Statistical Quality Control

Bowl of beads

The quality guru W. Edwards Deming conducted a simple experiment in his seminars with a bowl of beads. Many were colored white but a percentage of red beads was randomly mixed in the bowl. A participant from the seminar was provided a paddle made with indentations so that 50 beads at a time could be scooped from the bowl. The participant was allowed to use only the paddle and instructed to scoop only white beads (repeated multiple times with beads replaced). The red beads were considered to be defectives. Of course, this was difficult to do, and each scoop resulted in a count of red beads. Deming plotted the fraction of red beads from each scoop and used the results to make several points. As was clear from the scenario, this process was beyond the participant's ability to make simple improvements. The process needed to be changed (reduce the number of red beads), which is the responsibility of management. Furthermore, many business processes have this type of characteristic, and it is important to learn from the data whether the variability is common, intrinsic to the process, or is the result of some special cause. This distinction is important for the type of process control or improvements to be applied. Refer to the example of control adjustments in Chapter 1. Control charts are primary tools to understand process variability, and that is main topic of this chapter.

15-1 Quality Improvement and Statistics

The quality of products and services is a major decision factor in most businesses today. Regardless of whether the consumer is an individual, a corporation, a military defense program, or a retail store, a consumer making purchase decisions is likely to consider quality of equal importance to cost and schedule. Consequently, quality improvement is a major concern to many U.S. corporations.

Quality means fitness for use. For example, we purchase automobiles that we expect to be free of manufacturing defects and that should provide reliable and economical transportation, a retailer buys finished goods with the expectation that they are properly packaged and arranged for easy storage and display, or a manufacturer buys raw material and expects to process it with no rework or scrap. In other words, all customers expect that the products and services they buy meet their requirements. Those requirements define fitness for use.

Quality or fitness for use is determined through the interaction of quality of design and quality of conformance. By quality of design, we mean the different grades or levels of performance, reliability, serviceability, and function that are the result of deliberate engineering and management decisions. By quality of conformance, we mean the systematic reduction of variability and elimination of defects until every unit produced is identical and defect free.

Some confusion exists in our society about quality improvement; some people still think that it means gold-plating a product or spending more money to develop a product or process. This thinking is wrong. Quality improvement means better quality of design through improved knowledge of customers' requirements and the systematic elimination of waste. Examples of waste include scrap and rework in manufacturing, inspection and testing, errors on documents (such as engineering drawings, checks, purchase orders, and plans), customer complaint hotlines, warranty costs, and the time required to do things over again that could have been done right the first time. A successful quality-improvement effort can eliminate much of this waste and lead to lower costs, higher productivity, increased customer satisfaction, increased business reputation, higher market share, and ultimately higher profits for the company.

Statistical methods play a vital role in quality improvement. Some applications are outlined here:

1. In product design and development, statistical methods, including designed experiments, can be used to compare different materials, components, or ingredients, and to help determine both system and component tolerances. This application can significantly lower development costs and reduce development time.
2. Statistical methods can be used to determine the capability of a manufacturing process. Statistical process control can be used to systematically improve a process by reducing variability.
3. Experimental design methods can be used to investigate improvements in the process. These improvements can lead to higher yields and lower manufacturing costs.
4. Life testing provides reliability and other performance data about the product. This can lead to new and improved designs and products that have longer useful lives and lower operating and maintenance costs.
5. Regression methods are often used to determine key process indicators that link with customer satisfaction. This enables organizations to focus on the most important process measurements.

Some of these applications have been illustrated in earlier chapters of this book. It is essential that engineers, scientists, and managers have an in-depth understanding of these statistical tools in any industry or business that wants to be a high-quality, low-cost producer. In this chapter, we provide an introduction to the basic methods of statistical quality control that, along with experimental design, form the basis of a successful quality-improvement effort.

15-1.1 STATISTICAL QUALITY CONTROL

This chapter is about statistical quality control, a collection of tools that are essential in quality-improvement activities. The field of statistical quality control can be broadly defined as the use of those statistical and engineering methods in measuring, monitoring, controlling, and improving quality. Statistical quality control is a field that dates back to the 1920s. Dr. Walter A. Shewhart of Bell Telephone Laboratories was one of the early pioneers of the field. In 1924, he wrote a memorandum showing a modern control chart, one of the basic tools of statistical process control. Harold F. Dodge and Harry G. Romig, two other Bell System employees, provided much of the leadership in the development of statistically based sampling and inspection methods. The work of these three men forms much of the basis of the modern field of statistical quality control. The widespread introduction of these methods to U.S. industry occurred during World War II. Dr. W. Edwards Deming and Dr. Joseph M. Juran were instrumental in spreading statistical quality-control methods since World War II.

The Japanese have been particularly successful in deploying statistical quality-control methods and have used such methods to gain significant advantage over their competitors. In the 1970s, U.S. industry suffered extensively from Japanese (and other foreign) competition; that has led, in turn, to renewed interest in statistical quality-control methods in the United States. Much of this interest focuses on statistical process control and experimental design. Many U.S. companies have implemented these methods in their manufacturing, engineering, and other business organizations.

15-1.2 STATISTICAL PROCESS CONTROL

It is impractical to inspect quality into a product; the product must be built right the first time. The manufacturing process must therefore be stable or repeatable and capable of operating with little variability around the target or nominal dimension. Online statistical process control is a powerful tool for achieving process stability and improving capability through the reduction of variability.

It is customary to think of statistical process control (SPC) as a set of problem-solving tools that may be applied to any process. The major tools of SPC^* are

1. Histogram
2. Pareto chart
3. Cause-and-effect diagram
4. Defect-concentration diagram
5. Control chart
6. Scatter diagram
7. Check sheet

Although these tools are an important part of SPC, they compose only the technical aspect of the subject. An equally important element of SPC is attitude—the desire of all individuals in the organization for continuous improvement in quality and productivity through the systematic reduction of variability. The control chart is the most powerful of the SPC tools.

15-2 Introduction to Control Charts

15-2.1 BASIC PRINCIPLES

In any production process, regardless of how well-designed or carefully maintained it is, a certain amount of inherent or natural variability always exists. This natural variability or “background noise” is the cumulative effect of many small, essentially unavoidable causes. When the background noise in a process is relatively small, we usually consider it an acceptable level of process performance. In the framework of statistical quality control, this natural variability is often called a “stable system of chance causes.” A process that is operating with only chance causes of variation present is said to be in statistical control. In other words, the chance causes are an inherent part of the process.

Other kinds of variability may occasionally be present in the output of a process. This variability in key quality characteristics usually arises from three sources: improperly adjusted machines, operator errors, or defective raw materials. Such variability is generally large when compared to the background noise, and it usually represents an unacceptable level of process performance. We refer to these sources of variability that are not part of the chance cause pattern as assignable causes. A process that is operating in the presence of assignable causes is said to be out of control^†.

Production processes often operate in the in-control state, producing acceptable product for relatively long periods of time. Occasionally, however, assignable causes occur, seemingly at random, resulting in a “shift” to an out-of-control state in which a large proportion of the process output does not conform to requirements. A major objective of statistical process control is to quickly detect the occurrence of assignable causes or process shifts so that investigation of the process and corrective action can be undertaken before many nonconforming units are manufactured. The control chart is an online process-monitoring technique widely used for this purpose.

Recall the following from Chapter 1. Figure 1-11 illustrates that adjustments to common causes of variation increase the variation of a process whereas Fig. 1-12 illustrates that actions should be taken in response to assignable causes of variation. Control charts may also be used to estimate the parameters of a production process and, through this information, to determine the capability of a process to meet specifications. The control chart can also provide information that is useful in improving the process. Finally, remember that the eventual goal of statistical process control is the elimination of variability in the process. Although it may not be possible to eliminate variability completely, the control chart helps reduce it as much as possible.

A typical control chart is shown in Fig. 15-1, which is a graphical display of a quality characteristic that has been measured or computed from a sample versus the sample number or time. Often, the samples are selected at periodic intervals such as every few minutes or every hour. The chart contains a center line (CL) that represents the average value of the quality characteristic corresponding to the in-control state. (That is, only chance causes are present.) Two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL), are also shown on the chart. These control limits are chosen so that if the process is in control, nearly all of the sample points fall between them. In general, as long as the points plot within the control limits, the process is assumed to be in control, and no action is necessary. However, a point that plots outside of the control limits is interpreted as evidence that the process is out of control, and investigation and corrective action are required to find and eliminate the assignable cause or causes responsible for this behavior. The sample points on the control chart are usually connected with straight-line segments so that it is easier to visualize how the sequence of points has evolved over time.

