process capability - site.iugaza.edu.pssite.iugaza.edu.ps/.../02/dr....process_capability.pdf · 2...

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Islamic University, Gaza - Palestine

Department of Industrial Engineering

Statistical Quality Control

Presented by

Dr. Eng. Abed Schokry


Department of Industrial Engineering

Process Capability

Presented by

Dr. Eng. Abed Schokry


• Process capability refers to the ability of a process to produce products or provide services capable of meeting the specifications set by the customer or designer.

• Specification limits are set by management in response to customers’ expectations

• The upper specification limit (USL) is the largest value that can be obtained and still conform to customers’ expectations

• The lower specification limit (LSL) is the smallest value that is still conforming

Process Capability


• Tolerances or specifications

– Range of acceptable values established by engineering design or customer requirements

• Process variability

– Natural variability in a process

• Process capability

– Process variability relative to specification

Process Capability

2


Tolerance Limits vs. Process Capability

Actual Process Width

Specification Width

Specification Width

Actual Process Width


Process Capability

A capable process

PROCESS SPREAD

` SPECIFIED TOLERANCES

An incapable process

SPECIFIED TOLERANCES

` PROCESS SPREAD


• A process capability index is an aggregate measure of a process’s ability to meet specification limits

• The larger the value, the more capable a process is of meeting requirements

Process Capability


• Specification limits, the allowable spread of the individuals, are compared with the spread of the process to determine how capable the process is of meeting the specifications. Three different situations can exist when specifications and are compared:

(I) The process spread can be less than the spread of the specification limits;

(II) The process spread can be equal to the spread of the specification limits;

(III) The process spread can be greater than the spread of the specification limits.

Specification limits

3


• The control limits have been placed on the diagram, as well as the spread of the process averages (dotted line).

• The spread of the process individuals is shown by the solid line. As expected, the spread of the individual values is greater than the spread of the averages; however, the values are still within the specification limits.

• This allows for more room for process shifts while staying within the specifications.

• Notice that even if the process drifts out of control, the change must be dramatic before the parts are considered out of specification.

(I)The spread of the individuals is less than the spread of the specifications


• In this situation, is equal to the tolerance

• As long as the process remains in control and centered, with no change in process variation, the parts produced will be within specification.

• A shift in the process mean will result in the production of parts that are out of specification. An increase in the variation present in the process also creates an out-of-specification situation.

(II) The process spread can be equal to the spread of the specification limits


• Case III: The spread is greater than the tolerance spread.

• Even though the process is exhibiting only natural patterns of variation, it is incapable of meeting the specifications set by the customer.

• To correct this problem, management intervention will be necessary in order to change the process to decrease the variation.

• The capability of the process cannot be improved without changing the existing process.

(III) The spread is greater than the tolerance spread


• The capability index Cp is the ratio of tolerance (USL –LSL) and

Cp= USL - LSL

6s

6s

The capability index Cp

4


• Capability ratio Cr

Cr= 6

USL - LSL

s

Capability ratio Cr


• Cpk is the ratio that reflects how the process is performing in terms of a nominal, center, or target value:

Cpk

Z(min)

3

where Z(min) is the smaller of Z(USL) = USL - X

or Z(LSL) = X - LSL

s

s

Capability ratio Cpk


1. When Cp has a value of 1.0 or greater, the process is producing product capable of meeting specifications.

2. The Cp value does not reflect process centering.

3. When the process is centered, Cp = Cpk.

4. Cpk is always less than or equal to Cp.

5. When Cp is greater than or equal to 1.0 and Cpk has a value of 1.00 or more, it indicates the process is producing product that conforms to specifications.

The relationships between Cp and Cpk


6. When Cpk has a value less than 1.00, it indicates the process is producing product that does not conform to specifications.

7. A Cp value of less than 1.00 indicates that the process is not capable.

8. A Cpk value of zero indicates the process average is equal to one of the specification limits.

9. A negative Cpk value indicates that the average is outside the specification limits.

The relationships between Cp and Cpk (Cont.)

