Controls

A control chart employs a center line (denoted CNL) and two control limits—an upper control limit (denoted UCL) and a lower control limit (denoted LCL). The center line represents the average performance of the process when it is in a state of statistical control—that is, when only common cause variation exists. The upper and lower control limits are horizontal lines situated above and below the center line. These control limits are established so that, when the process is in control, almost all plot points will be between the upper and lower limits. In practice, the control limits are used as follows:

1 If all observed plot points are between the LCL and UCL (and if no unusual patterns of points exist—this will be explained later), we have no evidence that assignable causes exist and we assume that the process is in statistical control. In this case, only common causes of process variation exist, and no action to remove assignable causes is taken on the process. If we were to take such action, we would be unnecessarily tampering with the process.

2 If we observe one or more plot points outside the control limits, then we have evidence that the process is out of control due to one or more assignable causes. Here we must take action on the process to remove these assignable causes.

In the next section we begin to discuss how to construct control charts. Before doing this, however, we must emphasize the importance of documenting a process while the subgroups of data are being collected. The time at which each subgroup is taken is recorded, and the person who collected the data is also recorded. Any process changes (machine resets, adjustments, shift changes, operator changes, and so on) must be documented. Any potential sources of variation that may significantly affect the process output should be noted. If the process is not well documented, it will be very difficult to identify the root causes of unusual variations that may be detected when we analyze the subgroups of data.

17.4: and R Charts

Chapter 21

and
R
charts
are the most commonly used control charts for measurement data (such charts are often called
variables control charts
). Subgroup means are plotted versus time on the chart, while subgroup ranges are plotted on the R chart. The chart monitors the process mean or level (we wish to run near a desired target level). The R chart is used to monitor the amount of variability around the process level (we desire as little variability as possible around the target). Note here that we employ two control charts, and that it is important to use the two charts together. If we do not use both charts, we will not get all the information needed to improve the process.

Before seeing how to construct and R charts, we should mention that it is also possible to monitor the process variability by using a chart for subgroup standard deviations. Such a chart is called an
s chart. However, the overwhelming majority of practitioners use R charts rather than s charts. This is partly due to historical reasons. When control charts were developed, electronic calculators and computers did not exist. It was, therefore, much easier to compute a subgroup range than it was to compute a subgroup standard deviation. For this reason, the use of R charts has persisted. Some people also feel that it is easier for factory personnel (some of whom may have little mathematical background) to understand and relate to the subgroup range. In addition, while the standard deviation (which is computed using all the measurements in a subgroup) is a better measure of variability than the range (which is computed using only two measurements), the R chart usually suffices. This is because and R charts usually employ small subgroups—as mentioned previously, subgroup sizes are often between 2 and 6. For such subgroup sizes, it can be shown that using subgroup ranges is almost as effective as using subgroup standard deviations.

To construct and R charts, suppose we have observed rational subgroups of n measurements over successive time periods (hours, shifts, days, or the like). We first calculate the mean and range R for each subgroup, and we construct graphs of performance for the values and for the R values (as in Figure 17.2). In order to calculate center lines and control limits, let denote the mean of the subgroup of n measurements that is selected in a particular time period. Furthermore, assume that the population of all process measurements that could be observed in any time period is normally distributed with mean μ and standard deviation σ, and also assume successive process measurements are statistically independent.4 Then, if μ and σ stay constant over time, the sampling distribution of subgroup means in any time period is normally distributed with mean μ and standard deviation . It follows that (in any time period) 99.73 percent of all possible values of the subgroup mean are in the interval

This fact is illustrated in Figure 17.3. It follows that we can set a center line and control limits for the chart as

If an observed subgroup mean is inside these control limits, we have no evidence to suggest that the process is out of control. However, if the subgroup mean is outside these limits, we conclude that μ and/or σ have changed, and that the process is out of control. The chart limits are illustrated in Figure 17.3.

Figure 17.3: An Illustration of Chart Control Limits with the Process Mean μ and Process Standard Deviation σ Known

If the process is in control, and thus μ and σ stay constant over time, it follows that μ and σ are the mean and standard deviation of all possible process measurements. For this reason, we call μ the process mean and σ the process standard deviation. Since in most real situations we do not know the true values of μ and σ, we must estimate these values. If the process is in control, an appropriate estimate of the process mean μ is

( is pronounced “x double bar”). It follows that the center line for the chart is

To obtain control limits for the chart, we compute

It can be shown that an appropriate estimate of the process standard deviation σ is where d2 is a constant that depends on the subgroup size n. Although we do not present a development of d2 here, it intuitively makes sense that, for a given subgroup size, our best estimate of the process standard deviation should be related to the average of the subgroup ranges (). The number d2 relates these quantities. Values of d2 are given in Table 17.3 for subgroup sizes n = 2 through n = 25. At the end of this section we further discuss why we /d2 use to estimate the process standard deviation.

Table 17.3: Control Chart Constants for and R Charts

Substituting the estimate of μ and the estimate /d2 of σ into the limits

we obtain

Finally, we define

and rewrite the control limits as

Here we call A2 a
control chart constant. As the formula for A2 implies, this control chart constant depends on the subgroup size n. Values of A2 are given in Table 17.3 for subgroup sizes n = 2 through n = 25.

The center line for the R chart is

Furthermore, assuming normality, it can be shown that there are control chart constants D4 and D3 so that

Here the control chart constants D4 and D3 also depend on the subgroup size n. Values of D4 and D3 are given in Table 17.3 for subgroup sizes n = 2 through n = 25. We summarize the center lines and control limits for and R charts in the following box:

and R Chart Center Lines and Control Limits

EXAMPLE 17.3: The Hole Location Case

Consider the hole location data for air conditioner compressor shells that is given in Figure 17.1 (page 751). In order to calculate and R chart control limits for this data, we compute

Looking at Table 17.3, we see that when the subgroup size is n = 5, the control chart constants needed for and R charts are A2 = .577 and D4 = 2.114. It follows that center lines and control limits are

Since D3 is not listed in Table 17.3 for a subgroup size of n = 5, the R chart does not have a lower control limit. Figure 17.4 presents the MINITAB output of the and R charts for the hole location data. Note that the center lines and control limits that we have just calculated are shown on the and R charts.

Figure 17.4: MINITAB Output of and R Charts for the Hole Location Data

Control limits such as those computed in Example 17.3 are called trial control limits. Theoretically, control limits are supposed to be computed using subgroups collected while the process is in statistical control. However, it is impossible to know whether the process is in control until we have constructed the control charts. If, after we have set up the and R charts, we find that the process is in control, we can use the charts to monitor the process.

If the charts show that the process is not in statistical control (for example, there are plot points outside the control limits), we must find and eliminate the assignable causes before we can calculate control limits for monitoring the process. In order to understand how to find and eliminate assignable causes, we must understand how changes in the process mean and the process variation show up on and R charts. To do this, consider Figures 17.5 and 17.6. These figures illustrate that, whereas a change in the process mean shows up only on the chart, a change in the process variation shows up on both the and R charts. Specifically, Figure 17.5 shows that, when the process mean increases, the sample means plotted on the chart increase and go out of control. Figure 17.6 shows that, when the process variation (standard deviation, σ) increases,

Figure 17.5: A Shift of the Process Mean Shows Up on the Chart

Figure 17.6: An Increase in the Process Variation Shows Up on Both the and R Charts

1 The sample ranges plotted on the R chart increase and go out of control.

1 The sample means plotted on the chart become more variable (because, since σ increases, increases) and go out of control.

Since changes in the process mean and in the process variation show up on the chart, we do not begin by analyzing the chart. This is because, if there were out-of-control sample meanson the chart, we would not know whether the process mean or the process variation had changed. Therefore, it might be more difficult to identify the assignable causes of the out-of-control sample means because the assignable causes that would cause the process mean to shift could be very different from the assignable causes that would cause the process variation to increase. For instance, unwarranted frequent resetting of a machine might cause the process level to shift up and down, while improper lubrication of the machine might increase the process variation.

In order to simplify and better organize our analysis procedure, we begin by analyzing the R chart, which reflects only changes in the process variation. Specifically, we first identify and eliminate the assignable causes of the out-of-control sample ranges on the R chart, and then we analyze the chart. The exact procedure is illustrated in the following example.

EXAMPLE 17.4: The Hole Location Case

Consider the and R charts for the hole location data that are given in Figure 17.4. To develop control limits that can be used for ongoing control, we first examine the R chart. We find two points above the UCL on the R chart. This indicates that excess within-subgroup variability exists at these points. We see that the out-of-control points correspond to subgroups 7 and 17. Investigation reveals that, when these subgroups were selected, an inexperienced, newly hired operator ran the operation while the regular operator was on break. We find that the inexperienced operator is not fully closing the clamps that fasten down the compressor shells during the hole punching operation. This is causing excess variability in the hole locations. This assignable cause can be eliminated by thoroughly retraining the newly hired operator.

Since we have identified and corrected the assignable cause associated with the points that are out of control on the R chart, we can drop subgroups 7 and 17 from the data set. We recalculate center lines and control limits by using the remaining 18 subgroups. We first recompute (omitting and R values for subgroups 7 and 17)

Notice here that has not changed much (see Figure 17.4), but has been reduced from .085 to .0683. Using the new and values, revised control limits for the chart are

The revised UCL for the R chart is

Since D3 is not listed for subgroups of size 5, the R chart does not have a LCL. Here the reduction in has reduced the UCL on the R chart from .1797 to .1444 and has also narrowed the control limits for the chart. For instance, the UCL for the chart has been reduced from 3.0552 to 3.0457. The MINITAB output of the and R charts employing these revised center lines and control limits is shown in Figure 17.7.

Figure 17.7: MINITAB Output of and R Charts for the Hole Location Data: Subgroups 7 and 17 Omitted

We must now check the revised R chart for statistical control. We find that the chart shows good control: there are no other points outside the control limits or long runs of points on either side of the center line. Since the R chart is in good control, we can analyze the revised chart. We see that two plot points are above the UCL on the chart. Notice that these points were not outside our original trial control limits in Figure 17.4. However, the elimination of the assignable cause and the resulting reduction in has narrowed the chart control limits so that these points are now out of control. Since the R chart is in control, the points on the chart that are out of control suggest that the process level has shifted when subgroups 1 and 12 were taken. Investigation reveals that these subgroups were observed immediately after start-up at the beginning of the day and immediately after start-up following the lunch break. We find that, if we allow a five-minute machine warm-up period, we can eliminate the process level problem.

Since we have again found and eliminated an assignable cause, we must compute newly revised center lines and control limits. Dropping subgroups 1 and 12 from the data set, we recompute

Using the newest and values, we compute newly revised control limits as follows:

Again, the R chart does not have a LCL. We obtain the newly revised and R charts that are shown in the MINITAB output of Figure 17.8. We see that all the points on each chart are inside their respective control limits. This says that the actions taken to remove assignable causes have brought the process into statistical control. However, it is important to point out that, although the process is in statistical control, this does not necessarily mean that the process is capable of producing products that meet the customer’s needs. That is, while the control charts tell us that no assignable causes of process variation remain, the charts do not (directly) tell us anything about how much common cause variation exists. If there is too much common cause variability, the process will not meet customer or manufacturer specifications. We talk more about this later.

Figure 17.8: MINITAB Output of and R Charts for the Hole Location Data: Subgroups 1, 7, 12, and 17 Omitted. The Charts Show Good Control.

