Pre-control and Some Simple Alternatives

Stefan H. SteinerDept. of Statistics and Actuarial Sciences

University of WaterlooWaterloo, N2L 3G1 Canada

Pre-control, also called Stoplight control, is a quality monitoring scheme

similar to a control chart. However, the goal of Pre-control is the

prevention of nonconforming units. The appeal of Pre-control is due

mainly to its ease of implementation, lack of assumptions, and reported

successes. However, it has come under much criticism since many of its

decision rules appear ad hoc. In addition, there is some confusion

regarding different versions of Pre-control suggested in the literature. This

article compares the various Pre-control schemes and contrasts them with

more traditional control charts such as acceptance control charts and X

charts. Based on this analysis the conditions under which Pre-control is

appropriate are laid out. Also, alternatives that retain much of the simplicity

of Pre-control but have improved operating characteristics are suggested.

Key Words: Acceptance Control Charts; Grouped Data; Modified Pre-

control; Operating Characteristic; Process Capability; Stoplight Control

2

1. Introduction

Pre-control (PC), sometimes called Stoplight control, was developed to monitor the

proportion of nonconforming units produced in a manufacturing process. Implementation

is typically very straight-forward. All test units are classified into one of three groups:

green, yellow, or red, where the colors loosely correspond to good, questionable, and poor

quality products (see Figure 1). Decisions to stop and adjust the process or to continue

operation are made based on the number of green, yellow, and red units observed in a

small sample. The goal of PC is to detect when the proportion of nonconforming units

produced becomes too large. Thus, PC monitors the process to ensure that process

capability ( Cpk ) remains large.

PC was initial proposed in 1954 (see Shainin and Shainin, 1989 and Ermer and

Roepke, 1991 for more details) as an easier alternative to Shewhart charts. However, since

that time, at least three different versions of PC have been suggested in the literature. In

this article the three versions are called:

• Classical PC

• Two-stage PC

• Modified PC

Classical PC refers to the original formulation of Pre-control as described by

Shainin and Shainin (1989) and Traver (1985). Two-stage PC is a modification discussed

by Salvia (1988) that improves the method’s operating characteristics by taking an

additional sample if the initial sample yields ambiguous results. Modified PC was

suggested by Gurska and Heaphy (1991), and represents a departure from the philosophy

of Classical and Two-stage PC in that it compromises between the design philosophy of

Shewhart type charts and the simplicity of application of PC.

3

PC schemes are defined by two attributes; their group classification method and

their decision criteria. A third criteria, the qualification procedure, is the same for all three

PC versions, and is this not discussed in this article. The most important difference

between the three versions of PC is the group classification method. Classical PC and

Two-stage PC base the classification of units on engineering tolerance or specification

limits. A unit is classified as green if its quality dimension of interest falls into the central

half of the tolerance range. A yellow unit has a quality dimension that falls into the

remaining tolerance range, and a red unit falls outside the tolerance range. Assuming,

without loss of generality, that the upper and lower specification limits (USL and LSL) are

1 and –1 respectively, group classification is based on the endpoints: t = [–1, –.5, .5, 1].

The classification scheme is illustrated in Figure 1. The colored circle can be used for the

ease of the operators as a dial indictor overlay (Shainin and Shainin, 1989).

Red

Green

YellowYellow

Green YellowYellow RedRed

Measurement-1 -.5 .5 1

USLLSL

Figure 1: PC Classification Criteria

Let the quality dimension of interest be Y. Then, given a probability density

function for observations f y( ), the group probabilities for Classical and Two-stage PC are

given by (1.1).

P green( ) = Pg = f y( )dyy=−.5

.5

∫P yellow( ) = Py = f y( )dy

y=−1

−.5

∫ + f y( )dyy=.5

1

∫ (1.1)

P red( ) = Pr = f y( )dyy=−∞

−1

∫ + f y( )dyy=1

∞

∫

4

Using Modified PC, on the other hand, the group classification scheme is the same

as that used with Classical PC and Two-stage PC with one very important difference.

