CHAPTER-1 BASIC CONCEPTS OF PROCESS CAPABILITY ANALYSIS

Transcription

1 CHAPTER-1 BASIC CONCEPTS OF PROCESS CAPABILITY ANALYSIS

2 Manufacturing industries across the globe today face several challenges to meet international standards which are highly competitive. They also strive hard to meet customers expectations and forced to employ highly capable machines and processes. In this journey to meet the above objectives and to remain as a global player, many positive and proactive initiatives are to be taken to achieve zero defects by implementing Process Capability Analysis (PCA), Measurement System Analysis (MSA) with Statistical Process Control (SPC) in the organizations. The use of statistical methods for quality control started with the phenomenal paper Economic Control of Quality of Manufactured Product by Walter Andrew Shewhart in 1931 who introduced the concept of control chart and a brief outline of the early developments of control chart theory are discussed below. 1.1 THEORY OF SHEWHART CONTROL CHART The control chart is one of the powerful tools of SPC. The credit of applying statistical methods to the problem of quality control goes to Shewhart (1931) as he made the very first sketch of modern control chart. During 1930's the control charts & other SPC techniques for monitoring processes were expanded by Shewhart and his colleagues at Bell Telephone laboratories. Deming (1986) has observed that Shewhart's invention of the control chart in 1924 has been hailed as one of the greatest contributions to the philosophy of science History of control chart Let X be continuous random variable, indicating a measured quality characteristic and assume that X~N(µ,σ 2 ). According to the area property of normal distribution it is true that P[μ-3σ x μ+3σ] = which means that only 27% of the items fall outside the 3σ limits, when the process is not influenced by assignable causes. When X [μ-3σ, μ+3σ], it is an indication of out of control state. When the process has no evidence of assignable causes, we say that the process is in a state of control or incontrol state. Instead of examining the individual values of X, it is convenient to use a sample statistic t n = f(x 1, x 2,.., x n ). For instance X is the sample mean that can be called control statistic. Since, X~N μ, σ σ, it is true that Pμ 3 σ x μ 3 =

3 Suppose that m samples, each of size n are drawn from the process at regular intervals. Each sample is called a sub-group (or a rational sub-group). The size of sub-group is usually taken as n = 4 or 5 and the number of samples is usually taken as m = 30. For the i th sample, the value of the statistic x is plotted. The distribution of plotted points around the mean µ is a graphic display of dispersion in the process. Three horizontal lines are drawn for understanding and taking decision on the process as given below. a) Central Line (CL) at x μ b) Lower Control Limit (LCL) at x μ 3 σ c) Upper Control Limit (UCL) at x μ 3 σ If all the x values lie close to Central Line (CL), it means the process is operating at an average level µ. If all the plotted x values, lie within UCL and LCL without any trend then the process is said to be in a state of control. If one or two points lie outside the limits it indicates out of control situation, which means assignable causes are acting on the system. Under the influence of Central Limit Theorem, when x values are treated as independent and identically distributed random variables, the sampling distribution of x will be normal. According to Mittag and Rinne (1993) the most general form of the Central Limit Theorem is that of Lindeberg and Feller (1922) which specifies the necessary and sufficient conditions for the convergence of sequences of distributions to a normal distribution. The Lindeberg and Feller Central Limit Theorem: Let x 1,x 2,...,x n be arbitrary, but independently distributed random variables with finite expectations μ i =E(x i ) and finite variances σ =V(x i ). Let f n (z) denote the density of the standardized random variable Z n X µ σ with mean 0 and variance 1. Suppose that the series {X i } is 2

