Kate Gilland 10/2/13 IE 434 Homework 2 Process Capability 1. Figure 1: Capability Analysis σ = R = 4.642857 = 1.996069 P d 2 2.326 p = 1.80 C p = 2.17 These results are according to Method 2 in Minitab. Short-term process variability was used because the process is in a state of statistical control. Method 2 was used because the data has a sample size of n=5. With 1<n<10, an Xbar-R chart would be implemented which requires execution of Method 2 when calculating process variability. Since the process capability of 2.17 is greater than 1, the natural tolerance limits fall within the specification limits, meaning that we can conclude that this process would produce few nonconforming units.
2. Figure 2: XmR Chart 36 ± 12 LSL=24 USL=48
C p = 0.66 95% C.I. = (0.36, 0.95) P p = 0.65 95% C.I. = (0.36, 0.95) Figure 3: Capability Analysis σ = 1.047(moving R ) = 1.047(6.88) = 7.20336 Since the I-MR chart shows that the data is in a State of Statistical Control and there was limited data, only 10 observations, short-term process variation was implemented through Method 4 (XmR control chart). However, when calculating process capability indices, it is important to have both the process in control, and to have a long enough period of data. Its concerning in this problem, though, that there is only 10 data points and that a whole section of the histogram is not within the specification limits making the data off-centered with the displayed capability chart. It is also concerning that the minitab standard deviation value and the manually calculated standard deviation value differs by more than 1. Lastly, this problem is confusing because when there are no assignable causes, meaning the process is stable, essentially, the performance index and the process capability index can estimate the same thing. The calculated numbers prove this point; however, the question still arises whether or not short-term variation process should be used because of the limited data that is not even collected in a time sequence.
3. C p = 1.7 n=20 2 2 X (1 α 2 95% C. I. = C ),(n 1) X ( α 2 p C n 1 p C ),(n 1) p n 1 C. I. = 1.7(0.6847) C p 1.7(1.3149) C.I. = 1.16399 1.7 2.23533 4. The XmR chart for the original data, before any transformation, is shown below in figure 4. This control chart showed a calculated UCL = 22.347 and LCL = -3.441. Since there no given specification limits for the data, tolerance intervals with 95% confidence were calculated for the data set. These results are shown in figure 5. Figure 4: XmR Chart before Transformation
Figure 5: Tolerance Interval Plot for XmR Chart before Transformation The Upper and Lower Limits of the Tolerance Interval are [-3.441, 22.347]. These values were then set as the specification limits to be able to compute the capability analysis. The resulting capability analysis is shown in figure 6 below.
Figure 6: Capability Analysis before Transformation Since the displayed data, before the transformation, lacks normality, as seen through the scattered histogram, the above process capability and process performance, calculated in figure 6, are not applicable. Therefore, a Box-Cox Transformation with an optimal lambda (λ=0.50) was implemented to produce the XmR chart for the transformed data, which is displayed in figure 7.
Figure 7: XmR Chart after Transformation Using the stored data points from the transformation, a new tolerance interval plot was executed (figure 8), to again use for the specification limits. The new specs are [0.668, 5.239]. Using these specs, a capability analysis for the transformed data was executed and is shown in figure 9.
Figure 8: Tolerance Interval Plot for XmR Chart after Transformation Figure 9: Capability Analysis after Box-Cox Transformation
Table 1: Results for Before and After Transformation Spec Limits Capability Analysis Before Transformation After Transformation Lower -3.441 0.668 Upper 22.347 5.239 Sample Mean 9.45308 2.95319 StDev (Within) 4.83548 0.850244 StDev (Overall) 4.97066 0.881117 Cp 0.89 0.9 Pp 0.86 0.86 PPM Total (within) 7663.59 7187.04 PPM Total (overall) 9485.99 9490.31 This data shows that the Cp=0.89, from before the transformation, is less than the Cp=0.9, from after the transformation, meaning that the PPM nonconforming units is reduced after the transformation. This information could be used to validate the earlier assumption that the data before the transformation lacked normality, even though the original XmR chart didn t indicate any signals of out of control points.