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 $\phantom{\rule{0.166667em}{0ex}}7.9$% 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.

@article{RO_2001__35_1_37_0, author = {Roundy, Robin O. and Samorodnitsky, Gennady}, title = {Optimal and near-optimal ($s,S$) inventory policies for {Levy} demand processes}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {37--70}, publisher = {EDP-Sciences}, volume = {35}, number = {1}, year = {2001}, zbl = {0996.90003}, mrnumber = {1841813}, language = {en}, url = {http://www.numdam.org/item/RO_2001__35_1_37_0/} }

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 - Recherche Opérationnelle PY - 2001 DA - 2001/// SP - 37 EP - 70 VL - 35 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/item/RO_2001__35_1_37_0/ UR - https://zbmath.org/?q=an%3A0996.90003 UR - https://www.ams.org/mathscinet-getitem?mr=1841813 LA - en ID - RO_2001__35_1_37_0 ER -

`Roundy, Robin O.; Samorodnitsky, Gennady. Optimal and near-optimal ($s,S$) inventory policies for Levy demand processes. RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 1, pp. 37-70. http://www.numdam.org/item/RO_2001__35_1_37_0/`

[1] Using the Deterministic EOQ Formula in Stochastic Inventory Control. Management Sci. 42 (1996) 830-834. | Zbl 0880.90030

,[2] Average Cost Optimality in Inventory Models with Markovian Demands. J. Optim. Theory Appl. 92 (1997) 497-526. | MR 1432607 | Zbl 0873.90021

and ,[3] A simple heuristic for computing nonstationary $(s,S)$ policies. Oper. Res. 47 (1999) 576-585. | Zbl 1014.90001

,[4] Inventory Models with Continuous, Stochastic Demands. Ann. Appl. Probab. 1 (1991) 419-435. | MR 1111526 | Zbl 0732.60080

and ,[5] Inventory Policies with Quantized Ordering. Naval Res. Logist. 39 (1992) 654-665. | Zbl 0749.90023

and ,[6] A Continuous Review Inventory Model with a Time Discount. IEEE Trans. 30 (1998) 747-757.

,[7] Computing an Optimal $(s,S)$ Policy is as Easy as Evaluating a Single Policy. Oper. Res. 39 (1991) 654-665. | Zbl 0749.90024

and ,[8] Computational Issues in an Infinite-Horizon, Multi-Echelon Inventory Model. Oper. Res. 32 (1984) 818-835. | MR 865581 | Zbl 0546.90026

and ,[9] An Efficient Algorithm for Computing Optimal $(s,S)$ Policies. Oper. Res. 32 (1984) 1268-1285. | MR 775258 | Zbl 0553.90031

and ,[10] Computing Optimal $(s,S)$ Policies in Inventory Models with Continuous Demands. Adv. in Appl. Probab. 17 (1985) 424-442. | MR 789491 | Zbl 0566.90026

and ,[11] An Introduction to Probability and its Applications, Vol. II. Wiley, New York (1966). | MR 210154

,[12] Sample Path Derivatives for $(s,S)$ Inventory Systems. Oper. Res. 42 (1994) 351-364. | Zbl 0805.90038

,[13] A New Approach to ($s,S$) Inventory Systems. J. Appl. Probab. 30 (1993) 898-912. | MR 1242020 | Zbl 0820.90036

, and ,[14] New Bounds and Heuristics for $(Q,r)$ Policies. Management Sci. 44 (1988) 219-233. | Zbl 0989.90003

,[15] 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).

and ,[16] 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). | MR 1876976

and ,[17] 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).

and ,[18] Analysis of Inventory Systems. Prentice Hall, Englewood Cliffs, NJ (1963). | Zbl 0133.42901

and ,[19] Stationary Stochastic Processes. Princeton University Press, Princeton, NJ (1970). | MR 258106 | Zbl 0214.16401

,[20] Operations Research in Production Planning, Scheduling, and Inventory Control. John Wiley and Sons, New York (1974).

and ,[21] Production and Operations Analysis, Second Edition. Irwin, Homewood Illinois, 60430 (1993).

,[22] Stochastic Storage Processes. Springer-Verlag, New York (1980). | MR 602329 | Zbl 0453.60094

,[23] Adventures in Stochastic Processes. Birkhauser, Boston, MA (1992). | MR 1181423 | Zbl 0762.60002

,[24] Tractible ($Q,R$) Heuristic Models for Constrained Service Levels. Management Sci. 43 (1997) 951-965. | Zbl 0890.90051

,[25] 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).

and ,[26] On the Objective Function Behavior in ($s,S$) Inventory Models. Oper. Res. 82 (1982) 709-724. | MR 666361 | Zbl 0486.90034

,[27] Semi-Stationary Clearing Processes. Stochastic Process. Appl. 6 (1978) 165-178. | MR 478406 | Zbl 0372.60146

and ,[28] General Theory of Markov Porcesses. Academic Press, Boston Massachusetts (1988). | MR 958914 | Zbl 0649.60079

,[29] Inventory Control in a Fluctuating Demand Environment. Oper. Res. 41 (1993) 351-370. | Zbl 0798.90035

and ,[30] Manufacturing Planning and Control Systems, Third Edition. Irwin, Homewood Illinois (1992).

, and ,[31] Finding Optimal $(s,S)$ Policies is About as Simple as Evaluating a Single Policy. Oper. Res. 39 (1991) 654-665. | Zbl 0749.90024

and ,[32] On Properties of Stochastic Inventory Systems. Management Sci. 38 (1992) 87-103. | Zbl 0764.90029

,[33] Stochastic Lead Times in Continuous-Time Inventory Models. Naval Res. Logist. Quarterly 33 (1986) 763-774. | MR 860745 | Zbl 0632.90018

,[34] Foundations of Inventory Management. McGraw-Hill, Boston Massachusetts (2000).

,