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July 2018, 14(3): 857-876. doi: 10.3934/jimo.2017079

## Ergodic control for a mean reverting inventory model

 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 4 Naveen Jindal School of Management, University of Texas at Dallas, USA

Received  May 2015 Revised  August 2017 Published  September 2017

Fund Project: This research is supported by Natural and Science Natural Foundation of China(11771466,11301559,11471171), the 111 project(B17050), the National Science Foundation under grants DMS-1303775, DMS-1612880, and the Research Grants Council of the Hong Kong Special Administrative Region (CityU 500113, CityU 113 03 316). The second author is supported by PolyU grant G-YBKM

In this paper, an inventory control problem with a mean reverting inventory model is considered. The demand is assumed to follow a continuous diffusion process and a mean-reverting process which will take into account of the demand dependent of the inventory level. By choosing when and how much to stock, the objective is to minimize the long-run average cost, which consists of transaction cost for each replenishment, holding and shortage costs associated with the inventory level. An approach for deriving the average cost value of infinite time horizon is developed. By applying the theory of stochastic impulse control, we show that a unique (s, S) policy is indeed optimal. The main contribution of this work is to present a method to derive the (s, S) policy and hence the minimal long-run average cost.

Citation: 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
##### References:
 [1] S. Axsater, Inventory Control, Third edition. International Series in Operations Research & Management Science, 225. Springer, Cham, 2015. doi: 10.1007/978-3-319-15729-0. [2] 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. [3] 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. [4] A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, 2011. [5] 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. [6] C. Dellacherie and P. A. Meyer, Probabilites et Potentiel. Theorie des Martingales, Hermann, Paris, 1975. [7] P. L. Fackler and M. J. Livingston, Optimal storage by crop producers, American Journal of Agricultural Economics, 84 (2002), 645-659. doi: 10.1111/1467-8276.00325. [8] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. [9] B. Hogaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [10] 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 (2005), 4929-4944. doi: 10.1080/00207540500218967. [11] J. Jacod and A. N. Shiryaev, Limit theorems for Stochastic Processes, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5. [12] S. S. Ko, J. Kang and E. Y. Kwon, An (s, S) inventory model with level-dependent G/M/1-Type structure, Journal of Industrial and Management Optimization, 12 (2016), 609-624. doi: 10.3934/jimo.2016.12.609. [13] P. Kouvelis, R. Li and Q. Ding, Managing storable commodity risks: The role of inventory and financial hedge, Manufacturing & Service Operations Management, 15 (2013), 507-521. doi: 10.1002/9781118115800.ch6. [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. L. Porteus, Foundations of Stochastic Inventory Theory, Stanford Business Books, Stanford, 2002. [18] 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. [19] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. [20] L. Schwartz, The Economic Order-Quantity (EOQ) Model, D. Chhajed & T. J. Lowe (Ed. ), Building Intuition: Insights From Basic Operations Management Models and Principles, Springer US, 2008. [21] S. P. Sethi, W. Suo, M. I. Taksar and H. Yan, Optimal production planning in a multi-product stochastic manufacturing system with long-run average cost, Discrete Event Dynamic Systems, 8 (1998), 37-54. doi: 10.1023/A:1008256409920. [22] S. P. Sethi, H. Zhang and Q. Zhang, Minimum average cost production planning in stochastic manufacturing systems, Mathematical Models and Methods in Applied Sciences, 8 (1998), 1251-1276. doi: 10.1142/S0218202598000585. [23] A. Sulem, 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. [24] M. I. Taksar, Average optimal singular control and a related stopping problem, Mathematics of Operations Research, 10 (1985), 63-81. doi: 10.1287/moor.10.1.63. [25] 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. [26] C. D. J. Waters, Inventory Control and Management, $2^{nd}$ Ed. , John Wiley & Sons, Chichester, 2003. [27] T. Weston, Applying stochastic dynamic programming to the valuation of gas storage and generation assets, In E. Ronn (ed. ), Real Options and Energy Management Using Options Methodology to Enhance Capital Budgeting Decisions, Risk Publications, London, 2002. [28] T. Wild, Best Practice in Inventory Management, $2^{nd}$ Ed. , Butterworth Heinemann, Oxford, 2002. [29] J. C. Williams and B. D. Wright, Storage and Commodity Markets, Cambridge University Press, 1991. doi: 10.1017/CBO9780511571855. [30] H. L. Xu, P. Sui, G. L. Zhou and L. Caccetta, Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method, Numerical Algebra, Control and Optimization, 3 (2013), 655-664. doi: 10.3934/naco.2013.3.655. [31] 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. [32] 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. [33] 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. [34] P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill/Irwin, 2000.

