Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach

Javad Taheri; Abolfazl Mirzazadeh

doi:10.3934/jimo.2021068

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Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach

Javad Taheri and Abolfazl Mirzazadeh ^,

Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

^* Corresponding author: E-mail: mirzazadeh@khu.ac.ir, Tel: +98-21-8883-0891, Fax: +98-21-8883-0891, Address: Faculty of Engineering, Department of Industrial Engineering, Mofatteh Ave., Tehran, Iran

Received August 2020 Revised January 2021 Early access April 2021

In this paper, a new approach was applied to a single-item single-source ($ SISS $) system at the "$ EOQ-type $" mode considering imperfect items and uncertainty environment. The mentioned method was intended to produce an optimum order/production quantity as well as taking care of imperfect processes. The imperfect proportion of the received lot size was described by an imperfect inspection process. That is, two-way inspection errors may be committed by the inspector as separate items. Thus, this survey was aimed to maximize the benefit in the traditional inventory systems. The incorporation of both defects and defective classifications (Type-$ I\&II $ errors) was illustrated, in a way that the defects were returned by the consumers. Moreover, this inventory model had an extra step in the scope of inspection; which occurred after the rework process with no inspection error. To get closer to the practical circumstances and to consider the uncertainty, the model was formulated in the fuzzy environment. The demand, rework, and inspection rates of the inventory system were considered as the triangular fuzzy numbers where the output factors of the inventory system were obtained via nonlinear parametric programming and Zadeh's extension principle. Finally, this scenario was illustrated through a mathematical model. The concavity of the objective function was also calculated and the total profit function was presented to clarify the solution procedure by numerical examples.

Keywords: Imperfect item, fuzzy number, inspection errors, $ EOQ $, Zadeh's extension principle.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Javad Taheri, Abolfazl Mirzazadeh. Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021068

References:

