# 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 and Continuous Dynamical Systems - B, 2017, 22 (1) : 161-185. doi: 10.3934/dcdsb.2017008
##### References:
 [1] D. Bartmann and M. F. Bach, Inventory Control: Models and Methods Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-87146-7. [2] L. Benkherouf and M. Johnson, Optimality of (s, S) policies for jump inventory models, Mathematical Methods of Operations Research, 76 (2012), 377-393.  doi: 10.1007/s00186-012-0411-8. [3] A. Bensoussan, Dynamic Programming and Inventory Control IOS Press, 2011. [4] A. Bensoussan, R. H. Liu and S. P. Sethi, Optimality of an (s, S) policy with compound Poisson and diffusion demands: A quasi-variational inequalities approach, SIAM J. Control Optim., 44 (2005), 1650-1676.  doi: 10.1137/S0363012904443737. [5] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, France, 1984. [6] D. Beyer and S. P. Sethi, Average cost optimality in inventory models with markovian demands, Journal of Optimization Theory and Applications, 92 (1997), 497-526.  doi: 10.1023/A:1022651322174. [7] D. Beyer, S. P. Sethi and M. Taksar, Inventory models with Markovian demands and cost functions of polynomial growth, Journal of Optimization Theory and Applications, 98 (1998), 281-323.  doi: 10.1023/A:1022633400174. [8] A. Cadenillas, P. Lakner and M. Pinedo, Optimal control of a mean-reverting inventory, Operations research, 58 (2010), 1697-1710.  doi: 10.1287/opre.1100.0835. [9] G. Constantinides and S. Richard, Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time, Operations research, 26 (1978), 620-636.  doi: 10.1287/opre.26.4.620. [10] C. Dellacherie and P. -A. Meyer, Probabilites et Potentiel Theorie des Martingales. Paris: Hermann, 1987. [11] P. Finch, Some probability theorems in inventory control, Publ. Math. Debrecen, 8 (1961), 241-261. [12] W. H. Flemming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer, New York, 2006. [13] R. H. Hollier, K. L. Mak and K. F. C. Yiu, Optimal inventory control of lumpy demand items using (s, S) policies with a maximum issue quantity restriction and opportunistic replenishments, International Journal of Production Research, 43 (2007), 4929-4944.  doi: 10.1080/00207540500218967. [14] J. Z. Liu, K. F. C. Yiu and L. H. Bai, Minimizing the ruin probability with a risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547.  doi: 10.3934/jimo.2012.8.531. [15] K. L. Mak, K. K. Lai, W. C. Ng and K. F. C. Yiu, Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s, S) model with a maximum issue quantity restriction, European Journal of Operational Research, 166 (2005), 385-405.  doi: 10.1016/j.ejor.2002.05.001. [16] M. Ormeci, J. G. Dai and J. Vande Vate, Impulse control of Brownian motion: The constrained average cost case, Operations Research, 56 (2008), 618-629.  doi: 10.1287/opre.1060.0380. [17] E. Presman and S. P. Sethi, Stochastic inventory models with continuous and Poisson demands and discounted and average costs, Production and Operations Management, 15 (2004), 279-293. [18] H. Scarf, Stationary operating characteristics of an inventory model with time lag, Studies in the Mathematical Theory of Inventory and Production (eds. K. J. Arrow, S. Karlin, H. Scarf), Spring verlag. Stanford University Press, Stanford, CA. , 1958. [19] H. Scarf, A survey of analytic techniques in inventory theory, Multistage Inventory Models and Techniques(eds. H. Scarf, D.M. Gilford, M. Shelly), New York. Spring verlag.Stanford University Press, Stanford, CA., (1963), 185-225. [20] B. Sivazlian, A continuous review (s; S) inventory system with arbitrary interarrival distribution between unit demand, Operations Research, 22 (1974), 65-71.  doi: 10.1287/opre.22.1.65. [21] B. Sivazlian, A solvable one-dimensional model of a diffusion inventory system, Mathematics of Operations Research, 11 (1986), 125-133.  doi: 10.1287/moor.11.1.125. [22] S. Y. Wang, K. F. C. Yiu and K. L. Mak, Optimal inventory policy with fixed and proportional transaction costs under a risk constraint, Mathematical and Computer Modelling, 58 (2013), 1595-1614.  doi: 10.1016/j.mcm.2012.03.009. [23] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94.  doi: 10.3934/jimo.2008.4.81. [24] K. F. C. Yiu, L. L. Xie and K. L. Mak, Analysis of bullwhip effect in supply chains with heterogeneous decision models, Journal of Industrial and Management Optimization, 5 (2009), 81-94.  doi: 10.3934/jimo.2009.5.81. [25] K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [26] Y. S. Zheng, A Simple Proof for Optimality of (s; S) Policies in Infinite-Horizon Inventory Systems, Journal of Applied Probability, 28 (1991), 802-810.  doi: 10.1017/S0021900200042716. [27] P. H. Zipkin, Foundations of Inventory Management McGraw-Hill/Irwin, 2000.

