Skip to main content
SpringerLink
Log in

Modelling Economic Order Quantities, Considering Buy and Repair Options for Defective Items, and Allowing for Shortages and Inspection Errors

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

Nowadays, it is common for suppliers to be separated from their buyers by great distances. This means that, if some items in the lot turn out to be defective, the distance makes it uneconomical to order replacements from the original supplier. Moreover, both Type I and Type II errors may occur in the screening process for eliminating defective items and the combination of defective items and inspection errors may lead to shortages. Working with these assumptions, this paper develops two distinct models. Under the first model, defective items are repaired by a local repair shop subject to a repair charge and a mark-up margin. Under the second model, a local supplier replaces the defective with good ones, but at a higher cost. The expected total profit per cycle is developed, together with the expected cycle time, and, employing the renewal reward theorem, the objective function is derived, from which the optimum values are obtained for the order and shortage quantities. The paper presents numerical results and a discussion for both models. The study finds that repairing defective items generally leads to greater total profit than purchasing local replacements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Harris, F.W.: How many parts to make at once. Factory, Mag. Manag. 10(2), 135–136, 152 (1913): reprinted Oper. Res. 38, 947–950 (1990).

  2. Shih, W.: Optimal inventory policies when stockouts result from defective products. Int. J. Prod. Res. 18(6), 677–686 (1980)

    Article  Google Scholar 

  3. Porteus, E.L.: Optimal lot sizing, process quality improvement and setup cost reduction. Oper. Res. 34(1), 137–144 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Rosenblatt, M.J., Lee, H.L.: Economic production cycles with imperfect production processes. IIE Trans. 18(1), 48–55 (1986)

    Article  Google Scholar 

  5. Lee, H.L., Rosenblatt, M.J.: Simultaneous determination of production cycle and inspection schedules in a production system. Manag. Sci. 33(9), 1125–1136 (1987)

    Article  MATH  Google Scholar 

  6. Schwaller, R.L.: EOQ under inspection costs. Product. Inventory Manag. J. 29(3), 22–24 (1988)

    Google Scholar 

  7. Zhang, X., Gerchak, Y.: Joint lot sizing and inspection policy in an EOQ model with random yield. IIE Trans. 22(1), 41–47 (1990)

    Article  Google Scholar 

  8. Cheng, T.C.E.: An economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Trans. 23(1), 23–28 (1991)

    Article  Google Scholar 

  9. Ben-Daya, M., Hariga, M.: Economic lot scheduling problem with imperfect production processes. J. Oper. Res. Soc. 51(7), 875–881 (2000)

    Article  MATH  Google Scholar 

  10. Salameh, M.K., Jaber, M.Y.: Economic production quantity model for items with imperfect quality. Int. J. Prod. Econ. 64(1–3), 59–64 (2000)

    Article  Google Scholar 

  11. Cárdenas-Barrón, L.E.: Observation on: “Economic production quantity model for items with imperfect quality”. Int. J. Prod. Econ. 67(2), 201 (2000)

    Article  Google Scholar 

  12. Hayek, P.A., Salameh, M.K.: Production lot sizing with the reworking of imperfect quality items produced. Prod. Plan. Control 12, 584–590 (2001)

    Article  Google Scholar 

  13. Chiu, Y.P.: Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging. Eng. Optim. 35(4), 427–437 (2003)

    Article  Google Scholar 

  14. Chan, W.M., Ibrahim, R.N., Lochert, P.B.: A new EPQ model: integrating lower pricing, rework and reject situations. Prod. Plan. Control 14, 588–595 (2003)

    Article  Google Scholar 

  15. Papachristos, S., Konstantaras, I.: Economic ordering quantity models for items with imperfect quality. Int. J. Prod. Econ. 100(1), 148–154 (2006)

    Article  Google Scholar 

  16. Liao, G.L.: Optimal production correction and maintenance policy for imperfect process. Eur. J. Oper. Res. 182(3), 1140–1149 (2007)

    Article  MATH  Google Scholar 

  17. Wee, H.M., Yu, J., Chen, M.C.: Optimal inventory model for items with imperfect quality and shortage backordering. Omega 35(1), 7–11 (2007)

    Article  Google Scholar 

  18. Eroglu, A., Ozdemir, G.: An economic order quantity model with defective items and shortages. Int. J. Prod. Econ. 106(2), 544–549 (2007)

    Article  Google Scholar 

  19. Maddah, B., Jaber, M.Y.: Economic order quantity for items with imperfect quality: revisited. Int. J. Prod. Econ. 112(2), 808–815 (2008)

