SPC: using control charts to correct and improve a process

Title: SPC: Using Control Charts to Correct a Process

Authors:John D'Arcy, Matt Hagen, Adam Holewinski, and Alwin Ng Date Presented: 11-30-2006 /Date Revised:12/08/2006
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Introduction
Statistical process control (SPC) can be used to determine if a process is running within acceptable limits, or if it is out of control. The principle behind SPC is to create control charts that compare the means and ranges of subgroups of data to calculated operating limits. The methods for creating the charts are found on the Basic Control Charts page. However, they are of little use if one cannot determine the proper chart type and analysis method to apply to a scenario. This page outlines the rules and techniques most commonly used to correct a process using control charts.

Finding the Right Control Chart
Depending on the nature of the data being analyzed, different types of control charts may be more effective than others. Below are the most common chart types and the situations that they are used in.

Average and Range Charts
Standard average and range charts can be used to analyze data for which subgroup size and subgroup frequency are independent of each other. This is generally the case for continuous processes and essentially means that one can alter the group size of data subsets (adjacent points) to be compared without automatically changing the frequency at which the sets can be viewed.

XmR Charts
XmR charts plot the mean of a subgroup but use a moving range between subgroups, so they are the most convenient for a subgroup size of n=1, which would otherwise not have a range. They are often used for periodically collected data, such as in batch processes, where there is a high chance subgroups will not be homogeneous. Periodic measurements imply each value is unique to a particular period of time, meaning a change in subgroup size will change the time period as well. XmR charts allow one to plot a point every time one becomes available, making them the most sensitive in such scenarios.

Moving Average Charts
Moving average charts are often used to determine slowly evolving trends in a periodic process. They use moving averages and moving ranges for the data, which are computed by taking the average and range of the subgroup composed of a data point and the previous n points, where n is the subgroup size. This approach introduces a lag so that a problem must persist for n time steps before it is noticed. While such a control chart is not helpful in correcting a process with short term problems, the negation of short term errors can be useful in, say, finding an annual trend. By using a subgroup size of 12 and examining monthly data, seasonal variations would not hinder the determination of an annual change.

Approaching Control Chart Analysis
In order to correct a process, one needs to make sure they have a firm understanding of what the data in a control chart really indicates. The following questions should help guide your analysis of control charts:

1. What physical quantities do the numbers represent? What about these quantities makes them important to the process (e.g. does a certain temperature or concentration need to be maintained, etc.)?

2. Under what conditions are the measurements taken? With what instrument? By whom? Where? When and how often? Are there other possible sources of variation?

3. How is the data divided among subgroups? Given the organization, which sources of variation occur within subgroups and which occur between subgroups?

4. What behavior is expected? Are there any inherent limitations on values that are narrower than the statistical limits (e.g. physical limits on equipment, environmetal regulations, etc.)?

Determining if a Process is Out of Control
Before any additional analysis of the process can be done, the limits of process control (three sigma limits) must be established. These limits contain the grand average plus or minus three sigma limits. This operating range is created to use as a bench mark to compare future values to. The process being statistically out of control does not mean that the actual process is out of control. Only that there is no control over the statistical values being seen. The following rules are used to determine if a process is out of control based on these limits:

1.)If any data points that do not exhibit results within the “three sigma limits,” they should be disregarded and considered the result of an uncontrolled system.

2.)If two out of three consecutive data points (run of length 2) appear to be outside of two sigma limits, the system is determined to be out of control. These two entries must lie on the same side of the center line and be two sigma limits out of control.

3.)If four out of five data points (run of length 4) fall outside of one sigma value, the system is considered to be out of control. In the same manner explained in rule two, these points must also fall on the same side of the center line.

4.)If eight successive values (run of length 8) fall on the same side of the centerline, the system is considered out of control.

Exceptions
Because moving averages and ranges undermine the principles of run length analysis, it is recommended that only Rule 1 be used with XmR and moving average charts.

