Skip to main content
SpringerLink
Log in

A Fuzzy Random Continuous (Q, r, L) Inventory Model Involving Controllable Back-order Rate and Variable Lead-Time with Imprecise Chance Constraint

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 253))

Included in the following conference series:

  • 632 Accesses

Abstract

In this article, we analyze a fuzzy random continuous review inventory system with the mixture of back-orders and lost sales, where the annual demand is treated as a fuzzy random variable. The study under consideration assumes that the lead-time is a control variable and the lead-time crashing cost is being introduced as a negative exponential function of the lead-time. In a realistic situation, the back-order rate is dependent on the lead-time. Significantly large lead-times might lead to stock-out periods being longer. As a result, many customers may not be prepared to wait for back-orders. Instead of constant back-order rate, we introduce the back-order rate as a decision variable, which is a function of the lead-time throughout the amount of shortage. Moreover, a budgetary constraint is imposed on the model in the form of an imprecise chance constraint to capture the possible way of measuring the imprecisely defined uncertain information of the budget constraint. We develop a methodology to determine the optimum order quantity, reorder point, lead-time, and back-order rate such that the total cost is minimized in the fuzzy sense. Finally, a numerical example is presented to illustrate the proposed methodology.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abdel-Malek, Layek L., Montanari, R.: An analysis of the multi-product newsboy problem with a budget constraint. Int. J. Prod. Econ. 97(3), 296–307 (2005)

    Article  Google Scholar 

  2. Ben-Daya, M., Raouf, A.: Inventory models involving lead time as decision variable. J. Oper. Res. Soc. 45(5), 579–582 (1994)

    Article  Google Scholar 

  3. Carlsson, C., Fuller, R.: On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets Syst. 122(2), 315–326 (2001)

    Article  MathSciNet  Google Scholar 

  4. Chakraborty, D.: Redefining chance-constrained programming in fuzzy environment. Fuzzy Sets Syst. 125(3), 327–333 (2002)

    Article  MathSciNet  Google Scholar 

  5. Chang, H.C., Yao, J.S., Quyang, L.Y.: Fuzzy mixture inventory model involving fuzzy random variable lead time demand and fuzzy total demand. Eur. J. Oper. Res. 169(1), 65–80 (2006)

    Article  MathSciNet  Google Scholar 

  6. Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci. 6(1), 73–79 (1959)

    Article  MathSciNet  Google Scholar 

  7. Dey, O., Giri, B.C., Chakraborty, D.: A fuzzy random continuous review inventory model with a mixture of backorders and lost sales under imprecise chance constraint. Int. J. Oper. Res. 26(1), 34–51 (2016)

    Article  MathSciNet  Google Scholar 

  8. Dey, O., Chakraborty, D.: A fuzzy random periodic review system with variable lead-time and negative exponential crashing cost. Appl. Math. Model. 36(12), 6312–6322 (2012)

    Article  MathSciNet  Google Scholar 

  9. Dey, O., Chakraborty, D.: A fuzzy random periodic review system: a technique for real-life application. Int. J. Oper. Res. 13(4), 395–405 (2012)

    Article  MathSciNet  Google Scholar 

  10. Dey, O., Chakraborty, D.: Fuzzy periodic review system with fuzzy random variable demand. Eur. J. Oper. Res. 198(1), 9113–120 (2009)

    Article  Google Scholar 

  11. Dutta, P., Chakraborty, D.: Continuous review inventory model in mixed fuzzy and stochastic environment. Appl. Math. Comput. 188(1), 970–980 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Dutta, P., Chakrabortyand, D., Roy, A.R.: A single-period inventory model with fuzzy random variable demand. Math. Comput. Model. 41(8–9), 915–922 (2005)

    Article  MathSciNet  Google Scholar 

  13. Gen, M., Tsujimura, Y., Zheng, D.: An application of fuzzy set theory to inventory control models. Comput. Ind. Eng. 33(3), 553–556 (1997)

    Article  Google Scholar 

  14. Glock, C.H.: Lead time reduction strategies in a single-vendor-single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. Int. J. Prod. Econ. 136(1), 37–44 (2012)

    Article  Google Scholar 

  15. Hadley, G., Whitin, T.M.: Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, NJ (1963)

    MATH  Google Scholar 

  16. Kim, O.H., Park, K.S.: (Q, r) inventory model with a mixture of lost sales and weighted back-orders. J. Oper. Res. Soc. 36, 231–238 (1985)

    Article  Google Scholar 

  17. Kumar, R.S., Goswami, A.: A continuous review production-inventory system in fuzzy random environment: minmax distribution free procedure. Comput. Ind. Eng. 79(1), 65–75 (2015)

    Article  Google Scholar 

  18. Kundu, A., Chakrabarti, T.: A multi-product continuous review inventory system in stochastic environment with budget constraint. Optim. Lett. 6(2), 299–313 (2012)