When both the and R charts are in statistical control, we can use the control limits for ongoing process monitoring. New and R values for subsequent subgroups are plotted with respect to these limits. Plot points outside the control limits indicate the existence of assignable causes and the need for action on the process. The appropriate corrective action can often be taken by local supervision. Sometimes management intervention may be needed. For example, if the assignable cause is out-of-specification raw materials, management may have to work with a supplier to improve the situation. The ongoing control limits occasionally need to be updated to include newly observed data. However, since employees often seem to be uncomfortable working with limits that are frequently changing, it is probably a good idea to update center lines and control limits only when the new data would substantially change the limits. Of course, if an important process change is implemented, new data must be collected, and we may need to develop new center lines and control limits from scratch.

Sometimes it is not possible to find an assignable cause, or it is not possible to eliminate the assignable cause even when it can be identified. In such a case, it is possible that the original (or partially revised) trial control limits are good enough to use; this will be a subjective decision. Occasionally, it is reasonable to drop one or more subgroups that have been affected by an assignable cause that cannot be eliminated. For example, the assignable cause might be an event that very rarely occurs and is unpreventable. If the subgroup(s) affected by the assignable cause have a detrimental effect on the control limits, we might drop the subgroups and calculate revised limits. Another alternative is to collect new data and use them to calculate control limits.

In the following box we summarize the most important points we have made regarding the analysis of and R charts:

Analyzing and R Charts to Establish Process Control

1 Remember that it is important to use both the chart and the R chart to study the process.

2 Begin by analyzing the R chart for statistical control.

a Find and eliminate assignable causes that are indicated by the R chart.

b Revise both the and R chart control limits, dropping data for subgroups corresponding to assignable causes that have been found and eliminated in 2a.

c Check the revised R chart for control.

d Repeat 2a, b, and c as necessary until the R chart shows statistical control.

3 When the R chart is in statistical control, the chart can be properly analyzed.

a Find and eliminate assignable causes that are indicated by the chart.

b Revise both the and R chart control limits, dropping data for subgroups corresponding to assignable causes that have been found and eliminated in 3a.

c Check the revised chart (and the revised R chart) for control.

d Repeat 3a, b, and c (or, if necessary, 2a, b, and c and 3a, b, and c) as needed until both the and R charts show statistical control.

4 When both the and R charts are in control, use the control limits for process monitoring.

a Plot and R points for newly observed subgroups with respect to the established limits.

b If either the chart or the R chart indicates a lack of control, take corrective action on the process.

5 Periodically update the and R control limits using all relevant data (data that describe the process as it now operates).

6 When a major process change is made, develop new control limits if necessary.

EXAMPLE 17.5: The Hole Location Case

We consider the hole location problem and the revised and R charts shown in Figure 17.8. Since the process has been brought into statistical control, we may use the control limits in Figure 17.8 to monitor the process. This would assume that we have used an appropriate subgrouping scheme and have observed enough subgroups to give potential assignable causes a chance to show up. In reality, we probably want to collect considerably more than 20 subgroups before setting control limits for ongoing control of the process.

We assume for this example that the control limits in Figure 17.8 are reasonable. Table 17.4 gives four subsequently observed subgroups of five hole location dimensions. The subgroup means and ranges for these data are plotted with respect to the ongoing control limits in the MINITAB output of Figure 17.9. We see that the R chart remains in control, while the mean for subgroup 24 is above the UCL on the chart. This tells us that an assignable cause has increased the process mean. Therefore, action is needed to reduce the process mean.

Table 17.4: Four Subgroups of Five Hole Location Dimensions Observed after Developing Control Limits for Ongoing Process Monitoring

Figure 17.9: MINITAB Output of and R Charts for the Hole Location Data: Ongoing Control

EXAMPLE 17.6: The Hot Chocolate Temperature Case

Consider the hot chocolate data given in Table 17.2 (page 752). In order to set up and R charts for these data, we compute

and

Looking at Table 17.3 (page 757), we see that the and R control chart constants for the subgroup size n = 3 are A2 = 1.023 and D4 = 2.574. It follows that we calculate center lines and control limits as follows:

Since D3 is not given in Table 17.3 for the subgroup size n = 3, the R chart does not have a lower control limit.

The and R charts for the hot chocolate data are given in the MegaStat output of Figure 17.10. We see that the R chart is in good statistical control, while the chart is out of control with three subgroup means above the UCL and with one subgroup mean below the LCL. Looking at the chart, we see that the subgroup means that are above the UCL were observed during lunch (note subgroups 4, 10, and 22). Investigation and process documentation reveal that on these days the hot chocolate machine was not turned off between breakfast and lunch. Discussion among members of the dining hall staff further reveals that, because there is less time between breakfast and lunch than there is between lunch and dinner or dinner and breakfast, the staff often fails to turn off the hot chocolate machine between breakfast and lunch. Apparently, this is the reason behind the higher hot chocolate temperatures observed during lunch. Investigation also shows that the dining hall staff failed to turn on the hot chocolate machine before breakfast on Thursday (see subgroup 19)—in fact, a student had to ask that the machine be turned on. This caused the subgroup mean for subgroup 19 to be far below the chart LCL. The dining hall staff concludes that the hot chocolate machine needs to be turned off after breakfast and then turned back on 15 minutes before lunch (prior experience suggests that it takes the machine 15 minutes to warm up). The staff also concludes that the machine should be turned on 15 minutes before each meal. In order to ensure that these actions are taken, an automatic timer is purchased to turn on the hot chocolate machine at the appropriate times. This brings the process into statistical control. Figure 17.11 shows and R charts with revised control limits calculated using the subgroups that remain after the subgroups for the out-of-control lunches (subgroups 3, 4, 9, 10, 21, and 22) and the out-of-control breakfast (subgroups 19 and 20) are eliminated from the data set. We see that these revised control charts are in statistical control.

Figure 17.10: MegaStat Output of and R Charts for the Hot Chocolate Temperature Data

Figure 17.11: MegaStat Output of Revised and R Charts for the Hot Chocolate Temperature Data. The Process Is Now in Control.

Having seen how to interpret and R charts, we are now better prepared to understand why we estimate the process standard deviation σ by /d2. Recall that when μ and σ are known, the chart control limits are . The standard deviation σ in these limits is the process standard deviation when the process is in control. When this standard deviation is unknown, we estimate σ as if the process is in control, even though the process might not be in control. The quantity /d2 is an appropriate estimate of σ because is the average of individual ranges computed from rational subgroups—subgroups selected so that the chances that important process changes occur within a subgroup are minimized. Thus each subgroup range, and therefore /d2, estimates the process variation as if the process were in control. Of course, we could also compute the standard deviation of the measurements in each subgroup, and employ the average of the subgroup standard deviations to estimate σ. The key is not whether we use ranges or standard deviations to measure the variation within the subgroups. Rather, the key is that we must calculate a measure of variation for each subgroup and then must average the separate measures of subgroup variation in order to estimate the process variation as if the process is in control.

Exercises for Sections 17.3 and 17.4

CONCEPTS

17.5 Explain (1) the purpose of an chart, (2) the purpose of an R chart, (3) why both charts are needed.

17.6 Explain why the initial control limits calculated for a set of subgrouped data are called “trial control limits.”

17.7 Explain why a change in process variability shows up on both the and R charts.

17.8 In each of the following situations, what conclusions (if any) can be made about whether the process mean is changing? Explain your logic.

a R chart out of control.

b R chart in control, chart out of control.

c Both and R charts in control.

METHODS AND APPLICATIONS

17.9 Table 17.5 gives five subgroups of measurement data. Use these data to Measure 1

a Find and R for each subgroup.

b Find and .

c Find A2 and D4.

d Compute and R chart center lines and control limits.

Table 17.5: Five Subgroups of Measurement Data Measure 1

17.10 In the book Tools and Methods for the Improvement of Quality, Gitlow, Gitlow, Oppenheim, and Oppenheim discuss a resort hotel’s efforts to improve service by reducing variation in the time it takes to clean and prepare rooms. In order to study the situation, five rooms are selected each day for 25 consecutive days, and the time required to clean and prepare each room is recorded. The data obtained are given in Table 17.6. RoomPrep

a Calculate the subgroup mean and range R for each of the first two subgroups.

b Show that minutes and that = 2.696 minutes.

c Find the control chart constants A2 and D4 for the cleaning and preparation time data. Does D3 exist? What does this say?

d Find the center line and control limits for the chart for this data.

e Find the center line and control limit for the R chart for this data.

f Set up (plot) the and R charts for the cleaning time data.

g Are the and R charts in control? Explain.

Table 17.6: Daily Samples of Five Room Cleaning and Preparation Times RoomPrep

17.11 A pizza restaurant monitors the size (measured by the diameter) of the 10-inch pizzas that it prepares. Pizza crusts are made from doughs that are prepared and prepackaged in boxes of 15 by a supplier. Doughs are thawed and pressed in a pressing machine. The toppings are added, and the pizzas are baked. The wetness of the doughs varies from box to box, and if the dough is too wet or greasy, it is difficult to press, resulting in a crust that is too small. The first shift of workers begins work at 4 p.m., and a new shift takes over at 9 p.m. and works until closing. The pressing machine is readjusted at the beginning of each shift. The restaurant takes five consecutive pizzas prepared at the beginning of each hour from opening to closing on a particular day. The diameter of each baked pizza in the subgroups is measured, and the pizza crust diameters obtained are given in Table 17.7. Use the pizza crust diameter data to do the following: PizzaDiam

Table 17.7: 10 Samples of Pizza Crust Diameters PizzaDiam

a Show that and = .84.

b Find the center lines and control limits for the and R charts for the pizza crust data.

c Set up the and R charts for the pizza crust data.

d Is the R chart for the pizza crust data in statistical control? Explain.

e Is the chart for the pizza crust data in statistical control? If not, use the chart and the information given with the data to try to identify any assignable causes that might exist.

f Suppose that, based on the chart, the manager of the restaurant decides that the employees do not know how to properly adjust the dough pressing machine. Because of this, the manager thoroughly trains the employees in the use of this equipment. Because an assignable cause (incorrect adjustment of the pressing machine) has been found and eliminated, we can remove the subgroups affected by this unusual process variation from the data set. We therefore drop subgroups 1 and 6 from the data. Use the remaining eight subgroups to show that we obtain revised center lines of and = .825.

g Use the revised values of and to compute revised and R chart control limits for the pizza crust diameter data. Set up and R charts using these revised limits. Be sure to omit subgroup means and ranges for subgroups 1 and 6 when setting up these charts.

h Has removing the assignable cause brought the process into statistical control? Explain.

17.12 A chemical company has collected 15 daily subgroups of measurements of an important chemical property called “acid value” for one of its products. Each subgroup consists of six acid value readings: a single reading was taken every four hours during the day, and the readings for a day are taken as a subgroup. The 15 daily subgroups are given in Table 17.8. AcidVal

Table 17.8: 15 Subgroups of Acid Value Measurements for a Chemical Process AcidVal

a Show that for these data and = 2.06.

b Set up and R charts for the acid value data. Are these charts in statistical control?

c On the basis of these charts, is it possible to draw proper conclusions about whether the mean acid value is changing? Explain why or why not.

d Suppose that investigation reveals that the out-of-control points on the R chart (the ranges for subgroups 1 and 10) were caused by an equipment malfunction that can be remedied by redesigning a mechanical part. Since the assignable cause that is responsible for the large ranges for subgroups 1 and 10 has been found and eliminated, we can remove subgroups 1 and 10 from the data set. Show that using the remaining 13 subgroups gives revised center lines of and = 1.4385.

e Use the revised values of and to compute revised and R chart control limits for the acid value data. Set up the revised and R charts, making sure to omit subgroup means and ranges for subgroups 1 and 10.

f Are the revised and R charts for the remaining 13 subgroups in statistical control? Explain. What does this result tell us to do?