Choosing Appropriate Subgroups
When attempting to analyze any type of event from which occurrences of data are generated, subgroups can be implemented and used. These subgroups will aid in drawing conclusive information from seemingly random data to determine any patterns that exist as a result of the event.

One method of analysis of the subgroups will enable the user to determine if the process is changing with time or not. This is done by randomly obtaining measurements at a certain time interval. This process is then repeated several times and the data obtained is treated as distinct sets of data for each occurrence of this process. Once these separate data sets are generated comparisons across subgroups begin. If these different subgroups show trends and behavior that is inconstant, then it can be assumed that the process displays uncontrolled variation. However if the process is not changing over time the trends within the subgroup will not vary when making comparisons across the subgroups.

The various uses for the subgroups makes it essential to establish criteria from which they are picked and how data is selected to be placed inside them. This is necessary because it is extremely important that the data inside each group be relatively similar so deviations within the subgroup become unimportant. This will help ensure that the most noticeable variances within the data being analyzed will exist between the subgroups. Trends like this become more obvious when following the rules listed below.

1.)Things that are known to be unlike cannot be grouped together

This will eliminate the capability of observing trends across data sets because the data within each set would be unalike, so comparisons cannot be drawn amongst different set of data.

2.)Minimize the variations that can arise inside of subgroups

The variations that exist within a subgroup can be attributed to the noise that is present within the system. Once these variations are accounted for any new data that is collected will have accounted for the background noise that is present in the system, and therefore the analysis will be easier and the results will be a more accurate representation of the true information. The more similar data within a subgroup is, the more easily it can reveal variation between groups.

3.)Maximize the opportunity for variations between subgroups 

To best compare differing data sets or points place them within different subgroups. Because these subgroups occur at separate time intervals the comparisons that can be drawn will be more concrete, because the time of occurrence cannot be a reason to attribute similarities to.

4.)Average across noise, not across signal

When the measurements are taken with respect to system noise background noise will be similar amongst data, as a result the true signal will be easier to detect because the error within each measurement of the data set will comparable.

5.)Establish operational standards from which sampling procedures are followed

If sampling is done by different operators unintended errors can arise out of inconsistency from sampling procedures. For example, one operator can sample a process relatively close to a system startup and as a result might obtain data that would not be representative of the system running at steady state. A different operator would not be justified in comparing this information to samples they took after the process had been running steadily for an extended period of time. For this sake comparisons can be made of data that is acquired under similar conditions.

6.)Treat the Chart in accordance with the use of the data

The amount of entries within each subgroup should be a reflection of the amount of time devoted to monitoring the process. If the process is being watched around the clock by multiple workers in the area the sample size of each subgroup should be much larger than if on worker samples the process every other day.

Fixing the Process
Once you have determined the specific cause of out of control variation, you generally want to fix it, remove it, or replace it--e.g. a bad temperature controller should be replaced. However, before making any hasty (and potentially costly) decisions, consider the following:

-Resample and make sure that it is not an error (a good idea even when things are running smooth)

-Revise the control limits: adjust the new calculated mean up or down as more data becomes available. Control limits should be recalculated periodically to verify where 3 sigma limits really lie.

-Do nothing. You might need to keep the process going, as it takes too much time, effort, or cost to correct. Wait until it happens again before taking action in situations when fixing it "the right way" is impractical.

Worked out Example 1
Consider the production of rubber parts, where a template is used for casting the rubber parts. The template is capable of making three pieces of the rubber parts at one time. As the quality control engineer, you decided to use control charts to determine the consistency in the products weight and whether the machine is operating within the acceptable limits or not. You would gather a set of data every 4 hours, which amounts to total 6 sets of data for one day (24 hours). During each measurement, you would collect 3 consecutive cycles of the press. Since each cycle would make 3 rubber parts, you would have 9 measurements to collect for every set of data. There are 3 sources of variation within these data. There is the variation due to collecting the measurements at different hours of the day (Hour-to-Hour variation). There is the variation due to collecting the measurements at different cycles (Cycle-to-Cycle variation). There is the variation simply due to the socket difference (Rubber-to-Rubber variation). Thus, there are several ways to organize these data into subgroups. Explore the possible subgroup organizations and see if any organization shows out of control variation. How could the process be improved? Solution The first organization is shown in the next table. The columns are used to define the subgroups. The sources of variation are allocated as follows: the Hour-to-Hour variation is shown between the subgroup, the Cycle-to-Cycle variation is shown between the subgroup, and the Rubber-to-Rubber variation is shown within the subgroup. The organization yields 18 subgroups of size n = 3. The Grand Average is 6.98 units, and the Average Range is 4.56 units. The control limits for the Average Chart are 11.64 units and 2.32 units. To calculate for the control limits, refer to SPC: Basic control charts. The average and range charts are shown in the next figure.

