01 Process Capability

Design for Manufacturing and Assembly Process Capability Analysis

PROCESS CAPABILITY ANALYSIS PROCESS CAPABILITY (CP):

Process capability is the repeatability and consistency of a manufacturing process relative to the customer requirements in terms of specification limits of a product parameter. This measure is used to objectively measure the degree to which your process is or is not meeting the requirements.

Process capability compares the output of an in-control process to the specification limits by using capability indices. indices . The comparison is made by forming the ratio of the spread between the process specifications (the specification "width") to the spread of the process values, as measured by 6 process standard deviation units (the process "width"). Cp = (USL - LSL) / 6 sigma Cp<1 means the process variation exceeds specification, and a significant number of defects are being made. Cp=1 means that the process is just meeting specifications. A minimum of .3% defects will be made and more if the process is not centered. Cp>1 means that the process variation is less than the specification, however, defects might be made if the process is not centered on the target value.

1


While Cp relates the spread of the process relative to the specification width, it does not address how well the process average, X, is centered to the target value. Cp is often referred to as process "potential".

We define process capability analysis as an engineering study to estimate process capability. The estimate of process capability may be in the form of a probability distribution having a specified shape, center (mean), and spread (standard deviation). For example, we may determine that the process output is normally distributed with mean

= 1.0 cm and standard deviation

σ = 0.001cm.

in this sense, a process capability

analysis may be performed without regard to specifications on the quality characteristic.

2


While Cp relates the spread of the process relative to the specification width, it does not address how well the process average, X, is centered to the target value. Cp is often referred to as process "potential".

We define process capability analysis as an engineering study to estimate process capability. The estimate of process capability may be in the form of a probability distribution having a specified shape, center (mean), and spread (standard deviation). For example, we may determine that the process output is normally distributed with mean

= 1.0 cm and standard deviation

σ = 0.001cm.

in this sense, a process capability

analysis may be performed without regard to specifications on the quality characteristic.

2


PROCESS CAPABILITY INDICES (CPK): A capable process is one where almost all the measurements fall inside the specification limits. This can be represented pictorially by the plot below:

The C p, C pk , and C pm statistics assume that the population of data values is normally distributed. Assuming a two-sided specification, if and are the mean and standard deviation, respectively, of the normal data and USL, LSL, and T are the upper and lower specification limits and the target value, respectively, then the population capability indices are defined as follows:

The estimator for C pk can also be expressed as C pk = C p(1-k), where k is a scaled distance between the midpoint of the specification range, m, and the process mean, . Denote the midpoint of the specification range by m = (USL+LSL)/2. The distance between the process mean,

, and the optimum, which is m, is

. The scaled distance is

3

- m, where


(the absolute sign takes care of the case when estimated value, , we estimate by . Note that

). To determine the .

The estimator for the C p index, adjusted by the k factor, is

Since

, it follows that

.

To get an idea of the value of the C p statistic for varying process widths, consider the following plot

This can be expressed numerically by the table below:

where ppm = parts per million and ppb = parts per billion. Note that the reject figures are based on the assumption that the distribution is centered at Process Capability Ratio (Cp) and Associated

4

.Values of the


We have discussed the situation with two spec. limits, the USL and LSL. This is known as the bilateral or two-sided case. There are many cases where only the lower or upper specifications are used. Using one spec limit is called unilateral or one-sided. The corresponding capability indices are

where and are the process mean and standard deviation, respectively Estimators of C pu and C pl are obtained by replacing and by and s, respectively. The following relationship holds C ) /2. p = (C pu + C pl

This can be represented pictorially by

Note that we also can write: C pk = min {C pl , C pu }.

5


CAPABILITY INDEX EXAMPLE: For a certain process the USL = 20 and the LSL = 8. The observed process average, = 16, and the standard deviation, s = 2. From this we obtain

This means that the process is capable as long as it is located at the midpoint, m = (USL + LSL)/2 = 14. But it doesn't, since = 16. The factor is found by

and

We would like to have

at least 1.0, so this is not a good process. If possible,

reduce the variability or/and center the process. We can compute the

From this we see that the

and

, which is the smallest of the above indices, is 0.6667.

Note that the formula

is the algebraic equivalent of the min{

,

} definition.

PROCESS CAPABILITY METRICS: For processes that are in statistical control and that are normally distributed, we can do a Process Capability Analysis. Last month's e-zine contained an explanation of process capability and introduced one metric (Cp) to measure process capability. This month's e-zine introduces a second metric to measure process capability - Cpk.

