Research Method

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asymmetric quadratic Taguchi loss function and a piecewise linear loss function to determine the specification limits and process mean for a nominal-the-best quality variable with a normal distribution by minimizing the expected production cost per unit item. However, inspection error usually occurs when the product is measured by operators. To solve this situation, Chen and Khoo (2008) presented a profit model with inspection error for determining the most profitable specification and process mean.

The above papers discussed how to maintain the quality of products or improve the yield of products with minimal production cost or maximal production profit, in order to make sure that customers received conforming products. However, part of above papers determined economic control charts by minimizing the expected costs per unit time or maximizing net profits per unit time. The others determined the specification limits of outgoing quality by minimizing the expected loss per unit item or maximizing expected net profits per unit item. If we monitor the process using an economic control chart and inspect products at the same time, then we expect significant improvement in the yield of products but with the almost same expected costs per unit time as compared to only using an

economic control chart or inspection plan. So far, no paper considers using economic control charts and an inspection plan simultaneously. In this study, we hence determine to use an economic control chart and inspection plan at the same time to maintain the quality and improve the yield of products.

1.3 Research Method

In this study, for monitoring production process quality and having high outgoing quality, an economic X control chart and a complete inspection plan are adopted. We assume the interested quality characteristic is the larger-the-better variable following a Weibull distribution. A cost model, the expected production cost per unit time, includes all cost of process quality monitoring and all cost of the inspection plan in a cycle time, is derived. To determine the specification limit and an economic X control chart with minimal expected production costs per unit time, an optimization technique is applied. In

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Chapter 2, we construct an economic X control chart but without considering inspection.

We derived a model for expected cycle costs per unit time of using an economic X control chart. Then we use the optimization technique, routine “DEoptim” in R program, to

minimize the expected cost to determine the optimal design parameter for the economic X control chart. In Chapter 3, we construct an economic X control chart and consider the cost of outgoing quality under a complete inspection plan. We derived a model for the expected cost per unit time of using an economic X control chart and an inspection plan.

Then, we use the optimization technique, routine “DEoptim” in R program, to minimize the expected cost to determine the design parameters of the economic X control chart and the optimal specification limit for the outgoing products. Finally, we provide an example to illustrate the proposed approach and its application. In Chapter 4, we derived a model for expected cycle costs using an economic statistical X control chart without considering inspection plan. Then, we use the optimization technique, routine “DEoptim” in R program, to minimize the expected cost to determine the design parameters of the economic statistical

X control chart. In Chapter 5, we derived a model for expected cost per unit time of using an economic statistical X control chart and an inspection plan for outgoing quality. Then, we use the optimization technique, routine “DEoptim” in R program, to minimize the expected cost to determine the design parameters of the economic statistical X control chart and the optimal specification limit for outgoing quality. Finally, we provide an example to illustrate the determination of the optimal specification limit and design parameters of the economic statistical X control chart and its application. In Chapter 6, we consider one model for the expected cycle cost per unit time that only using the economic statistical X chart, the other model for the expected cycle cost per unit time that only using the inspection plan for outgoing quality. We compared the costs of using the economic statistical X chart with that using the inspection plan for outgoing quality to determine which quality

improvement approach should be adopted. In Chapter 7, we give the summary of this research and discuss directions for future study.

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ECONOMIC DESIGN OF A X CONTROL CHART FOR A PROCESS WITH WEIBULL DATA

In order to construct an economic X control chart, we first have to derive the distribution of sample mean. Then we derived the expected cycle time and expected cycle cost for defining our cost model (expected cycle cost per unit time).