4 uniformly bounded with probability 1, this means that there exists a real number a > 0, such that P X a1 i=1,,n (Feller (1968)). Further, suppose that as n,. Then the sequence of the density function f n (z) converges, as n, to the density of the standard normal distributions. lim f z π exp σ A typical control chart is shown in Figure-1.1 Sample Mean mean CL LCL UCL Sample Mean mean CL LCL UCL Sample Number Sample Number In-control state Out-of-control state Figure-1.1: Shewhart control chart for means Therefore irrespective of the distribution of X (discrete/continuous) the sample statistic X follows normal distribution and hence the 3σ limits can be used to detect the presence of assignable causes on the process. Control charts are also plotted for different summary statistics of the process (Feller (1971)). 1) One sided X-charts for non-conformities per product unit 2) Two sided X-charts for non- conformities per product unit 3) u-charts for non-conformities per physical unit 4) p-charts for the fraction of non-conforming units 5) Median Charts (x-charts) 6) Extreme Value Charts (x-charts) 7) Single observation charts (x-charts) 8) Standard deviation charts (s-charts) 9) Range charts (r-charts) 10) Charts for Multiple non-conformities 11) C-chart (C) for Count of defects 12) np-chart for the number of defectives 3

5 1.1.2 Statistical inference from a control chart One can use a central chart to a) Estimate the process level and b) Estimate the process spread. There are several estimates of process level and spread. Mittag and Rinne (1993) discussed the following estimators assuming that X ~ N(µ 0,σ ). Let m be the number of samples and n be the subgroup size. a) First estimator (E 1 ): Total sample mean ( ) Let X is the mean of X m for m-subgroups of size n with expectation μ 0 and variance V(X = σ where σ is the process variance. Provided that the X ij are normally distributed, this estimator also has a normal distribution. b) Second estimator (E 2 ): Median of the sample medians ( If the pre-run variables X ij are N(µ 0, σ ) distributed, then the medians X 1n,, X mn of the individual samples have a normal distribution with expectation µ 0 and variance V(X in )= σ c, where c n is a constant. The median X of the sample medians is approximately normally distributed with expectation µ 0 and variance V(X σ c c. The values of the factors c ν can be found from the statistical table of special factors, for ν= 2, For ν>20, there is an approximation c ν π c) Third estimator (E 3 ): Mean of the sample medians ( The mean X of the sample medians given by, X = X, if m 2k 1 X, X, if m 2k is approximately normally distributed with expectation µ 0 and variance V(X = σ c. d) Fourth estimator (E 4 ): Median of the sample means ( The median X of the sample means is approximately normally distributed with 4

6 expectation µ 0 and variance V(X = σ c. Since c ν >1 for ν>2 and c ν+2 > c ν, it can be shown that, V(X VX VX V(X. Hence among all estimates of process mean, X is the best one. Consequently one would choose the total sample mean when efficiency is the decision criterion. However with respect to robustness, i.e., sensitivity towards outliers, the median of the sample medians (E 2 ) would be preferred. Values of these estimates are obtained from pre-run data which serves as a pilot study before fixing the limits. Again there are five different estimators of process spread as follows. 1) First estimator (E 1 ): Corrected total sample standard deviation (S mn ) The first candidate for an estimator is the total standard deviation S mn =S. The sum Q mn in S consists of the two terms Q 1 and Q 2. These two components reflect different sources of variation. The variation resulting from the spread of the means is contained in Q 1, whereas the variation of the individual values with respect to their corresponding sample mean is measured by Q 2. For the variation of the process spread one can use only the latter component. Since Q 1 >0 usually, S is an unbiased estimate of the actual process variance only when the process level is constant. Thus, one will use the F-test in order to justify the assumption of a time constant process level. If this is the case, it follows that σ S is an unbiased estimator for σ. However, it does not follow from this that S mn is also an unbiased estimator of σ 0. Analogous to E S σ, one has to divide S mn by a correction term a m.n in order to achieve unbiasedness. Thus, the corrected total sample standard deviation S a results in an unbiased estimator of σ 0. The variance of this estimator is given by, V S 1 a. σ = b σ with b v =1 a ν. The values of the terms a ν and b ν are taken from the statistical table of Special factors for ν = 2, For ν = 20, one can use the approximations; a ν 1 ; b ν ν. ν 2) Second estimator (E 2 ): Corrected mean of the sample standard deviations For normally distributed pre-run variables X ij the m standard deviations S 1n,,S mn of 5

