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Lost-Sales Problems with Stochastic Lead Times: Convexity Results for Base-Stock Policies

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  • Ganesh Janakiraman

    (IOMS-OM Group, Stern School of Business, New York University, 44 West 4th Street, New York, New York 10012-1126)

  • Robin O. Roundy

    (School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853)

Abstract

We consider a single-location inventory system with periodic review and stochastic demand. It places replenishment orders to raise the inventory position—that is, inventory on hand plus inventory in transit—to exactly S at the beginning of every period. The lead time associated with each of these orders is random. However, the lead-time process is such that these orders do not cross. Demand that cannot be met with inventory available on hand is lost permanently. We state and prove some sample-path properties of lost sales, inventory on hand at the end of a period, and inventory position at the end of a period as functions of S . The main result is the convexity of the expected discounted sum of holding and lost-sales costs as a function of S . This result justifies the use of common search procedures or linear programming methods to determine optimal base-stock levels for inventory systems with lost sales and stochastic lead times. It should be noted that the class of base-stock policies is suboptimal for such systems, and we are primarily interested in them because of their widespread use.

Suggested Citation

  • Ganesh Janakiraman & Robin O. Roundy, 2004. "Lost-Sales Problems with Stochastic Lead Times: Convexity Results for Base-Stock Policies," Operations Research, INFORMS, vol. 52(5), pages 795-803, October.
    • Handle: RePEc:inm:oropre:v:52:y:2004:i:5:p:795-803
    DOI: 10.1287/opre.1040.0130
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    References listed on IDEAS

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    Cited by:

    1. Erica L. Plambeck & Amy R. Ward, 2007. "Note: A Separation Principle for a Class of Assemble-to-Order Systems with Expediting," Operations Research, INFORMS, vol. 55(3), pages 603-609, June.
    2. Jinzhi Bu & Xiting Gong & Dacheng Yao, 2019. "Technical Note—Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic," Operations Research, INFORMS, vol. 68(4), pages 1063-1073, July.
    3. Kouki, Chaaben & Babai, M. Zied & Jemai, Zied & Minner, Stefan, 2019. "Solution procedures for lost sales base-stock inventory systems with compound Poisson demand," International Journal of Production Economics, Elsevier, vol. 209(C), pages 172-182.
    4. Silver, Edward A. & Robb, David J., 2008. "Some insights regarding the optimal reorder period in periodic review inventory systems," International Journal of Production Economics, Elsevier, vol. 112(1), pages 354-366, March.
    5. Bijvank, Marco & Vis, Iris F.A., 2011. "Lost-sales inventory theory: A review," European Journal of Operational Research, Elsevier, vol. 215(1), pages 1-13, November.
    6. Xiuli Chao & Xiting Gong & Cong Shi & Chaolin Yang & Huanan Zhang & Sean X. Zhou, 2018. "Approximation Algorithms for Capacitated Perishable Inventory Systems with Positive Lead Times," Management Science, INFORMS, vol. 64(11), pages 5038-5061, November.
    7. Qi Feng & Sirong Luo & J. George Shanthikumar, 2020. "Integrating Dynamic Pricing with Inventory Decisions Under Lost Sales," Management Science, INFORMS, vol. 66(5), pages 2232-2247, May.
    8. Li, Zhaolin (Erick) & Liang, Guitian & Fu, Qi (Grace) & Teo, Chung-Piaw, 2023. "Base-Stock Policies with Constant Lead Time: Closed-Form Solutions and Applications," Working Papers BAWP-2023-01, University of Sydney Business School, Discipline of Business Analytics.
    9. Schmitt, Thomas G. & Kumar, Sanjay & Stecke, Kathryn E. & Glover, Fred W. & Ehlen, Mark A., 2017. "Mitigating disruptions in a multi-echelon supply chain using adaptive ordering," Omega, Elsevier, vol. 68(C), pages 185-198.
    10. Yanyi Xu & Sang-Phil Kim & Arnab Bisi & Maqbool Dada & Suresh Chand, 2018. "Base-Stock Models for Lost Sales: A Markovian Approach," Purdue University Economics Working Papers 1305, Purdue University, Department of Economics.
    11. Bijvank, Marco & Vis, Iris F.A., 2012. "Lost-sales inventory systems with a service level criterion," European Journal of Operational Research, Elsevier, vol. 220(3), pages 610-618.
    12. Woonghee Tim Huh & Ganesh Janakiraman & John A. Muckstadt & Paat Rusmevichientong, 2009. "An Adaptive Algorithm for Finding the Optimal Base-Stock Policy in Lost Sales Inventory Systems with Censored Demand," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 397-416, May.
    13. Huanan Zhang & Xiuli Chao & Cong Shi, 2020. "Closing the Gap: A Learning Algorithm for Lost-Sales Inventory Systems with Lead Times," Management Science, INFORMS, vol. 66(5), pages 1962-1980, May.
    14. Retsef Levi & Robin Roundy & Van Anh Truong & Xinshang Wang, 2017. "Provably Near-Optimal Balancing Policies for Multi-Echelon Stochastic Inventory Control Models," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 256-276, January.
    15. Woonghee Tim Huh & Mahesh Nagarajan, 2010. "Technical note ---Linear Inflation Rules for the Random Yield Problem: Analysis and Computations," Operations Research, INFORMS, vol. 58(1), pages 244-251, February.
    16. Woonghee Tim Huh & Ganesh Janakiraman & John A. Muckstadt & Paat Rusmevichientong, 2009. "Asymptotic Optimality of Order-Up-To Policies in Lost Sales Inventory Systems," Management Science, INFORMS, vol. 55(3), pages 404-420, March.
    17. Hill, R.M., 2007. "Continuous-review, lost-sales inventory models with Poisson demand, a fixed lead time and no fixed order cost," European Journal of Operational Research, Elsevier, vol. 176(2), pages 956-963, January.
    18. Anshul Sheopuri & Ganesh Janakiraman & Sridhar Seshadri, 2010. "New Policies for the Stochastic Inventory Control Problem with Two Supply Sources," Operations Research, INFORMS, vol. 58(3), pages 734-745, June.
    19. Qiushi Chen & Lei Zhao & Jan C. Fransoo & Zhe Li, 2019. "Dual-mode inventory management under a chance credit constraint," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(1), pages 147-178, March.
    20. Cai, Gangshu (George) & Chiang, Wen-Chyuan & Chen, Xiangfeng, 2011. "Game theoretic pricing and ordering decisions with partial lost sales in two-stage supply chains," International Journal of Production Economics, Elsevier, vol. 130(2), pages 175-185, April.
    21. Kumar Muthuraman & Sridhar Seshadri & Qi Wu, 2015. "Inventory Management with Stochastic Lead Times," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 302-327, February.
    22. David A. Goldberg & Dmitriy A. Katz-Rogozhnikov & Yingdong Lu & Mayank Sharma & Mark S. Squillante, 2016. "Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 898-913, August.

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