6


Cpk

EXAMPLE FOR CPK :

7


Cp values are not the best indicators of process capability. As shown last month, Cp is the ratio of the engineering tolerance (USL - LSL) to the natural tolerance (6s). The value of Cp does not take into account where the process is centered. Just knowing that a process is capable (Cp > 1.0) does not ensure that all the product or service being received is within the specifications. A process can have a Cp > 1.0 and produce no product or service within specifications. In addition, Cp values can't be calculated for one- sided specifications. A better measure of process capability is Cpk. Cpk takes into account where the process is centered. The value of Cpk is the minimum of two process capability indices. One process capability is Cpu, which is the process capability based on the upper specification limit. The other is Cpl, which the process capability is based on the lower specification limit. Algebraically, Cpk is defined as shown in the figure.

Both Cpu and Cpl take into account where the process is centered. The value of Cpk is the difference between the process average and the nearest specification limit divided by three times the standard deviation. It should be noted that the standard deviation is the standard deviation based on a R or s chart - not the standard deviation of

the

individual

measurements.

Cpk values above 1.0 are desired. This means that essentially no product or service is being produced above USL or below LSL. The figure above shows how the Cpk values are calculated. If Cpk is less than 1.0, this means that there is some product being produced out of specification. If there is only one specification, the value of Cpk is either Cpu or Cpl, whichever is appropriate for the specification.

A bagging operation is designed to place 50 pounds of sand into each bag. The specifications for the operation are a minimum of 49.5 pounds and a maximum of 50.5 pounds. So, the lower specification limit (LSL) is 49.5. The upper specification limit (USL)

is

50.5 8

pounds.


The operation is being monitored using a Xbar-R chart with a subgroup size of 4. Each hour, four consecutive bags are weighed. The subgroup average and range are calculated and plotted on an Xbar-R chart. The control chart is shown in the figure and is in statistical control. The standard deviation (from the range chart data) is 0.212. The average bag weight is 50.05. The calculations for Cpk are shown below. Since Cpk is the minimum of Cpu and Cpl, Cpk = 0.71.

9


Definitions Mean ( x ): The arithmetic mean of a set of ‘n’ numbers is the sum of the numbers divided by ‘n’. Mean is expressed algebrically, X + X 2 + X 3 + ......... + X n x = 1 , n Where the symbol x represents the arithmetic mean. X 1 , X 2 , X 3 ,..... X n , are the n values of the variate X

x =

i.e.,

∑ x

n If X1 occurs f 1 times, X2 occurs f 2 times, etc and finally X n occurs f n times, then, n = f 1 + f 2 + f 3 + .......... f n x =

Then

f 1 X 1 + f 2 X 2 + f 3 X 3 + ......... + f n X n

f 1 + f 2 + f 3 + ......... + f n The mean is used to report average size, average yield, average percent defective etc.

Median: When all the observations are arranged in ascending or descending order, then the median is the magnitude of the middle case. n +1 If n is odd, Median = 2  n   n  If n is even, Median is average of   th and  + 1 th value.  2   2  Where n = No of observations.

Mode: The mode of a set of data is the value which occurs most frequently.

Range(R): In the control chart, the range is difference between the largest observed value and the smallest observed value.

Variance ( σ 2 ): It is defined as the sum of the squares of the deviations from the arithmetic mean divided by the number of observations ‘n’. Variance ( σ

2

( x )=

1

− x ) + ( x 2 − x ) + ( x n − x ) 2

2

2

n

Sample problems 10


Example 1: The no of orders received for a particular item on each day for five days are as follows. Calculate the mode and variance. 1, 2, 0, 3, 2 Solution: Mode = 2 (it occurs more than the other values) This can be put more succinctly using the summation notation as; 2 1 2 Variance σ = ∑ x − x n It is possible to rearrange this formula in a way which makes the calculation of the variance much easier in general. 2 1 2 Variance σ = ∑ x 2 − x , To calculate x ,we use n 1 + 2 + +0 + 3 + 2 Mean x = = 1.6 5 Using this formula, the variance for the data used above is calculated as follows: 1 2 2 Variance σ = (1 + 22 + 02 + 32 + 22 ) - 1.62 5 18 = - 2.56 5 σ 2 = 1.04

(

)

()

Standard deviation =

(σ ) 2

Standard deviation = (1.04) = 1.02

Example 2: Calculate mean, variance, standard deviation for the given order size data. Order size range 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90

Class mark(x) 5.5 15.5 25.5 35.5 45.5 55.5 65.5 75.5 85.5

Frequency (f)

fx

1 2 4 12 13 8 8 1 1

∑ f = 50

5.5 31.0 102.0 426.0 591.5 444.0 524.0 75.5 85.5

∑ fx = 2285.0

11

fx2

30.25 480.50 2601.00 15123.00 26913.25 24642.00 34322.00 5700.25 7310.25

∑ fx 2 = 117122.50

Design for Manufacturing and Assembly Process Capability Analysis Mean x

=

2285

50 = 45.7

Variance σ = 2

117122.5

50 = 253.96

Standard deviation =

- (45.7) 2

253.96

= 15.9

Skewness: The curve, which does not follow the shape of the normal curve. These generally represent a purely temporary process condition, and serve as a guide to detecting the presence of some unusual factor like defective material, or abnormal machining conditions. (e.g.) tool chatter, tool vibration, etc. These curves are like normal curves in that the frequencies decrease continuously from the centre to extreme values, but unlike the normal curve they are not symmetrical.