7 S in =+S have the expectations E(S in )=a n σ 0 and the variance V(S in )= 1a σ, i=1,,m. Thus the corrected mean of the sample standard deviations S σ σ is an unbiased estimator of σ 0. Its variance is V S. For n = 2 20 the ratios b n /a n are taken from the table of a statistical table for Special factors. 3) Third estimator (E 3 ): Corrected square root of the average sample variance As an alternative to S, we also present S a. This estimator, which is based on S = S, is also unbiased and has variance: S V = b σ. 4) Fourth estimator (E 4 ): Corrected mean of the sample ranges If the pre-run variables X ij pre-run are normally distributed, then the sample ranges R 1n,,R mn of the individual samples have a distribution with expectation ER =d n σ 0 and variance VR = e σ ; i=1,2,,m. The corrected mean of the sample ranges R d is then unbiased and has variance V R = σ. The values of the factors e n and d n as well as those of the ratios e n /d n can be found from the statistical table of Special factors. 5) Fifth estimator (E 5 ): Corrected median of the sample ranges Instead of the corrected mean R d, one can also use the corrected median of the sample ranges R d for estimating σ 0. Its variance is V R σ. The conversion factors d and e can also be found from the statistical table of Special factors. When comparing the variances of these five unbiased estimators, the following ranking can be verified for n>2: S V S V V S V R V R. 6

8 Hence S mn is the best of all the four estimators of spread. Thus, the corrected total sample standard deviation has the highest efficiency, but the difference from the other four estimators are small for 2<n 12 (m arbitrary). The Table-1.1 shows the Estimation of the target values for the process level and process spread based on pre-run data. Θ Estimator of process terms Variance of the estimator Rank μ 0 µ X µ X µ X µ X σ S a σ S a Vµ VX σ Vµ V X σ Vµ VX σ Vµ VX σ 1 c c 4 c 2 c 3 Vσ b σ 1 Vσ σ 3 σ S σ 0 a Vσ b σ 2 σ R d Vσ σ 4 σ R d Vσ σ 5 Table-1.1: Estimation of the target values for the process level and process spread based on pre-run data Uses of control chart Control charts are essential tools of continuous quality control help in reducing the variability (Engineering Statistics Handbook. (2003)). Theses charts monitor performance overtime. Allow process corrections to prevent rejections. Trends and out-of-control conditions are immediately detected. These may be used in making judgment about the process, such as establishing whether the process was in a state of control at a given time. These charts are also useful in determining the capability of the process. 7

9 Again, they may be used in an ongoing effort to maintain the centering and spread of the process, implies maintaining control. Furthermore, these charts are used to detect clues for process change. This is at the heart of process improvement and troubleshooting. Additionally, there are 5 reasons for their popularity, they are; 1. Control charts are a proven technique for improving productivity. 2. Control charts are effective in defect prevention. 3. Control charts prevent unnecessary process adjustments. 4. Control charts provide diagnostic information. 5. Control charts provide information about process capability Hypothesis test There is a close connection between control charts and hypothesis testing. Essentially the control chart is a test of the hypothesis that the process is in a state of statistical control. A point plotting within the control limits is equivalent to failing to reject the hypothesis of statistical control, and a point plotting outside the control limits is equivalent to rejecting the hypothesis of statistical control. Thus as in hypothesis testing, we may consider the probability of type-i and type-ii error of the control chart. It is occasionally helpful to use the Operating Characteristic (OC) curve to display its probability of a type-ii error which implies the ability of the control chart to detect process shifts of different magnitudes (Gupta (2002)) Types of control chart Control charts may be classified into two types. They are; (a) Attribute control chart (b) Variable control chart Attribute control chart Many quality characteristics are unable to measure on a continuous scale or even a quantitative scale. In this case, we may judge each unit of product as either confirming (non-defective) or non-confirming (defective) on the basis of whether or not it possesses certain attributes, or we may count the number of non-conformities (defects) appearing on a unit of product. Control chart for such quality characteristics are called attributes control charts. 8