The second organization is shown in the next table. The rows are used to define the subgroups. The sources of variation are allocated as follows: the Hour-to-Hour variation is shown between the subgroup, the Cycle-to-Cycle variation is shown within the subgroup, and the Rubber-to-Rubber variation is shown between the subgroup. The organization yields 18 subgroups of size n = 3. The Grand Average remains the same as 6.98 units, and the Average Range is now 1.78 units. The control limits for the Average Chart are 8.80 units and 5.16 units. The average and range charts are shown in the next figure.

Although the two control charts represent the same weight data, they appear completely different from one another. The difference between the two charts can be attributed to how the subgroups are defined. From the second control chart, it is easily noticeable that the weight for Rubber 1 is always out of control, being higher than Rubber 2 and Rubber 3. However, such irregularity does not appear in the first control chart. In the first control chart, the Rubber-to-Rubber variation is shown within the subgroup, meaning that we are only checking whether the Rubber-to-Rubber differences are consistent going from one subgroup to the next one. In contrast, in the second control chart, the Rubber-to-Rubber variation is shown between the subgroup, meaning that we are checking whether there are any detectable Rubber-to-Rubber differences or not. It is also possible to draw the average control charts for each of the three rubber parts. This chart will be the most sensitive and also be able to detect the irregularity with Rubber 1. However, with this case, this control chart will not provide additional variation information and thus for the sake of space, will not be provided. Therefore, using control charts, we can now identify and fix the irregularity (Assignable Cause) in the template for Rubber 1, resulting in rubber parts with higher weight consistency.

Worked out Example 2
Now consider a process where, for the past 13 years, the average reactor temperature (degrees C) was recorded each season. Find out if the system is within statistical control limits. If the average temperature is changing with an upward or downward trend each year, what might be done to correct this?





Solution

Looking at the data given it is very difficult to see any annual trends. To help look at only the annual trends and average out the short term seasonal variability, a moving average and moving range chart are used (for equations and instructions for making these charts see Basic control charts). The subgroups in this case are most easily found by placing all the data points in a line. The first point is the first subgroup. The first two points are the second subgroup. The first three the third, and the first four the fourth. Thereafter, points 2-5 are the next subgroup, 3-6 the next subgroup, and so on. The mean and range of these groups represent the moving values.



The system seems to be out of statistical control because the moving average chart has more than 8 consecutive points on the same side of the mean range. Also, the moving range is outside the upper critical limit (three sigma UCL shown in red). Again looking at the average chart, it can be seen that the average annual temperature is increasing each year. The most probable assignable cause would be a defective temperature controller, so this should be replaced

Multiple Choice Question 1
A run of length 4 indicates a lack of control when it occurs outside______.

A. The critical density of states B. One sigma limit C. The first Brillouin zone D. The event horizon

Multiple Choice Question 2
A scientist is trying to determine if global warming is really happening. He takes the average monthly temperature for the past 100 years and wants to create a control chart to see if mother nature is out of control. What chart type and subgrouping would be most effective?

A. Standard mean and range with subgroup size 1 B. XmR with subgroup size 1 C. A geothermal phase plot with subgroup size 12 D. Moving average with subgroup size 12

Submitting answers to the multiple choice questions

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https://lessons.ummu.umich.edu/2k/che_466/SPCCorrect]
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