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

• * 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
• 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.

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

 Citation: • • 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

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

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]

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

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

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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