Modified PC uses control limits, as defined in Shewhart charts, rather than tolerance limits

to define the boundaries between green, yellow and red units. As shown later, this change

is fundamental and makes Modified PC much more like a Shewhart chart than like Classical

PC. Group probabilities for the Modified PC procedure are given in (1.2). Note that to

avoid confusion, through this article the current process mean and standard deviation are

denoted µ and σ , whereas estimates of the in-control mean and standard deviation used to

set the control limits for Modified PC and Shewhart type charts are denoted µc and σc .

Pm green( ) = f y( )dyµc −1.5σc

µc +1.5σc

∫Pm yellow( ) = f y( )dy

µc −3σc

µc −1.5σc

∫ + f y( )dyµc +1.5σc

µc +3σc

∫ (1.2)

Pm red( ) = f y( )dy−∞

µc −3σc

∫ + f y( )dyµc +3σc

∞

∫

The group probabilities given by Equations (1.1) and (1.2) are the same when µc =

0 and σc = 1/3. This makes sense because in that case control limits and tolerance limits

are the same. It has been suggested that Classical PC and Two-stage PC are only

applicable if the current process spread (six process standard deviations) covers less than

88% of the tolerance range (Traver, 1985). With specification limits at ±1, as defined

previously, this condition corresponds to the constraint σ < 0.29333.

The second important distinction between the three PC versions is their decision

criteria. With Classical PC the decision to continue operation or to adjust the process is

based on only one or two sample units. The decision rules are given below:

Classical PC Decision Procedure

Sample two consecutive parts A and B

• If part A is green continue operation (no need to measure B).

5

• If part A is yellow measure part B. If part B is green continue operation,

otherwise stop and adjust process.

• If part A is red stop and adjust process (no need to measure B).

The Two-stage PC and Modified PC decision criteria are more complicated since

they are based on a two-stage procedure. If the initial two observations do not provide

clear evidence regarding the state of the process, additional observations (up to three more)

are taken. The decision criteria for Two-stage and Modified PC is given below:

Two-stage and Modified PC Decision Procedure

Sample two consecutive parts.

• If either part is red stop process and adjust.

• If both parts are green continue operation.

• If either or both of the parts are yellow continue to sample, up to an additional

three units. In the combined sample, if three greens are obtained continue

operation; and if three yellows or a single red are observed stop the process.

The advantage of this more complicated decision procedure is that more information

regarding the state of the process is obtained and thus decision errors are less likely. The

disadvantage is that, on average, larger sample sizes are needed, and thus implementation is

more demanding. Table 1 summarizes the comparison of the three versions of PC.

Table 1: Comparison of Pre-control Versions

Pre-controlVersion

GroupClassification

DecisionCriteria

Classic tolerance limits two observations

Two-stage tolerance limits five observations

Modified control limits five observations

Based on these comparisons it can be concluded that the purpose of the various

versions of PC are quite different. By design, Classical PC, and Two-stage PC tolerate

6

some deviation in the target mean of a process so long as the proportion nonconforming

does not become too large. Thus, process stability is not required, and Classical and Two-

stage PC can be quickly implemented since there is no need to estimate the current process

mean and standard deviation and show stability. Thus Classical and Two-stage PC are

very similar to Acceptance Control Charts. By contrast, the goal of Modified PC is to

monitor the process for deviations from stability. As a result, mean drifts are not tolerated,

and Modified PC is very similar to Shewhart type charts such as X charts.

2. Comparison with Traditional Control Charting Methods

In this section the various PC methods are compared with appropriate traditional

control charting techniques. In addition, the performance of PC under some special

situations is explored. Classical and Two-stage PC are compared with Acceptance Control

Charts (ACCs), and Modified PC is compared with an X chart. Previous comparisons of

Classical and Two-stage PC with more traditional control charts were made with X & R

charts (Mackertich, 1990, Ermer and Roepke, 1991), however this is the wrong

comparison. For both ACC and Classical and Two-stage PC the goal is not statistical

control, but rather producing parts within specification. An ACC is designed to monitor a

process when the process variability is much smaller than the specification (tolerance)

spread. Under that assumption moderate drifts in the mean (from the target value) are

tolerable since they do not yield a significant increase in the proportion of nonconforming

units. Like Classical and Two-stage PC, ACCs are based on specification limits.

