Chapter 6 Process Capability Analysis

8/18/2019 Chapter 6 Process Capability Analysis

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CHAPTER 6: PROCESS CAPABILITY ANALYSISSTATE OF CONTROL

Process in Control

When the assignable causes have been eliminated from the process to the extent the points

plotted on the control chart remain within the control limits, the process is in a state of

control. No higher degree of uniformity can be attained with the existing process. Greateruniformity can, however can, however, be attained through a change in the basic process

through quality improvement ideas.

When a process is in control, there occurs a normal pattern of variation which is

illustrated in Figure 3-7. This natural pattern of variation has (1) about two thirds of the

points near the central line, (2) a few points closer to the control limits, (3) points located

back and forth across the central line, (4) points balanced on both sides of the central line,

and (5) no points beyond the control limits. The natural pattern of the points or subgroup

values forms its own frequency distribution, which follows or subgroup values forms its own

frequency distribution, which follows a normal curve. As the number of plotted points

increases, the frequency distribution will take on the appearance of a smooth polygon. The

dashed normal curve at the left of Figure 3-7 represents the distribution of the points when a

process is in control.

Figure 3-7 Natural pattern of variation of a control chart.

Control limits are usually established at three standard deviations from the central

line. They are used as a basis to judge whether there is evidence of lack of control. The

choice of 3σ limits is an economic one with respect to two types of errors that can occur. One

error, called Type I by statisticians, occurs when looking for an assignable cause of variation

when in reality a chance cause is present. When the limits are set at three standard deviations,

a Type I error will occur 0.27% (3 out of 1,000) of the time. In other words, when a point is

outside the control limits, it is assumed to be due to an assignable cause even though it would

be due to a chance cause 0.27% of the time. The other type error, called Type II, occurs when

assuming that a chance cause of variation is present when in reality there is an assignable

cause. In other words, when a point is inside the control limits, it assumed to be due to a

chance cause even though it might be an assignable cause. Abundant experience since 1930

in all types of industry indicates that 3σ limits provide an economic balance between the costs

resulting from the two types of errors. Unless there are strong practical reasons for doingotherwise, the ±3 standard deviation limits should be used.

When a process is in control, only chance causes of variation are present. Small

variations in machine performance, operator performance, and material characteristics are

expected and are considered to be part of a stable process.

When a process is in control, certain practical advantages accrue to the producer and

consumer.

1.

Individual units of the product will be more uniform-or, stated another way, there will

be less variation.

2. Since the product is more uniform, fewer samples are needed to judge the quality.

Therefore, the cost of inspection can be reduced to a minimum. This advantage is extremely

important when 100% conformance to specifications in not essential.


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3. The process capability or spread of the process is easily attained from 6σ. With a

knowledge of the process capability, a number of reliable decisions relative to specifications

can be made, such as:

a. To decide the product specifications.

b. To decide the amount of rework or scrap when there is insufficient tolerance.

c.

To decide whether to produce the product to tight specifications and permitinterchangeability of components or to produce the product to loose specifications and

use selective matching of components.

4. The percentage of product that falls within any pair of values may be predicted with

the highest degree of assurance. For example, this advantage can be very important when

adjusting filling machines to obtain different percentage of items below, between, or above

particular values.

5. It permits the consumer as a check on the producer’s data and, therefore, to test only a

few subgroups as a check on the producer’s records.

6. The operator is performing satisfactorily from a quality viewpoint.

Process Out of ControlWhen a point (subgroup value) falls outside its control limits, the process is out of control.

This means that an assignable cause of variation is present. Another way of viewing the out-

of-control point is to think of the subgroup value as coming from a different population than

the one from which the control limits were obtained. Figure 3-8 shows a frequency

distribution of subgroup averages for cereal boxes which was developed from a large number

of subgroups and, therefore, represents the population mean, μ = 450 g, and the population

standard deviation for the averages, x

= 8 g. the frequency distribution for subgroup

averages is shown by a dashed line, which represents a smooth polygon. For instructional

purposes the individual dots represents the number of subgroup averages at particular values.

