Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity - the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.

[1]  David F. Pyke,et al.  Inventory management and production planning and scheduling , 1998 .

[2]  W. D. Ray,et al.  Stochastic Models: An Algorithmic Approach , 1995 .

[3]  Awi Federgruen,et al.  Computational Issues in an Infinite-Horizon, Multiechelon Inventory Model , 1984, Oper. Res..

[4]  Edward A. Silver,et al.  Inventory management with periodic ordering and minimum order quantities , 1998, J. Oper. Res. Soc..

[5]  Jr. Arthur F. Veinott.  The Status of Mathematical Inventory Theory , 1966 .

[6]  Herbert E. Scarf,et al.  Optimal Policies for a Multi-Echelon Inventory Problem , 1960, Manag. Sci..

[7]  M. Katehakis,et al.  Effective control policies for stochastic inventory systems with a minimum order quantity and linear costs , 2007 .

[8]  D. Zalkind.  Order-Level Inventory Systems with Independent Stochastic Leadtimes , 1978 .

[9]  Marshall L. Fisher,et al.  Reducing the Cost of Demand Uncertainty Through Accurate Response to Early Sales , 1996, Oper. Res..

[10]  Awi Federgruen,et al.  Capacitated Multi-Item Inventory Systems with Random and Seasonally Fluctuating Demands: Implications for Postponement Strategies , 2001, Manag. Sci..

[11]  H. Scarf.  THE OPTIMALITY OF (S,S) POLICIES IN THE DYNAMIC INVENTORY PROBLEM , 1959 .

[12]  Daniel P. Heyman,et al.  Stochastic models in operations research , 1982 .

[13]  M. Katehakis,et al.  ON THE STRUCTURE OF OPTIMAL ORDERING POLICIES FOR STOCHASTIC INVENTORY SYSTEMS WITH MINIMUM ORDER QUANTITY , 2006, Probability in the Engineering and Informational Sciences.

[14]  Cyrus Derman,et al.  Finite State Markovian Decision Processes , 1970 .

[15]  S. Karlin,et al.  Mathematical Methods in the Social Sciences , 1962 .

[16]  Awi Federgruen,et al.  Finding Optimal (s, S) Policies Is About As Simple As Evaluating a Single Policy , 1991, Oper. Res..

[17]  Fangruo Chen,et al.  Lower Bounds for Multi-Echelon Stochastic Inventory Systems , 1994 .

[18]  Lawrence W. Robinson,et al.  Improved Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover , 2005, Manuf. Serv. Oper. Manag..

[19]  R. Kaplan.  A Dynamic Inventory Model with Stochastic Lead Times , 1970 .