Optimal Replenishment Rate for Inventory Systems with Compound Poisson Demands and Lost-Sales: A Direct Treatment of Time-Average Cost

39 Pages Posted: 8 Oct 2015

See all articles by Michael N. Katehakis

Michael N. Katehakis

Rutgers, The State University of New Jersey - Management Science & Information Systems

Benjamin Melamed

Rutgers, The State University of New Jersey, Rutgers Business School - Newark & New Brunswick, Supply Chain Management Department

Jim (Junmin) Shi

New Jersey Institute of Technology

Date Written: October 6, 2014

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.

Keywords: Compound Poisson, continuous replenishment, integro-differential equation, production-inventory systems, lost-sales, replenishment rate, time-average cost

Suggested Citation

Katehakis, Michael N. and Melamed, Benjamin and Shi, Jim (Junmin), Optimal Replenishment Rate for Inventory Systems with Compound Poisson Demands and Lost-Sales: A Direct Treatment of Time-Average Cost (October 6, 2014). Annals of Operation Research, 2015, Available at SSRN: https://ssrn.com/abstract=2670109

Michael N. Katehakis

Rutgers, The State University of New Jersey - Management Science & Information Systems ( email )

180 University Ave.
Newark, NJ 07102-1895
United States

Benjamin Melamed

Rutgers, The State University of New Jersey, Rutgers Business School - Newark & New Brunswick, Supply Chain Management Department ( email )

South Tower, Room 3149
100 Rockafeller Rd.
Piscataway, NJ 08854
United States

Jim (Junmin) Shi (Contact Author)

New Jersey Institute of Technology ( email )

University Heights
NJIT
Newark, NJ US 07102
United States
9736427027 (Phone)

HOME PAGE: http://https://web.njit.edu/~jshi/

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