Abstract
We consider a model of a supply chain consisting of n production facilities in tandem and producing a single product class. External demand is met from the finished goods inventory maintained in front of the most downstream facility (stage 1); unsatisfied demand is backlogged. We adopt a base-stock production policy at each stage of the supply chain, according to which the facility at stage i produces if inventory falls below a certain level w i and idles otherwise. We seek to optimize the hedging vector w=(w 1,...,w n ) to minimize expected inventory costs at all stages subject to maintaining the stockout probability at stage 1 below a prescribed level (service level constraint). We make rather general modeling assumptions on demand and production processes that include autocorrelated stochastic processes. We solve this stochastic optimization problem by combining analytical (large deviations) and sample path-based (perturbation analysis) techniques. We demonstrate that there is a natural synergy between these two approaches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abate, J., G. Choudhury, and W. Whitt. (1995). "Exponential Approximations for Tail Probabilities in Queues, I: Waiting Times." Operations Research 43(5), 885–901.
Baccelli, F. and Z. Liu. (1992). "On a Class of Stochastic Recursive Sequences Arising in Queueing Theory." Annals of Probability 20, 350–374.
Bertsekas, D. (1999). Nonlinear Programming, 2nd edn. Belmont, MA: Athena Scientific.
Bertsimas, D. and I.C. Paschalidis. (2001). "Probabilistic Service Level Guarantees in Make-to-Stock Manufacturing Systems." Operations Research 49(1), 119–133.
Bertsimas, D., I.C. Paschalidis, and J.N. Tsitsiklis. (1998). "On the Large Deviations Behaviour of Acyclic Networks of G/G/1 Queues." The Annals of Applied Probability 8(4), 1027–1069.
Cassandras, C. and S. Lafortune. (1999). Introduction to Discrete Event Systems. Dordrecht: Kluwer.
Cassandras, C., Y. Wardi, B. Melamed, G. Sun, and C. Panayiotou. (2002). "Perturbation Analysis for On-Line Control and Optimization of Stochastic Fluid Models." IEEE Transactions on Automatic Control 47(8), 1234–1248.
Clark, A. and H. Scarf. (1960). "Optimal Policies for a Multi-Echelon Inventory Problem." Management Science 6, 475–490.
Dembo, A. and O. Zeitouni. (1993). Large Deviations Techniques and Applications. Jones & Bartlett.
Federgruen, A. and P. Zipkin. (1986). "An Inventory Model with Limited Production Capacity and Uncertain Demands I. The Average Cost Criterion." Mathematics of Operations Research 11(2), 193–207.
Glasserman, P. (1997). "Bounds and Asymptotics for Planning Critical Safety Stocks." Operations Research 45(2), 244–257.
Glasserman, P. and S. Tayur. (1995). "Sensitivity Analysis for Base-Stock Levels in Multiechelon Production-Inventory Systems." Management Science 45(2), 263–281.
Gokbayrak, K. and C.G. Cassandras. (2001). "An On-Line 'Surrogate Problem'Methodology for Stochastic Discrete Resource Allocation Problems."Journal of Optimization Theory and Applications 108(2), 349–376.
Kapuściński, R. and S. Tayur. (1999). "Optimal Policies and Simulation Based Optimization for Capacitated Production Inventory Systems." In S. Tayur, R. Ganeshan, and M. Magazine (eds.), Quantitative Models for Supply Chain Management, chapter 2, pp. 7–40. Dordrecht: Kluwer.
Kushner, H. and D. Clark. (1978). Stochastic Approximation for Constrained and Unconstrained Systems. Berlin: Springer.
Liu, Y. and W. Gong. (1999). "Perturbation Analysis for Stochastic Fluid Queueing Systems." In Proceedings of 38th IEEE Conference on Decision and Control, pp. 4440–4445.
National Research Council. (1998). "Visionary Manufacturing Challenges for 2000." Washington, DC: National Academy Press.
Panayiotou, C.G. and C.G. Cassandras. (1999). "Optimization of Kanban-Based Manufacturing Systems." Automatica 35, 1521–1533.
Panayiotou, C.G., C.G. Cassandras, and P. Zhang. (2002). "On-Line Inventory Cost Minimization for Make-to-StockManufacturing Systems." In Proceedings of the American Control Conference, Vol.6, pp. 4469–4474.
Paschalidis, I. and Y. Liu. (2003). "Large Deviations-Based Asymptotics for Inventory Control in Supply Chains." Operations Research 51(3), 437–460.
Paschalidis, I., Y. Liu, C. Cassandras, and P. Zhang. (2001). "Threshold-Based Control for Make-to-Stock Models: A Synergy between Large Deviations and Perturbation Analysis." In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, pp. 4523–4528.
Rockafellar, R. (1970). Convex Analysis. Princeton University Press.
Shi, L. and S. Olafsson. (2000). "Nested Partitions Method for Global Optimization." Operations Research 48, 390–407.
The Economist (June 20th, 1998). "A Survey of Manufacturing: Meet the Global Factory."
Wardi, Y., B. Melamed, C. Cassandras, and C. Panayiotou. (2001). "IPA Gradient Estimators in Single-Node Stochastic Fluid Models." Journal of Optimization Theory and Applications 115(2), 369–406.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Paschalidis, I.C., Liu, Y., Cassandras, C.G. et al. Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis. Annals of Operations Research 126, 231–258 (2004). https://doi.org/10.1023/B:ANOR.0000012283.36012.e8
Issue Date:
DOI: https://doi.org/10.1023/B:ANOR.0000012283.36012.e8