[1]	M. Ben-Daya, M.A. Darwish and A. Rahim, Two-stage imperfect production systems with inspection errors, Int. J. Oper. Quant. Manag., 9 (2003), 117-131. Google Scholar
[2]	B. Bharani, Fuzzy economic production quantity model for a sustainable system via geometric programming, J. Glob. Res. Math. Arch., 5 (2018), 26-33. Google Scholar
[3]	L. E. Cárdenas-Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Appl. Math. Model., 35 (2011), 2394-2407. doi: 10.1016/j.apm.2010.11.053. Google Scholar
[4]	L. E. Cárdenas-Barrón, An easy method to derive EOQ and EPQ inventory models with backorders, Comput. Math. with Appl., 59 (2010), 948-952. doi: 10.1016/j.camwa.2009.09.013. Google Scholar
[5]	L. E. Cárdenas-Barrón, A simple method to compute economic order quantities: Some observations, Appl. Math. Model., 34 (2010), 1684-1688. doi: 10.1016/j.apm.2009.08.024. Google Scholar
[6]	L. E. Cárdenas-Barrón, Optimal manufacturing batch size with rework in a single-stage production system-a simple derivation, Comput. Ind. Eng., 55 (2008), 758-765. Google Scholar
[7]	L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Comput. Ind. Eng., 57 (2009), 1105-1113. doi: 10.1016/j.cie.2009.04.020. Google Scholar
[8]	L. E. Cárdenas-Barrón, Observation on: "Economic production quantity model for items with imperfect quality", Int. J. Production Economics, 64 (2000), 59-64. Google Scholar
[9]	L. E. Cárdenas-Barrón, The economic production quantity (EPQ) with shortage derived algebraically, Int. J. Prod. Econ., 70 (2001), 289-292. Google Scholar
[10]	H. C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Comput. Oper. Res., 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. Google Scholar
[11]	H.-C. Chang and C.-H. Ho, Exact closed-form solutions for "optimal inventory model for items with imperfect quality and shortage backordering", Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. Google Scholar
[12]	S.-P. Chen, Parametric nonlinear programming approach to fuzzy queues with bulk service, Eur. J. Oper. Res., 163 (2005), 434-444. doi: 10.1016/j.ejor.2003.10.041. Google Scholar
[13]	S.-P. Chen, Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method, Eur. J. Oper. Res., 177 (2007), 445-457. doi: 10.1016/j.ejor.2005.09.040. Google Scholar
[14]	T. C. E. Cheng, An economic order quantity model with demand-dependent unit production cost and imperfect production processes, IIE Trans., 23 (1991), 23-28. doi: 10.1080/07408179108963838. Google Scholar
[15]	S. W. Chiu, Robust planning in optimization for production system subject to random machine breakdown and failure in rework, Comput. Oper. Res., 37 (2010), 899-908. doi: 10.1016/j.cor.2009.03.016. Google Scholar
[16]	Y. P. Chiu, Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging, Eng. Optim., 35 (2003), 427-437. doi: 10.1080/03052150310001597783. Google Scholar
[17]	S. W. Chiu, S.-L. Wang and Y.-S.P. Chiu, Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns, Eur. J. Oper. Res., 180 (2007), 664-676. doi: 10.1016/j.ejor.2006.05.005. Google Scholar
[18]	S. W. Chiu, Y. P. Chiu and B. P. Wu, An economic production quantity model with the steady production rate of scrap items, J. Chaoyang Univ. Technol., 8 (2003), 225-235. Google Scholar
[19]	S. W. Chiu, Production lot size problem with failure in repair and backlogging derived without derivatives, Eur. J. Oper. Res., 188 (2008), 610-615. doi: 10.1016/j.ejor.2007.04.049. Google Scholar
[20]	Y.-S. P. Chiu, K.-K. Chen, F.-T. Cheng and M.-F. Wu, Optimization of the finite production rate model with scrap, rework and stochastic machine breakdown, Comput. Math. with Appl., 59 (2010), 919-932. doi: 10.1016/j.camwa.2009.10.001. Google Scholar
[21]	Y.-S. P. Chiu, K.-K. Chen and C.-K. Ting, Replenishment run time problem with machine breakdown and failure in rework, Expert Syst. Appl., 39 (2012), 1291-1297. doi: 10.1016/j.eswa.2011.08.005. Google Scholar
[22]	Y.-S. P. Chiu, S.-C. Liu, C.-L. Chiu and H.-H. Chang, Mathematical modeling for determining the replenishment policy for EMQ model with rework and multiple shipments, Math. Comput. Model., 54 (2011), 2165-2174. doi: 10.1016/j.mcm.2011.05.025. Google Scholar
[23]	S. W. Chiu, C.-K. Ting and Y.-S.P. Chiu, Optimal production lot sizing with rework, scrap rate, and service level constraint, Math. Comput. Model., 46 (2007), 535-549. doi: 10.1016/j.mcm.2006.11.031. Google Scholar
[24]	K. J. Chung, C. C. Her and S. D. Lin, A two-warehouse inventory model with imperfect quality production processes, Comput. Ind. Eng., 56 (2009), 193-197. doi: 10.1016/j.cie.2008.05.005. Google Scholar
[25]	K.-J. Chung and Y.-F. Huang, Retailer's optimal cycle times in the EOQ model with imperfect quality and a permissible credit period, Qual. Quant., 40 (2006), 59-77. doi: 10.1007/s11135-005-5356-z. Google Scholar
[26]	K.-J. Chung, Bounds for production lot sizing with machine breakdowns, Comput. Ind. Eng., 32 (1997), 139-144. doi: 10.1016/S0360-8352(96)00207-0. Google Scholar
[27]	L. R. A. Cunha, A. P. S. Delfino, K. A. dos Reis and A. Leiras, Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches, Int. J. Prod. Res., (2018), 1–15. doi: 10.1080/00207543.2018.1445878. Google Scholar
