# American Institute of Mathematical Sciences

January  2017, 22(1): 161-185. doi: 10.3934/dcdsb.2017008

## Optimality of (s, S) policies with nonlinear processes

 1 School of Insurance, Central University of Finance and Economics, Beijing 100081, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China 3 Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Hong Kong, China and Naveen Jindal School of Management, University of Texas at Dallas, USA

* Corresponding author

Received  August 2015 Revised  May 2016 Published  December 2016

Fund Project: J.L. Liu's research is supported by the support of Natural and the Science National Foundation of China (11301559), K. F. C. Yiu's research is supported by the research committee of Hong Kong Polytechnic University, and A. Bensoussan is supported by the National Science Foundation Under Grant DMS-1303775, the National Science Foundation Grant DMS-1612880, and the Research Grants Council of the Hong Hong Special Administrative Region (CityU 500113 and CityU 11303316).

It is observed empirically that mean-reverting processes are more realistic in modeling the inventory level of a company. In a typical mean-reverting process, the inventory level is assumed to be linearly dependent on the deviation of the inventory level from the long-term mean. However, when the deviation is large, it is reasonable to assume that the company might want to increase the intensity of interference to the inventory level significantly rather than in a linear manner. In this paper, we attempt to model inventory replenishment as a nonlinear continuous feedback process. We study both infinite horizon discounted cost and the long-run average cost, and derive the corresponding optimal (s, S) policy.

Citation: Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Optimality of (s, S) policies with nonlinear processes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 161-185. doi: 10.3934/dcdsb.2017008
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