10 Variable control chart If the quality characteristic can be measured and expressed as a number on some continuous scale of measurement, it is usually called a variable. In such cases, it is convenient to describe the quality characteristic with a measure of central tendency and a measure of variability. Control charts for central tendency X and variability R are collectively called continuous random-variable or measurement control charts. Simply the charts based upon measurements of quality characteristics are called as control charts for variables. We consider these charts elaborately in this research since these are more economical means of controlling quality than attribute control chart. Some of the measurements like diameter, electric resistance, thickness will come under this measurement control chart. 1.2 CONTROL CHARTS FOR MEASUREMENTS ( AND R ) Assume that X~N(μ, σ 2 ) and let X be the sample mean based on a sub group of size n. Then it follows that X~N(μ, σ ). The corresponding control limits are exhibited under the conditions explained below; a) When μ and σ are known Suppose (μ, σ 2 ) are known from past data. Then the control limits for mean chart (X) are computed by definition we have, UCL = μ + Z α/2.σ X = μ + Z α/2 σ n CL = μ LCL = μ - Z α/2.σ X = μ - Z α/2 σ n To employ 3σ limits, we get, (i.e. put Z α/2 = 3) UCL = μ + 3σ n CL = μ LCL = μ - 3σ n The factor 3 in 3σ n is obtained on the Standard Normal Deviate corresponding to α = which mean α/2 = on each side, due to symmetry. Hence, Z α/2 = 3. 9

11 b) When μ and σ are unknown Since the parameters μ and σ are unknown, we consider μ and σ are their estimates and then the 3σ control limits are given by, UCL = μ+ 3σ/ n CL = μ LCL = μ - 3σ/ n Now these unknown parameters are estimated by using the following procedure. Estimation of unknown parameters We consider the use of at least 20 to 25 preliminary samples. Suppose these m preliminary samples are available, each of size n. Typically, n will be 4, 5 or 6; these relatively small samples often arise from the construction of rational subgroups. They represent samples drawn from the process at regular time intervals. Estimation of Process mean Let the sample mean for i th sample be X. Then we estimate the mean of the process, μ, by using the grand mean,x X where X x. Thus we get μ X. Estimation of Process variance To estimate the variance σ 2 one method is to use σ = R where R R and also R i = (Max Min) i for i th sample and the constant d 2 is found from the statistical tables for given value of n. Thus the resultant control limits for the mean chart are given by, UCL = μ + 3σ/ n = X R CL = μ = X LCL = μ - 3σ/ n = X R We denote the quantity A 2 by possible to rewrite the above the control limits as, that depends only on the sample size, so it is 10

12 UCL = X A 2 R CL = X LCL = X A 2 R The constant A 2 is tabulated for various sample sizes in the statistical table. These limits become valid only if R is a reliable estimate of the process variance. From the above we say that the sample range is related to the process standard deviation. Therefore, process variability may be controlled by plotting values of R from successive samples on a control chart. Control chart for R chart: To determine the control limits for R chart we need an estimate of σ R. Assuming that the quality characteristic is normally distributed, σ R can be found the distribution of the relative range W=R/σ. The standard deviation of W say, σ W = d 3 is a known function of n. Thus the computation is given by, Consider the range W= R/σ R = Wσ The standard deviation of R is given by, σ R = σ W σ σ R = d 3 σ Since σ is unknown, we may estimate σ R by, σ R = d 3 R Consequently, we expect the sample ranges which can be used to design the R-chart as follows, by definition: UCL = R + 3 σ R = R + 3 d R = R 1 3 d R = D 4 R, where D 4 = 1 3 d R 11