Future explanations will use only the dashed line to represent the frequency distribution ofaverages and will use a solid line for the frequency distribution of individual values. The out-

of-control point has a value of 483 g. This point is so far away from the 3σ limits (99.73%)

that it can only be considered to have come from another population. In other words, the

process that produced the subgroup average of 483 g is a different process than the stable

process from which the 3σ control limits were developed. Therefore, something has gone

wrong with the process; some assignable cause of variation is present. This assignable cause

must be found and corrected before a normal, stable process can continue.

A process can also be considered out of control even the point’s fall inside the 3σ

limits. This situation occurs when unnatural patterns of variation are present in the process. It

is not normal for seven or more consecutive points to be above or below the central line. Also

when 10 out of 11 points or 12 out of 14 points, etc., are located on one side of the centralvalue, it is an unnatural pattern. These unnatural patterns are shown in Figure 3-9. The

chance that these unnatural patterns or runs will occur is the same chance that a point will fall

outside the 3σ control limits.


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Figure 3-8 Frequency distribution of subgroup averages with control limits.

Figure 3-9 Some unnatural patterns of variation-process out of control.

There are many other unnatural patterns, and it is important to remember the

conditions of normality described in the preceding section. One technique for recognizing

unnatural patterns is to divide the control chart into six equal imaginary bands. Three equals

bands are between the central line, and the lower control limit and three equal bands are

between the central line and the upper control limit. A normal pattern of variation occurs

when (1) about 34% of the points fall in each of the two bands adjacent to the center value,

(2) about 13½ % of the points fall in each of the two middle bands, and (3) 2½ % of the

points fall in each of the three outer bands. Any significant divergence from the normal

pattern, such as 2 out of 3 consecutive points in the outer band, would be an unnatural pattern

and would be classified as an out-of-control condition.

Analysis of Out-of-Control ConditionWhen a process is out of control, the assignable cause responsible for the condition must be

found. The detective work necessary to locate the cause of the out-of-control condition can be

minimized by a knowledge of the types of out-of-control patterns and their assignable causes.

Types of out-of-control X and R patterns are (1) change or jump in level, (2) trend or steady

change in level, (3) recurring cycles, (4) two populations, and (5) mistakes.

1. Change or jump in level. This type is connected with a sudden change in level to the

X chart, to the R chart, or to both charts. Figure 3-10 illustrates the change in level. For an

X chart, the change in the process average can be due to:

(a)

An intentional or unintentional change in the process setting.

(b)

A new or inexperienced operator.(c) A different raw material.


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(d) A minor failure of a machine part.

Some causes for a sudden change in the process spread or variability as shown on the R chart

are:

(a) Inexperienced operator.

(b) Sudden increase in gear play.

(c)

Greater variation in incoming material.Sudden changes in level can occur on both the X and the R charts. This situation iscommon during the beginning of control chart activity prior to the attainment of a state of

control. There may be more than one assignable cause, or it may be a cause that could affect

both charts, such as an inexperienced operator.

Figure 3-10 Out-of-control pattern: change or jump in level.

2. Trend or steady change in level. Steady changes in control chart level are a very

common industrial phenomena. Figure 3-11 illustrates a trend or steady change that is

occurring in the upward direction; the trend could have been illustrated in the downward

direction. Some causes of steady progressive changes on an X chart are:

(a) Tool or die wear

(b) Gradual deterioration of equipment

(c) Gradual change in temperature or humidity

(d) Viscosity in a chemical process

(e)

Buildup of chips in a work-holding device

A steady change in level or trend on the R chart is not as common as the X chart. It does,however, occur and some possible causes are:

(a) An improvement in worker skill (downward trend)

(b) A decrease in worker skill due to fatigue, boredom, inattention, and so on

(c) A gradual improvement in the homogeneity of incoming material.

Figure 3-11 Out-of-control pattern: trend or steady change in level.

