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. BensoussanR. 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. BeyerS. 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. CadenillasP. 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. HollierK. 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. LiuK. 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. MakK. K. LaiW. 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. OrmeciJ. 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. WangK. 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. YiuS. 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. YiuL. 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. YiuJ. Z. LiuT. 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.

show all references

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. BensoussanR. 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. BeyerS. 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. CadenillasP. 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. HollierK. 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. LiuK. 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. MakK. K. LaiW. 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. OrmeciJ. 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. WangK. 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. YiuS. 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. YiuL. 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. YiuJ. Z. LiuT. 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.

[1]

Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027

[2]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial and Management Optimization, 2018, 14 (3) : 857-876. doi: 10.3934/jimo.2017079

[3]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143

[4]

Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1415-1433. doi: 10.3934/jimo.2019009

[5]

Edward Allen. Environmental variability and mean-reverting processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2073-2089. doi: 10.3934/dcdsb.2016037

[6]

Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403

[7]

Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control and Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026

[8]

Weiwei Wang, Ping Chen. A mean-reverting currency model with floating interest rates in uncertain environment. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1921-1936. doi: 10.3934/jimo.2018129

[9]

Kun-Jen Chung, Pin-Shou Ting. The inventory model under supplier's partial trade credit policy in a supply chain system. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1175-1183. doi: 10.3934/jimo.2015.11.1175

[10]

Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2703-2714. doi: 10.3934/jimo.2020090

[11]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

[12]

Shoude Li. A dynamic analysis of a monopolist's product and process innovation with nonlinear demand and expected quality effects. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022056

[13]

Mondal Hasan Zahid, Christopher M. Kribs. Ebola: Impact of hospital's admission policy in an overwhelmed scenario. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1387-1399. doi: 10.3934/mbe.2018063

[14]

Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial and Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609

[15]

Raina Raj, Vidyottama Jain. Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022044

[16]

Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations and Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015

[17]

Genlong Guo, Shoude Li. A dynamic analysis of a monopolist's quality improvement, process innovation and goodwill. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022014

[18]

Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial and Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022

[19]

Christine Burggraf, Wilfried Grecksch, Thomas Glauben. Stochastic control of individual's health investments. Conference Publications, 2015, 2015 (special) : 159-168. doi: 10.3934/proc.2015.0159

[20]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (182)
  • HTML views (128)
  • Cited by (1)

[Back to Top]