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Chapter 6 – Part 4. Process Capability. 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.

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chapter 6 part 4

Chapter 6 – Part 4

Process Capability

meaning of process capability
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 0.0027, 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.
control limits vs spec limits
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
control limits vs spec limits1

Mean=

Target

Control Limits vs. Spec. Limits

Samplingdistribution, X-bar

Processdistribution, X

USL

LSL

Lowercontrollimit

Uppercontrollimit

requirements for assessing process capability
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.
c p index
Cp Index

USL = upper specification limit

LSL = Lower specification limit

c p index1
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 ????
slide8

Cp Index

USL

LSL

Width of spec limits = USL - LSL

Processdistribution, X

Spread of Process = USL - LSL

slide9

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.

process barely capable if cp 1
Process Barely Capable if Cp = 1

If , what does this imply regarding the spec. limits?

Cp=1

LSL =

USL =

slide12

Process is Capable if Cp > 1

>.9973

< .00135

< .00135

X

LSL

USL

Spread of process is less than the width of specs.

More than 99.73% of output is within the spec. limits.

slide13

Process is Not Capable if Cp < 1

< .9973

> .00135

> .00135

X

LSL

USL

Spread of process is greater than the width of specs.

Less than 99.73% of output is within the spec. limits.

capability of sugar process
Capability of Sugar Process

USL = 20 grams

LSL = 10 grams

capability of sugar process1
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 .0027.
  • 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.
slide20

LSL

USL

F1

F2

Mean

Estimated Process Fraction Defective

  • What is the estimated process fraction defective -- the percentage of product out of spec?

p-bar = F1 + F2

estimated process fraction defective
Estimated Process Fraction Defective
  • We can then use Cp to determine the p-barbecause 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

estimated process fraction defective1
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.

slide23

Process Fraction Defective

Area(z) = Area(1.88) = 0.4699

LSL

USL

F2

0

z =1.88

F2 = % above USL = .5000 - 0.4699 = .0301

process fallout

LSL

USL

F1

F2

0

Process Fallout

p-bar =

2[.5 – Area(z)] = F1 + F2

0.4699

z =1.88

p-bar = 2(.5 – .4699) = 2(.0301)=.0602

slide29

Soft Drink Example

Cp =0.33

z = 3Cp = 3(0.33) = 0.99

Area(z) = Area(0.99) = 0.3389

p-bar = 2[.5 - Area(0.99)]

= 2[.5 - 0.3389]

= 0.3222

capability index based on target
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

c t capability index
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.
c t capability index2
CT Capability Index

If process is centered on target,

If process is off target,

example of c t
Example of CT

LSL = 10, USL = 20, estimated standard deviation =

5.0 and estimate process mean = 15.33. Compute

CT.

slide35

CT Capability Index

If process mean is adjusted to target,

c t capability index3
CT Capability Index
  • Cpis the largest value that CTcan equal.
  • Since Cp = 2.2 and CT = .44, the difference

is the maximum amount by which we can increase CTby adjusting the mean to the target value.