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Polynomial-Time Solvability of Dynamic Lot Size Problems

Author

Listed:
  • Chung-Lun Li

    (Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong)

  • Qingying Li

    (Glorious Sun School of Business and Management, Donghua University, 1882 West Yan’an Road, Shanghai, China, 200051, China)

Abstract

There has been a lot of research on dynamic lot sizing problems with different nonlinear cost structures due to capacitated production, minimum order quantity requirements, availability of quantity discounts, etc. Developing optimal solutions efficiently for dynamic lot sizing models with nonlinear cost functions is a challenging topic. In this paper, we present a set of sufficient conditions such that if a single-item dynamic lot sizing problem satisfies these conditions, then the existence of a polynomial-time solution method for the problem is guaranteed. Several examples are presented to demonstrate the use of these sufficient conditions.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:apjorx:v:33:y:2016:i:03:n:s0217595916500184
    DOI: 10.1142/S0217595916500184
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    1. Salah E. Elmaghraby & Vishwas Y. Bawle, 1972. "Optimization of Batch Ordering under Deterministic Variable Demand," Management Science, INFORMS, vol. 18(9), pages 508-517, May.
    2. Albert Wagelmans & Stan van Hoesel & Antoon Kolen, 1992. "Economic Lot Sizing: An O(n log n) Algorithm That Runs in Linear Time in the Wagner-Whitin Case," Operations Research, INFORMS, vol. 40(1-supplem), pages 145-156, February.
    3. Steven A. Lippman, 1969. "Optimal Inventory Policy with Multiple Set-Up Costs," Management Science, INFORMS, vol. 16(1), pages 118-138, September.
    4. 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.
    5. Sohn, Kwon-Ik & Hwang, Hark, 1987. "A dynamic quantity discount lot size model with resales," European Journal of Operational Research, Elsevier, vol. 28(3), pages 293-297, March.
    6. Chung-Lun Li & Vernon Ning Hsu & Wen-Qiang Xiao, 2004. "Dynamic Lot Sizing with Batch Ordering and Truckload Discounts," Operations Research, INFORMS, vol. 52(4), pages 639-654, August.
    7. Hark‐Chin Hwang & Wilco van den Heuvel, 2012. "Improved algorithms for a lot‐sizing problem with inventory bounds and backlogging," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(3‐4), pages 244-253, April.
    8. Hellion, Bertrand & Mangione, Fabien & Penz, Bernard, 2012. "A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs," European Journal of Operational Research, Elsevier, vol. 222(1), pages 10-16.
    9. Okhrin, Irena & Richter, Knut, 2011. "An O(T3) algorithm for the capacitated lot sizing problem with minimum order quantities," European Journal of Operational Research, Elsevier, vol. 211(3), pages 507-514, June.
    10. Harvey M. Wagner & Thomson M. Whitin, 1958. "Dynamic Version of the Economic Lot Size Model," Management Science, INFORMS, vol. 5(1), pages 89-96, October.
    11. Gabriel R. Bitran & Horacio H. Yanasse, 1982. "Computational Complexity of the Capacitated Lot Size Problem," Management Science, INFORMS, vol. 28(10), pages 1174-1186, October.
    12. Ng, C.T. & Kovalyov, Mikhail Y. & Cheng, T.C.E., 2010. "A simple FPTAS for a single-item capacitated economic lot-sizing problem with a monotone cost structure," European Journal of Operational Research, Elsevier, vol. 200(2), pages 621-624, January.
    13. Willard I. Zangwill, 1969. "A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System--A Network Approach," Management Science, INFORMS, vol. 15(9), pages 506-527, May.
    14. Okhrin, Irena & Richter, Knut, 2011. "The linear dynamic lot size problem with minimum order quantity," International Journal of Production Economics, Elsevier, vol. 133(2), pages 688-693, October.
    15. Jans, Raf & Degraeve, Zeger, 2007. "Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1855-1875, March.