[28]	T. Garai, D. Chakraborty and T. K. Roy, Multi-objective Inventory Model with Both Stock-Dependent Demand Rate and Holding Cost Rate Under Fuzzy Random Environment, Ann. Data Sci., 6 (2019), 61-81. doi: 10.1007/s40745-018-00186-0. Google Scholar
[29]	S. K. Goyal, Economic ordering policy for a product with periodic price changes, in Proceeding Third Int. Symp. Invent., Budapest, Hungry, 1984. Google Scholar
[30]	S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality–a practical approach, Int. J. Prod. Econ., 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1. Google Scholar
[31]	R. W. Grubbström, Material requirements planning and manufacturing resource planning, Int. Encycl. Bus. Manag., 4 (1996) 3400–3420. Google Scholar
[32]	R. W. Grubbström and A. Erdem, The EOQ with backlogging derived without derivatives, Int. J. Prod. Econ., 59 (1999), 529-530. Google Scholar
[33]	S. Harbi, M. Bahroun and H. Bouchriha, How to estimate the supplier fill rate when the supply order and the supply lead-time are uncertain?, Int. J. Supply Oper. Manag., 5 (2018), 197-206. Google Scholar
[34]	P. A. Hayek and M. K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Prod. Plan. Control., 12 (2001), 584-590. doi: 10.1080/095372801750397707. Google Scholar
[35]	J.-S. Hu, R.-Q. Xu and C.-Y. Guo, Fuzzy economic production quantity models for items with imperfect quality, Int. J. Inf. Manag. Sci., (2011), 43–58. Google Scholar
[36]	M. W. Iqbal and B. Sarkar, A model for imperfect production system with probabilistic rate of imperfect production for deteriorating products, DJ J. Eng. Appl. Math., 4 (2018), 1-12. Google Scholar
[37]	A. M. M. Jamal, B. R. Sarker and S. Mondal, Optimal manufacturing batch size with rework process at a single-stage production system, Comput. Ind. Eng., 47 (2004), 77-89. doi: 10.1016/j.cie.2004.03.001. Google Scholar
[38]	P. Jawla and S. R. Singh, A production reliable model for imperfect items with random machine breakdown under learning and forgetting, in Optim. Invent. Manag., Springer, 2020, 93–117. Google Scholar
[39]	C. Kahraman, B. Öztayşi, İ U. Sarı and E. Turanoğlu, Fuzzy analytic hierarchy process with interval type-2 fuzzy sets, Knowledge-Based Syst., 59 (2014), 48-57. doi: 10.1016/j.knosys.2014.02.001. Google Scholar
[40]	M. Khan, M. Y. Jaber and M. I. M. M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96. doi: 10.1016/j.ijpe.2009.10.011. Google Scholar
[41]	M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items, Int. J. Prod. Econ., 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. Google Scholar
[42]	A. Kundu, P. Guchhait, B. Das and M. Maiti, A Multi-item EPQ Model with Variable Demand in an Imperfect Production Process, in Inf. Technol. Appl. Math., Springer Singapore, Singapore, 2019,217–233. doi: 10.1007/978-981-13-2402-4_1. Google Scholar
[43]	H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycle and inspection schedules in a production system, Manage. Sci., 33 (1987), 1125-1136. doi: 10.1287/mnsc.33.9.1125. Google Scholar
[44]	T.-Y. Lin and M.-T. Chen, An economic order quantity model with screening errors, returned cost, and shortages under quantity discounts, African J. Bus. Manag., 5 (2011), 1129-1135. doi: 10.5897/AJBM10.376. Google Scholar
[45]	J. Liu and H. Zheng, Fuzzy economic order quantity model with imperfect items, shortages and inspection errors, Syst. Eng. Procedia., 4 (2012), 282-289. doi: 10.1016/j.sepro.2011.11.077. Google Scholar
[46]	M. Mizumoto and K. Tanaka, Fuzzy sets and type 2 under algebraic product and algebraic sum, Fuzzy Sets Syst., 5 (1981) 277–290. doi: 10.1016/0165-0114(81)90056-7. Google Scholar
[47]	M. Najafi, A. Ghodratnama and H. R. Pasandideh, Solving a deterministic multi product single machine EPQ model withpartial backordering, scrapped products and rework, Int. J. Supply Oper. Manag., 5 (2018), 11-27. Google Scholar
[48]	A. H. Nobil, S. Tiwari and F. Tajik, Economic production quantity model considering warm-up period in a cleaner production environment, Int. J. Prod. Res., (2018), 1–14. doi: 10.1080/00207543.2018.1518608. Google Scholar
[49]	H. Öztürk, Optimal production run time for an imperfect production inventory system with rework, random breakdowns and inspection costs, Oper. Res., (2018), 1–38. Google Scholar
[50]	S. Pal and G. S. Mahapatra, A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages, Comput. Ind. Eng., 106 (2017), 299-314. doi: 10.1016/j.cie.2017.02.003. Google Scholar
[51]	S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154. doi: 10.1016/j.ijpe.2004.11.004. Google Scholar
[52]	R. Patro, M. M. Nayak and M. Acharya, An EOQ model for fuzzy defective rate with allowable proportionate discount, OPSEARCH., 56 (2019), 1-25. doi: 10.1007/s12597-018-00352-1. Google Scholar
[53]	S. Priyan, P. Mala and R. Gurusamy, Optimal inventory strategies for two-echelon supply chain system involving carbon emissions and fuzzy deterioration, Int. J. Logist. Syst. Manag., 37 (2020), 324. doi: 10.1504/IJLSM.2020.111386. Google Scholar
