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Provably near-optimal sampling-based algorithms for Stochastic inventory control models

Published: 21 May 2006 Publication History

Abstract

We consider two fundamental stochastic optimization problems that arise in the context of supply-chain models, the single-period newsvendor problem and its multiperiod extension with independent demands. These problems are among the most well-studied stochastic optimization problems in the Operations Research literature. Most commonly, these problems are studied from the perspective that the input probability distributions are given in terms of specific probability distribution functions that are computationally tractable; under this assumption, both problems can be solved efficiently. Unfortunately, this information is unlikely to be available in practice, and hence we make the more realistic assumption that the probability distribution is given by a "black box" from which independent samples can be drawn. We give the first fully polynomial randomized approximation schemes for these two problems in this sampling-based model.Our work provides new insights into the power of two of the most often-used approaches to solving stochastic optimization problems, the sample average approximation (SAA) and stochastic dynamic programming. For the newsvendor problem, we show that by taking a polynomial number of samples and then solving the newsvendor problem with respect to the resulting approximation to the true distribution, we obtain provably near-optimal solution. This significantly extends the class of problems for which the SAA is known to yield a scheme. Finally, we show how to adapt the framework of stochastic dynamic programming to yield an approximation scheme for the multiperiod newsvendor problem with independent demands. We believe that this is an interesting first step towards the goal of providing a mechanism for deriving efficient approximate stochastic dynamic programming methods for a wide range of multistage stochastic optimization problems.

References

[1]
D. Bertsimas and A. Thiele. A robust optimization approach to supply chain management. In Proceedings of 14th IPCO, pages 86--100, 2004.
[2]
D. Bertsimas and A. Thiele. A data-driven approach to newsvendor problems. Working Paper, 2005.
[3]
P. Billingsley. Probability and Measure. John Wiley & Sons, 1995. Third edition.
[4]
M. Charikar, C. Chekuri, and M. Pál. Sampling bounds for stochastic optimization. In Proceedings of APPROX-RANDOM 2005, pages 257--269, 2005.
[5]
R. Chase, R. Jacobs, and N. Aquilano. Production and Operations Management. McGraw-Hill, 2000.
[6]
B. C. Dean, M. X. Goemans, and J. Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. In Proceedings of the 45th Annual IEEE Symposium on the Foundations of Computer Science, pages 208--217, 2004.
[7]
B. C. Dean, M. X. Goemans, and J. Vondrák. Adaptivity and approximation for stochastic packing problems. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 395--404, 2005.
[8]
G. Gallego, J. K. Ryan, and D. Simchi-Levi. Minimax analysis for discrete finite horizon inventory models. IIE Transactions, pages 861--874, 2001.
[9]
P. Glasserman and Y. C. Ho. Gradient estimation via pertubation analysis. Kluwer Academic Publishers, 1991.
[10]
P. Glasserman and S. Tayur. Sensitivity analysis for base-stock levels in multiechelon production-inventory systems. Management Science, 41:263--282, 1995.
[11]
A. Gupta, M. Pal, R. Ravi, and A. Sinha. Boosted sampling: approximation algorithms for stochastic optimization. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 265--274, 2004.
[12]
F. Hillier and G. J. Lieberman. Introduction to Operations Research. McGraw-Hill, 2005.
[13]
W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13--30, 1963.
[14]
W. T. Huh and P. Rusmevichientong. A non-parametric approach to stochastic inventory planning with lost-sales and censored demand. Technical Report 1427, School of OR&IE, Cornell University, 2006. Submitted to OR.
[15]
N. Immorlica, D. R. Karger, M. Minkoff, and V. S. Mirrokni. On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 691--700, 2004.
[16]
L. Johnson and D. C. Montgomery. Operations Research in Production Planning, Scheduling, and Inventory Control. John Wiley & Sons, New York, 1974.
[17]
R. Kapuscinski and S. Tayur. Optimal policies and simulation based optimization for capacitated production inventory systems. In Quantative models for supply chain management, chapter 2. Kluwer Academic Publisher, 1998.
[18]
A. J. Kleywegt, A. Shapiro, and T. Homem-de-Mello. The sample average approximation method for stochastic discrete optimization. SIAM Journal of Optimization, 12:479--502, 2001.
[19]
R. Levi, G. Janakiraman, and M. Nagarajan. Provably near-optimal balancing policies for stochastic inventory control models with lost-sales. Working Paper, 2005.
[20]
R. Levi, M. Pal, R. O. Roundy, and D. B. Shmoys. Approximation algorithms for stochastic inventory control models. Technical Report TR1412, ORIE Department, Cornell University, 2004. Submitted to MOR.
[21]
R. Levi, R. O. Roundy, and D. B. Shmoys. Sample-based optimization algorithms for stochastic inventory control models. Full version of this extended abstract, 2005.
[22]
R. Levi, R. O. Roundy, and V. A. Truong. Provably near-optimal balancing policies for multi-echelon stochastic inventory control models. Working paper, 2005.
[23]
R. Levi, R. O. Roundy, D. B. Shmoys, and V. A. Truong. Approximation algorithms for capacitated stochastic inventory control models. Submitted to OR, 2005.
[24]
L. H. Liyanage and J. G. Shanthikumar. A practical inventory control using operational statistics. Operations Research Letters, 33:341--348, 2005.
[25]
A. Nemirovski and A. Shapiro, October, 2004. Personal communication.
[26]
G. Perakis and G. Roels. The distribution-free newsvendor: Inventory management with limited demand information. Unpublished manuscript, 2005.
[27]
R. T. Rockafellar. Convex Analysis. Princeton University Press, 1972.
[28]
A. Shapiro and T. Homem-de-Mello. On the rate of convergence of Monte Carlo approximations of stochastic programs. SIAM Journal of Optimization, 11:70--86, 2000.
[29]
A. Shapiro, T. Homem-de-Mello, and J. Kim. Conditioning of convex piecewise linear stochastic programs. Mathematical Programming, 94:1--19, 2002.
[30]
A. Shapiro and A. Nemirovski. On the complexity of stochastic programming problems. E-print available at: http://www.optimization-online.org, 2004.
[31]
D. B. Shmoys and C. Swamy. Stochastic optimization is (almost) as easy as deterministic optimization. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 228--237, 2004.
[32]
J. Si, A. G. Barto, P. W. B, and D. W. II. Handbook of Learning and Approximate Dynamic Programming. Wiley&Sons, INC., Publications, 2004.
[33]
C. Swamy and D. B. Shmoys. Sampling-based approximation algorithms for multi-stage stochastic optimization. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 357--366.
[34]
P. H. Zipkin. Foundations of inventory management. The McGraw-Hill Companies, Inc, 2000.

