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Statistical Process Control Prof Statistical Process Control Prof. Robert Leachman IEOR 130 Fall, 2020 Introduction • Quality control is about controlling manufacturing or service operations such that the output of those operations conforms to specifications of acceptable quality. • Statistical process control (SPC), also known as statistical quality control (SQC), dates back to the early 1930s and is primarily the invention of one man. The chief developer was Walter Shewhart, a scientist employed by Bell Telephone Laboratories. • Control charts (discussed below) are sometimes termed Shewhart Control Charts in recognition of his contributions. Introduction (cont.) • W. Edwards Deming, the man credited with exporting statistical quality control methodology to the Japanese and popularizing it, was an assistant to Shewhart. • Deming stressed in his teachings that understanding the statistical variation of manufacturing processes is a precursor to designing an effective quality control system. That is, one needs to quantitatively characterize process variation in order to know how to produce products that conform to specifications. Introduction (cont.) • Briefly, a control chart is a graphical method for detecting if the underlying distribution of variation of some measurable characteristic of the product seems to have undergone a shift. Such a shift likely reflects a subtle drift or change to the desired manufacturing process that needs to be corrected in order to maintain good quality output. • Control charts are very practical and easy to use, yet they are grounded in rigorous statistical theory. In short, they are a fine example of excellent industrial engineering. History • Despite their great value, initial use of control charts in the late 1930s and early 1940s was mostly confined to Western Electric factories making telephony equipment. (Western Electric was a manufacturing subsidiary of AT&T.) Evidently, the notion of using formal statistics to manage manufacturing was too much to accept for many American manufacturing managers at the time. • Following World War II, Japanese industry was decimated and in urgent need of rebuilding. It may be hard to imagine today, but in the 1950s, Japanese products had a low-quality reputation in America. • W. Edwards Deming went to Japan in the 1950s, and his SPC teachings were quickly embraced by Japanese management. History (cont.) • Through the 1960s, 1970s and 1980s, many Japanese-made products were improved dramatically and eventually surpassed competing American- made products in terms of quality, cost and consumer favor. • Many American industries lost substantial domestic market share or were driven completely out of business. • This led to a “quality revolution” in US industries during the 1980s and 1990s featuring widespread implementation and acceptance of SPC and other quality management initiatives. Important additions were made to quality control theory and practice, especially Motorola’s Six Sigma controllability methodology (to be discussed). • It is ironic that a brilliant American invention was not accepted by American industries until threatened by Japanese competition making good use of that invention. Control Charts • Control charts provide a simple graphical means of monitoring a process in real time. Today, they have gained wide acceptance in industry – you would be hard-pressed to find a volume manufacturing plant producing technologically advanced products anywhere in the world that is not using SPC extensively. • A control chart maps the output of a production process over time and signals when a change in the probability distribution generating observations seems to have occurred. To construct a control chart one uses information about the probability distribution of process variation and fundamental results from probability theory. Central Limit Theorem • Most types of control charts are based on the Central Limit Theorem of statistics. Roughly speaking, the central limit theorem says that the distribution of a sum of independent and identically distributed (IID) random variables approaches the normal distribution as the number of terms in the sum increases. • Generally, the distribution of the sum converges very quickly to a normal distribution. For example, consider a random variable with a uniform distribution on the interval (0, 1). See Figure 1. Figure 1 Probability density of a uniform variate on (0, 1) Central Limit Theorem • Now assume the three random variables X1, X2, and X3 are independent, each of which has the uniform distribution on the interval (0, 1). Consider the random variable W = X1 + X2 + X3. If one plots the distribution of W, it tracks remarkably close to a normal distribution with the same mean and variance. See Figure 2. If we were to continue to add