IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v56y2008i3p618-629.html
   My bibliography  Save this article

Impulse Control of Brownian Motion: The Constrained Average Cost Case

Author

Listed:
  • Melda Ormeci

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • J. G. Dai

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • John Vande Vate

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Abstract

When a manufacturer places repeated orders with a supplier to meet changing production requirements, he faces the challenge of finding the right balance between holding costs and the operational costs involved in adjusting the shipment sizes. We consider an inventory whose content fluctuates as a Brownian motion in the absence of control. At any moment, a controller can adjust the inventory level by any positive or negative quantity, but incurs both a fixed cost and a cost proportional to the magnitude of the adjustment. The inventory level must be nonnegative at all times and continuously incurs a linear holding cost. The objective is to minimize long-run average cost. We show that control band policies are optimal for the average cost Brownian control problem and explicitly calculate the parameters of the optimal control band policy. This form of policy is described by three parameters { q,Q,S }, 0 q (le) Q S . When the inventory falls to zero (rises to S ), the controller expedites (curtails) shipments to return it to q ( Q ). Employing apparently new techniques based on methods of Lagrangian relaxation, we show that this type of policy is optimal even with constraints on the size of adjustments and on the maximum inventory level. We also extend these results to the discounted cost problem.

Suggested Citation

  • Melda Ormeci & J. G. Dai & John Vande Vate, 2008. "Impulse Control of Brownian Motion: The Constrained Average Cost Case," Operations Research, INFORMS, vol. 56(3), pages 618-629, June.
  • Handle: RePEc:inm:oropre:v:56:y:2008:i:3:p:618-629
    DOI: 10.1287/opre.1060.0380
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1060.0380
    Download Restriction: no

    File URL: https://libkey.io/10.1287/opre.1060.0380?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. M. I. Taksar, 1985. "Average Optimal Singular Control and a Related Stopping Problem," Mathematics of Operations Research, INFORMS, vol. 10(1), pages 63-81, February.
    2. Elena V. Krichagina & Sheldon X. C. Lou & Michael I. Taksar, 1994. "Double Band Policy for Stochastic Manufacturing Systems in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 560-596, August.
    3. J. Michael Harrison & Thomas M. Sellke & Allison J. Taylor, 1983. "Impulse Control of Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 454-466, August.
    4. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    5. George M. Constantinides, 1976. "Stochastic Cash Management with Fixed and Proportional Transaction Costs," Management Science, INFORMS, vol. 22(12), pages 1320-1331, August.
    6. Plambeck, Erica L., 2005. "Asymptotically Optimal Control for an Assemble-to-Order System with Capacitated Component Production and Fixed Transport Costs," Research Papers 1893, Stanford University, Graduate School of Business.
    7. Agnès Sulem, 1986. "A Solvable One-Dimensional Model of a Diffusion Inventory System," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 125-133, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jingchen Wu & Xiuli Chao, 2014. "Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 163-189, February.
    2. Zhen Xu & Jiheng Zhang & Rachel Q. Zhang, 2019. "Instantaneous Control of Brownian Motion with a Positive Lead Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 943-965, August.
    3. Perera, Sandun & Gupta, Varun & Buckley, Winston, 2020. "Management of online server congestion using optimal demand throttling," European Journal of Operational Research, Elsevier, vol. 285(1), pages 324-342.
    4. Sandun C. Perera & Suresh P. Sethi, 2023. "A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 154-169, January.
    5. Haolin Feng & Kumar Muthuraman, 2010. "A Computational Method for Stochastic Impulse Control Problems," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 830-850, November.
    6. Abel Cadenillas & Peter Lakner & Michael Pinedo, 2010. "Optimal Control of a Mean-Reverting Inventory," Operations Research, INFORMS, vol. 58(6), pages 1697-1710, December.
    7. Korn, Ralf & Melnyk, Yaroslav & Seifried, Frank Thomas, 2017. "Stochastic impulse control with regime-switching dynamics," European Journal of Operational Research, Elsevier, vol. 260(3), pages 1024-1042.
    8. Gurjeet Dhesi & Bilal Shakeel & Marcel Ausloos, 2021. "Modelling and forecasting the kurtosis and returns distributions of financial markets: irrational fractional Brownian motion model approach," Annals of Operations Research, Springer, vol. 299(1), pages 1397-1410, April.
    9. Jiankui Yang & David D. Yao & Heng-Qing Ye, 2020. "Technical Note—On the Optimality of Reflection Control," Operations Research, INFORMS, vol. 68(6), pages 1668-1677, November.
    10. Shuangchi He & Dacheng Yao & Hanqin Zhang, 2017. "Optimal Ordering Policy for Inventory Systems with Quantity-Dependent Setup Costs," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 979-1006, November.
    11. Jinbiao Wu, 2019. "Optimal exchange rates management using stochastic impulse control for geometric Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 257-280, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shuangchi He & Dacheng Yao & Hanqin Zhang, 2017. "Optimal Ordering Policy for Inventory Systems with Quantity-Dependent Setup Costs," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 979-1006, November.
    2. Abel Cadenillas & Peter Lakner & Michael Pinedo, 2010. "Optimal Control of a Mean-Reverting Inventory," Operations Research, INFORMS, vol. 58(6), pages 1697-1710, December.
    3. Haolin Feng & Kumar Muthuraman, 2010. "A Computational Method for Stochastic Impulse Control Problems," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 830-850, November.
    4. Jinbiao Wu, 2019. "Optimal exchange rates management using stochastic impulse control for geometric Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 257-280, April.
    5. Fernando Alvarez & Francesco Lippi & Roberto Robatto, 2019. "Cost of Inflation in Inventory Theoretical Models," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 32, pages 206-226, April.
    6. Alain Bensoussan & Benoît Chevalier-Roignant, 2019. "Sequential Capacity Expansion Options," Operations Research, INFORMS, vol. 67(1), pages 33-57, January.
    7. Zhen Xu & Jiheng Zhang & Rachel Q. Zhang, 2019. "Instantaneous Control of Brownian Motion with a Positive Lead Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 943-965, August.
    8. Cadenillas, Abel & Zapatero, Fernando, 1999. "Optimal Central Bank Intervention in the Foreign Exchange Market," Journal of Economic Theory, Elsevier, vol. 87(1), pages 218-242, July.
    9. Andrew W. Lo & Harry Mamaysky & Jiang Wang, 2004. "Asset Prices and Trading Volume under Fixed Transactions Costs," Journal of Political Economy, University of Chicago Press, vol. 112(5), pages 1054-1090, October.
    10. Jingchen Wu & Xiuli Chao, 2014. "Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 163-189, February.
    11. Ben A. Chaouch, 2018. "Analysis of the stochastic cash balance problem using a level crossing technique," Annals of Operations Research, Springer, vol. 271(2), pages 429-444, December.
    12. Daniel Mitchell & Haolin Feng & Kumar Muthuraman, 2014. "Impulse Control of Interest Rates," Operations Research, INFORMS, vol. 62(3), pages 602-615, June.
    13. Jiankui Yang & David D. Yao & Heng-Qing Ye, 2020. "Technical Note—On the Optimality of Reflection Control," Operations Research, INFORMS, vol. 68(6), pages 1668-1677, November.
    14. Weerasinghe, Ananda & Zhu, Chao, 2016. "Optimal inventory control with path-dependent cost criteria," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1585-1621.
    15. Wolosewicz, Cathy & Dauzère-Pérès, Stéphane & Aggoune, Riad, 2015. "A Lagrangian heuristic for an integrated lot-sizing and fixed scheduling problem," European Journal of Operational Research, Elsevier, vol. 244(1), pages 3-12.
    16. Ricardo J. Caballero & Eduardo M.R.A. Engel, 2004. "A Comment on the Economics of Labor Adjustment: Mind the Gap: Reply," American Economic Review, American Economic Association, vol. 94(4), pages 1238-1244, September.
    17. Jukka Isohätälä & Alistair Milne & Donald Robertson, 2020. "The Net Worth Trap: Investment and Output Dynamics in the Presence of Financing Constraints," Mathematics, MDPI, vol. 8(8), pages 1-32, August.
    18. M Diaby & A L Nsakanda, 2006. "Large-scale capacitated part-routing in the presence of process and routing flexibilities and setup costs," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(9), pages 1100-1112, September.
    19. Ogbe, Emmanuel & Li, Xiang, 2017. "A new cross decomposition method for stochastic mixed-integer linear programming," European Journal of Operational Research, Elsevier, vol. 256(2), pages 487-499.
    20. Mutsunori Yagiura & Toshihide Ibaraki & Fred Glover, 2004. "An Ejection Chain Approach for the Generalized Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 133-151, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:56:y:2008:i:3:p:618-629. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.