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We summarize the data analysis in Chapter 3 to Table 3-7. In Table 3-7, we use the same notation as we introduced in Table 2-5.
Table 3- 7. The Significant Parameters and Their Relationship of Each Design Parameter and EAL Optimal Design
parameter and EAL
Parameters
k (IC, Rn) e T1T2 c d W Y (1,2)
h × ○+ × × × × × × × Q
UCL × × × × × × × × × ×
LCL × × × × × × × × × ×
LSL × × × × × × × × × ×
EA L ○+ ○+ ○+ × ○+ × × × × Q
× × × × × × × × × ×
Here, we provide an example to illustrate the determination of the optimal design parameters of the economic X control chart and the optimal lower specification limit and its applications. We first explain the data we used, and then construct the economic X control chart and inspection plan.
3.4 An Example 3.4.1 Data
We used data from Erto, Pallotta and Park (2008). These data pertain to the breaking stress of carbon fibers: 10 samples of size n = 5 under an in-control state. We constructed the economic X control chart for sample size n = 4; however, the data provided from the article were intended for a sample size of n = 5. For our purposes, we moved the #1-#4 data in the last column (Column 5) to the #11 sample, and moved the #5-#8 data in last column to the #12 sample to obtained data for 12 samples of size n = 4.
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Table 3- 8. Breaking Stress of Carbon Fibres.
j-th sample Stress X
3.4.2 Estimating the in-control parameters of the Weibull distribution
We calculate the maximum likelihood estimators (MLEs) of the Weibull(aI,bI) distribution aˆI 4.895 and bˆI 3.235 by using the routine “mle” in the R program. In
hypothesis test and draw the Weibull probability plot for checking whether the data is following Weibull(4.895,3.235) or not.
0:
H data is sampling from Weibull(4.895,3.235)
v.s. H1:data is not sampling from Weibull(4.895,3.235)
We use the Kolmogorov-Smirnov test (K-S test) and obtain the p-value=0.3176.
Because the p-value is 0.3176 is larger than 0.05, so we do not reject the null hypothesis. That means we do not reject that the data in table 3-8 is following
)
Secondly, Fig. 3-2 shows the Weibull probability plot for the sample data in Table 3-8.
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We can find that the p-value is greater than 0.250, so we do not reject that the data is following Weibull(4.895,3.235).
Figure 3- 2. The Probability Plot for Weibull Distribution
3.4.3 Simulation Data for Out-of-control Distribution
The simulated data from the out-of-control Weibull distribution was (a0.9,b0.726) and 10 samples of a size of n = 4. The out-of-control mean = 0.764 and variance = 0.723.
Table 3- 9. Simulated Data From Out-of-Control Distribution
j-th sample Simulated data X
13 3.03 0.23 1.04 0.09 1.10
14 1.98 0.27 1.14 0.58 0.99
15 0.53 0.33 0.04 0.05 0.24
16 0.07 0.12 1.26 0.07 0.38
17 1.95 0.47 0.45 0.82 0.92
18 0.18 0.84 0.57 0.09 0.42
19 2.85 0.71 1.15 0.40 1.28
20 0.08 0.26 1.06 0.66 0.52
21 0.39 0.02 2.94 0.29 0.91
22 0.03 0.56 2.27 0.70 0.89
According to our discussion in Section 3.4, we find that the expected cost per unit time for the economic X control chart with inspection plan is much smaller than the expected
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cycle cost per unit time for the economic X control chart without inspection plan. So, we show how to determine the design parameters and the optimal lower specification limit simultaneously to obtain the minimal expected cost per unit time.
3.4.4 Constructing the economic X control chart and inspection plan
We use the same assumptions as those presented in Section 2.6. Let n = 4, A = 20, IC = 0.05, Rn = 1,000, 0.01, e = 0.05, T1T2= 3, c = 0.5, d = 0.1, W = 500,
Y = 250, aˆ4.895,bˆ3.235, (ˆ1,ˆ2)(3.179, 1.227), and 0.0027. Using the routine “DEoptim” in the R program to minimize the EAL subject to 0.5h8 and
LSL 001 .
0 . We obtained h* = 0.5, LCL* = 1.875, UCL * = 3.939, EAL*
= 666.6, = 0.0200, LSL* = 0.308, and yield = 0.99999. In addition, if we use the same data to construct the economic X control chart without inspection plan, we obtained h* = 0.5, LCL* = 1.875, UCL * = 3.939, EA* = 110665.6, and = 0.020. We can find out that we can save numerous costs with the inspection plan. Hence, we construct the economic X control chart to
monitor the process and develop the inspection plan, and then plot the statistics of the in-control data.
Figure 3- 3. Economic X Control Chart (Phase I)
Figure 3-3 shows that no points are out of the control limits for the in-control state (i.e., no false alarms occurred in Phase I). We then plotted the statistics of the out-of-control data as in Fig. 3-4.
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Figure 3- 4. Economic X Control Chart (Phase II)
Figure 3-4 shows that No.13 to 22 are out of control limits, and the first true alarm is No. 13. All out-of-control data were detected.
Figure 3- 5. The Relationship Between the In-control and Out-of-control Distribution and the Lower Specification Limit
According to Fig. 3-5, we can find the yield for in-control state is 0.9999, and the yield for out-of-control state is 0.6299. Hence, we can improve the outgoing quality by conducting inspection, and detect the out-of-control state by monitoring the process using economic X control chart.
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ECONOMIC STATISTICAL DESIGN OF THE X CHART FOR THE PROCESS USING WEIBULL DATA
In Chapter 2 and 3, we consider the economic design of the X chart without or with considering inspection plan. We find that the expected cost per unit time will decrease if we consider conducting inspection plan. However, the EAL* and * is still too large. Hence,
we consider the economic statistical design of X control chart, to reduce the EAL* and *. In economic statistical X control chart, we determine the optimal design parameters by minimizing the cost model like we presented in Chapter 2 and 3, but we modify the
determination of the UCL* and LCL*. In the economic X control chart, we determine the UCL* and LCL* by the approximated CDF of XI with given aI, b , and I , but in the economic statistical X control chart, we determine the UCL* and LCL* by minimizing EA* (or EAL*) with constraints of and . Therefore, we can expect the reduction of EA* (or EAL*) and * for sure.
To obtain the expected cycle cost per unit time, we must derive the expected cycle time and the expected cycle cost. However, the expected cycle time is the same as Equation (2-14) and the expected cycle cost is the same as Equation (2-20). Hence, the expected cycle cost per unit time is the same as Equation (2-21).
4.1 Construction of the Economic Statistical X Chart Based on the X Sampling