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Solving a constrained economic lot size problem by ranking efficient production policies
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
##### References:

show all references

##### References:
The relationship between different levels of the inventory system over time
Four possibilities in the first stage inspection process (lot size)
Two possibilities in the second stage inspection process (re-workable items)
The flow diagram in the regular ordering process
The flow diagram in the regular rework process
Schematic behavior of the objective function
The relationship between the fraction of defectives and the annual profit
The relationship between the fraction of defectives and the order/lot size
The relationship between the fraction of defectives and the time cycle
The relationship between the fraction of Type-I error and the annual profit
The relationship between the fraction of Type-I error and the order/lot size
The relationship between the fraction of Type-I error and the time cycl
The relationship between the fraction of Type-II error and the annual profit
The relationship between the fraction of Type-II error and the order/lot size
The relationship between the fraction of Type-II error and the cycle 17:33:11
The relationship between the fraction of rework and the annual profit
The relationship between the fraction of rework and the order/lot size
The relationship between time cycle and the fraction of rework
Schematic display of the optimal point at $\alpha = 0$
Schematic display of the optimal point at $\alpha = 0.1$
Schematic display of the optimal point at $\alpha = 0.2$
Schematic display of the optimal point at $\alpha = 0.3$
Schematic display of the optimal point at $\alpha = 0.4$
Schematic display of the optimal point at $\alpha = 0.5$
Schematic display of the optimal point at $\alpha = 0.6$
Schematic display of the optimal point at $\alpha = 0.7$
Schematic display of the optimal point at $\alpha = 0.8$
Schematic display of the optimal point at $\alpha = 0.9$
Schematic display of the optimal point at $\alpha = 1$
Membership function of output parameters - optimal solution
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
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
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]
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
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
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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