13 CL LCL = R = R - 3 σ R = R - 3 d R = R 1 3 d R = D 3 R, where D 3 = 1 3 d R 1.3 OPERATING CHARACTERISTIC CURVE AND AVERAGE RUN LENGTH OF CHART Definition: The curve that describes the ability of the charts in detecting shifts in process quality is said to be Operating Characteristic (OC) curve OC for - chart Consider the OC curve for an x-bar chart such that the quality characteristic (X) is said to follow N(μ,σ 2 ), where the standard deviation σ is a known constant. Assuming that X~N(μ 0, σ, Let the process mean undergoes a shift from μ 0 to μ 1. Let μ = μ 0 be the initial in-control mean value shifts to another value due to some assignable cause and is given by μ= μ 1 = μ 0 +kσ, where k is a constant. Let Φ(x) denotes the cumulative standard normal distribution at x. Then the probability of detecting this shift whose magnitude is kσ is said to be β-risk and is given by, β = P {LCL X-bar UCL μ = μ 0 +kσ}. Since x-bar follows N(μ,σ 2 ), we can write β as, β Φ UCLµ σ σ LCLµ σ σ By substituting UCL µ k σ and LCL µ k σ and solving we get, β = Φ3 k n - Φ3 k n (1.1) A plot of β as a function of k gives the operation characteristic curve, which appears as shown in Figure

14 OC-Curve Φ n=5 n=10 n= Shift (k) Figure-1.2: OC curve for x-bar chart Thus β is said to be Operating Characteristic curve at different values of k and is also called as probability of not detecting a shift. Moreover, the probability that a shift will be detected on the first subsequent sample is (1- β) ARL for - chart The average number of sample points that are plotted within limits before getting an out of control condition is said to be Average Run Length (ARL). The length of run is a random variable following Geometric Distribution (Montgomery (1991)). In other words, ARL is an expected number of samples taken before the shift is detected. Thus, the ARL for the shewhart control chart when the process is out of control is given by the following theorem (Duncan (1955)). Theorem: Let Y is the random variable such that Y = r(1,2, ) implies that the r th sample mean is the first to exceed UCL then Y follows the Geometric Distribution with the probability function given by; f(r) = P(Y =r) = q p for r = 1,2, Proof: Let Y denote the number of sample means lying within UCL and LCL. Define p n = P{Detecting a shift with sample size n} E(Y) = r fr = r q p = p 1 2q 3q 13

15 Putting 1 2q 3q = S gives q Sq 2q 3q 1 q S = 1+ q q q = S = = Therefore, ARL kβ 1 β β If the process is in control then, ARL k1α α α In general ARL =, where p is the probability that a single point exceeds the control limits. For the X-chart with the usual 3σ limits, p = is the probability that a single point falls outside the control limits usually when the process is in control. So the ARL of the X-chart is given by, ARL = P = α = β = (when the process is in control) (when the process is out of control). 370 This implies, the process is really in control. Moreover, it is the point or signal where the process is going out of control called as false-alarm. This false-alarm is also defined as, the out-of-control signal about every 370 samples. 1.4 SPECIFICATION LIMITS AND MODIFIED CONTROL LIMITS For every dimension of the product there will be lower and upper specification limits denoted by LSL and USL. Each individual component is supposed to have a dimension within these limits failing which the item become defective. Thus the proportion of defective items can be estimated as p = P[X < LSL] + P[X > USL] which can be computed using the distribution of X. 14

