IDEAS home Printed from https://ideas.repec.org/p/zbw/euvwdp/284.html
   My bibliography  Save this paper

An O(Tˆ3) algorithm for the capacitated lot sizing problem with minimum order quantities

Author

Listed:
  • Okhrin, Irena
  • Richter, Knut

Abstract

This paper explores a single-item capacitated lot sizing problem with minimum order quantity, which plays the role of minor set-up cost. We work out the necessary and suffcient solvability conditions and apply the general dynamic programming technique to develop an O(T³) exact algorithm that is based on the concept of minimal sub-problems. An investigation of the properties of the optimal solution structure allows us to construct explicit solutions to the obtained sub-problems and prove their optimality. In this way, we reduce the complexity of the algorithm considerably and confirm its efficiency in an extensive computational study.

Suggested Citation

  • Okhrin, Irena & Richter, Knut, 2010. "An O(Tˆ3) algorithm for the capacitated lot sizing problem with minimum order quantities," Discussion Papers 284, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    • Handle: RePEc:zbw:euvwdp:284
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/41404/1/623985543.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Musalem, Eric Porras & Dekker, Rommert, 2005. "Controlling inventories in a supply chain: A case study," International Journal of Production Economics, Elsevier, vol. 93(1), pages 179-188, January.
    2. Pichtchoulov, Grigori & Richter, Knut, 2005. "Economic effects of mobile technologies on operations of sales agents," Discussion Papers 226, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    3. William W. Trigeiro & L. Joseph Thomas & John O. McClain, 1989. "Capacitated Lot Sizing with Setup Times," Management Science, INFORMS, vol. 35(3), pages 353-366, March.
    4. Robinson, Powell & Narayanan, Arunachalam & Sahin, Funda, 2009. "Coordinated deterministic dynamic demand lot-sizing problem: A review of models and algorithms," Omega, Elsevier, vol. 37(1), pages 3-15, February.
    5. Jonathan Tan & Friedel Bolle, 2006. "On the Relative Strengths of Altruism and Fairness," Theory and Decision, Springer, vol. 60(1), pages 35-67, February.
    6. Alexander Kritikos & Frank Wießner, 2000. "Ein zweiter Kreditmarkt für eine zweite Chance," Perspektiven der Wirtschaftspolitik, Verein für Socialpolitik, vol. 1(3), pages 357-378, August.
    7. Zhou, Bin & Zhao, Yao & Katehakis, Michael N., 2007. "Effective control policies for stochastic inventory systems with a minimum order quantity and linear costs," International Journal of Production Economics, Elsevier, vol. 106(2), pages 523-531, April.
    8. Anderson, Edward J. & Cheah, Boon Soon, 1993. "Capacitated lot-sizing with minimum batch sizes and setup times," International Journal of Production Economics, Elsevier, vol. 30(1), pages 137-152, July.
    9. Kamath B, Narasimha & Bhattacharya, Subir, 2007. "Lead time minimization of a multi-product, single-processor system: A comparison of cyclic policies," International Journal of Production Economics, Elsevier, vol. 106(1), pages 28-40, March.
    10. Jan Winiecki, 2000. "Successes of trade reorientation and expansion in post-communist transition: an enterprise-level approach," Banca Nazionale del Lavoro Quarterly Review, Banca Nazionale del Lavoro, vol. 53(213), pages 187-223.
    11. Tan, Jonathan H.W., 2006. "Religion and social preferences: An experimental study," Economics Letters, Elsevier, vol. 90(1), pages 60-67, January.
    12. Miguel Constantino, 1998. "Lower Bounds in Lot-Sizing Models: A Polyhedral Study," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 101-118, February.
    13. 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.
    14. Alan S. Manne, 1958. "Programming of Economic Lot Sizes," Management Science, INFORMS, vol. 4(2), pages 115-135, January.
    15. Bolle, Friedel & Kritikos, Alexander S., 2004. "Altruistic Behavior Under Incomplete Information," Discussion Papers 229, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    16. 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.
    17. Kritikos, Alexander & Bolle, Friedel, 2001. "Distributional concerns: equity- or efficiency-oriented?," Economics Letters, Elsevier, vol. 73(3), pages 333-338, December.
    18. Porras, Eric & Dekker, Rommert, 2006. "An efficient optimal solution method for the joint replenishment problem with minimum order quantities," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1595-1615, November.