    16. van den Heuvel, W. & Wagelmans, A.P.M., 2007. "Four equivalent lot-sizing models," Econometric Institute Research Papers EI 2007-30, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    17. C. P. M. van Hoesel & A. P. M. Wagelmans, 2001. "Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 339-357, May.
    18. Yves Pochet & Laurence A. Wolsey, 1993. "Lot-Sizing with Constant Batches: Formulation and Valid Inequalities," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 767-785, November.
    19. Akbalik, A. & Pochet, Y., 2009. "Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs," European Journal of Operational Research, Elsevier, vol. 198(2), pages 412-434, October.
    20. Dong X. Shaw & Albert P. M. Wagelmans, 1998. "An Algorithm for Single-Item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs," Management Science, INFORMS, vol. 44(6), pages 831-838, June.
    21. Lap Mui Ann Chan & Ana Muriel & Zuo-Jun Shen & David Simchi-Levi, 2002. "On the Effectiveness of Zero-Inventory-Ordering Policies for the Economic Lot-Sizing Model with a Class of Piecewise Linear Cost Structures," Operations Research, INFORMS, vol. 50(6), pages 1058-1067, December.
    22. Michael Florian & Morton Klein, 1971. "Deterministic Production Planning with Concave Costs and Capacity Constraints," Management Science, INFORMS, vol. 18(1), pages 12-20, September.
    23. Archetti, Claudia & Bertazzi, Luca & Grazia Speranza, M., 2014. "Polynomial cases of the economic lot sizing problem with cost discounts," European Journal of Operational Research, Elsevier, vol. 237(2), pages 519-527.
    24. Willard I. Zangwill, 1966. "A Deterministic Multi-Period Production Scheduling Model with Backlogging," Management Science, INFORMS, vol. 13(1), pages 105-119, September.
    25. Stephen F. Love, 1973. "Bounded Production and Inventory Models with Piecewise Concave Costs," Management Science, INFORMS, vol. 20(3), pages 313-318, November.
    26. Akbalik, Ayse & Rapine, Christophe, 2013. "The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases," European Journal of Operational Research, Elsevier, vol. 229(2), pages 353-363.
    27. Esra Koca & Hande Yaman & M. Selim Aktürk, 2014. "Lot Sizing with Piecewise Concave Production Costs," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 767-779, November.
    28. POCHET, Yves & WOLSEY, Laurence A., 1993. "Lot-sizing with constant batches: formulation and valid inequalities," LIDAM Reprints CORE 1066, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    29. Mathieu Van Vyve, 2007. "Algorithms for Single-Item Lot-Sizing Problems with Constant Batch Size," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 594-613, August.
    30. Cary Swoveland, 1975. "A Deterministic Multi-Period Production Planning Model with Piecewise Concave Production and Holding-Backorder Costs," Management Science, INFORMS, vol. 21(9), pages 1007-1013, May.
    31. Brahimi, Nadjib & Dauzere-Peres, Stephane & Najid, Najib M. & Nordli, Atle, 2006. "Single item lot sizing problems," European Journal of Operational Research, Elsevier, vol. 168(1), pages 1-16, January.
    32. Awi Federgruen & Michal Tzur, 1991. "A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0(n log n) or 0(n) Time," Management Science, INFORMS, vol. 37(8), pages 909-925, August.
    33. Alok Aggarwal & James K. Park, 1993. "Improved Algorithms for Economic Lot Size Problems," Operations Research, INFORMS, vol. 41(3), pages 549-571, June.
    34. Liu, Tieming, 2008. "Economic lot sizing problem with inventory bounds," European Journal of Operational Research, Elsevier, vol. 185(1), pages 204-215, February.
    35. Sang-Bum Lee & Paul H. Zipkin, 1989. "A Dynamic Lot-Size Model with Make-or-Buy Decisions," Management Science, INFORMS, vol. 35(4), pages 447-458, April.
    36. M. Florian & J. K. Lenstra & A. H. G. Rinnooy Kan, 1980. "Deterministic Production Planning: Algorithms and Complexity," Management Science, INFORMS, vol. 26(7), pages 669-679, July.
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