17.13 The data in Table 17.9 consist of 30 subgroups of measurements that specify the location of a “tube hole” in an air conditioner compressor shell. Each subgroup contains the tube hole dimension measurement for five consecutive compressor shells selected from the production line. The first 15 subgroups were observed on March 21, and the second 15 subgroups were observed on March 22. As indicated in Table 17.9, the die press used in the hole punching operation was changed after subgroup 5 was observed, and a die repair was made after subgroup 25 was observed. TubeHole

Table 17.9: 30 Subgroups of Tube Hole Location Dimensions for Air Conditioner Compressor Shells TubeHole

a Show that for the first 15 subgroups (observed on March 21) we have and = 6.1333.

b Set up and R charts for the 15 subgroups that were observed on March 21 (do not use any of the March 22 data). Do these and R charts show statistical control?

c Using the control limits you computed by using the 15 subgroups observed on March 21, set up and R charts for all 30 subgroups. That is, add the subgroup means and ranges for March 22 to your and R charts, but use the limits you computed from the March 21 data.

d Do the and R charts obtained in part c show statistical control? Explain.

e Does it appear that changing to die press #628 is an assignable cause? Explain. (Note that the die press is the machine that is used to punch the tube hole.)

f Does it appear that making a die repair is an assignable cause? Explain.

17.14 A company packages a bulk product in bags with a 50-pound label weight. During a typical day’s operation of the fill process, 22 subgroups of five bag fills are observed. Using the observed data, and are calculated to be 52.9364 pounds and 1.6818 pounds, respectively. When the 22 ’s and 22 R’s are plotted with respect to the appropriate control limits, the first 6 subgroups are found to be out of control. This is traced to a mechanical start-up problem, which is remedied. Using the remaining 16 subgroups, and are calculated to be 52.5875 pounds and 1.2937 pounds, respectively.

a Calculate appropriate revised and R chart control limits.

b When the remaining 16 ’s and 16 R’s are plotted with respect to the appropriate revised control limits, they are found to be within these limits. What does this imply?

17.15 In the book Tools and Methods for the Improvement of Quality, Gitlow, Gitlow, Oppenheim, and Oppenheim discuss an example of using and R charts to study tuning knob diameters. In their problem description the authors say this:

A manufacturer of high-end audio components buys metal tuning knobs to be used in the assembly of its products. The knobs are produced automatically by a subcontractor using a single machine that is supposed to produce them with a constant diameter. Nevertheless, because of persistent final assembly problems with the knobs, management has decided to examine this process output by requesting that the subcontractor keep an x-bar and R chart for knob diameter.

On a particular day the subcontractor selects four knobs every half hour and carefully measures their diameters. Twenty-five subgroups are obtained, and these subgroups (along with their subgroup means and ranges) are given in Table 17.10. KnobDiam

Table 17.10: 25 Subgroups of Tuning Knob Diameters KnobDiam

a For these data show that and = 5.16. Then use these values to calculate control limits and to set up and R charts for the 25 subgroups of tuning knob diameters. Do these and R charts indicate the existence of any assignable causes? Explain.

b An investigation is carried out to find out what caused the large range for subgroup 23. The investigation reveals that a water pipe burst at 7:25 p.m. and that the mishap resulted in water leaking under the machinery used in the tuning knob production process. The resulting disruption is the apparent cause for the out-of-control range for subgroup 23. The water pipe is mended, and since this fix is reasonably permanent, we are justified in removing subgroup 23 from the data set. Using the remaining 24 subgroups, show that revised center lines are and = 4.88.

c Use the revised values of and to set up revised and R charts for the remaining 24 subgroups of diameters. Be sure to omit the mean and range for subgroup 23.

d Looking at the revised R chart, is this chart now in statistical control? What does your answer say about whether we can use the chart to decide if the process mean is changing?

e Looking at the revised chart, is this chart in statistical control? What does your answer tell us about the process mean?

f An investigation is now undertaken to find the cause of the very high values for subgroups 10, 11, 12, and 13. We again quote Gitlow, Gitlow, Oppenheim, and Oppenheim:

The investigation leads to the discovery that … a keyway wedge had cracked and needed to be replaced on the machine. The mechanic who normally makes this repair was out to lunch, so the machine operator made the repair. This individual had not been properly trained for the repair; for this reason, the wedge was not properly aligned in the keyway, and the subsequent points were out of control. Both the operator and the mechanic agree that the need for this repair was not unusual. To correct this problem it is decided to train the machine operator and provide the appropriate tools for making this repair in the mechanic’s absence. Furthermore, the maintenance and engineering staffs agree to search for a replacement part for the wedge that will not be so prone to cracking.

Since the assignable causes responsible for the very high values for subgroups 10, 11, 12, and 13 have been found and eliminated, we remove these subgroups from the data set. Show that removing subgroups 10, 11, 12, and 13 (in addition to the previously removed subgroup 23) results in the revised center lines and = 5.25. Then use these revised values to set up revised and R charts for the remaining 20 subgroups.

g Are all of the subgroup means and ranges for these newly revised and R charts inside their respective control limits?

17.5: Pattern Analysis

Chapter 21

When we observe a plot point outside the control limits on a control chart, we have strong evidence that an assignable cause exists. In addition, several other data patterns indicate the presence of assignable causes. Precise description of these patterns is often made easier by dividing the control band into zones—designated A, B, and C. Zone boundaries are set at points that are one and two standard deviations (of the plotted statistic) on either side of the center line. We obtain six zones—each zone being one standard deviation wide—with three zones on each side of the center line. The zones that stretch one standard deviation above and below the center line are designated as
C zones. The zones that extend from one to two standard deviations away from the center line are designated as
B zones. The zones that extend from two to three standard deviations away from the center line are designated as
A zones. Figure 17.12(a) illustrates a control chart with the six zones, and Figure 17.12(b) shows how the zone boundaries for an chart and an R chart are calculated. Part (b) of this figure also shows the values of the zone boundaries for the hole location and R charts shown in Figure 17.9 (page 763). In calculating these boundaries, we use and = .0675, which we computed from subgroups 1 through 20 with subgroups 1, 7, 12, and 17 removed from the data set; that is, we are using and when the process is in control. For example, the upper A–B boundary for the chart has been calculated as follows:

Figure 17.12: Zone Boundaries

Finally, Figure 17.12(b) shows (based on a normal distribution of plot points) the percentages of points that we would expect to observe in each zone when the process is in statistical control. For instance, we would expect to observe 34.13 percent of the plot points in the upper portion of zone C.

For an chart, if the distribution of process measurements is reasonably normal, then the distribution of subgroup means will be approximately normal, and the percentages shown in Figure 17.12 apply. That is, the plotted subgroup means for an “in control” chart should look as if they have been randomly selected from a normal distribution. Any distribution of plot points that looks very different from the expected percentages will suggest the existence of an assignable cause.

Various companies (for example, Western Electric [AT&T] and Ford Motor Company) have established sets of rules for identifying assignable causes; use of such rules is called
pattern analysis. We now summarize some commonly accepted rules. Note that many of these rules are illustrated in Figures 17.13, 17.14, 17.15, and 17.16, which show several common out-of-control patterns.

Figure 17.13: A Plot Point outside the Control Limits

Figure 17.14: Two Out of Three Consecutive Plot Points in Zone A (or beyond)

Figure 17.15: Four Out of Five Consecutive Plot Points in Zone B (or beyond)

Source: H. Gitlow, S. Gitlow, A. Oppenheim, and R. Oppenheim, Tools and Methods for the Improvement of Quality, pp. 191–93, 209–211. Copyright © 1989. Reprinted by permission of McGraw-Hill Companies, Inc.

Figure 17.16: Other Out-of-Control Patterns

Pattern Analysis for and R Charts

If one or more of the following conditions exist, it is reasonable to conclude that one or more assignable causes are present:

1 One plot point beyond zone A (that is, outside the three standard deviation control limits)—see Figure 17.13.

2 Two out of three consecutive plot points in zone A (or beyond) on one side of the center line of the control chart. Sometimes a zone boundary that separates zones A and B is called a two standard deviation warning limit. It can be shown that, if the process is in control, then the likelihood of observing two out of three plot points beyond this warning limit (even when no points are outside the control limits) is very small. Therefore, such a pattern signals an assignable cause. Figure 17.14 illustrates this pattern. Specifically, note that plot points 5 and 6 are two consecutive plot points in zone A and that plot points 19 and 21 are two out of three consecutive plot points in zone A.

3 Four out of five consecutive plot points in zone B (or beyond) on one side of the center line of the control chart. Figure 17.15 illustrates this pattern. Specifically, note that plot points 2, 3, 4, and 5 are four consecutive plot points in zone B and that plot points 12, 13, 15, and 16 are four out of five consecutive plot points in zone B (or beyond).

4 A run of at least eight plot points. Here we define a
run
to be a sequence of plot points of the same type. For example, we can have a run of points on one side of (above or below) the center line. Such a run is illustrated in part (a) of Figure 17.16, which shows a run above the center line. We might also observe a run of steadily increasing plot points (a
run up
) or a run of steadily decreasing plot points (a
run down
). These patterns are illustrated in parts (b) and (c) of Figure 17.16. Any of the above types of runs consisting of at least eight points is an out-of-control signal.

5 A nonrandom pattern of plot points. Such a pattern might be an increasing or decreasing trend, a fanning-out or funneling-in pattern, a cycle, an alternating pattern, or any other pattern that is very inconsistent with the percentages given in Figure 17.12 (see parts (d) through (h) of Figure 17.16).

If none of the patterns or conditions in 1 through 5 exists, then the process shows good statistical control—or is said to be “in control.” A process that is in control should not be tampered with. On the other hand, if one or more of the patterns in 1 through 5 exist, action must be taken to find the cause of the out-of-control pattern(s) (which should be eliminated if the assignable cause is undesirable).

It is tempting to use many rules to decide when an assignable cause exists. However, if we use too many rules we can end up with an unacceptably high chance of a false out-of-control signal (that is, an out-of-control signal when there is no assignable cause present). For most control charts, the use of the rules just described will yield an overall probability of a false signal in the range of 1 to 2 percent.

EXAMPLE 17.7: The Hole Location Case

Figure 17.17 shows ongoing and R charts for the hole location problem. Here the chart includes zone boundaries with zones A, B, and C labeled. Notice that the first out-of-control condition (one plot point beyond zone A) exists. Looking at the last five plot points on the chart, we see that the third out-of-control condition (four out of five consecutive plot points in zone B or beyond) also exists.

Figure 17.17: Ongoing and R Charts for the Hole Location Data—Zones A, B, and C Included

Exercises for Section 17.5

CONCEPTS

17.16 When a process is in statistical control:

a What percentage of the plot points on an chart will be found in the C zones (that is, in the middle 1/3 of the chart’s “control band”)?

b What percentage of the plot points on an chart will be found in either the C zones or the B zones (that is, in the middle 2/3 of the chart’s “control band”)?

c What percentage of the plot points on an chart will be found in the C zones, the B zones, or the A zones (that is, in the chart’s “control band”)?

