# A Fuzzy Production Inventory Model for Deteriorating Items with Shortages

### Abstract

In this paper we have developed a supply chain production inventory model for deteriorating items with shortage under Fuzzy environment. The formulae for the optimal average system cost, stock level, backlog level and production cycle time are derived when the deterioration rate is very small. In reality it is seen that we cannot define all parameters precisely due to imprecision or uncertainty in the environment. So, we have defined the inventory parameter deterioration rate as triangular fuzzy numbers. The signed distance method and graded mean integration method have been used for defuzzification. Numerical examples are taken to illustrate the procedure of finding the optimal total inventory cost, stock level and backlog level. Sensitivity analysis is carried out to demonstrate the effects of changing parameter values on the optimal solution of the system.

### Downloads

### References

Bellman R E, Zadeh LA., (1970). “Decision-Making in a Fuzzy Environment”. Manage Sci. 17(4), (pp. 141-164).

Dubois D, Prade H., (1978). “Operations on fuzzy numbers”. Int. J. Sys.t Sci. 9(6), (pp. 613-26).

He W., Wei H. E., Fuyuan X. U., (2013). “Inventory model for deteriorating items with time-dependent partial backlogging rate and inventory level-dependent demand rate”. Journal of Computer Applications. 33(8), (pp. 2390–3).

Wu k, Yao J-S., (2003).” Fuzzy inventory with backorder for fuzzy order quantity and fuzzy shortage quantity”. Eur. J. Oper. Res. 150(20), (pp. 320-52).

Wang X, Zhao R., (2007). “Fuzzy economic order quantity inventory models without backordering”. Tsinghua Sci. Technol. 12(1), (pp. 91-6).

Jinsong Hu, Hu J, Guo C, Xu R, Ji Y., (2010). “Fuzzy economic order quantity model with imperfect quality and service level”. In: 2010 Chinese Control and Decision Conference [Internet]. Available from: http://dx.doi.org/10.1109/ccdc.2010.5498441

N. K. Duari, T Chakrabarti. (2012), A Marketing Decision Problem in a Periodic Review Model with Exponential Demand and Shortages. IOSR Journal of Mathematics (IOSRJM). Vol. 1, Issue 6, (pp. 35-38).

Dutta D., Kumar P., (2015). “A partial backlogging inventory model for deteriorating items with time-varying demand and holding cost”. Int. J. Math. Oper. Res. 7(3), (pp. 281).

Roy A., Samanta G. P., (2009). “Fuzzy continuous review inventory model without backorder for deteriorating items”. Electronic Journal of Applied Statistical Analysis. 2, (pp. 58–66).

Jaggi K., et al., (2013). “Fuzzy inventory model for deteriorating items with time-varying demand and shortages”. American Journal of Operational Research. 2(6), (pp. 81– 92).

S. Saha, T. Chakrabarti. (2016) “A buyer vendor EOQ model with time varying holding cost involving lead time as a decision variable under an integrated Supply chain system”. International Journal of Sciences & Engineering Research. Vol. 7, Issue 1, (pp. 352-36)

N. K. Duari, T Chakrabarti. (2014), “An order level EOQ model for deteriorating items in a single warehouse system with price dependent demand and shortages”. American Journal of Engineering Research. Vol. 03, Issue 04, (pp. 11-16).

S. Shee, T. Chakrabarti. (2020), “A Fuzzy Two-Echelon Supply Chain model for deteriorating items with time varying holding cost involving lead time as a decision variable”. Book title: Optimization and Inventory Management. Springer Nature Singapore Pte Ltd. Chapter 21. Copyright year: 2020.

S. Shee, T. Chakrabarti. (2020), “Fuzzy Inventory Model for Deteriorating Items in a Supply Chain System with Time Dependent Demand Rate”. 2nd International Journal of Engineering Applied Sciences and Technology, Vol. 5, Issue 1, Page: 558-569, 2020.

Roy A., Samanta G. P., (2009). “Fuzzy continuous review inventory model without backorder for deteriorating items”. Electronic Journal of Applied Statistical Analysis. 2, (pp. 58–66).

Jaggi K., et al., (2013). “Fuzzy inventory model for deteriorating items with time-varying demand and shortages”. American Journal of Operational Research. 2(6), (pp. 81– 92).

Yao J. S. and Chiang J., (2003). “Inventory without backorder with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance”. European Journal of Operational Research. 148(2), (pp. 401–409).

Wang X., Tang W. and Zhao R., (2007). “Fuzzy economic order quantity inventory models without backordering”. Tsinghua Science and Technology. 12(1), (pp. 91–96).

Kao C. and Hsu W. K., (2002). “A single-period inventory model with fuzzy demand”. Computers & Mathematics with Applications. 43(6-7), (pp. 841–848).

Dutta P., Chakraborty D. and Roy A. R., (2005). “A single-period inventory model with fuzzy random variable demand”. Mathematical and Computer Modelling. 41(8-9), (pp. 915–922).

Bera U. K., Bhunia A. K., Maiti M., (2013). “Optimal partial backordering two-storage inventory model for deteriorating items with variable demand”. Int. J. Oper. Res. 16(1), (pp. 96).

He W., Wei H. E., Fuyuan X. U., (2013). “Inventory model for deteriorating items with time-dependent partial backlogging rate and inventory level-dependent demand rate”. Journal of Computer Applications. 33(8), (pp. 2390–3).

Dutta D., Kumar P., (2015). “A partial backlogging inventory model for deteriorating items with time-varying demand and holding cost”. Int. J. Math. Oper. Res. 7(3): (pp. 281).

Mishra N., Mishra S. P., Mishra S., Panda J., Misra U. K., (2015). “Inventory model of deteriorating items for linear holding cost with time dependent demand”. Mathematical Journal of Interdisciplinary Sciences. 4(1), (pp. 29–36).

Priyan S., Manivannan P., (2017). “Optimal inventory modelling of supply chain system involving quality inspection errors and fuzzy effective rate”. Opsearch. 54, (pp. 21-43)

*International Journal for Research in Applied Sciences and Biotechnology*,

*8*(5), 140-146. https://doi.org/10.31033/ijrasb.8.5.20

Copyright (c) 2021 International Journal for Research in Applied Sciences and Biotechnology

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.