[54]	F. Rahmanniyay, J. Razmi and A. J. Yu, An interactive multi-objective fuzzy linear programming model for hub location problems to minimise cost and delay time in a distribution network, Int. J. Logist. Syst. Manag., 37 (2020), 79-105. doi: 10.1504/IJLSM.2020.109649. Google Scholar
[55]	A. Raouf, J. K. Jain and P. T. Sathe, A cost-minimization model for multicharacteristic component inspection, AIIE Trans., 15 (1983), 187-194. doi: 10.1080/05695558308974633. Google Scholar
[56]	S. Rani, R. Ali and A. Agarwal, Fuzzy inventory model for deteriorating items in a green supply chain with carbon concerned demand, OPSEARCH., 56 (2019) 91–122. doi: 10.1007/s12597-019-00361-8. Google Scholar
[57]	M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Trans., 18 (1986), 48-55. doi: 10.1080/07408178608975329. Google Scholar
[58]	J. Sadeghi, S. T. A. Niaki, M. R. Malekian and Y. Wang, A Lagrangian relaxation for a fuzzy random EPQ problem with shortages and redundancy allocation: two tuned meta-heuristics, Int. J. Fuzzy Syst., 20 (2018), 515-533. doi: 10.1007/s40815-017-0377-z. Google Scholar
[59]	M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. Google Scholar
[60]	R. Saranya and R. Varadarajan, A fuzzy inventory model with acceptable shortage using graded mean integration value method, in J. Phys. Conf. Ser., IOP Publishing, 2018, 12009. doi: 10.1088/1742-6596/1000/1/012009. Google Scholar
[61]	B. Sarkar, L. E. Cárdenas-Barrón, M. Sarkar and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, J. Manuf. Syst., 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001. Google Scholar
[62]	B. Sinha, A. Das and U. K. Bera, Profit maximization solid transportation problem with trapezoidal interval type-2 fuzzy numbers, Int. J. Appl. Comput. Math., 2 (2015), 41-56. doi: 10.1007/s40819-015-0044-8. Google Scholar
[63]	J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai, An inventory model for imperfect items under inflationary conditions with considering inspection errors, Comput. Math. with Appl., 63 (2012), 1007-1019. doi: 10.1016/j.camwa.2011.09.050. Google Scholar
[64]	J. Taheri-Tolgari, M. Mohammadi, B. Naderi, A. Arshadi-Khamseh and A. Mirzazadeh, An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment, J. Ind. Manag. Optim., (2018), 275–285. doi: 10.3934/jimo.2018097. Google Scholar
[65]	J. Taheri-Tolgari and A. Mirzazadeh, Determining Economic Order Quantity (EOQ) with Increase in a Known Price under Uncertainty through Parametric Non-Linear Programming approach, J. Qual. Eng. Prod. Optim., 4 (2019), 197-207. doi: 10.22070/jqepo.2019.4126.1098. Google Scholar
[66]	J. Taheri-Tolgari and A. Mirzazadeh, Multi-item single-source (MISS) production quantity model for imperfect items with rework failure, inspection errors, scraps, and backordering, Int. J. Ind. Syst. Eng., 1 (2020), 1. doi: 10.1504/IJISE.2020.10027983. Google Scholar
[67]	A. A. Taleizadeh, H.-M. Wee and S. J. Sadjadi, Multi-product production quantity model with repair failure and partial backordering, Comput. Ind. Eng., 59 (2010), 45-54. doi: 10.1016/j.cie.2010.02.015. Google Scholar
[68]	A. A. Taleizadeh, S. T. A. Niaki and A. A. Najafi, Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint, J. Comput. Appl. Math., 233 (2010), 1834-1849. doi: 10.1016/j.cam.2009.09.021. Google Scholar
[69]	A. A. Taleizadeh, H. M. Wee and S. G. Jalali-Naini, Economic production quantity model with repair failure and limited capacity, Appl. Math. Model., 37 (2013), 2765-2774. doi: 10.1016/j.apm.2012.06.006. Google Scholar
[70]	M. Tayyab and B. Sarkar, Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate, J. Clean. Prod., 139 (2016), 922-934. doi: 10.1016/j.jclepro.2016.08.062. Google Scholar
[71]	G. Treviño-Garza, and K. K. Castillo-Villar and L. E. Cárdenas-Barrón, Joint determination of the lot size and number of shipments for a family of integrated vendor-buyer systems considering defective products, Int. J. Syst. Sci., 46 (2015), 1705-1716. doi: 10.1080/00207721.2014.886750. Google Scholar
[72]	M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note, Comput. Ind. Eng., 58 (2010), 186-190. doi: 10.1016/j.apm.2010.02.004. Google Scholar
[73]	S. Wang Chiu, Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging, Appl. Stoch. Model. Bus. Ind., 23 (2007), 165-178. doi: 10.1002/asmb.664. Google Scholar
[74]	H. M. Wee, J. Yu and M. C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega, 35 (2007), 7-11. doi: 10.1016/j.omega.2005.01.019. Google Scholar
[75]	I. Yazici and C. Kahraman, VIKOR method using interval type two fuzzy sets, J. Intell. Fuzzy Syst., 29 (2015), 411-421. doi: 10.3233/IFS-151607. Google Scholar
[76]	S. H. Yoo, D. S. Kim and M.-S. S. Park, Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return, Int. J. Prod. Econ., 121 (2009), 255-265. doi: 10.1016/j.ijpe.2009.05.008. Google Scholar
[77]	L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inf. Sci. (Ny), 8 (1975), 199-249. doi: 10.1016/0020-0255(75)90036-5. Google Scholar
[78]	L. A. Zadeh, Is there a need for fuzzy logic?, Inf. Sci. (Ny)., 178 (2008), 2751-2779. doi: 10.1016/j.ins.2008.02.012. Google Scholar
[79]	H.-J. Zimmermann, Fuzzy set theory-and its applications, Springer Science & Business Media, 2011. doi: 10.1007/978-94-010-0646-0. Google Scholar