2.1 Approximated In-Control Sampling Distribution of the X under Weibull Distribution

Filho and Yacoub (2006) proposed an approximate distribution to the sum of

independent random variables (X ), each _i X follows Weibull distribution. They used the _i moment-generating function to derive the approximate distribution of



 their derivation to obtain the approximate distribution of sample mean, X , for a sample with Weibull data. They used the numerical integration to calculate the exact sum pdf and cdf. Sample numerical examples are given to illustrate the excellent performance of the proposed approximations. The approximation is proved excellent by showing the

corresponding pdf and cdf curves for exact and approximate pdfs of the sum of two, three, and four i.i.d. unity-power Weibull variates, respectively, for different values of fading parameter. In the study, we use the simulation to show the adequacy of the approximate distribution of X . The procedure is shown as follows:

Step 1. Let X_I be the in-control Weibull distribution )

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Step 3. The derivation of moment-generating function of Z . _Ii



Step 4. The derivation of jth moment of XI

[ ] [ ¹] [ ^{1 2}] [ ¹]

The moment-based estimators for  and  can be obtained by solving Equations

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(2-5), (2-6), and (2-7).

We use the simulated data and the goodness of fit test to investigate whether our sampling mean distribution differed significantly from the approximate distribution of X .

The procedure is illustrate as follows.

Step 1. We simulate 12000 samples from Weibull (2, 2) as shown in Table 2-1. We divide X into 7 groups and count the observed frequency for each group (O ). Then calculate _i the probability in each group (P ) by using the approximate cdf of X in order to _i obtain the expected frequency counts (E ) at each group (see Table 2-2). _i

Step 2. Let H₀:the data meet the approximate X distribution v.s. H₁:not H . ₀ Step 3. We use the goodness of fit and calculate the test statistic using Equation 2-8.





 ⁷ 

1

2

2 ( )

i i

i i

E E

 O . (2-8)

Step 4. However, because the value of test statistic ² 10.867 is not fall in the rejection region (10.867₀²_.05,(6)12.592) at  0.05, so we do not reject H . ₀

It means the approximation distribution of Weibull’s sample mean is good.

Table 2- 1. The Simulated Data for Weibul(2, 2)

i-th sample Xi,1 X_i,2 X_i,3 X_i,4 X _i

1 1.754 1.145 1.127 0.765 1.198

2 0.442 2.290 3.793 2.761 2.321

3 0.984 2.203 1.247 2.453 1.722

     

1200 3.561 3.575 1.736 1.484 2.589

1201 2.690 3.249 2.198 1.112 2.312

     

2999 1.173 0.531 0.930 4.188 1.706

3000 0.541 3.741 2.389 1.377 2.012

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Table 2- 2. The Summary Table for Goodness of Fit

<0.8 0.8~1.2 1.2~1.6 1.6~2 2~2.4 2.4~2.8 >2.8

O i 21 280 772 985 665 209 68

P i 0.0076 0.0932 0.2754 0.3276 0.2016 0.0738 0.0209 E i 22.67 279.45 826.14 982.90 604.72 221.55 62.57

2.2 Approximated Out-of-Control Sampling Distribution of X under Weibull Distribution

Because we are considering the larger-the-better quality characteristic, the out-of-control distributions should have smaller expectation and the larger variance compared to those of in-control distributions.

Let X be the out-of-control Weibull distribution, that is _O ) variance also shifts. Therefore, the X control chart can detect both the shifts in process mean and variance.

2.3 Construction of Economic X Probability Chart Based on X Sampling Distribution

The X control chart is constructed by the approximated cdf, ()

XI

F , of XI. The control limits of the X probability control chart are

2)

‧

where UCL is the upper control limit, LCL is the lower control limit, and  is the probability of type I error or the false alarm rate. That is,

1 2

Then, we can obtain the solution of UCL and LCL of the economic X control chart by using the routine “uniroot” in the R program to find the root in Equation (2-9) and (2-10).

2.4 The Calculation of  and  approximated pdf of out-of-control sample mean, XO.

After the discussion about the determination of the control limits of the economic X control chart, we continued to implement how to determine the optimal sampling interval of economic X control chart. Hence, we have to derive the expected cycle time and the expected cycle cost.

2.5 Derivation of Expected Cycle Time

In Sections 2.5 and 2.6, we derive the cost model in a cycle by referring to Panagos, Heikes, and Montgomery (1985) and Lorenzen and Vance (1986).

Some assumptions in the model are as follows:

(1) There is only one assignable cause in the process.