Even if all the points plot inside the control limits, if they behave in a systematic or nonrandom manner, this is an indication that the process is out of control. For example, if 18 of the last 20 points plotted above the center line but below the upper control limit, and only two of these points plotted below the center line but above the lower control limit, we would be very suspicious that something was wrong. If the process is in control, all plotted points should have an essentially random pattern. Methods designed to find sequences or nonrandom patterns can be applied to control charts as an aid in detecting out-of-control conditions. A particular nonrandom pattern usually appears on a control chart for a reason, and if that reason can be found and eliminated, process performance can be improved.

FIGURE 15-1 A typical control chart.

A close connection exists between control charts and hypothesis testing. Essentially, the control chart is a series of tests 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.

We give a general model for a control chart. Let W be a sample statistic that measures some quality characteristic of interest, and suppose that the mean of W is μ_W and the standard deviation of W is σ_W.*. Then the center line, the upper control limit, and the lower control limit become

where k is the “distance” of the control limits from the center line expressed in standard deviation units. A common choice is k = 3. Dr. Walter A. Shewhart first proposed this general theory of control charts, and those developed according to these principles are often called Shewhart control charts.

The control chart is a device for describing exactly what statistical control means; as such, it may be used in a variety of ways. In many applications, the control chart is used for online process monitoring. That is, sample data are collected and used to construct the control chart, and if the sample values of (say) fall within the control limits and do not exhibit any systematic pattern, we say the process is in control at the level indicated by the chart. Note that we may be interested here in determining both whether the past data came from a process that was in control and whether future samples from this process indicate statistical control.

The most important use of a control chart is to improve the process. We have found that, generally

1. Most processes do not operate in a state of statistical control.
2. Consequently, the routine and attentive use of control charts identifies assignable causes. If these causes can be eliminated from the process, variability is reduced and the process is improved.

   This process-improvement activity using the control chart is illustrated in Fig. 15-2. Notice that:

3. The control chart only detects assignable causes. Management, operator, and engineering action usually are necessary to eliminate the assignable cause. An action plan for responding to control chart signals is vital.

In identifying and eliminating assignable causes, it is important to find the underlying root cause of the problem and to attack it. A cosmetic solution does not result in any real, long-term process improvement. Developing an effective system for corrective action is an essential component of an effective SPC implementation.

We may also use the control chart as an estimating device. That is, from a control chart that exhibits statistical control, we may estimate certain process parameters, such as the mean, standard deviation, and fraction nonconforming or fallout. These estimates may then be used to determine the capability of the process to produce acceptable products. Such process capability studies have considerable impact on many management decision problems that occur over the product cycle, including make-or-buy decisions, plant and process improvements that reduce process variability, and contractual agreements with customers or suppliers regarding product quality. Such estimates are discussed in a later section.

FIGURE 15-2 Process improvement using the control chart.

Control charts may be classified into two general types. Many quality characteristics can be measured and expressed as numbers on some continuous scale of measurement. 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 and variability are collectively called variables control charts. The chart is the most widely used chart for monitoring central tendency, and charts based on either the sample range or the sample standard deviation are used to control process variability. Many quality characteristics are not measured on a continuous scale or even a quantitative scale. In these cases, we 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 a unit of product. Control charts for such quality characteristics are called attributes control charts.

Control charts have had a long history of use in industry. There are at least five reasons for their popularity:

1. Control charts are a proven technique for improving productivity. A successful control chart program reduces scrap and rework, which are the primary productivity killers in any operation. If you reduce scrap and rework, productivity increases, cost decreases, and production capacity (measured in the number of good parts per hour) increases.
2. Control charts are effective in defect prevention. The control chart helps keep the process in control, which is consistent with the “do it right the first time” philosophy. It is never cheaper to sort out the “good” units from the “bad” later on than it is to build them correctly initially. If you do not have effective process control, you are paying someone to make a nonconforming product.
3. Control charts prevent unnecessary process adjustments. A control chart can distinguish between background noise and abnormal variation; no other device, including a human operator, is as effective in making this distinction. If process operators adjust the process based on periodic tests unrelated to a control chart program, they often overreact to the background noise and make unneeded adjustments. These unnecessary adjustments can result in a deterioration of process performance. In other words, the control chart is consistent with the “if it isn't broken, don't fix it” philosophy.
4. Control charts provide diagnostic information. Frequently, the pattern of points on the control chart contains information that is of diagnostic value to an experienced operator or engineer. This information allows the operator to implement a change in the process that improve its performance.
5. Control charts provide information about process capability. The control chart provides information about the value of important process parameters and their stability over time. This allows an estimate of process capability to be made. This information is of tremendous use to product and process designers.

Control charts are among the most effective management control tools, and they are as important as cost controls and material controls. Modern computer technology has made it easy to implement control charts in any type of process because data collection and analysis can be performed in real time, online at the work center.

15-2.2 DESIGN OF A CONTROL CHART

To illustrate these ideas, we give a simplified example of a control chart. In manufacturing automobile engine piston rings, the inside diameter of the rings is a critical quality characteristic. The process mean inside ring diameter is 74 millimeters, and it is known that the standard deviation of ring diameter is 0.01 millimeters. A control chart for average ring diameter is shown in Fig. 15-3. Every few minutes a random sample of five rings is taken, the average ring diameter of the sample (say ) is computed, and is plotted on the chart. Because this control chart utilizes the sample mean to monitor the process mean, it is usually called an control chart. Note that all the points fall within the control limits, so the chart indicates that the process is in statistical control.

Consider how the control limits were determined. The process average is 74 millimeters, and the process standard deviation is σ = 0.01 millimeters. Now if samples of size n = 5 are taken, the standard deviation of the sample average is

Therefore, if the process is in control with a mean diameter of 74 millimeters, by using the central limit theorem to assume that is approximately normally distributed, we would expect approximately 100(1 − α)% of the sample mean diameters to fall between 74 + z_α/2(0.0045) and 74 − z_α/2(0.0045). As discussed previously, we customarily choose the constant z_α/2 to be 3, so the upper and lower control limits become

and

FIGURE 15-3 control chart for piston ring diameter.

as shown on the control chart. These are the 3-sigma control limits referred to earlier. Note that the use of 3-sigma limits implies that α = 0.0027; that is, the probability that the point plots outside the control limits when the process is in control is 0.0027. The width of the control limits is inversely related to the sample size n for a given multiple of sigma. Choosing the control limits is equivalent to setting up the critical region for testing the hypotheses

where σ = 0.01 is known. Essentially, the control chart tests this hypothesis repeatedly at different points in time.

In designing a control chart, we must specify both the sample size to use and the frequency of sampling. In general, larger samples make it easier to detect small shifts in the process. When choosing the sample size, we must keep in mind the size of the shift that we are trying to detect. If we are interested in detecting a relatively large process shift, we use smaller sample sizes than those that would be employed if the shift of interest were relatively small.

We must also determine the frequency of sampling. The most desirable situation from the of view of detecting shifts would be to take large samples very frequently; however, this is usually not economically feasible. The general problem is one of allocating sampling effort. That is, either we take small samples at short intervals or larger samples at longer intervals. Current industry practice tends to favor smaller, more frequent samples, particularly in high-volume manufacturing processes or when a great many types of assignable causes can occur. Furthermore, as automatic sensing and measurement technology develops, it is becoming possible to greatly increase frequencies. Ultimately, every unit can be tested as it is manufactured. This capability does not eliminate the need for control charts because the test system cannot prevent defects. The increased data expand the effectiveness of process control and improve quality.

When preliminary samples are used to construct limits for control charts, these limits are customarily treated as trial values. Therefore, the sample statistics should be plotted on the appropriate charts, and any points that exceed the control limits should be investigated. If assignable causes for these points are discovered, they should be eliminated and new limits for the control charts determined. In this way, the process may be eventually brought into statistical control and its inherent capabilities assessed. Other changes in process centering and dispersion may then be contemplated.

15-2.3 RATIONAL SUBGROUPS

A fundamental idea in the use of control charts is to collect sample data according to what Shewhart called the rational subgroup concept. Generally, this means that subgroups or samples should be selected so that to the extent possible, the variability of the observations within a subgroup should include all the chance or natural variability and exclude the assignable variability. Then, the control limits represent bounds for all the chance variability, not the assignable variability. Consequently, assignable causes tend to generate points that are outside of the control limits, and chance variability tends to generate points that are within the control limits.

When control charts are applied to production processes, the time order of production is a logical basis for rational subgrouping. Even though time order is preserved, it is still possible to form subgroups erroneously. If some of the observations in the subgroup are taken at the end of one eight-hour shift and the remaining observations are taken at the start of the next eight-hour shift, any differences between shifts are handled as chance variability when, instead, it should be considered as assignable variability. This makes detecting differences between shifts more difficult. Still, in general, time order is frequently a good basis for forming subgroups because it allows us to detect assignable causes that occur over time.