However, ACC limits are derived based on a distributional assumption and require a

known and constant process standard deviation. ACC limits are usually set assuming a

normal process, although if justified other assumptions could be made. For more

information on the design of ACCs see Ryan (1989). Modified PC, on the other hand,

bases the grouping criteria on the control limits, and thus Modified PC has the same goal as

a Shewhart X chart, namely statistical control of a process.

7

PC and traditional control charting techniques differ in a number of ways. The first

obvious difference is that PC uses only information from the classified (or grouped)

observations, whereas traditional control charts ( X charts and ACCs) use variables data.

Grouping data results in decision rules that are easy to implement, but clearly, the grouping

also discards information. The loss of information can be quantified by calculating the

expected statistical (Fisher) information available in the PC grouped observations.

Calculating the Fisher information available in grouped data is discussed in Kulldorff

(1961) and Steiner, Geyer and Wesolowsky (1995). Figure 2 shows the expected

information about the mean and standard deviation for various parameter values. The plots

are scaled so that variables data (infinite number of groups) would provide an expected

information content of unity for all values of µ and σ . The expected information graph

about µ is symmetric about µ = 0, thus negative mean values are not shown. Figure 2

shows that the group classification scheme used in PC is not very efficient when µ is close

to zero and/or σ is small. The small amount of information available when µ equals zero

may not be a major concern if at that mean value it is very easy to conclude that the process

should continue operation. This would be the case, for example, if the process standard

deviation were very small compared with the specification range. This issue is explored

further in Section 3.

Exp

ecte

d In

form

atio

n

µ0 0.5 1 1.50.3

0.4

0.5

0.6

0.7

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.80.3

0.4

0.5

0.6

0.7

0.8

0.9

σ

Exp

ecte

d In

form

atio

n

Figure 2: Expected Information about µ (left, assume σ =0.2933)and σ (right, assume µ =0), t = [–1, –.5, .5, 1]

8

PC and traditional control charts also differ in their decision rules. All control

charts require some decision rules to determine whether the process should be allowed to

continue operation or if it should be stopped. It is difficult to evaluate the effectiveness of

the decision rules in isolation from the grouping criteria. However, the overall

effectiveness of various control charts can be compared through their Operating

Characteristic (OC) curve. An OC curve shows the probability a control chart signals (or

fails to signal) for different parameter values. OC curves for ACCs and X charts can be

derived from results given in Ryan (1989). The probability Classical and Two-stage PC

schemes continue operating, called the probability of acceptance of the process, can be

found by calculating the probability of each combination of green and yellow units that

leads to acceptance. For Classical and Two-stage PC the results are given by equations

(2.1) and (2.2), see Salvia (1988).

Paccept Classical( ) = Pg + PyPg (2.1)

Paccept Two − stage( ) = Pg + Py( )2 − 2PgPy(1 − Pg3 − 3Pg2Py) − Py2 1 − Pg3( ) (2.2)

where Pg and Py are given by Equations (1.1).

Figure 3 shows the OC curves for Classical PC, Two-stage PC, as derived from

(2.1) and (2.2), and an ACC with sample size n = 2. Figure 3 shows that an ACC with

sample size n = 2 is superior to Classical PC since it has significantly better power to detect

mean shifts. Similarly, the Two-stage PC scheme is superior to an ACC with n = 2. Note

that different σ values simply shift the OC curves horizontally without changing the

ranking. Also, a process whose quality characteristic is approximately normally distributed

is assumed. A different underlying distribution for the quality characteristic may

dramatically change the probabilities of acceptance, but generally all the considered

procedure are similarly effected.

9

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mean Shift in Sigma Units

Prob

abili

ty o

f A

ccep

tanc

e Classical Precontrol

2-stage Precontrol

ACC n = 2

Figure 3: OC curve Classical and Two-stage PC versus ACC with n=2N( µ , σ =.2933) process

Note however, that on average the Two-stage PC scheme requires a larger sample

size. Two-stage PC uses a partially sequential procedure and requires sample sizes of

between one and five units. In the appendix, the average sampling number of the Two-

stage PC procedure, denoted E n( ) is derived. E n( ) depends on the group probabilities,

given by (1.1), that are in turn dependent on the process parameters µ and σ . Figure 4

shows plots of E n( ) versus the process mean assuming normality. For example,

E n ;µ = 0,σ =.29333( ) = 2.4, and E n ;µ = 2σ,σ =.29333( ) = 3.5,

0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

E n( )

