Approximate and exact formulas for the (Q,r) inventory model

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January 2015, 11(1): 135-144. doi: 10.3934/jimo.2015.11.135

Approximate and exact formulas for the $(Q,r)$ inventory model

Zvi Drezner ^1, and Carlton Scott ^2,

1.	Steven G. Mihaylo College of Business and Economics, California State University-Fullerton, Fullerton, CA 92634, United States
2.	The Paul Merage School of Business, University of California, Irvine, CA 92697, United States

Received January 2012 Revised January 2014 Published May 2014

Abstract
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In this paper, new results are derived for the $(Q,r)$ stochastic inventory model. We derive approximate formulas for the optimal solution for the particular case of an exponential demand distribution. The approximate solution is within 0.29% of the optimal value. We also derive simple formulas for a Poisson demand distribution. The original expression involves double summation. We simplify the formula and are able to calculate the exact value of the objective function in $O(1)$ time with no need for any summations.

Keywords: Inventory models, exponential distribution, poisson distribution..

Mathematics Subject Classification: Primary: 90B05; Secondary: 33F05, 34K2.

Citation: Zvi Drezner, Carlton Scott. Approximate and exact formulas for the $(Q,r)$ inventory model. Journal of Industrial and Management Optimization, 2015, 11 (1) : 135-144. doi: 10.3934/jimo.2015.11.135

References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 7th printing, Applied Mathematics Series, National Bureau of Standards, Washington, DC., 1968. doi: 10.1119/1.1972842.
[2]	R. B. S. Brooks and J. Y. Lu, On the convexity of the backorder function for an E.O.Q policy, Management Science, 15 (1969), 453-454.
[3]	A. Federgruen and Y. -S. Zheng, An efficient algorithm for computing an optimal $(r,Q)$ policy in continuous review stochastic inventory systems, Operations Research, 40 (1992), 808-813. doi: 10.1287/opre.40.4.808.
[4]	G. Gallego, New bounds and heuristics for ($Q,r$) policies, Management Science, 44 (1998), 219-233. doi: 10.1287/mnsc.44.2.219.
[5]	R. Loxton and Q. Lin, Optimal fleet composition via dynamic programming and golden section search, Journal of Industrial and Management Optimization, 7 (2011), 875-890. doi: 10.3934/jimo.2011.7.875.
[6]	J. O. Parr, Formula approximations to Brown's service function, Production and Inventory Management, 13 (1972), 84-86.
[7]	D. E. Platt, L. W. Robinson and R. B. Freund, Tractable ($Q,R$) heuristic models for constrained service levels, Management Science, 43 (1997), 951-965.
[8]	P. Zipkin, Foundations of Inventory Management, McGraw-Hill, New York, 2000.

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References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 7th printing, Applied Mathematics Series, National Bureau of Standards, Washington, DC., 1968. doi: 10.1119/1.1972842.
[2]	R. B. S. Brooks and J. Y. Lu, On the convexity of the backorder function for an E.O.Q policy, Management Science, 15 (1969), 453-454.
[3]	A. Federgruen and Y. -S. Zheng, An efficient algorithm for computing an optimal $(r,Q)$ policy in continuous review stochastic inventory systems, Operations Research, 40 (1992), 808-813. doi: 10.1287/opre.40.4.808.
[4]	G. Gallego, New bounds and heuristics for ($Q,r$) policies, Management Science, 44 (1998), 219-233. doi: 10.1287/mnsc.44.2.219.
[5]	R. Loxton and Q. Lin, Optimal fleet composition via dynamic programming and golden section search, Journal of Industrial and Management Optimization, 7 (2011), 875-890. doi: 10.3934/jimo.2011.7.875.
[6]	J. O. Parr, Formula approximations to Brown's service function, Production and Inventory Management, 13 (1972), 84-86.
[7]	D. E. Platt, L. W. Robinson and R. B. Freund, Tractable ($Q,R$) heuristic models for constrained service levels, Management Science, 43 (1997), 951-965.
[8]	P. Zipkin, Foundations of Inventory Management, McGraw-Hill, New York, 2000.

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