Two general approaches to constructing rational subgroups are used. In the first approach, each subgroup consists of units that were produced at the same time (or as closely together as possible). This approach is used when the primary purpose of the control chart is to detect process shifts. It minimizes variability due to assignable causes within a sample, and it maximizes variability between samples if assignable causes are present. It also provides better estimates of the standard deviation of the process in the case of variables control charts. This approach to rational subgrouping essentially gives a “snapshot” of the process at each point in time when a sample is collected.

In the second approach, each sample consists of units of product that are representative of all units that have been produced since the last sample was taken. Essentially, each subgroup is a random sample of all process output over the sampling interval. This method of rational subgrouping is often used when the control chart is employed to make decisions about the acceptance of all units of product that have been produced since the last sample. In fact, if the process shifts to an out-of-control state and then back in control again between samples, it is sometimes argued that the first method of rational subgrouping defined earlier is ineffective against these types of shifts, and so the second method must be used.

When the rational subgroup is a random sample of all units produced over the sampling interval, considerable care must be taken in interpreting the control charts. If the process mean drifts between several levels during the interval between samples, the range of observations within the sample may consequently be relatively large. It is the within-sample variability that determines the width of the control limits on an chart, so this practice results in wider limits on the chart. This makes it more difficult to detect shifts in the mean. In fact, we can often make any process appear to be in statistical control just by stretching out the interval between observations in the sample. It is also possible for shifts in the process average to cause points on a control chart for the range or standard deviation to plot out of control even though no shift in process variability has taken place.

Other bases for forming rational subgroups can be used. For example, suppose that a process consists of several machines that pool their output into a common stream. If we sample from this common stream of output, it is very difficult to detect whether or not some of the machines are out of control. A logical approach to rational subgrouping here is to apply control chart techniques to the output for each individual machine. Sometimes this concept needs to be applied to different heads on the same machine, different workstations, different operators, and so forth.

The rational subgroup concept is very important. The proper selection of samples requires careful consideration of the process with the objective of obtaining as much useful information as possible from the control chart analysis.

15-2.4 ANALYSIS OF PATTERNS ON CONTROL CHARTS

A control chart may indicate an out-of-control condition either when one or more points fall beyond the control limits, or when the plotted points exhibit some nonrandom pattern of behavior. For example, consider the chart shown in Fig. 15-4. Although all 25 points fall within the control limits, the points do not indicate statistical control because their pattern is very nonrandom in appearance. Specifically, we note that 19 of the 25 points plot below the center line, but only 6 of them plot above. If the points are truly random, we should expect a more even distribution of points above and below the center line. We also observe that following the fourth point, five points in a row increase in magnitude. This arrangement of points is called a run. Because the observations are increasing, we could call it a run up; similarly, a sequence of decreasing points is called a run down. This control chart has an unusually long run up (beginning with the 4th point) and an unusually long run down (beginning with the 18th point).

In general, we define a run as a sequence of observations of the same type. In addition to runs up and runs down, we could define the types of observations as those above and below the center line, respectively, so two points in a row above the center line would be a run of length 2.

FIGURE 15-4 An control chart.

A run of length 8 or more points has a very low probability of occurrence in a random sample of points. Consequently, any type of run of length 8 or more is often taken as a signal of an out-of-control condition. For example, 8 consecutive points on one side of the center line indicate that the process is out of control.

Although runs are an important measure of nonrandom behavior on a control chart, other types of patterns may also indicate an out-of-control condition. For example, consider the chart in Fig. 15-5. Note that the plotted sample averages exhibit a cyclic behavior, yet they all fall within the control limits. Such a pattern may indicate a problem with the process, such as operator fatigue, raw material deliveries, and heat or stress buildup. The yield may be improved by eliminating or reducing the sources of variability causing this cyclic behavior (see Fig. 15-6). In Fig. 15-6, LSL and USL denote the lower and upper specification limits of the process, respectively. These limits represent bounds within which acceptable product must fall, and they are often based on customer requirements.

The problem is one of pattern recognition, that is, recognizing systematic or nonrandom patterns on the control chart and identifying the reason for this behavior. The ability to interpret a particular pattern in terms of assignable causes requires experience and knowledge of the process. That is, we must not only know the statistical principles of control charts, but we must also have a good understanding of the process.

The Western Electric Handbook (1956) suggests a set of decision rules for detecting nonrandom patterns on control charts. Specifically, the Western Electric rules conclude that the process is out of control if either

1. One point plots outside 3-sigma control limits.
2. Two of three consecutive points plot beyond a 2-sigma limit.

   

   FIGURE 15-5 An chart with a cyclic pattern.

   

   FIGURE 15-6 (a) Variability with the cyclic pattern. (b) Variability with the cyclic pattern eliminated.

   

   FIGURE 15-7 The Western Electric zone rules.

3. Four of five consecutive points plot at a distance of 1 sigma or beyond from the center line.
4. Eight consecutive points plot on one side of the center line.

We have found these rules very effective in practice for enhancing the sensitivity of control charts. Rules 2 and 3 apply to one side of the center line at a time. That is, a point above the upper 2-sigma limit followed immediately by a point below the lower 2-sigma limit would not signal an out-of-control alarm.

Figure 15-7 shows an control chart for the piston ring process with the 1-sigma, 2-sigma, and 3-sigma limits used in the Western Electric procedure. Notice that these inner limits (sometimes called warning limits) partition the control chart into three zones A, B, and C on each side of the center line. Consequently, the Western Electric rules are sometimes called the run rules for control charts. Notice that the last four points fall in zone B or beyond. Thus, because four of five consecutive points exceed the 1-sigma limit, the Western Electric procedure concludes that the pattern is nonrandom and the process is out of control.

15-3 and R or S Control Charts

When dealing with a quality characteristic that can be expressed as a measurement, monitoring both the mean value of the quality characteristic and its variability is customary. Control over the average quality is exercised by the control chart for averages, usually called the chart. Process variability can be controlled by either a range chart (R chart) or a standard deviation chart (S chart), depending on how the population standard deviation is estimated.

Suppose that the process mean and standard deviation μ and σ are known and that we can assume that the quality characteristic has a normal distribution. Consider the chart. As discussed previously, we can use μ as the center line for the control chart, and we can place the upper and lower 3-sigma limits at

When the parameters μ and σ are unknown, we usually estimate them on the basis of preliminary samples taken when the process is thought to be in control. We recommend the use of at least 20 to 25 preliminary samples. Suppose that m preliminary samples are available, each of size n. Typically, n is 4, 5, or 6; these relatively small sample sizes are widely used and often arise from the construction of rational subgroups. Let the sample mean for the ith sample be _i. Then we estimate the mean of the population, μ, by the grand mean

Thus, we may take as the center line on the control chart.

We may estimate σ from either the standard deviation or the range of the observations within each sample. The sample size is relatively small, so there is little loss in efficiency in estimating σ from the sample ranges.

and R Charts

The relationship between the range R of a sample from a normal population with known parameters and the standard deviation of that population is needed. Because R is a random variable, the quantity W = R/σ, called the relative range, is also a random variable. The mean and standard deviation of the distribution of W are called d₂ and d₃, respectively. The values for d₂ and d₃ depend on the subgroup size n. They are computed numerically and available in tables or computer software. Because R = σW,

Let R_i be the range of the ith sample, and let

be the average range. Then is an estimator of μ_R and from Equation 15-4 we obtain the following.

Therefore, once we have computed the sample values and , we may use as our upper and lower control limits for the chart

Define the constant

Now, the control chart may be defined as follows.

The parameters of the R chart may also be easily determined. The center line is . To determine the control limits, we need an estimate of σ_R, the standard deviation of R. Once again, assuming that the process is in control, the distribution of the relative range, W, is useful. We may estimate σ_R from Equation 15-4 as

and the upper and lower control limits on the R chart are

Setting D₃ = 1 − 3d₃/d₂ and D₄ = 1 + 3d₃/d₂ leads to the following definition.

The LCL for an R chart can be a negative number. In that case, it is customary to set LCL to zero. Because the points plotted on an R chart are non-negative, no points can fall below an LCL of zero. Also, we often study the R chart first because if the process variability is not constant over time, the control limits calculated for the chart can be misleading.

and S Charts

Rather than base control charts on ranges, a more modern approach is to calculate the standard deviation of each subgroup and plot these standard deviations to monitor the process standard deviation σ. This is called an S chart. When an S chart is used, it is common to use these standard deviations to develop control limits for the chart. Typically, the sample size used for subgroups is small (fewer than 10) and in that case there is usually little difference in the chart generated from ranges or standard deviations. However, because computer software is often used to implement control charts, S charts are quite common. Details to construct these charts follow.