    19. Chia-Shin Chung & Chien-Hua Mike Lin, 1988. "An O(T 2 ) Algorithm for the NI/G/NI/ND Capacitated Lot Size Problem," Management Science, INFORMS, vol. 34(3), pages 420-426, March.
    20. Michael Florian & Morton Klein, 1971. "Deterministic Production Planning with Concave Costs and Capacity Constraints," Management Science, INFORMS, vol. 18(1), pages 12-20, September.
    21. Friedel Bolle & Alexander Kritikos, 2006. "Reciprocity, Altruism, Solidarity: A Dynamic Model," Theory and Decision, Springer, vol. 60(4), pages 371-394, June.
    22. repec:bpj:pewipo:v:1:y:2000:i:3:p:357-378:n:7 is not listed on IDEAS
    23. Stephen F. Love, 1973. "Bounded Production and Inventory Models with Piecewise Concave Costs," Management Science, INFORMS, vol. 20(3), pages 313-318, November.
    24. Merce, C. & Fontan, G., 2003. "MIP-based heuristics for capacitated lotsizing problems," International Journal of Production Economics, Elsevier, vol. 85(1), pages 97-111, July.
    25. Jan Winiecki, 2000. "Successes of trade reorientation and expansion in post-communist transition: an enterprise-level approach," BNL Quarterly Review, Banca Nazionale del Lavoro, vol. 53(213), pages 187-223.
    26. Kritikos, Alexander S. & Bolle, Friedel, 2006. "Utility versus Income-Based Altruism," Discussion Papers 249, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    27. Narayanan, Arunachalam & Robinson, Powell, 2010. "Efficient and effective heuristics for the coordinated capacitated lot-size problem," European Journal of Operational Research, Elsevier, vol. 203(3), pages 583-592, June.
    28. M Perugini & J H W Tan & D J Zizzo, 2010. "Which is the More Predictable Gender? Public Good Contribution and Personality," Economic Issues Journal Articles, Economic Issues, vol. 15(1), pages 83-110, March.
    29. Vogel, Claudia & Heimel, Jana & Bolle, Friedel, 2004. "Crowding out and imitation behavior in the solidarity game," Discussion Papers 223, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    30. Krohn, Michael, 2004. "Die virtuelle Falle - Konfliktpotentiale der Informationsgesellschaft und ihre Überwindung durch Investitionen in Sozialkapital," Discussion Papers 222, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    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. Baier, Antje & Bolle, Friedel, 2005. "Cyclical price fluctuations caused by information inertia : evidence from the German call-by-call telephone market," Discussion Papers 225, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    33. Tan, Jonathan H.W. & Zizzo, Daniel John, 2008. "Groups, cooperation and conflict in games," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 37(1), pages 1-17, February.
    34. Gröppel-Klein, Andrea & Germelmann, Claas Christian, 2005. "The Impact of Terminal Values and Yellow Press on Consumer Behavior," Discussion Papers 224, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bunn, Kevin A. & Ventura, José A., 2023. "A dynamic programming approach for the two-product capacitated lot-sizing problem with concave costs," European Journal of Operational Research, Elsevier, vol. 307(1), pages 116-129.
    2. Ou, Jinwen, 2017. "Improved exact algorithms to economic lot-sizing with piecewise linear production costs," European Journal of Operational Research, Elsevier, vol. 256(3), pages 777-784.
    3. 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.
    4. Brahimi, Nadjib & Absi, Nabil & Dauzère-Pérès, Stéphane & Nordli, Atle, 2017. "Single-item dynamic lot-sizing problems: An updated survey," European Journal of Operational Research, Elsevier, vol. 263(3), pages 838-863.
    5. Muriel, Ana & Chugh, Tammana & Prokle, Michael, 2022. "Efficient algorithms for the joint replenishment problem with minimum order quantities," European Journal of Operational Research, Elsevier, vol. 300(1), pages 137-150.
    6. Sazvar, Z. & Mirzapour Al-e-hashem, S.M.J. & Govindan, K. & Bahli, B., 2016. "A novel mathematical model for a multi-period, multi-product optimal ordering problem considering expiry dates in a FEFO system," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 93(C), pages 232-261.
    7. 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.
    8. Chen, Yujing & Zhong, Yuanguang & Cheng, T.C.E., 2023. "Impacts of the minimum quantity contract on an online retail platform," European Journal of Operational Research, Elsevier, vol. 306(3), pages 1236-1247.
    9. Hong, Zhaofu & Chu, Chengbin & Yu, Yugang, 2016. "Dual-mode production planning for manufacturing with emission constraints," European Journal of Operational Research, Elsevier, vol. 251(1), pages 96-106.
