A Fuzzy Inventory Model with Imperfect Items and Backorder with Allowable Proportionate Discount

A Fuzzy Inventory Model with Imperfect Items and Backorder with Allowable Proportionate Discount

Rojalini Patro Sujit Acharya Mitali Madhusmita NayakMilu Acharya 

Department of Mathematics, Siksha ‘O’ Anusandhan Deemed to be University, Bhubaneswar-751030, Odisha, India

Department of Management Studies, DDCE, Utkal University, Bhubaneswar – 751007, Odisha, India

Corresponding Author Email: 
mitalinayak7@gmail.com
Page: 
39-46
|
DOI: 
https://doi.org/10.18280/mmc_d.390106
Received: 
23 January 2018
|
Accepted: 
20 December 2018
|
Published: 
31 December 2018
| Citation

OPEN ACCESS

Abstract: 

This paper presents both crisp and fuzzy EOQ models for defective items present in each lot when shortages are allowed and backorder takes place. The aim of the work is to first construct an optimal order quantity for the crisp case and then to develop the corresponding fuzzy model. In contrast to the previous inventory models, an allowable proportionate discount is incorporated for the defective items present in each lot to provide a general framework to the model. The aim of the present paper is to find the optimal order size and the expected shortage level so as to obtain the optimum total profit for both the models. The necessary and sufficient conditions for the existence and uniqueness of the optimal solutions are derived and it is also shown that under certain conditions the crisp model boils down the traditional EOQ backorder formula. For the fuzzy case, triangular fuzzy numbers are used for the defective rates and for defuzzification signed distance method is used. Finally, numerical example is provided to illustrate the solution procedure and sensitivity analysis is performed on the results to analyze the effect of the variations taken place for the parameters involved in the model.

Keywords: 

inventory, imperfect quality, proportionate discount, backorder, EOQ

1. Introduction
2. The Mathematical Model
3. Numerical Results
4. Observations
5. Conclusion
  References

[1] Chang HC. (2004). An implication of fuzzy sets theory to the EOQ model with imperfect quality items. Computers and Operations Research 31(12): 2079-2092. https://doi.org/10.1016/S0305-0548(03)00166-7

[2] Chang HC, Ho CH. (2010). Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering. Omega 38(3-4): 233-237. https://doi.org/10.1016/j.omega.2009.09.006

[3] Chen SH, Chang SM. (2008). Optimization of fuzzy production inventory model with unrepairable defective products. International Journal of Production Economics 113(2): 887–894. https://doi.org/10.1016/j.ijpe.2007.11.004

[4] De PK, Rawat A. (2011). A fuzzy inventory model without shortages using triangular fuzzy number. Fuzzy Information and Engineering 3: 59-68. https://doi.org/10.1007/s12543-011-0066-9

[5] Eroglu A, Ozdemir, A. (2007). An economic order quantity model with defective items and shortages. International Journal of Production Economics 106: 544–49. https://doi.org/10.1016/j.ijpe.2006.06.015

[6] Goyal SK, Cardenas-Barron LE. (2002). ‘Note on: Economic production quantity model for items with imperfect quality- a practical approach. International Journal of Production Economics 77: 85-87. https://doi.org/10.1016/S0925-5273(01)00203-1.

[7] Harris FW. (1913). Operations and costs (Factory Management Series). A.W. Shaw Co, Chicago, pp. 18-52.

[8] Hsu JT, Hsu LF. (2012). A note on: ‘Optimal inventory model for items with imperfect quality and shortage backordering. International Journal of Industrial Engineering Computations 3(5): 939–948. https://doi.org/10.5267/j.ijiec.2012.05.007.

[9] Jagadeeswari J, Chenniappan PK. (2014). An order level inventory model for deteriorating items with time–quadratic demand and partial backlogging. Journal of Business and Management Sciences. 2(3): 79-82. https://doi.org/10.12691/jbms-2-3-3

[10] Khana A, Gautam P, Jaggi CK. (2017). Inventory Modelling for deteriorating imperfect quality items with selling price dependent demand and shortage backordering under credit financing. International Journal of Mathematical, Engineering and Management Sciences 2(2): 110-124.

[11] Kumar RS, Goswami A. (2013). Fuzzy stochastic EOQ inventory model for items with imperfect quality and shortages are backlogged. AMO - Advanced Modeling and Optimization 15(2): 261-279.

[12] Lee HL, Rossenblatt MJ. (1987). Simultaneously determination of production cycles and inspection schedules in a production system. Management Science 33: 1125-1137.

[13] Lee HM, Yao JS. (1998). Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operational Research 109(1): 203-211. https://doi.org/10.1016/S0377-2217(97)00200-2

[14] Maddah B, Jaber MY. (2008). Economic order quantity for items with imperfect quality: revisited. International Journal of Production Economics 112(2): 808-815. https://doi.org/10.1016/j.ijpe.2007.07.003

[15] Patro R, Acharya M, Nayak MM, Patnaik S. (2017). A fuzzy inventory model with time dependent Weibull deterioration, quadratic demand and partial backlogging. International Journal Management and Decision Making 16(3): 243-279. https://doi.org/10.1504/IJMDM.2017.085636

[16] Patro R, Acharya M, Nayak MM, Patnaik S. (2017). A fuzzy imperfect quality inventory model with proportionate discount under learning effect. International Journal of Intelligent Enterprise 4(4): 303-327.

[17] Porteus EL. (1986). Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research 34: 137-144. https://doi.org/10.1287/opre.34.1.137

[18] Rezaei J. (2005). Economic order quantity model with backorder for imperfect quality items. Proceeding of IEEE International Engineering Management Conference, pp. 466-470.

[19] Rossenblatt MJ, Lee HL. (1986). Economic production cycles with imperfect production processes. IIE Transactions 18: 48-55. https://doi.org/10.1080/07408178608975329

[20] Salameh MK, Jaber MY. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics Vol. 64: 59-64. https://doi.org/10.1016/S0925-5273(99)00044-4

[21] Sujatha J, Parvathi P. (2015). Fuzzy EOQ model for deteriorating items with weibull demand and partial backlogging under trade credit. International Journal of Informative and Futuristic Research 3(2): 526-537.

[22] Wahab MIM, Jaber MY. (2010). Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note. Computers and Industrial Engineering 58(1): 186-190. https://doi.org/10.1016/j.ijpe.2010.01.023.

[23] Wee HW, Yu J, Chen MC. (2007). Optimal inventory model for items with imperfect quality and shortage backordering. Omega 35(1): 7-11, 13. https://doi.org/10.5267/j.ijiec.2012.05.007

[24] Zadeh LA, Bellman RE. (1970). Decision making in a fuzzy environment. Management Science 17: 140-164. https://doi.org/10.1287/mnsc.17.4.B141