17.17 Discuss how a sudden increase in the process mean shows up on the chart.

17.18 Discuss how a sudden decrease in the process mean shows up on the chart.

17.19 Discuss how a steady increase in the process mean shows up on the chart. Also, discuss how a steady decrease in the process mean shows up on the chart.

17.20 Explain what we mean by a “false out-of-control signal.”

METHODS AND APPLICATIONS

17.21 In the June 1991 issue of Quality Progress, Gunter presents several control charts. Four of these charts are reproduced in Figure 17.18. For each chart, find any evidence of a lack of statistical control (that is, for each chart identify any evidence of the existence of one or more assignable causes). In each case, if such evidence exists, clearly explain why the plot points indicate that the process is not in control.

Figure 17.18: Charts for Exercise 17.21

Source: B. Gunter, “Process Capability Studies Part 3: The Tale of the Charts,” Quality Progress (June 1991), pp. 77–82. Copyright © 1991. American Society for Quality. Used with permission.

17.22 In the book Tools and Methods for the Improvement of Quality, Gitlow, Gitlow, Oppenheim, and Oppenheim present several control charts in a discussion and exercises dealing with pattern analysis. These control charts, which include appropriate A, B, and C zones, are reproduced in Figure 17.19. For each chart, identify any evidence of a lack of statistical control (that is, for each chart identify any evidence suggesting the existence of one or more assignable causes). In each case, if such evidence exists, clearly explain why the plot points indicate that the process is not in control.

Figure 17.19: Charts for Exercise 17.22

Source: H. Gitlow, S. Gitlow, A. Oppenheim, and R. Oppenheim, Tools and Methods for the Improvement of Quality, pp. 191–93, 209–11. Copyright © 1989. Reprinted by permission of McGraw-Hill Companies, Inc.

17.23 Consider the tuning knob diameter data given in Table 17.10 (page 770). Recalling that the subgroup size is n = 4 and that and = 5.16 for these data, KnobDiam

a Calculate all of the zone boundaries for the chart.

b Calculate all of the R chart zone boundaries that are either 0 or positive.

17.24 Given what you now know about pattern analysis, examine each of the following and R charts for evidence of lack of statistical control. In each case, explain any evidence indicating the existence of one or more assignable causes.

a The pizza crust diameter and R charts of Exercise 17.11 (page 766). PizzaDiam

b The acid value and R charts of Exercise 17.12 (page 767). AcidVal

c The tube hole location and R charts of Exercise 17.13 (page 768). TubeHole

17.6: Comparison of a Process with Specifications: Capability Studies

If we have a process in statistical control, we have found and eliminated the assignable causes of process variation. Therefore, the individual process measurements fluctuate over time with a constant standard deviation σ
around a constant mean μ. It follows that we can use the individual process measurements to estimate μ and σ. Doing this lets us determine if the process is capable of producing output that meets specifications. Specifications are based on fitness for use criteria—that is, the specifications are established by design engineers or customers. Even if a process is in statistical control, it may exhibit too much common cause variation (represented by σ) to meet specifications.

As will be shown in Example 17.9, one way to study the capability of a process that is in statistical control is to construct a histogram from a set of individual process measurements. The histogram can then be compared with the product specification limits. In addition, we know that if all possible individual process measurements are normally distributed with mean μ and standard deviation σ, then 99.73 percent of these measurements will be in the interval [μ − 3σ, μ + 3σ]. Estimating μ and σ by and, /d2, we obtain the natural tolerance limits6 for the process.

Natural Tolerance Limits

The natural tolerance limits for a normally distributed process that is in statistical control are

where d2 is a constant that depends on the subgroup size n. Values of d2 are given in Table 17.3 (page 757) for subgroup sizes n = 2 to n = 25. These limits contain approximately 99.73 percent of the individual process measurements.

If the natural tolerance limits are inside the specification limits, then almost all (99.73 percent) of the individual process measurements are produced within the specification limits. In this case we say that the process is
capable
of meeting specifications. Furthermore, if we use and R charts to monitor the process, then as long as the process remains in statistical control, the process will continue to meet the specifications. If the natural tolerance limits are wider than the specification limits, we say that the process is not capable. Here some individual process measurements are outside the specification limits.

EXAMPLE 17.8: The Hot Chocolate Temperature Case

Consider the and R chart analysis of the hot chocolate temperature data. Suppose the dining hall staff has determined that all of the hot chocolate it serves should have a temperature between 130°F and 150°F. Recalling that the and R charts of Figure 17.11 (page 765) show that the process has been brought into control with and with = 4.75, we find that is an estimate of the mean hot chocolate temperature, and that /d2 = 4.75/1.693 = 2.81 is an estimate of the standard deviation of all the hot chocolate temperatures. Here d2 = 1.693 is obtained from Table 17.3 (page 757) corresponding to the subgroup size n = 3. Assuming that the temperatures are approximately normally distributed, the natural tolerance limits

tell us that approximately 99.73 percent of the individual hot chocolate temperatures will be between 132.31°F and 149.15°F. Since these natural tolerance limits are inside the specification limits (130°F to 150°F), almost all the temperatures are within the specifications. Therefore, the hot chocolate–making process is capable of meeting the required temperature specifications. Furthermore, if the process remains in control on its and R charts, it will continue to meet specifications.

EXAMPLE 17.9: The Hole Location Case

Again consider the hole punching process for air conditioner compressor shells. Recall that we were able to get this process into a state of statistical control with and = .0675 by removing several assignable causes of process variation.

Figure 17.20 gives a relative frequency histogram of the 80 individual hole location measurements used to construct the and R charts of Figure 17.8 (page 761). This histogram suggests that the population of all individual hole location dimensions is approximately normally distributed.

Figure 17.20: A Relative Frequency Histogram of the Hole Location Data (Based on the Data with Subgroups 1, 7, 12, and 17 Omitted)

Since the process is in statistical control, = 3.0006 is an estimate of the process mean, and /d2 = .0675/2.326 = .0290198 is an estimate of the process standard deviation. Here d2 = 2.326 is obtained from Table 17.3 (page 757) corresponding to the subgroup size n = 5. Furthermore, the natural tolerance limits

tell us that almost all (approximately 99.73 percent) of the individual hole location dimensions produced by the hole punching process are between 2.9135 inches and 3.0877 inches.

Suppose a major customer requires that the hole location dimension must meet specifications of 3.00 ± .05 inches. That is, the customer requires that every individual hole location dimension must be between 2.95 inches and 3.05 inches. The natural tolerance limits, [2.9135, 3.0877], which contain almost all individual hole location dimensions, are wider than the specification limits [2.95, 3.05]. This says that some of the hole location dimensions are outside the specification limits. Therefore, the process is not capable of meeting the specifications. Note that the histogram in Figure 17.20 also shows that some of the hole location dimensions are outside the specification limits.

Figure 17.21 illustrates the situation, assuming that the individual hole location dimensions are normally distributed. The figure shows that the natural tolerance limits are wider than the specification limits. The shaded areas under the normal curve make up the fraction of product that is outside the specification limits. Figure 17.21 also shows the calculation of the estimated fraction of hole location dimensions that are out of specification. We estimate that 8.55 percent of the dimensions do not meet the specifications.

Figure 17.21: Calculating the Fraction out of Specification for the Hole Location Data. Specifications Are 3.00 ± .05.

Since the process is not capable of meeting specifications, it must be improved by removing common cause variation. This is management’s responsibility. Suppose engineering and management conclude that the excessive variation in the hole locations can be reduced by redesigning the machine that punches the holes in the compressor shells. Also suppose that after a research and development program is carried out to do this, the process is run using the new machine and 20 new subgroups of n = 5 hole location measurements are obtained. The resulting and R charts (not given here) indicate that the process is in control with and = .0348. Furthermore, a histogram of the 100 hole location dimensions used to construct the and R charts indicates that all possible hole location measurements are approximately normally distributed. It follows that we estimate that almost all individual hole location dimensions are contained within the new natural tolerance limits

As illustrated in Figure 17.22, these tolerance limits are within the specification limits 3.00 ± .05. Therefore, the new process is now capable of producing almost all hole location dimensions inside the specifications. The new process is capable because the estimated process standard deviation has been substantially reduced (from /d2 = .0675/2.326 = .0290 for the old process to /d2 = .0348/2.326 = .0149613 for the redesigned process).

Figure 17.22: A Capable Process: The Natural Tolerance Limits Are within the Specification Limits

Next, note that (for the improved process) the z value corresponding to the lower specification limit (2.95) is

This says that the lower specification limit is 3.36 estimated process standard deviations below . Since the lower natural tolerance limit is 3 estimated process standard deviations below , there is a leeway of .36 estimated process standard deviations between the lower natural tolerance limit and the lower specification limit (see Figure 17.22). Also, note that the z value corresponding to the upper specification limit (3.05) is

This says that the upper specification limit is 3.33 estimated process standard deviations above . Since the upper natural tolerance limit is 3 estimated process standard deviations above , there is a leeway of .33 estimated process standard deviations between the upper natural tolerance limit and the upper specification limit (see Figure 17.22). Because some leeway exists between the natural tolerance limits and the specification limits, the distribution of process measurements (that is, the curve in Figure 17.22) can shift slightly to the right or left (or can become slightly more spread out) without violating the specifications. Obviously, the more leeway, the better.

To understand why process leeway is important, recall that a process must be in statistical control before we can assess the capability of the process. In fact:

In order to demonstrate that a company’s product meets customer requirements, the company must present

1 and R charts that are in statistical control.

2 Natural tolerance limits that are within the specification limits.

However, even if a capable process shows good statistical control, the process mean and/or the process variation will occasionally change (due to new assignable causes or unexpected recurring problems). If the process mean shifts and/or the process variation increases, a process will need some leeway between the natural tolerance limits and the specification limits in order to avoid producing out-of-specification product. We can determine the amount of process leeway (if any exists) by defining what we call the
sigma level capability
of the process.

Sigma Level Capability

The sigma level capability of a process is the number of estimated process standard deviations between the estimated process mean, , and the specification limit that is closest to .

For instance, in the previous example the lower specification limit (2.95) is 3.36 estimated standard deviations below the estimated process mean, , and the upper specification limit (3.05) is 3.33 estimated process standard deviations above . It follows that the upper specification limit is closest to the estimated process mean , and because this specification limit is 3.33 estimated process standard deviations from , we say that the hole punching process has 3.33 sigma capability.

If a process has a sigma level capability of three or more, then there are at least three estimated process standard deviations between and the specification limit that is closest to . It follows that, if the distribution of process measurements is normally distributed, then the process is capable of meeting the specifications. For instance, Figure 17.23(a) illustrates a process with three sigma capability. This process is just barely capable—that is, there is no process leeway. Figure 17.23(b) illustrates a process with six sigma capability. This process has three standard deviations of leeway. In general, we see that if a process is capable, the sigma level capability expresses the amount of process leeway. The higher the sigma level capability, the more process leeway. More specifically, for a capable process, the sigma level capability minus three gives the number of estimated standard deviations of process leeway. For example, since the hole punching process has 3.33 sigma capability, this process has 3.33 − 3 = .33 estimated standard deviations of leeway.

