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On a General Solution of the Deterministic Lot Size Problem with Time-Proportional Demand

Author

Listed:
  • Michael Resh

    (Israel Ministry of Communications, Tel-Aviv, Israel)

  • Moshe Friedman

    (Arizona State University, Tempe, Arizona)

  • Lineu C. Barbosa

    (IBM Research Laboratory, San Jose, California)

Abstract

We reconsider the classical lot-size model with the assumption that demand rate is deterministic, starts at the origin, and linearly increases with time. Two relevant costs are involved: carrying cost and replenishment cost. The planning horizon is finite and known. The problem is to find the optimal schedule of replenishments, i.e., their number and the schedule of time intervals between consecutive orders. Mathematically, we have to find an integer m and the values of m continuous variables so as to minimize the total carrying and replenishments costs. We prove that for a givers number of replenishments m , there exists a unique vector of m time intervals that minimizes the total cost function. It is further shown that the total carrying cost obtained after substituting the optimal value of that vector is a convex function of m . The algorithm that determines the unique optimal value of m and the unique optimal scheduling of replenishments for m employs these mathematical results. Finally, we investigate the asymptotic properties of the model when the planning horizon is infinite.

Suggested Citation

  • Michael Resh & Moshe Friedman & Lineu C. Barbosa, 1976. "On a General Solution of the Deterministic Lot Size Problem with Time-Proportional Demand," Operations Research, INFORMS, vol. 24(4), pages 718-725, August.
    • Handle: RePEc:inm:oropre:v:24:y:1976:i:4:p:718-725
    DOI: 10.1287/opre.24.4.718
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    Cited by:

    1. Sicilia, Joaquín & González-De-la-Rosa, Manuel & Febles-Acosta, Jaime & Alcaide-López-de-Pablo, David, 2014. "Optimal policy for an inventory system with power demand, backlogged shortages and production rate proportional to demand rate," International Journal of Production Economics, Elsevier, vol. 155(C), pages 163-171.
    2. Dye, Chung-Yuan & Chang, Horng-Jinh & Teng, Jinn-Tsair, 2006. "A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging," European Journal of Operational Research, Elsevier, vol. 172(2), pages 417-429, July.
    3. Hill, Roger M., 1996. "Batching policies for a product life cycle," International Journal of Production Economics, Elsevier, vol. 45(1-3), pages 421-427, August.
    4. Hung, Kuo-Chen, 2011. "An inventory model with generalized type demand, deterioration and backorder rates," European Journal of Operational Research, Elsevier, vol. 208(3), pages 239-242, February.
    5. Abbott, Harish & Palekar, Udatta S., 2008. "Retail replenishment models with display-space elastic demand," European Journal of Operational Research, Elsevier, vol. 186(2), pages 586-607, April.
    6. Hill, Roger M. & Omar, Mohd & Smith, David K., 1999. "Stock replenishment policies for a stochastic exponentially-declining demand process," European Journal of Operational Research, Elsevier, vol. 116(2), pages 374-388, July.
    7. Chang, Horng-Jinh & Teng, Jinn-Tsair & Ouyang, Liang-Yuh & Dye, Chung-Yuan, 2006. "Retailer's optimal pricing and lot-sizing policies for deteriorating items with partial backlogging," European Journal of Operational Research, Elsevier, vol. 168(1), pages 51-64, January.
    8. Massonnet, G. & Gayon, J.-P. & Rapine, C., 2014. "Approximation algorithms for deterministic continuous-review inventory lot-sizing problems with time-varying demand," European Journal of Operational Research, Elsevier, vol. 234(3), pages 641-649.
    9. Skouri, K. & Konstantaras, I. & Papachristos, S. & Ganas, I., 2009. "Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate," European Journal of Operational Research, Elsevier, vol. 192(1), pages 79-92, January.
    10. Urban, Timothy L. & Baker, R. C., 1997. "Optimal ordering and pricing policies in a single-period environment with multivariate demand and markdowns," European Journal of Operational Research, Elsevier, vol. 103(3), pages 573-583, December.
    11. Balkhi, Zaid T., 2001. "On a finite horizon production lot size inventory model for deteriorating items: An optimal solution," European Journal of Operational Research, Elsevier, vol. 132(1), pages 210-223, July.
    12. Narayan Singh & Bindu Vaish & Shiv Ray Singh, 2014. "A collaborative strategy for a three echelon supply chain with ramp type demand, deterioration and inflation," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 24(3), pages 77-100.
    13. Hong, Jae-Dong & Kim, Seung-Lae & Hayya, Jack C., 1996. "Dynamic setup reduction in production lot sizing with nonconstant deterministic demand," European Journal of Operational Research, Elsevier, vol. 90(1), pages 182-196, April.
    14. Konstantaras, Ioannis & Skouri, Konstantina & Benkherouf, Lakdere, 2021. "Optimizing inventory decisions for a closed–loop supply chain model under a carbon tax regulatory mechanism," International Journal of Production Economics, Elsevier, vol. 239(C).
    15. Yang, Hui-Ling & Teng, Jinn-Tsair & Chern, Maw-Sheng, 2002. "A forward recursive algorithm for inventory lot-size models with power-form demand and shortages," European Journal of Operational Research, Elsevier, vol. 137(2), pages 394-400, March.
    16. Hill, Roger M., 1996. "Batching policies for linearly increasing demand with a finite input rate," International Journal of Production Economics, Elsevier, vol. 43(2-3), pages 149-154, June.

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