16 When the process is under control it is sometimes possible to allow shift in the mean to some extent so that the process continues to produce no defectives even if the mean gets shifted. These limits are known as modified control limits as discussed below Modified (Rejection) control limits Modified control limits give the relationship between specification limits and X values in X-chart used to allow a shift in process levels within permissible limits. To specify the control limits for this chart, assume the process output is normally distributed. For the sample size n, the maximum and minimum value of upper and lower control limits for X -chart are also known as Upper Rejection Limit (URL) and Lower Rejection Limit (LRL) respectively. This chart is concerned only with detecting whether the true process mean μ is located or not. For the process fraction nonconfirming to be <δ a specified value, we must show μ L μ μ U, here μ L and μ U indicates the smallest and largest permissible value of μ, then we get (Montgomery (1991)), μ L = LSL + Z δ σ μ L = USL - Z δ σ where Z δ is the upper [100(1-δ)]% point of the standard normal deviation given by, Z α ΦZdz = α, which implies the type-i error of α, then we get, i) URL UCL µ U Z ασ USLZ δ σ Z ασ USL Z δ Z α σ ii) LRL LCL µ L Z ασ LSLZ δ σ Z ασ LSLZ δ Z α σ Here Z α σ is said to be distance between μ L and LSL or μ U and USL. This modified control chart is equivalent to testing the hypothesis that the process mean lies between μ L and μ U. which is written as H 0 : μ L μ μ U. 15

17 Also, a good estimate of σ helps to design this modified control chart. If there is a shift in the process variability then modified control limits are not appropriate. 1.5 CONTROL CHARTS WITH MEMORY A major disadvantage of any Shewhart control chart is that it only uses the information about the process contained in the last plotted point and it ignores any information given by the entire sequence of points. This feature makes the Shewhart control chart relatively insensitive to small shifts in the process say for 1.5σ or less. The distribution of run length on Shewhart control charts has geometric distribution which has memory less property. This occurs because the successively plotted points are statistically independent. Such charts will have long ARL which indirectly means that the weighting time to detect a shift will be quiet long. An alternative method is to device a chart that plots a cumulative statistic instead of absolute value of the statistic from a sample. There are three popularly used control charts which have memory as given below. 1) Cumulative sum (CUSUM) control chart 2) Moving average (MA) control chart 3) Exponentially weighted moving average (EWMA) control chart These charts take into account the information on the previously plotted points. Mittag and Rinne (1993) discussed such nomenclature. Further for Shewhart chart the Run length has geometric distribution which is memory-less, when these charts do not have this property Cumulative sum (CUSUM) control chart These charts were first proposed by Page (1954). This CUSUM chart is a good alternative to Shewhart control chart when small shifts are important. It directly incorporates all the information in the sequence of sample values by plotting the CUSUM of the deviation of the sample values from a target value μ 0. Suppose the 16

18 process is designed to produce output with mean μ 0 (target). Then the test statistic at time t is called CUSUM and is given by, S t = x μ, j = 1, 2 n. S i = x μ + x μ S i = S i-1 + x μ Thus CUSUM chart is a plot of S i against the sample number. Here S i could be either positive or negative. To test whether the process has undergone a shift, we apply the V- mask. The occurrence of sudden upward trend in the process of the CUSUM indicates a shift. The amount of shift in the process between j th and i th sample is estimated by solving; μ μ S S i j If the mean shifts upward to some value say, μ 1 > μ 0. Then the shift developed is positive else negative when μ 1 < μ 0 and μ 1 = μ 0 indicates no shift. The occurrence of positive or negative shift indicates the effect of assignable causes. V-mask V-mask is a geometric procedure developed by Barnard (1959) to verify whether the process has undergone the shift or not. It is a V-shaped notch in a plane that can be placed at different points on the CUSUM chart to measure the shifts. In this device, θ is a parameter represents the half angle of the V-mask and d is a parameter represents lead distance where these parameters depend on the magnitude of the shift used to determine the performance of the CUSUM chart. To detect the shift we use, = δσ or δ = σ where σ is the SD of X. θ = tan, where A is a scale factor normally lie between σ A and 2σ drawn on the Y-axis. Lead distance d = δ ln β, whereα, β are type-i and type-ii risks. α 17