Figure 17.23: Sigma Level Capability and Process Leeway

The difference between three sigma and six sigma capability is dramatic. To illustrate this, look at Figure 17.23(a), which shows that a normally distributed process with three sigma capability produces 99.73 percent good quality (the area under the distribution curve between the specification limits is .9973). On the other hand, Figure 17.23(b) shows that a normally distributed process with six sigma capability produces 99.9999998 percent good quality. Said another way, if the process mean is centered between the specifications, and if we produce large quantities of product, then a normally distributed process with three sigma capability will produce an average of 2,700 defective products per million, while a normally distributed process with six sigma capability will produce an average of only .002 defective products per million.

In the long run, however, process shifts due to assignable causes are likely to occur. It can be shown that, if we monitor the process by using an chart that employs a typical subgroup size of 4 to 6, the largest sustained shift of the process mean that might remain undetected by the chart is a shift of 1.5 process standard deviations. In this worst case, it can be shown that a normally distributed three sigma capable process will produce an average of 66,800 defective products per million (clearly unacceptable), while a normally distributed six sigma capable process will produce an average of only 3.4 defective products per million. Therefore, if a six sigma capable process is monitored by and R charts, then, when a process shift occurs, we can detect the shift (by using the control charts), and we can take immediate corrective action before a substantial number of defective products are produced.

This is, in fact, how control charts are supposed to be used to prevent the production of defective product. That is, our strategy is

Prevention Using Control Charts

1 Reduce common cause variation in order to create leeway between the natural tolerance limits and the specification limits.

2 Use control charts to establish statistical control and to monitor the process.

3 When the control charts give out-of-control signals, take immediate action on the process to reestablish control before out-of-specification product is produced.

Since 1987, a number of U.S. companies have adopted a six sigma philosophy. In fact, these companies refer to themselves as six sigma companies. It is the goal of these companies to achieve six sigma capability for all processes in the entire organization. For instance, Motorola, Inc., the first company to adopt a six sigma philosophy, began a five-year quality improvement program in 1987. The goal of Motorola’s companywide defect reduction program is to achieve six sigma capability for all processes—for instance, manufacturing processes, delivery, information systems, order completeness, accuracy of transactions records, and so forth. As a result of its six sigma plan, Motorola claims to have saved more than $1.5 billion. The corporation won the Malcolm Baldrige National Quality Award in 1988, and Motorola’s six sigma plan has become a model for firms that are committed to quality improvement. Other companies that have adopted the six sigma philosophy include IBM, Digital Equipment Corporation, and General Electric.

To conclude this section, we make two comments. First, it has been traditional to measure process capability by using what is called the
C p k
index. This index is calculated by dividing the sigma level capability by three. For example, since the hole punching process illustrated in Figure 17.22 has a sigma level capability of 3.33, the C p k for this process is 1.11. In general, if C p k is at least 1, then the sigma level capability of the process is at least 3 and thus the process is capable. Historically, C p k has been used because its value relative to the number 1 describes the process capability. We prefer using sigma level capability to characterize process capability because we believe that it is more intuitive.

Second, when a process is in control, then the estimates /d2 and s of the process standard deviation will be very similar. This implies that we can compute the natural tolerance limits by using the alternative formula For example, since the mean and standard deviation of the 80 observations used to construct the and R charts in Figure 17.8 (page 761) are and s = .028875, we obtain the natural tolerance limits

These limits are very close to those obtained in Example 17.9, [2.9135, 3.0877], which were computed by using the estimate /d2 = .0290198 of the process standard deviation. Use of the alternative formula is particularly appropriate when there are long-run process variations that are not measured by the subgroup ranges (in which case /d2 underestimates the process standard deviation). Since statistical control in any real application of SPC will not be perfect, some people believe that this version of the natural tolerance limits is the most appropriate.

Exercises for Section 17.6

CONCEPTS

17.25 Write a short paragraph explaining why a process that is in statistical control is not necessarily capable of meeting customer requirements (specifications).

17.26 Explain the interpretation of the natural tolerance limits for a process. What assumptions must be made in order to properly make this interpretation? How do we check these assumptions?

17.27 Explain how the natural tolerance limits compare to the specification limits when

a A process is capable of meeting specifications.

b A process is not capable of meeting specifications.

17.28 For each of the following, explain

a Why it is important to have leeway between the natural tolerance limits and the specification limits.

b What is meant by the sigma level capability for a process.

c Two reasons why it is important to achieve six sigma capability.

METHODS AND APPLICATIONS

17.29 Consider the room cleaning and preparation time situation in Exercise 17.10 (page 766). We found that and R charts based on subgroups of size 5 for this data are in statistical control with minutes and = 2.696 minutes. RoomPrep

a Assuming that the cleaning and preparation times are approximately normally distributed, calculate a range of values that contains almost all (approximately 99.73 percent) of the individual cleaning and preparation times.

b Find reasonable estimates of the maximum and minimum times needed to clean and prepare an individual room.

c Suppose the resort hotel wishes to specify that every individual room should be cleaned and prepared in 20 minutes or less. Is this upper specification being met? Explain. Note here that there is no lower specification, since we would like cleaning times to be as short as possible (as long as the job is done properly).

d If the upper specification for room cleaning and preparation times is 20 minutes, find the sigma level capability of the process. If the upper specification is 30 minutes, find the sigma level capability.

17.30 Suppose that and R charts based on subgroups of size 3 are used to monitor the moisture content of a type of paper. The and R charts are found to be in statistical control with percent and = .4 percent. Further, a histogram of the individual moisture content readings suggests that these measurements are approximately normally distributed.

a Compute the natural tolerance limits (limits that contain almost all the individual moisture content readings) for this process.

b If moisture content specifications are 6.0 percent ± .5 percent, is this process capable of meeting the specifications? Why or why not?

c Estimate the fraction of paper that is out of specification.

d Find the sigma level capability of the process.

17.31 A grocer has a contract with a produce wholesaler that specifies that the wholesaler will supply the grocer with grapefruit that weigh at least .75 pounds each. In order to monitor the grapefruit weights, the grocer randomly selects three grapefruit from each of 25 different crates of grapefruit received from the wholesaler. Each grapefruit’s weight is determined and, therefore, 25 subgroups of three grapefruit weights are obtained. When and R charts based on these subgroups are constructed, we find that these charts are in statistical control with and = .11. Further, a histogram of the individual grapefruit weights indicates that these measurements are approximately normally distributed.

a Calculate a range of values that contains almost all (approximately 99.73 percent) of the individual grapefruit weights.

b Find a reasonable estimate of the maximum weight of a grapefruit that the grocer is likely to sell.

c Suppose that the grocer’s contract with its produce supplier specifies that grapefruits are to weigh a minimum of .75 lb. Is this lower specification being met? Explain. Note here that there is no upper specification, since we would like grapefruits to be as large as possible.

d If the lower specification of .75 lb. is not being met, estimate the fraction of grapefruits that weigh less than .75 lb. Hint: Find an estimate of the standard deviation of the individual grapefruit weights.

17.32 Consider the pizza crust diameters for 10-inch pizzas given Exercise 17.11 (pages 766–767). We found that, by removing an assignable cause, we were able to bring the process into statistical control with and = .825. PizzaDiam

a Recalling that the subgroup size for the pizza crust and R charts is 5, and assuming that the pizza crust diameters are approximately normally distributed, calculate the natural tolerance limits for the diameters.

b Using the natural tolerance limits, estimate the largest diameter likely to be sold by the restaurant as a 10-inch pizza.

c Using the natural tolerance limits, estimate the smallest diameter likely to be sold by the restaurant as a 10-inch pizza.

d Are all 10-inch pizzas sold by this restaurant really at least 10 inches in diameter? If not, estimate the fraction of pizzas that are not at least 10 inches in diameter.

17.33 Consider the bag fill situation in Exercise 17.14 (page 769). We found that the elimination of a start-up problem brought the filling process into statistical control with and = 1.2937.

a Recalling that the fill weight and R charts are based on subgroups of size 5, and assuming that the fill weights are approximately normally distributed, calculate the natural tolerance limits for the process.

b Suppose that management wishes to reduce the mean fill weight in order to save money by “giving away” less product. However, since customers expect each bag to contain at least 50 pounds of product, management wishes to leave some process leeway. Therefore, after the mean fill weight is reduced, the lower natural tolerance limit is to be no less than 50.5 lb. Based on the natural tolerance limits, how much can the mean fill weight be reduced? If the product costs $2 per pound, and if 1 million bags are sold per year, what is the yearly cost reduction achieved by lowering the mean fill weight?

17.34 Suppose that a normally distributed process (centered at target) has three sigma capability. If the process shifts 1.5 sigmas to the right, show that the process will produce defective products at a rate of 66,800 per million.

17.35 Suppose that a product is assembled using 10 different components, each of which must meet specifications for five different quality characteristics. Therefore, we have 50 different specifications that potentially could be violated. Further suppose that each component possesses three sigma capability (process centered at target) for each quality characteristic. Then, if we assume normality and independence, find the probability that all 50 specifications will be met.

17.7: Charts for Fraction Nonconforming

Sometimes, rather than collecting measurement data, we inspect items and simply decide whether each item conforms to some desired criterion (or set of criteria). For example, a fuel tank does or does not leak, an order is correctly or incorrectly processed, a batch of chemical product is acceptable or must be reprocessed, or plastic wrap appears clear or too hazy. When an inspected unit does not meet the desired criteria, it is said to be
nonconforming
(or defective). When an inspected unit meets the desired criteria, it is said to be
conforming
(or nondefective). Traditionally, the terms defective and nondefective have been employed. Lately, the terms nonconforming and conforming have become popular.

The control chart that we set up for this type of data is called a
p chart. To construct this chart, we observe subgroups of n units over time. We inspect or test the n units in each subgroup and determine the number d of these units that are nonconforming. We then calculate for each subgroup

and we plot the values versus time on the p chart. If the process being studied is in statistical control and producing a fraction p of nonconforming units, and if the units inspected are independent, then the number of nonconforming units d in a subgroup of n units inspected can be described by a binomial distribution. If, in addition, n is large enough so that np is greater than 2,7 then both d and the fraction of nonconforming units are approximately described by normal distributions. Furthermore, the population of all possible values has mean = p and standard deviation

Therefore, if p is known we can compute three standard deviation control limits for values of by setting

However, since it is unlikely that p will be known, we usually must estimate p from process data. The estimate of p is

Substituting for p, we obtain the following:

Center Line and Control Limits for a p Chart

The control limits calculated using these formulas are considered to be trial control limits. Plot points above the upper control limit suggest that one or more assignable causes have increased the process fraction nonconforming. Plot points below the lower control limit may suggest that an improvement in the process performance has been observed. However, plot points below the lower control limit may also tell us that an inspection problem exists. Perhaps defective items are still being produced, but for some reason the inspection procedure is not finding them. If the chart shows a lack of control, assignable causes must be found and eliminated and the trial control limits must be revised. Here data for subgroups associated with assignable causes that have been eliminated will be dropped, and data for newly observed subgroups will be added when calculating the revised limits. This procedure is carried out until the process is in statistical control. When control is achieved, the limits can be used to monitor process performance. The process capability for a process that is in statistical control is expressed using
, the estimated process fraction nonconforming. When the process is in control and is too high to meet internal or customer requirements, common causes of process variation must be removed in order to reduce . This is a management responsibility.

