average-Cost Criterion.” Operations Research 46, 899–911.
Karlin, S. and A. Fabens. (1959). “The (s, S) Inventory Model under Markovian Demand Process.” In K.J. Arrow, S. Karlin, and P. Suppes (eds.), Mathematical Methods in the Social Sciences. Stanford, CA: Stanford University Press.
Lee, H.L. (1996). “Input Control for Serial Production Lines Consisting of Processing and Assembly Operations with Random Yields.” Operations Research 44, 464–468.
Lee, H.L. and M.J. Rosenblatt. (1985). “Optimal Inspection and Ordering Polices for Products with Imperfect Quality.” IIE Transactions 17, 284–289.
Lee, H.L. and C.A. Yano. (1988). “Production Control in Multistage Systems with Variable Yield Losses.” Operations Research 36, 269–278.
Lovejoy, W. (1990). “Myopic Policies for Some Inventory Problem with Uncertain Demand Distribution.” Management Science 36, 688–707.
Parlar, M., Y. Wang, and Y. Gerchak. (1995). “A Periodic Review Inventory Model with Markovian Supply Availability.” International Journal of Production Economics 42, 131–136.
Portues, E. (1990). “Stochastic Inventory Theory.” In D. Heyman and M. Sobel (eds.), Handbooks in Operations Research and Management Science: Stochastic Models, Vol. 2. Amsterdam: Elsevier Science.
Rockafellar, R.T. (1997). Convex Analysis. Princeton, NJ: Princeton University Press.
Scarf, H.E. (1960). “The Optimalities of (s, S) Policies in the Dynamic Inventory Problem.” In K.J. Arrow, S. Karlin, and P. Suppes (eds.), Mathematical Methods in the Social Sciences. Stanford, CA: Stanford University Press.
Sethi, S. and F. Cheng. (1997). “Optimality of (s, S) Policies in Inventory Models with Markovian Demand.” Operations Research 45, 931–939.
Silver, E.A. (1976). “Establishing the Order Quantity when the Amount Received Is Uncertain.” INFOR 14, 29–32.
Song, G., P. Evans, and L. Shulman. (1992). “Competing on Capabilities: The New Rules of Corporate Strategy.” Harvard Business Review 70, 57–69.
Song, J.-S. and P. Zipkin. (1993). “Inventory Control in a Fluctuating Demand Environment.” Operations Research 41, 351–370.
Strichartz, R. (1995). The Way of Analysis. Boston, MA: Jones & Bartlett.
Tyndall, G. (1988). “Supply Chain Management Innovations Spur Long-Term Strategies Retail Alliances.” Marketing News, December 19.
Wang, Y. and Y. Gerchak. (1996). “Periodic Review Production Models with Variable Capacity, Random Yield and Uncertain Demand.” Management Science 42, 130–137.
Yano, C.A. and H.L. Lee. (1995). “Lot Sizing with Random Yields: A Review.” Operations Research 43, 311–334.
Yao, D. (1988). “Optimal Run Quantities for an Assembly System with Random Yields.” IIE Transactions 20, 399–403.
Zipkin, P. (1986). “Inventory Service-Level Measures: Convexity and Approximation.” Management Science 32, 975–981.
Zipkin, P. (1989). “Critical Number Policies for Inventory Models with Periodic Data.” Management Science 35, 71–80.
Zipkin, P. (2000). Foundations of Inventory Management. Boston, MA: McGraw-Hill.
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Gallego, G., Hu, H. Optimal Policies for Production/Inventory Systems with Finite Capacity and Markov-Modulated Demand and Supply Processes. Annals of Operations Research 126, 21–41 (2004). https://doi.org/10.1023/B:ANOR.0000012274.69117.90
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DOI: https://doi.org/10.1023/B:ANOR.0000012274.69117.90