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July  2022, 18(4): 2289-2318. doi: 10.3934/jimo.2021068

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

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 Published  July 2022 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.

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 and Management Optimization, 2022, 18 (4) : 2289-2318. doi: 10.3934/jimo.2021068
References:
[1]

M. Ben-DayaM.A. Darwish and A. Rahim, Two-stage imperfect production systems with inspection errors, Int. J. Oper. Quant. Manag., 9 (2003), 117-131. 

[2]

B. Bharani, Fuzzy economic production quantity model for a sustainable system via geometric programming, J. Glob. Res. Math. Arch., 5 (2018), 26-33. 

[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.

[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.

[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.

[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. 

[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.

[8]

L. E. Cárdenas-Barrón, Observation on: "Economic production quantity model for items with imperfect quality", Int. J. Production Economics, 64 (2000), 59-64. 

[9]

L. E. Cárdenas-Barrón, The economic production quantity (EPQ) with shortage derived algebraically, Int. J. Prod. Econ., 70 (2001), 289-292. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

S. W. ChiuS.-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.

[18]

S. W. ChiuY. 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. 

[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.

[20]

Y.-S. P. ChiuK.-K. ChenF.-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.

[21]

Y.-S. P. ChiuK.-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.

[22]

Y.-S. P. ChiuS.-C. LiuC.-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.

[23]

S. W. ChiuC.-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.

[24]

K. J. ChungC. 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.

[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.

[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.

[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.

[28]

T. GaraiD. 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.

[29]

S. K. Goyal, Economic ordering policy for a product with periodic price changes, in Proceeding Third Int. Symp. Invent., Budapest, Hungry, 1984.

[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.

[31]

R. W. Grubbström, Material requirements planning and manufacturing resource planning, Int. Encycl. Bus. Manag., 4 (1996) 3400–3420.

[32]

R. W. Grubbström and A. Erdem, The EOQ with backlogging derived without derivatives, Int. J. Prod. Econ., 59 (1999), 529-530. 

[33]

S. HarbiM. 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. 

[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.

[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.

[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. 

[37]

A. M. M. JamalB. 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.

[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.

[39]

C. KahramanB. Öztayşiİ 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.

[40]

M. KhanM. 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.

[41]

M. KhanM. Y. JaberA. 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.

[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.

[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.

[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.

[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.

[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.

[47]

M. NajafiA. 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. 

[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.

[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.

[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.

[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.

[52]

R. PatroM. 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.

[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.

[54]

F. RahmanniyayJ. 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.

[55]

A. RaoufJ. K. Jain and P. T. Sathe, A cost-minimization model for multicharacteristic component inspection, AIIE Trans., 15 (1983), 187-194.  doi: 10.1080/05695558308974633.

[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.

[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.

[58]

J. SadeghiS. T. A. NiakiM. 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.

[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.

[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.

[61]

B. SarkarL. E. Cárdenas-BarrónM. 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.

[62]

B. SinhaA. 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.

[63]

J. Taheri-TolgariA. 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.

[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.

[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.

[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.

[67]

A. A. TaleizadehH.-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.

[68]

A. A. TaleizadehS. 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.

[69]

A. A. TaleizadehH. 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.

[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.

[71]

G. Treviño-Garzaand 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.

[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.

[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.

[74]

H. M. WeeJ. 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.

[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.

[76]

S. H. YooD. 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.

[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.

[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.

[79]

H.-J. Zimmermann, Fuzzy set theory-and its applications, Springer Science & Business Media, 2011. doi: 10.1007/978-94-010-0646-0.

show all references

References:
[1]

M. Ben-DayaM.A. Darwish and A. Rahim, Two-stage imperfect production systems with inspection errors, Int. J. Oper. Quant. Manag., 9 (2003), 117-131. 

[2]

B. Bharani, Fuzzy economic production quantity model for a sustainable system via geometric programming, J. Glob. Res. Math. Arch., 5 (2018), 26-33. 

[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.

[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.

[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.

[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. 

[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.

[8]

L. E. Cárdenas-Barrón, Observation on: "Economic production quantity model for items with imperfect quality", Int. J. Production Economics, 64 (2000), 59-64. 

[9]

L. E. Cárdenas-Barrón, The economic production quantity (EPQ) with shortage derived algebraically, Int. J. Prod. Econ., 70 (2001), 289-292. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

S. W. ChiuS.-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.

[18]

S. W. ChiuY. 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. 

[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.

[20]

Y.-S. P. ChiuK.-K. ChenF.-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.

[21]

Y.-S. P. ChiuK.-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.

[22]

Y.-S. P. ChiuS.-C. LiuC.-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.

[23]

S. W. ChiuC.-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.

[24]

K. J. ChungC. 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.

[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.

[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.

[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.

[28]

T. GaraiD. 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.

[29]

S. K. Goyal, Economic ordering policy for a product with periodic price changes, in Proceeding Third Int. Symp. Invent., Budapest, Hungry, 1984.

[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.

[31]

R. W. Grubbström, Material requirements planning and manufacturing resource planning, Int. Encycl. Bus. Manag., 4 (1996) 3400–3420.

[32]

R. W. Grubbström and A. Erdem, The EOQ with backlogging derived without derivatives, Int. J. Prod. Econ., 59 (1999), 529-530. 

[33]

S. HarbiM. 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. 

[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.

[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.

[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. 

