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A weak convergence approach to inventory control using a long-term average criterion

Part of: Stochastic systems and control Operations research and management science

Published online by Cambridge University Press:  29 November 2018

K. L. Helmes ,
R. H. Stockbridge  and
C. Zhu
K. L. Helmes*
Affiliation:
Humboldt University of Berlin
R. H. Stockbridge*
Affiliation:
University of Wisconsin-Milwaukee
C. Zhu*
Affiliation:
University of Wisconsin-Milwaukee
*
* Postal address: Institute for Operations Research, Humboldt University of Berlin, Spandauer Street 1, 10178, Berlin, Germany. Email address: [email protected]
** Postal address: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA.
** Postal address: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA.
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Abstract

In this paper we continue the examination of inventory control in which the inventory is modeled by a diffusion process and a long-term average cost criterion is used to make decisions. The class of such models under consideration has general drift and diffusion coefficients, and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. The conditions on the cost functions are greatly relaxed from those in Helmes et al. (2017). Characterization of the cost of a general (s, S) policy as a function of two variables naturally leads to a nonlinear optimization problem over the ordering levels s and S. Existence of an optimizing pair (s*, S*) is established for these models under very weak conditions; nonexistence of an optimizing pair is also discussed. Using average expected occupation and ordering measures and weak convergence arguments, weak conditions are given for the optimality of the (s*, S*) ordering policy in the general class of admissible policies. The analysis involves an auxiliary function that is globally C2 and which, together with the infimal cost, solves a particular system of linear equations and inequalities related to but different from the long-term average Hamilton‒Jacobi‒Bellman equation. This approach provides an analytical solution to the problem rather than a solution involving intricate analysis of the stochastic processes. The range of applicability of these results is illustrated on a drifted Brownian motion inventory model, both unconstrained and reflected, and on a geometric Brownian motion inventory model under two different cost structures.

Keywords

Inventoryimpulse controllong-term average costgeneral diffusion model(𝑠,𝑆) policyweak convergence

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 90B05: Inventory, storage, reservoirs
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1032 - 1074
Copyright
Copyright © Applied Probability Trust 2018 

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/apr.2018.50.

References

Bather, J. A. (1966).A continuous time inventory model.J. Appl. Prob. 3,538–549.Google Scholar
Bensoussan, A. (2011).Dynamic Programming and Inventory Control (Stud. Prob. Optimization Statist. 3).IOS Press,Amsterdam.Google Scholar
Borodin, A. N. and Salminen, P. (2002).Handbook of Brownian Motion–Facts and Formulae,2nd edn.Birkhäuser,Basel.Google Scholar
Chen, H.,Wu, O. Q. and Yao, D. D. (2010).On the benefit of inventory-based dynamic pricing strategies.Prod. Operat. Manag. 19,249–260.Google Scholar
Christensen, S. (2014).On the solution of general impulse control problems using superharmonic functions.Stoch. Process. Appl. 124,709–729.Google Scholar
Chung, K. L. (2001).A Course in Probability Theory,3rd edn.Academic Press,San Diego, CA.Google Scholar
Dai, J. G. and Yao, D. (2013).Brownian inventory models with convex holding cost, part 1: average-optimal controls.Stoch. Systems 3,442–499.Google Scholar
He, S.,Yao, D. and Zhang, H. (2017).Optimal ordering policy for inventory systems with quantity-dependent setup costs.Math. Operat. Res. 42,979–1006.Google Scholar
Helmes, K. L.,Stockbridge, R. H. and Zhu, C. (2017).Continuous inventory models of diffusion type: long-term average cost criterion.Ann. Appl. Prob. 27,1831–1885.Google Scholar
Helmes, K. L.,Stockbridge, R. H. and Zhu, C. (2018). Feller's branching diffusion inventory model. Supplementary material. Available at http://doi.org/10.1017/apr.2018.50.Google Scholar
Karlin, S. and Taylor, H. M. (1981).A Second Course in Stochastic Processes.Academic Press,New York.Google Scholar
Shiryaev, A. N. (1996).Probability (Graduate Texts Math. 95),2nd edn.Springer,New York.Google Scholar
Sigman, K. and Wolff, R. W. (1993).A review of regenerative processes.SIAM Rev. 35,269–288.Google Scholar
Sulem, A. (1986).A solvable one-dimensional model of a diffusion inventory system.Math. Operat. Res. 11,125–133.Google Scholar
Yao, D.,Chao, X. and Wu, J. (2015).Optimal control policy for a Brownian inventory system with concave ordering cost.J. Appl. Prob. 52,909–925.Google Scholar
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