5


Meanings of Cpk Measures

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1


Limitations of Capability Indexes

1. Process may not be stable

2. Process output may not be normally distributed

3. Process not centered but Cp is used

http://elsmar.com/Cp_vs_Cpk.html


Estimating Process Capability

• Must first have an in-control process

• Estimate the percentage of product or service within

specification

• Assume the population of X values is approximately normally

distributed with mean estimated by and standard deviation

estimated by

X

2/ dR


Process Capability Ratio

Process capability ratio, Cp = specification widthprocess width

upper specification – lower specification6

Cp =

3

X-UTLor

3

LTLXmin=C pk

If the process is centered use Cp

If the process is not centered use Cpk

6


Cp Index

• A measure of potential process performance is the Cp

index

– Cp > 1 implies a process has the potential of having more than 99.73% of outcomes within specifications

spreadprocess

spreadionspecificat

)d/R(6

LSLUSLC

2

p


CPL and CPU

• To measure capability in terms of actual process performance:

– CPL (CPU) > 1 implies that the process mean is more than 3standard deviation away from the lower (upper) specification limit

)d/R(3

XUSLCPU

)d/R(3

LSLXCPL

2

2


CPL and CPU

• Used for one-sided specification limits

– Use CPU when a characteristic only has a USL

– Use CPL when a characteristic only has an LSL


Cpk Index

• The most commonly used capability index is the Cpk index

• Measures actual process performance for characteristics with two-sided specification limits

Cpk = min(CPL, CPU)

– Cpk = 1 indicates that the process average is 3 standard deviation away from the closest specification limit

– Larger Cpk indicates greater capability of meeting the requirements

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Process Capability & Tolerance

•When spec. established without knowing whether process capable of meeting it or not serious situations can result

•Process capable or not – actually looking at process spread, which is called process capability (6)

•Let’s define specification limit as tolerance (T) : T = USL –LSL

•3 types of situation can resultthe value of 6 < USL-LSLthe value of 6 = USL - LSLthe value of 6 > USL - LSL


Process Capability

LowerSpecification

UpperSpecification

A. Process variability matches specifications

LowerSpecification

UpperSpecification

B. Process variabilitywell within specifications

LowerSpecification

UpperSpecification

C. Process variability exceeds specifications

Process Capability


Processmean

Lowerspecification

Upperspecification

1350 ppm 1350 ppm

1.7 ppm 1.7 ppm

+/- 3 Sigma

+/- 6 Sigma

33 Sigma and Sigma and 66 Sigma QualitySigma Quality


Case I situation

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Case II situation


Case 3 situation


Both Cp and Cpk are identical because process mean is at the center of the specification spread

As the process mean starts to drift away from the center of the specification spread, value of Cpk starts getting smaller (although Cp does not change)


Process Capability (6) And Tolerance

• Cp - Capability IndexT = U-LCp = 1 Case II 6 = TCp > 1 Case I 6 < TCp < 1 Case III 6 > TUsually Cp = 1.33 (de facto std.)• Measure of process performance• Shortfall of Cp - measure

not in terms of nominal or target value >>> must use Cpk

Formulas

Cp = (T)/6

Cpk = 3

Z(min)

Z (USL) =σ

xUSL

LSLx

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Example

Determine Cp and Cpk for a

process with average 6.45,

= 0.030, having USL =

6.50 , LSL = 6.30 -- T = 0.2

SolutionCp= T/6= 0.2/6(0.03)=1.11Cpk = Z(min)/3Z(U) = (USL -x)/ = 6.50-

6.45)/0.03 = 1.67Z(L) = (x –LSL)/ = 6.45-6.30)/0.03 =

5.00Cpk = 1.67/3 = 0.56

Process NOT capable since not centered. Cp > 1 doesn’t mean capable. Have to check Cpk

UL T

6.30 6.506.45= x


Interpreting the Process Capability Index

Cpk < 1 Not Capable

Cpk > 1 Capable at 3

Cpk > 1.33 Capable at 4

Cpk > 1.67 Capable at 5

Cpk > 2 Capable at 6


You are the manager of a 500-room hotel. You have instituted a policy that all luggage deliveries must be completed within ten minutes or less. For seven days, you collect data on five deliveries per day. Compute an appropriate capability index for the delivery process.

