Optimal and Near-Optimal (s,S) Inventory Policies for Levy Demand Processes

Robin O. Roundy; Gennady Samorodnitsky

RAIRO - Operations Research (2010)

  • Volume: 35, Issue: 1, page 37-70
  • ISSN: 0399-0559

Abstract

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A Levy jump process is a continuous-time, real-valued stochastic process which has independent and stationary increments, with no Brownian component. We study some of the fundamental properties of Levy jump processes and develop (s,S) inventory models for them. Of particular interest to us is the gamma-distributed Levy process, in which the demand that occurs in a fixed period of time has a gamma distribution. We study the relevant properties of these processes, and we develop a quadratically convergent algorithm for finding optimal (s,S) policies. We develop a simpler heuristic policy and derive a bound on its relative cost. For the gamma-distributed Levy process this bound is 7.9% if backordering unfilled demand is at least twice as expensive as holding inventory. Most easily-computed (s,S) inventory policies assume the inventory position to be uniform and assume that there is no overshoot. Our tests indicate that these assumptions are dangerous when the coefficient of variation of the demand that occurs in the reorder interval is less than one. This is often the case for low-demand parts that experience sporadic or spiky demand. As long as the coefficient of variation of the demand that occurs in one reorder interval is at least one, and the service level is reasonably high, all of the polices we tested work very well. However even in this region it is often the case that the standard Hadley–Whitin cost function fails to have a local minimum.

How to cite

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Roundy, Robin O., and Samorodnitsky, Gennady. "Optimal and Near-Optimal (s,S) Inventory Policies for Levy Demand Processes." RAIRO - Operations Research 35.1 (2010): 37-70. <http://eudml.org/doc/197783>.

@article{Roundy2010,
abstract = {A Levy jump process is a continuous-time, real-valued stochastic process which has independent and stationary increments, with no Brownian component. We study some of the fundamental properties of Levy jump processes and develop (s,S) inventory models for them. Of particular interest to us is the gamma-distributed Levy process, in which the demand that occurs in a fixed period of time has a gamma distribution. We study the relevant properties of these processes, and we develop a quadratically convergent algorithm for finding optimal (s,S) policies. We develop a simpler heuristic policy and derive a bound on its relative cost. For the gamma-distributed Levy process this bound is 7.9% if backordering unfilled demand is at least twice as expensive as holding inventory. Most easily-computed (s,S) inventory policies assume the inventory position to be uniform and assume that there is no overshoot. Our tests indicate that these assumptions are dangerous when the coefficient of variation of the demand that occurs in the reorder interval is less than one. This is often the case for low-demand parts that experience sporadic or spiky demand. As long as the coefficient of variation of the demand that occurs in one reorder interval is at least one, and the service level is reasonably high, all of the polices we tested work very well. However even in this region it is often the case that the standard Hadley–Whitin cost function fails to have a local minimum. },
author = {Roundy, Robin O., Samorodnitsky, Gennady},
journal = {RAIRO - Operations Research},
keywords = {backordering; inventory position; service levels; Levy processes; Gamma distribution},
language = {eng},
month = {3},
number = {1},
pages = {37-70},
publisher = {EDP Sciences},
title = {Optimal and Near-Optimal (s,S) Inventory Policies for Levy Demand Processes},
url = {http://eudml.org/doc/197783},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Roundy, Robin O.
AU - Samorodnitsky, Gennady
TI - Optimal and Near-Optimal (s,S) Inventory Policies for Levy Demand Processes
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 1
SP - 37
EP - 70
AB - A Levy jump process is a continuous-time, real-valued stochastic process which has independent and stationary increments, with no Brownian component. We study some of the fundamental properties of Levy jump processes and develop (s,S) inventory models for them. Of particular interest to us is the gamma-distributed Levy process, in which the demand that occurs in a fixed period of time has a gamma distribution. We study the relevant properties of these processes, and we develop a quadratically convergent algorithm for finding optimal (s,S) policies. We develop a simpler heuristic policy and derive a bound on its relative cost. For the gamma-distributed Levy process this bound is 7.9% if backordering unfilled demand is at least twice as expensive as holding inventory. Most easily-computed (s,S) inventory policies assume the inventory position to be uniform and assume that there is no overshoot. Our tests indicate that these assumptions are dangerous when the coefficient of variation of the demand that occurs in the reorder interval is less than one. This is often the case for low-demand parts that experience sporadic or spiky demand. As long as the coefficient of variation of the demand that occurs in one reorder interval is at least one, and the service level is reasonably high, all of the polices we tested work very well. However even in this region it is often the case that the standard Hadley–Whitin cost function fails to have a local minimum.
LA - eng
KW - backordering; inventory position; service levels; Levy processes; Gamma distribution
UR - http://eudml.org/doc/197783
ER -

