MODIFIED CONTROLLED CHARTS

MODIFIED CONTROLLED CHARTS

In the traditional charting process the assumption of normality is present and the variability is considered to be less than the spread between the specifications. In fact, quite often it is much less than the specification. However, that is not always the case.

When the variability is much smaller than the specifications, we use modified control limits, especially for the Xbar chart. The modified Xbar chart is concerned only in detecting whether the true process mean ¼ is located such that the process is producing a fraction nonconforming in excess of some specified value . In effect, ¼ is allowed to vary over the interval ¼ _L ‰ ¼ ‰ ¼ _U , where ¼ _L and ¼ _U are chosen as the smallest and largest permissible values of ¼ , respectively, consistent with producing a fraction nonconforming of at most 8. The assumption of course is that variability ƒ is in control. For a very thorough discussion, see Hill (1956).

Thus, to construct the Xbar control limits, we assume normality and that the true mean is in the interval of ¼ _L ‰ ¼ ‰ ¼ _U , so that the process fraction nonconforming is less than . Consequently,

¼ _L = LSL + Z ƒ

and

¼ _U = USL - Z ƒ

where Z is the upper 100(1 - ) percentage point of the standard normal distribution. Now if we apply a type I error of a, the upper and lower control limits, respectively, are as follows :

and

Now, instead of specifying a type I error, one may use the following:

and

There is an alternative to this modified chart and the control limits that is based on the ² risk. A good discussion on this approach may be found in Duncan (1986), Freund (1957), and Montgomery (1985). This approach, however, is not very common and therefore is not discussed here. Some quality professionals recommend the use of 2-sigma limits on modified control charts, arguing that the tighter control limits afford better protection because they provide for smaller ² risk against critical shifts in the mean at little loss in the ± risk. The reader may want to read Freund (1957) for a more detailed explanation.

A final warning: When the modified charts are used, a good estimate of sigma is necessary, because if the variability shifts, then the modified limits are inappropriate. To make sure that the variability is consistent, both the R and s charts should be used.

TOOL WEAR AND TREND ANALYSIS

Control charts of the type we have been discussing can be made highly useful when associated with those processes subject to tool wear. Developing an appropriate control chart under these circumstances, it should be noted, requires a thorough understanding of the basic theory of control charts and process control and the relationship of these areas to capability analysis. In fact, all of these elements are applied together in the analysis and development of control charts associated with process change due to wear.

When a process is subject to tool wear, the natural spread of the process at any one point in time will generally be much less than the spread over the life of the tool (i.e., the spread of grouped output). In fact, if one plotted the points on regular control charts, there would be trends and cycles on regular bases. For this reason, in this type of case, special control limits may be developed. The control chart in Figure 10.16 reflects this relationship.

Figure 10.16: A tool wear control chart.

It should also be noted that (1) the trend may be downward and (2) the life of the tool is dependent on two factors other than simple wear:

1. The percentage of defective items tolerated for any short interval of production and

2. The natural dispersion or distribution of the process over the short term ( )

Both of these factors may be used in examining and planning for process control in considering tool life. As an example of how this analysis is performed, let us examine a sample problem presented by Burr (1976). In this industrial situation (actual), a quality control inspector had taken readings of the outside diameters of spacers from an automatic machine. The samples drawn were in sample sizes of 5 and were taken every 30 minutes. The specifications for the spacers were identified as 0.1250,0.0000, -0.0015.

If the measures were to be recorded on the regular Xbar and R chart, we would find that there are quite a few problems, that is, out-of-control conditions. But these out-of-control conditions are only due to tool wear.

In this type of situation, a revised Xbar chart can be developed to maximize tool life and guide the operator. It should be noted, however, that we would not be able to proceed further with this technique had the R chart not shown process control.

The steps in developing a revised X chart, reflecting process changes over time due to tool wear, are based on the work of Duncan (1986) and Manuele (1945); the regression approach is based on the work of Mandel (1969). The specific steps are as follows:

• Step 1: Draw a line of best fit through the points on the Xbar chart.