    Article  Google Scholar 

  20. Cárdenas-Barrón, L.E.: Economic production quantity with rework process at a single-stage manufacturing system with planned backorders. Comput. Ind. Eng. 57(3), 1105–1113 (2009)

    Article  Google Scholar 

  21. Sana, S.S.: An economic production lot size model in an imperfect production system. Eur. J. Oper. Res. 201(1), 158–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maddah, B., Salameh, M.K., Moussawi-Haidar, L.: Order overlapping: A practical approach for preventing shortages during screening. Comput. Ind. Eng. 58(4), 691–695 (2010)

    Article  Google Scholar 

  23. Chang, H.C., Ho, C.H.: Exact closed-form solutions for “optimal inventory model for items with imperfect quality and shortage backordering. Omega 38(3–4), 233–237 (2010)

    Article  Google Scholar 

  24. Chung, K.J.: The economic production quantity with rework process in supply chain management. Comput. Math. Appl. 62(6), 2547–2550 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liao, G.L., Sheu, S.H.: Economic production quantity model for randomly failing production process with minimal repair and imperfect maintenance. Int. J. Prod. Econ. 130(1), 118–124 (2011)

    Article  Google Scholar 

  26. Hsu, J.-T., Hsu, L.-F.: A note on “Optimal inventory model for items with imperfect quality and shortage backordering. Int. J. Ind. Eng. Comput. 3(5), 939–948 (2012)

    Google Scholar 

  27. Khan, M., Jaber, M.Y., Guiffrida, A.L., Zolfaghari, S.: A review of the extensions of a modified EOQ model for imperfect quality items. Int. J. Prod. Econ. 132(1), 1–12 (2011)

    Article  Google Scholar 

  28. Jaber, M.Y., Zanoni, S., Zavanella, L.E.: Economic order quantity models for imperfect items with buy and repair options. Int. J. Prod. Econ. 155, 126–131 (2014)

    Article  Google Scholar 

  29. Taleizadeh, A.A., Khanbaglo, M.P.S., Cárdenas-Barrón, L.E.: An EOQ inventory model with partial backordering and reparation of imperfect products. Int. J. Prod. Econ. 182, 418–434 (2016)

    Article  Google Scholar 

  30. Li, N., Chan, F.T., Chung, S.H., Tai, A.H.: A stochastic production-inventory model in a two-state production system with inventory deterioration, rework process, and backordering. IEEE Trans. Syst. Man Cybern. Syst. 47(6), 916–926 (2017)

    Article  Google Scholar 

  31. Skouri, K.: An EOQ model with backlog-dependent demand. Oper. Res. Int. Journal 18(2), 561–574 (2018)

    Article  Google Scholar 

  32. Ross, S.: Probability and statistics for engineers and scientists. Elsevier Academic Press, Canada (2004)

    MATH  Google Scholar 

  33. Yoo, S.H., Kim, D., Park, M.S.: Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return. Int. J. Prod. Econ. 121(1), 255–265 (2009)

    Article  Google Scholar 

  34. Khan, M., Jaber, M.Y., Bonney, M.: An economic order quantity (EOQ) for items with imperfect quality and inspection errors. Int. J. Prod. Econ. 133(1), 113–118 (2011)

    Article  Google Scholar 

  35. Hsu, L.-F.: A note on An economic order quantity (EOQ) for items with imperfect quality and inspection errors. Int. J. Ind. Eng. Comput. 3(4), 695–702 (2012)

    Google Scholar 

  36. Hsu, L.-F.: Erratum to “An economic order quantity (EOQ) for items with imperfect quality and inspection errors”. Int. J. Prod. Econ. 136(1), 253 (2012)

    Article  Google Scholar 

  37. Hsu, J.-T., Hsu, L.-F.: Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Comput. Ind. Eng. 64(1), 389–402 (2013)

    Article  Google Scholar 

  38. Glock, C.H., Jaber, M.Y.: A multi-stage production-inventory model with learning and forgetting effects, rework and scrap. Comput. Ind. Eng. 64(2), 708–720 (2013)

    Article  Google Scholar 

  39. Khan, M., Jaber, M.Y., Ahmad, A.-R.: An integrated supply chain model with errors in quality inspection and learning in production. Omega 42(1), 16–24 (2014)

    Article  Google Scholar 

  40. Mohammadi, B., Taleizadeh, A.A., Noorossana, R., Samimi, H.: Optimizing integrated manufacturing and products inspection policy for deteriorating manufacturing system with imperfect inspection. J. Manuf. Syst. 37(1), 299–315 (2015)

