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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

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.

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
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References:
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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.   Google Scholar

[2]

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[3]

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[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]

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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]

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[13]

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[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]

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[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.   Google Scholar

[19]

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[20]

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[21]

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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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