1. Chapter 6 – Part 4
Process Capability

2. Meaning of Process Capability
The capability of a process is the ability of the process to meet the specifications.
A process is capability of meeting the specification limits if at least 99.73% of the product falls within the specification limits.
This means that the fraction of product that falls outside the specification limits is no greater than , or that no more that 3 out of 1,000 units is “out of spec.”
Our method of computing process capability assumes that the process is normally distributed.

3. Control Limits vs. Spec. Limits
Control limits apply to sample means, not individual values.
Mean diameter of sample of 5 parts, X-bar
Spec limits apply to individual values
Diameter of an individual part, X

4. Control Limits vs. Spec. Limits
Sampling distribution, X-bar
Process distribution, X
Mean=
Target
USL
LSL
Lower control limit
Upper control limit

5. Requirements for Assessing Process Capability
To assess capability of a process, the process must be in statistical control.
That is, all special causes of variation must be removed prior to assessing capability.
Also, process performance characteristic (e.g., diameter, bake time) must be normally distributed.

6. Cp Index USL = upper specification limit
LSL = Lower specification limit

7. Cp Index We want the spread (variability) of the process to be as ???
If the spread of the process is very ????, the
capability of the process will be very ????

8. Cp Index Process distribution, X USL LSL
Width of spec limits = USL - LSL
Spread of Process = USL - LSL

9. Process is Barely Capable if Cp = 1
.9973
.00135
.00135 X
LSL
USL
Spread of process matches the width of specs.
99.73% of output is within the spec. limits.

10. Process Barely Capable if Cp = 1
If , what does this imply regarding the spec. limits?
Cp=1
LSL =
USL =

11. Process Barely Capable if Cp = 1

12. Process is Capable if Cp > 1
>.9973
<
< X
LSL
USL
Spread of process is less than the width of specs.
More than 99.73% of output is within the spec. limits.

13. Process is Not Capable if Cp < 1
< .9973
>
> X
LSL
USL
Spread of process is greater than the width of specs.
Less than 99.73% of output is within the spec. limits.

14. Estimating the Standard Deviation

15. Estimating the Standard Deviation

16. Sugar Example Ch. 6 - 3 Day Hour X1 X2 X3 R 1 10 am 17 13 6 36/3 =1211
1 pm 15 12 24
51/3 =17
4 pm 21
48/3 =16 9 2
42/3 =14 5 18
54/3 =18 10
45/3 =15 8
= 92/6
= 15.33
= 51/6
= 8.5 X

18. Capability of Sugar Process
USL = 20 grams
LSL = 10 grams

19. Capability of Sugar Process
Since Cp <1, the process is not capability of meeting the spec limits.
The fraction of defective drinks (drinks with either too much or not enough sugar) will exceed
That is, more than 3 out of every 1000 drinks produced can be expected to be too sweet or not sweet enough.
We now estimate the process fraction defective, p-bar.

20. Estimated Process Fraction Defective
What is the estimated process fraction defective -- the percentage of product out of spec?
p-bar = F1 + F2
LSL
USL F2 F1
Mean

21. Estimated Process Fraction Defective
We can then use Cp to determine the p-bar because there is a simple relationship between Cp and z:
z = 3Cp
(See last side for deviation of this result.)
Suppose, Cp =0.627
z = 3(0.627) =1.88

22. Estimated Process Fraction Defective
The z value tells us how many standard deviations the specification limits are away from the mean.
A z value of 1.88 indicates that the USL is 1.88 standard deviations above the mean.
The negative of z, -1.88, indicates that the LSL is 1.88 standard deviations below the mean.
We let
Area(z)
be the area under the standard normal curve between 0 and z.

23. Process Fraction Defective
Area(z) = Area(1.88) =
LSL
USL F2
z =1.88
F2 = % above USL = = .0301

24. z Table (Text, p. 652) z
.00
.01
.02 .
.08
.09
0.0
0.1
0.2
1.8
.4699

25. Process Fallout p-bar = 2[.5 – Area(z)] = F1 + F2
0.4699
LSL
USL F2 F1
z =1.88
p-bar = 2(.5 – .4699) = 2(.0301)=.0602

26. Process Fallout – Two Sided Spec.Cp
z = 3Cp
Fallout =
2[.5 – Area(z)]
Defect Rate in PPM (parts per million)
0.25
0.75
2[ ] = .4532
453,200 PPM
0.80
2.40
2[ ] = .0164
16,400 PPM
1.0 3
2[ ] = .0026
2,600 PPM
1.5
-4.5
From Excel
2[Area(-z)]=
2[ ] =
7 PPM

27. Recommended Minimum Cp
Process Cp
z = 3Cp
Fallout
PPM
Existing process
1.25
3.75
2[Area(-z)]=
2[ ] =
176.9
New process
1.45
4.35
2[ ] =
13.6

28. Recommended Minimum Cp
Process Cp
z = 3Cp
Fallout
PPM
Safety, existing process
1.45
4.35
2[Area(-z)]=
2[ ] =
13.6
Safety, new process
1.60
4.80
2[ ] =
1.6

29. Soft Drink Example Cp =0.33 z = 3Cp = 3(0.33) = 0.99
Area(z) = Area(0.99) =
p-bar = 2[.5 - Area(0.99)]
= 2[ ] =

30. Capability Index Based on Target
Limitation of Cp is that it assumes that the process is mean is on target.
Process Mean = Target Value = (LSL + USL)/2

31. CT Capability Index
With Cp, capability value is the same whether the process is centered on target or is way off.
Cp is not affected by location of mean relative to target.
We need capability index that accounts for location of the mean relative to the target as well as the variance.
CT is an index that accounts for the location of mean relative to target.

32. CT Capability Index

33. CT Capability Index If process is centered on target,
If process is off target,

34. Example of CT LSL = 10, USL = 20, estimated standard deviation =
5.0 and estimate process mean = Compute
CT.

35. CT Capability Index
If process mean is adjusted to target,

36. CT Capability Index Cp is the largest value that CT can equal.
Since Cp = 2.2 and CT = .44, the difference
is the maximum amount by which we can increase CT by adjusting the mean to the target value.

37. Conclusion?

38. Derivation of z = 3Cp