Section 7-3 stated that S is a biased estimator of σ. That is, E(S) = c₄σ where c₄ is a constant that is near, but not equal to, 1. Furthermore, a calculation similar to the one used for E(S) can derive the standard deviation of the statistic S with the result σ. Therefore, the center line and 3-sigma control limits for S are

Assume that there are m preliminary samples available, each of size n, and let S_i denote the standard deviation of the ith sample. Define

Because E() = c₄σ, we obtain the following.

When an S chart is used, the estimate for σ in Equation 15-15 is commonly used to calculate the control limits for an chart. This produces the following control limits for an chart.

A control chart for standard deviations follows.

The LCL for an S chart can be a negative number; in that case, it is customary to set LCL to zero.

Computer Construction of and R Control Charts

Many computer programs construct and R control charts. Figures 15-8 and 15-10 show charts similar to those produced by computer software for the vane-opening data. Software usually allow the user to select any multiple of sigma as the width of the control limits and use the Western Electric rules to detect out-of-control points. The software also prepares a summary report as in Table 15-2 and excludes subgroups from the calculation of the control limits.

TABLE • 15-2 Summary Report from Computer Software for the Vane-Opening Data

15-4 Control Charts for Individual Measurements

In many situations, the sample size used for process control is n = 1; that is, the sample consists of an individual unit. Some examples of these situations follow:

1. Automated inspection and measurement technology is used, and every unit manufactured is analyzed.
2. The production rate is very slow, and it is inconvenient to allow sample sizes of n > 1 to accumulate before being analyzed.
3. Repeat measurements on the process differ only because of laboratory or analysis error as in many chemical processes.
4. In process plants, such as papermaking, measurements on some parameters such as coating thickness across the roll differ very little and produce a standard deviation that is much too small if the objective is to control coating thickness along the roll.

In such situations, the individuals control chart (also called an X chart) is useful. The control chart for individuals uses the moving range of two successive observations to estimate the process variability. The moving range is defined as MR_i = |X_i − X_i−1| and for m observations the average moving range is m

An estimator of σ is

where the value for d₂ corresponds to n = 2 because each moving range is the range between two consecutive observations. Note that there are only m − 1 moving ranges. It is also possible to establish a control chart on the moving range using D₃ and D₄ for n = 2. The parameters for these charts are defined as follows.

Note that LCL for this moving range chart is always zero because D₃ = 0 for n = 2. The procedure is illustrated in the following example.

FIGURE 15-11 Control charts for individuals and the moving range from computer software for the chemical process concentration data.

The chart for individuals can be interpreted much like an ordinary control chart. A shift in the process average results in either a point (or points) outside the control limits or a pattern consisting of a run on one side of the center line.

Some care should be exercised in interpreting patterns on the moving-range chart. The moving ranges are correlated, and this correlation may often induce a pattern of runs or cycles on the chart. The individual measurements are assumed to be uncorrelated, however, and any apparent pattern on the individuals' control chart should be carefully investigated.

The control chart for individuals is not very sensitive to small shifts in the process mean. For example, if the size of the shift in the mean is 1 standard deviation, the average number of points to detect this shift is 43.9. This result is shown later in the chapter. Although the performance of the control chart for individuals is much better for large shifts, in many situations the shift of interest is not large and more rapid shift detection is desirable. In these cases, we recommend time-weighted charts such as the cumulative sum control chart or an exponentially weighted moving-average chart (discussed in Section 15-8).

Some individuals have suggested that limits narrower than 3-sigma be used on the chart for individuals to enhance its ability to detect small process shifts. This is a dangerous suggestion, for narrower limits dramatically increase false alarms and the charts may be ignored and become useless. If you are interested in detecting small shifts, consider the time-weighted charts.

15-5 Process Capability

It is usually necessary to obtain some information about the process capability, that is, the performance of the process when it is operating in control. Two graphical tools, the tolerance chart (or tier chart) and the histogram, are helpful in assessing process capability. The tolerance chart for all 20 samples from the vane-manufacturing process is shown in Fig. 15-12. The specifications on vane opening are 0.5030 ± 0.0010 in. In terms of the coded data, the upper specification limit is USL = 40 and the lower specification limit is LSL = 20, and these limits are shown on the chart in Fig. 15-12. Each measurement is plotted on the tolerance chart. Measurements from the same subgroup are connected with lines. The tolerance chart is useful in revealing patterns over time in the individual measurements, or it may show that a particular value of or r was produced by one or two unusual observations in the sample. For example, note the two unusual observations in sample 9 and the single unusual observation in sample 8. Note also that it is appropriate to plot the specification limits on the tolerance chart because it is a chart of individual measurements. It is never appropriate to plot specification limits on a control chart or to use the specifications in determining the control limits. Specification limits and control limits are unrelated. Finally, note from Fig. 15-12 that the process is running off-center from the nominal dimension of 30 (or 0.5030 in).

The histogram for the vane-opening measurements is shown in Fig. 15-13. The observations from samples 6, 8, 9, 11, and 19 (corresponding to out of-control points on either the or R chart) have been deleted from this histogram. The general impression from examining this histogram is that the process is capable of meeting the specification but that it is running off-center.

Another way to express process capability is in terms of an index that is defined as follows.

The numerator of PCR is the width of the specifications. The limits 3σ on either side of the process mean are sometimes called natural tolerance limits, for these represent limits that an in-control process should meet with most of the units produced. Consequently, 6σ is often referred to as the width of the process. For the vane opening, where our sample size is 5, we could estimate σ as

Therefore, the PCR is estimated to be

The PCR has a natural interpretation: (1/PCR)100% is just the percentage of the specifications' width used by the process. Thus, the vane-opening process uses approximately (1/1.55)100% = 64.5% of the specifications' width.

FIGURE 15-12 Tolerance diagram of vane openings.

Figure 15-14(a) shows a process for which the PCR exceeds unity. Because the process natural tolerance limits lie inside the specifications, very few defective or nonconforming units are produced. If PCR = 1, as shown in Fig. 15-14(b), more nonconforming units result. In fact, for a normally distributed process, if PCR = 1, the fraction nonconforming is 0.27%, or 2700 parts per million. Finally, when the PCR is less than unity, as in Fig. 15-14(c), the process is very yield-sensitive and a large number of nonconforming units is be produced.

FIGURE 15-13 Histogram for vane opening.

FIGURE 15-14 Process fallout and the process capability ratio (PCR).

The definition of the PCR given in Equation 15-20 implicitly assumes that the process is centered at the nominal dimension. If the process is running off-center, its actual capability is less than indicated by the PCR. It is convenient to think of PCR as a measure of potential capability, that is, capability with a centered process. If the process is not centered, a measure of actual capability is often used. This ratio, called PCR_k, is defined next.

In effect, PCR_k is a one-sided process capability ratio that is calculated relative to the specification limit nearest to the process mean. For the vane-opening process, we find that the estimate of the process capability ratio PCR_k (after deleting the samples corresponding to out-of-control points) is

Note that if PCR = PCR_k, the process is centered at the nominal dimension. Because PCR_K = 1.05 for the vane-opening process and PCR = 1.55, the process is obviously running off-center, as was first noted in Figs. 15-10 and 15-13. This off-center operation was ultimately traced to an oversized wax tool. Changing the tooling resulted in a substantial improvement in the process.

The fractions of nonconforming output (or fallout) below the lower specification limit and above the upper specification limit are often of interest. Suppose that the output from a normally distributed process in statistical control is denoted as X. The fractions are determined from

We emphasize that the fraction-nonconforming calculation assumes that the observations are normally distributed and the process is in control. Departures from normality can seriously affect the results. The calculation should be interpreted as an approximate guideline for process performance. To make matters worse, μ and σ need to be estimated from the data available, and a small sample size can result in poor estimates that further degrade the calculation.

Table 15-4 relates fallout in parts per million (PPM) for a normally-distributed process in statistical control to the value of PCR. The table shows PPM for a centered process and for one with a 1.5σ shift in the process mean. Many U.S. companies use PCR = 1.33 as a minimum acceptable target and PCR = 1.66 as a minimum target for strength, safety, or critical characteristics.

