chapter 35 control charting alternatives

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Chapter 35

Control Charting Alternatives

Introduction

• This chapter discusses the three-way control chart, the

cumulative sum (CUSUM) control chart, and the zone chart.

• The three-way (or Shewhart) control chart is useful for

tracking both within- and between-part variability.

• CUSUM charts can detect small process shifts faster than

Shewhart control charts.

• A zone chart is a hybrid between an 𝑥 or 𝑋 chart and a

CUSUM chart.

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35.1 S4/IEE Application Examples:

Three-way Control Chart

• Manufacturing 30,000-foot-level metric: An S4/IEE project

was to improve the process capability/performance metrics

for the diameter of a plastic part from an injection-molding

machine. A three-way control chart could track over time

not only the part-to-part variability but within part variability

over the three positions within the length of the part.

• Transaction 30,000-foot-level metric: An S4/IEE project was

to improve the process capability/performance metrics for

the duration of phone calls received within a call center. A

three-way control chart could track over time within-person

and between-person variability by randomly selecting three

calls for five people for each subgroup.

35.2 Three-way Control Chart (Monitoring Within- and Between-Part Variability)

• Consider that a part is sampled once every hour, where five

readings are made at specific locations within the part. Not

only can there be hour-to-hour part variability, but the

measurements at the five locations within a part can be

consistently different in all parts. One particular location, for

example, might consistently produce either the largest or

smallest measurement.

• For this situation the within-sample standard deviation no

longer estimates random error. Instead this standard

deviation is estimating both random error and location effect.

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• The result from this is an inflated standard deviation, which

causes control limits that are too wide, and the plot position

of most points are very close to the centerline.

• An XmR-R(between/within) chart can solve this problem

through the creation of three separate evaluations of

process variation.

• The first two charts are an individuals chart and a moving-

range chart of the mean from each sample. Moving ranges

between the consecutive means are used to determine the

control limits. The distribution of the sample means will be

related to random error.


• The moving range will estimate the standard deviation of the

sample means, which is similar to estimating the random

error component alone.

• Using only the between-sample component of variation,

these two charts in conjunction track both process location

and process variation.

• The third chart is an R chart of the original measurements.

This chart tracks the within-sample component of variation.

• The combination of the three charts provides a method of

assessing the stability of process location. between-sample

component of variation, and within-sample component of

variation.

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35.3 Example 35.1:


Roll Number

Position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Near side 269 274 268 280 288 278 306 303 306 283 279 285 274 265 269

Middle 306 275 291 277 288 288 284 292 292 303 300 279 278 278 276

Far side 279 302 308 306 298 313 308 307 307 297 299 293 297 282 286

• Plastic film is coated onto paper. A set of three samples is taken at the end of each roll. The coating weight for each of

these samples is shown in Table 35.1 (Wheeler 1995a).

35.3 Example 35.1:


Minitab: Stat

Control Charts

Variable Charts

for Subgroups I-MR-R/S

(between/within)

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35.3 Example 35.1:


• The individuals chart for subgroup means shows roll-to-roll coating weights to be out of control. Something in this process

allows film thickness to vary excessively.

• The sawtooth pattern on the moving-range chart suggests that

larger changes in film thickness have a tendency to occur

every other roll.

• The sample-range chart indicates stability between rolls, but the magnitude of this positional variation is larger than the

average moving range between rolls (an opportunity for

improvement).

35.4 Cumulative Sum (CUSUM) Chart

• CUSUM charts can detect small process shifts faster than

Shewhart control charts.

• The form of a CUSUM chart can be “v mask” or “decision

intervals.”

• The following scenario is for the consideration where smaller

numbers are better (single-sided case). For a double-sided

situation, two single-sided intervals are run concurrently.

• The three parameters considered in CUSUM analyses are

𝑛, 𝑘, and ℎ, where 𝑛 is the sample size of the subgroup, 𝑘 is

the reference value, and ℎ is the decision interval.

• The two parameters h and k are often referred to as the

“CUSUM plan.”

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• Consider a situation where it is desirable for a process to

operate at a target value 𝜇. [an acceptable quality level

(AQL)] with desired protection against an upper shift of the

mean to a reject quality level (RQL).

• The chart is obtained by plotting 𝑠𝑚 the CUSUM value for

the 𝑚th subgroup:

𝑠𝑚 = (𝑥 𝑖 − 𝑘)

𝑚

𝑖=1

where 𝑥𝑖 is the average of the 𝑖th subgroup of a total 𝑚

subgroup and 𝑘 is the reference value, which is usually

chosen to be halfway between the AQL and RQL values.


• This equation is written in a one-sided form in which only

high numbers can cause problems. The process is assumed

to be in control, and no chart even needed if the values for

𝑥 𝑖 are less than 𝑘.