Figure 1 Skewed curves Their extreme values occur more frequently in one direction from the centre than in the other. They appear like “disturbed normal” curves and hence are called “skewed curves”. The normal distribution is the most commonly occurring symmetrical frequency distribution. Positive skewness is also quite common, as for instance the shape of the distribution of personal incomes. Another example is the distribution of the time intervals between randomly occurring events, such as the arrival of customers at the ends of a queue. Negative skewness is less common, but occurs, for instance, in the distribution of times to failure of certain types of equipment. Coefficient of Skewness=

Mean − Mode

σ Several measures of skewness have been proposed, but are rarely used in practice. The simplest way of describing skewness is to quote the mean, the median, and,

12

Design for Manufacturing and Assembly Process Capability Analysis where possible the mode. For symmetrical distributions, these three measures will approximately coincide. For positively skewed distributions, the mode will be less than the median, which will in turn be less than the mean. This is very noticeable for the distribution of personal incomes. For negatively skewed distributions, these three measures will be in the reverse order. The differences between the measures give some indication of the extent of the skewness. When the distribution is moderately, there is an approximate relationship between the three measures, expressed as Mean-Mode=3(Mean-Median).

Measure Of Skewness: (A) Absolute Skeweness (a) Absolute Sk = Mean-Mode (when mode is not ill-defined) (b) Absolute Sk = 3(Mean-Median) (B) Relative Skewness (a) Karl Pearson’s coefficient of Skewness. Coefficient of Skewness=

Mean − Mode σ

When mode is ill-defined Coefficient of Skewness=

3( Mean − Mode) σ

(b) Measure of Skewness based on moments : With the help of moments Skewness can be determined, Karl Pearson suggested β 1 as Measure of Skewness.

β 1 =

For a symmetrical distribution

µ 3

2

µ 2

3

β 1 = 0.

Moments (i) Moments about Mean

µ 1 = µ 2

∑ ( X − X ) N

∑ ( X − X ) =

2

N

13

2

= σ or σ =

µ 2

Design for Manufacturing and Assembly Process Capability Analysis µ 3

∑ ( X − X ) =

µ 4

∑ ( X − X ) =

3

N

4

N

In case of frequency distribution µ 1 =

∑ f ( X − X ) =

µ 2

∑ f ( X − X )

=0

N

2 2

= σ , etc.

N (ii) Moments about arbitrary origin A ( X − A) 1 µ 1 = N

∑

µ 2 = 1

∑ ( X − A)

2

N

∑ ( X − A)

3

µ 3 = 1

µ 4 = 1

N

∑ ( X − A)

4

N

In a frequency distribution the moments about an arbitrary origin will be calculated as follows:

µ 1 = 1

∑ fd × i

or

N

∑ fd =

×i

2

∑ fd =

×i

3

∑ fd =

×i

4

2

µ 2

1

N

or

µ 3

N

or

4

µ 4

1

N

N

∑ f ( X − A)

2

×i2

N

∑ f ( X − A)

3

3

1

∑ f ( X − A) × i

or

× i3

N

∑ f ( X − A) N

4

×i4

 X − A  .  i 

Where ‘ i ’ is the class interval and l= 

Order to simplify calculations the moments are first calculated about an origin A. They can then be converted with the help of the following relationships to obtain moments about mean. µ1= µ11-µ11 =0 µ2 = µ21-( µ11)2 µ3 = µ31-(3µ11 µ21)+2 ( µ11)3 µ4

=

µ41-(4µ11 µ31)+6 ( µ11)2 µ21 -3( µ11)4

14


Example 3: Calculate any measure of skewness from the following data: X f

0 12

1 27

2 29

3 19

4 8

5 4

6 1

7 0

Solution: Since the question is to calculate any measure of skewness, we should prefer Karl Pearson’s coefficient of skewness because it is considered to be the best measure for calculating skewness. The formula is: Coefficient of Skewness=