References

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  1. S. Asxater, Using the Deterministic EOQ Formula in Stochastic Inventory Control. Management Sci.42 (1996) 830-834.  
  2. D. Beyer and S. Sethi, Average Cost Optimality in Inventory Models with Markovian Demands. J. Optim. Theory Appl.92 (1997) 497-526.  
  3. S. Bollapragada, A simple heuristic for computing nonstationary (s, S)policies. Oper. Res.47 (1999) 576-585.  
  4. S. Browne and P. Zipkin, Inventory Models with Continuous, Stochastic Demands. Ann. Appl. Probab.1 (1991) 419-435.  
  5. F. Chen and Y. Zheng, Inventory Policies with Quantized Ordering. Naval Res. Logist.39 (1992) 654-665.  
  6. K. Cheung, A Continuous Review Inventory Model with a Time Discount. IEEE Trans.30 (1998) 747-757.  
  7. A. Federgruen and Y. Zheng, Computing an Optimal (s,S) Policy is as Easy as Evaluating a Single Policy. Oper. Res.39 (1991) 654-665.  
  8. A. Federgruen and P. Zipkin, Computational Issues in an Infinite-Horizon, Multi-Echelon Inventory Model. Oper. Res.32 (1984) 818-835.  
  9. A. Federgruen and P. Zipkin, An Efficient Algorithm for Computing Optimal (s,S) Policies. Oper. Res.32 (1984) 1268-1285.  
  10. A. Federgruen and P. Zipkin, Computing Optimal (s,S) Policies in Inventory Models with Continuous Demands. Adv. in Appl. Probab.17 (1985) 424-442.  
  11. W. Feller, An Introduction to Probability and its Applications, Vol. II. Wiley, New York (1966).  
  12. M. Fu, Sample Path Derivatives for (s,S) Inventory Systems. Oper. Res.42 (1994) 351-364.  
  13. J. Hu, S. Nananukul and W. Gong, A New Approach to (s,S) Inventory Systems. J. Appl. Probab.30 (1993) 898-912.  
  14. G. Gallego, New Bounds and Heuristics for (Q,r) Policies. Management Sci.44 (1988) 219-233.  
  15. G. Gallego and T. Boyaci, Managing Waiting Time Related Service Levels in Single-Stage (Q,r) Inventory Systems, Working paper. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY (2000).  
  16. G. Gallego and T. Boyaci, Minimizing Holding and Ordering Costs subject to a Bound on Backorders is as Easy as Solving a Single Backorder Cost Model, Working paper. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY (2000).  
  17. G. Gallego and T. Boyaci, Minimizing Average Ordering and Holding Costs subject to Service Constraints, Working paper. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY (2000).  
  18. G.J. Hadley and T.M. Whitin, Analysis of Inventory Systems. Prentice Hall, Englewood Cliffs, NJ (1963).  
  19. T. Hida, Stationary Stochastic Processes. Princeton University Press, Princeton, NJ (1970).  
  20. L.A. Johnson and D.C. Montgomery, Operations Research in Production Planning, Scheduling, and Inventory Control. John Wiley and Sons, New York (1974).  
  21. S. Nahmias, Production and Operations Analysis, Second Edition. Irwin, Homewood Illinois, 60430 (1993).  
  22. N.U. Prabhu, Stochastic Storage Processes. Springer-Verlag, New York (1980).  
  23. S. Resnick, Adventures in Stochastic Processes. Birkhauser, Boston, MA (1992).  
  24. L. Robinson, Tractible (Q,R) Heuristic Models for Constrained Service Levels. Management Sci.43 (1997) 951-965.  
  25. R. Roundy and G. Samorodnitsky, Optimal and Heuristic (s,S) Inventory Policies for Levy Demand Processes, Technical Report. School of Opeations Research and Industrial Engineering, Cornell University, Ithaca NY 14853 (1996).  
  26. I. Sahin, On the Objective Function Behavior in (s,S) Inventory Models. Oper. Res.82 (1982) 709-724.  
  27. R. Serfozo and S. Stidham, Semi-Stationary Clearing Processes. Stochastic Process. Appl.6 (1978) 165-178.  
  28. M. Sharpe, General Theory of Markov Porcesses. Academic Press, Boston Massachusetts (1988).  
  29. J. Song and P. Zipkin, Inventory Control in a Fluctuating Demand Environment. Oper. Res.41 (1993) 351-370.  
  30. T.E. Vollman, W.L. Berry and D.C. Whybark, Manufacturing Planning and Control Systems, Third Edition. Irwin, Homewood Illinois (1992).  
  31. Y. Zheng and A. Federgruen, Finding Optimal (s,S) Policies is About as Simple as Evaluating a Single Policy. Oper. Res.39 (1991) 654-665.  
  32. Y. Zheng, On Properties of Stochastic Inventory Systems. Management Sci.38 (1992) 87-103.  
  33. P. Zipkin, Stochastic Lead Times in Continuous-Time Inventory Models. Naval Res. Logist. Quarterly33 (1986) 763-774.  
  34. P. Zipkin, Foundations of Inventory Management. McGraw-Hill, Boston Massachusetts (2000).  

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