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
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. Google Scholar

[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. Google Scholar

[3]

A. Bensoussan, Dynamic Programming and Inventory Control IOS Press, 2011. Google Scholar

[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. Google Scholar

[5]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, France, 1984. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

C. Dellacherie and P. -A. Meyer, Probabilites et Potentiel Theorie des Martingales. Paris: Hermann, 1987. Google Scholar

[11]

P. Finch, Some probability theorems in inventory control, Publ. Math. Debrecen, 8 (1961), 241-261. Google Scholar

[12]

W. H. Flemming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer, New York, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

P. H. Zipkin, Foundations of Inventory Management McGraw-Hill/Irwin, 2000.Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

A. Bensoussan, Dynamic Programming and Inventory Control IOS Press, 2011. Google Scholar

[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. Google Scholar

[5]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, France, 1984. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

C. Dellacherie and P. -A. Meyer, Probabilites et Potentiel Theorie des Martingales. Paris: Hermann, 1987. Google Scholar

[11]

P. Finch, Some probability theorems in inventory control, Publ. Math. Debrecen, 8 (1961), 241-261. Google Scholar

[12]

W. H. Flemming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer, New York, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

P. H. Zipkin, Foundations of Inventory Management McGraw-Hill/Irwin, 2000.Google Scholar

[1]

Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & 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 & Management Optimization, 2018, 14 (3) : 857-876. doi: 10.3934/jimo.2017079

[3]

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 & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019009

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial & Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105

[14]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067

[15]

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

[16]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401

[17]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

[18]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[19]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607

[20]

Philippe Ciarlet. Korn's inequalities: The linear vs. the nonlinear case. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 473-483. doi: 10.3934/dcdss.2012.5.473

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (16)
  • HTML views (61)
  • Cited by (1)

[Back to Top]