# American Institute of Mathematical Sciences

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

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

 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

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.
Citation: Zvi Drezner, Carlton Scott. Approximate and exact formulas for the $(Q,r)$ inventory model. Journal of Industrial & 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, (1968).  doi: 10.1119/1.1972842.  Google Scholar [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.   Google Scholar [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.  doi: 10.1287/opre.40.4.808.  Google Scholar [4] G. Gallego, New bounds and heuristics for ($Q,r$) policies,, Management Science, 44 (1998), 219.  doi: 10.1287/mnsc.44.2.219.  Google Scholar [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.  doi: 10.3934/jimo.2011.7.875.  Google Scholar [6] J. O. Parr, Formula approximations to Brown's service function,, Production and Inventory Management, 13 (1972), 84.   Google Scholar [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.   Google Scholar [8] P. Zipkin, Foundations of Inventory Management,, McGraw-Hill, (2000).   Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, 7th printing, (1968).  doi: 10.1119/1.1972842.  Google Scholar [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.   Google Scholar [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.  doi: 10.1287/opre.40.4.808.  Google Scholar [4] G. Gallego, New bounds and heuristics for ($Q,r$) policies,, Management Science, 44 (1998), 219.  doi: 10.1287/mnsc.44.2.219.  Google Scholar [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.  doi: 10.3934/jimo.2011.7.875.  Google Scholar [6] J. O. Parr, Formula approximations to Brown's service function,, Production and Inventory Management, 13 (1972), 84.   Google Scholar [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.   Google Scholar [8] P. Zipkin, Foundations of Inventory Management,, McGraw-Hill, (2000).   Google Scholar
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