    10. Ou, Jinwen & Feng, Jiejian, 2019. "Production lot-sizing with dynamic capacity adjustment," European Journal of Operational Research, Elsevier, vol. 272(1), pages 261-269.
    11. Zhu, Han & Liu, Xing & Chen, Youhua (Frank), 2015. "Effective inventory control policies with a minimum order quantity and batch ordering," International Journal of Production Economics, Elsevier, vol. 168(C), pages 21-30.
    12. Absi, Nabil & Dauzère-Pérès, Stéphane & Kedad-Sidhoum, Safia & Penz, Bernard & Rapine, Christophe, 2016. "The single-item green lot-sizing problem with fixed carbon emissions," European Journal of Operational Research, Elsevier, vol. 248(3), pages 849-855.
    13. Froese, Sarah & Kunz, Nadja C. & Ramana, M.V., 2020. "Too small to be viable? The potential market for small modular reactors in mining and remote communities in Canada," Energy Policy, Elsevier, vol. 144(C).
    14. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Okhrin, Irena & Richter, Knut, 2010. "The linear dynamic lot size problem with minimum order quantities," Discussion Papers 283, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
    2. 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.
    3. Brahimi, Nadjib & Absi, Nabil & Dauzère-Pérès, Stéphane & Nordli, Atle, 2017. "Single-item dynamic lot-sizing problems: An updated survey," European Journal of Operational Research, Elsevier, vol. 263(3), pages 838-863.
    4. 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.
    5. Melega, Gislaine Mara & de Araujo, Silvio Alexandre & Jans, Raf, 2018. "Classification and literature review of integrated lot-sizing and cutting stock problems," European Journal of Operational Research, Elsevier, vol. 271(1), pages 1-19.
    6. 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.
    7. Jans, R.F. & Degraeve, Z., 2005. "Modeling Industrial Lot Sizing Problems: A Review," ERIM Report Series Research in Management ERS-2005-049-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    8. Muriel, Ana & Chugh, Tammana & Prokle, Michael, 2022. "Efficient algorithms for the joint replenishment problem with minimum order quantities," European Journal of Operational Research, Elsevier, vol. 300(1), pages 137-150.
    9. Berk, Emre & Toy, Ayhan Ozgur & Hazir, Oncu, 2008. "Single item lot-sizing problem for a warm/cold process with immediate lost sales," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1251-1267, June.
    10. 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.
    11. 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.
    12. Absi, Nabil & Kedad-Sidhoum, Safia, 2008. "The multi-item capacitated lot-sizing problem with setup times and shortage costs," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1351-1374, March.
    13. Laurence A. Wolsey, 2002. "Solving Multi-Item Lot-Sizing Problems with an MIP Solver Using Classification and Reformulation," Management Science, INFORMS, vol. 48(12), pages 1587-1602, December.
    14. 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.
    15. Önal, Mehmet & van den Heuvel, Wilco & Dereli, Meryem Merve & Albey, Erinç, 2023. "Economic lot sizing problem with tank scheduling," European Journal of Operational Research, Elsevier, vol. 308(1), pages 166-182.
    16. Kerem Akartunalı & Andrew Miller, 2012. "A computational analysis of lower bounds for big bucket production planning problems," Computational Optimization and Applications, Springer, vol. 53(3), pages 729-753, December.
    17. Stan van Hoesel & H. Edwin Romeijn & Dolores Romero Morales & Albert P. M. Wagelmans, 2005. "Integrated Lot Sizing in Serial Supply Chains with Production Capacities," Management Science, INFORMS, vol. 51(11), pages 1706-1719, November.
    18. Toy, Ayhan Özgür & Berk, Emre, 2013. "Dynamic lot sizing for a warm/cold process: Heuristics and insights," International Journal of Production Economics, Elsevier, vol. 145(1), pages 53-66.
    19. Awi Federgruen & Joern Meissner & Michal Tzur, 2007. "Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems," Operations Research, INFORMS, vol. 55(3), pages 490-502, June.
    20. Wang, Lin & He, Jing & Wu, Desheng & Zeng, Yu-Rong, 2012. "A novel differential evolution algorithm for joint replenishment problem under interdependence and its application," International Journal of Production Economics, Elsevier, vol. 135(1), pages 190-198.

    More about this item

    Keywords

    production planning; capacitated lot sizing problem; single item; minimum order quantities; capacity constraints; dynamic programming;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:euvwdp:284. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/fwffode.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.