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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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

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 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.

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

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References:

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[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.
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[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.
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[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.
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[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. 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.
[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.
[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. 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.
[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.
[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.
[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.
[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.
[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. 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.
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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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American Institute of Mathematical Sciences

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

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