show all references

References:

[1]	M. Ben-Daya, M.A. Darwish and A. Rahim, Two-stage imperfect production systems with inspection errors, Int. J. Oper. Quant. Manag., 9 (2003), 117-131. Google Scholar
[2]	B. Bharani, Fuzzy economic production quantity model for a sustainable system via geometric programming, J. Glob. Res. Math. Arch., 5 (2018), 26-33. Google Scholar
[3]	L. E. Cárdenas-Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Appl. Math. Model., 35 (2011), 2394-2407. doi: 10.1016/j.apm.2010.11.053. Google Scholar
[4]	L. E. Cárdenas-Barrón, An easy method to derive EOQ and EPQ inventory models with backorders, Comput. Math. with Appl., 59 (2010), 948-952. doi: 10.1016/j.camwa.2009.09.013. Google Scholar
[5]	L. E. Cárdenas-Barrón, A simple method to compute economic order quantities: Some observations, Appl. Math. Model., 34 (2010), 1684-1688. doi: 10.1016/j.apm.2009.08.024. Google Scholar
[6]	L. E. Cárdenas-Barrón, Optimal manufacturing batch size with rework in a single-stage production system-a simple derivation, Comput. Ind. Eng., 55 (2008), 758-765. Google Scholar
[7]	L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Comput. Ind. Eng., 57 (2009), 1105-1113. doi: 10.1016/j.cie.2009.04.020. Google Scholar
[8]	L. E. Cárdenas-Barrón, Observation on: "Economic production quantity model for items with imperfect quality", Int. J. Production Economics, 64 (2000), 59-64. Google Scholar
[9]	L. E. Cárdenas-Barrón, The economic production quantity (EPQ) with shortage derived algebraically, Int. J. Prod. Econ., 70 (2001), 289-292. Google Scholar
[10]	H. C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Comput. Oper. Res., 31 (2004), 2079-2092. doi: 10.1016/S0305-0548(03)00166-7. Google Scholar
[11]	H.-C. Chang and C.-H. Ho, Exact closed-form solutions for "optimal inventory model for items with imperfect quality and shortage backordering", Omega, 38 (2010), 233-237. doi: 10.1016/j.omega.2009.09.006. Google Scholar
[12]	S.-P. Chen, Parametric nonlinear programming approach to fuzzy queues with bulk service, Eur. J. Oper. Res., 163 (2005), 434-444. doi: 10.1016/j.ejor.2003.10.041. Google Scholar
[13]	S.-P. Chen, Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method, Eur. J. Oper. Res., 177 (2007), 445-457. doi: 10.1016/j.ejor.2005.09.040. Google Scholar
[14]	T. C. E. Cheng, An economic order quantity model with demand-dependent unit production cost and imperfect production processes, IIE Trans., 23 (1991), 23-28. doi: 10.1080/07408179108963838. Google Scholar
[15]	S. W. Chiu, Robust planning in optimization for production system subject to random machine breakdown and failure in rework, Comput. Oper. Res., 37 (2010), 899-908. doi: 10.1016/j.cor.2009.03.016. Google Scholar
[16]	Y. P. Chiu, Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging, Eng. Optim., 35 (2003), 427-437. doi: 10.1080/03052150310001597783. Google Scholar
[17]	S. W. Chiu, S.-L. Wang and Y.-S.P. Chiu, Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns, Eur. J. Oper. Res., 180 (2007), 664-676. doi: 10.1016/j.ejor.2006.05.005. Google Scholar
[18]	S. W. Chiu, Y. P. Chiu and B. P. Wu, An economic production quantity model with the steady production rate of scrap items, J. Chaoyang Univ. Technol., 8 (2003), 225-235. Google Scholar
[19]	S. W. Chiu, Production lot size problem with failure in repair and backlogging derived without derivatives, Eur. J. Oper. Res., 188 (2008), 610-615. doi: 10.1016/j.ejor.2007.04.049. Google Scholar
[20]	Y.-S. P. Chiu, K.-K. Chen, F.-T. Cheng and M.-F. Wu, Optimization of the finite production rate model with scrap, rework and stochastic machine breakdown, Comput. Math. with Appl., 59 (2010), 919-932. doi: 10.1016/j.camwa.2009.10.001. Google Scholar
[21]	Y.-S. P. Chiu, K.-K. Chen and C.-K. Ting, Replenishment run time problem with machine breakdown and failure in rework, Expert Syst. Appl., 39 (2012), 1291-1297. doi: 10.1016/j.eswa.2011.08.005. Google Scholar
[22]	Y.-S. P. Chiu, S.-C. Liu, C.-L. Chiu and H.-H. Chang, Mathematical modeling for determining the replenishment policy for EMQ model with rework and multiple shipments, Math. Comput. Model., 54 (2011), 2165-2174. doi: 10.1016/j.mcm.2011.05.025. Google Scholar
[23]	S. W. Chiu, C.-K. Ting and Y.-S.P. Chiu, Optimal production lot sizing with rework, scrap rate, and service level constraint, Math. Comput. Model., 46 (2007), 535-549. doi: 10.1016/j.mcm.2006.11.031. Google Scholar
[24]	K. J. Chung, C. C. Her and S. D. Lin, A two-warehouse inventory model with imperfect quality production processes, Comput. Ind. Eng., 56 (2009), 193-197. doi: 10.1016/j.cie.2008.05.005. Google Scholar
[25]	K.-J. Chung and Y.-F. Huang, Retailer's optimal cycle times in the EOQ model with imperfect quality and a permissible credit period, Qual. Quant., 40 (2006), 59-77. doi: 10.1007/s11135-005-5356-z. Google Scholar
[26]	K.-J. Chung, Bounds for production lot sizing with machine breakdowns, Comput. Ind. Eng., 32 (1997), 139-144. doi: 10.1016/S0360-8352(96)00207-0. Google Scholar
[27]	L. R. A. Cunha, A. P. S. Delfino, K. A. dos Reis and A. Leiras, Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches, Int. J. Prod. Res., (2018), 1–15. doi: 10.1080/00207543.2018.1445878. Google Scholar