• In this step, what we are trying to develop is the mean or central line that represents the manner in which our process average runs along the trend (note that we are using what previously might have been considered to be an out-of-control condition). To help visualize the steps that we are taking in this procedure, the final chart, as it will appear when we are done, is shown in Figure 10.17. There are two ways to do this: as Burr suggests, that is, trying visually to draw the line, or as Mandel recommends. We prefer the Mandel approach of regression. To do this, we must transform the original data for convenience, as in Table 10.2.

  Table 10.2: Transformed Data from Burr's Example

  Subgroup Revised (X)	Y	XY	X Squared
  1	-10.80	-10.80	1.00
  2	-11.60	-23.20	4.00
  3	-10.40	-31.20	9.00
  4	-9.60	-38.40	16.00
  5	-9.80	-49.00	25.00
  6	-9.40	-56.40	36.00
  7	-7.40	-51.80	49.00
  8	-7.80	-62.40	64.00
  9	-7.40	-66.60	81.00
  10	-5.40	-54.00	100.00
  11	-7.00	-77.00	121.00
  12	-5.80	-69.60	144.00
  13	-3.80	-49.40	169.00
  14	-4.60	-64.40	196.00
  15	-4.20	-63.00	225.00
  16	-2.80	-44.80	256.00
  Totals 136	-117.80	-812.00	1496.00

  
  Figure 10.17: A tool wear process control chart.

• The equation of a line is Y = a + bX. The slope, b, is what we want for the average trend for our data, and that is equal to

  

• The Y-axis intercept is . With these two equations, we can find any Y for a value of X.

• Step 2: Develop the control limits for the Xbar chart around the trend line. To accomplish this, we would use the same formulas previously employed for this purpose, specifically ,

  

  We would then take this value and create the control limits around the trend line by measuring vertically a number of points (at least 2) on the trend line and plotting points A ₂ units above and below the trend line. Connecting these points will provide us with the control limits for the trend line. (If we use the regression approach, the is the b value that is the slope of the line.)

• Step 3: Determine the minimum and maximum safe process average settings:

  • Calculate the short-term process distribution ( ):

    

  • Add 3 to the LSL.

    Draw this horizontal line on your Xbar control chart. Note that this value represents the answer to the question, "What value should be used for the initial setting of the tool?" Therefore, in the future, we would use this minimum value at the beginning of each process run (i.e., after tool changes). In our example, the minimum is ‰ˆ -11.13.

  • Subtract 3 from the USL:

    USL - 3 = maximum safe process average

  • Draw this horizontal line on your Xbar chart. This line will help us determine when we should stop the process run and change the tooling. Normally, we would stop the process run when the trend line (line of best fit) intersects the maximum safe process average.

  • For this example, then, we would stop the machine and change or sharpen the tooling when the trend line reached a value of ‰ˆ -3.87. Further, a study of this type would allow us to determine the standard tool life for this machine, or class. In our example, the life of the tool, in terms of time and output, for this machine is ‰ˆ 8.5 hours.

One more note before we leave this topic. We would normally reset the tool when the trend line intersects the maximum safe process average. Once repetitive studies yield a secure tool life estimate, we would normally change or sharpen the tooling after that particular period had elapsed. However, note that we would also cease operations for tooling changes or other modifications whenever a single Xbar value was measured out of specification.

Finally, two points should be made to summarize this unit. First, trends can also be downward. This can easily occur, for example, when the process is associated with turning an inside diameter. In this case, the procedure used would still be the same as that employed with upward trends. Second, there are numerous examples of where this technique may be appropriately employed. Examples include:

• Grinder/polisher wear

• Moisture content in lumber

• Die wear

• Mold wear

• Human fatigue

• Environmental (ambient characteristics) control

• Viscosity

• Acid and alkaline content

Remember, however, that this technique requires that the chart reflect process control in the following:

1. The percentage of defective items tolerated for any short interval of production and

2. The natural dispersion or distribution of the process over the short term ( )

In addition, whereas Burr (1976) and Duncan (1986) have accounted for the modification of the limits on a fairly conservative approach, there are others (Manuele, 1945; Montgomery, 1985; Quesenberry, 1991) that provide a more liberal interpretation. On the other hand, Aerne et al. (1991) and Davis and Woodhall (1988) explore problems associated with interpreting charts when the process average is undergoing sustained linear shifts.