EXAMPLE 17.10

To improve customer service, a corporation wishes to study the fraction of incorrect sales invoices that are sent to its customers. Every week a random sample of 100 sales invoices sent during the week is selected, and the number of sales invoices containing at least one error is determined. The data for the last 30 weeks are given in Table 17.11. To construct a p chart for these data, we plot the fraction of incorrect invoices versus time. Since the true overall fraction p of incorrect invoices is unknown, we estimate p by (see Table 17.11)

Table 17.11: Sales Invoice Data—100 Invoices Sampled Weekly Invoice

Since = 100(.023) = 2.3 is greater than 2, the population of all possible values has an approximate normal distribution if the process is in statistical control. Therefore, we calculate the center line and control limits for the p chart as follows:

Since the LCL calculates negative, there is no lower control limit for this p chart. The MegaStat output of the p chart for these data is shown in Figure 17.24. We note that none of the plot points is outside the control limits, and we fail to see any nonrandom patterns of points. We conclude that the process is in statistical control with a relatively constant process fraction nonconforming of = .023. That is, the process is stable with an average of approximately 2.3 incorrect invoices per each 100 invoices processed. Since no assignable causes are present, there is no reason to believe that any of the plot points have been affected by unusual process variations. That is, it will not be worthwhile to look for unusual circumstances that have changed the average number of incorrect invoices per 100 invoices processed. If an average of 2.3 incorrect invoices per each 100 invoices is not acceptable, then management must act to remove common causes of process variation. For example, perhaps sales personnel need additional training or perhaps the invoice itself needs to be redesigned.

Figure 17.24: MegaStat Output of a p Chart for the Sales Invoice Data

In the previous example, subgroups of 100 invoices were randomly selected each week for 30 weeks. In general, subgroups must be taken often enough to detect possible sources of variation in the process fraction nonconforming. For example, if we believe that shift changes may significantly influence the process performance, then we must observe at least one subgroup per shift in order to study the shift-to-shift variation. Subgroups must also be taken long enough to allow the major sources of process variation to show up. As a general rule, at least 25 subgroups will be needed to estimate the process performance and to test for process control.

We have said that the size n of each subgroup should be large enough so that np (which is usually estimated by ) is greater than 2 (some practitioners prefer np to be greater than 5). Since we often monitor a p that is quite small (.05 or .01 or less), n must often be quite large. Subgroup sizes of 50 to 200 or more are common. Another suggestion is to choose a subgroup size that is large enough to give a positive lower control limit (often, when employing a p chart, smaller subgroup sizes give a calculated lower control limit that is negative). A positive LCL is desirable because it allows us to detect opportunities for process improvement. Such an opportunity exists when we observe a plot point below the LCL. If there is no LCL, it would obviously be impossible to obtain a plot point below the LCL. It can be shown that

A condition that guarantees that the subgroup size is large enough to yield a positive lower control limit for a p chart is

where
p0 is an initial estimate of the fraction nonconforming produced by the process. This condition is appropriate when three standard deviation control limits are employed.

For instance, suppose experience suggests that a process produces 2 percent nonconforming items. Then, in order to construct a p chart with a positive lower control limit, the subgroup size employed must be greater than

As can be seen from this example, for small values of p0 the above condition may require very large subgroup sizes. For this reason, it is not crucial that the lower control limit be positive.

We have thus far discussed how often—that is, over what specified periods of time (each hour, shift, day, week, or the like)—we should select subgroups. We have also discussed how large each subgroup should be. We next consider how we actually choose the items in a subgroup. One common procedure—which often yields large subgroup sizes—is to include in a subgroup all (that is, 100 percent) of the units produced in a specified period of time. For instance, a subgroup might consist of all the units produced during a particular hour. When employing this kind of scheme, we must carefully consider the independence assumption. The binomial distribution assumes that successive units are produced independently. It follows that a p chart would not be appropriate if the likelihood of a unit being defective depends on whether other units produced in close proximity are defective. Another procedure is to randomly select the units in a subgroup from all the units produced in a specified period of time. This was the procedure used in Example 17.10 to obtain the subgroups of sales invoices. As long as the subgroup size is small relative to the total number of units produced in the specified period, the units in the randomly selected subgroup should probably be independent. However, if the rate of production is low, it could be difficult to obtain a large enough subgroup when using this method. In fact, even if we inspect 100 percent of the process output over a specified period, and even if the production rate is quite high, it might still be difficult to obtain a large enough subgroup. This is because (as previously discussed) we must select subgroups often enough to detect possible assignable causes of variation. If we must select subgroups fairly often, the production rate may not be high enough to yield the needed subgroup size in the time in which the subgroup must be selected.

In general, the large subgroup sizes that are required can make it difficult to set up useful p charts. For this reason, we sometimes (especially when we are monitoring a very small p) relax the requirement that np be greater than 2. Practice shows that even if np is somewhat smaller than 2, we can still use the three standard deviation p chart control limits. In such a case, we detect assignable causes by looking for points outside the control limits and by looking for runs of points on the same side of the center line. In order for the distribution of all possible values to be sufficiently normal to use the pattern analysis rules we presented for charts, must be greater than 2. In this case we carry out pattern analysis for a p chart as we do for an chart (see Section 17.5), and we use the following zone boundaries:

Here, when the LCL calculates negative, it should not be placed on the control chart. Zone boundaries, however, can still be placed on the control chart as long as they are not negative.

Exercises for Section 17.7

CONCEPTS

17.36 In your own words, define a nonconforming unit.

17.37 Describe two situations in your personal life in which you might wish to plot a control chart for fraction nonconforming.

17.38 Explain why it can sometimes be difficult to obtain rational subgroups when using a control chart for fraction nonconforming.

METHODS AND APPLICATIONS

17.39 Suppose that = .1 and n = 100. Calculate the upper and lower control limits, UCL and LCL, of the corresponding p chart.

17.40 Suppose that = .04 and n = 400. Calculate the upper and lower control limits, UCL and LCL, of the corresponding p chart.

17.41 In the July 1989 issue of Quality Progress, William J. McCabe discusses using a p chart to study a company’s order entry system. The company was experiencing problems meeting the promised 60-day delivery schedule. An investigation found that the order entry system frequently lacked all the information needed to correctly process orders. Figure 17.25 gives a p chart analysis of the percentage of orders having missing information.

a From Figure 17.25 we see that = .527. If the subgroup size for this p chart is n = 250, calculate the upper and lower control limits, UCL and LCL.

b Is the p chart of Figure 17.25 in statistical control? That is, are there any assignable causes affecting the fraction of orders having missing information?

c On the basis of the p chart in Figure 17.25, McCabe says,

The process was stable and one could conclude that the cause of the problem was built into the system. The major cause of missing information was salespeople not paying attention to detail, combined with management not paying attention to this problem. Having sold the product, entering the order into the system was generally left to clerical people while the salespeople continued selling.

Can you suggest possible improvements to the order entry system?

Figure 17.25: A p Chart for the Fraction of Orders with Missing Information

Source: W. J. McCabe, “Examining Processes Improves Operations,” Quality Progress (July 1989), pp. 26–32. Copyright © 1989 American Society for Quality. Used with permission.

17.42 In the book Tools and Methods for the Improvement of Quality, Gitlow, Gitlow, Oppenheim, and Oppenheim discuss a data entry operation that makes a large number of entries every day. Over a 24-day period, daily samples of 200 data entries are inspected. Table 17.12 gives the number of erroneous entries per 200 that were inspected each day. DataErr

a Use the data in Table 17.12 to compute . Then use this value of to calculate the control limits for a p chart of the data entry operation, and set up the p chart. Include zone boundaries on the chart.

Table 17.12: The Number of Erroneous Entries for 24 Daily Samples of 200 Data Entries DataErr

b Is the data entry process in statistical control, or are assignable causes affecting the process? Explain.

c Investigation of the data entry process is described by Gitlow, Gitlow, Oppenheim, and Oppenheim as follows:

In our example, to bring the process under control, management investigated the observations which were out of control (days 8 and 22) in an effort to discover and remove the special causes of variation in the process. In this case, management found that on day 8 a new operator had been added to the workforce without any training. The logical conclusion was that the new environment probably caused the unusually high number of errors. To ensure that this special cause would not recur, the company added a one-day training program in which data entry operators would be acclimated to the work environment.

A team of managers and workers conducted an investigation of the circumstances occurring on day 22. Their work revealed that on the previous night one of the data entry consoles malfunctioned and was replaced with a standby unit. The standby unit was older and slightly different from the ones currently used in the department. The repairs on the regular console were not expected to be completed until the morning of day 23. To correct this special source of variation, the team recommended purchasing a spare console that would match the existing equipment and disposing of the outdated model presently being used as the backup. Management then implemented the suggestion.

Since the assignable causes on days 8 and 22 have been found and eliminated, we can remove the data for these days from the data set. Remove the data and calculate the new value of . Then set up a revised p chart for the remaining 22 subgroups.

d Did the actions taken bring the process into statistical control? Explain.

17.43 In the July 1989 issue of Quality Progress, William J. McCabe discusses using a p chart to study the percentage of errors made by 21 buyers processing purchase requisitions. The p chart presented by McCabe is shown in Figure 17.26. In his explanation of this chart, McCabe says,

The causes of the errors … could include out-of-date procedures, unreliable office equipment, or the perceived level of management concern with errors. These causes are all associated with the system and are all under management control.

Focusing on the 21 buyers, weekly error rates were calculated for a 30-week period (the data existed, but weren’t being used). A p-chart was set up for the weekly department error rate. It showed a 5.2 percent average rate for the department. In week 31, the manager called the buyers together and made two statements: “I care about errors because they affect our costs and delivery schedules,” and “I am going to start to count errors by individual buyers so I can understand the causes.” The p-chart … shows an almost immediate drop from 5.2 percent to 3.1 percent.

The explanation is that the common cause system (supervision, in this case) had changed; the improvement resulted from eliminating buyer sloppiness in the execution of orders. The p-chart indicates that buyer errors are now stable at 3.1 percent. The error rate will stay there until the common cause system is changed again.

Figure 17.26: p Chart for the Weekly Department Error Rate for 21 Buyers Processing Purchase Requisitions

Source: W. J. McCabe, “Examining Processes Improves Operations,” Quality Progress (July 1989), pp. 26–32. Copyright © 1989 American Society for Quality. Used with permission.

a The p chart in Figure 17.26 shows that = .052 for weeks 1 through 30. Noting that the subgroup size for this chart is n = 400, calculate the control limits UCL and LCL for the p chart during weeks 1 through 30.

b The p chart in Figure 17.26 shows that after week 30 the value of is reduced to .031. Assuming that the process has been permanently changed after week 30, calculate new control limits based on = .031. If we use these new control limits after week 30, is the improved process in statistical control? Explain.

17.44 The customer service manager of a discount store monitors customer complaints. Each day a random sample of 100 customer transactions is selected. These transactions are monitored, and the number of complaints received concerning these transactions during the next 30 days is recorded. The numbers of complaints received for 20 consecutive daily samples of 100 transactions are, respectively, 2, 5, 10, 1, 5, 6, 9, 4, 1, 7, 1, 5, 7, 4, 5, 4, 6, 3, 10, and 5. Complaints

a Use the data to compute . Then use this value of to calculate the control limits for a p chart of the complaints data. Set up the p chart.

b Are the customer complaints for this 20-day period in statistical control? That is, have any unusual problems caused an excessive number of complaints during this period? Explain why or why not.

c Suppose the discount store receives 13 complaints in the next 30 days for the 100 transactions that have been randomly selected on day 21. Should the situation be investigated? Explain why or why not.