    Article  MathSciNet  Google Scholar 

  19. Kwakernaak, H.: Fuzzy random variables—I. Definitions and theorems. Inform. Sci. 15(1), 1–29 (1978)

    Article  MathSciNet  Google Scholar 

  20. Lee, W.C.: Inventory model involving controllable back-order rate and variable lead time demand with the mixture of distribution. Appl. Math. Comput. 160(3), 701–717 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Liao, C.J., Shyu, C.H.: An analytical determination of lead time with normal demand. Int. J. Oper. Prod. Manage. 11(9), 72–78 (1991)

    Article  Google Scholar 

  22. Lin, H.J.: A stochastic periodic review inventory model with back-order discounts and ordering cost dependent on lead time for the mixtures of distributions. Top 23(2), 386–400 (2015)

    Article  MathSciNet  Google Scholar 

  23. Montgomery, D.C., Bazaraa, M.S., Keswani, A.K.: Inventory models with a mixture of backorders and lost sales. Naval Res. Logistics Q. 20(2), 255–263 (1973)

    Article  Google Scholar 

  24. Moon, I., Silver, E.A.: The multi-item newsvendor problem with a budget constraint and fixed ordering costs. J. Oper. Res. Soc. 51(5), 602–608 (2000)

    Article  Google Scholar 

  25. Moon, I., Choi, S.: TECHNICAL NOTEA note on lead time and distributional assumptions in continuous review inventory models. Comput. Oper. Res. 25(11), 1007–1012 (1998)

    Article  MathSciNet  Google Scholar 

  26. Naddor, E.: Inventory Systems. Wiley, New York (1966)

    MATH  Google Scholar 

  27. Ouyang, L.Y., Yao, J.S.: A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand. Comput. Oper. Res. 29(5), 471–487 (2002)

    Article  Google Scholar 

  28. Ouyang, L.Y., Chuang, B.R.: Mixture inventory model involving variable lead time and controllable back-order rate. Comput. Ind. Eng. 40(4), 339–348 (2001)

    Article  Google Scholar 

  29. Ouyang, L.Y., Chang, H.C.: Impact of investing in quality improvement on (Q, r, L) model involving the imperfect production process. Prod. Plan. Control 11(6), 598–607 (2000)

    Article  Google Scholar 

  30. Ouyang, L.Y., Yeh, N.C., Wu, K.S.: Mixture inventory model with back-orders and lost sales for variable lead time. J. Oper. Res. Soc. 47, 829–832 (1996)

    Article  Google Scholar 

  31. Padmanabhan, G., Vrat, P.: Inventory models with a mixture of backorders and lost sales. Int. J. Syst. Sci. 21(8), 1721–1726 (1990)

    Article  Google Scholar 

  32. Park, K.S.: Fuzzy-set theoretic interpretation of economic order quantity. IEEE Trans. Syst. Man Cybern. 17(6), 1082–1084 (1987)

    Article  Google Scholar 

  33. Sarkar, B., Moon, I.: Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process. Int. J. Prod. Econ. 155, 204–213 (2014)

    Article  Google Scholar 

  34. Shekarian, E., Kazemi, N., Rashid, S.H.A., Olugu, E.U.: Fuzzy inventory models: a comprehensive review. Appl. Soft Comput. 45(2–3), 260–264 (2017)

    Google Scholar 

  35. Tersine, R.J.: Principles of Inventory and Materials Management. Prentice Hall, Englewood Cliffs, NJ (1994)

    Google Scholar 

  36. Tütüncü, G.Y., Aköz, O., Apaydın, A., Petrovic, D.: Continuous review inventory control in the presence of fuzzy costs. Int. J. Prod. Econ 113(2), 775–784 (2008)

    Article  Google Scholar 

  37. Vijayan, T., Kumaran, M.: Inventory models with mixture of back-orders and lost sales under fuzzy cost. Eur. J. Oper. Res. 189(1), 105–119 (2008)

    Article  Google Scholar 

  38. Zimmermann, H.J.: Fuzzy Set Theory and its Applications. Springer Science & Business Media (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, West Bengal, India

    Debjani Chakraborty & Sushil Kumar Bhuiya

  2. Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, 221005, Uttar Pradesh, India

    Debdas Ghosh

Authors
  1. Debjani Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sushil Kumar Bhuiya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Debdas Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debjani Chakraborty .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India

    Debdas Ghosh

  2. Department of Computer Science and Engineering, Haldia Institute of Technology, Haldia, West Bengal, India

    Debasis Giri

  3. Department of Mathematics, University of Central Florida, Orlando, FL, USA

    Ram N. Mohapatra

  4. Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan

    Kouichi Sakurai

  5. Uşak University, Uşak, Turkey

    Ekrem Savas

  6. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India

    Tanmoy Som

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chakraborty, D., Bhuiya, S.K., Ghosh, D. (2018). A Fuzzy Random Continuous (Q, r, L) Inventory Model Involving Controllable Back-order Rate and Variable Lead-Time with Imprecise Chance Constraint. In: Ghosh, D., Giri, D., Mohapatra, R., Sakurai, K., Savas, E., Som, T. (eds) Mathematics and Computing. ICMC 2018. Springer Proceedings in Mathematics & Statistics, vol 253. Springer, Singapore. https://doi.org/10.1007/978-981-13-2095-8_21

Download citation

Publish with us

Policies and ethics

Search

Navigation