Statistical Process Control is a powerful collection of problem solving tools useful in achieving process stability and improving capability through the reduction of variability. 

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

1. Shewhart control chart is probably the most technically sophisticated.
2. It was developed in the 1920s by Walter A. Shewhart of the Bell Telephone Laboratories.

In any production process, regardless of how well designed or carefully maintained it is, a certain amount of inherent or natural variability will always exist. This natural variability or “background noise” is the cumulative effect of many small, essentially unavoidable causes. 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.

Variability may occasionally be present usually arising from three sources:

improperly adjusted or controlled machines,

operator errors, or defective raw material.

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 of variation. A process that is operating in the presence of assignable causes is said to be an out-of-control process.

Elimination of variability is the eventual goal of Statistical Control Process.

[image]w = a sample statistic that measures some quality of interest. 

µ_{w =} Mean of W

σ_{w =} Standard Deviation

L = Distance of the control limits from the center line, expressed in Standard deviation units.

1. Most processes do not operate in a state of statistical control, and 

2. Consequently, the routine and attentive use of control charts will identify assignable causes. If these causes can be eliminated from the process, variability will be reduced and the process will be improved. 

3. The control chart will only detect assignable causes. Management, operator, and engineering action will usually be necessary to eliminate the assignable causes.

• An OCAP is a flow chart or text-based description of the sequence of activities that must take place following the occurrence of an activating event.
• These are usually out-of-control signals from the control chart.
• The OCAP consists of checkpoints, which are potential assignable causes, and terminators, which are actions taken to resolve the out-of-control condition, preferably by eliminating the assignable cause.
• It is very important that the OCAP specify as complete a set as possible of checkpoints and terminators, and that these be arranged in an order that facilitates process diagnostic activities.
• An OCAP is a living document in the sense that it will be modified over time as more knowledge and understanding of the process is gained.
• Consequently, when a control chart is introduced, an initial OCAP should accompany it. Control charts without an OCAP are not likely to be useful as a process improvement tool.

1. Variable control charts
2. Attributes control charts

• Quality characteristic measured and expressed as a number on some contimuous scale of measurement.
• Quality characterstic described with a measure of central tendency and measure of variability.
• Xbar chart is the most widely used variable control chart.

• Describes quality characteristics measured in a qualitative manner.
• 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 non-conformities(defects) appearing on a unit of product.

1. Control charts are a proven technique for improving productivity.
2. Control charts are effective in defect prevention.
3. Control charts prevent unnecessary process adjustment.
4. Control charts provide diagnostic information.
5. Control charts provide information about process capability.

Choice of control limits and its impact of type I and type II errors

1. Moving the control limits farther away from the center line decreases 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.
2. Widening the control limits will also increase 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.
3. Moving the control limits closer to the center line, the opposite effect is obtained: The risk of type I error is increased, while the risk of type II error is decreased. 

• Probability of type I error is only 0.0027
• Probability in one direction is only 0.00135
• The 3.09 σ control limites are called 0.001 probability limits.
• These are also action limits; that is, when a point plots outside of this limit, a search for an assignable cause is made and corrective action is taken if necessary.

• These lines are plotted at 2σ limits.
• If one or more points fall between the warning limits and the control limits, or very close to the warning limit, we should be suspicious that the process may not be operating properly.
• One possible action to take when this occurs is to increase the sampling frequency and/or the sample size so that more information about the process can be obtained quickly.
• The use of warning limits can increase the sensitivity of the control chart; that is, it can allow the control chart to signal a shift in the process more quickly.
• They may be confusing to operating personnel.
• They also result in an increased risk of false alarms.

Process control schemes that change the sample size and/or the sampling frequency depending on the position of the current sample value are called adaptive or variable sampling interval (or variable sample size, etc.) schemes.

• In general, larger samples will make it easier to detect small shifts in the process.
• The probability of detecting a shift increases as the sample size n increases.
• When choosing the sample size, we must keep in mind the size of the shift that we are trying to detect.
• If the process shift is relatively large, then we use smaller sample sizes than those that would be employed if the shift of interest were relatively small.

Determining the Sampling Frequency for a Control Chart

• The most desirable situation from the point of view of detecting shifts would be to take large samples very frequently.
• however, this is usually not economically feasible.
• 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 where a great many types of assignable causes can occur.
• As automatic sensing and measurement technology develops, it is becoming possible to greatly increase sampling frequencies.

ARL

(Average Run Length)

Essentially, the ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. If the process observations are uncorrelated, then for any Shewhart control chart, the ARL can be calculated easily from [image]where p is the probability that any point exceeds the control limits. This equation can be used to evaluate the performance of the control chart.

