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Base-Stock Models for Lost Sales: A Markovian Approach

Author

Listed:
  • Yanyi Xu
  • Sang-Phil Kim
  • Arnab Bisi
  • Maqbool Dada
  • Suresh Chand

Abstract

We consider the lost sales model with discrete demand that is filled only from inventory on-hand. The inventory is reviewed every T periods and an order is placed to bring the inventory position back to a target base-stock level R. The order is received after a lead time of L periods. Based on the outstanding orders in the pipeline, we represent the state of the system as a Markov chain for a given base-stock level. We develop structural properties of the best stationary base-stock policy. These properties include a stopping rule and a method for finding strictly improving lower-bounds that facilitate the development of an optimization algorithm. We also show that the structure of the transition probability matrix is recursive in R and L and that the matrix is sparse with special structure. This special structure is used to facilitate computation of the stationary distribution. In turn this distribution is used to compute the long-run average cost for a given base-stock from which a search yields the best, minimum cost, base-stock value. Analytical results complemented by numerical examples reveal that neither the best stationary base-stock nor its average cost is monotone in L for a given T.

Suggested Citation

  • 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.
    • Handle: RePEc:pur:prukra:1305
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    References listed on IDEAS

    as
    1. 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.
    2. Brian Downs & Richard Metters & John Semple, 2001. "Managing Inventory with Multiple Products, Lags in Delivery, Resource Constraints, and Lost Sales: A Mathematical Programming Approach," Management Science, INFORMS, vol. 47(3), pages 464-479, March.
    3. Linwei Xin & David A. Goldberg, 2016. "Optimality Gap of Constant-Order Policies Decays Exponentially in the Lead Time for Lost Sales Models," Operations Research, INFORMS, vol. 64(6), pages 1556-1565, December.
    4. Marco Bijvank & Woonghee Tim Huh & Ganesh Janakiraman & Wanmo Kang, 2014. "Robustness of Order-Up-to Policies in Lost-Sales Inventory Systems," Operations Research, INFORMS, vol. 62(5), pages 1040-1047, October.
    5. Nir Halman & Diego Klabjan & Mohamed Mostagir & Jim Orlin & David Simchi-Levi, 2009. "A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 674-685, August.
    6. Retsef Levi & Ganesh Janakiraman & Mahesh Nagarajan, 2008. "A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 351-374, May.
    7. Thomas E. Morton, 1971. "The Near-Myopic Nature of the Lagged-Proportional-Cost Inventory Problem with Lost Sales," Operations Research, INFORMS, vol. 19(7), pages 1708-1716, December.
    8. Steven Nahmias, 1979. "Simple Approximations for a Variety of Dynamic Leadtime Lost-Sales Inventory Models," Operations Research, INFORMS, vol. 27(5), pages 904-924, October.
    9. 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.
    10. Paul Zipkin, 2008. "On the Structure of Lost-Sales Inventory Models," Operations Research, INFORMS, vol. 56(4), pages 937-944, August.
    11. Bijvank, Marco & Johansen, Søren Glud, 2012. "Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length," European Journal of Operational Research, Elsevier, vol. 220(1), pages 106-114.
    12. Paul Zipkin, 2008. "Old and New Methods for Lost-Sales Inventory Systems," Operations Research, INFORMS, vol. 56(5), pages 1256-1263, October.
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