µ 0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

E n( )

µ

Figure 4: Average sample size E n( ) for Two-stage PC as function of µσ = 0.29333 on left, σ = 0.2 on right, N( µ , σ ) process

10

These results suggest a comparison of Two-stage PC with ACCs having larger

sample sizes. OC curves of Two-stage PC and ACCs with sample sizes of 3, 4 and 5 are

shown in Figure 5. Clearly, of the compared curves the PC procedure has least power to

detect mean shifts. In addition ACCs are likely easier to implement since they require fixed

sample sizes. On the other hand, the power of Two-stage PC is fairly competitive with an

ACC with n = 3, and ACCs have the disadvantage of requiring an estimate of σ , an

assumption that σ does not change, and precise or variables measurements rather than

grouped data. In addition, Two-stage PC requires smaller sample size on average when the

process is centered at µ = 0. In conclusion, Two-stage PC appears a reasonable

alternative to ACCs when little process knowledge is available or if estimating the process

mean and standard deviation are expensive.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abili

ty A

ccep

t

Mean Shift in Sigma Units

2-stage Precontrol

ACC n=3

ACC n=5

ACC n=4

Figure 5: Comparison of Two-stage PC and ACC with n=3, n=4, n=5N( µ , σ =.29333) process

Classical and Two-stage PC schemes are designed to prevent defectives, but they

do not adapt to changes in process variability. Figures 6 and 7 explore the relation between

the probability Classical and Two-stage PC signal and the probability of a defect. Figure 6

11

shows a contour plot of the probability of a defect for various combinations of µ and σ ,

assuming USL = 1 and LSL = –1. The probability of a defect, or a nonconforming unit, is

given by Pr in Equations (1.1). Figure 7 shows contour plots of probability that Classical

and Two-stage PC signal (one minus the probability of acceptance as given by (2.1) and

(2.2). Ideally Figures 6 and 7 would look similar, however, this is not the case. The PC

methods are too conservative when σ is small. When σ is small compared with the

specification limits a significant drift in the process mean can be tolerated before the

proportion defective becomes large. However, using PC, large mean values together with

small σ values are likely to lead to a signal since then many yellow units would be

observed. As a result, PC is not applicable when σ is very small compared with the

specification limits.

-1 -0.5 0 0.5 10.1

0.15

0.2

0.25

0.3

0.35

0.4

1e-05

0.001

0.005

0.02

0.02

0.05

0.05

0.1

0.1

Mean

Stan

dard

Dev

iatio

n

Figure 6: Contour plot of the probability of a defect

12

-1 -0.5 0 0.5 10.1

0.15

0.2

0.25

0.3

0.35

0.4

1e-05

0.001

0.005

0.02

0.05

0.1

0.1

0.5

0.5

0.9

0.9

Mean

Stan

dard

Dev

iatio

n

-1 -0.5 0 0.5 10.1

0.15

0.2

0.25

0.3

0.35

0.4

1e-05

0.001

0.005

0.02

0.05 0.1

0.1

0.5

0.5

0.9 0.9

Mean

Stan

dard

Dev

iatio

n

Figure 7: Contour plot of P(signal) for Classical & Two-stage PC

ACCs do not suffer from this shortcoming since they adjust their control limits for

different σ values. Table 2 gives specific values shown on the contour plots and

equivalent values from an ACC with n = 5 for a direct comparison. Most enlightening are

the two cases: µ = 0, σ = .2, and µ = .6, σ = .1. In both cases the probability of a

defect is fairly small and approximately the same. However, when µ = 0, σ = .2 the

probability the PC scheme signals is very small, whereas when µ = .6, σ = .1 the

probability of a signal using PC is very large. Clearly, when σ is small, PC rejects some

processes that yield very few defects. This is appropriate if the goal is also to keep the

mean centered on the target value. However, the goal of PC is defect detection and

prevention. Thus, PC is too conservative when σ is small, say when 6σ covers less than

40% of the tolerance range. As shown in Table 2 an ACC with n = 5 does not suffer from

this problem to the same degree.

