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Learning Objectives Chapter 7 Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery. Copyright (c) 2009 John Wiley & Sons, Inc. 3 7.1 Introduction Many quality characteristics are not measured on a continuous scale or even a quantitative scale. In such cases, one may judge each unit of product as either conforming or nonconforming on the basis of whether or not it possesses certain attributes or we may count the number of nonconformities (defects) appearing on the unit of product. Control charts for such quality characteristics are called attributes control charts. 1 7.2 The Control Chart for Fraction nonconforming The population fraction nonconforming (FNC) is defined as the ratio of the number of nonconforming items in a population to the total number of items in that population. Similarly, the sample fraction nonconforming is defined as the ratio of the number of nonconforming items in a sample to the number of items in the corresponding sample. The item may have several quality characteristics that are examined simultaneously by the inspector. If the item does not conform to standard on one or more of these characteristics, it is classified as nonconforming. The statistical principles underlying the control chart for FNC are based on the binomial distribution. We assume that the process is operating in a stable manner, such that the probability that any unit will not conform to specifications is p and successive units produced are independent. Then each unit produced is a realization of Bernoulli random variable with parameter p . Suppose that each subgroup is of the same size n , and D denotes the random variable that counts the number of nonconforming items in a subgroup. Then D can be modeled as a binomial random variable with parameters n and p . n P ( D = x) = p x (1 p ) n x ; x = 0,1,2, , n x That is Then D = np, D2 = np(1 p) The sample fraction nonconforming corresponding to population nonconforming is obtained as: pˆ = D . n It can be shown that D np E ( pˆ ) = pˆ = E = =p n n and np(1 p) p (1 p ) 1 Var ( D) = = . 2 n n2 n So the mean and standard deviation of p̂ are respectively Var ( pˆ ) = p2ˆ = pˆ = p, and pˆ = 2 p (1 p ) . n (1) 7.2.1 Development and Operation of the Control Chart Let w be a sample statistic that measures some quality characteristic of interest, and suppose, E ( w) = w V(w) = w2 , SD( w) = w and Then the UCL, center line, and LCL become: UCL = w L w (2) CL = w LCL = w L w , where L (usually 3) is the ``distance'' of the control limits from the center line, expressed in standard deviation units. This is called Shewhart (Dr. Walter A. Shewhart) Control chart. Fraction Nonconforming Control Chart: Standard Given UCL = p 3 p(1 p) n (3) CL = p LCL = p 3 p(1 p ) n The actual operation of this chart would consist of taking subsequent samples of n units. Computing the sample fraction nonconforming p̂i and plotting the statistic p̂i versus its subgroup number i on the chart. As long as p̂ remains within the control limits and the sequence of plotted points does not exhibit any systematic nonrandom pattern, we may conclude that the process is in control at the level p . If a point plots outside of the control limits, or if a nonrandom pattern in the plotted points is observed, we may conclude that the process fraction nonconforming most likely shifted to a new level and the process is out of control. 3 Fraction Nonconforming (FNC) Control Chart: No Standard Given UCL = p 3 p (1 p ) n (4) CL = p LCL = p 3 p (1 p ) n The control limits in (4) should be considered as trial control limits. Estimation of p When the process fraction nonconforming p is not known, then it must be estimated from observed data. The usual procedure is to select m preliminary samples (20 to 30), each of size n (5 to 10). Case 1: Suppose there are Di nonconforming items in the sample i , then we compute the fraction nonconforming for the i th sample as pˆ i = Di , n i = 1,2, , m. And the average of these individual sample FNC is m m D pˆ i i p= i =1 i =1 = nm m Case 2: When the subgroup sizes n1 , n2 ,..., nm are not all equal, then the p can be calculated as D D2 ,..., Dm p= 1 n1 n 2 ,..., n m D D D = 1 2 ... m N N N n n n = 1 pˆ 1 2 pˆ 2 ... m pˆ m N N N m = wi pˆ i , i =1 n where wi = i and n1 n2 ... nm = N and p is a weighted average of N the subgroup statistics. 