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2013, 3(4): 655-664. doi: 10.3934/naco.2013.3.655

## Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method

 1 School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China 2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China 3 Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia, Australia

Received  March 2013 Revised  October 2013 Published  October 2013

In this paper, we consider the bullwhip effect problem of an Order-Up-To (OUT) inventory strategy for a supply chain system. We firstly establish a new discrete-time dynamical model which is suitable to describe the OUT inventory strategy. Then, we analyze the bullwhip effect for the dynamical model of the supply chain system. We thus transform the bullwhip effect's dampening problem to a discrete-time optimal control problem. By using the Pontryagin's maximum principle, we compute the corresponding optimal control and obtain the optimal manufacturer productivity of goods. Finally, we carry out numerical simulation experiments to show that the devised optimal control strategy is useful to dampen the bullwhip effect which always happens in the supply chain system.
Citation: Honglei Xu, Peng Sui, Guanglu Zhou, Louis Caccetta. Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 655-664. doi: 10.3934/naco.2013.3.655
##### References:
 [1] R. B. Chase and N. J. Aquilano, "Production and Operations Management," Irwin. 7th ed, 1995. [2] F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information, Management Science, 46 (2000), 436-443. [3] Y. F. Chen and S. M. Disney, The myopic order-up-to policy with a proportional feedback controller, Int J Prod Res, 45 (2007), 351-368. [4] S. Eiamkanchanalai and A. Banerjee, Production lot sizing with variable production rate and explicit idle capacity cost, International Journal of Production Economics, 59 (1999), 251-259. [5] J. Forrestor, "Industrial Dynamics," New York: MIT Press and John Wily &Sons, Inc, 1961. [6] L. L. Hau, V. Padmanabhan and S. Whang, Information distortion in a supply chain: the bullwhip effect, Management Science, 43 (1997), 546-558. [7] S. Karlin and H. Scarf, "One Stage Inventory Models with Uncertainty. In: Studies in the Mathematical Theory of Inventory and Production," Stanford University Press, (1958), 109-134. [8] S. Makridakis, S. C. Wheelwright and V. E. McGee, "Forecasting: Methods and Applications," John Wiley & Sons, New York, 1978. [9] E. Ricard and K. Baradia, Evaluation of supply chain structures through modularization and postponement, European Journal Of Operational Research, 124 (2000), 495-510. [10] J. K. Ryan, "Analysis of Inventory Models with Limited Demand Information," Ph.D. Dissertation, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, 1997. [11] S. P. Sethi and G. L. Thompson, "Optimal Control Theory-Applications to Management Science," Martinus Nijhoff, Boston, 1981. [12] J. D. Sterman, Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment, Management Science, 35 (1989), 321-339. [13] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Longman Scientific & Technical, New York, 1991. [14] D. R. Towill and M. M. Naim, Industrial dynamics simulation models in the design of supply chains, International Journal of Physical Distribution and Logistics Management, 22 (1992), 3-12. [15] 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, 2 (2009), 81-94. doi: 10.3934/jimo.2009.5.81.

show all references

##### References:
 [1] R. B. Chase and N. J. Aquilano, "Production and Operations Management," Irwin. 7th ed, 1995. [2] F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information, Management Science, 46 (2000), 436-443. [3] Y. F. Chen and S. M. Disney, The myopic order-up-to policy with a proportional feedback controller, Int J Prod Res, 45 (2007), 351-368. [4] S. Eiamkanchanalai and A. Banerjee, Production lot sizing with variable production rate and explicit idle capacity cost, International Journal of Production Economics, 59 (1999), 251-259. [5] J. Forrestor, "Industrial Dynamics," New York: MIT Press and John Wily &Sons, Inc, 1961. [6] L. L. Hau, V. Padmanabhan and S. Whang, Information distortion in a supply chain: the bullwhip effect, Management Science, 43 (1997), 546-558. [7] S. Karlin and H. Scarf, "One Stage Inventory Models with Uncertainty. In: Studies in the Mathematical Theory of Inventory and Production," Stanford University Press, (1958), 109-134. [8] S. Makridakis, S. C. Wheelwright and V. E. McGee, "Forecasting: Methods and Applications," John Wiley & Sons, New York, 1978. [9] E. Ricard and K. Baradia, Evaluation of supply chain structures through modularization and postponement, European Journal Of Operational Research, 124 (2000), 495-510. [10] J. K. Ryan, "Analysis of Inventory Models with Limited Demand Information," Ph.D. Dissertation, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, 1997. [11] S. P. Sethi and G. L. Thompson, "Optimal Control Theory-Applications to Management Science," Martinus Nijhoff, Boston, 1981. [12] J. D. Sterman, Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment, Management Science, 35 (1989), 321-339. [13] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Longman Scientific & Technical, New York, 1991. [14] D. R. Towill and M. M. Naim, Industrial dynamics simulation models in the design of supply chains, International Journal of Physical Distribution and Logistics Management, 22 (1992), 3-12. [15] 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, 2 (2009), 81-94. doi: 10.3934/jimo.2009.5.81.
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