Mean − Mode σ

Hence for calculating skewness we have to determine the values of mean, mode and standard deviation. Calculation of Coefficient of Skewness

x

X

f

0 1 2 3 4 5 6 7 N=100

12 27 29 19 8 4 1 0

= A+

x-2 d -2 -1 0 +1 +2 +3 +4 +5

fd

fd2

-24 -27 0 +19 +16 +12 +4 0

48 27 0 12 32 36 16 0

∑ fd = 0

∑ fd = 2 + (0 / 100) = 2 N

  ∑ fd 2   ∑ fd  2   − Standard deviation σ =     N   N       178   0  2   = 1.78 = 1.334 σ =   −   100   100     15

∑ fd = 178 2

Design for Manufacturing and Assembly Process Capability Analysis Mode: Since the highest frequency is 29, by inspection the mode is the value corresponding to the frequency 29 i.e. 2.

x =2 , Mo =2, σ =1.334 Substituting these values in the formula, Coefficient of Skewness =

Mean − Mode σ

=

2−2 1.334

= 0.

Example: 4 Calculate Karl Pearson’s co-efficient of skewness from the following data: Size

1

2

3

4

5

6

7

Frequency

10

18

30

25

12

3

2

Solution: Calculation of Karl Pearson’s Coefficient of Skewness Size x 1 2 3 4 5 6 7

Frequency f 10 18 30 25 12 3 2 N=100

x-4 d -3 -2 -1 0 +1 +2 +3

fd

fd2

-30 -36 -30 0 +12 +6 +6

90 72 30 0 12 12 18

∑ fd = −72

∑ fd = 234 2

Coefficient of Skewness= Mean − Mode Mean: x = A+

∑ fd N

σ = 4 - (72/100) =3.28

  ∑ fd 2   ∑ fd  2   − Standard deviation σ =     N   N     σ =

  232   − 72  2   = −   100   100    

2 .28856

= 1.152 Mode: Since the maximum frequency is 30, by inspection the mode is the value corresponding to the frequency 30 i.e. 3.

16


x =3.28 ,

Mo =3, σ =1.518

Substituting these values in the formula, Coefficient of Skewness =

Mean − Mode σ

= 3.28 − 3 = 0.184

1.518

Kurtosis: The fourth momentum will provide a numerical value associated with the peakedness or flatness of the data as it is a distributed about the mean also known as “kurtosis”. The following equation incorporates the fourth moment about the mean and the fourth power of the samples standard deviation to measure kurtosis.

µ 4 =

1

n

x ( ∑ n

− x )

4

i

i =1

Kurtosis =

µ

4

s

4

The following equation is commonly used to calculate the zero based kurtosis in statistical analysis computer programming. Zero-based kurtosis=

4

 x i − x    −3 ∑  n i =1  s  1

n

Note that the value of 3 is subtracted from the kurtosis value. This force the value to be zero based, as opposed to be centered around the number 3. The common approach to quantity kurtosis is that the normal peak distribution is centered about the value 3. As the kurtosis deviates above or below 3. The peakedness or flatness begins to take a numerical significance as described below.

Mesokurtic: They are three general distributions types used to define nature of kurtosis. The first is mesokurtic distribution as shown in the Figure 2. In it the data is normal distributed about the mean the kurtosis will be equal to 3.

Figure 2 Mesokurtic distribution 17


Platykurtic: The second is platykurtic distribution, shown in figure 3. In it the data is dispersed bout the mean in a manner that is flat in nature: the kurtosis will be less than 3.

Figure 3 Platykurtic distribution

Leptokurtic: The third is leptokurtic distribution, shown in figure 4. In the data is dispersed about the mean in a manner that is very peaked in nature; the kurtosis will be greater than 3.

Figure 4 Leptokurtic distribution

18


Measures Of Kurtosis: Kurtosis are measured by the coefficient µ β 2 = 42 µ 2

γ 2 = β 2 − 3

or

For normal distribution β 2 =3. If β 2 is more than 3 the curve is leptokurtic and if it less than 3 the curve is platykurtic.

Example: 5 Calculate first four moments from the following data: Also calculate the values of β 1 and β 2 and comment on the nature of the distribution X Y

0 5

1 10

2 15

3 20

4 25

5 20

6 15

7 10

8 5

Solution: Calculation of moments X

f

fX

(X-4)

f(X-4)

0 1 2 3 4 5 6 7 8

5 10 15 20 25 20 15 10 5

0 10 30 60 100 100 90 70 40

-4 -3 -2 -1 0 +1 +2 +3 +4

-20 -30 -30 -20 0 +20 +30 +30 +20

N=125

∑fX = 500

f(x-4)2 80 90 60 20 0 20 60 90 80

∑f(X-4)