(2) The time until the assignable cause appears is the exponential distribution with mean

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(3) The process is in an in-control state from the beginning.

(4) The process is not self-correcting; that is, if the assignable cause occurs and the process changes to an out-of-control state, the quality engineer would be required to perform an action that enables the process to return to the in-control condition.

(5) If an assignable cause appears, the two parameters of the Weibull distribution a_I and

bI change to a and _O b , respectively. _O

(6) A sample size n is obtained for every h of unit time, and the sample mean is plotted on the X control chart.

(7) The manufacturing might continue or stop when the assignable cause is located. We set two dummy variables to determine whether the process continues during locate and repair. However, in Chapter 2, 3, 4, and 5, we examine the process in which

manufacturing continues (₁₂ 1).

A cycle time is the sum of the in-control cycle time and out-of-control cycle time. The

in-control time includes the time until an assignable cause appears and the sampling and interpreting time. The out-of-control cycle time includes the expected time after the occurrence of the assignable cause and before the appearance of true alarm, the time to test and interpret the results, the time to discover the assignable cause, and the time to repair the process.

The expected cycle time can be divide into four parts, and is shown as follows:

(1) The expected time until the assignable cause occurs is



 ⁽¹ ^{1) 0}

1   sT

If production continues during searches If production ceases during searches

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(2) The expected time between the occurrence of the assignable cause and the next sample

is h, where (4) The expected time to detect a shift, discover the assignable cause, and repair the process

is

discover the assignable cause, and T₂ is the time to repair the process.

Hence, the expected cycle time for process control is shown in Equation 2-15 and the cycle time for continuous process is shown in Fig. 2-1.

2

Figure 2- 1.Continuous Process Cycle.

2.6 Derivation of the Expected Cycle Cost

In the cost model, we calculate the expected cost first, then divide it by the expected cycle time to obtain the expected cycle cost per unit time. Our target is to determine the

True Alarm Out-of-control In-control

h h h τ

Assignable Cause Search Repair

Production continues Occurs

‧

optimal sampling interval which minimizes the expected cycle cost per unit time given

4

n and  0.0027. In this study, we use the quadratic loss function to evaluate the expected quality cost for the in-control and out-of-control period.

The quadratic Taguchi loss function for the larger-the-better quality variable (See Taguchi (1984)) is

X2

L k , (2-16)

where X is the quality variable and k is the coefficient of the loss function.

The expected cycle cost is the sum of the expected in-control cycle cost and the expected out-of-control cycle cost. The expected in-control cycle cost includes the expected cost before the occurrence of the assignable cause, the expected sampling cost in the in-control cycle period, and the expected cost of investigating a false alarm. The expected out-of-control cycle cost includes the expected cost after the occurrence of the assignable cause and before the appearance of true alarm, the expected sampling cost in the out-of-control cycle time, and the expected cost of locating and repairing the assignable cause.

The expected cycle cost can be divided into five parts, and is shown as follows:

(1) The expected cost per unit item in the in-control period if the manager decides not to inspect the products is:



In the in-control period, the expected quality cost per unit time is

I nl R ,

where R is the number of products produced per unit of time. _n Hence, the expected quality cost for the in-control period is:

 1

I nl

R . (2-17)

(2) The expected cost per unit item in out-of-control period if the manager decides not to

‧

inspect the products is:



In the out-of-control period, the expected quality cost per unit of time is

O nl R .

Hence, the expected quality cost for the out-of-control period is:



where c is the fixed cost per unit sample, and d is the variable cost per unit sampled.

Hence, the expected sampling and testing cost per cycle time is

 

(4) The cost of investigating false alarms per cycle time is the expected number of false

alarm times the expected number of samples taken before the occurrence of the assignable cause and the cost of investigating a false alarm. That is,

λh

Y α . (2-20)

(5) The cost to locate and repair the assignable cause is

W. (2-21)

Therefore, the expected cycle cost is

 

Hence, the expected cycle cost per unit time is

ET

EA EC. (2-23)

2.7 Determination of the Optimum Sampling Interval of the Economic X Control Chart

The control limits of the economic X control chart and the optimal sampling interval

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are obtained by the following steps:

Step 1. Given k, R_n, , e, T₁T₂, c, d, W, Y, and (₁,₂).