[28]	T. Garai, D. Chakraborty and T. K. Roy, Multi-objective Inventory Model with Both Stock-Dependent Demand Rate and Holding Cost Rate Under Fuzzy Random Environment, Ann. Data Sci., 6 (2019), 61-81. doi: 10.1007/s40745-018-00186-0. Google Scholar
[29]	S. K. Goyal, Economic ordering policy for a product with periodic price changes, in Proceeding Third Int. Symp. Invent., Budapest, Hungry, 1984. Google Scholar
[30]	S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality–a practical approach, Int. J. Prod. Econ., 77 (2002), 85-87. doi: 10.1016/S0925-5273(01)00203-1. Google Scholar
[31]	R. W. Grubbström, Material requirements planning and manufacturing resource planning, Int. Encycl. Bus. Manag., 4 (1996) 3400–3420. Google Scholar
[32]	R. W. Grubbström and A. Erdem, The EOQ with backlogging derived without derivatives, Int. J. Prod. Econ., 59 (1999), 529-530. Google Scholar
[33]	S. Harbi, M. Bahroun and H. Bouchriha, How to estimate the supplier fill rate when the supply order and the supply lead-time are uncertain?, Int. J. Supply Oper. Manag., 5 (2018), 197-206. Google Scholar
[34]	P. A. Hayek and M. K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Prod. Plan. Control., 12 (2001), 584-590. doi: 10.1080/095372801750397707. Google Scholar
[35]	J.-S. Hu, R.-Q. Xu and C.-Y. Guo, Fuzzy economic production quantity models for items with imperfect quality, Int. J. Inf. Manag. Sci., (2011), 43–58. Google Scholar
[36]	M. W. Iqbal and B. Sarkar, A model for imperfect production system with probabilistic rate of imperfect production for deteriorating products, DJ J. Eng. Appl. Math., 4 (2018), 1-12. Google Scholar
[37]	A. M. M. Jamal, B. R. Sarker and S. Mondal, Optimal manufacturing batch size with rework process at a single-stage production system, Comput. Ind. Eng., 47 (2004), 77-89. doi: 10.1016/j.cie.2004.03.001. Google Scholar
[38]	P. Jawla and S. R. Singh, A production reliable model for imperfect items with random machine breakdown under learning and forgetting, in Optim. Invent. Manag., Springer, 2020, 93–117. Google Scholar
[39]	C. Kahraman, B. Öztayşi, İ U. Sarı and E. Turanoğlu, Fuzzy analytic hierarchy process with interval type-2 fuzzy sets, Knowledge-Based Syst., 59 (2014), 48-57. doi: 10.1016/j.knosys.2014.02.001. Google Scholar
[40]	M. Khan, M. Y. Jaber and M. I. M. M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96. doi: 10.1016/j.ijpe.2009.10.011. Google Scholar
[41]	M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items, Int. J. Prod. Econ., 132 (2011), 1-12. doi: 10.1016/j.ijpe.2011.03.009. Google Scholar
[42]	A. Kundu, P. Guchhait, B. Das and M. Maiti, A Multi-item EPQ Model with Variable Demand in an Imperfect Production Process, in Inf. Technol. Appl. Math., Springer Singapore, Singapore, 2019,217–233. doi: 10.1007/978-981-13-2402-4_1. Google Scholar
[43]	H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycle and inspection schedules in a production system, Manage. Sci., 33 (1987), 1125-1136. doi: 10.1287/mnsc.33.9.1125. Google Scholar
[44]	T.-Y. Lin and M.-T. Chen, An economic order quantity model with screening errors, returned cost, and shortages under quantity discounts, African J. Bus. Manag., 5 (2011), 1129-1135. doi: 10.5897/AJBM10.376. Google Scholar
[45]	J. Liu and H. Zheng, Fuzzy economic order quantity model with imperfect items, shortages and inspection errors, Syst. Eng. Procedia., 4 (2012), 282-289. doi: 10.1016/j.sepro.2011.11.077. Google Scholar
[46]	M. Mizumoto and K. Tanaka, Fuzzy sets and type 2 under algebraic product and algebraic sum, Fuzzy Sets Syst., 5 (1981) 277–290. doi: 10.1016/0165-0114(81)90056-7. Google Scholar
[47]	M. Najafi, A. Ghodratnama and H. R. Pasandideh, Solving a deterministic multi product single machine EPQ model withpartial backordering, scrapped products and rework, Int. J. Supply Oper. Manag., 5 (2018), 11-27. Google Scholar
[48]	A. H. Nobil, S. Tiwari and F. Tajik, Economic production quantity model considering warm-up period in a cleaner production environment, Int. J. Prod. Res., (2018), 1–14. doi: 10.1080/00207543.2018.1518608. Google Scholar
[49]	H. Öztürk, Optimal production run time for an imperfect production inventory system with rework, random breakdowns and inspection costs, Oper. Res., (2018), 1–38. Google Scholar
[50]	S. Pal and G. S. Mahapatra, A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages, Comput. Ind. Eng., 106 (2017), 299-314. doi: 10.1016/j.cie.2017.02.003. Google Scholar
[51]	S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154. doi: 10.1016/j.ijpe.2004.11.004. Google Scholar
[52]	R. Patro, M. M. Nayak and M. Acharya, An EOQ model for fuzzy defective rate with allowable proportionate discount, OPSEARCH., 56 (2019), 1-25. doi: 10.1007/s12597-018-00352-1. Google Scholar
[53]	S. Priyan, P. Mala and R. Gurusamy, Optimal inventory strategies for two-echelon supply chain system involving carbon emissions and fuzzy deterioration, Int. J. Logist. Syst. Manag., 37 (2020), 324. doi: 10.1504/IJLSM.2020.111386. Google Scholar