Cited By

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  • (2019)Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control ModelsMathematics of Operations Research10.1287/moor.2018.094044:2(668-692)Online publication date: 1-May-2019
  • (2014) Modified Echelon ( r, Q ) Policies with Guaranteed Performance Bounds for Stochastic Serial Inventory Systems Operations Research10.1287/opre.2014.129162:4(812-828)Online publication date: Aug-2014
  • (2012)Newsvendor-type models with decision-dependent uncertaintyMathematical Methods of Operations Research10.1007/s00186-012-0396-376:2(189-221)Online publication date: 1-Aug-2012
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    cover image ACM Conferences
    STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
    May 2006
    786 pages
    ISBN:1595931341
    DOI:10.1145/1132516
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Publication History

    Published: 21 May 2006

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    Author Tags

    1. approximation algorithms
    2. black box
    3. inventory problems
    4. sampling-based algorithms

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    May 21 - 23, 2006
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    View all
    • (2019)Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control ModelsMathematics of Operations Research10.1287/moor.2018.094044:2(668-692)Online publication date: 1-May-2019
    • (2014) Modified Echelon ( r, Q ) Policies with Guaranteed Performance Bounds for Stochastic Serial Inventory Systems Operations Research10.1287/opre.2014.129162:4(812-828)Online publication date: Aug-2014
    • (2012)Newsvendor-type models with decision-dependent uncertaintyMathematical Methods of Operations Research10.1007/s00186-012-0396-376:2(189-221)Online publication date: 1-Aug-2012
    • (2011)Constraint programming for stochastic inventory systems under shortage costAnnals of Operations Research10.1007/s10479-011-0936-x195:1(49-71)Online publication date: 30-Jul-2011
    • (2010)Online algorithms for the newsvendor problem with and without censored demandsProceedings of the 4th international conference on Frontiers in algorithmics10.5555/1881195.1881218(234-249)Online publication date: 11-Aug-2010
    • (2008)Solving operational statistics via a Bayesian analysisOperations Research Letters10.1016/j.orl.2007.04.01036:1(110-116)Online publication date: 1-Jan-2008
    • (2006)Approximation algorithms for 2-stage stochastic optimization problemsACM SIGACT News10.1145/1122480.112249337:1(33-46)Online publication date: 1-Mar-2006

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