independent random variables, the agreement would be even closer. Figure 2 Density of the sum of three uniform random variables Central Limit Theorem (cont.) • If (X1, X2, … , Xn) is a random sample, the sample mean is defined as 1 n X = ∑ X i . n i=1 • The Central Limit Theorem tells us that, regardless of the distribution of Xi, the sample mean will have a normal distribution, provided the variables are IID. • Suppose that a random variable Z has the standard normal distribution (i.e., mean 0 and variance 1). Then, according to the table of the unit normal distribution (see Table A-1 in the SPC notes), P{− 3 ≤ Z ≤ 3}= 0.9974. Central Limit Theorem (cont.) • In other words, the likelihood of obtaining a value of Z either larger than 3 or less than -3 is 0.0026, or roughly 3 chances in 1,000. If such a value of Z is encountered, it is more likely that the IID assumption has been violated, i.e., there has been a drift or shift of the process that needs to be corrected. • This is the basis of the so-called three-sigma limits that have become the de facto standard in SPC. Back to Control Charts • Consider the average of n samples of a process variable. The Central Limit Theorem tells us it should be (approximately) normally distributed. Suppose the mean of each� sample is µ and the standard deviation is σ. The mean of is expressed as 1 n 1 n 1 n 1 n 1 �EX = E ∑ X i = E∑ X i = ∑ EX i = ∑ µ = (n)µ = µ. n i=1 n i=1 n i=1 n i=1 n • The variance of is derived as follows: 1 n 1 n �= = Var X Var ∑ X i 2 Var ∑ X i . n i=1 n i=1 • Now n 2 Var∑ X i = n Var(X i ) = nσ , i=1 Control Charts (cont.) • Therefore, = . 2 • Hence the standard deviation of is� σ � . n • Therefore, the standardized variate X − µ Z = σ n has (approximately) the normal distribution with mean zero and unit variance. Control Charts (cont.) • It follows that X − µ P − 3 ≤ ≤ 3 = 0.9974, σ n or equivalently, 3σ 3σ Pµ − ≤ X ≤ µ + = 0.9974. n n • That is, the likelihood of observing a value of either larger than + or 3 less than is 0.0026. Such an event is sufficiently� rare that if it were to occur, it is more3 likely to have been caused by a shift in the population mean µ, than to have− been the result of chance. • This is the basis of the theory of control charts. Control Charts (cont.) • A manufacturing process is said to be in statistical control if a stable system of chance causes is operating. That is, the underlying probability distribution generating observations of the process variable is not changing with time. • When the observed value of the sample mean of a group of observations falls outside the appropriate three-sigma limits, it is likely that there has been a change in the probability distribution generating observations. When an observed value falls outside these limits, it is customary to say the process is out-of-control, i.e., it is out of statistical control. Control Charts for Continuous Variables • For a continuous variable describing the quality of the process, control charts may be set up to track both the mean of the variable over fixed-size samples (the -chart) and its range (maximum minus minimum) over fixed- size samples (the R-chart). � • An out-of-control signal on the -chart indicates that the process mean has shifted; an out-of-control signal on the R-chart indicates that the process variance has changed. Either out�-of-control signal should trigger a halt of the process. • There should be an investigation to ascertain if and why the process is no longer in statistical control. (An alternative explanation is that the observations were not measured correctly.) • The investigation culminates in corrective action to restore the process to statistical control, whereupon manufacturing is resumed. In this way, quality losses can be kept to a minimum. and Charts • An -chart requires that collection of data on the process variable be �broken down into subgroups of fixed size. The most common size of subgroups� in industrial practice is n = 5. The subgroup size n ought to be at least four in order to have an accurate application of the Central Limit Theorem. • To construct an -chart, it is necessary to estimate the sample mean and the sample variance of the process variable. This could be done using standard statistical� estimates from an initial population of N measurements (ideally, N much larger than n) of the variable: 1 1 = , = . 1 2 2 � � � − � =1 − =1 and Charts • However, it is not recommended that one use the sample standard �deviation s as an estimator of σ when constructing an -chart. For s to be an accurate estimator of σ, it is necessary that the underlying mean of the sample population be constant. Because the� purpose of an -chart is to determine whether a shift in the mean has occurred, we should not assume a priori that the mean is constant when estimating� σ.
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