Step 2. Set n4 and obtain the approximated in-control X distribution using the method proposed in Section 2.1.

Step 3. Set 0.0027 to solve the UCL and LCL of the economic X control chart using the method proposed in Section 2.3. Hence,

) 99865 . 0

1(

  XI

F

UCL (2-24)

 ^1(0.00135)

XI

F

LCL (2-25)

Step 4. We determine the optimal sampling interval by minimizing EA using the routine

“DEoptim” in the R program under the constraint 0.5h8.

2.8 Data Analysis and Resulting Comparison to Different Out-of-Control Distributions In order to investigate how the process parameters affect the design parameter (h) of the economic X control chart and the expected cycle cost, we conduct data analysis to find the significant process parameters for the design parameter of economic X control chart and the expected cycle cost. We consider 9 parameters each with two levels, and one parameter with three levels (see Table 2-3). We put these 10 parameters in each column of orthogonal array L32(2³¹) with 32 level combinations for 10 parameters (see Table 2-4).

Table 2- 3. The Level of Each Parameter

k R_n (₁,₂)  e T₁T₂ c d W Y

level 1 5 1000 (1.33, 2.41) 0.01 0.05 3 0.5 0.1 500 250

level 2 10 200 (1.43, 0.25) 0.05 0.5 20 5 1 100 70

level 3 (-49.89,430.35)

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Table 2- 4. Parameters for Each Level Combination Level

combination k Rn (₁,₂)  e T₁T₂ c d W Y 1 5 1000 (1.33, 2.41) 0.01 0.05 3 0.5 0.1 500 250 2 5 1000 (1.33, 2.41) 0.01 0.05 3 0.5 0.1 100 70 3 5 1000 (1.33, 2.41) 0.01 0.5 20 5 1 500 250 4 5 1000 (1.33, 2.41) 0.01 0.5 20 5 1 100 70 5 5 1000 (1.43, 0.25) 0.05 0.05 3 5 1 500 250 6 5 1000 (-49.89,430.35) 0.05 0.05 3 5 1 100 70 7 5 1000 (1.43, 0.25) 0.05 0.5 20 0.5 0.1 500 250 8 5 1000 (-49.89,430.35) 0.05 0.5 20 0.5 0.1 100 70 9 5 200 (1.33, 2.41) 0.05 0.05 20 0.5 1 500 70 10 5 200 (1.33, 2.41) 0.05 0.05 20 0.5 1 100 250 11 5 200 (1.33, 2.41) 0.05 0.5 3 5 0.1 500 70 12 5 200 (1.33, 2.41) 0.05 0.5 3 5 0.1 100 250 13 5 200 (1.43, 0.25) 0.01 0.05 20 5 0.1 500 70 14 5 200 (-49.89,430.35) 0.01 0.05 20 5 0.1 100 250 15 5 200 (1.43, 0.25) 0.01 0.5 3 0.5 1 500 70 16 5 200 (-49.89,430.35) 0.01 0.5 3 0.5 1 100 250 17 10 1000 (1.33, 2.41) 0.05 0.05 20 5 0.1 500 70 18 10 1000 (1.33, 2.41) 0.05 0.05 20 5 0.1 100 250 19 10 1000 (1.33, 2.41) 0.05 0.5 3 0.5 1 500 70 20 10 1000 (1.33, 2.41) 0.05 0.5 3 0.5 1 100 250 21 10 1000 (-49.89,430.35) 0.01 0.05 20 0.5 1 500 70 22 10 1000 (1.43, 0.25) 0.01 0.05 20 0.5 1 100 250 23 10 1000 (-49.89,430.35) 0.01 0.5 3 5 0.1 500 70 24 10 1000 (1.43, 0.25) 0.01 0.5 3 5 0.1 100 250 25 10 200 (1.33, 2.41) 0.01 0.05 3 5 1 500 250 26 10 200 (1.33, 2.41) 0.01 0.05 3 5 1 100 70 27 10 200 (1.33, 2.41) 0.01 0.5 20 0.5 0.1 500 250 28 10 200 (1.33, 2.41) 0.01 0.5 20 0.5 0.1 100 70 29 10 200 (-49.89,430.35) 0.05 0.05 3 0.5 0.1 500 250 30 10 200 (1.43, 0.25) 0.05 0.05 3 0.5 0.1 100 70 31 10 200 (-49.89,430.35) 0.05 0.5 20 5 1 500 250 32 10 200 (1.43, 0.25) 0.05 0.5 20 5 1 100 70