• However, whenever 𝑠𝑚 exceeds ℎ, the process should be

investigated to determine if a problem exists.

• Note that for a two-sided decision interval, two single-sided

intervals are examined concurrently (one for the upper limit

and one for the lower limit).

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• CUSUM usually tests the process to a quantifiable shift in

the process mean or directly to acceptable/rejectable limits.

• When samples are taken frequently and tested against a

criterion, there are two types of sampling problems.

• First, when the threshold for detecting problems is large,

small process shifts can take a long time to detect.

• Second, when many samples are taken, false alarms will

eventually occur because of chance.

• The design of a CUSUM chart addresses these problems

directly using average run length (ARL) as a design input,

where 𝐿𝑟, is the ARL at the reject quality level (RQL) and 𝐿𝑎 is the ARL at the accept quality level (AQL).


• 𝐿𝑟 and 𝐿𝑎 are selected by considering:

• High frequency of sampling: 𝐿𝑟 should usually be higher for a high-

volume process that is checked hourly than for a process that is

checked monthly.

• Low frequency of sampling: 𝐿𝑎 should usually be lower for a sampling

plan that has infrequent sampling than that which has frequent

sampling.

• Other process charting: 𝐿𝑟 should usually be higher when a product

has many process control charts that have frequent test intervals

because the overall chance of false alarms can increase

dramatically.

• Importance of specification: 𝐿𝑎 should usually be lower when the

specification limit is important for product safety and reliability.

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• The final input requirement to the design of a CUSUM chart

is the process standard deviation (𝜎). A preproduction

experiment is a possible source for this information.

• Note that after the CUSUM test is begun, it may be

necessary to readjust the sampling plan because of an

erroneous assumption or an improvement, with time, of the

parameter of concern.


• The nomogram can now be used

to design the sampling plan.

• By placing a ruler across the

nomogram corresponding to 𝐿𝑟 and 𝐿𝑎, values for the following

can be determined:

|𝜇 − 𝑘|𝑛

𝜎

ℎ 𝑛

𝜎

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• At RQL, 𝜇 − 𝑘 is a known parameter; 𝑛 can be determined

from the first equation.

• With this 𝑛 value, the second equation can then be used to

yield the value of ℎ.

• The data are then plotted with the control limit ℎ using the

equation presented above:

𝑠𝑚 = (𝑥 𝑖 − 𝑘)

𝑚

𝑖=1

35.5 Example 35.2: CUSUM Chart

• An earlier example described a preproduction DOE of the

settle-out time of a selection motor. In this experiment it was

determined that an algorithm change was important, along with an inexpensive reduction in a motor adjustment tolerance. As a

result of this work and other analyses, it was determined that there would be no functional problems unless the motors

experienced a settle-out time greater than 8 msec. A CUSUM

control charting scheme was desired to monitor the production

process to assess this criterion on a continuing basis.

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• If we consider the 8-msec tolerance to be a single-sided 3 upper limit, we need to subtract the expected 3 variability from

8 to get an upper mean limit for our CUSUM chart. Given an expected production 3 value of 2, the upper accepted mean

criterion could be assigned a value of 6. However, because of

the importance of this machine criterion and previous test

results, the designers decided to set the upper AQL at 5 and the RQL at 4. The value for 𝑘 is then determined to be 4.5,

which is the midpoint between these extremes.


• Given 𝐿𝑟 of 3 and 𝐿𝑎 of 800,

𝜇 − 𝑘𝑛

𝜎= 1.11

5.0 − 4.5𝑛

2/3= 1.11,

𝑛 = 2.19 ≅ 3

ℎ 𝑛

𝜎= 2.3

ℎ 3

2/3= 2.3, ℎ = 0.89

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• ln summary, the overall design is as follows. The inputs were an RQL of 4.0 with an associated 𝐿𝑟 of 3, an AQL of 4.0 with an

associated 𝐿𝑎 of 800, and a process standard deviation of 2/3. Rational subgroup samples of size 3 should be taken where a

change in the process is declared whenever the cumulative

sum above 4.5 exceeds 0.89—that is, whenever

𝑠𝑚 = (𝑥 𝑖 − 4.5)

𝑚

𝑖=1

> 0.89


• A typical conceptual plot of the CUSUM chart is

• In time, enough data can be collected to yield a more precise

estimate for , which can be used to adjust the preceding

procedural computations.

• In addition, it may be appropriate to adjust the 𝑘 value to the mean

of the sample that is being

assessed, which can yield an

earlier indicator to determine when a process change is

occurring.

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• Other supplemental tests can be useful when the process is stable to understand the data better and to yield earlier problem

detection.