3. Recurring cycles. When the plotted points on an X or R chart show a wave or periodic high and low points, it is called a cycle. A typical recurring out-of-control pattern is

shown in Figure 3-12. For an X chart, some of the causes of recurring cycles are: (a)

The seasonal effects of incoming material

(b)

Any daily or weekly chemical, mechanical, or psychological event(c)

The periodic rotation of operators


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Periodic cycles on an R chart are not as common as for an X chart. Some affecting the Rchart are due to:

(a) Operator fatigue and rejuvenation (upgrading) resulting from morning, noon, and

afternoon breaks

(b) Lubrication cycles

The out-of-control pattern of a recurring cycle sometimes goes unreported because of theinspection cycle. Thus, a cycle pattern of variation that occurs approximately every month 2

hours could coincide with the inspection frequency. Therefore, only the low points on the

cycle are reported, and there is no evidence that a cyclic event is present.

Figure 3-12 Out-of-control pattern: recurring cycles.

4.

Two populations. When there are a large number of points near or outside the control

limits, a two-population situation may be present. This type of out-of-control pattern is

illustrated in Figure 3-13. For an X chart the out-of-control pattern can be due to:(a) Large differences in material quality

(b) Two or more machines on the same chart

(c)

Large differences in test method or equipment

Some causes for an out-of-control pattern on an R chart are due to:

(a)

Different workers using the same chart

(b) Materials from different suppliers.

Figure 3-13 Out-of-control pattern: two populations.

5. Mistakes. Mistakes can be very embarrassing to the quality assurance operation. Some

causes of out-of-control patterns resulting from mistakes are:

(a)

Measuring equipment out of calibration(b) Errors in calculations

(c) Errors in using test equipment

(d) Taking samples from different populations

Many of the out-of-control patterns that have been described can be attributed to some sort of

inspection error.

The causes given for the different types of out-of-control patterns are suggested possibilities

and are not meant to be all-inclusive. These causes will give production and quality personnel

ideas for the solution of particular industrial problems. They can be a start toward the

development of an assignable cause checklist which is applicable to a particular

manufacturing entity.


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When out-of-control patterns occur in relation to the lower control limits of the R chart, it is

the result of outstanding performance. The cause should be determined so that the

outstanding performance can continue.

The preceding discussion has used the R chart as the measure of the dispersion. Information

on patterns and causes also pertains to an s chart.

SPECIFICATIONS

Individual Values Compared to Averages

Before discussing specifications and their relationship with control charts, it appears

desirable, at this time, to obtain a better understanding of individual values and average

values. Figure 3-14 shows a tally of the subgroup values or individual values ( X ’s) and a

tally of the subgroup averages ( X ’s) for the data on keyway widths given in Table 3-2. The

four out-of-control subgroups were not used in the two tallys; therefore, there are 84

individual values and 21 averages. It is observed that the averages are grouped much closer to

the center than the individual values. This is true because when we average four values, the

affect of an extreme value is minimized, since the chance of four extremely high or four

extremely low values in one subgroup is slight.

Table 3-2 Data on the Depth of the Keyway (millimeters)

Subgroup

Number Date TimeMeasurements Average

X

Range

R Comment1

X 2

X 3

X 4

X

1 23/12 8:50 6.35 6.40 6.32 6.33 6.35 0.08

2 11:30 6.46 6.37 6.36 6.41 6.40 0.10

3 1:45 6.34 6.40 6.34 6.36 6.36 0.06

4 3:45 6.69 6.64 6.68 6.59 6.65 0.10 New, temporaryoperator

5 4:20 6.38 6.34 6.44 6.40 6.39 0.10

6 27/12 8:35 6.42 6.41 6.43 6.34 6.40 0.097 9:00 6.44 6.41 6.41 6.46 6.43 0.05

8 9:40 6.33 6.41 6.38 6.36 6.37 0.08

9 1:30 6.48 6.52 6.49 6.51 6.50 0.04

10 2:50 6.47 6.43 6.36 6.42 6.42 0.11

11 28/12 8:30 6.38 6.41 6.39 6.38 6.39 0.03

12 1:35 6.37 6.37 6.41 6.37 6.38 0.04

13 2:25 6.40 6.38 6.47 6.35 6.40 0.12

14 2:35 6.38 6.39 6.45 6.42 6.41 0.07

15 3:55 6.50 6.42 6.43 6.45 6.45 0.08

16 29/12 8:25 6.33 6.35 6.29 6.39 6.34 0.10

17 9:25 6.41 6.40 6.29 6.34 6.36 0.1218 11:00 6.38 6.44 6.28 6.58 6.42 0.30 Damaged oil line