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Table 2-5 shows the optimal solutions for these 32 combinations of 10 parameters that we set n = 4, 0.0027, a_I 2, and b_I 2.

Table 2- 5. Optimal Solutions for Each Level Combination Level

Combination h^* LCL* UCL* EA^* β^*

1 0.5 0.613 3.332 297625142.1 0.2063 2 0.5 0.613 3.332 297625137.3 0.2063 3 0.5 0.613 3.332 1574327284.3 0.2063 4 0.5 0.613 3.332 1574327280.3 0.2063 5 0.5 0.613 3.332 118455.9 0.0788 6 0.5 0.613 3.332 825909026.5 0.1987 7 0.5 0.613 3.332 212100.0 0.0788 8 0.5 0.613 3.332 2875586117.2 0.1987 9 0.5 0.613 3.332 873176187.5 0.2063 10 0.5 0.613 3.332 873176178.1 0.2063 11 0.5 0.613 3.332 365027564.5 0.2063 12 0.5 0.613 3.332 365027549.5 0.2063 13 0.5 0.613 3.332 24755.9 0.0788 14 0.5 0.613 3.332 185864840.0 0.1987 15 0.5 0.613 3.332 18823.3 0.0788 16 0.5 0.613 3.332 55564626.4 0.1987 17 0.5 0.613 3.332 8731761683.5 0.2063 18 0.5 0.613 3.332 8731761674.1 0.2063 19 0.5 0.613 3.332 3650275366.2 0.2063 20 0.5 0.613 3.332 3650275351.2 0.2063 21 0.5 0.613 3.332 1858648285.5 0.1987 22 0.5 0.613 3.332 247416.9 0.0788 23 0.5 0.613 3.332 555646167.7 0.1987 24 0.5 0.613 3.332 188104.7 0.0788 25 0.5 0.613 3.332 119050077.8 0.2063 26 0.5 0.613 3.332 119050073.0 0.2063 27 0.5 0.613 3.332 629730911.4 0.2063 28 0.5 0.613 3.332 629730907.4 0.2063 29 0.5 0.613 3.332 330363625.7 0.1987 30 0.5 0.613 3.332 47372.6 0.0788 31 0.5 0.613 3.332 1150234475.6 0.1987 32 0.5 0.613 3.332 84854.8 0.0788

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Based on the results presented in Table 2-5, we learn the following things:

(1) All parameters are not significant to the optimal sampling interval h*. The value of h* is equal to the lower bound of its range, 0.5, for all levels of combinations.

(2) All parameters are not significant to the optimal control limits, UCL* and LCL*. This is because in the data analysis in Table 2-5, we set a_I 2, b_I 2, and 0.0027, it’s all the same in all 32 level combinations.

We consider the parameter as significant to EA* if the difference of the average of EA*

between two (or three) levels is greater than 10⁹ (See Table 2-6.).With this criterion, Table 2-6 shows that k, Rn, λ, T₁T₂, and (₁,₂) are the significant parameters, and it comes the following findings:

(1) A larger k indicates a larger EA*; because if the quality loss per unit item increases, then the expected cycle cost per unit item, EA*, will increase.

(2) A larger R_n indicates a larger EA*; because if the number of products per unit time, R_n, increases under the expected quality cost per unit item fixed, then the expected cycle cost, EC, will increase. Thus, EA* increases when R_n increases.

(3) When  increases, the average of EA* increases. This is because the expected time for the out-of-control period, 

 



ET 1 , increases if  increases. It leads the expected

quality cost for the out-of-control period ( 

 

 ET 1 l

R_{n O} ) to increase.