Some companies require that internal processes and those at suppliers achieve a PCR_k = 2.0. Figure 15-15 illustrates a process with PCR = PCR_k = 2.0. Assuming a normal distribution, the calculated fallout for this process is 0.0018 parts per million. A process with PCR_k = 2.0 is referred to as a 6-sigma process because the distance from the process mean to the nearest specification is 6 standard deviations. The reason that such a large process capability is often required is that maintaining a process mean at the center of the specifications for long periods of time is difficult. A common model that is used to justify the importance of a 6-sigma process is illustrated by referring to Fig. 15-15. If the process mean shifts off-center by 1.5 standard deviations, the PCR_k decreases to

TABLE • 15-4 PCR Related to PPM for a Normally Distributed Process

FIGURE 15-15 Mean of a 6-sigma process shifts by 1.5 standard deviations.

Assuming a normally distributed process, the fallout of the shifted process is 3.4 parts per million. Consequently, the mean of a 6-sigma process can shift 1.5 standard deviations from the center of the specifications and still maintain a minimal fallout.

In addition, some U.S. companies, particularly the automobile industry, have adopted the terminology C_p = PCR and C_pk = PCR_k. Because C_p has another meaning in statistics (in multiple regression), we prefer the traditional notation PCR and PCR_k.

We repeat that process capability calculations are meaningful only for stable processes; that is, processes that are in control. A process capability ratio indicates whether or not the natural or chance variability in a process is acceptable relative to the specifications.

15-6 Attribute Control Charts

15-6.1 P CHART (CONTROL CHART FOR PROPORTIONS)

Often it is desirable to classify a product as either defective or nondefective on the basis of comparison with a standard. This classification is usually done to achieve economy and simplicity in the inspection operation. For example, the diameter of a ball bearing may be checked by determining whether it passes through a gauge consisting of circular holes cut in a template. This kind of measurement would be much simpler than directly measuring the diameter with a device such as a micrometer. Control charts for attributes are used in these situations. Attribute control charts often require a considerably larger sample size than do their variable measurements counterparts. In this section, we discuss the fraction-defective control chart, or P chart. Sometimes the P chart is called the control chart for fraction nonconforming.

At each sample time, a random sample of n units is selected. Suppose that D is the number of defective units in the sample. We assume that D is a binomial random variable with unknown parameter p. The fraction defective

of each sample is plotted on the chart. Furthermore, the binomial distribution, the variance of the statistic P, is

Therefore, a P chart for fraction defective could be constructed using p as the center line and control limits at

However, the true process fraction defective is almost always unknown and must be estimated using the data from preliminary samples.

Suppose that m preliminary samples each of size n are available, and let D_i be the number of defectives in the ith sample. Then P_i = D_i/n is the sample fraction defective in the ith sample. The average fraction defective is

Now may be used as an estimator of p in the center line and control limit formulas.

These control limits are based on the normal approximation to the binomial distribution. When p is small, the normal approximation may not always be adequate. In such cases, we may use control limits obtained directly from a table of binomial probabilities. If is small, the lower control limit obtained from the normal approximation may be a negative number. If this should occur, it is customary to consider zero as the lower control limit.

Computer software also produces an NP chart. This is just a control chart of nP = D, the number of defectives in a sample. The points, center line, and control limits for this chart are simply multiples (times n) of the corresponding elements of a P chart. The use of an NP chart avoids the fractions in a P chart, but it is otherwise equivalent.

15-6.2 U CHART (CONTROL CHART FOR DEFECTS PER UNIT)

It is sometimes necessary to monitor the number of defects in a unit of product rather than the fraction defective. For example, a hospital might record the number of cases of infection per month, or a semiconductor manufacturer might record the number of large contamination particles per wafer. In these situations, we may use the defects-per-unit chart or U chart. If each subgroup consists of n units and there are C total defects in the subgroup, then,

is the average number of defects per unit. A U chart may be constructed for such data.

Many defects-per-unit situations can be modeled by the Poisson distribution. Suppose that the number of defects in a unit is a Poisson random variable with mean λ. The variance also equals λ. Each point on the chart is an observed value of U, the average number of defects per unit from a sample of n units. The mean of U is λ, and the variance of U is λ/n. Therefore, the control limits for the U chart with known λ are:

If there are m preliminary samples, and the number of defects per unit in these samples are U₁, U₂,..., U_m, the estimator of the average number of defects per unit is

Now is used as an estimator of λ in the center line and control limit formulas.

These control limits are based on the normal approximation to the Poisson distribution. When λ is small, the normal approximation may not always be adequate. In such cases, we may use control limits obtained directly from a table of Poisson probabilities. If is small, the lower control limit obtained from the normal approximation may be a negative number. If this should occur, it is customary to use zero as the lower control limit.

Computer software also produces a C chart. This is just a control chart of C, the total of defects in a sample. The points, center line, and control limits for this chart are simply multiples (times n) of the corresponding elements of a U chart. The use of a C chart avoids the fractions that can occur in a U chart, but it is otherwise equivalent.

15-7 Control Chart Performance

Specifying the control limits is one of the critical decisions that must be made in designing a control chart. By moving the control limits farther from the center line, we decrease the risk of a type I error—that is, the risk of a point falling beyond the control limits, indicating an out-of-control condition when no assignable cause is present. However, widening the control limits also increases the risk of a type II error—that is, the risk of a point falling between the control limits when the process is really out of control. If we move the control limits closer to the center line, the opposite effect is obtained: The risk of type I error increases, and the risk of type II error decreases.

The control limits on a Shewhart control chart are customarily located a distance of plus or minus 3 standard deviations of the statistic plotted on the chart from the center line. That is, the constant k in Equation 15-1 should be set equal to 3. These limits are called 3-sigma control limits.

A way to evaluate decisions regarding sample size and sampling frequency is through the average run length (ARL) of the control chart. Essentially, the ARL is the average number of points plotted to signal an out-of-control condition. For any Shewhart control chart, the ARL can be calculated from the mean of a geometric random variable. Suppose that p is the probability that any point exceeds the control limits. Then

Thus, for an chart with 3-sigma limits, p = 0.0027 is the probability that a normally distributed point falls outside the limits when the process is in control, so

is the average run length of the chart when the process is in control. That is, even if the process remains in control, an out-of-control signal is generated every 370 points on average.

Consider the piston ring process discussed in Section 15-2.2, and suppose that we are sampling every hour. Thus, we have a false alarm about every 370 hours on average. Suppose that we are using a sample size of n = 5 and that when the process goes out of control, the mean shifts to 74.0135 millimeters. Then, the probability that falls between the control limits of Fig. 15-3 is equal to

Therefore, p in Equation 15-28 is 0.50, and the out-of-control ARL is

That is, the control chart will require two samples to detect the process shift, on the average, so two hours will elapse between the shift and its detection (again, on the average). Suppose that this approach is unacceptable because production of piston rings with a mean diameter of 74.0135 millimeters results in excessive scrap costs and delays final engine assembly. How can we reduce the time needed to detect the out-of-control condition? One method is to sample more frequently. For example, if we sample every half hour, only one hour elapses (on the average) between the shift and its detection. The second possibility is to increase the sample size. For example, if we use n = 10, the control limits in Fig. 15-3 narrow to 73.9905 and 74.0095. The probability of falling between the control limits when the process mean is 74.0135 millimeters is approximately 0.1, so p = 0.9, and the out-of-control ARL is

TABLE • 15-7 Average Run Length (ARL) for an Chart with 3-SigmaControl Limits

FIGURE 15-18 Process mean shift of 2σ.

Thus, the larger sample size would allow the shift to be detected about twice as quickly as the smaller one. If it became important to detect the shift in approximately the first hour after it occurred, two control chart designs would work:

Table 15-7 provides average run lengths for an chart with 3-sigma control limits. The average run lengths are calculated for shifts in the process mean from 0 to 3.0σ and for sample sizes of n = 1 and n = 4 by using 1/p, where p is the probability that a point plots outside of the control limits (based on a normal distribution). Figure 15-18 illustrates a shift in the process mean of 2σ.

15-8 Time-Weighted Charts

In Sections 15-3 and 15-4 we presented basic types of Shewhart control charts. A major disadvantage of any Shewhart control chart is that it is relatively insensitive to small shifts in the process, say, on the order of about 1.5σ or less. One reason for this relatively poor performance in detecting small process shifts is that the Shewhart chart uses only the information in the last plotted point and ignores the information in the sequence of points. This problem can be addressed to some extent by adding criteria such as the Western Electric rules to a Shewhart chart, but the use of these rules reduces the simplicity and ease of interpretation of the chart. These rules would also cause the in-control average run length of a Shewhart chart to drop below 370. This increase in the false alarm rate can have serious practical consequences.