4 (5) Example 7.1, page 292 From Table 7.1, we calculate the following preliminary control limits. m D i p= i =1 = nm 347 = 0.2313 30 50 The upper control limit, center line and the lower control limits are p (1 p ) = 0.4102 n CL = p = 0.2313 UCL = p 3 LCL = p 3 (6) p (1 p ) = 0.0524 n Figure 7.1 Initial phase I fraction nonconforming control chart The control chart has shown in Figure 7.1. We observed that samples 15 and 23 plot above the upper control limit. Therefore, the process is not in control. Since the samples 15 and 23 are out of control limits they are eliminated and the revised control limits are as follows: m D i p= i =1 = nm 301 = 0.2150 28 50 p (1 p ) = 0.3893 n CL = p = 0.2150 UCL LCL = p 3 = p 3 p (1 p ) = 0.0407 n 5 (7) The revised control chart has shown in Figure 7.2. We observed that samples 15 and 23 plot above the upper control limit, even they have been excluded from the calculation of the control limits. In the revised control limits, the sample number 21 is out of the limit. However, there is no assignable causes related to that sample. So we conclude that the process is in control at level p = 0.2150 and the revised control limits should be used for monitoring current production. Figure 7.2 Revised control limits for the data in Table 7.1 (page 293) During the next three shifts following the machine adjustments and the introduction of the control chart, an additional 24 samples of size n = 50 observations each are collected and provided them in Table 7.2, page 295. The sample fraction nonconforming are plotted on the control chart in Fig 7.3. Figure 7.3: Continution of fraction nonconforming control chart. 6 From Fig 7.3, we see that the process is now operating at a new level which is lower than the present level p = 0.2150 . Now, we are interested for the following hypothesis H 0 : p1 = p2 H1 : p1 > p2 We have pˆ 1 = 0.2150 , and 54 D i pˆ 2 = i = 31 50 24 = 133 = 0.1108 50 24 The approximate Z-test (see more on page 296) is Z0 = Z0 = pˆ 1 pˆ 2 pˆ (1 pˆ )1/n1 1/n2 0.2150 0.1108 = 7.10 (0.1669)(0.8331)1/1400 1/1200 Since Z 0 = 7.10 > 1.645 , we do reject the null hypothesis and conclude that there has been a significant decrease in the process fallout. The revised control limits based on the last 24 samples (numbers 31-54) are p (1 p ) = 0.2440 n CL = p = 0.1108 UCL = p 3 LCL = p 3 p (1 p ) = 0.0224 = 0 n The new revised control chart has shown in Figure 7.4. Figure 7.4: New control limits on FNC control chart, Example 7.1 7 (8) The new control limits will have only upper control limit. All points are inside the revised control limits. Therefore the process is in control at this new level. The continued operation of this control chart for the next five shifts (data in Table 7.3, page 298) is shown in Fig 7.5. Figure 7.5: Completed fraction nonconforming CC for Example 7.1 The process in control. However, the fraction nonconforming is still high. You need Experimental Design to reduce the process fraction nonconforming. See more on page 297. Design of the Fraction Nonconforming Control Chart (FNCC) The FNCC has three parameters: the sample size, the frequency of sampling and the width of the control limits. Various rules have been suggested for the choice of sample size n . When p is small, larger subgroup sizes are necessary, and for larger p , smaller subgroups are necessary. We can choose the sample size n so that the probability of finding at least one nonconforming unit per sample is at least . If D denotes the number of NCF items in the sample, then we want to find n such that p{D 1} . 8 Using Poisson approximation to binomial, we find that n= ln(1 ) . p (9) Example, if P( D 1) 0.95 , & p = 0.01 , then sample size should be 300. Example, if P( D 1) 0.99 , & p = 0.01 , then sample size should be 461. Duncan (1986) suggested that the sample size should be large enough such that we have approximately 50% chance of detecting a process shift of some specified amount. If is the magnitude of the process shift, then 2 L n = p(1 p) (10) For example, p = 0.01 , we want to detect a shift when p = 0.05 . Then = 0.05 0.01 = 0.04 . And the sample size for 3-sigma control limit would be 2 3 n= 0.01(1 0.01) = 56 0.04 For example, p = 0.01 , we want to detect a shift when p = 0.02 . Then = 0.02 0.01 = 0.01 . And the sample size for 3-sigma control limit would be 2 3 n= 0.01(1 0.01) = 891 0.01 For a smaller shift, you need a bigger sample size. What is the smallest sample size that would give a positive lower limit? LCL p L n> p(1 p ) >0 n (1 p ) 2 L . p For example, if p = 0.05 , and 3-sigma limits are used, the sample size must be n 171 . For example, if p = 0.01 , and 3-sigma limits are used, the sample size must be n 892 . 9 The np Control Chart limits are as follows: The number nonconforming or np control UCL = np 3 np(1 p ) (11) CL = np LCL = np 3 np (1 p) If p is unknown, use p to estimate p . Example 7.2, page 300. Consider the data in Table 7.1 of example 7.1. The control limits for np chart would be UCL = np 3 np (1 p ) = 20.51 = 20 CL = np = 11.57 = 12 (12) LCL = np 3 np (1 p ) = 2.62 = 2 From Table 7.1, the sample number 15 and 23 are out of control. 7.2.2 Variable Sample Size There are three approaches to constructing and operating a control chart. 