A Model Inventory / Production of Single Vendor and Single Buyer by Considering Shortage and Rate of Deterioration, Lead Time Uncertainty Based Approach the Dempster-Shafer Theory

Document Type : Research Paper


Department of Industrial Engineering, Kharazmi University, Tehran, Iran


In the real world, there is much uncertainty. The stochastic, fuzzy, fuzzy-stochastic, and evince methods, have been used for considering these uncertain conditions. The fuzzy method, as the most common one, has been broadly used in this direction. Yet fuzzy may not consider all of the uncertain conditions, such as unassigned, incomplete and interval data. Therefore, using evidence theory has been considered as the proposed approach to issues that are the interval basis, and also it is suitable for low data. In this paper, a continuous inventory model is presented with a single vendor, and a single buyer in a state of shortage, where deterioration rate is considered for the goods, and the demand is used as a log-normal. Also, lead time and deterioration rate, are considered uncertain based on Dempster-Shafer theory. The proposed model objective is minimizing the inventory system’s total cost. The model is solved by several numerical examples, and finally, the sensitivity problem is analyzed.


Main Subjects

  1. Dempster, A. P. (1967). “Upper and lower probabilities induced by a multivalued mapping”, The annals of mathematical statistics, Vol. 38 No. 2 PP. 325-339.
  2. Shafer, G. (1976). A mathematical theory of evidence, Princeton University press Princeton, Vol. 1.
  3. Kohlas, J. and P. A. Monney. (2013). “A mathematical theory of hints: An approach to the Dempster-Shafer theory of evidence”, Springer Science & Business Media, Vol. 425, No.1, PP-1-47.
  4. Smets, P. and R. Kennes. (1994). “The transferable belief model”, Artificial intelligence, Vol. 66, No. 2, PP. 191-234.
  5. Troiano, L., Rodríguez-Muñiz, L. J. and Díaz, I. (2015). “Discovering user preferences using Dempster–Shafer theory”, Fuzzy Sets and Systems, Vol. 278, PP. 98-117.
  6. Nodoust, S., Mirzazadeh, A. and Mohammadi, M. (2016). “A Genetic Algorithm for an inventory system under belief structure inflationary conditions”, RAIRO-Operations Research, Vol. 50, No. 4-5, PP. 1027-1041.
  7. Ouyang, L. Y. and Wu, K. S. (1997). “Mixture inventory model involving variable lead time with a service level constraint”, Computers & Operations Research, Vol. 24, No. 9, PP. 875-882.
  8. Tersine, R. J. (1994). Principles of inventory and materials management. IIE transactions, Vol. 26, No. 2, PP 97-101.
  9. Tadikamalla, P. R. (1979). “The lognormal approximation to the lead time demand in inventory control”, Omega, Vol. 7, No. 6, PP. 553-556.
  10. Ghare, P. and G. Schrader. (1963). “A model for exponentially decaying inventory”, Journal of Industrial Engineering, Vol. 14, No. 5, PP. 238-243.
  11. Yang, P. C. and H. M. Wee. (2000). “Economic ordering policy of deteriorated item for vendor and buyer: An integrated approach”, Production Planning & Control, Vol. 11, No. 5, PP. 474-480.
  12. Wu, M. Y. and Wee, H. M. (2001). “Buyer-seller joint cost model for deteriorating items with multiple lot-size deliveries”, Journal of the Chinese Institute of Industrial Engineers, Vol. 18, No. 1, PP. 109-119.
  13. Zhou, Y. W. and S. D. Wang. (2007). “Optimal production and shipment models for a single-vendor–single-buyer integrated system”, European Journal of Operational Research, Vol. 180, No. 1, PP. 309-328.
  14. July, F., Nojavan, F. and Qysryha, A. (2011). “Inventory control of deterioration items in two-level supply chain”, Journal of Industrial Engineering, Vol. 45, PP. 69-77.
  15. Mahdavi, M. (2013). “Development of certain inventory control models for deterioration items by considering backlog shortage and discounts”, Journal of Industrial Engineering, Vol. 47, No. 1, PP. 69-80
  16. Lo, S. T., Wee, H. M. and Huang, W. C. (2007). “An integrated production-inventory model with imperfect production processes and Weibull distribution deterioration under inflation”, International Journal of Production Economics, Vol. 106, No. 1, PP. 248-260.