For the chart with three-sigma limits, p = 0.0027 is the probability that a single point falls outside the limits when the process is in control. Therefore, the average run length of the chart when the process is in control (called ARL₀) is

[image]

Average Time to Signal

(ATS)

It is also occasionally convenient to express the performance of the control chart in terms of its average time to signal (ATS). If samples are taken at fixed intervals of time that are h hours apart, then

[image]

A larger sample size would allow the shift to be detected more quickly than with the smaller one.

• Collection of sample data should be according to what Shewhart called the rational subgroup concept.
• When we use a control chart to detect changes in the process mean. Then the rational subgroup concept means that subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized.
• When control charts are applied to production processes, the time order of production is a logical basis for rational subgrouping. Time order is frequently a good basis for forming subgroups because it allows us to detect assignable causes that occur over time.

Constructing Rational Subgroups:

Taking consecutive units of production

Picking Consecutive Units:

1. Each sample consists of units that were produced at the same time (or as closely together as possible). Ideally, we would like to take consecutive units of production.
2. This approach is used when the primary purpose of the control chart is to detect process shifts.
3. It minimizes the chance of variability due to assignable causes within a sample, and
4. it maximizes the chance of variability between samples if assignable causes are present.
5. It also provides a better estimate of the standard deviation of the process in the case of variables control charts.
6. This approach to rational subgrouping essentially gives a snapshot of the process at each point in time where a sample is collected.

Constructing Rational Subgroups:

Picking Random sample of all

process output over

the sampling interval

Each sample consists of units of product that are representative of all units that have been produced since the last sample was taken.

1. Essentially, each subgroup is a random sample of all process output over the sampling interval.
2. 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.
3. 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 snapshot method of rational subgrouping will be ineffective against these types of shifts, and so the random sample method must be used.

Decision Rules for detecting non-random patterns on control charts

(Western Electric Handbook - 1956)

1. One point plots outside the three-sigma control limits,
2. Two out of three consecutive points plot beyond the two-sigma warning limits,
3. Four out of five consecutive points plot at a distance of one-sigma or beyond from the center line, or
4. Eight consecutive points plot on one side of the center line.

Those rules apply to one side of the center line at a time. Therefore, a point above the upper warning limit followed immediately by a point below the lower warning limit would not signal an out-of-control alarm. These are often used in practice for enhancing the sensitivity of control charts. That is, the use of these rules can allow smaller process shifts to be detected more quickly than would be the case if our only criterion was the usual three-sigma control limit violation.

1. One or more points outside of the control limits.
2. Two of three consecutive points outside the two-sigma warning limits but still inside the control limits.
3. Four of five consecutive points beyond the one-sigma limits.
4. A run of eight consecutive points on one side of the center line.
5. Six points in a row steadily increasing or decreasing.
6. Fifteen points in a row in zone C (both above and below the center line).
7. Fourteen points in a row alternating up and down.
8. Eight points in a row on both sides of the center line with none in zone C.
9. An unusual or nonrandom pattern in the data.
10. One or more points near a warning or control limit.

The first four are Western Electric Rules

Control Chart Application:

Phase I

Standard control chart usage involves phase I and phase II applications, with two different and distinct objectives.

• In phase I, a set of process data is gathered and analyzed all at once in a  retrospective analysis, constructing trial control limits to determine if the process has been in control over the period of time where the data were collected, and to see if reliable control limits can be established to monitor future production.
• Control charts in phase I primarily assist operating personnel in bringing the process into a state of statistical control.
• Thus, in phase I we are comparing a collection of, say, m points to a set of control limits computed from those points. Typically m = 20 or 25 subgroups are used in phase I.
• It is fairly typical in phase I to assume that the process is initially out of control, so the objective of the analyst is to bring the process into a state of statistical control.
• Control limits are calculated based on the m subgroups and the data plotted on the control charts. Points that are outside the control limits are investigated, looking for potential assignable causes.
• Points outside the control limits are then excluded and a new set of revised control limits calculated. Then new data are collected and compared to these revised limits.
• Sometimes this type of analysis will require several cycles in which the control chart is employed, any assignable causes that are identified are worked on by engineering and operating personnel in an effort to eliminate them, revised control limits are calculated, and the out-of-control action plan is updated and expanded.
• Eventually the process is stabilized, and a clean set of data that represents in-control process performance is obtained for use in phase II.

Control Chart Application

(Phase II)

Phase II begins after we have a “clean” set of process data gathered under stable conditions and representative of in-control process performance.

• In phase II, we use the control chart to monitor the process by comparing the sample statistic for each successive sample as it is drawn from the process to the control limits.
• In phase II, we usually assume that the process is reasonably stable.
• Often, the assignable causes that occur in phase II result in smaller process shifts, because sources of variability have been systematically removed during phase I.
• Our emphasis is now on process monitoring, not on bringing an unruly process into control.
• Average run length is a valid basis for evaluating the performance of a control chart in phase II.
• Shewhart control charts are much less likely to be effective in phase II because they are not very sensitive to small to moderate size process shifts; that is, their ARL performance is relatively poor.
• The cumulative sum and EWMA control charts discussed in Chapter 9 are much more likely to be effective in phase II.