Process Capability ExampleIslamic University, Gaza - Palestine

Process Capability: Hotel Example Solution

326.2d 894.3R 813.5X 5n 2

833672.)326.2/894.3(3

813.510

)d/R(3

XUSLCPU

2

Since there is only the upper specification limit, we need only to compute CPU. The capability index for the luggage delivery process is .8337, which is less than 1. The upper specification limit is less than 3standard deviation above the mean.

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Comments On Cp, Cpk

• Cp does not change when process center (avg.) changes• Cp = Cpk when process is centered• Cpk Cp always this situation• Cpk = 1.00 de facto standard• Cpk < 1.00 process producing rejects• Cp < 1.00 process not capable• Cpk = 0 process center is at one of spec. limit (U or L)• Cpk < 0 i.e. – value, avg. outside of limits


Process Capability: The Control Chart Method for Variables Data

1. Construct the control chart and remove all special causes.NOTE: special causes are “special” only in that they come and go, not because their impact is either “good” or “bad”.

2. Estimate the standard deviation. The approach used depends onwhether a R or S chart is used to monitor process variability.

= R / d2 = S / c4

Several capability indices are provided on the following slide.


Process Capability Indices: Variables Data

CP = (engineering tolerance)/6 = (USL – LSL) / 6

This index is generally used to evaluate machine capability.tolerance to the engineering requirements. Assuming that the process is (approximately) normally distributed and that the process average is centered between the specifications, an index value of “1” is considered to represent a “minimally capable” process. HOWEVER … allowing for a drift, a minimum value of 1.33 is ordinarily sought … bigger is better. A true “Six Sigma” process that allows for a 1.5 shift will have Cp = 2.



CR = 100*6 / (Engineering Tolerance) = 100* 6 /(USL –LSL)

This is called the “capability ration”. Effectively this is the reciprocal of Cp so that a value of less than 75% is generally needed and a Six Sigma process (with a 1.5 shift) will lead to a CR of 50%.

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CM = (engineering tolerance)/8 = (USL – LSL) / 8

This index is generally used to evaluate machine capability.Note … this is only MACHINE capability and NOT the capability of the full process. Given that there will be additional sources of variation (tooling, fixtures, materials, etc.) CM uses an 8 spread, rather than 6. For a machine to be used on a Six Sigma process, a 10 spread would be used.



ZU = (USL – X) / ZL = (X – LSL) /

Zmin = Minimum (ZL , ZU)

Cpk = Zmin / 3

This index DOES take into account how well or how poorly centered a process is. A value of at least +1 is required with a value of at least +1.33 being preferred.

Cp and Cpk are closely related. In some sense Cpk represents the current capability of the process whereas Cp represents the potential gain to be had from perfectly centering the process between specifications.


Limitations of Capability Indexes

1. Process may not be stable

2. Process output may not be normally distributed

3. Process not centered but Cp is used


Process Capability: Example

Assume that we have conducted a capability analysis using X-bar and R charts with subgroups of size n = 5. Also assume the process is in statistical control with an average of 0.99832 and an average range of 0.02205. A table of d2 values gives d2 = 2.326 (for n = 5). Suppose LSL = 0.9800 and USL = 1.0200

= R / d2 = 0.02205/2.326 = 0.00948

Cp = (1.0200 – 0.9800) / 6(.00948) = 0.703

CR = 100*(6*0.00948) / (1.0200 – 0.9800) = 142.2%

CM = (1.0200 – 0.9800) / (8*(0.00948)) = 0.527

T,ZL = (.99832 - .98000)/(.00948) = 1.9

T,ZU = (1.02000 – .99832)/(.00948) = 2.3 so that L, Zmin = 1.9

Cpk = Zmin / 3 = 1.9 / 3 = 0.63

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Process Capability: Interpretation

Cp = 0.703 … since this is less than 1, the process is not regarded as being capable.