  17. Goyal, S. (1977). “An integrated inventory model for a single supplier-single customer problem”, The International Journal of Production Research, Vol. 15, No. 1, PP. 107-111.
  18. Pan, J. C. H. and Yang, J. S. (2002). “A study of an integrated inventory with controllable lead time”, International Journal of Production Research, Vol. 40, No. 5, PP. 1263-1273.
  19. Goyal, S. (2003). “A note on: On controlling the controllable lead time component in the integrated inventory models”, INT.J.PROD.RES ,Vol. 41, No. 12, PP. 24-61.
  20. Ouyang, L. Y., Wu, K. S. and Ho, C. H. (2004). “Integrated vendor–buyer cooperative models with stochastic demand in controllable lead time”, International Journal of Production Economics, Vol. 92, No. 3, PP. 255-266.
  21. Hoque, M. A. and S. K. Goyal. (2006). “A heuristic solution procedure for an integrated inventory system under controllable lead-time withequal or unequal sized batch shipments between a vendor and a buyer”, International Journal of Production Economics, Vol. 102, No. 2, PP. 217-225.
  22. Ouyang, L. Y., Wu, K. S. and Ho. C. H. (2007). “An integrated vendor–buyer inventory model with quality improvement and lead time reduction”, International Journal of Production Economics, Vol. 108, No. 1, PP. 349-358.
  23. Yang, J. S. and Pan. J. C. H. (2004). “Just-in-time purchasing: an integrated inventory model involving deterministic variable lead time and quality improvement investment”, International Journal of Production Research, Vol. 42, No. 5, PP. 853-863.
  24. Mirzazadeh, A. (2011). “A Comparison of the Mathematical Modeling Methods in the Inventory Systems under Uncertain Inflationary Conditions”, International Journal of Engineering Science and Technology, No. 3, PP. 6131-6142.
  25. Jha, J. and Shanker, K. (2009). “A single-vendor single-buyer production-inventory model with controllable lead time and service level constraint for decaying items”, International Journal of Production Research, Vol. 47, No. 24, PP. 6875-6898.
  26. Lin, H. J. (2012). “An integrated supply chain inventory model with imperfect-quality items, controllable lead time and distribution-free demand”, Yugoslav Journal of Operations Research, Vol. 23, No. 1, PP-87-109
  27. Jha, J. and Shanker, K. (2013). “Single-vendor multi-buyer integrated production-inventory model with controllable lead time and service level constraints”, Applied Mathematical Modelling, Vol. 37, No. 4, PP. 1753-1767.
  28. Shahpouri, S. et al. (2013). “Integrated vendor–buyer cooperative inventory model with controllable lead time, ordering cost reduction, and service-level constraint”, The International Journal of Advanced Manufacturing Technology, Vol. 65, No. 5-8, PP. 657-666.
  29. Mandal, P. and Giri, B. (2015). “A single-vendor multi-buyer integrated model with controllable lead time and quality improvement through reduction in defective items”, International Journal of Systems Science: Operations & Logistics, Vol. 2, No. 1, PP. 1-14.
  30. Fauza, G. et al. (2016). “An integrated single-vendor multi-buyer production-inventory policy for food products incorporating quality degradation”, International Journal of Production Economics, Vol. 182, PP. 409-417.
  31. Lin, H. J. (2016). “Investing in lead-time variability reduction in a collaborative vendor–buyersupply chain model with stochastic lead time”, Computers & Operations Research, Vol. 72, No.1, PP. 43-49.
  32. Chan, C. K. et al. (2017). “An Integrated Production-Inventory Model for Deteriorating Items with Consideration of Optimal Production Rate and Deterioration during Delivery”, International Journal of Production Economics, Vol. 188, No. 1, PP. 1-13.
  33. Hossain, M. S. J., Ohaiba, M. M. and Sarker, B. R. (2017). “An optimal vendor-buyer cooperative policy under generalized lead-time distribution with penalty cost for delivery lateness”, International Journal of Production Economics, Vol. 188, No. 1, PP. 50-62.