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gail_massari

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Feb 27, 2013

Designing Insightful Process Behavior Charts

This article originally appeared in JMPer Cable, Issue 29 Summer 2014.

by José G. Ramirez zeneos  , PhD and Annie D. Zangi annie.zangi , JMP Research Statistician Developer, SAS

The Control Chart Builder platform was introduced in JMP® 10 as an easy and fun way to design process behavior charts. This becomes really useful in situations where different subgroups are possible, depending on how the data is organized. In this article we show how Control Chart Builder can be used to investigate different chart designs, helping select the one that reveals hidden differences in the data.

Injection Molding Example

Four hundred observations were collected in an injection molding operation that manufactures ball joint sockets, as shown in Figure 1 below.

Figure 1.jpg

Figure 1 Scatterplot with exploratory fits

Figure 1 was adapted from Figure 3.9a of Ramírez and Ramirez (2010)1.


The response of interest is the effective thickness of the ball joint socket (in 100th mm) in excess of 12 mm. Because this is a new process, management wants to know whether the process is ready for production. In other words, is the process producing ball joint sockets in a predictable way? We can use a process behavior chart to determine whether the process is in control or predictable.


It is easy to generate a process behavior chart. For example, in order to generate an individual and moving range chart we select Analyze > Quality and Process > Control Chart > IR and assign Thickness to Process. Figure 2 does not show any points outside the control limits, but it does not look quite in “control.” Is the information displayed in this chart enough for us to qualify the process?

 

Figure 2.jpg

Figure 2 Individual Measurement and Moving Range chart of Thickness


In this case, the individual and moving range charts do not reflect the structure of data, that is, the different sources of variation present in the data. So, no, this chart does not give us enough information to qualify the process.

 

Process Behavior Chart Design

A process behavior chart needs to be designed so that it can answer the questions of interest. What are these questions? They depend on which type of chart we are using. Observations in a process behavior chart represent “rational” subgroups of like things. For example, the XBar and Range (or R) charts plot the range of subgroup values on the R chart to monitor variation. They plot the average of the subgroup values on the XBar chart to monitor location. In their book, Understanding Statistical Process Control, Wheeler and Chambers clearly present the questions that the XBar and R process behavior charts pose.

 

 

• The R chart asks the question: “Making allowance for the average amount of variation within the subgroups, are the within subgroups differences consistent?”

• The XBar chart asks the question: “Making allowance for the amount of variation within the subgroups, are there detectable differences between the subgroup averages?”

 

As you can see, each chart answers a different question in terms of consistency within subgroups, and the ability to detect changes between subgroups. As we show below, the answers to these questions depend on how the data is organized into rational subgroups.


Rational Subgroups

Rational subgroups define the process of organizing the data into groups of like things that reflect the context and sources of variation present in the data. Let us revisit the data in Figure 2. What is the context of the ball joint socket data? The injection molding process produces sockets four at a time that come from a mold that has four cavities. To qualify the process, data was collected four times a day for five days. At each of those four periods during a day, five cycles of the press were performed, giving 20 parts per hour. This gave a total of 400 readings, as shown in Figure 2.

There are sources of variation in this data, as shown in Table 1.

 

Table 1. Sources of variation for ball joint socket data

8700_Table 1.JPG

The design of the process behavior chart requires us to think about the allocation of these sources of variation to the rational subgroups, and the questions that the charts answer. The different allocations can be easily explored by means of Control Chart Builder.

 

Control Chart Builder

Control Chart Builder works much like Graph Builder with a drag-and-drop interface. It displays Individual and Moving Range charts the instant that you drag a variable onto the y-axis. When  you drag a subgroup column onto the x-axis, it switches to an XBar and R chart. Let us look at three different organizations of the ball joint socket data, according to how the sources of variation in Table 1 are allocated to the subgroups.


First Organization of the Data

The first organization of the thickness data allocates data to the different sub­groups as shown in Table 2.

 

Table 2. First organization of the data

8702_Table 2.JPG

Note that due to this allocation the XBar chart confounds the Hour-to-Hour and the Cycle-to-Cycle information.

 

To generate the Individual and Moving Range charts shown in Figure 2 in Control Chart Builder, follow these steps:

 

1. Open Socket Thickness.jmp from the JMP File Exchange. The data table has columns for Day, Time, Hour, Cycle, Cavity and Thickness.

2. Select Analyze > Quality and Process > Control Chart Builder.

3. Drag the Thickness column onto the graph.

Now follow these steps to generate the XBar and R charts shown in Figure 3.

4. Drag the Hour column to the x-axis area of the chart.

5. Drag the Cycle column to the dropzone just above the x-axis to nest Cycle within Hour.

Figure 3.jpg

Figure 3 XBar and R chart showing the first organization of the thickness data


The charts in Figure 3 answer the following questions:

 

 

• The R chart asks the question: “Are the Cavity-to-Cavity differences consistent?”