    Article  Google Scholar 

  41. Zhou, Y., Chen, C., Li, C., Zhong, Y.: A synergic economic order quantity model with trade credit, shortages, imperfect quality and inspection errors. Appl. Math. Model. 40, 1012–1028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sarkar, B., Saren, S.: Product inspection policy for an imperfect production system with inspection errors and warranty cost. Eur. J. Oper. Res. 248(1), 263–271 (2016)

    Article  MATH  Google Scholar 

  43. Cheikhrouhou, N., Sarkar, B., Ganguly, B., Malik, A.I., Batista, R., Lee, Y.H.: Optimization of sample size and order size in an inventory model with quality inspection and return of defective items. Ann. Oper. Res. 271(2), 445–467 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yang, C.T., Ho, C.H., Lee, H.M., Ouyang, L.Y.: Supplier-retailer production and inventory models with defective items and inspection errors in non-cooperative and cooperative environments. RAIRO Oper. Res. 52(2), 453–471 (2018)

    Article  MathSciNet  Google Scholar 

  45. Dey, O., Giri, B.C.: A new approach to deal with learning in inspection in an integrated vendor-buyer model with imperfect production process. Comput. Ind. Eng. 131, 515–523 (2019)

    Article  Google Scholar 

  46. Flapper, S.D.P., Fransoo, J.C., Broekmeulen, R.A.C.M., Inderfurth, K.: Planning and control of rework in the process industries: a review. Prod. Plan. Control 13(1), 26–34 (2002)

    Article  Google Scholar 

  47. Duffuaa, S.O., Khan, M.: A general repeat inspection plan for dependent multicharacteristic critical components. Eur. J. Oper. Res. 191, 374–385 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Raouf, A., Jain, J.K., Sathe, P.T.: A cost-minimization model for multi characteristic component inspection. IIE Trans. 15(3), 187–194 (1983)

    Article  Google Scholar 

  49. Duffuaa, S.O., Khan, M.: An optimal repeat inspection plan with several classifications. J. Oper. Res. Soc. 53(9), 1016–1026 (2002)

    Article  MATH  Google Scholar 

  50. Duffuaa, S.O., Khan, M.: Impact of inspection errors on the performance measures of a general repeat inspection plan. Int. J. Prod. Res. 43(23), 4945–4967 (2005)

    Article  MATH  Google Scholar 

  51. Khan, M., Ahmad, A.-R., Hussain, M.: Integrated decision models for a vendor–buyer supply chain with inspection errors and purchase and repair options. Int. J. Adv. Manuf. Technol. 104, 3221–3228 (2019)

    Article  Google Scholar 

  52. Ross, S.M.: Introduction to Probability Models. Academic Press Inc., Cambridge (1989)

    MATH  Google Scholar 

  53. Duffuaa, S.O., Nadeem, I.A.: A complete inspection plan for dependent multicharacteristic critical components. Int. J. Prod. Res. 32(8), 1897–1907 (1994)

    Article  MATH  Google Scholar 

  54. Lee, H.L.: On the optimality of a simplified multicharacteristic component inspection model. IIE Trans. 20(4), 392–398 (1988)

    Article  Google Scholar 

  55. Buscher, U., Lindner, G.: Optimizing a production system with rework and equal sized batch shipments. Comput. Oper. Res. 34, 515–535 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Dorris, A. L., Hoag, L. L., Kasiviswanathan, G.: Inspector error models: an empirical investigation. In: Proceedings of the Human Factors Society 21st Annual Meeting, pp. 137–141 (1977). https://doi.org/10.1177/107118137702100211

  57. Reddy, J.N.: An Introduction to the Finite Element Method. McGraw-Hill, New York (1993)

    Google Scholar 

  58. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  59. Rao, S.S.: The Finite Element Method in Engineering. Elsevier Inc., Amsterdam (2011)

    MATH  Google Scholar 

  60. Lopes, R.: Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system. Comput. Ind. Eng. 126, 650–656 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

The author is grateful for the valuable comments from the Editors and two anonymous reviewers.

Author information

Authors and Affiliations

  1. Department of Business Administration, Faculty of Economics and Administrative Sciences, Süleyman Demirel University, Çünür, 32260, Isparta, Turkey

    Harun Öztürk

Authors
  1. Harun Öztürk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harun Öztürk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Öztürk, H. Modelling Economic Order Quantities, Considering Buy and Repair Options for Defective Items, and Allowing for Shortages and Inspection Errors. J. Oper. Res. Soc. China 9, 757–795 (2021). https://doi.org/10.1007/s40305-021-00339-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-021-00339-8

Keywords

Mathematics Subject Classification

Search

Navigation