∑ f(x-4) 2

=0

=500

x =

∑ fX = 500 = 4

µ 1 =

N

125

∑ f ( X − X ) N 19

f(X-4)3

f(X-4)4

-320 -270 -120 -20 0 +20 +120 +270 +320

1280 810 240 20 0 2 40 10 280

∑ f(X-4)3

∑ f(X-4) 4

=0

=4700


∑ f ( X − X ) = 0, N = 125 µ 1 =

0 125

=0

µ 2

∑ f ( X − X ) =

µ 3

3 f ( X − X ) ∑ = =

µ 4

f ( X − X ) 4 4700 ∑ = = = 37.6

N

N

β 1 =

N

µ 3

2

µ 2

3

=

02 43

2

=

500 125 0 125

=4 =0

125

=0

Since β 1 is zero, the distribution is symmetrical

β 2 =

µ 4

=

37.6

= 2.35 16 µ 2 Since β 2 is less than 3, the distribution is platykurtic. 2

Example: 6 Using moments, calculate a measure of relative skewness and a measure of relative kurtosis for the following distribution and comment on the result obtained:

Weekly Wages (Rs) 70 but below 90 “ 110 “ 130 “ 150 “

No. of Workers

90 110 130 150 170

8 11 18 9 4

20

Design for Manufacturing and Assembly Process Capability Analysis Solution:

Weekly wages (Rs) 70-90 90-110 110-130 130-150 150-170

f

m.p

d

fd

fd2

fd3

fd4

8 11 18 9 4

80 100 120 140 160

-2 -1 0 1 2

-16 -11 0 9 8

32 11 0 9 16

-64 -11 0 9 32

128 11 0 9 64

∑fd =-10

∑fd2=68

∑ fd3=-34

∑fd4=212

N=50

µ 1 = 1

µ 2 = 1

µ 3 = 1

µ 4 = 1

∑ fd × i = − 10 × 20 = −4 N fd 2

∑

N fd 3

∑

N fd 4

∑

N

50

× i2 = × i3 = × i4 =

68 50

× 400 = 544

− 34 50 212 50

× 8000 = −5440 × 160000 = 678400

Moment about Mean µ2 = µ21-( µ11)2 =544-(-4) 2=528 µ3 = µ31-3(µ11 µ21)+2 ( µ11)3 = -5440-3(-4) (544) +2(-4) 3 = 960. µ4

=

µ41-4(µ11 µ31)+6 ( µ11)2 µ21 -3( µ11)4

= 674800-4(-4) (-5440) +6(-4) 2(544)-3(-4) 4 = 642816

21

Design for Manufacturing and Assembly Process Capability Analysis Skewness= β 1 =

β 2 =

µ 4 µ 2

2

=

µ 3

2

µ 2

3

960 2

=

642816 278784

528 3

= 0.08

= 2.306

Since β 2 is less than 3, the distributions platykurtic.

Example: 7 Calculate coefficient of skewness by Karl Pearson’s method and the values of β 1 and β 2 from the following data: Profits(Rs. Lakhs)

10-20 0 18

No. of companies

20-30

30-40

40-50

50-60

20

30

22

10

Solution: Calculation of Karl Pearson’s Coefficient of Skewness β 1 and β 2 Profits No. m.p (m(Rs. of 45)/10 fd fd2 fd3 Lakhs) cos m d f 10-20 18 15 -2 -36 72 -144 20-30 20 25 -1 -20 20 -20 30-40 30 35 0 0 0 0 40-50 22 45 +1 +22 22 +22 50-60 10 55 +2 +20 40 +80 2 N=100 fd = −14 fd = 154 fd 3 = −62

∑

Karl Pearson’s Coefficient of Skewness=

Mean:

x

= A+

σ

N

= 35+

Mode: Mode= L +

ean −

∑ fd × i

A=35,

x

∑

∑ fd =-14 , N=100,

14 100

i=10

× 10 = 35 − 1.4 = 33.6

∆1 ×i ∆1 − ∆ 2

By inspection mode lies in the class 30-40

22

∑

ode

fd4

288 20 0 22 160

∑ fd

4

= 490

Design for Manufacturing and Assembly Process Capability Analysis L=30 ∆ 1 = 30 − 20 = 10, ∆ 2 = 30 − 22 = 8, i = 10 Mode= 30+

10 10 + 8

× 10 = 30 + 5.56 = 35.56

  ∑ fd 2   ∑ fd  2  − ×i Standard deviation σ =     N   N       154   − 14  2  σ =   − × 10  = 12.33     100   100     Karl Pearson’s Coefficient of Skewness= Calculation of β 1 =

µ 3

33.6 − 35.56 12.33

=-0.159

2 3

µ 2 We will have to calculate moments µ 1 = 1

µ 2 = 1

µ 3 = 1

µ 4 = 1

∑ fd × i = − 14 × 10 = −1.4 N fd 2

∑

N fd 3

∑

N fd 4

∑

N

100

× i2 = × i3 = × i4 =

154 100

− 62 110 490 100

× 100 = 154 × 1000 = −620 × 10000 = 49000

µ2 = µ21-( µ11)2 =154-(-1.4) 2=152.04 µ3 = µ31-3(µ11 µ21)+2 ( µ11)3 = -620-3(-1.4) (154) +2(-1.4) 3 = 20.32 µ4