19 V-mask construction Place the V-shaped notch on the CUSUM chart with the point O on the last value of S i such that the horizontal line OP is parallel to X-axis. Verify the two limbs of the V-mask carefully. So that if all the previous cumulative sums s 1,s 2, s 2,,s i-1 lie or fall within the two arms or limbs of the device, then the process is said to be in control. If any of the previously plotted means fall or lie below the lower arm of the mask, then upward shift is indicated. Similarly if they fall above the upper arm of the mask, then downward shift in the mean is indicated. Since V-mask device moves from point to point, fixed control limits will not exist Moving Average (MA) control chart It is an extension over the Shewhart control chart, in the sense of its memory. This is the chart based on a simple, un-weighted moving average. Suppose that, sample size n is collected with the corresponding sample means x,x,..,x,...,x. Define M t a moving average of w observations at time t given by; M t =,,.., = x,x,...,x, for t = 1,2, At time period t, the old sample is dropped and the new one is added to the set, is given by, as a result for w=3, we get, M x x x 3 M x x x 3 M x x x 3 Thus, M t, t gives the summary of the most recent history values. The process mean is given by, E(M t ) =μ. The process variance is given by, VM V X 18

20 VX σ σ σ Then the 3σ control limits are given by, UCL = X + 3 σ CL = X LCL = X - 3 σ Exponentially Weighted Moving Average (EWMA) control chart It is one of the charts with memory introduced by Roberts (1959). The statistic Z t to be plotted is a linear combination of current mean x and Z t-1 with damping fact λ. The sum of weights in Z t decrease geometrically and hence this chart for Z t is also known as Geometric Moving Average (GMA) chart. It is a good alternative to the Shewhart control chart in detecting small shifts. Its performance is approximately equivalent to that of the CUSUM control chart. Moreover this control chart is easier to setup and operate in some ways Construction of EWMA Consider all the n observations sequentially in constructing the moving average chart defined by a recursive formula expressed as, Z t = λ x + (1-λ) Z t-1, for Z 0 = x and 0 < λ 1 Thus at any point t, the test statistic Z t can be evaluated. In general it gives, Z t = λ x + λ (1-λ) x + (1- λ) 2 Z t-2 Z t = λ 1 λ x (1- λ) t Z 0 19

21 Here the values 1,1 λ, 1 λ,, 1 λ are said to be the weights of x. Further we get, the sum of weights will be, λ 1 λ = 1-(1-λ) t as t increases, σ Z tends to limited value. Hence Z t is a statistic, where sum of weights is unity. Control limits Consider Z t = λ x + (1-λ) Z t-1, t = 1,2,,n. We obtain Z 1,Z 2,...,Z n values where 0 λ 1 is a constant and Z 0 is the starting value of EWMA which is given by, Z 0 = μ 0. The limits are given by, UCL = μ 0 + 3σ λ λ 1 1λ CL = μ 0 LCL = μ 0-3σ λ λ 1 1λ Thus among the control charts with memory, the CUSUM, MA & EWMA (or GMA) charts are treated as effective when the process data undergoes changes over time. When the process is stabilized (steady state) then a Shewhart control chart can be used. 1.6 PROCESS CAPABILITY MEASURES The Process capability has the following definitions; 1. It is a measure of the relationship between the actual process performance and the manufacturing specifications. 2. It is also defined as the ratio of the distance from the process center to the nearest specification limit to a measure of the process variability. 3. The natural behavior of the process after un-natural disturbances is eliminated. 4. Shortly it can be defined as the uniformity of the process. 5. The minimum spread of a specific measurement variation which will include 99.73% of the measurement from the given process. 20

