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Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems


  • Awi Federgruen

    () (Graduate School of Business, Columbia University, 101 Uris Hall, New York, New York 10027)

  • Joern Meissner

    () (Department of Management Science, Lancaster University Management School, Room A48, Lancaster, LA1 4YX, United Kingdom)

  • Michal Tzur

    () (Department of Industrial Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel)


We consider a family of N items that are produced in, or obtained from, the same production facility. Demands are deterministic for each item and each period within a given horizon of T periods. If in a given period an order is placed, setup costs are incurred. The aggregate order size is constrained by a capacity limit. The objective is to find a lot-sizing strategy that satisfies the demands for all items over the entire horizon without backlogging, and that minimizes the sum of inventory-carrying costs, fixed-order costs, and variable-order costs.All demands, cost parameters, and capacity limits may be time dependent. In the basic joint setup cost (JS) model, the setup cost of an order does not depend on the composition of the order. The joint and item-dependent setup cost (JIS) model allows for item-dependent setup costs in addition to the joint setup costs. We develop and analyze a class of so-called progressive interval heuristics. A progessive interval heuristic solves a JS or JIS problem over a progressively larger time interval, always starting with period 1, but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations. Different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm.For the JS-model and the two basic implementations of the progressive interval heuristics, we show under some mild parameter conditions that the heuristics can be designed to be (epsilon)-optimal for any desired value of (epsilon) > 0 with a running time that is polynomially bounded in the size of the problem. They can also be designed to be simultaneously asymptotically optimal and polynomially bounded.A numerical study covering both the JS and JIS models shows that a progressive interval heuristic generates close-to-optimal solutions with modest computational effort and that it can be effectively used to solve large-scale problems.

Suggested Citation

  • 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.
  • Handle: RePEc:inm:oropre:v:55:y:2007:i:3:p:490-502

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    References listed on IDEAS

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    3. 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.
    4. 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.
    5. Kristin Uggen & Marte Fodstad & Vibeke Nørstebø, 2013. "Using and extending fix-and-relax to solve maritime inventory routing problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 355-377, July.
    6. Jenny Carolina Saldaña Cortés, 2011. "Programación semidefinida aplicada a problemas de cantidad económica de pedido," DOCUMENTOS CEDE 008735, UNIVERSIDAD DE LOS ANDES-CEDE.
    7. Marc Le Menestrel & Luk N. Wassenhove, 2016. "Subjectively biased objective functions," EURO Journal on Decision Processes, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 73-83, June.
    8. 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.
    9. Saravanan Venkatachalam & Arunachalam Narayanan, 2016. "Efficient formulation and heuristics for multi-item single source ordering problem with transportation cost," International Journal of Production Research, Taylor & Francis Journals, vol. 54(14), pages 4087-4103, July.
    10. Narayanan, Arunachalam & Robinson, Powell, 2010. "Evaluation of joint replenishment lot-sizing procedures in rolling horizon planning systems," International Journal of Production Economics, Elsevier, vol. 127(1), pages 85-94, September.

    More about this item


    inventory; production; multi-item; echelon; approximation; heuristic;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis


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