Table 2: Probability of defect and signals

µ σ P(defect)P(signal)

Classical PCP(signal)

Two-stage PCP(signal)

ACC n = 5

0 .1 ~ 0 3.2 e–13 1.7 e–18 ~ 0

0 .2 5.7 e–5 1.5 e–4 1.8 e–5 2.6 e–7

0 .3 8.6 e–4 .0099 .0088 not applicable

.5 .1 2.9 e–7 .25 .4688 1.3 e–7

.6 .1 3.2 e–5 .7079 .9540 .0018

.7 .1 .0013 .9550 .9994 .2489

13

Modified PC, unlike Classical and Two-stage PC, classifies units based on control

limits. As a result, Modified PC is the same in philosophy as an X control chart, since its

goal is to monitor the stability of a process. With Modified PC and X charts ideally any

drift in the process mean would be detected. The operating characteristics of Modified PC

can be determined using Equation (2.2) and (1.2) with µ = 0, σ = 1/3. Modified PC is

compared with traditional X control charts in Table 3. The average sample size of

Modified PC can be determined through the result in the appendix setting µ = 0 and σ =

1/3. A plot of the result would be similar to those shown in Figure 4.

Table 3: Modified PC compared with X control charts

µ

Psignal

ModifiedPC

Psignal

X chartn = 3

Psignal

X chartn = 4

Psignal

X chartn = 5

0 .0238 .0027 .0027 .0027

±1σ .2097 .1024 .1587 .2225

±2σ .8370 .6787 .8413 .9295

Table 3 shows that Modified PC has a very high false alarm rate that is an order of

magnitude larger than that for X charts. When the process is stable ( µ = 0) a Modified

PC chart signals a problem on average almost 2.4% of the time. This false alarm rate is too

high, since searching for assignable causes is usually time consuming and expensive, and

too many false alarms greatly reduces the credibility of the control chart and may lead to it

being ignored.

In summary, Classical and Two-stage PC are fairly competitive with ACCs, except

under certain circumstances. When the process variation is very small compared with the

tolerance range ( 6σ < .4T , where T is the tolerance range), Classical and Two-stage PC

lead to undesirable results such as rejecting a process that is producing virtually all parts

within specification. On the other hand, when the process variation is relatively large

compared to the tolerance range ( 6σ > .88T ) Classical and Two-stage PC yield

14

excessively large false alarm rates. However, if 6σ lies between 40% and 88% of the

tolerance range (with specification limits of –1 and 1, .29333 > σ > .13333) Classical or

Two-stage PC yield good results. Modified PC, on the other hand, is designed to have the

same goal as a Shewhart chart. Thus, Modified PC is similar to Two-stage PC with 6σ

equal to 100% of the tolerance range. As a result, Modified PC is not a good method since

it yields too many false alarms.

3. Alternatives to Traditional Pre-control

As discussed in Section 2, PC has a number of important advantages over

traditional control charts, mainly in terms of simplicity of implementation. However, its

design choices, in particular the grouping criteria and the decision rules, appear quite ad

hoc. We may ask if the performance of PC could be improved while retaining its

simplicity? In this section, three simple variations of PC called Ten Unit PC, Mean Shift

PC, and Simplified PC are considered. Each variation is very similar to Two-stage PC,

and requires only minor modifications in the design. The goal is not to develop the optimal

approach under particular assumptions, but rather to consider simple changes that retain the

flavor of PC. If optimal charts are desired, and more process information is available, the

grouped data ACCs and Shewhart type charts developed by Steiner, Geyer, and

Wesolowsky (1994, 1995) should be considered.

One reason why Two-stage PC is fairly competitive with ACCs in terms of power

is due to its partially sequential decision procedure. As a result, one possible improvement

approach is to make the decision process more sequential. A totally sequential procedure is

theoretically feasible, but would be difficult to implement since it would occasionally

require large sample sizes and defining the acceptance/rejection criteria would be difficult.

A version of PC that allows a maximum of ten units at each decision point is a

compromise. The proposed decision rules for Ten Unit PC are given below. This

sampling procedure should be repeated each time the process is to be monitored.

15

Ten Unit PC Decision Process

Take sample units one at a time defining #G and #Y as the number of green and

yellow units observed so far respectively.