22 1.6.1 Process Capability Ratios The formula that expresses the process capability in the quantitative way is said to be Process Capability Ratio, given by (Montgomery (1991)), PCR USLLSL σ N (1.1) where USL and LSL are the upper and lower specification limit respectively. This PCR implicitly assumes that the process is centered at the nominal dimensions. Since the value of process S.D. (σ) is unknown, replace it by an estimate, σ, where σ = R considered when the quality characteristic of interest is continuous. The estimate of Process Capability Ratio is given by, PCR = USLLSL, for sample SD it is given that, PCR = USLLSL. The PCR in the form of percentage can be converted by using the S equation P= 100 called as percentage of the specifications band. PCR accounts PCR only the process standard deviation but not the process mean relative to the specifications. PCR considers concentrates and measures the spread of the specifications specifically 6σ (Montgomery (2011)) Process Capability Ratios (PCR) for an off-center process It is convenient to think of PCR as a measure of potential capability. I.e. Capability with a centered process. In case if the process is not centered, then the resultant ratio often used is known as measure of actual capability given by, PCR k = C pk = min USLμ σ, μlsl σ = minpcr U,PCR L = minc U, C L (1.2) σ Here PCR U = C U = USLμ defines only one sided upper specifications. Similarly, σ we get PCR L = C L = μlsl defines only one sided lower specifications. σ Therefore, estimates one-sided specifications are given by the following equations, C U = USLμ σ & C L = μlsl. σ 21

23 Comparison of C p & C pk Some of the measures of C p relative to C pk is as follows, 1. If C p = C pk, then the process is said to be centered at the nominal dimensions. 2. If C P C pk, then the process is said to be off-centered. Since it is not operating at the midpoint of the interval between the specifications. Comparing C p with unity 1. If C p = 1, more fraction nonconforming units will be produced (AIAG (1995)). The process is said to be normally distributed, which means the fraction nonconforming is 0.27% or 2700 ppm. Here natural tolerance limits lies on the specifications. 2. If C p < 1, the process is very yield-sensitive and a large number of non-confirming units will be produced. 3. If C p > 1, very few defective or nonconforming units will be produced. Here the process natural tolerance limits lies inside the specifications. Comparing C pk with zero 1. If C pk = 0, then the process mean is exactly equal to one of the specifications. 2. If C pk < 0, then the process mean lie outside the specifications. 3. If C pk > 0, then the process mean lie inside the specifications Confidence interval estimation of C p To produce a point estimate of C p, put σ S in C, such that, C USLLSL. Then a 1001 α % confidence interval estimate of C p can be obtained from S USLLSL S, χ α USLLSL σ USLLSL χ α S, = C χ α, C C χ α, where χ α, and χ α, are the lower α/2 and upper α/2 percentage points of the χ 2 distribution with (n-1) degrees of freedom. These percentage points will be found in the statistical tables. 22

24 1.6.4 Six sigma approach and PCI Sigma is used to designate the distribution or spread of any process about the mean. Sigma measures the capability of the process to perform defect free work. As σ increases, costs go down, cycle time goes down and customer satisfaction goes up. It is a quality improvement programme to reduce the number of defects to as low as 3.4ppm. Approach to 6σ Quality in Mahajan (2006) gives that the Motorola originally developed the 6σ, which was a quality improvement program that aimed to reduce the number of defects to as low as 3.4ppm in It uses the normal distribution and the strong relationships between product non-conformities or defects and product yield, relationship, cycle time, inventory, schedule, and so on. Statistically, 6σ refers to a process in which the range between the mean of a process quality measurement and the nearest specification limit is at least six times the standard deviation of the process. The statistical objectives of 6σ are to centre the process on the target and reduce process variation. A 6σ process width approach zero defects with only 3.4 defects per million opportunities (dpmo) whereas the 4σ width has 6210dpmo. In comparison, the goal of many quality initiatives was to obtain a C pk of at least 1.0, which approximately transform to 3σ. However, this level of quality still produces a defect rate of 66,810dpmo. Some of the uses of 6σ can be listed below; 1. 6σ level is closest to zero defects. 2. Total customer satisfaction can be achieved. 3. Reduction of cost is possible. 4. Gives higher yield. 5. Gives improved reliability. Thus Process Capability Analysis is a vital area of SPC. The role of normality and non-normality of X has to be discussed in order to develop new procedures. In the next chapter a related area called Measurement System Analysis is discussed. 23