19 2:35 6.33 6.32 6.37 6.38 6.35 0.06

20 3:15 6.56 6.55 6.45 6.48 6.51 0.11 Bad material

21 30/12 9:35 6.38 6.40 6.45 6.37 6.40 0.08

22 10:20 6.39 6.42 6.35 6.40 6.39 0.07

23 11:35 6.42 6.39 6.39 6.36 6.39 0.06

24 2:00 6.43 6.36 6.35 6.38 6.38 0.08

25 4:25 6.39 6.38 6.43 6.44 6.44 0.06

Sum 160.25 2.19


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Figure 3-14 Comparison of individual values and averages using the same data.

Calculations of the average for both the individual values and for the subgroup

averages are the same, X = 38.9. However, the sample standard deviation of the individual

values ( s ) is 4.16, while the sample standard deviation of the subgroup average ( X s ) is 2.77.

If there are a large number of individual values and subgroup averages, the smooth

polygons of Figure 3-14 would represent their frequency distribution if the distribution is

normal. The curve for the frequency distribution of the averages has a dashed line while the

curve for the frequency distribution of individual values has a solid line; this convention will

be followed throughout the text. In comparing the two distributions it is observed that both

distributions are normal in shape; in fact, even if the curve for individual values was not quite

normal, the curve for averages would be close to a normal shape. The base of the curve for

individual values is about twice as large as the base of the curve for averages. When

population values are available for the standard deviation for individual values ( ) and for

the standard deviation for averages ( X ), there is a definite relationship between them, as

given by the formulan

X

, where

X = population standard deviation of subgroup

averages ( X ’ s ), = population standard deviation of individual values, n= subgroup size.

Thus, for a subgroup of size 5, X

=0.45 , and for a subgroup of size 4, X

=0.50 .

If we assume normality (which may or may not be true), the population standard

deviation can be estimated from4

cs

,where

is the “estimate” of the population

standard deviation and4c is “appr oximately equal to (

) 0.996997 for n=84. Thus,4cs

= 996997.016.4 =4.17 and n X

= 417.4 =2.09. Note that X s , which was calculated


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from sample data, and X

which was calculated above, are different. This difference is due

to sample variation or the small number of samples, which was only 21, or some combination

thereof. The difference would not be caused by a non-normal population of X ’s.

Since the height of the curve is a function of the frequency, the curve for individual

values is higher. This is easily verified by comparing the tally sheet in Figure 3-14 and is a

true relationship when making comparisons using frequencies from the sample data.However, if the curves represent relative or percentage frequency distributions, then the area

under the curve must be equal to 100%. Therefore, the percentage frequency distribution

curve for averages, with its smaller base, would need to be much higher to enclose the same

area as the percentage frequency distribution curve for individual values.

Central Limit Theorem

Now that you are aware of the difference between the frequency distribution of individual

values, X ’s, and the frequency distribution of averages, X ’s, the central limit theorem can

be discussed. In simple terms it is:

If the population from which samples are taken is not normal, the distribution of sampleaverages will tend toward normality provided that the sample size,n , is at least 4. This

tendency gets better and better as the sample size gets larger. Furthermore, the standardized

normal can be used for the distribution of averages with the modification,

n

X X Z

X

This theorem was illustrated by W.A. Shewhart for a uniform population distribution and a

triangular population distribution of individual values as shown in Figure 3-15. Obviously,

the distribution of X ’s is approximately normal. The central limit theorem is the reason the X chart works, in that we do not need to be

concerned if the distribution of X

’s is not normal provided that the sample size is 4 or more.