• The XBar chart asks the question: “Are there detectable differences from Hour-to-Hour and Cycle-to- Cycle?”


The R chart shows that the Cavity-to- Cavity differences are consistent and centered around 0.08 mm. Although no points are outside the control limits in the XBar chart, the pattern does not seem random. We are not sure, then, if there are detectable differences Cycle-to-Cycle and Hour-to-Hour.


Second Organization of the Data

The second organization of the thick­ness data allocates data to the different subgroups as shown in Table 3.

 

Table 3. Second organization of the data

8704_Table 3.JPG

Note that, due to this allocation, the XBar chart confounds the Hour-to-Hour and the Cavity-to-Cavity information.

 

Rearrange the variables to update the XBar and R charts as shown in Figure 4.

 

 

6. Drag Cycle off of the x-axis.

7. Drag the Cavity column to the drop zone just above the x-axis to nest Cavity within Hour.

 

The charts in Figure 4 answer the following questions:

• The R chart asks the question: “Are the Cycle-to-Cycle differences consistent?”

• The XBar chart asks the question: “Are there detectable differences from Hour-to-Hour and Cavity-to-Cavity?”

Figure 4.jpg 

Figure 4 XBar and R chart showing the second organization of the thickness data

From the R chart in Figure 4, we can see that the cycle-to-cycle differences are consistent. This time, however, we see many points out of control in the XBar chart, indicating the presence of detect­able differences Cavity-to-Cavity and Hour-to-Hour. Why didn’t we see these points in Figure 3? The XBar chart in the first organization did not ask whether there were detectable differences between the cavities, just between Cycle-to-Cycle and Hour-to-Hour.


Selecting the high points in the XBar chart and looking back at the data table reveals that all the high-thickness read­ings come from Cavity I. This is useful information because it shows that even though the Cavity-to-Cavity differences are consistent, Cavity I in general pro­duces sockets with a higher thickness. This is an indication that the cavities behave differently and their data should not be combined in a single process behavior chart.

 

Third Organization

To further investigate the differences between cavities discovered in Figure 4, we generate separate process behavior charts per cavity. Table 4 shows the reallocated data.

 

Table 4. Third organization of the data

8707_Table 4.JPG

Rearrange the variables to update the XBar and R charts as shown in Figure 5.

 

8. Drag Cavity from the x-axis to the Phase drop zone above the graph.

Note: This example essentially makes separate control charts for each cavity.

 

Voilà! Figure 5 clearly shows how the thickness readings coming from Cavity I are higher than the other three cavities. Because there is only one source of variation allocated to each chart, we can confidently answer these questions:

 

 

“For any given cavity, are the Cycle-to-Cycle differences consistent?” Yes, and they are consistent for each of the cavities. The average ranges, green lines, are similar.

“For any given cavity, are there detectable differences from Hour-to- Hour?” Yes. Sockets are thicker one hour, thinner the next.

 Figure 5.jpg

Figure 5 XBar and R chart showing the third organization of the thickness data


What Did We Learn?

The careful design of the process behavior chart can reveal patterns that a “default” software chart might mask. However, even when the charts are designed carefully, it is important to rationally think about the allocation of the different sources of variation to subgroups. For our three allocations, the first process behavior chart did not signal the Cavity I difference because it was not designed to detect differences between cavities. The second process behavior chart did signal the Cavity I difference because it was specifically designed to answer the question: Are there detectable differences from Cavity-to-Cavity? But it was the third allocation that clearly revealed not only the Cavity-to-Cavity differences, but also the Hour-to-Hour differences that the other allocations missed. Control Chart Builder made it very easy to design the charts, and helped reveal the hidden features in our data.

 

The Rest of the Story

The engineer in charge of the molding process sent the mold to the tool shop to have a 3-1,000th inch shim put behind Cavity I to solve the thickness problem shown in Figure 4. After he got the mold back, he asked the toolmaker if he had done anything else to the mold. The toolmaker said, “I did clean it up real good – there was a wax build-up on the face of the mold. I cleaned that off for you.” It was then that the process engineer realized that the process was out of control because the operators were not cleaning off the wax build-up often enough.

 

REFERENCES

Wheeler, D.J., and D.S. Chambers. (1992) Understanding Statistical Process Control. Second Edition. Knoxville, TN: SPC Press. The data and example come from Section 5.6.

 

Ramírez, José G., and Ramírez, Brenda S. (2009) Analyzing and Interpreting Continuous Data Using JMP: A Step-by-Step Guide. Cary, NC: SAS Institute Inc.

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