(4) When T₁T₂ increases, the average of EA* increases. This is because the expected quality cost for out-of-control period increases if T₁T₂ increases. Thus, EA* increases

when T₁T₂ increases.

(5) When the shift of variance increases, the average of EA* increases. This is because the expected quality cost for out-of-control period increases if the shift of variance

increases.

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Table 2- 6. The Values of EA^* for Each Parameter.

k Rn  e T₁T₂ c

level1 635225692 2164033412 493604365 1452778121 664488279 1001393972 level2 1884818522 356010801 2026439849 1067266093 1855555935 1518650242 difference 1249592830 1808022611 1532835484 385512028 1191067656 517256270

d W Y (₁,₂)

level1 1499763978 1258515057 1122735488 2030121773 level2 1020280235 1261529157 1397308725 117735.5

level3 979727146

difference 479483743 3014100 274573237 2030004038

We summarize the above data analysis in Table 2-7. In Table 2-7, if the parameter is significant to optimal design parameter and they have a positive linear relationship, we use the notation “○⁺ ”. If the parameter is significant to optimal design parameter but we cannot define their relationship, we use the notation “Q”. If parameter is not significant, we use the notation “×”.

Table 2- 7. The Significant Parameters and Their Relationship of Each Design Parameter and EA Optimal Design

parameter and EA

Parameters

k Rn  e T₁T₂ c d W Y (₁,₂)

h ^{× × × × × × × × × ×}

UCL ^{× × × × × × × × × ×}

LCL ^{× × × × × × × × × ×}

EA ○⁺ ○⁺ ○⁺ × ○⁺ × × × × Q

 × × × × × × × × × ×

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CHAPTER 3.

DESIGN OF ECONOMIC X CHART AND INSPECTION SPECIFICATION LIMIT FOR A PROCESS WITH WEIBULL DATA

According to the data analysis performed in Section 2.8, we find that EA* is very large in all 32 level combinations and varies extremely between each combination (10⁴~10⁹). So, we consider adding specification limits in order to improve the outgoing quality and reduce the expected cycle cost per unit item. To obtain the expected cost per unit time, we have to derive the expected cycle time and the expected cycle cost. The expected cycle time is the same as Equation (2-14).

3.1 Derivation of the Expected Cycle Cost

Again, we consider the larger-the-better quality variable following Weibull distribution;

thus, we must determine the lower specification limit if we have the inspection plan. We consider the Taguchi quadratic loss function to be









 

LSL X

if A

LSL X X if

k L_Ins

,

,

2 , (3-1)

where LSL is the lower specification limit, and A is the scrap cost per unit item.

Fig. 3-1 shows that the relationship between the pdf of the Weibull distribution,

Taguchi’s quadratic loss function, scrap cost per unit item, and the lower specification limit.

Figure 3- 1. The Weibull Distribution and Taguchi’s Quadratic Loss Function With Inspection Specification Limit

‧

The expected cost can be divided into five parts; three of them are the same as we showed in Section 2.6. The expected sampling and testing cost per cycle time is the same as Equation (2-19), the cost of investigating false alarms per cycle time is the same as Equation (2-20), and the cost to locate and repair the assignable cause is the same as Equation (2-21).

The others are described as follows.

(1) The expected cost per unit item using the Taguchi quadratic loss function in the in-control period if the manager decides to inspect the products includes the expected quality cost when the product meets the specification limit, the scrap cost when the product fail to conform to the specification, and the inspection cost. Thus, the expected cost per unit item in the in-control period is:

IC

where IC is the inspection cost per unit item.

In the in-control period, the expected quality cost per unit time is

I nL R .

Hence, the expected quality cost for the in-control period is:

 1

I nL

R . (3-2)

(2) The expected cost per unit item using the Taguchi quadratic loss function in the out-of-control period if the manager decides to inspect the products is:

(2) The expected cost per unit item using the Taguchi quadratic loss function in the out-of-control period if the manager decides to inspect the products is:

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