17.8: Cause-and-Effect and Defect Concentration Diagrams (Optional)

We saw in Chapter 2 that Pareto charts are often used to identify quality problems that require attention. When an opportunity for improvement has been identified, it is necessary to examine potential causes of the problem or defect (the undesirable effect). Because many processes are complex, there are often a very large number of possible causes, and it may be difficult to focus on the important ones. In this section we discuss two diagrams that can be employed to help uncover potential causes of process variation that are resulting in the undesirable effect.

The
cause-and-effect diagram
was initially developed by Japanese quality expert Professor Kaoru Ishikawa. In fact, these diagrams are often called Ishikawa diagrams; they are also called fishbone charts for reasons that will become obvious when we look at an example. Cause-and-effect diagrams are usually constructed by a quality team. For example, the team might consist of product designers and engineers, production workers, inspectors, supervisors and foremen, quality engineers, managers, sales representatives, and maintenance personnel. The team will set up the cause-and-effect diagram during a brainstorming session. After the problem (effect) is clearly stated, the team attempts to identify as many potential causes (sources of process variation) as possible. None of the potential causes suggested by team members should be criticized or rejected. The goal is to identify as many potential causes as possible. No attempt is made to actually develop solutions to the problem at this point. After beginning to brainstorm potential causes, it may be useful to observe the process in operation for a period of time before finishing the diagram. It is helpful to focus on finding sources of process variation rather than discussing reasons why these causes cannot be eliminated.

The causes identified by the team are organized into a cause-and-effect diagram as follows:

1 After clearly stating the problem, write it in an effect box at the far right of the diagram. Draw a horizontal (center) line connected to the effect box.

2 Identify major potential cause categories. Write them in boxes that are connected to the center line. Various approaches can be employed in setting up these categories. For example, Figure 17.27 is a cause-and-effect diagram for “why tables are not cleared quickly” in a restaurant. This diagram employs the categories

Figure 17.27: A Cause-and-Effect Diagram for “Why Tables Are Not Cleared Quickly” in a Restaurant

Source: M. Gaudard, R. Coates, and L. Freeman, “Accelerating Improvement,” Quality Progress (October 1991), pp. 81–88. Copyright © 1991. American Society for Quality. Used with permission.

3 Identify subcauses and classify these according to the major potential cause categories identified in step 2. Identify new major categories if necessary. Place subcauses on the diagram as branches. See Figure 17.27.

4 Try to decide which causes are most likely causing the problem or defect. Circle the most likely causes. See Figure 17.27.

After the cause-and-effect diagram has been constructed, the most likely causes of the problem or defect need to be studied. It is usually necessary to collect and analyze data in order to find out if there is a relationship between likely causes and the effect. We have studied various statistical methods (for instance, control charts, scatter plots, ANOVA, and regression) that help in this determination.

A
defect concentration diagram
is a picture of the product. It depicts all views—for example, front, back, sides, bottom, top, and so on. The various kinds of defects are then illustrated on the diagram. Often, by examining the locations of the defects, we can discern information concerning the causes of the defects. For example, in the October 1990 issue of Quality Progress, The Juran Institute presents a defect concentration diagram that plots the locations of chips in the enamel finish of a kitchen range. This diagram is shown in Figure 17.28. If the manufacturer of this range plans to use protective packaging to prevent chipping, it appears that the protective packaging should be placed on the corners, edges, and burners of the range.

Figure 17.28: Defect Concentration Diagram Showing the Locations of Enamel Chips on Kitchen Ranges

Source: “The Tools of Quality Part V: Check Sheets,” from QI Tools: Data Collection Workbook, p. 11. Copyright © 1989. Juran Institute, Inc. Reprinted with permission from Juran Institute, Inc.

Exercises for Section 17.8

CONCEPTS

17.45 Explain the purpose behind constructing (a) a cause-and-effect diagram and (b) a defect concentration diagram.

17.46 Explain how to construct (a) a cause-and-effect diagram and (b) a defect concentration diagram.

METHODS AND APPLICATIONS

17.47 In the January 1994 issue of Quality Progress, Hoexter and Julien discuss the quality of the services delivered by law firms. One aspect of such service is the quality of attorney–client communication. Hoexter and Julien present a cause-and-effect diagram for “poor client–attorney telephone communications.” This diagram is shown in Figure 17.29.

a Using this diagram, what (in your opinion) are the most important causes of poor client–attorney telephone communications?

b Try to improve the diagram. That is, try to add causes to the diagram.

Figure 17.29: A Cause-and-Effect Diagram for “Poor Client–Attorney Telephone Communications”

Source: R. Hoexter and M. Julien, “Legal Eagles Become Quality Hawks,” Quality Progress (January 1994), pp. 31–33. Copyright © 1994 American Society for Quality. Used with permission.

17.48 In the October 1990 issue of Quality Progress, The Juran Institute presents an example that deals with the production of integrated circuits. The article describes the situation as follows:

The manufacture of integrated circuits begins with silicon slices that, after a sequence of complex operations, will contain hundreds or thousands of chips on their surfaces. Each chip must be tested to establish whether it functions properly. During slice testing, some chips are found to be defective and are rejected. To reduce the number of rejects, it is necessary to know not only the percentage but also the locations and the types of defects. There are normally two major types of defects: functional and parametric. A functional reject occurs when a chip does not perform one of its functions. A parametric reject occurs when the circuit functions properly, but a parameter of the chip, such as speed or power consumption, is not correct.

Figure 17.30 gives a defect concentration diagram showing the locations of rejected chips within the integrated circuit. Only those chips that had five or more defects during the testing of 1,000 integrated circuits are shaded. Describe where parametric rejects tend to be, and describe where functional rejects tend to be.

Figure 17.30: Defect Concentration Diagram Showing the Locations of Rejected Chips on Integrated Circuits

Source: “The Tools of Quality Part V: Check Sheets,” from QI Tools: Data Collection Workbook, p. 12. Copyright © 1989 Juran Institute, Inc. Used with permission.

17.49 In the September 1994 issue of Quality Progress, Franklin P. Schargel presents a cause-and-effect diagram for the “lack of quality in schools.” We present this diagram in Figure 17.31.

a Identify and circle the causes that you feel contribute the most to the “lack of quality in schools.”

b Try to improve the diagram. That is, see if you can add causes to the diagram.

Figure 17.31: A Cause-and-Effect Diagram on the “Lack of Quality in Schools”

Source: F. P. Schargel, “Teaching TQM in an Inner City High School,” Quality Progress (September 1994), pp. 87–90. Copyright © 1994 American Society for Quality. Used with permission.

Chapter Summary

In this chapter we studied how to improve business processes by using control charts. We began by considering several meanings of quality, and we discussed the history of the quality movement in the United States. We saw that Walter Shewhart introduced statistical quality control while working at Bell Telephone Laboratories during the 1920s and 30s, and we also saw that W. Edwards Deming taught the Japanese how to use statistical methods to improve product quality following World War II. When the quality of Japanese products surpassed that of American-made goods, and when, as a result, U.S. manufacturers lost substantial shares of their markets, Dr. Deming consulted and lectured extensively in the United States. This sparked an American reemphasis on quality that continues to this day. We also briefly presented Deming’s 14 Points, a set of management principles that, if followed, Deming believed would enable a company to improve quality and productivity, reduce costs, and gain competitive advantage.

We next learned that processes are influenced by common cause variation (inherent variation) and by assignable cause variation (unusual variation), and we saw that a control chart signals when assignable causes exist. Then we discussed how to sample a process. In particular, we explained that effective control charting requires rational subgrouping. Such subgroups minimize the chances that important process variations will occur within subgroups, and they maximize the chances that such variations will occur between subgroups.

Next we studied and R charts in detail. We saw that charts are used to monitor and stabilize the process mean (level), and that R charts are used to monitor and stabilize the process variability. In particular, we studied how to construct and R charts by using control chart constants, how to recognize out-of-control conditions by employing zone boundaries and
pattern analysis, and how to use and R charts to get a process into statistical control.

While it is important to bring a process into statistical control, we learned that it is also necessary to meet the customer’s or manufacturer’s requirements (or specifications). Since statistical control does not guarantee that the process output meets specifications, we must carry out a capability study after the process has been brought into control. We studied how this is done by computing
natural tolerance limits, which are limits that contain almost all the individual process measurements. We saw that, if the natural tolerance limits are inside the specification limits, then the process is
capable
of meeting the specifications. We also saw that we can measure how capable a process is by using
sigma level capability, and we learned that a number of major businesses now orient their management philosophy around the concept of six sigma capability. In particular, we learned that, if a process is in statistical control and if the process has six sigma or better capability, then the defective rate will be very low (3.4 per million or less).

We continued by studying
p charts, which are charts for fraction nonconforming. Such charts are useful when it is not possible (or when it is very expensive) to measure the quality characteristic of interest.

We concluded this chapter with an optional section on how to construct
cause-and-effect diagrams
and
defect concentration diagrams. These diagrams are used to identify opportunities for process improvement and to discover sources of process variation.

It should be noted that two useful types of control charts not discussed in this chapter are individuals charts and c charts. These charts are discussed in Appendix L of the CD-ROM included with this book.

Glossary of Terms

acceptance sampling:

A statistical sampling technique that enables us to accept or reject a quantity of goods (the lot) without inspecting the entire lot. (page 744)

assignable causes (of process variation):

Unusual sources of process variation. Also called special causes or specific causes of process variation. (page 748)

capable process:

A process that has the ability to produce products or services that meet customer or manufacturer requirements (specifications). (page 776)

cause-and-effect diagram:

A diagram that enumerates (lists) the potential causes of an undesirable effect. (page 789)

common causes (of process variation):

Sources of process variation that are inherent to the process design—that is, sources of usual process variation. (page 747)

conforming unit (nondefective):

An inspected unit that meets a set of desired criteria. (page 783)

control chart:

A graph of process performance that includes a center line and two control limits—an upper control limit, UCL, and a lower control limit, LCL. Its purpose is to detect assignable causes. (page 754)

C p k index:

A process’s sigma level capability divided by 3. (page 781)

defect concentration diagram:

An illustration of a product that depicts the locations of defects that have been observed. (page 790)

ISO 9000:

A series of international standards for quality assurance management systems. (page 746)

natural tolerance limits:

Assuming a process is in statistical control and assuming process measurements are normally distributed, limits that contain almost all (approximately 99.73 percent) of the individual process measurements. (page 775)

nonconforming unit (defective):

An inspected unit that does not meet a set of desired criteria. (page 783)

pattern analysis:

Looking for patterns of plot points on a control chart in order to find evidence of assignable causes. (page 771)

p chart:

A control chart on which the proportion nonconforming (in subgroups of size n) is plotted versus time. (page 783)

quality of conformance:

How well a process is able to meet the requirements (specifications) set forth by the process design. (page 743)

quality of design:

How well the design of a product or service meets and exceeds the needs and expectations of the customer. (page 743)

quality of performance:

How well a product or service performs in the marketplace. (page 743)

rational subgroups:

Subgroups of process observations that are selected so that the chances that process changes will occur between subgroups is maximized. (page 750)

R chart:

A control chart on which subgroup ranges are plotted versus time. It is used to monitor the process variability (or spread). (page 754)

run:

A sequence of plot points on a control chart that are of the same type—for instance, a sequence of plot points above the center line. (page 772)

sigma level capability:

The number of estimated process standard deviations between the estimated process mean, , and the specification limit that is closest to . (page 779)

statistical process control (SPC):

A systematic method for analyzing process data in which we monitor and study the process variation. The goal is continuous process improvement. (page 747)

subgroup:

A set of process observations that are grouped together for purposes of control charting. (page 750)

total quality management (TQM):

Applying quality principles to all company activities. (page 745)

variables control charts:

Control charts constructed by using measurement data. (page 754)

chart (x-bar chart):

A control chart on which subgroup means are plotted versus time. It is used to monitor the process mean (or level). (page 754)

Important Formulas

Center line and control limits for an chart: page 757

Center line and control limits for an R chart: page 757

Zone boundaries for an chart: page 771

Zone boundaries for an R chart: page 771

Natural tolerance limits for normally distributed process measurements: page 775

Sigma level capability: page 779

Cpk index: page 781

Center line and control limits for a p chart: page 784

Zone boundaries for a p chart: page 786

Supplementary Exercises

Exercises 17.50 through 17.53 are based on a case study adapted from an example presented in the paper “Managing with Statistical Models” by James C. Seigel (1982). Seigel’s example concerned a problem encountered by Ford Motor Company.