• Stop the process

– if there are any red units, or

– if at any time #Y – #G 2 together with #Y 3, or

– if at any time #Y 5.

• Continue operation of the process and stop sampling if at anytime #G – #Y 2.

• Otherwise, continue to sample.

By enumerating all the possible paths to acceptance and rejection of Ten Unit PC it

is possible to derive expressions for the probability of a signal and the average sample size.

Table 4 shows a comparison of the operating characteristics of the proposed Ten Unit PC

scheme and Two-stage PC for various combinations of process mean and standard

deviation. Table 4 shows that Ten Unit PC has lower false alarm rates, more power to

detect two-sigma shifts in the mean, and requires approximately the same average number

of units as Two-stage PC.

Table 4: Comparison of Ten Unit and Two-stage PC

µ σP(signal)

Two-stage PCE n( )

Two-stage PCP(signal)

Ten Unit PCE n( )

Ten Unit PC

0 .29333 .0069 2.37 .0045 2.41

±1σ .29333 .1058 2.96 .0895 3.27

±2σ .29333 .7155 3.32 .7427 3.74

0 1/3 .0238 2.55 .0174 2.65

±1σ 1/3 .2097 3.11 .1959 3.52

±2σ 1/3 .8370 2.95 .8513 3.17

A second improvement idea stems from the realization that using PC, units are

classified into one of five groups (see Figure 1) but only three groups (green, yellow and

red) are used during the decision process. As a result, important information pertaining to

16

the process mean is ignored. The proposed Mean Shift PC utilizes this information. Mean

Shift PC is a very simple adaptation of Two-stage PC since it does not requiring any

changes in the data collection process. The Mean Shift PC method classifies observations

into one of four groups: Green, Yellow+, Yellow–, and Red. Red units above or below

the specification limits are not distinguished because they are very rare and automatically

lead to a signal in any case. The Mean Shift PC decision rules, given below, are similar to

those used in Two-stage PC.

Mean Shift PC Decision Process

Sample two consecutive parts.

• If either unit is red stop the process.

• If both units are green continue operation.

• If either or both units are yellow, sample an additional three units. If in the

combined sample three greens are obtained continue the process. If three

Yellow+ or three Yellow– or a single Red are observed then stop the process.

Table 5: Comparison of Two-stage PC and Mean Shift PC

µ σPsignal Two-stage PC

Psignal MeanShift PC

0 1/3 .0238 .0120

±1σ 1/3 .2097 .2031

±2σ 1/3 .8370 .8369

0 .29333 .0069 .0031

±1σ .29333 .1058 .1029

±2σ .29333 .7155 .7154

Table 5 compares the power of the Mean Shift PC and Two-stage PC. The table

shows that Mean Shift PC has a much smaller false alarm rate than Two-stage PC, and

virtually identical power to detect two sigma mean shifts. This advantage in terms of

operating characteristics will typically outweigh the slightly more complicated decision

17

process. Thus, Mean Shift PC is a good alternative to Two-stage PC, though the process

standard deviation should also be monitored in some way.

The third variation, called Simplified PC uses only green and yellow units. The

group probabilities and decision process for Simplified PC are given below:

P green( ) = Pg = f y( )dyy=−.5

.5

∫P yellow( ) = Py = f y( )dy

y=−∞

−.5

∫ + f y( )dyy=.5

∞

∫ (3.1)

Simplified PC Decision Process

Sample five consecutive parts

• If three or more units are yellow, then stop the process

• Otherwise continue operation.

The loss of efficiency using Simplified PC rather than Two-stage PC is very slight

when the process variation is in the range where PC is appropriate. This only small loss of

efficiency arises because the probability of a red unit is usually very small, and thus the red

classification provides very little information. In fact, as designed, Simplified PC has a

slightly smaller false alarm rate, though not quite as much power to detect mean shifts.

Table 6 shows a comparison. Compared with Two-stage PC, Simplified PC has the

advantage of requiring less effort in the group classification, and the decision rules are

more straight forward. In addition, although Simplified PC requires a larger sample size

on average often the required sample sizes are the same and implementation is usually

easier if the sample size is fixed.