=

µ41-4(µ11 µ31)+6 ( µ11)2 µ21 -3( µ11)4

= 49000-{4(-1.4) (-620)} +{6(-1.4) 2(154)-{3(-1.4) 4 } = 47327.516

23

Design for Manufacturing and Assembly Process Capability Analysis β 1 = β 2 =

µ 3

2

µ 2

2

µ 4 µ 2

2

454.54 (21.32 )2 = = = 0.00013 (152.04 )2 3514581.1

=

47327.516

(152.04 )2

= 2.047

Process Capability Analysis: Statistical techniques can be helpful throughout the product cycle, including development activities prior to manufacturing, in quantifying process variability, in analyzing this variability relative to product requirements or specifications, and in assisting development and manufacturing in eliminating or greatly reducing this variability. This general activity is called process capability analysis. Product capability refers to the uniformity of the process. Obviously, the variability in the process is a measure of the uniformity of output. There are two ways to think of this variability: 1. The natural or inherent variability at a specified time; that is, “Instantaneous” variability. 2. The variability over time. , We present methods for investigating and assessing both aspects of process capability. It is customary to take the 6-sigma spread in the distribution of the product quality characteristic as a measure of process capability. Figure 5 shows a process for which the quality characteristic has a normal distribution with mean µ and standard deviation σ . The upper and lower “natural tolerance limits” (UNTL & LNTL) of the process fall at µ +3 and0 µ −3 , respectively. That is,

UNTL = LNTL =

+ 3σ − 3σ

For a normal distribution, the natural tolerance limits include 99.73% of the variable, or put another way, only 0.27% of the process output will fall outside the natural tolerance limits. Two points should be remembered: 1.0.27% outside the natural tolerances sounds small, but this corresponds to 2700 nonconforming parts per million. 2. If the distribution of process output is nonnormal, then the percentage of output falling outside µ ±3 may differ considerably from 0.27%. We define process capability analysis as an engineering study to estimate process capability. The estimate of process capability may be in the form of a probability distribution having a specified shape, center (mean), and spread (standard deviation). For example, we may determine that the process output is normally distributed with mean = 1.0 cm and standard deviation σ = 0.001 cm. in this sense, a process capability analysis may be performed without regard to specifications on the quality characteristic. Alternatively, we may express process capability as a percentage outside of specifications. However, specifications are not necessary to perform a process capability analysis.

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Figure 5 Upper and Lower natural tolerance limits in the normal distribution. A process capability study usually measures functional parameters on the product, not the process itself. When the analyst can directly observe the process and can control or monitor the data-collection activity, the study is a true process capability study, because by controlling the data collection and knowing the time sequence of the data, interferences can be made about the stability of the process over time. However, when we have available only sample units of products, perhaps supplied by the vendor or obtained via receiving inspection, and there is no direct observation of the process or time history of production, then the study is more properly called product characterization. In a characteristic or the process yield (fraction conforming to specifications); we can say nothing about the dynamic behavior of the process or its state of statistical control. Process capability analysis is a vital part of an overall quality-improvement program. Among the major uses of data from a process capability analysis are the following:

1. Predicting how well the process will hold the tolerances. 2. Assisting product developers/designers in selecting or modifying a process. 3. Assisting in establishing an interval between sampling for process monitoring. 4. Specifying performance requirements for few equipment. 5. Selecting between competing vendors. 6. Planning the sequence of production process when there is an interactive effect of process on tolerances. 7. Reducing the variability in a manufacturing process. Thus, process capability analysis is a technique that has application in many segments of the product cycle, including product and process design, vendor sourcing, production or manufacturing planning, and manufacturing. Three primary techniques are used in process capability analysis: histograms or probability plots, control charts, and designed experiments.

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Six-Sigma: Sigma ( σ ) is a character of the Greek alphabet which is used in mathematical statistics to define standard deviation. The standard deviation indicates how tightly all the various examples are clustered around the mean in a set of data. Six Sigma is a business method for improving quality by removing defects and their causes in business process activities. It concentrates on those outputs which are important to customers. The method uses various statistical tools to measure business processes. In technical terms, Six Sigma means that there are 3.4 defects per million events. The main goal is continuous improvement. Six Sigma is carried out as projects. Most common type is the DMAIC method (Define, Measure, Analyze, Improve, and Control). First, the project and the process to be improved are defined after which the performance of the process is measured. The data is then analyzed and bottle-necks and problems identified. After analysis, improvement program is defined and defects removed. This development program is controlled by a management group. After DMAIC circle it is time to define a new project.