show all references

##### References:
 [1] S. Axsater, Inventory Control, Third edition. International Series in Operations Research & Management Science, 225. Springer, Cham, 2015. doi: 10.1007/978-3-319-15729-0. [2] 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. [3] 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. [4] A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, 2011. [5] 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. [6] C. Dellacherie and P. A. Meyer, Probabilites et Potentiel. Theorie des Martingales, Hermann, Paris, 1975. [7] P. L. Fackler and M. J. Livingston, Optimal storage by crop producers, American Journal of Agricultural Economics, 84 (2002), 645-659. doi: 10.1111/1467-8276.00325. [8] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. [9] B. Hogaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [10] 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 (2005), 4929-4944. doi: 10.1080/00207540500218967. [11] J. Jacod and A. N. Shiryaev, Limit theorems for Stochastic Processes, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5. [12] S. S. Ko, J. Kang and E. Y. Kwon, An (s, S) inventory model with level-dependent G/M/1-Type structure, Journal of Industrial and Management Optimization, 12 (2016), 609-624. doi: 10.3934/jimo.2016.12.609. [13] P. Kouvelis, R. Li and Q. Ding, Managing storable commodity risks: The role of inventory and financial hedge, Manufacturing & Service Operations Management, 15 (2013), 507-521. doi: 10.1002/9781118115800.ch6. [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. L. Porteus, Foundations of Stochastic Inventory Theory, Stanford Business Books, Stanford, 2002. [18] 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. [19] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. [20] L. Schwartz, The Economic Order-Quantity (EOQ) Model, D. Chhajed & T. J. Lowe (Ed. ), Building Intuition: Insights From Basic Operations Management Models and Principles, Springer US, 2008. [21] S. P. Sethi, W. Suo, M. I. Taksar and H. Yan, Optimal production planning in a multi-product stochastic manufacturing system with long-run average cost, Discrete Event Dynamic Systems, 8 (1998), 37-54. doi: 10.1023/A:1008256409920. [22] S. P. Sethi, H. Zhang and Q. Zhang, Minimum average cost production planning in stochastic manufacturing systems, Mathematical Models and Methods in Applied Sciences, 8 (1998), 1251-1276. doi: 10.1142/S0218202598000585. [23] A. Sulem, 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. [24] M. I. Taksar, Average optimal singular control and a related stopping problem, Mathematics of Operations Research, 10 (1985), 63-81. doi: 10.1287/moor.10.1.63. [25] 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. [26] C. D. J. Waters, Inventory Control and Management, $2^{nd}$ Ed. , John Wiley & Sons, Chichester, 2003. [27] T. Weston, Applying stochastic dynamic programming to the valuation of gas storage and generation assets, In E. Ronn (ed. ), Real Options and Energy Management Using Options Methodology to Enhance Capital Budgeting Decisions, Risk Publications, London, 2002. [28] T. Wild, Best Practice in Inventory Management, $2^{nd}$ Ed. , Butterworth Heinemann, Oxford, 2002. [29] J. C. Williams and B. D. Wright, Storage and Commodity Markets, Cambridge University Press, 1991. doi: 10.1017/CBO9780511571855. [30] H. L. Xu, P. Sui, G. L. Zhou and L. Caccetta, Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method, Numerical Algebra, Control and Optimization, 3 (2013), 655-664. doi: 10.3934/naco.2013.3.655. [31] 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. [32] 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. [33] 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. [34] P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill/Irwin, 2000.
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