[37]

A. M. M. JamalB. 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.

[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.

[39]

C. KahramanB. Öztayşiİ 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.

[40]

M. KhanM. 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.

[41]

M. KhanM. Y. JaberA. 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.

[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.

[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.

[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.

[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.

[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.

[47]

M. NajafiA. 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. 

[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.

[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.

[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.

[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.

[52]

R. PatroM. 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.

[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.

[54]

F. RahmanniyayJ. 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.

[55]

A. RaoufJ. K. Jain and P. T. Sathe, A cost-minimization model for multicharacteristic component inspection, AIIE Trans., 15 (1983), 187-194.  doi: 10.1080/05695558308974633.

[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.

[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.

[58]

J. SadeghiS. T. A. NiakiM. 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.

[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.

[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.

[61]

B. SarkarL. E. Cárdenas-BarrónM. 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.

[62]

B. SinhaA. 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.

[63]

J. Taheri-TolgariA. 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.

[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.

[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.

[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.

[67]

A. A. TaleizadehH.-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.

[68]

A. A. TaleizadehS. 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.

[69]

A. A. TaleizadehH. 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.

[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.

[71]

G. Treviño-Garzaand 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.

[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.

[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.

[74]

H. M. WeeJ. 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.

[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.

[76]

S. H. YooD. 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.

[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.

[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.

[79]

H.-J. Zimmermann, Fuzzy set theory-and its applications, Springer Science & Business Media, 2011. doi: 10.1007/978-94-010-0646-0.

Figure 1.  The relationship between different levels of the inventory system over time
Figure 2.  Four possibilities in the first stage inspection process (lot size)
Figure 3.  Two possibilities in the second stage inspection process (re-workable items)
Figure 4.  The flow diagram in the regular ordering process
Figure 5.  The flow diagram in the regular rework process
Figure 6.  Schematic behavior of the objective function
Figure 7.  The relationship between the fraction of defectives and the annual profit
Figure 8.  The relationship between the fraction of defectives and the order/lot size
Figure 9.  The relationship between the fraction of defectives and the time cycle
Figure 10.  The relationship between the fraction of Type-I error and the annual profit
Figure 11.  The relationship between the fraction of Type-I error and the order/lot size
Figure 12.  The relationship between the fraction of Type-I error and the time cycl
Figure 13.  The relationship between the fraction of Type-II error and the annual profit
Figure 14.  The relationship between the fraction of Type-II error and the order/lot size
Figure 15.  The relationship between the fraction of Type-II error and the cycle 17:33:11
Figure 16.  The relationship between the fraction of rework and the annual profit
Figure 17.  The relationship between the fraction of rework and the order/lot size
Figure 18.  The relationship between time cycle and the fraction of rework
Figure 19.  Schematic display of the optimal point at $ \alpha = 0 $
Figure 20.  Schematic display of the optimal point at $ \alpha = 0.1 $
Figure 21.  Schematic display of the optimal point at $ \alpha = 0.2 $
Figure 22.  Schematic display of the optimal point at $ \alpha = 0.3 $
Figure 23.  Schematic display of the optimal point at $ \alpha = 0.4 $
Figure 24.  Schematic display of the optimal point at $ \alpha = 0.5 $
Figure 25.  Schematic display of the optimal point at $ \alpha = 0.6 $
Figure 26.  Schematic display of the optimal point at $ \alpha = 0.7 $
Figure 27.  Schematic display of the optimal point at $ \alpha = 0.8 $
Figure 28.  Schematic display of the optimal point at $ \alpha = 0.9 $
Figure 29.  Schematic display of the optimal point at $ \alpha = 1 $
Figure 30.  Membership function of output parameters - optimal solution
Table 2.  Optimal solution of Illustrated example
Parameter/unit $ y^{*} $ $ T_{1}^{*} $ $ T_{2}^{*} $ $ T_{3}^{*} $ $ T^{*} $ $ E[TPU] $
unit/cycle day day day day $/year
Value 1611 5.88 0.03 5.77 11.68 1079591
Parameter/unit $ y^{*} $ $ T_{1}^{*} $ $ T_{2}^{*} $ $ T_{3}^{*} $ $ T^{*} $ $ E[TPU] $
unit/cycle day day day day $/year
Value 1611 5.88 0.03 5.77 11.68 1079591
Table 3.  General data - input parameters
Fuzzy number:$ [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
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
Table 4.  Summary of the model results presented in fuzzy mode
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]
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 $level 0 [102631,2434893] [1586,1980]
0.1 [183152,2282247] [1587,1918]
0.2 [267488,2133404] [1588,1865]
0.3 [355640,1988365] [1590,1865]
0.4 [447608,1847131] [1592,1778]
0.5 [543393,1709699] [1594,1742]
0.6 [642996,1576071] [1596,1710]
0.7 [746417,1446247] [1599,1682]
0.8 [853656,1320226] [1602,1656]
0.9 [964714,1198007] [1606,1632]
1 [1079592,1079592] [1611,1611]
Table 5.  Summary of the model results presented in crisp mode
Defuzzification method
Variable Signed distance/$ 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
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
Table 6.  Summary of the model results presented in crisp mode
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
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
Table 7.  General data -comparing the results obtained at the fuzzy mode and the crisp mode
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 $: Almost equal to).
Type-1 fuzzy number
Crisp mode Signed distance/$ 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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