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Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle

Author

Listed:
  • Nir Halman

    (Jerusalem School of Business Administration, The Hebrew University, 91905 Jerusalem, Israel; Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • James B. Orlin

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • David Simchi-Levi

    (Department of Civil and Environmental Engineering and Division of Engineering Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon. A closely related problem is the single-period newsvendor model. It is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement (ordering) cost is fixed plus linear. The optimal policy for the single-item lot-sizing model is also well known under similar assumptions.In this paper we show that the classical (single-period) newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle. We provide a fully polynomial time approximation scheme for the nonlinear single-item stochastic lot-sizing problem, when demand distribution is given by an oracle, procurement costs are provided as nondecreasing oracles, holding/backlogging/disposal costs are linear, and lead time is positive. Similar results exist for the nonlinear newsvendor problem. These approximation schemes are designed by extending the technique of K -approximation sets and functions.

Suggested Citation

  • Nir Halman & James B. Orlin & David Simchi-Levi, 2012. "Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle," Operations Research, INFORMS, vol. 60(2), pages 429-446, April.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:2:p:429-446
    DOI: 10.1287/opre.1110.1031
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    References listed on IDEAS

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    1. Retsef Levi & Martin Pál & Robin O. Roundy & David B. Shmoys, 2007. "Approximation Algorithms for Stochastic Inventory Control Models," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 284-302, May.
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    3. Retsef Levi & Robin O. Roundy & David B. Shmoys & Van Anh Truong, 2008. "Approximation Algorithms for Capacitated Stochastic Inventory Control Models," Operations Research, INFORMS, vol. 56(5), pages 1184-1199, October.
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    Cited by:

    1. 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.
    2. Halman, Nir & Kellerer, Hans & Strusevich, Vitaly A., 2018. "Approximation schemes for non-separable non-linear boolean programming problems under nested knapsack constraints," European Journal of Operational Research, Elsevier, vol. 270(2), pages 435-447.
    3. Levi DeValve & Saša Pekeč & Yehua Wei, 2020. "A Primal-Dual Approach to Analyzing ATO Systems," Management Science, INFORMS, vol. 66(11), pages 5389-5407, November.
    4. Liu, Congzheng & Letchford, Adam N. & Svetunkov, Ivan, 2022. "Newsvendor problems: An integrated method for estimation and optimisation," European Journal of Operational Research, Elsevier, vol. 300(2), pages 590-601.
    5. 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.
    6. Nir Halman, 2020. "Provably Near-Optimal Approximation Schemes for Implicit Stochastic and Sample-Based Dynamic Programs," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1157-1181, October.
    7. Hanzhang Qin & David Simchi-Levi & Li Wang, 2022. "Data-Driven Approximation Schemes for Joint Pricing and Inventory Control Models," Management Science, INFORMS, vol. 68(9), pages 6591-6609, September.
    8. Chung-Lun Li & Qingying Li, 2016. "Polynomial-Time Solvability of Dynamic Lot Size Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(03), pages 1-20, June.

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