American Institute of Mathematical Sciences

Advanced
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

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 & 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

show all references

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

[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 & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036
[2]

Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34.

[3]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
[4]

King-Yeung Lam, Daniel Munther. Invading the ideal free distribution. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3219-3244. doi: 10.3934/dcdsb.2014.19.3219
[5]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[6]

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 & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020221
[7]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437
[8]

I-Lin Wang, Ju-Chun Lin. A compaction scheme and generator for distribution networks. Journal of Industrial & Management Optimization, 2016, 12 (1) : 117-140. doi: 10.3934/jimo.2016.12.117
[9]

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
[10]

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
[11]

Ginestra Bianconi, Riccardo Zecchina. Viable flux distribution in metabolic networks. Networks & Heterogeneous Media, 2008, 3 (2) : 361-369. doi: 10.3934/nhm.2008.3.361
[12]

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
[13]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237
[14]

Pieter Moree. On the distribution of the order over residue classes. Electronic Research Announcements, 2006, 12: 121-128.

[15]

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 & Management Optimization, 2009, 5 (3) : 525-540. doi: 10.3934/jimo.2009.5.525
[16]

Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219
[17]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[18]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[19]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023
[20]

Orazio Muscato, Wolfgang Wagner, Vincenza Di Stefano. Properties of the steady state distribution of electrons in semiconductors. Kinetic & Related Models, 2011, 4 (3) : 809-829. doi: 10.3934/krm.2011.4.809

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences