IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v241y2016i1d10.1007_s10479-014-1679-2.html
   My bibliography  Save this article

A make-to-stock production/inventory model with MAP arrivals and phase-type demands

Author

Listed:
  • Yonit Barron

    (University of Haifa)

  • David Perry

    (University of Haifa)

  • Wolfgang Stadje

    (University of Osnabrück)

Abstract

We consider a make-to-stock production/inventory model in a random environment with finite storage capacity and restricted backlogging possibility. Our aim is to demonstrate that all cost quantities of interest can be derived in closed form under quite general assumptions on the demand arrival process and on the switches in the production rates. Specifically, the demands arrive according to a Markov additive process governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. The production process switches between predetermined rates which depend on the state of the environment and on the presence or absence of backlogs. Four types of costs are considered: the holding cost for the stock, the cost of lost production due to the finite storage capacity, the shortage cost for the backlogged demand and the cost due to unsatisfied demand. We obtain explicit formulas for these cost functionals in the discounted case and under the long-run average criterion. The derivations are based on optional sampling of a multi-dimensional martingale and on fluid flow techniques.

Suggested Citation

  • Yonit Barron & David Perry & Wolfgang Stadje, 2016. "A make-to-stock production/inventory model with MAP arrivals and phase-type demands," Annals of Operations Research, Springer, vol. 241(1), pages 373-409, June.
    • Handle: RePEc:spr:annopr:v:241:y:2016:i:1:d:10.1007_s10479-014-1679-2
    DOI: 10.1007/s10479-014-1679-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-014-1679-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-014-1679-2?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. V. Ramaswami, 2006. "Passage Times in Fluid Models with Application to Risk Processes," Methodology and Computing in Applied Probability, Springer, vol. 8(4), pages 497-515, December.
    2. Michael Katehakis & Laurens Smit, 2012. "On computing optimal (Q,r) replenishment policies under quantity discounts," Annals of Operations Research, Springer, vol. 200(1), pages 279-298, November.
    3. B. T. Doshi & F. A. van der Duyn Schouten & J. J. Talman, 1978. "A Production-Inventory Control Model with a Mixture of Back-Orders and Lost-Sales," Management Science, INFORMS, vol. 24(10), pages 1078-1087, June.
    4. Frostig, Esther, 2005. "The expected time to ruin in a risk process with constant barrier via martingales," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 216-228, October.
    5. Perry, David & Berg, M. & Posner, M. J. M., 2001. "Stochastic models for broker inventory in dealership markets with a cash management interpretation," Insurance: Mathematics and Economics, Elsevier, vol. 29(1), pages 23-34, August.
    6. Junmin Shi & Michael Katehakis & Benjamin Melamed, 2013. "Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands," Annals of Operations Research, Springer, vol. 208(1), pages 593-612, September.
    7. David Perry & Wolfgang Stadje & Shelemyahu Zacks, 2005. "Sporadic and Continuous Clearing Policies for a Production/Inventory System Under an M / G Demand Process," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 354-368, May.
    8. Dickson,David C. M., 2005. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521846400.
    9. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
    10. Jim (Junmin) Shi & Michael N. Katehakis & Benjamin Melamed & Yusen Xia, 2014. "Production-Inventory Systems with Lost Sales and Compound Poisson Demands," Operations Research, INFORMS, vol. 62(5), pages 1048-1063, October.
    11. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    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. Dhanya Shajin & A. Krishnamoorthy & A. N. Dudin & Varghese C. Joshua & Varghese Jacob, 2020. "On a queueing-inventory system with advanced reservation and cancellation for the next K time frames ahead: the case of overbooking," Queueing Systems: Theory and Applications, Springer, vol. 94(1), pages 3-37, February.
    2. Pablo Azcue & Esther Frostig & Nora Muler, 2023. "Optimal Strategies in a Production Inventory Control Model," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-43, March.
    3. Baek, Jung Woo & Bae, Yun Han, 2022. "A queuing-inventory model for manufacturing systems with fluid-type inventory," Omega, Elsevier, vol. 111(C).
    4. Barron, Yonit, 2022. "The continuous (S,s,Se) inventory model with dual sourcing and emergency orders," European Journal of Operational Research, Elsevier, vol. 301(1), pages 18-38.
    5. Alexander Moiseev & Maria Shklennik & Evgeny Polin, 2023. "Infinite-server queueing tandem with Markovian arrival process and service depending on its state," Annals of Operations Research, Springer, vol. 326(1), pages 261-279, July.
    6. Sally McClean, 2021. "Using Markov Models to Characterize and Predict Process Target Compliance," Mathematics, MDPI, vol. 9(11), pages 1-12, May.
    7. Walid W. Nasr, 2022. "Inventory systems with stochastic and batch demand: computational approaches," Annals of Operations Research, Springer, vol. 309(1), pages 163-187, February.
    8. Barron, Yonit, 2023. "A stochastic card balance management problem with continuous and batch-type bilateral transactions," Operations Research Perspectives, Elsevier, vol. 10(C).

    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. Onno Boxma & David Perry & Wolfgang Stadje & Shelley Zacks, 2022. "A compound Poisson EOQ model for perishable items with intermittent high and low demand periods," Annals of Operations Research, Springer, vol. 317(2), pages 439-459, October.
    2. Apostolos Burnetas & Odysseas Kanavetas, 2018. "Inventory policies for two products under Poisson demand: Interaction between demand substitution, limited storage capacity and replenishment time uncertainty," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(8), pages 676-698, December.
    3. Wen Chen & Adam J. Fleischhacker & Michael N. Katehakis, 2015. "Dynamic pricing in a dual‐market environment," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(7), pages 531-549, October.
    4. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    5. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    6. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    7. Zeng, Xudong, 2010. "Optimal reinsurance with a rescuing procedure," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 397-405, April.
    8. Yonit Barron & Dror Hermel, 2017. "Shortage decision policies for a fluid production model with MAP arrivals," International Journal of Production Research, Taylor & Francis Journals, vol. 55(14), pages 3946-3969, July.
    9. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    10. Kartikeya Puranam & Michael Katehakis, 2014. "On optimal bidding and inventory control in sequential procurement auctions: the multi period case," Annals of Operations Research, Springer, vol. 217(1), pages 447-462, June.
    11. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    12. Chi, Yichun & Lin, X. Sheldon, 2011. "On the threshold dividend strategy for a generalized jump-diffusion risk model," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 326-337, May.
    13. Pablo Azcue & Esther Frostig & Nora Muler, 2023. "Optimal Strategies in a Production Inventory Control Model," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-43, March.
    14. Eugene A. Feinberg & Yan Liang, 2022. "On the optimality equation for average cost Markov decision processes and its validity for inventory control," Annals of Operations Research, Springer, vol. 317(2), pages 569-586, October.
    15. Feng, Runhuan & Shimizu, Yasutaka, 2014. "Potential measures for spectrally negative Markov additive processes with applications in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 11-26.
    16. Kim, Bara & Kim, Hwa-Sung & Kim, Jeongsim, 2008. "A risk model with paying dividends and random environment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 717-726, April.
    17. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    18. Eric C. K. Cheung & David Landriault, 2012. "On a Risk Model with Surplus-dependent Premium and Tax Rates," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 233-251, June.
    19. Frostig, Esther, 2010. "Asymptotic analysis of a risk process with high dividend barrier," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 21-26, August.
    20. Dassios, Angelos & Wu, Shanle, 2009. "On barrier strategy dividends with Parisian implementation delay for classical surplus processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 195-202, October.

    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:spr:annopr:v:241:y:2016:i:1:d:10.1007_s10479-014-1679-2. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.