# Modeling and Solving a Multi-product Economic Order Quantity Problem with Imperfect Items and Emergency Buy or Repair Options

Document Type : Research Paper

Authors

Department of Industrial Engineering, Kharazmi University, Tehran, Iran

Abstract

In this paper, it has been proposed a multi-product economic order quantity (EOQ) model, assuming the presence of some imperfect items. They are inspected 100 percent as they are delivered, and defective items are withdrawn. We do not access to supplier to replace the imperfect items with perfect one. So for meeting the demand, there are two options for imperfect items: buy or repair. In buy option, we sell the imperfect items in second market with lower price, and then buy the imperfect items from local supplier with higher price; but in repair option, we will send the imperfect items to repairing workshop, and after they are delivered from repair facility, they are sold as perfect items. There are also some limitations in available budget and warehouse space, and thus we should determine the economic order quantity for each item by considering these limitations. Additional in repairing option, we have some constraints for repairing rate. In the proposed model, the shortage is not allowed, and we have considered this issue in employing alternatives. This paper extends the supposed assumes by Jaber et al. (سال؟؟؟). The Proposed problem is a Mixed Integer Non-Linear problem (MINLP). This field of problems are NP-Hard, and they are solved by Meta heuristic algorithms, but they can be solved by exact methods in small scale. So, the proposed problem in this paper is solved by exact method. We also solved numerical examples, and applied the sensitivity analysis for important parameters.

Keywords

Main Subjects

#### References

Biswajit, .S, Cárdenas-Barrón, L.E, Sarkar, .M, Singgih, .M.L (2014). "An economic production quantity model with random defective rate", Journal of Manufacturing Systems,  Vol. 33, No. 3 , PP. 423-435.
2. Harris, F. W. (1913). "How many parts to make at once", Factory Magaz Manage,Vol. 10, No.2, PP. 135-136.
3. Porteus, E. L. (1986). "Optimal lot sizing, process quality improvement and setup cost reduction", Operations Research, Vol. 34, No. 1, PP. 137-144.
4. Rosenblatt, Hau L. Lee, (1984). "Economic production cycles with imperfect production processs", IIE Transactions,Vol. 18, No. 1, PP. 48-55.
5. Salameh, M. K. and Jaber, M. Y. (2000). "Economic production quantity model for items with imperfect quality", International Journal of Production Economics,Vol. 64, No. 1-3, PP. 59-64, 200.
6. Cárdenas-Barrón, L. E. (2000). "Observation on: Economic production quantity model for items with imperfect quality" International Journal of Production Economics, Vol. 67, No. 2, P. 201-201
7. Goyal, S. K and Cárdenas-Barrón, L. E. (2002). "Note on: Economic production quantity model for items with imperfect quality – a practical approach", International Journal of Production Economics,Vol. 77, No. 1, PP. 85-87.
8. Chang, H. C. (2004). "An application of fuzzy sets theory to the EOQ model with imperfect quality items", Computers & Operations Research,Vol. 31, No. 12, PP. 2079-2092.
9. Wee, H. M., Yu, J., Chen, M. C.  (2007). "Optimal inventory model for items with imperfect quality and shortage backordering", Omega, Vol. 1, No. 35, PP. 7-11.
10. Maddah, B., Jaber, M. Y. (2008). "Economic order quantity for items with imperfect quality: Revisited", International Journal of Production Economics,Vol. 112, No. 2, PP. 808-815.
11. Wahab, M. I. M. and Jaber, M. Y. (2010). "Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note", Computers & Industrial Engineering,Vol. 58, No. 1, PP. 186-190.
12. Khan, M.,  Jaber, M. Y., Bonney, M.  (2011). "An economic order quantity (EOQ) for items with imperfect quality and inspection errors", International Journal of Production Economics,Vol. 133, No. 1, PP. 113-118.
13. Khan, M., Jaber, M. Y., Guiffrida, A. L. and Zolfaghari, S. (2011). "A review of the extensions of a modified EOQ model for imperfect quality items", Int. J. Prod. Econ.,Vol. 132, No. 1, PP. 1-12.
14. Rezaei, J. and Salimi, N. (2012). "Economic order quantity and purchasing price for items with imperfect quality when inspection shifts from buyer to supplier", International Journal of Production Economics, Vol. 137, No. 1, PP. 11-18.
15. Taheri-Tolgari, J., Mirzazadeh, A. and Jolai, F. (2012). "An inventory model for imperfect items under inflationary conditions with considering inspection errors", Computers & Mathematics with Applications,Vol. 63, No. 6, PP. 1007-1019.
16. Hsu, J. T., Hsu, L. F (2013). "An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns", International Journal of Production Economics,Vol. 143, No. 1, PP. 162-170.
17. Moussawi-Haidar, L., Salameh, M. and Nasr, W. (2013). "An instantaneous replenishment model under the effect of a sampling policy for defective items", Applied Mathematical Modelling, Vol. 37, No. 3, PP. 719-727.
18. Jaber, M. Y., Zanoni, S., Zavanell, L. E. (2014). "Economic order quantity models for imperfect items with buy and repair options", Int. J. Production Economics, Vol. 155, PP. 126-131.
19. Teimoury, E. and Kazemi, S. M. M. (2015). "Development of pricing model for deteriorating items with constant deterioration rate considering replacement", School of Inddustrial Engineering, Iran University of Science & Technology,Vol. 49, No. 1, PP. 1-9.
20. Zhou, Y.W., Chen, J.,  Yongzhong,  W., Zhou, W. (2015). "EPQ models for items with imperfect quality and one-time-only discount", Applied Mathematical Modelling,Vol. 39, No. 3-4, PP. 1000-1018.
21. Kumar, R. Sh., Goswami, A. (2015). "A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints", Applied Soft Computing, Vol. 34,  PP. 838-850.
22. Taleizadeh, A. and Hasani, M. (2015). "A multi-product inventory control model with imperfect production process and rework under delayed payment policy", Journal of Industerial Engineering, Vol. 49, No. 2, PP. 223-235.
23. Pasandideh, S. H. R., Akhavan Niaki, S. T., Nobil, A. H., Cárdenas-Barrón, L. E. (2015). "A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost", International Journal of Production Economics, Vol. 169, PP. 203-214.
24. Kazemi, A., Malekian, M. R. and Sarrafha, K. (2012). "Presenting a new model for inventory control of multi-item Economic Production Quantity (EPQ) with fuzzy random demand", Journal of Industerial Engineering, Vol. 46, No. 1, PP. 53-62.
25. Chen, T. H. (2016). "Optimizing pricing, replenishment and rework decision for imperfect and deteriorating items in a manufacturer-retailer channel", International Journal of Production Economics, Vol. 183, Part B, PP. 539-550
26. Rezaei, J. (2016). "Economic order quantity and sampling inspection plans for imperfect items", Computers & Industrial Engineering,Vol. 96, PP. 1-7.
27, Khan, M., Jaber, M. Y., Zanoni, S., Zavanella, L.  (2016). "Vendor managed inventory with consignment stock agreement for a supply chain with defective items", Department of Mechanical and Industrial Engineering, Vol. 183, No. 15-16, PP. 7102-7114.
28. Aghaei, A. and Zandi, F. (2010). "Analyzing and optimization a two echelon supply chain with uncertainly returned product", Journal of Industerial Engineering,Vol. 46, No. 2, PP. 119-132.
29. Gen, M. and  Cheng, R. (1997). Genetic algorithms and engineering design, John Wiley & sons, New York.