CR = 142.2% implies that the “natural tolerance” consumes 142% of the specifications (not a good situation at all).

CM = 0.527 = Being less than 1.33, this implies that – if we were dealing with amachine, that it would be incapable of meeting requirements.

ZL = 1.9 … This should be at least +3 and this value indicates that approximately 2.9% of product will be undersized.

ZU = 2.3 should be at least +3 and this value indicates that approximately 1.1% of product will be oversized.

Cpk = 0.63 … since this is only slightly less that the value of Cp the indication is that there is little to be gained by centering and that the need is to reduce process variation.


The Cereal Box Example

• Consumer Reports has just published an article that shows that we frequently have less than 15 grams of cereal in a box.

• Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.

• Upper Tolerance Limit = 16 + 0.05(16) = 16.8 grams

• Lower Tolerance Limit = 16 – 0.05(16) = 15.2 grams

• We go out and buy 1,000 boxes of cereal and find that they weight an average of 15.875 grams with a standard deviation of 0.529 grams.


Cereal Box Process Capability

• Specification or Tolerance Limits

– Upper Spec = 16.8 grams,

– Lower Spec = 15.2 grams

• Observed Weight

– Mean = 15.875 grams, Std Dev = 0.529 grams

– What does a Cpk of 0.4253 mean?

• Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!

3

;3

XUTLLTLXMinCpk


Process Capability

• Tolerance (specification, design) Limits– bearing width 1.250 +- 0.005 cm

– LTL = 1.245 cm UTL = 1.255 cm

• Process Limits – The actual distribution from the process– Run the process to make 100 bearings, compute the mean

and std. dev. (and plot/graph the complete results)

– Suppose, mean = 1.250, std. dev = 0.002

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Design Specs: Bearing diameter 1.250 +- 0.005 cm’s

– LTL = cm’s inches UTL = 1.255 cm’s

• The actual distribution from the process mean = 1.250, s = 0.002– +- 3s limits 1.250 +- 3(0.002) [1.244, 1.256]



– Anew process, std. dev. = 0.00083



Process Capability Index, Cpk

A process has a mean of 45.5 and a standard deviation of 0.9. The product has a specification of 45.0 ± 3.0.

Find the Cpk .


Process Capability Index, Cpk

3

X-UTLor

3

LTLXmin=Cpk

= min { (45.5 – 42.0)/3(0.9) or (48.0-45.5)/3(0.9) }

= min { (3.5/2.7) or (2.5/2.7) }

= min { 1.30 or 0.93 } = 0.93 (Not capable!)

However, by adjusting the mean, the process can become capable.

Example problem solution:

14


Process Capability Indices

• Consider our bags of sugar:

m – 10 kg

LSL, USL – 9.5, 10.5 kg

m – 10.1 kg

s – 0.1 kg

67.1)1.0(6

5.95.10Cp

• The results look ok, but the results are misleading since Cp is target insensitive


Process Capability Example

• Specification– Nominal (target) dimension: 30 mm

– Tolerance: + 1 mm, - 0.5 mm

• Process standard deviation: 0.2531 – 30 30 – 29.5

cpu = ----------- = 1.33 cpl = ------------- = 0.67

3 (0.25) 3(0.25)

and therefore Cpk = 0.67


Process Capability Ratios

Centered process (special case):specification width

cp = ----------------------------process width

Upper Spec Limit– Lower Spec Limit= -----------------------------------------

6


Process Capability Example

• Specification– Nominal (target) dimension: 30 mm

– Tolerance: ± 1 mm

• Process standard deviation: 0.2531 – 29

cp = ----------- = 1.33

6 (0.25)

15


Process Capability

• Cp = (design tolerance width)/(process width) = (max-spec – min-spec)/ /6x

• Example:

– Plane is “on time” if it arrives between T – 15min and T + 15min.

– Design tolerance width is therefore 30 minutes

– x of arrival time is 12 min

– Cp = 30/6*12 = 30/72 = 0.42

• A “capable” process can still miss target if there is a shift in the mean.