15-8.1 CUMULATIVE SUM CONTROL CHART

A very effective alternative to the Shewhart control chart is the cumulative sum control chart (CUSUM). This chart has much better performance (in terms of ARL) for detecting small shifts than the Shewhart chart, but it does not cause the in-control ARL to drop significantly. This section illustrates the use of the CUSUM for sample averages and individual measurements. CUSUM charts for other sample statistics are also available.

The CUSUM chart plots the cumulative sums of the deviations of the sample values from a target value. For example, suppose that samples of size n ≥ 1 are collected, and _j is the average of the jth sample. Then if μ₀ is the target for the process mean, the cumulative sum control chart is formed by plotting the quantity

against the sample number i. Now S_i is called the cumulative sum up to and including the ith sample. Because they combine information from several samples, cumulative sum charts are more effective than Shewhart charts for detecting small process shifts. Furthermore, they are particularly effective with samples of n = 1. This makes the cumulative sum control chart a good candidate for use in the chemical and process industries in which rational subgroups are frequently of size 1, as well as in discrete parts manufacturing with automatic measurement of each part and online control using a computer directly at the work center.

If the process remains in control at the target value μ₀, the cumulative sum defined in Equation 15-29 should fluctuate around zero. However, if the mean shifts upward to some value μ₁ > μ₀, for example, an upward or positive drift develops in the cumulative sum S_i. Conversely, if the mean shifts downward to some μ₁ < μ₀, a downward or negative drift in S_i develops. Therefore, if a trend develops in the plotted points either upward or downward, we should consider this as evidence that the process mean has shifted, and a search for the assignable cause should be performed.

This theory can easily be demonstrated by applying the CUSUM to the chemical process concentration data in Table 15-3. Because the concentration readings are individual measurements, we would take _j = X_j in computing the CUSUM. Suppose that the target value for the concentration is μ₀ = 99. Then the CUSUM is

Table 15-8 shows the computing values, s_i's for this CUSUM, for which the starting value of the CUSUM, s₀, is taken to be zero. Figure 15-19 plots the CUSUM from the last column of Table 15-8. Notice that the CUSUM fluctuates around the value of 0.

The graph in Fig. 15-19 is not a control chart because it lacks control limits. There are two general approaches to devising control limits for CUSUMs. The older of these two methods is the V-mask procedure. A typical V mask is shown in Fig. 15-20(a). It is a V-shaped notch in a plane that can be placed at different locations on the CUSUM chart. The decision procedure consists of placing the V mask on the cumulative sum control chart with the point O on the last value of s_i and the line OP parallel to the horizontal axis. If all the previous cumulative sums, s₁, s₂,..., s_i−1, lie within the two arms of the V mask, the process is in control. The arms are the lines that make angles θ with segment OP in Figure 15-20(a) and are assumed to extend infinitely in length. However, if any s_i lies outside the arms of the mask, the process is considered to be out of control. In actual use, the V mask would be applied to each new point on the CUSUM chart as soon as it was plotted. The example in Fig. 15-20(b) indicates an upward shift in the mean because at least one of the points that occurred earlier than sample 22 now lies below the lower arm of the mask when the V mask is centered on sample 30. If the point lies above the upper arm, a downward shift in the mean is indicated. Thus, the V mask forms a visual frame of reference similar to the control limits on an ordinary Shewhart control chart. For the technical details of designing the V mask, see Montgomery (2013).

TABLE • 15-8 CUSUM Computations for the Chemical Process Concentration Data in Table 15-3

FIGURE 15-19 Plot of the cumulative sums for the concentration data.

Although some computer programs plot CUSUMs with the V-mask control scheme, we believe that the other approach to CUSUM control, the tabular CUSUM, is superior. The tabular procedure is particularly attractive when the CUSUM is implemented on a computer.

FIGURE 15-20 The cumulative sum control chart. (a) The V-mask and scaling. (b) The cumulative sum control chart in operation.

Let S_H(i) be an upper one-sided CUSUM for period i and S_L(i) be a lower one-sided CUSUM for period i. These quantities are calculated from

In Equations 15-30 and 15-30, K is called the reference value, which is usually chosen about halfway between the target μ₀ and the value of the mean corresponding to the out-of-control state, μ₁ = μ₀ + Δ. That is, K is about one-half the magnitude of the shift we are interested in, or

Notice that S_H(i) and S_L(i) accumulate deviations from the target value that are greater than K with both quantities reset to zero upon becoming negative. If either S_H(i) or S_L(i) exceeds a constant H, the process is out of control. This constant H is usually called the decision interval.

When the tabular CUSUM indicates that the process is out of control, we should search for the assignable cause, take any corrective actions indicated, and restart the CUSUMs at zero. It may be helpful to have an estimate of the new process mean following the shift. This can be computed from

Also, an estimate of the time at which the assignable cause occurred is often taken as the sample time at which the upper or lower CUSUM (whichever one signaled) was last equal to zero.

It is also useful to present a graphical display of the tabular CUSUMs, which are sometimes called CUSUM status charts. They are constructed by plotting s_H(i) and s_L(i) versus the sample number. Figure 15-21 shows the CUSUM status chart for the data in Example 15-6. Each vertical bar represents the value of s_H(i) and s_L(i) in period i. With the decision interval plotted on the chart, the CUSUM status chart resembles a Shewhart control chart. We have also plotted the sample statistics x_i for each period on the CUSUM status chart as the solid dots. This frequently helps the user of the control chart to visualize the actual process performance that has led to a particular value of the CUSUM.

The tabular CUSUM is designed by choosing values for the reference value K and the decision interval H. We recommend that these parameters be selected to provide good average run-length values. There have been many analytical studies of CUSUM ARL performance. Based on them, we may give some general recommendations for selecting H and K. Define H = h and K = k where is the standard deviation of the sample variable used in forming the CUSUM (if n = 1, = σ_X). Using h = 4 or h = 5 and k = 1/2 generally provide a CUSUM that has good ARL properties against a shift of about 1 (or 1σ_X) in the process mean. If much larger or smaller shifts are of interest, set k = δ/2 where δ is the size of the shift in standard deviation units.

To illustrate how well the recommendations of h = 4 or h = 5 with k = 1/2 work, consider these average run lengths in Table 15-10. Notice that a shift of 1 would be detected in either 8.38 samples (with k = 1/2 and h = 4) or 10.4 samples (with k = 1/2 and h = 5). By comparison, Table 15-7 shows that an chart would require approximately 43.9 samples, on the average, to detect this shift.

FIGURE 15-21 The CUSUM status chart for Example 15-6.

TABLE • 15-10 Average Run Lengths for a CUSUM Control Chart with k = 1/2

These design rules were used for the CUSUM in Example 15-6. We assumed that the process standard deviation σ = 2. (This is a reasonable value; see Example 15-2.) Then with k = 1/2 and h = 5, we would use

in the tabular CUSUM procedure.

Finally, we should note that supplemental procedures such as the Western Electric rules cannot be safely applied to the CUSUM because successive values of S_H(i) and S_L(i) are not independent. In fact, the CUSUM can be thought of as a weighted average, where the weights are stochastic or random. In effect, all CUSUM values are highly correlated, thereby causing the Western Electric rules to produce too many false alarms.

15-8.2 EXPONENTIALLY WEIGHTED MOVING-AVERAGE CONTROL CHART

Data collected in time order are often averaged over several time periods. For example, economic data are often presented as an average over the last four quarters. That is, at time t, the average of the last four measurements can be written as

This average places weight of 1/4 on each of the most recent observations and zero weight on older observations. It is called a moving average and in this, case a window of size 4 is used. An average of the recent data is used to smooth the noise in the data to generate a better estimate of the process mean than only the most recent observation. However, in a dynamic environment in which the process mean may change, the number of observations used to construct the average is kept to a modest size so that the estimate can adjust to any change in the process mean. Therefore, the window size is a compromise between a better statistical estimate from an average and a response to a mean change. If a window of size 10 were used in a moving average, the statistic _t(10) would have lower variability, but it would not adjust as well to a mean change.

For statistical process control rather than use a fixed window size it is useful to place the most weight on the most recent observation or subgroup average and then gradually decrease the weights on older observations. An average of this type can be constructed by a multiplicative decrease in the weights. Let 0 < λ ≤ 1 denote a constant and μ₀ denote the process target or historical mean. Suppose that samples of size n ≥ 1 are collected and _t is the average of the sample at time t. The exponentially weighted moving-average (EWMA) is

Each older observation has its weight decreased by the factor (1 − λ). The weight on the starting value μ₀ is selected so that the weights sum to 1. An EWMA is also sometimes called a geometric average.