Example: 7 This sample example at GE, illustrates how the concept of Six Sigma affects different people. Average Vs Variation Customer expectations: 8 day order to Delay Cycle.

Internal Look Existing Process Delay Cycle (days) 20 15 30 10 5

After conventional improvements (days) 17 2 5 12 4

16 Days (Average)

8Days (Average)

“Internal Calibration “= 16 – 8 = 8 Therefore improvement is 50%

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Design for Manufacturing and Assembly Process Capability Analysis GE employees claimed that they had achieved Six Sigma capability after improving the delivery time for a medical product by 50% (brining it from an average of 16 days to average of 8 days).But this effect was not reflected on the customer’s side as they were still getting their products delivered at random as seen from the Figure 6 CUSTOMER LOOK

Figure 6 From the above Figure Customer feels no change And once the feedback from the customer was heard, they modified the process to reflect Six Sigma delivery for the customer which resulted in the following: 6Sigma Internal Process 7 9 9 8 7 8 Days (Average) Here the internal look is same. But the customer feels Six Sigma (Figure 7). CUSTOMER LOOK

Figure 7

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Process Capability Ratios: Use and interpretation of Cp It is frequently convenient to have a sample, quantitative way to express process capability.

Cp =

USL − LSL 6 σ

-----------------------------------------

(2-1)

Where USL and LSL are the upper and lower specification limits, respectively. Usually, the process standard deviation σ is unknown and must be replaced by an estimate σ . To estimate σ we typically use either the sample standard deviation S or

R /d2 (when variables control charts are used in the capability study). This results in an estimate of the Cp, say USL − LSL Cp = --------------------------------------------- (2-2) 6 σ To illustrate the calculation of the Cp, The specifications on piston-ring diameter are USL=74.05mm and LSL=73.95mm, and σ =0.0099. Thus, our estimate of the Cp is USL − LSL Cp = 6 σ 74 . 05 − 73 . 95

=

6 ( 0 . 0099 ) = 1 . 68

We assumed that piston-ring diameter is approximately normally distributed and the cumulative normal distribution table in the appendix was used to estimate that the process produces approximately 20PPM (Parts Per Million) defective. The Cp in equation (2-1) has a useful practical interpretation, namely

 1 P =   Cp

 100 

----------------------------------------------

(2-3)

Is the percentage of specification band used up by the process. The piston-ring process uses  1  P =   100 1 . 68   = 59 . 5 percent of the specification band. Equation (2-1) and (2-2) assume that the process has both upper and lower specification limits. For one-sided specifications, we define the Cp as follows.

C pu =

USL − µ 3σ

(upper specification only)---

-------------- (2-4) 28


C pl =

µ − LSL 3σ

(lower specification only) --

--------------- (2-5) Estimate CpU and Cp L would be obtained by replacing µ and σ in equation (2-4) and (2-5) by estimate µ and σ , respectively.

The process capability ratio is a measure of the ability of the process to manufacture product that means specification table 2.1 shows several values of Cp along with the associated values of process fallout, expressed in defective are non-conforming parts per million. This process fallout were calculated assuming a normal distribution of the quality characteristics, and the case of two sides specification, assuming the process mean is centered between the upper and lower specification limits. These assumptions are essential to the accuracy of the reported numbers, and if they are not true,

Table 2.1 Values of the Process Capability Ratio (Cp) and Associated Process Fallout for a normal distribution process (in defective PPM) A process fallout(in defective PPM) Cp One side Specifications Two-sided specifications 0.25 226,628 453,255 050 66,807 133,614 0.60 35,931 71,861 0.70 17,865 35,729 0.80 8,198 16,395 0.90 3,467 6,934 1.00 1,350 2,700 1.10 484 967 1.20 159 318 1.30 48 96 1.40 14 27 1.50 4 7 1.60 1 2 1.70 0.17 0.34 1.80 0.03 0.06 2.00 0.0009 0.0018 Then the table is invalid. To illustrate the use of table, notice, that the Cp of one implies a fallout rate of 2700 PPM of two sides specifications, while the Cp of 1.5 implies fallout rate of 4 PPM of one side specification.