Figure 3-15 Illustration of central limit theorem.Control Limits and Specifications


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Control limits are established as a function of the averages; in other words, control limits are

for averages. Specifications, on the other hand, are the permissible variation in the size of the

part and are, therefore, for individual values. The specification or tolerance limits are

established by product engineers to meet a particular function. Figure 3-16 shows that the

location of the specification is optional and is not related to any other features in the figure.

The control limits, process spread, distribution of averages, and distribution of individualvalues are interdependent, since n

X .

Figure 3-16 Relation of limits, specifications, and distributions.

Process Spread and Specifications

While specifications can be established by the product engineer without regard for the spread

of the process, serious situations can result when this type of action is adopted. There arethree situations: (1) when the process spread is less than the difference between

specifications, (2) when the process spread is equal to the difference between specifications,

(3) and when the process spread is greater than the difference between specifications.

Case I: LU 6 . This situation, where the spread of the process ( 6 ) is less than the

difference between specifications ( LU ), is the most desirable case. Figure 3-17

illustrates this ideal relationship by the distribution of individual values labeled A. Since

the specifications are appreciably greater than the process spread, no difficulty isencountered even when there is a substantial shift in the process average, as shown by thedistributions at B. At C a shift in the dispersion is illustrated, and all the individual values

are between specifications. Case I is economically advantageous since an out-of-control

condition, as illustrated at B and C, does not produce defective product. Therefore,frequent machine adjustments or searches for assignable causes are not necessary. In fact,this satisfactory state of affairs suggests that the control chart may be discontinued.


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Figure 3-17 Changes in the process average and dispersion when LU 6 .

Case II: LU 6 . Figure 3-18 illustrates this case where the spread of the process, or

process capability, is equal to the difference between specifications. The frequency

distribution at A represents a natural pattern of variation. However, when there is a shiftin the process average, as indicated at B, or a change in the dispersion, as indicated at C,the individual values exceed the specifications. As long as the process remains in control

as indicated at A, no defective is produced; however, when the process is out of control asindicated at B and C, defective is being produced. Therefore, assignable causes of

variation must be corrected as soon as they occur.


Case III: LU 6 . When the spread of the process or process capability is greater than

the difference between specifications, an undesirable situation exists. Figure 3-19

illustrates this case. Even though a natural pattern of variation is occurring, as shown bythe frequency distribution at A, some of the individual values are greater than the upper

specification and are less than the lower specification. This case presents the uniquesituation where the process is in control, but defective product is produced. In otherwords, the process is not capable of manufacturing a product that will meet the

specifications.



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One solution is to discuss with the product engineer the possibility of increasing the

difference between the upper and lower specifications. This solution may require reliability

studies with mating parts to determine if the product can function with increased specification

differences.

Another solution is to leave the process and the specifications alone and perform 100%

inspection to eliminate the defective parts. This is not an attractive solution, but it may be themost economical or only one.

A third possibility to change the process dispersion so that a more peaked distribution

occurs, as illustrated by frequency distribution B. to obtain such a substantial shift in the

standard deviation might require new material, a more experienced operator or retraining, a

new or overhauled machine, or possibly automatic in-process control.

Another solution is to shift the process average so that all of the defective product occurs

at one tail of the frequency distribution as indicated at C of Figure 3-19. To illustrate, assume

that a shaft is being ground to tight specifications. If too much metal is removed, the part is

scraped; if too little is removed, the part must be reworked. By shifting the process average

the amount of scarp is eliminated and the amount of rework is increased. A similar situation

exists for an internal member such as a hole or keyway except that scrap occurs above theupper specification and rework occurs below the lower specification. This type of solution is

feasible when the cost of the part is sufficient economically to justify the reworking

operation. Note that the crosshatched area at C is much more than that at A.