[image]

The constant A2 is tabulated for various sample sizes in Appendix Table VI.

[image]

The constants D3 and D4 are tabulated for various values of n in Appendix Table VI. 

Interpretation of Xbar and R charts:

[image]

1. Cyclic patterns occasionally appear on the control chart.
2. Such a pattern on the chart may result from systematic environmental changes such as temperature, operator fatigue, regular rotation of operators and/or machines, or fluctuation in voltage or pressure or some other variable in the production equipment.
3. R charts will sometimes reveal cycles because of maintenance schedules, operator fatigue, or tool wear resulting in excessive variability.

Interpretation of Xbar and R charts:

[image]

1. A mixture is indicated when the plotted points tend to fall near or slightly outside the control limits, with relatively few points near the center line.
2. A mixture pattern is generated by two (or more) overlapping distributions generating the process output (shown in figure).
3. The severity of the mixture pattern depends on the extent to which the distributions overlap.
4. Sometimes mixtures result from “overcontrol,” where the operators make process adjustments too often,  responding to random variation in the output rather than systematic causes.
5. A mixture pattern can also occur when output product from several sources (such as parallel machines) is fed into a common stream which is then sampled for process monitoring purposes.

Interpretation of Xbar and R charts:

[image]

1. A shift in process level may result from the introduction of new workers; changes in methods, raw materials, or machines; a change in the inspection method or standards; or a change in either the skill, attentiveness, or motivation of the operators.
2. Sometimes an improvement in process performance is noted following introduction of a control chart program, simply because of motivational factors influencing the workers.

Interpretation of Xbar and R charts:

[image]

1. A trend, or continuous movement in one direction, is usually due to a gradual wearing out or deterioration of a tool or some other critical process component.
2. In chemical processes they often occur because of settling or separation of the components of a mixture.
3. They can also result from human causes, such as operator fatigue or the presence of supervision.
4. Trends can result from seasonal influences, such as temperature.
5. When trends are due to tool wear or other systematic causes of deterioration, this may be directly incorporated into the control chart model.
6. A device useful for monitoring and analyzing processes with trends is the regression control chart.
7. The modified control chart, is also used when the process exhibits tool wear.

Interpretation of Xbar and R charts:

[image]

1. Stratification, is a tendency for the points to cluster artificially around the center line.
2. There is a marked lack of natural variability in the observed pattern.
3. One potential cause of stratification is incorrect calculation of control limits.
4. This pattern may also result when the sampling process collects one or more units from several different underlying distributions within each subgroup.

For example, suppose that a sample of size 5 is obtained by taking one observation from each of five parallel processes. If the largest and smallest units in each sample are relatively far apart because they come from two different distributions, then R will be incorrectly inflated, causing the limits on the Xbar chart to be too wide. In this case R incorrectly measures the variability between the different underlying distributions, in addition to the chance cause variation that it is intended to measure.

What is control chart for Attributes?

3 Common types of Attribute control chart

Many quality characteristics cannot be conveniently represented numerically. In such cases,we usually classify each item inspected as either conforming or nonconforming to the specifications on that quality characteristic. Quality characteristics of this type are called attributes.
Three widely used attributes control charts are:

1. control chart for fraction nonconforming, or p chart
2. control chart for nonconformities, or the c chart,
3. control chart for nonconformities per unit, or the u chart

Control chart for fraction nonconforming,

p chart

The fraction nonconforming is defined as the ratio of the number of nonconforming items in a population to the total number of items in that population. 

The statistical principles underlying the control chart for fraction nonconforming are based on the binomial distribution.

Suppose the production process is operating in a stable manner, such that the probability that any unit will not conform to specifications is p, and that successive units produced are independent. Then each unit produced is a realization of a Bernoulli random variable with parameter p.

[image]

If a random sample of n units of product is selected, and if D is the number of units of product that are nonconforming, then D has a binomial distribution with parameters n and p; that is,

the mean and variance of the random variable D are

np and np(1 − p), respectively.

Fraction Nonconforming Control Chart:

P Chart

Standard Given

Suppose that the true fraction nonconforming p in the production process is known or is a specified standard value. The the center line and control limits of the fraction nonconforming control chart would be as follows:

[image]

Fraction Nonconforming Control Chart:

P Chart

Standard Not Given

When the process fraction nonconforming p is not known, then it must be estimated from observed data. The usual procedure is to select m preliminary samples, each of size n. As a general rule, m should be at least 20 or 25. Then if there are Di nonconforming units in sample i, we compute the fraction nonconforming in the ith sample as:

[image]and the average of these individual sample fractions nonconforming is:

[image]The statistic p bar estimates the unknown fraction nonconforming p.

The center line and control limits of the control chart for fraction nonconforming are computed as follows:

[image]

This control chart is also often called the p-chart

.