[54]	F. Rahmanniyay, J. Razmi and A. J. Yu, An interactive multi-objective fuzzy linear programming model for hub location problems to minimise cost and delay time in a distribution network, Int. J. Logist. Syst. Manag., 37 (2020), 79-105. doi: 10.1504/IJLSM.2020.109649. Google Scholar
[55]	A. Raouf, J. K. Jain and P. T. Sathe, A cost-minimization model for multicharacteristic component inspection, AIIE Trans., 15 (1983), 187-194. doi: 10.1080/05695558308974633. Google Scholar
[56]	S. Rani, R. Ali and A. Agarwal, Fuzzy inventory model for deteriorating items in a green supply chain with carbon concerned demand, OPSEARCH., 56 (2019) 91–122. doi: 10.1007/s12597-019-00361-8. Google Scholar
[57]	M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Trans., 18 (1986), 48-55. doi: 10.1080/07408178608975329. Google Scholar
[58]	J. Sadeghi, S. T. A. Niaki, M. R. Malekian and Y. Wang, A Lagrangian relaxation for a fuzzy random EPQ problem with shortages and redundancy allocation: two tuned meta-heuristics, Int. J. Fuzzy Syst., 20 (2018), 515-533. doi: 10.1007/s40815-017-0377-z. Google Scholar
[59]	M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. Google Scholar
[60]	R. Saranya and R. Varadarajan, A fuzzy inventory model with acceptable shortage using graded mean integration value method, in J. Phys. Conf. Ser., IOP Publishing, 2018, 12009. doi: 10.1088/1742-6596/1000/1/012009. Google Scholar
[61]	B. Sarkar, L. E. Cárdenas-Barrón, M. Sarkar and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, J. Manuf. Syst., 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001. Google Scholar
[62]	B. Sinha, A. Das and U. K. Bera, Profit maximization solid transportation problem with trapezoidal interval type-2 fuzzy numbers, Int. J. Appl. Comput. Math., 2 (2015), 41-56. doi: 10.1007/s40819-015-0044-8. Google Scholar
[63]	J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai, An inventory model for imperfect items under inflationary conditions with considering inspection errors, Comput. Math. with Appl., 63 (2012), 1007-1019. doi: 10.1016/j.camwa.2011.09.050. Google Scholar
[64]	J. Taheri-Tolgari, M. Mohammadi, B. Naderi, A. Arshadi-Khamseh and A. Mirzazadeh, An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment, J. Ind. Manag. Optim., (2018), 275–285. doi: 10.3934/jimo.2018097. Google Scholar
[65]	J. Taheri-Tolgari and A. Mirzazadeh, Determining Economic Order Quantity (EOQ) with Increase in a Known Price under Uncertainty through Parametric Non-Linear Programming approach, J. Qual. Eng. Prod. Optim., 4 (2019), 197-207. doi: 10.22070/jqepo.2019.4126.1098. Google Scholar
[66]	J. Taheri-Tolgari and A. Mirzazadeh, Multi-item single-source (MISS) production quantity model for imperfect items with rework failure, inspection errors, scraps, and backordering, Int. J. Ind. Syst. Eng., 1 (2020), 1. doi: 10.1504/IJISE.2020.10027983. Google Scholar
[67]	A. A. Taleizadeh, H.-M. Wee and S. J. Sadjadi, Multi-product production quantity model with repair failure and partial backordering, Comput. Ind. Eng., 59 (2010), 45-54. doi: 10.1016/j.cie.2010.02.015. Google Scholar
[68]	A. A. Taleizadeh, S. T. A. Niaki and A. A. Najafi, Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint, J. Comput. Appl. Math., 233 (2010), 1834-1849. doi: 10.1016/j.cam.2009.09.021. Google Scholar
[69]	A. A. Taleizadeh, H. M. Wee and S. G. Jalali-Naini, Economic production quantity model with repair failure and limited capacity, Appl. Math. Model., 37 (2013), 2765-2774. doi: 10.1016/j.apm.2012.06.006. Google Scholar
[70]	M. Tayyab and B. Sarkar, Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate, J. Clean. Prod., 139 (2016), 922-934. doi: 10.1016/j.jclepro.2016.08.062. Google Scholar
[71]	G. Treviño-Garza, and K. K. Castillo-Villar and L. E. Cárdenas-Barrón, Joint determination of the lot size and number of shipments for a family of integrated vendor-buyer systems considering defective products, Int. J. Syst. Sci., 46 (2015), 1705-1716. doi: 10.1080/00207721.2014.886750. Google Scholar
[72]	M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note, Comput. Ind. Eng., 58 (2010), 186-190. doi: 10.1016/j.apm.2010.02.004. Google Scholar
[73]	S. Wang Chiu, Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging, Appl. Stoch. Model. Bus. Ind., 23 (2007), 165-178. doi: 10.1002/asmb.664. Google Scholar
[74]	H. M. Wee, J. Yu and M. C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega, 35 (2007), 7-11. doi: 10.1016/j.omega.2005.01.019. Google Scholar
[75]	I. Yazici and C. Kahraman, VIKOR method using interval type two fuzzy sets, J. Intell. Fuzzy Syst., 29 (2015), 411-421. doi: 10.3233/IFS-151607. Google Scholar
[76]	S. H. Yoo, D. S. Kim and M.-S. S. Park, Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return, Int. J. Prod. Econ., 121 (2009), 255-265. doi: 10.1016/j.ijpe.2009.05.008. Google Scholar
[77]	L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inf. Sci. (Ny), 8 (1975), 199-249. doi: 10.1016/0020-0255(75)90036-5. Google Scholar
[78]	L. A. Zadeh, Is there a need for fuzzy logic?, Inf. Sci. (Ny)., 178 (2008), 2751-2779. doi: 10.1016/j.ins.2008.02.012. Google Scholar
[79]	H.-J. Zimmermann, Fuzzy set theory-and its applications, Springer Science & Business Media, 2011. doi: 10.1007/978-94-010-0646-0. Google Scholar