18

Table 6: Comparison of Two-stage PC and Simplified PC

µ σPsignal Two-stage PC

Psignal

Simplified PC

0 1/3 .0238 .0193

±1σ 1/3 .2097 .1831

±2σ 1/3 .8370 .8258

0 .29333 .0069 .006

±1σ .29333 .1058 .0972

±2σ .29333 .7155 .7103

Simplified PC is also appealing since it can fairly easily be adapted when the

process variation is much smaller than the tolerance range. As shown in Section 2, PC is

not appropriate if the process standard deviation is less than around 1 15th of the tolerance

range. When σ is small, observing yellow units with the traditional grouping criterion as

defined by (1.1), is not necessarily evidence of a problem. See Figures 6 and 7. In this

circumstance the Simplified PC group limits [–.5, .5] can be replaced with the group limits

[–c, c], where c is closer to the tolerance limits (.5 < c < 1). By adjusting the group limits

it is possible to only detecting process mean shifts that yield an excessive proportion of

non-conforming units. The decision process remains unchanged.

The best value for c depends on the process standard deviation and the acceptable

and rejectable process levels (APL and RPL). A good choice for c is the midpoint between

the APL mean and the RPL mean. APL and RPL mean values can be estimated from

acceptable and unacceptable process capability values. Process capability is defined as Cpk

= min USL − µ( ) 3σ, µ − LSL( ) 3σ( ), where LSL and USL is the lower and upper

specification limits respectively. For example, assume the process standard deviation is σ

= .1, and that an acceptable process capability value is Cpk = 4/3 (an average of 6 defects

out of 100,000 units) whereas Cpk = 2/3 (an average of 94 defects out of 10,000 units) is

unacceptable. Then looking only at the upper specification limit (the problem is symmetric

on the lower limit) the corresponding APL and RPL mean values are µ = .6 and µ = .8

respectively. In this case choosing c = .7 is reasonable. Figure 8 illustrates the operating

19

characteristics obtained in this example for various parameter values. For example, the

probability of a signal when µ = .6 is .031 and the probability of a signal when µ = .8 is

.969. If this false alarm rate is too large adjusting the decision barrier c slightly higher may

be appropriate. An OC curve for Two-stage PC would show a much great chance of

rejecting the process at the acceptable mean value µ = .6.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

µ

Psignal

Figure 8: OC curve of the adapted Simplified PC procedure

4. Conclusions

This article compared and contrasts previously recommended PC approaches and

suggests simple variations that improve performance. Classical and Two-stage PC are

compared with acceptance control charts rather than X charts since this is a more

appropriate comparison. It is concluded that Classical and Two-stage PC are good

methods when the process standard deviation σ lies in the range T 15 ≤ σ ≤ 11T 75 ,

where T is the tolerance range. Two-stage PC is preferred over Classical PC unless the

additional sampling required is very onerous. Modified PC, on the other hand, has the

same goal as an X chart, but is shown to have an excessively large false alarm rate, and is

thus not recommended.

Three alternatives to PC are suggested that retain the great advantage of very simple

implementation. The Ten Unit approach is recommended when improved operating

20

characteristics are desired and taking some additional observations is not difficult. The

Mean Shift approach yields better results than Two-stage PC when the goal is the detection

of process mean shifts, and Simplified PC utilizes a simplified classification and decision

procedure while providing similar operating characteristics. Simplified PC can also easily

be adapted to handle situations when the process standard deviation is smaller than T 15.

Appendix

In this appendix an expression for the average sampling number E n( ) of the Two-

stage PC scheme, discussed in Section 2, is derived. E n( ) is determined by calculating

the probability of each possible path to either acceptance or rejection of the process. Below

in (A.1), pi is the probability that the Two-stage PC reaches a decision in i units. For

example, Two-stage PC would stop after two units if we observe either two green units

(process acceptable), or if the first observation is yellow or green and the second

observation is red (process rejectable).

p1 = pr , p2 = pg2 + pr py + pg( ), p3 = pr py

2 + 2 pg py pr + py3 (A.1)

p4 = 3py2 pg pr + 2 pg

2 py pr + 2 pg3 py + 3py

3 pg , p5 = 5py2 pg

2 pr + 5py3 pg

2 + 5pg3 py

2

E n( ) = ipii=1

5∑

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21

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