The Camshaft Case Camshaft

An automobile manufacturer produces the parts for its vehicles in many different locations and transports them to assembly plants. In order to keep the assembly operations running efficiently, it is vital that all parts be within specification limits. One important part used in the assembly of V6 engines is the engine camshaft, and one important quality characteristic of this camshaft is the case hardness depth of its eccentrics. A camshaft eccentric is a metal disk positioned on the camshaft so that as the camshaft turns, the eccentric drives a lifter that opens and closes an engine valve. The V6 engine camshaft and its eccentrics are illustrated in Figure 17.32. These eccentrics are hardened by a process that passes the camshaft through an electrical coil that “cooks” or “bakes” the camshaft. Studies indicate that the hardness depth of the eccentric labeled in Figure 17.32 is representative of the hardness depth of all the eccentrics on the camshaft. Therefore, the hardness depth of this representative eccentric is measured at a specific location and is regarded to be the hardness depth of the camshaft. The optimal or target hardness depth for a camshaft is 4.5 mm. In addition, specifications state that, in order for the camshaft to wear properly, the hardness depth of a camshaft must be between 3.0 mm and 6.0 mm.

Figure 17.32: A Camshaft and Related Parts

The automobile manufacturer was having serious problems with the process used to harden the camshaft. This problem was resulting in 12 percent rework and 9 percent scrap, or a total of 21 percent out-of-specification camshafts. The hardening process was automated. However, adjustments could be made to the electrical coil employed in the process. To begin study of the process, a problem-solving team selected 30 daily subgroups of n = 5 hardened camshafts and measured the hardness depth of each camshaft. For each subgroup, the team calculated the mean and range R of the n = 5 hardness depth readings. The 30 subgroups are given in Table 17.13. The subgroup means and ranges are plotted in Figure 17.33. These means and ranges seem to exhibit substantial variability, which suggests that the hardening process was not in statistical control; we will compute control limits shortly.

Table 17.13: Hardness Depth Data for Camshafts (Coil #1) Camshaft

Figure 17.33: Graphs of Performance ( and R) for Hardness Depth Data (Using Coil #1)

Although control limits had not yet been established, the problem-solving team took several actions to try to stabilize the process while the 30 subgroups were being collected:

1 At point A, which corresponds to a low average and a high range, the power on the coil was increased from 8.2 to 9.2.

2 At point B the problem-solving team found a bent coil. The coil was straightened, although at point B the subgroup mean and range do not suggest that any problem exists.

3 At point C, which corresponds to a high average and a high range, the power on the coil was decreased to 8.8.

4 At point D, which corresponds to a low average and a high range, the coil shorted out. The coil was straightened, and the team designed a gauge that could be used to check the coil spacing to the camshaft.

5 At point E, which corresponds to a low average, the spacing between the coil and the camshaft was decreased.

6 At point F, which corresponds to a low average and a high range, the first coil (Coil #1) was replaced. Its replacement (Coil #2) was a coil of the same type.

17.50 Using the data in Table 17.13: Camshaft

a Calculate and and then find the center lines and control limits for and R charts for the camshaft hardness depths.

b Set up the and R charts for the camshaft hardness depth data.

c Are the and R charts in statistical control? Explain.

Examining the actions taken at points A through E (in Figure 17.33), the problem-solving team learned that the power on the coil should be roughly 8.8 and that it is important to monitor the spacing between the camshaft and the coil. It also learned that it may be important to check for bent coils. The problem-solving team then (after replacing Coil #1 with Coil #2) attempted to control the hardening process by using this knowledge. Thirty new daily subgroups of n = 5 hardness depths were collected. The and R charts for these subgroups are given in Figure 17.34.

Figure 17.34: and R Charts for Hardness Depth Data Using Coil #2 (Same Type as Coil #1)

17.51 Using the values of and in Figure 17.34: Camshaft

a Calculate the control limits for the chart in Figure 17.34.

b Calculate the upper control limit for the R chart in Figure 17.34.

c Are the and R charts for the 30 new subgroups using Coil #2 (which we recall was of the same type as Coil #1) in statistical control? Explain.

17.52 Consider the and R charts in Figure 17.34. Camshaft

a Calculate the natural tolerance limits for the improved process.

b Recalling that specifications state that the hardness depth of each camshaft must be between 3.0 mm. and 6.0 mm., is the improved process capable of meeting these specifications? Explain.

c Use and to estimate the fraction of hardness depths that are out of specification for the improved process.

Since the hardening process shown in Figure 17.34 was not capable, the problem-solving team redesigned the coil to reduce the common cause variability of the process. Thirty new daily subgroups of n = 5 hardness depths were collected using the redesigned coil, and the resulting and R charts are given in Figure 17.35.

Figure 17.35: and R Charts for Hardness Depth Data Using a Redesigned Coil

17.53 Using the values of and given in Figure 17.35: Camshaft

a Calculate the control limits for the and R charts in Figure 17.35.

b Is the process (using the redesigned coil) in statistical control? Explain.

c Calculate the natural tolerance limits for the process (using the redesigned coil).

d Is the process (using the redesigned coil) capable of meeting specifications of 3.0 mm. to 6.0 mm.? Explain. Also find and interpret the sigma level capability.

17.54 A bank officer wishes to study how many credit cardholders attempt to exceed their established credit limits. To accomplish this, the officer randomly selects a weekly sample of 100 of the cardholders who have been issued credit cards by the bank, and the number of cardholders who have attempted to exceed their credit limit during the week is recorded. The numbers of cardholders who exceeded their credit limit in 20 consecutive weekly samples of 100 cardholders are, respectively, 1, 4, 9, 0, 4, 6, 0, 3, 8, 5, 3, 5, 2, 9, 4, 4, 3, 6, 4, and 0. Construct a control chart for the data and determine if the data are in statistical control. If 12 cardholders in next week’s sample of 100 cardholders attempt to exceed their credit limit, should the bank regard this as unusual variation in the process? CredLim

Appendix 17.1: Control Charts Using MINITAB

The instruction blocks in this section each begin by describing the entry of data into the Minitab Data window. Alternatively, the data may be loaded directly from the data disk included with the text. The appropriate data file name is given at the top of each instruction block. Please refer to Appendix 1.1 for further information about entering data, saving data, and printing results when using MINITAB.

p control chart
similar to Figure 17.24 on page 785 (data file: Invoice.MTW):

• In the Data window, enter the 30 weekly error counts from Table 17.11 (page 785) into column C1 with variable name Invoice.

• Select Stat: Control Charts: Attributes Charts: p.

• In the p Chart dialog box, enter Invoice into the Variables window.

• Enter 100 in the “Subgroup size” window to indicate that each error count is based on a sample of 100 invoices.

• Click OK in the p Chart dialog box.

• The p control chart will be displayed in a graphics window.

Appendix 17.2: Control Charts Using MegaStat

The instructions in this section begin by describing the entry of data into an Excel worksheet. Alternatively, the data may be loaded directly from the data disk included with the text. The appropriate data file name is given at the top of each instruction block. Please refer to Appendix 1.2 for further information about entering data, saving data, and printing results in Excel. Please refer to Appendix 1.3 for more information about using MegaStat.

X-bar and R charts in Figure 17.10 on page 764 (data file: HotChoc.xlsx):

• In cells A1, A2, and A3, enter the column labels Temp1, Temp2, and Temp3.

• In columns A, B, and C, enter the hot chocolate temperature data as 24 rows of 3 measurements, as laid out in the columns headed 1, 2, and 3 in Table 17.2 on page 752. When entered in this way, each row is a subgroup (sample) of three temperatures. Calculated means and ranges (as in Table 17.2) need not be entered—only the raw data is needed.

• Select Add-Ins: MegaStat: Quality Control Process Charts.

• In the “Quality Control Process Charts” dialog box, click on “Variables (Xbar and R).”

• Use the autoexpand feature to select the range A1.C25 into the Input Range window. Here each row in the selected range is a subgroup (sample) of measurements.

• Click OK in the “Quality Control Process Charts” dialog box.

• The requested control charts are placed in an output file and may be edited using standard Excel editing features. See Appendix 1.3 (page 41) for additional information about editing Excel graphics.

p control chart
in Figure 17.24 on page 785 (data file: Invoice.xlsx):

• Enter the 30 weekly error counts from Table 17.11 (page 785) into Column A with the label Invoice in cell A1.

• Select Add-Ins: MegaStat: Quality Control Process Charts.

• In the “Quality Control Process Charts” dialog box, select “Proportion nonconforming (p).”

• Use the autoexpand feature to enter the range A1.A31 into the Input Range window.

• Enter the subgroup (sample) size (here equal to 100) into the Sample size box.

• Click OK in the “Quality Control Process Charts” dialog box.

A c chart for nonconformities (discussed in Appendix X of this book) can be obtained by entering data as for the p chart and by selecting “Number of defects (c).”

1 Source: CEEM Information Services, “Is ISO 9000 for You?” 1993.

2 The data for this case were obtained from a metal fabrication plant located in the Cincinnati, Ohio, area. For confidentiality, we have agreed to withhold the company’s name.

3 The data for this case were collected for a student’s term project with the cooperation of the Food Service at Miami University, Oxford, Ohio.

4 Basically, statistical independence here means that successive process measurements do not display any kind of pattern over time.

5 When D3 is not listed, the theoretical lower control limit for the R chart is negative. In this case, some practitioners prefer to say that the LCLR equals 0. Others prefer to say that the LCLR does not exist because a range R equal to 0 does not indicate that an assignable cause exists and because it is impossible to observe a negative range below LCLR. We prefer the second alternative. In practice, it makes no difference.

6 There are a number of alternative formulas for the natural tolerance limits. Here we give the version that is the most clearly related to using and R charts. At the end of this section we present an alternative formula.

7 Some statisticians believe that this condition should be np > 5. However, for p charts many think np > 2 is sufficient.

(Bowerman 742)

Bowerman, Bruce L. Business Statistics in Practice, 5th Edition. McGraw-Hill Learning Solutions, 022008. .

CHAPTER <<<<<<<<<<<<<<<<<<<<<b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>b>13: Simple Linear Regression Analys

is

Chapter Outline

13.1

The Simple Linear Regression Model

and

the Least Squares Point Estimates

13.2
Model Assumptions and the Standard Error

13.3
Testing the Significance of the Slope and

y

-Intercept

13.4
Confidence and Prediction Intervals

13.5
Simple Coefficients of Determination and Correlation (This section may be read anytime after reading Section 13.1)