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Optimal replenishment rate for inventory systems with compound Poisson demands and lost sales: a direct treatment of time-average cost

Author

Listed:
  • Michael N. Katehakis

    (Rutgers Business School – Newark and New Brunswick)

  • Benjamin Melamed

    (Rutgers Business School – Newark and New Brunswick)

  • Jim Junmin Shi

    (New Jersey Institute of Technology)

Abstract

Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes, we exhibit a closed form expression for the optimal replenishment rate and its corresponding cost. Finally, a numerical study is carried out to illustrate the results.

Suggested Citation

  • Michael N. Katehakis & Benjamin Melamed & Jim Junmin Shi, 2022. "Optimal replenishment rate for inventory systems with compound Poisson demands and lost sales: a direct treatment of time-average cost," Annals of Operations Research, Springer, vol. 317(2), pages 665-691, October.
    • Handle: RePEc:spr:annopr:v:317:y:2022:i:2:d:10.1007_s10479-015-1998-y
    DOI: 10.1007/s10479-015-1998-y
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    References listed on IDEAS

    as
    1. SPRINGAEL, Johan & VAN NIEUWENHUYSE, Inneke, 2005. "A lost sales inventory model with a compound poisson demand pattern," Working Papers 2005017, University of Antwerp, Faculty of Business and Economics.
    2. Izzet Sahin, 1979. "On the Stationary Analysis of Continuous Review ( s , S ) Inventory Systems with Constant Lead Times," Operations Research, INFORMS, vol. 27(4), pages 717-729, August.
    3. Awi Federgruen & Zvi Schechner, 1983. "Technical Note—Cost Formulas for Continuous Review Inventory Models with Fixed Delivery Lags," Operations Research, INFORMS, vol. 31(5), pages 957-965, October.
    4. Minner, Stefan & Silver, Edward A., 2007. "Replenishment policies for multiple products with compound-Poisson demand that share a common warehouse," International Journal of Production Economics, Elsevier, vol. 108(1-2), pages 388-398, July.
    5. Stephen C. Graves & Julian Keilson, 1981. "The Compensation Method Applied to a One-Product Production/Inventory Problem," Mathematics of Operations Research, INFORMS, vol. 6(2), pages 246-262, May.
    6. Stephen C. Graves, 1982. "The Application of Queueing Theory to Continuous Perishable Inventory Systems," Management Science, INFORMS, vol. 28(4), pages 400-406, April.
    7. Blyth C. Archibald & Edward A. Silver, 1978. "(s, S) Policies Under Continuous Review and Discrete Compound Poisson Demand," Management Science, INFORMS, vol. 24(9), pages 899-909, May.
    8. Ivo Adan & Onno Boxma & David Perry, 2005. "The G/M/1 queue revisited," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(3), pages 437-452, December.
    9. Junmin Shi & Michael Katehakis & Benjamin Melamed, 2013. "Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands," Annals of Operations Research, Springer, vol. 208(1), pages 593-612, September.
    10. Dong Li & Kevin D. Glazebrook, 2010. "An approximate dynamic programing approach to the development of heuristics for the scheduling of impatient jobs in a clearing system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(3), pages 225-236, April.
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