1. Variable-width Control Limits The control limits for the i th sample are p (1 p ) ni UCL = p 3 (13) CL = p p (1 p ) ni LCL = p 3 m Replace p with p , where, p = D i i =1 m n . i i =1 10 Consider data in Table 7.4, page 302 we calculate 25 D i p= i =1 25 n = 234 = 0.096 2450 i i =1 then the center line is 0.096. The control limits are UCL = 0.096 3 0.096 0.904 0.884 = 0.096 ni ni (14) CL = 0.096 LCL = 0.096 3 0.096 0.904 0.884 = 0.096 ni ni The control chart for fraction nonconforming with variable sample size is provided in Fig 7.6. Figure 7.6: CC for FNC with variable sample size 11 2. Control limits based on an average sample size m n i n= i =1 m Then the approximate control limits are p (1 p) n UCL = p 3 (15) CL = p p (1 p ) n LCL = p 3 Consider data in Table 7.4, page 302 we calculate 25 D i n= i =1 25 n = 2450 = 98 25 i i =1 Then the control limits are UCL = 0.096 3 0.096 0.904 = 0.185 98 (16) CL = 0.096 LCL = 0.096 3 0.096 0.904 = 0.007 98 The resulting control chart is shown in Fig 7.8. Figure 7.8: CC for FNC based on the average sample size 12 3. The standardized control chart The standardized control chart has center line at 0, upper and lower limits of +3 and -3 respectively. The variable plotted on the chart is Zi = pˆ i p p(1 p) ni where p (or p , if standard is not given) is the process fraction nonconforming in the in-control state. The standardized control chart for fraction nonconforming is presented in Fig 7.9 for the data in Table 7.5, page 305. Figure 7.9: Standardized CC for FNC 7.2.3 Application in Transactional and Service Business: pages 304306 7.2.4 The Operating Characteristic (OC) Function and Average Run Length (ARL) The OC function of the fraction nonconforming control chart is a graphical display of the probability of incorrectly accepting the hypothesis of statistical control (i.e. type II error or -error) against the process fraction nonconforming. The OC curve provides a measure of the sensitivity of the control chart, that is, its ability to detect a shift in the process fraction nonconforming from the nominal value p to some other value. 13 The probability of type II error is = P{LCL < pˆ < UCL | p} D < UCL | p} n = P{D < n UCL | p} P{D n LCL | p} P{LCL < Since D is the binomial random variable with parameters n and p , the error defined in (17) can be obtained from binomial table. Table 7.6, page 307 illustrates the calculations of a OC curve for n = 50 , LCL = 0.0303 and UCL = 0.3697 . The corresponding OC curve has shown in Fig 7.11. 14 (17) The average run length (ARL) for any Shewhart control chart is defined as ARL = 1 Sample point plots out of control (18) If the process is in control, ARL0 is defined as ARL0 = 1 (19) For example, if the process is in control with p = 0.20 , the probability of a point plotting in control is 0.9973 (from Table 7.6). Then ARL0 = 1 = 370 1 0.9973 Then, even the process is in control, we will experience a false alarm out of control signal about every 370 samples. However, if the process is out of control, then ARL1 = 1 1 (20) For example, if the process shift to p = 0.30 , Table 7.6 indicates that = 0.8594 . Then ARL1 = 1 =7 1 0.8594 and it will take about 7 samples, on average, to detect this shift with a point outside of the control limits. Exercise 7.7, page 336. 15 Exercise 7.10, page 336. Exercise 7.18, page 337. 7.3 Control Charts for Nonconformities (Defects) A nonconforming item is a unit of product that does not satisfy one or more of the specifications for that product. Each specific point at which a specification is not satisfied results in a defect or nonconformity. Consequently, a nonconforming item will contain at least one nonconformity. It is possible to construct control chart based on the total number of nonconformities in a unit or the average number of nonconformities per unit. 7.3.1 Procedures with Constant Sample Size Consider the occurences of nonconformities in an inspection unit of product. The inspection unit is simply an entry for which it is convenient to keep records. Suppose that defects or nonconformities occur in the inspection unit according to the Poisson distribution. That is, p ( x) = ecc x , x! where x is the number of nonconformities and c is the parameter of Poisson distribution. The control chart for nonconformities with 3sigma control limits are defined as follows: Control chart for Nonconformities: Standard Given (c-chart) UCL = c 3 c (21) CL = c LCL = c 3 c If the LCL is a negative value, consider as LCL=0. 16 Control chart for Nonconformities: No Standard Given UCL = c 3 c (22) CL = c LCL = c 3 c When no standard is given, the control limits in equation (22) should be regarded as trial control limits. Example 7.3, page 310 Table 7.7 (page 310) represents the number of nonconformities observed in 26 successive samples of 100 printed circuit boards. For reasons of convenience, the inspection unit is defines as 100 boards. Construct control limits for the c-Chart. Since 26 samples contain 516 total nonconformities, we have c= 516 = 19.85. 