Process Capability

• Motorola “Six Sigma” is defined as Cp = 2.0

– I.e., design tolerance width is +/- 6x or 12 x

3 3

process width

min acceptable max acceptableDesign tolerance width


Process Capability Requirements

• Process must be normally distributed• Process must be in control• Process capability result:

– < 1.00 = not capable– < 1.33 = capable, but not acceptable– > 1.33 = capable and acceptable (generally)– > 2.00 = capable and acceptable (6Σ)– > 5 or 10 is “overkill”, excessive resource use


Capability Versus Control

• Process capability (Cp or Cpk)– Measure of variability against design specifications– Specs set by customer or design engineer– Spec width: USL & LSL (or UTL & LTL)

• Statistical process control (SPC)– Measure of variability against control limits– Control limits calculated from sample data– UCL and LCL

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Capability Versus Control

Control

Capability

Capable

Not Capable

In Control Out of Control

IDEAL


Process Control vs. Capability

The difference between capability and stability (control)

A process is capable if individual products consistently meet specification

A process is stable (in control) only if common variation is present in the process


Example 1

MachineStandard Deviation

Machine Capability Cp

A 0.13 0.78 0.80/0.78 = 1.03

B 0.08 0.48 0.80/0.48 = 1.67

C 0.16 0.96 0.80/0.96 = 0.83

Cp > 1.33 is desirableCp = 1.00 process is just capableCp < 1.00 process is not capable


Improving Process Capability

• Simplify

• Standardize

• Mistake-proof

• Upgrade equipment

• Automate

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Relationship of limits, specifications, and distributions

Control Limits & Specifications


The control limits are established as a function of the average

Specifications are the permissible variation in the size of the part and are, therefore, for individual values

The specifications or tolerance limits are established by

design engineers to meet a particular function

Control Limits & Specifications


The process spread will be referred to as the process capability and is equal to 6σ

The difference between specifications is called the tolerance

When the tolerance is established by the design

engineer without regard to the spread of the process, undesirable situations can result



Three situations are possible:

Case I: When the process capability is less than the tolerance 6σ<USL-LSL

Case II: When the process capability is equal to the tolerance 6σ=USL-LSL

Case III: When the process capability is greater than the tolerance 6σ >USL-LSL

Process Capability & ToleranceProcess Capability & Tolerance

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Case I: When the process capability is less than the tolerance 6σ<USL-LSL


Case I 6σ<USL-LSL


Case II: When the process capability is less than the tolerance 6σ=USL-LSL


Case I 6σ=USL-LSL


Case III: When the process capability is less than the tolerance 6σ>USL-LSL


Case I 6σ>USL-LSL


The range over which the natural variation of a process occurs as determined by the system of common causes

Measured by the proportion of output that can be produced within design specifications

Process CapabilityProcess Capability

19


This following method of calculating the process capability assumes that the process is stable or in statistical control:

Take 25 (g) subgroups of size 4 for a total of 100 measurements Calculate the range, R, for each subgroup Calculate the average range, RBar= ΣR/g Calculate the estimate of the population standard deviation Process capability will equal 6σ0

Process Capability

0

2

R

d


The process capability can also be obtained by using the standard deviation:

Take 25 (g) subgroups of size 4 for a total of 100 measurements Calculate the sample standard deviation, s, for each subgroup Calculate the average sample standard deviation, sbar = Σs/g Calculate the estimate of the population standard deviation Process capability will equal 6σo

Process CapabilityProcess Capability

0

4

s

c


Process capability and tolerance are combined to form the capability index.

Capability Index

0

0

6

6

p

p

USL LSLC

where C capabilityindex

USL LSL tolerance

process capability


The capability index does not measure process performance in terms of the nominal or target value. This measure is accomplished by Cpk.

Capability Index

0

{( ) ( )

3

6

pk

p

Min USL X or X LSLC

where C capabilityindex

USL LSL tolerance

process capability

20


Cp = USL - LSL

6 ơ

(USL- ¯X), (¯X-LSL)} Cpk = min{

The Capability Index does not measureprocess performance in terms of the nominal or target

Capability IndexCapability Index

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