The value of λ determines the compromise between noise reduction and response to a mean change. For example, the series of weights when λ = 0.8 are

and when λ = 0.2, the weights are

When λ = 0.8, the weights decrease rapidly. Most of the weight is placed on the most recent observation with modest contributions to the EWMA from older measurements. In this case, the EWMA does not average noise much, but it responds quickly to a mean change. However, when λ = 0.2, the weights decrease much more slowly and the EWMA has substantial contributions from the more recent observations. In this case, the EWMA averages noise more, but it responds more slowly to a change in the mean. Fig. 15-22 displays a series of observations with a mean shift in the middle on the series. Notice that the EWMA with λ = 0.2 smooths the data more but that the EWMA with λ = 0.8 adjusts the estimate to the mean shift more quickly.

It appears that it is difficult to calculate an EWMA because at every time t a new weighted average of all previous data is required. However, there is an easy method to calculate z_t based on a simple recursive equation. Let z₀ = μ₀. Then it can be shown that

Consequently, only a brief computation is needed at each time t.

To develop a control chart from an EWMA, control limits are needed for Z_t. The control limits are defined in a straightforward manner. They are placed at 3 standard deviations around the mean of the plotted statistic Z_t. This follows the general approach for a control chart in Equation 15-1. An EWMA control chart may be applied to individual measurements or to subgroup averages. Formulas here are developed for the more general case with an average from a subgroup of size n. For individual measurements, n = 1.

Because Z_t is a linear function of the independent observations X₁, X₂,..., X_t (and μ₀), the results from Chapter 5 can be used to show that

FIGURE 15-22 EWMAs with λ = 0.8 and λ = 0.2 show a compromise between a smooth curve and a response to a shift.

where n is the subgroup size. Therefore, an EWMA control chart uses estimates of μ₀ and σ in the following formulas:

Note that the control limits are not of equal width about the center line. The control limits are calculated from the variance of Z_t and that changes with time. However, for large t, the variance of Z_t converges to

so that the control limits tend to be parallel lines about the center line as t increases.

The parameters μ₀ and σ are estimated by the same statistics used in or X charts. That is, for subgroups

and for n = 1

Similar to a CUSUM chart, the points plotted on an EWMA control chart are not independent. Therefore, run rules should not be applied to an EWMA control chart. Information in the history of the data that is considered by run rules is to a large extent incorporated into the EWMA that is calculated at each time t.

The value of λ is usually chosen from the range 0.1 < λ < 0.5. A common choice is λ = 0.2. Smaller values for λ provide more sensitivity for small shifts and larger values better tune the chart for larger shifts. This performance can be seen in the (ARLs) in Table 15-11. These calculations are more difficult than those used for Shewhart charts, and details are omitted. Here, λ = 0.1 and 0.5 are compared. The multiplier of the standard deviation, denoted L in the table, is adjusted so that the ARL equals 500 for both choices for λ. That is, the control limits are placed at E(Z_t) ± L, and L is chosen so the ARL without a mean shift is 500 in both cases.

The EWMA ARLs in the table indicate that the smaller value for λ is preferred when the magnitude of the shift is small. Also, the EWMA performance is in general much better than results for a Shewhart control chart (in Table 15-7), and the results are comparable to a CUSUM control chart (in Table 15-10). However, these are average results. At the time of an increase in the process mean, z_t might be negative, and there would be some performance penalty to first increase z_t to near zero and then further increase it to a signal above the UCL. Such a penalty provides an advantage to CUSUM control charts that is not accounted for in these ARL tables. A more refined analysis can be used to quantify this penalty, but the conclusion is that the EWMA penalty is moderate to small in most applications.

TABLE • 15-11 Average Run Lengths for an EWMA Control Chart

15-9 Other SPC Problem-Solving Tools

Although the control chart is a very powerful tool for investigating the causes of variation in a process, it is most effective when used with other SPC problem-solving tools. In this section, we illustrate some of these tools, using the printed circuit board defect data in Example 15-5.

Figure 15-17 shows a U chart for the number of defects in samples of five printed circuit boards. The chart exhibits statistical control, but the number of defects must be reduced. The average number of defects per board is 8/5 = 1.6, and this level of defects would require extensive rework.

FIGURE 15-24 Pareto diagram for printed circuit board defects.

FIGURE 15-25 Cause-and-effect diagram for the printed circuit board flow solder process.

The first step in solving this problem is to construct a Pareto diagram of the individual defect types. The Pareto diagram shown in Fig. 15-24, indicates that insufficient solder and solder balls are the most frequently occurring defects, accounting for (109/160)100 = 68% of the observed defects. Furthermore, the first five defect categories on the Pareto chart are all solder-related defects. This points to the flow solder process as a potential opportunity for improvement.

To improve the surface mount process, a team consisting of the operator, the shop supervisor, the manufacturing engineer responsible for the process, and a quality engineer meets to study potential causes of solder defects. They conduct a brainstorming session and produce the cause-and-effect diagram in Fig. 15-25. The cause-and-effect diagram is widely used to display the various potential causes of defects in products and their interrelationships. They are useful in summarizing knowledge about the process.

As a result of the brainstorming session, the team tentatively identifies the following variables as potentially influential in creating solder defects:

1. Flux specific gravity
2. Reflow temperature
3. Squeegee speed
4. Squeegee angle
5. Paste height
6. Reflow temperature
7. Board loading method

A statistically designed experiment could be used to investigate the effect of these seven variables on solder defects.

In addition, the team constructed a defect concentration diagram for the product; it is just a sketch or drawing of the product with the most frequently occurring defects shown on the part. This diagram is used to determine whether defects occur in the same location on the part. The defect concentration diagram for the printed circuit board is shown in Fig. 15-26. This diagram indicates that most of the insufficient solder defects are near the front edge of the board. Further investigation showed that one of the pallets used to carry the boards was bent, causing the front edge of the board to make poor contact with the squeegee.

FIGURE 15-26 Defect concentration diagram for a printed circuit board.

When the defective pallet was replaced, a designed experiment was used to investigate the seven variables discussed earlier. The results of this experiment indicated that several of these factors were influential and could be adjusted to reduce solder defects. After the results of the experiment were implemented, the percentage of solder joints requiring rework was reduced from 1% to under 100 parts per million (0.01%).

15-10 Decision Theory

Quality improvement requires decision making in the presence of uncertainty as do many other management and engineering decisions. Consequently, a framework to characterize the decision problem in terms of actions, possible states, and associated probabilities is useful to quantitatively compare alternatives. Decision theory is the study of mathematical models for decision making. Actions are evaluated and selected based on the model and quantitative criteria.

15-10.1 DECISION MODELS

A simple way to characterize decisions is in terms of actions, states with probabilities, and outcomes as costs (or profits). A decision usually involves a set of possible actions

For example, one might purchase an extended warranty with the purchase of a vehicle (action a₁) or not (action a₂).

The possible future situations are represented with a collection of states

One state occurs, but we are not certain of the future state at the time of our decision. For example, a possible state is that a major repair is required during the extended warranty; another is that no repair is required.

We often associate a probability, for example, p_m with each state s_m, so that p₁ + p₂ + ... + p_M = 1. The probabilities are important and can be difficult to estimate. Sometimes we have historical data from past performance and can derive estimates. In other cases, we might have to rely on the subjective belief of a collection of experts.

The outcome is often expressed in terms of economic cost (or profit) that depends on the actions and the state that occurs. That is, let

For example, if we purchases an extended warranty and no repair is required, our loss is only the cost of the warranty. Clearly, if we do not purchase the warranty and a major repair is required, our loss is the cost of the repair.

15-10.2 DECISION CRITERIA

The numerical summary of the decision problem is presented with actions, states, and probabilities as illustrated in Example 15-8. However, the question regarding the “best” action still needs to be answered. Because the state is not known at the time of the decision, different costs are possible. Different criteria based on these costs (and the associated probabilities) can be used to select an action, but, as we show, these criteria do not always lead to the same action.

We might be pessimistic and focus on the worst-possible state for each action. In this approach, we compares actions based on the maximum cost that can occur. For example, from Table 15-12, the maximum cost that occurs for action a₁ is max_mC_1m = $200 (regardless of the state). For action a₂, we have max_m C_2m = max{1200,300,0} = $1200. A reasonable criterion is to select the action that minimizes this maximum cost, and in this case, the choice is a₁. In general, this approach selects the action a_k to minimize max_m C_km.

The minimax criterion focuses on the worst-case (pessimistic) scenario and ignores the probabilities associated with the states. A state with a high cost for a specific action, even if the state is very unlikely, can penalize and eliminate the action.

An alternative criterion is to focus on the best-case (optimistic) scenario among the states and order the actions based on minimum cost min_mC_km.