26 The trial control limits are UCL = c 3 c = 33.22 (23) CL = c = 19.85 LCL = c 3 c = 6.48 The control chart based on limits in (23) is shown in Fig 7.12. Figure 7.12: CC for NConformities for Example 7.3 17 From Fig 7.12, we observe that Samples 6 and 20 plot outside the control limits. Then exclude these two samples and revised control limits which are obtained as follows: c= 472 = 19.67. 24 The revised control limits are UCL = c 3 c = 32.97 (24) CL = c = 19.67 LCL = c 3 c = 6.37 The above control limits become the standard values against which production in the next period can be compared. Twenty new samples are collected and given them in Table 7.8, page 311. Using revised control limits in (24), these points are plotted in Fig 7.13. It is evident that this process in statistical control. Figure 7.13: Continution of the CC for NConformities, Example 7.3 Exercise 7.36, 338. 18 Control chart for average number of nonconformities per unit (uchart) To account for the variable numbers of inspection units, c charts are replaced by the u charts. The u chart monitors nonconformities per inspection unit, when the number of inspection units are vary from sample to sample. If a total of x nonconformities are found in the subgroup of size n inspection units, then the average number of nonconformities per unit can be estimated as u= x , n where x is a Poisson random variable. The control limits for u chart are: UCL = u 3 u n (25) CL = u LCL = u 3 u , n where m u u= i =1 m i . Example 7.4, page 315 Data are presented in Table 7.10 (page 316). 20 u u= i i =1 20 = 1.48 = 0.0740 20 The upper control limit, center line and lower control limits are UCL = u 3 u = 0.1894 n (26) CL = u = 1.93 LCL = u 3 u = 0.0414 0 n The control chart for nonconformities is presented in Fig 7.16. The preliminary data do not exhibit lack of statistical control. Therefore, the trial limits in (26) would be adopted for current control purposes. 19 Figure 7.16: The CC for NConformities per unit for Example 7.4 Exercise 7.38, Page 338. 6-3.2 Procedures with Variable Sample Size There are three approaches to constructing and operating a control chart. 1. Variable-width Control Limits: The control limits for the i th sample are UCL = u 3 u ni (27) CL = u LCL = u 3 u ni Consider data in Table 7.11, page 319 we calculate u= 153 = 1.42 107.5 The upper control limit, center line and lower control limits are UCL = 1.42 3 1.42 3.575 = 1.42 ni ni CL = 1.42 LCL = 1.42 3 1.42 3.575 1.42 ni ni 20 The control chart for fraction nonconforming with variable sample size is provided in Fig 7.17. Figure 7.17: CC for Example 7.5 2. Control limits based on an average sample size The approximate control limits are UCL = u 3 u n (28) CL = u LCL = u 3 u , n where m n i n= i =1 m . 3. The standardized control chart The standardized control chart has center line at 0, upper and lower limits of +3 and -3 respectively. The variable plotted on the chart is Zi = uˆi u u ni (29) The standardized control chart for nonconformities per unit is presented in Fig 7.18 for the data in Table 7.11. 21 Figure 7.18: Standardized CC for NConformities per unit, Example 7.5 22 7.3.4 The Operating -Characteristic (OC) Function The OC curves for both the c and u charts can be obtained from the Poisson distribution. For c chart, the OC curve plots the probability of type II error against the true mean number of defects c . The expression for is = P{LCL < x < UCL | c} (30) = P{x < UCL | c} P{x LCL | c} where x is a Poisson random variable with parameter c . Table 7.13, page 323 illustrates the calculations for a OC curve for example 7.3, LCL = 6.48 and UCL = 33.22 . The corresponding is (31) = P{x 33 | c} P{x 6 | c} The OC curve has shown in Fig 7.19, page 324. Figure 7.19: OC Curve of a c chart with LCL=6.48 and UCL=33.22 The OC curve for u chart is = P{x < UCL | u} P{x LCL | u} = P{c < n UCL | u} P{c n LCL | u} Exercise 7.56, Page 340. 23 (32) 7.3.5 Dealing with Low Defect Levels When defect levels (or count rates) in a process become very low (less than 1000 per million), there will be very long periods of time between the occurrence of nonconforming unit. In these situations many samples will have zero defects and therefore, conventional c and u chart become ineffective. See more in Example 7.6, page 324. 7.4 Choice between attributes and variables control charts Read page 326. Example 7.7, page 327, demonstrates the economic advantage of variable control chart. Example 7.8, page 328, demonstrates a misapplication of x and R charts. 7.5 Guidelines for implementing control charts (pages 330-334) 1. Determining which process characteristics to control 2. Determining where the charts should be implemented in the process 3. Choosing the proper type of control charts 4. Taking actions to improve processes as the result of SPC/ control chart analysis. 5. Selecting data-collection systems and computer software 24