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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2022, Volume 18, Issue 4: 2289-2318. 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
^* 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

Published: July 2022

Abstract / Introduction Full Text(HTML) Figure(30) / Table(6) Related Papers Cited by

Abstract

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:

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

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

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Figure 2. Four possibilities in the first stage inspection process (lot size)

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Figure 3. Two possibilities in the second stage inspection process (re-workable items)

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Figure 4. The flow diagram in the regular ordering process

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Figure 5. The flow diagram in the regular rework process

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Figure 6. Schematic behavior of the objective function

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Figure 7. The relationship between the fraction of defectives and the annual profit

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Figure 8. The relationship between the fraction of defectives and the order/lot size

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Figure 9. The relationship between the fraction of defectives and the time cycle

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Figure 10. The relationship between the fraction of Type-I error and the annual profit

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Figure 11. The relationship between the fraction of Type-I error and the order/lot size

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Figure 12. The relationship between the fraction of Type-I error and the time cycl

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Figure 13. The relationship between the fraction of Type-II error and the annual profit

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Figure 14. The relationship between the fraction of Type-II error and the order/lot size

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Figure 15. The relationship between the fraction of Type-II error and the cycle 17:33:11

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Figure 16. The relationship between the fraction of rework and the annual profit

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Figure 17. The relationship between the fraction of rework and the order/lot size

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Figure 18. The relationship between time cycle and the fraction of rework

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Figure 19. Schematic display of the optimal point at $ \alpha = 0 $

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Figure 20. Schematic display of the optimal point at $ \alpha = 0.1 $

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Figure 21. Schematic display of the optimal point at $ \alpha = 0.2 $

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Figure 22. Schematic display of the optimal point at $ \alpha = 0.3 $

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Figure 23. Schematic display of the optimal point at $ \alpha = 0.4 $

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Figure 24. Schematic display of the optimal point at $ \alpha = 0.5 $

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Figure 25. Schematic display of the optimal point at $ \alpha = 0.6 $

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Figure 26. Schematic display of the optimal point at $ \alpha = 0.7 $

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Figure 27. Schematic display of the optimal point at $ \alpha = 0.8 $

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Figure 28. Schematic display of the optimal point at $ \alpha = 0.9 $

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Figure 29. Schematic display of the optimal point at $ \alpha = 1 $

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Figure 30. Membership function of output parameters - optimal solution

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Table 2. Optimal solution of Illustrated example

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

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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

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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]

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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

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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

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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).

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