For the warranty example, min_mC_1m = 200 and min_mC_2m = 0 so that a₂ is selected with this criterion.

The previous criteria ignore the probabilities associated with the states. The most probable criterion evaluates an action based on the cost of the state with the most likely probability.

For the warranty example, the most probable state is a minor repair. Based on this state alone, the cost associated with a₁ is $200, and the cost associated with a₂ is $300. Consequently, a₁ is chosen with this criterion.

An obvious criterion is the expected cost. Here we associate a random variable X_k with each action a_k. The distribution of the discrete random variable X_k consists of the costs and the associated probabilities for action a_k. The expected cost of a_k is defined simply as the expected value E(X_k).

For the warranty example, E(X₁) = 200 and E(X₂) = 1200(0.1) + 300(0.5) + 0(0.4) = 270. Consequently, the minimum expected cost is produced by action a₁.

A decision problem is often represented with a graph known as a decision tree. See Fig. 15-27 for a decision tree of the warranty example. A square denotes a decision node where an action is selected. Each arc from a decision node represents an action. Each action arc terminates with a circular node to indicate that a state has been chosen (outside the control of the decision maker), and the states are represented by the arcs from the circular nodes. Each arc is labeled with the probability of the state. The cost is shown at the end of these arcs. This is the basic structure of the decision problem. Fig. 15-27 computed from Equation 15-38 also shows the expected cost above each circle. Based on the expected costs, action a₁ is clearly preferred. We could replace the expected cost with other criterion discussed in this section to summarize the alternative actions.

In more complex problems, there is a series of actions and states (represented as rectangles and circles, respectively, in the decision tree). Probabilities are associated with each state, and a cost is associated with each path through the tree. Fig 15-28 provides an illustration. Given a criterion, we start at the end of a path and apply the criterion to determine the action. We continue the analysis from the end of a path until we reach the initial action node of the tree. This is illustrated in the following example.

FIGURE 15-27 Decision tree for extended warranty example, expected cost of each action shown above circular modes.

FIGURE 15-28 Decision tree for the develop or contract example.

15-11 Implementing SPC

The methods of statistical process control can provide significant payback to those companies that can successfully implement them. Although SPC seems to be a collection of statistically based problem-solving tools, there is more to its successful use than simply learning and using these tools. Management involvement and commitment to the quality-improvement process is the most vital component of SPC's potential success. Management is a role model, and others in the organization look to management for guidance and as an example. A team approach is also important, for it is usually difficult for one person alone to introduce process improvements. Many of the “magnificent seven” problem-solving tools are helpful in building an improvement team, including cause-and-effect diagrams, Pareto charts, and defect concentration diagrams. The basic SPC problem-solving tools must become widely known and widely used throughout the organization. Continuous training in SPC and quality improvement is necessary to achieve this widespread knowledge of the tools.

The objective of an SPC-based quality-improvement program is continuous improvement on a weekly, quarterly, and annual basis. SPC is not a one-time program to be applied when the business is in trouble and later abandoned. Quality improvement must become part of the organization's culture.

The control chart is an important tool for process improvement. Processes do not naturally operate in an in-control state, and the use of control charts is an important step that must be taken early in an SPC program to eliminate assignable causes, reduce process variability, and stabilize process performance. To improve quality and productivity, we must begin to manage with facts and data, not just rely on judgment. Control charts are an important part of this change in management approach.

In implementing a companywide SPC program, we have found that the following elements are usually present in all successful efforts:

1. Management leadership
2. Team approach
3. Education of employees at all levels
4. Emphasis on continuous improvement
5. Mechanism for recognizing success

We cannot overemphasize the importance of management leadership and the team approach. Successful quality improvement is a “top-down” management-driven activity. It is also important to measure progress and success and to spread knowledge of this success throughout the organization. Communicating successful improvements throughout the company can provide motivation and incentive to improve other processes and to make continuous improvement a normal part of the way of doing business.

The philosophy of W. Edwards Deming provides an important framework for implementing quality and productivity improvement. Deming's philosophy is summarized in his 14 points for management. The adherence to these management principles has been an important factor in Japan's industrial success and continues to be the catalyst in that nation's quality- and productivity-improvement efforts. This philosophy has also now spread rapidly in the West. Deming's 14 points are as follows.

1. Create a constancy of purpose focused on the improvement of products and services. Constantly try to improve product design and performance. Investment in research, development, and innovation will have a long-term payback to the organization.
2. Adopt a new philosophy of rejecting poor workmanship, defective products, or bad service. It costs as much to produce a defective unit as it does to produce a good one (and sometimes more). The cost of dealing with scrap, rework, and other losses created by defectives is an enormous drain on company resources.
3. Do not rely on mass inspection to “control quality.” All inspection can do is sort out defectives, and at this point, it is too late because we have already paid to produce these defectives. Inspection occurs too late in the process, is expensive, and is often ineffective. Quality results from the prevention of defectives through process improvement, not inspection.
4. Do not award business to suppliers on the basis of price alone, but also consider quality. Price is a meaningful measure of a supplier's product only if it is considered in relation to a measure of quality. In other words, the total cost of the item, not jus the purchase price must be considered. When quality is considered, the lowest bidder is frequently not the low-cost supplier. Preference should be given to suppliers who use modern methods of quality improvement in their business and who can demonstrate process control and capability.
5. Focus on continuous improvement. Constantly try to improve the production and service system. Involve the workforce in these activities and use statistical methods, particularly the SPC problem-solving tools discussed previously.
6. Practice modern training methods and invest in training for all employees. All employees should be trained in the technical aspects of their job, as well as in modern quality- and productivity-improvement methods. The training should encourage all employees to practice these methods every day.
7. Practice modern supervision methods. Supervision should not consist merely of passive surveillance of workers but also should be focused on helping the employees improve the system in which they work. The first goal of supervision should be to improve the work system and the product.
8. Drive out fear. Many workers are afraid to ask questions, report problems, or point out conditions that are barriers to quality and effective production. In many organizations, the economic loss associated with fear is large; only management can eliminate fear.
9. Break down the barriers between functional areas of the business. Teamwork among different organizational units is essential for effective quality and productivity improvement to take place.
10. Eliminate targets, slogans, and numerical goals for the workforce. A target such as “zero defects” is useless without a plan as to how to achieve it. In fact, these slogans and “programs” are usually counterproductive. Work to improve the system and provide information on that.
11. Eliminate numerical quotas and work standards. These standards have historically been set without regard to quality. Work standards are often symptoms of management's inability to understand the work process and to provide an effective management system focused on improving this process.
12. Remove the barriers that discourage employees from doing their jobs. Management must listen to employee suggestions, comments, and complaints. The person who is doing the job is the one who knows the most about it, and usually has valuable ideas about how to make the process work more effectively. The workforce is an important participant in the business, not just an opponent in collective bargaining.
13. Institute an ongoing program of training and education for all employees. Education in simple, powerful statistical techniques should be mandatory for all employees. Use of the basic SPC problem-solving tools, particularly the control chart, should become widespread in the business. As these charts become widespread, and as employees understand their uses, they are more likely to look for the causes of poor quality and to identify process improvements. Education is a way of making everyone partners in the quality-improvement process.
14. Create a structure in top management that vigorously advocates the first 13 points.

As we read Deming's 14 points, we notice two things. First, there is a strong emphasis on change. Second, the role of management in guiding this change process is of dominating importance. But what should be changed, and how should this change process be started? For example, if we want to improve the yield of a semiconductor manufacturing process, what should we do? It is in this area that statistical methods most frequently come into play. To improve the semiconductor process, we must determine which controllable factors in the process influence the number of defective units produced. To answer this question, we must collect data on the process and see how the system reacts to changes in the process variables. Statistical methods, including the SPC and experimental design techniques in this book, can contribute to this knowledge.

Important Terms and Concepts

Assignable causes

Attributes control charts

Average run length (ARL)

C chart

Cause-and-effect diagram

Center line

Chance causes

Control chart

Control limits

Cumulative sum control chart (CUSUM)

Decision theory

Decision tree

Defect concentration diagram

Defects-per-unit chart

Deming's 14 points

Exponentially weighted moving-average control chart (EWMA)

False alarm

Fraction-defective control chart

Implementing SPC

Individuals control chart (X chart)

Moving range

NP chart

Natural tolerance limits

P chart

Pareto diagram

Problem-solving tools

Process capability

Process capability ratio (PCR, PCR_k)

Quality improvement

R chart

Rational subgroup

Run rules

S chart

Shewhart control chart

6-sigma process

Specification limits

Statistical process control (SPC)

Statistical quality control

U chart

Variables control charts

Warning limits

Western Electric rules

chart