Process and Measurement System Capability Analysis

Transcription

1 Process and Measurement System aability Analysis Process caability is the uniformity of the rocess. Variability is a measure of the uniformity of outut. Assume that a rocess involves a quality characteristic that follows a normal distribution with mean, and standard deviation,. The uer and lower natural tolerance limits of the rocess are UNTL = + 3 LNTL = - 3 Process caability refers to the uniformity of the rocess. The variability of critical-to-quality characteristics in the rocess is a measure of the uniformity of outut. There are instantaneous variability and variability over time. The nature tolerance is defined as: 79

2 Techniques used in rocess caability analysis Histograms or robability lots ontrol harts Designed Exeriments Histogram The histogram along with the samle mean and samle standard deviation rovides information about rocess caability. Examle 7.1. Data for 100 units shown in Table 7-1. Samle mean: x Samle standard deviation: S 3. 0 Estimated rocess caability: x 3S (3.0)

3 Process aability Ratios USL LSL 6 where LSL and USL are the lower and uer secification limits, resectively. The estimate of is given by ˆ USL LSL 6ˆ Where the estimate ˆ can be calculated using the samle standard deviation, S, or R d. For the data in Examle 5-1, we have: 81

4 R 5 R i i R ˆ d For this examle roblem, assume that the secs are , so USL=.0 and LSL =1.0. The estimate of is ˆ USL LSL 6ˆ (0.1398) The ercentage of the secification band used by the rocess is 1 P ( )100% 1 For this examle, P ( )100% 83.89% 1.19 The one-sided secifications are defined by: ˆ u ˆ l USL 3 ˆ LSL 3 ˆ 8

5 Examle 7. For x 64 and 3 ˆ l LSL 3 ˆ S, if the lower secs limit is 00, then: (3) To calculate the ercentage of the roducts not meeting the sec, use: LSL Z. The estimated fallout is.8% ˆ 3 3 assuming normal distribution. These quantities, u and l have very critical assumtions: The quality characteristic has a normal distribution. The rocess is in statistical control In the case of two-sided secifications, the rocess mean is centered between the lower and uer secification limits. 83

6 Recommended caability ratios for different rocesses Process aability Ratio for an Off-enter Process does not take into account where the rocess mean is located relative to the secifications. A rocess caability ratio that does take into account centering is k defined as More about Process entering min( k u, l ) k should not be used alone as an measure of rocess centering. It may be used together with 84

7 k deends inversely on and becomes large as aroaches zero. (That is, a large value of k does not necessarily reveal anything about the location of the mean in the interval (LSL, USL) Examle. Two rocesses shown in Figure Testing Hyotheses about PRs Demonstrate that the rocess meets certain target value, 0. Use hyotheses testing as: Null hyothesis H 0 : 0 (rocess is not caable) Alternative hyothesis H 1 : 0 (rocess is caable) Table 8-4 rovides samle sizes and critical values for rejecting H : 0 (High) rocess caability to accet with rob. 0 1 It means that we do not accet the rocess with robability of although its caability is higher than (High) (Low) rocess caability to reject with rob. 1 85

8 It means that we do not reject the rocess with robability while its caability is not higher than (Low) Examle 8-6. For a sulier to demonstrate that his rocess caability exceeds H : H : We want to have 90% robability to reject the rocess if rocess caability is below 1.33 or (Low)=1.33. Also we want to have 90% robability to accet the rocess if the rocess caability exceeds (High) =1.66. That is, we have =0.10. We can use the information in Table 8-4 to decide the value of rocess caability ratio. We first calculate that: ( High) ( Low) Ĉ, the estimated 86

9 From Table 8-4, we find that the samle size should be 70. Then we use the second column to find that ( Low) 1.10 or the estimated critical is: ˆ ( Low) So the rocess must have a ˆ The sulier must take a samle of n = 70 arts, and the samle rocess caability ratio must exceed Process aability Analysis Using a ontrol hart The rocess is in statistical control. PRs can be calculated using the rocess mean and rocess standard deviation estimates. Examle. Table 7-5. The same data for Examle 7.. LSL=00 87

10 For the R chart, enter Line: R UL = D R (.1145)(77.3) , LL = D R (0)(77.3) 0 4 For x chart, enter Line: x UL = x A R (0.577)(77.3) , LL = The control charts show that the rocess was in control at the time these samles were taken. However, since x , and the estimated rocess standard R 77.3 deviation is: ˆ 33. 3, the one sided lower rocess caability d.36 ratio is: ˆ l The rocess is not caable. LSL 3 ˆ (33.3)

11 and Measurement System aability Determine how much of the total observed variability is due to the gauge Isolate the comonents of variability in the measurement system Assess whether the instrument or gauge is caable for the intended alication. A simle model: y x where y is the total observed measurement, x is the true value of the measurement and is the measurement error. We assume that x and are normally and indeendently distributed random variables with means 0 and variances and Examle 8.7, resectively. The variance of y is Total A gauge will be used in a SP rocess. The gauge caability should be assessed. 0 units of the roducts are obtained for this urose. Since it is to determine the caability of the measurement oeration by the oerator using the given device, the oerator measured each unit of the roduct twice. The data of the measurement are shown in Table 8-6. The control charts are shown in Figure 8-14 below. 89

12 The x chart, not served as a control chart in this case, shows that the gauge can be used for a certain range of the measurement. The R chart shows that the oints are within control limits. The oerator is good in using this instrument to do the measurement. The standard deviation of the measurement error can be estimated by: ˆ d R Estimating the Variance of the omonents From the data in Table 8-6, we can calculate that the standard deviation of the 40 measurements is s=3.17. Then: ˆTotal s (3.17) Since we have: Total and ˆ , we have (0.887) So the variance of the rocess can be estimated by: ˆ ˆ ˆ ˆ P Total and the estimate of the rocess standard deviation of the rocess can be calculated by: ˆ P P/T Ratio The P/T ratio, or the recision to-tolerance ratio, is to comare the estimate of gauge caability to the width of the secs: USL-LSL for the art that is measured. kˆ Guage P / T USL LSL 90

13 In most cases, k=6 is used. For the revious examle, if the art has USL=60, LSL=5, then 6(0.887) P / T Then the gauge caability is accetable as the P/T ratio is less than 0.1. Other related concets Precision and accuracy These two concets can be illustrate by the charts in Figure 7-15 Reeatability and reroducibility ( R&R Analysis) o Reeatability reflects the recision of the gauge o Reroducibility reflects the different conditions where the gauge is used to do the measurement such as oerator, time, environments, etc. Measurement Error Reeatability Reroducibility 91