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Design for Manufacturing and Assembly Process Capability Analysis Table2.2 Recommended Minimum Values of the Process Capability Ratio (Cp) Two-sided specifications Existing processes New processes Safety, strength, or critical parameter, existing process Safety, strength, or critical parameter, new process

One-sided specifications

1.33 1.50

1.25 1.45

1.50

1.45

1.67

1.60

Table2.2 represents some recommended guidelines for minimum values of Cp the bottle strength characteristics a parameter closely related to the safety of product, bottles with inadequate pressure strength may fail and injury customers. This implies that the Cp should be atleast 1.45 perhaps one way the Cp could be improved would be increasing the mean strength of the bottles, say by pouring more glass in the mould. We point out that the values in the table 2.2 are only recommended minimum. In recent years, many companies have adopted criteria for evaluating their processes that include process capability objectives that are more stringent that those of table2.2. For example, Motorola’s “six-sigma” program essentially requires that when the process mean in control, it will not be closer that six standard deviations from the nearest specification limit. This, in effect, requires that the process capability ratio will be least 2.0. Within Motorola, this has become a corporate quality objective. Many other organizations, including their suppliers and customers, have adopted similar criteria.

Process Capability Ratio For An Off-Center Process The process capability ratio (Cp) does not take into account where the process mean is located relative to the specifications. Cp simply measures the spread of the specifications relative to the 6-sigma spread in the process. For example the top two normal distributions in figure 2.8 both have Cp =2.0, but the process in panel (b) of the figure clearly has lower capability than the process in panel (a) because it is not operating at the midpoint of the interval between the specifications.

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Figure 8 Relationship of Cp and Cpk

This situation may be more accurately reflected by defining a new process capability ratio that takes process centering into account. This quantity is C pk = min( C pu , C pl ) ---------------------------------------------- (2-6)

Notice that Cpk is just the one-sided Cp for the specification limit nearest to the process average. For the process shown in figure 8a, we would have

C pk = min( C pu , C pl ) USL − µ µ − LSL , C pl = ) 3 σ 3σ 62 − 53 53 − 38 = min( C pu = 1 . 5 , C pl = = 2 .5 ) 3( 2 ) 3( 2 ) Generally, if Cp= Cp k, the process is centered at the midpoint of the specifications, and when Cpk < Cp the process is off-center.

= min( C pu =

The magnitude of Cp k relative to Cp is a direct measure of how off-center the process is operating. Several commonly encountered cases are illustrated in figure 2.8. Note in panel (c) of figure 2.8 that Cp k =1.0 while Cp=2.0. One can use table 2.1 to get a

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Design for Manufacturing and Assembly Process Capability Analysis quick estimate of potential improvement that would be possible by centering the process. If we take Cp =1.0 in table 2.1 and read the fallout from the one-sided specifications column, we can estimate the estimate the actual fallout as 1350 PPM. However, if we can center the process, then Cp=2.0 can be achieved, and table 2.1 (using Cp=2.0 and two sided specifications) suggests that the potential fallout is 0.0018 PPM, an improvement of several orders of magnitude in process performance. Thus, we usually say that PCR measures potential capability in the process, while Cpk measures actual capability. Panel (d) of figure 8 illustrates the case in which the process mean is exactly equal to one of the specification limits, leading to Cp k = 0. As panel (e) illustrates, when Cp k < 0 the implication is that the process mean lies outside the specifications. Clearly, if Cpk < -1, the entire process lies outside the specification limits. Some authors define Cpk to be nonnegative, so that values less than zero are defined as zero. Many quality engineering authorities have advised against the routine use of process capability ratios such as Cp and Cp k (or the others discussed later in this section) on the grounds that they are an oversimplification of a complex phenomenon. Certainly, any statistic that combines information about both location (the mean and process centering) and variability, and which requires the assumption of normality for its meaning full interpolation is likely to be misused (or abused). Furthermore, as we will see, point estimates of process capability ratios are virtually useless if they are computed from small samples. Clearly, these ratios need to be used and interpreted very carefully.

Manufacturing Process Capability Metrics: Tolerances are always related to manufacturing processes or to materials used in the manufacture of a product, and they must be designed in conjunction with the application of the specific manufacturing process. If a tolerance band is determined without considering a manufacturing process, there is great risk in having a mismatch between the required tolerance and the capability of a given process- when the engineer finally gets around to selecting one. It is a fundamental precept in concurrent engineering to develop technology concepts or product-design concepts in simultaneously with the necessary manufacturing processes to support the timely and economic commercialization of the desired product. Often during technology development it is necessary to invent and co develop the manufacturing technology required to make the product. It is unwise to wait until the tolerance design phase of a product-commercialization process to select or optimize a manufacturing process. Capable manufacturing processes must be aligned with the product concept as early as possible. Only in this way will there be enough time develops necessary relationship between tolerances and manufacturing processes. It is also essential to perform manufacturing-process parameter optimization just as one would for design-component parameters. The design engineer and the manufacturing engineer have to define a common metric that quantifies the relationship that exists between the nominal design specifications, their tolerances, and the variability associated with the measurable output from the manufacturing process. The manufacturing engineer must also provide tolerances on the manufacturing process parameters and raw materials to help the team stay well within the tolerances assigned to the component being

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01 Process Capability

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