# 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 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.

show all references

##### 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.
 [1] Abraão D. C. Nascimento, Leandro C. Rêgo, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems and Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036 [2] François Golse. The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles. Kinetic and Related Models, 2022, 15 (3) : 517-534. doi: 10.3934/krm.2022001 [3] Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34. [4] Yuli Zhang, Lin Han, Xiaotian Zhuang. Distributionally robust front distribution center inventory optimization with uncertain multi-item orders. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1777-1795. doi: 10.3934/dcdss.2022006 [5] Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110 [6] King-Yeung Lam, Daniel Munther. Invading the ideal free distribution. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3219-3244. doi: 10.3934/dcdsb.2014.19.3219 [7] Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949 [8] Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221 [9] Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks and Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437 [10] I-Lin Wang, Ju-Chun Lin. A compaction scheme and generator for distribution networks. Journal of Industrial and Management Optimization, 2016, 12 (1) : 117-140. doi: 10.3934/jimo.2016.12.117 [11] Yvo Desmedt, Niels Duif, Henk van Tilborg, Huaxiong Wang. Bounds and constructions for key distribution schemes. Advances in Mathematics of Communications, 2009, 3 (3) : 273-293. doi: 10.3934/amc.2009.3.273 [12] Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433 [13] Ginestra Bianconi, Riccardo Zecchina. Viable flux distribution in metabolic networks. Networks and Heterogeneous Media, 2008, 3 (2) : 361-369. doi: 10.3934/nhm.2008.3.361 [14] Robert Stephen Cantrell, Chris Cosner, Yuan Lou. Evolution of dispersal and the ideal free distribution. Mathematical Biosciences & Engineering, 2010, 7 (1) : 17-36. doi: 10.3934/mbe.2010.7.17 [15] Pieter Moree. On the distribution of the order over residue classes. Electronic Research Announcements, 2006, 12: 121-128. [16] R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial and Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237 [17] Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. On the weight distribution of the cosets of MDS codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021042 [18] Hanqing Jin, Shige Peng. Optimal unbiased estimation for maximal distribution. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 189-198. doi: 10.3934/puqr.2021009 [19] Eunju Hwang, Kyung Jae Kim, Bong Dae Choi. Delay distribution and loss probability of bandwidth requests under truncated binary exponential backoff mechanism in IEEE 802.16e over Gilbert-Elliot error channel. Journal of Industrial and Management Optimization, 2009, 5 (3) : 525-540. doi: 10.3934/jimo.2009.5.525 [20] Ross Callister, Duc-Son Pham, Mihai Lazarescu. Using distribution analysis for parameter selection in repstream. Mathematical Foundations of Computing, 2019, 2 (3) : 215-250. doi: 10.3934/mfc.2019015

2020 Impact Factor: 1.801