• For example, a probability plot of data by lots could be used to

assess visually whether there appears to be a percentage of

population differences that can be detrimental. lf no differences are noted, one probability plot might be made of all collected

data over time to determine the percentage of population as a

function of a control parameter.

35.6 Example 35.3:

CUSUM Chart of Bearing Diameter

• The data (Wheeler l995b; Hunter 1995) are

the bearing diameters of 50 camshafts

collected over time. The data is analyzed using CUSUM techniques.

• Target = 50, ℎ = 4.0, 𝑘 = 0.5

Dia. 1 50 2 51 3 50.5 4 49 5 50 6 43 7 42 8 45 9 47

10 49 11 46 12 50 13 52 14 52.5 15 51 16 52 17 50

Dia. 18 49 19 54 20 51 21 52 22 46 23 42 24 43 25 45 26 46 27 42 28 44 29 43 30 46 31 42 32 43 33 42 34 45

Dia.

35 49

36 50

37 51

38 52

39 54

40 51

41 49

42 50

43 49.5

44 51

45 50

46 52

47 50

48 48

49 49.5

50 49

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35.6 Example 35.3:


Minitab:

Stat

Control Charts

Time-Weighted Charts CUSUM

35.6 Example 35.3:


• This chart tracks the cumulative sums of the deviations of each

sample value from the target value.

• The two one-sided CUSUM chart shown uses the upper

CUSUM to detect upward shifts in the process, while the lower

CUSUM detects downward shifts.

• The UCL and LCL determine out-of-control conditions.

• This chart is based on individual observations; however, plots

can be based on subgroup mean.

• For an in-control/predictable process, the CUSUM chart is good at detecting small shifts from the target.

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35.6 Example 35.3:


• With this plot we note:

• Two one-sided CUSUMs, where the upper CUSUM is to detect upward shifts in the level of the process and the lower

CUSUM is for detecting downward shifts.

• ℎ is the number of standard deviations between the centerline and control limits for the one-sided CUSUM.

• 𝑘 is the allowable “stack” in the process.

• Constant magnitude region when the slope is the same.

• Process has shifted at slope change, steeper slope indicates

larger change.

35.7 Zone Chart

• A zone chart is a hybrid between an 𝑥 or 𝑋 chart and a

CUSUM chart.

• In this chart the cumulative score is plotted, based on 1, 2,

and 3 sampling standard deviation zones from the centerline.

• An advantage of zone charts over 𝑥 and 𝑋 charts is its

simplicity.

• A point is out of control simply, by default, if its score is

greater than or equal to 8.

• With zone charts there is no need to recognize patterns

associated with nonrandom behavior as on a Shewhart chart.

• The zone chart methodology is equivalent to using four of the

standard tests for special causes 𝑥 or X chart.

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35.8 Example 35.4: Zone Chart

• In Example 10.3, individual viscosity readings from Table

10.2 were plotted to create the XmR chart shown in Figure

10.10. Compare this figure to the zone chart of the data,

which is shown in Figure 35.5.

10.13 Example 10.3: 𝑋𝑚𝑅 Charts

• The viscosity of a

chemical mixing process

has the centipoise (cP) measurements for 20

batches.

Batch # Viscosity MR 1 70.10 2 75.20 5.10 3 74.40 0.80 4 72.07 2.33 5 74.70 2.63 6 73.80 0.90 7 72.77 1.03 8 78.17 5.40 9 70.77 7.40 10 74.30 3.53 11 72.90 1.40 12 72.50 0.40 13 74.60 2.10 14 75.43 0.83 15 75.30 0.13 16 78.17 2.87 17 76.00 2.17 18 73.50 2.50 19 74.27 0.77 20 75.05 0.78

74.20 2.267

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Fig 10.10 𝑋𝑚𝑅 control charts


Minitab: Stat

Control Charts

Variable Charts

for Subgroups Zone Chart

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• Weights are assigned to each zone, which typically equate to

0, 2, 4, and 8.

• Each circle contains the cumulative score for each subgroup

or observation.

• The cumulative score is set to zero when the next plotted

point crosses over the centerline.

• Weights for points on the same size of the centerline are

added.

• With the above-noted weights, the cumulative score of 8

indicates an out-of-control process.

35.9 S4/IEE Assessment

Additional considerations for Shewhart versus CUSUM chart

selection are the following:

• When choosing a control chart strategy using Shewhart

techniques, variable sample sizes can be difficult to manage

when calculating control limits, while this is not a concern with

CUSUM techniques.

• Shewhart charts handle a number of nonconfomnng items via p

charts, while CUSUM can consider this a continuous variable

(e.g., number of “good” samples selected before a “bad” sample is found).

• CUSUM charting does not use many rules to define an out-of-

control condition. while, in the case of Shewhart charting it can lead to an increase in false alarms (Yashchin 1989).

chapter 35 control charting alternatives

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