Example ProblemLocation pins for work holding devices are ground to a diameter of 12.50 mm (approximately

½ in.), with a tolerance of 0.05 mm. if the process is centered at 12.50 mm ( ) and the

dispersion is 0.02 mm ( ), what percent of the product must be scrapped and what percent

can be reworked? How can be process center be changed to eliminate the scrap? What is the

rework percentage?Solution

55.1205.050.1205.0 U mm

45.1205.050.1205.0 L mm

50.202.0

50.1245.12

t

X Z

From Table A of the Appendix for a Z value of -2.50: Area1= 0.0062 or 0.62% scrap


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Since the process is centered between the specifications and a symmetrical distribution is

assumed, the rework percentage will be equal to the scrap percentage of 0.62%. The second

part of the problem is solved using the following sketch:

If the amount of scrap is to be zero, then Area1=0. From Table A, the closest value to an

Area1 value of zero is 0.00017, which has a Z value of -3.59. Thus,

52.12,02.0

45.1259.3,

i

X Z mm

The percentage of rework is obtained by first determining Area3.

50.102.0

52.1255.12

i

X Z

The Table A, Area3 = 0.9332 and Area2 = AreaTotal- Area3 =1.0000-0.9332=0.0668 or 6.68%

The amount of rework is 6.68%, which, incidentally, is considerably more than the combined

rework and scrap percentage (1.24%) when the process is centered.

The preceding analysis of the process spread and the specifications was made utilizing an

upper and a lower specification. Many times there is only one specification and it may beeither upper or lower. A similar and much simpler analysis could be for a single specification

limit.


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PROCESS CAPABILITY

The true process capability cannot be determined until the X and R charts have achieved theoptimal quality improvement without a substantial investment for new equipment or

equipment modification. Process capability or spread of the process is equal to 6 standard

deviations, which is0

6 or 6 if the population standard deviation is known

In the example problem for the X and R charts, the quality improvement process began in

January with 038.00 . The process capability is for 6 = (6) (0.038) = 0.228 mm or 0.038

mm or 0.114mm. By July,0

=0.030, which gives a process capability of 0.180 mm or

0.090 mm. This is a 20% improvement in the process capability, which in most situations

would be sufficient to solve a quality problem.

It is frequently necessary to obtain the process capability by a quick method rather than by

using the X and R charts. The procedure is:


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1. Take 20 subgroups of size 4 for a total of 80 measurements. These subgroups should

be selected at random to minimize any bias.

2.

Calculate the sample standard deviation, s , for the each subgroup.

3. Calculate the average sample standard deviation, 20// sgss .4. Calculate the estimate of the population standard deviation.

40 cs

5. Process capability will be0

6 .

Remember that this technique does not give the true process capability and should be used

only if circumstances require its use.

Process capability and the tolerance are combined to form a capability index, defined as

06

LU C p

where pC = capability index

LU = upper specification – lower specification, or tolerance

06 = process capability

If the capability index is 1.00, we have the case II situation discussed in the preceding

section; if the ratio is greater than 1.00, we have the case I situation, which is desirable; and if

the ratio is less than 1.00, we have the case III situation, which is undesirable.

Example ProblemAssume that the specifications are 6.50 and 6.30 in the depth of the keyway problem.

Determine the capability index before and after improvement.

88.0)038.0(6

30.650.6

60

LU C p 11.1

)030.0(6

30.650.6

60

LU C p

In the example problem the improvement in quality resulted in a desirable capability index.

The minimum capability index is frequently established at 1.33. Below this value, design

engineers have to seek approval from manufacturing before the product can be released for

production.

In Chapter 1, quality was defined as conformance to specifications. Using the capability

index concept, we can measure quality provided the process is centered correctly. The larger

the capability index, the better the quality. We should strive to make the capability index as

large as possible. This is accomplished by having realistic specifications and continual

striving to improve the process capability.

Table A The equivalent Cp value corresponding to capability percentage.

Equivalent Cp Capability in percentage

Equivalent Cp Capability in percentage

0.50 86.64 0.86 99.00

0.62 93.50 0.91 99.35

0.68 96.00 1.00 99.73

0.75 97.50 1.33 99.994

0.81 98.50


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Table B Interpreting the process capability index.

Cp < 1 Not capable

Cp > 1 Capable at 3

Cp > 1.33 Capable at 4

Cp > 1.67 Capable at 5

Cp > 2 Capable at 6 Table C Six sigma value chart.

Sigma DPMO COPQ Capability

6 Sigma 3.4

Chapter 6 Process Capability Analysis

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