Developing a Mathematical Model for Vendor Managed Inventory Considering Deterioration and Amelioration Items in a Three-Level Supply Chain

Document Type: Research Paper


Department of Industrial Engineering, Yazd University, Yazd, Iran


Todays, improving the competitiveness of companies and rapidly responding to diverse customers’ demands requires efficient management of the supply chain. Integration is one of the effective indices in a supply chain, and vendor managed inventory (VMI) is one of the innovative approaches in this field. In this paper, an integrated model based on a VMI system for deteriorating and ameliorating items in a three-level supply chain is developed. To coordinate retailers and producers in the supply chain, the common replenishment cycle is used which is a technique of a VMI system. The assumptions of this model consist of a deterministic demand rate for each retailer, certain production and deterioration rates. Furthermore, in this paper the amelioration of items is mentioned that follow the Weibull distribution. First, a mixed-integer nonlinear mathematical model is presented. Then to validate the proposed model, a numerical example is given. To solve the model, a genetic algorithm method is presented.


Main Subjects

1-  Goyal, S.K. (1976). “An integrated inventory model for a single supplier–single customer problem.” International Journal of Production Research, PP. 107–111.

2- Rau, H., Wu, M.Y. and Wee, H.M. (2003). “Integrated inventory model for deteriorating items under a multi-echelon supply chain environment.” Int. J. Production Economics, No. 86, PP. 155–168.

3- Li, L.F. and Huang, P.Q. (2005). “Study of production-inventory cooperation for deteriorating items on supply chain.” Journal of Shanghai Jiaotong University, Vol. 39, No. 3, PP. 464–467.

4-Yang, P.C. and Wee, H.M. (2003). “An integrated multi-lot-size production inventory model for deteriorating item.” Computers& Operations Research, No. 30, PP. 671-682.

5- Shah, N.H., Gor, A. S. and Jhaveri, C. (2011). “An integrated inventory policy with deterioration single vendor and multiple buyers in supply chain when demand is quadratic.” Revista investigacion operacional, Vol. 32, No.2, PP. 93–106.

6- Magee, J. F. (1958). “Production planning and inventory control.” New York, Ny, USA: McGraw-Hill book Company.

7-Clark, A.J. and Scarf, H. (1960). “Optimal policies for a multi echel on inventory problem.” Management Science, No.6, PP. 475–490.

8- Whitin, T. M. (1957).“Theory of inventory management.” Princeton University Press, PP.62–72.

9- Ghare, P. M. and Schrader, G. P. (1963). “A model for an exponentially decaying inventory.” International Journal of Production Research, Vol. 21, PP. 449–460.

10- Covert, R. P. and Philip, G. C. (1973). “An EOQ model for items with weibull distribution deterioration.” AIIE Trans., PP.159-162.

11- Tadikamalla, P. R. (1978). “An EOQ model for items with gamma distributed deterioration.” AIIE Trans,100-103.

12-Yan Chou, S., Chouhuang, W.T., Lin, J.S.J. and Chu., P. (2008). “An analytic solution approach for the economic order quantity model with Weibull ameliorating items.” Mathematical and Computer Modeling, No. 48, PP. 1868–1874.

13- Hwang, H.S. (1997)." A study on an inventory model for items with Weibull ameliorating". Computers and Industrial Engineering, No. 33, PP. 701–704.

14- Hwang, H.S. (1999). “Inventory models for both deteriorating and ameliorating items.” Computers and Industrial Engineering, No. 37, PP. 257–260.

15- Mondal, B., Bhunia, A.K. and Maiti, M. (2003). “An inventory system of ameliorating items for price dependent demand rate.” Computers and Industrial Engineering, No. 45, PP. 443–456.

16-Yu, Y., Huang, G.Q., Hong, Z. and Zhang, X. (2011). “An integrated pricing and deteriorating model and a hybrid algorithm for a VMI supply chain.” IEEE Transactions on Automation Science and Engineering, Vol. 8, No. 4, PP. 673–682.

17- Yu, Y., Wang, Zh. and Liang, L. (2012). “A vendor managed inventory supply chain with deteriorating raw materials and products.” Int. J. Production Economics, No. 136, PP. 266–274.

18- Law, S.T. and Wee, H.M. (2006). “An integrated production-inventory model for ameliorating and deteriorating items taking account of time discounting.” Mathematical and Computer Modelling, No. 43, PP. 673–685.

19- Taguchi, G. (1986). Introduction to quality engineering: designing quality into products and processes.