Figure 1. The relationship between different levels of the inventory system over time

Parameter/unit	$ y^{*} $	$ T_{1}^{*} $	$ T_{2}^{*} $	$ T_{3}^{*} $	$ T^{*} $	$ E[TPU] $
Parameter/unit	unit/cycle	day	day	day	day	＄/year
Value	1611	5.88	0.03	5.77	11.68	1079591

		Fuzzy number:$ [a_{1}, a_{2}, a_{3}] $
General Data	Symbol	$ a_{1} $	$ a_{2} $	$ a_{3} $
Rework and inspection rate re-workable items	$ P_{R} $	65000	800000	1000000
Demand rate	$ D $	40000	50000	60000
Rework and inspection rate of serviceable items	$ P_{RS} $	520000	640000	800000
Screening rate for order size	$ x $	80000	100000	150000

		Left and right bound of $ E[TPU] $	Left and right bound of $ y $
		$ [E[TPU]_{\alpha}^{l}, E[TPU]_{\alpha}^{r}] $	$ [y_{\alpha}^{l}, y_{\alpha}^{r}] $
$ \alpha $level	0	[102631,2434893]	[1586,1980]
	0.1	[183152,2282247]	[1587,1918]
	0.2	[267488,2133404]	[1588,1865]
	0.3	[355640,1988365]	[1590,1865]
	0.4	[447608,1847131]	[1592,1778]
	0.5	[543393,1709699]	[1594,1742]
	0.6	[642996,1576071]	[1596,1710]
	0.7	[746417,1446247]	[1599,1682]
	0.8	[853656,1320226]	[1602,1656]
	0.9	[964714,1198007]	[1606,1632]
	1	[1079592,1079592]	[1611,1611]

	Defuzzification method
Variable	Signed distance/$ SD $	Centroid/$ C $
E[TPU]	1174177	1205705
$ y $	1697	1726
$ T $	12.67	13
$ T_{1} $	6.16	6.26
$ T_{2} $	0.031	0.032
$ T_{3} $	6.48	6.72

	Fuzzy mode: defuzzification method		Crisp mode
Variable	Signed distance/$ SD $	Centroid/$ C $
E[TPU]	1174177	1205705	1079591
$ y $	1697	1726	1611
E[TPU]/$ y $	692	699	670

	Type-1 fuzzy number
Crisp mode	Signed distance/$ SD $	Centroid/$ C $
$ E[TPU]=1079591 $	$ \uparrow $	$ \uparrow $
$ y=1611 $	$ \uparrow $	$ \uparrow $
$ T_{1}=5.88 $	$ \uparrow $	$ \uparrow $
$ T_{2}=0.03 $	$ \approx $	$ \downarrow $
$ T_{3}=5.77 $	$ \uparrow $	$ \approx $
$ T=11.68 $	$ \uparrow $	$ \uparrow $
Components: ($ \uparrow $: Increase, $ \downarrow $: Decrease and $ \approx $: Almost equal to).

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Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach

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