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##### References:
 [1] D. Bartmann and M. F. Bach, Inventory Control: Models and Methods Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-87146-7. [2] L. Benkherouf and M. Johnson, Optimality of (s, S) policies for jump inventory models, Mathematical Methods of Operations Research, 76 (2012), 377-393.  doi: 10.1007/s00186-012-0411-8. [3] A. Bensoussan, Dynamic Programming and Inventory Control IOS Press, 2011. [4] A. Bensoussan, R. H. Liu and S. P. Sethi, Optimality of an (s, S) policy with compound Poisson and diffusion demands: A quasi-variational inequalities approach, SIAM J. Control Optim., 44 (2005), 1650-1676.  doi: 10.1137/S0363012904443737. [5] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, France, 1984. [6] D. Beyer and S. P. Sethi, Average cost optimality in inventory models with markovian demands, Journal of Optimization Theory and Applications, 92 (1997), 497-526.  doi: 10.1023/A:1022651322174. [7] D. Beyer, S. P. Sethi and M. Taksar, Inventory models with Markovian demands and cost functions of polynomial growth, Journal of Optimization Theory and Applications, 98 (1998), 281-323.  doi: 10.1023/A:1022633400174. [8] A. Cadenillas, P. Lakner and M. Pinedo, Optimal control of a mean-reverting inventory, Operations research, 58 (2010), 1697-1710.  doi: 10.1287/opre.1100.0835. [9] G. Constantinides and S. Richard, Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time, Operations research, 26 (1978), 620-636.  doi: 10.1287/opre.26.4.620. [10] C. Dellacherie and P. -A. Meyer, Probabilites et Potentiel Theorie des Martingales. Paris: Hermann, 1987. [11] P. Finch, Some probability theorems in inventory control, Publ. Math. Debrecen, 8 (1961), 241-261. [12] W. H. Flemming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer, New York, 2006. [13] R. H. Hollier, K. L. Mak and K. F. C. Yiu, Optimal inventory control of lumpy demand items using (s, S) policies with a maximum issue quantity restriction and opportunistic replenishments, International Journal of Production Research, 43 (2007), 4929-4944.  doi: 10.1080/00207540500218967. [14] J. Z. Liu, K. F. C. Yiu and L. H. Bai, Minimizing the ruin probability with a risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547.  doi: 10.3934/jimo.2012.8.531. [15] K. L. Mak, K. K. Lai, W. C. Ng and K. F. C. Yiu, Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s, S) model with a maximum issue quantity restriction, European Journal of Operational Research, 166 (2005), 385-405.  doi: 10.1016/j.ejor.2002.05.001. [16] M. Ormeci, J. G. Dai and J. Vande Vate, Impulse control of Brownian motion: The constrained average cost case, Operations Research, 56 (2008), 618-629.  doi: 10.1287/opre.1060.0380. [17] E. Presman and S. P. Sethi, Stochastic inventory models with continuous and Poisson demands and discounted and average costs, Production and Operations Management, 15 (2004), 279-293. [18] H. Scarf, Stationary operating characteristics of an inventory model with time lag, Studies in the Mathematical Theory of Inventory and Production (eds. K. J. Arrow, S. Karlin, H. Scarf), Spring verlag. Stanford University Press, Stanford, CA. , 1958. [19] H. Scarf, A survey of analytic techniques in inventory theory, Multistage Inventory Models and Techniques(eds. H. Scarf, D.M. Gilford, M. Shelly), New York. Spring verlag.Stanford University Press, Stanford, CA., (1963), 185-225. [20] B. Sivazlian, A continuous review (s; S) inventory system with arbitrary interarrival distribution between unit demand, Operations Research, 22 (1974), 65-71.  doi: 10.1287/opre.22.1.65. [21] B. Sivazlian, A solvable one-dimensional model of a diffusion inventory system, Mathematics of Operations Research, 11 (1986), 125-133.  doi: 10.1287/moor.11.1.125. [22] S. Y. Wang, K. F. C. Yiu and K. L. Mak, Optimal inventory policy with fixed and proportional transaction costs under a risk constraint, Mathematical and Computer Modelling, 58 (2013), 1595-1614.  doi: 10.1016/j.mcm.2012.03.009. [23] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94.  doi: 10.3934/jimo.2008.4.81. [24] K. F. C. Yiu, L. L. Xie and K. L. Mak, Analysis of bullwhip effect in supply chains with heterogeneous decision models, Journal of Industrial and Management Optimization, 5 (2009), 81-94.  doi: 10.3934/jimo.2009.5.81. [25] K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [26] Y. S. Zheng, A Simple Proof for Optimality of (s; S) Policies in Infinite-Horizon Inventory Systems, Journal of Applied Probability, 28 (1991), 802-810.  doi: 10.1017/S0021900200042716. [27] P. H. Zipkin, Foundations of Inventory Management McGraw-Hill/Irwin, 2000.
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