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Six Sigma Black Belt –
Study Guides
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Process Capability Analysis
Causes of variation
• No process can be variation free. There will be some sort of variation in a process. Major causes of variation in a process can be classified into two categories. They are:
Common (chance) causes of variation
• Inherent or natural in a process • Small in magnitude • Difficult to identify or eliminate from the process
Special causes of variation
• Variation due to some special causes • Large in magnitude • Easy to identify and eliminate from the process
Stable process
• A process is said to be a stable process if there exists no special causes of variation. Hence, a stable process runs under common causes only.
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Process Capability Analysis
Process Capability study:
• Process capability study gives numerical assessment of the capability of the process. It compares the voice of the customer with the voice of the process
The different steps involved in the capability study are
Step 1:
Check whether the process is stable – Stability of the process can be checked by using control charts.
Step 2:
Check whether the process data is taken from a normal distribution – This can be done by constructing a histogram using the original reading from the control chart.
Different techniques should be adopted to find the process capability in the case of normal and non-normal data
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Process Capability Analysis
Determined by the Customer
Determined by the Process
LSL USL
LCL TARGET UCL
-3σ +3σ VOP = UCL – LCL = 6σ LSL – Lower Spec Limit
VOC = USL - LSL USL – Upper Spec Limit
LCL – Lower Control Limit
GOOD CAPABILITY UCL – Upper Control Limit
LSL USL
LCL TARGET UCL
-3σ +3σ VOP = UCL – LCL = 6σ
VOC = USL - LSL
BAD CAPABILITY
Allowed Variation
Actual Variation
Voice of Customer
Voice of Process R
EJE
CT
S
R
EJE
CT
S
R
EJE
CT
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R
EJE
CT
S
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Process Capability Analysis
Process capability indices:
• Process capability indices are used to evaluate the capability of a process in a single number.
• Three important process capability indices are Cp, Cpk and CR. • The important requirements that these measures to be effective are
• Process should be stable. • Data should be normal.
Potential capability Cp:
• Cp is a measure, which measures the ability of the process to meet the customer specifications. Customer requirements can be expressed as lower specification limit (LSL) and upper specification limit (USL).
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Process Capability Analysis
Potential capability Cp:
• In the formula for Cp, ‘σ’ refers to the point estimate for the process standard deviation and it is given by the formula
here, R is the average range and d2 is a constant depends on the subgroup size.
The different values of d2 corresponding to the subgroup size is given in the following table
Subgroup Size
2 3 4 5 6 7 8 9 10
d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078
• Allowed variation. 6 SD < (USL – LSL) Cp > 1. Hence, Cp should be greater than 1 for a process in order to meet the customer requirements.
• Limitations of Cp: It checks only the potentiality of the process to meet the customer requirements. But it never checks whether the process is actually meeting customer requirements.
σ = R
d2
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Process Capability Analysis
Cpk:
• Cpk gives an idea about the position of the mean comparing with the
nearest specification limit
• It checks centering of the process.
• The formula for Cpk is given by
x= is the process average
LSL and USL are the lower and upper specification limits respectively.
Please note: If Cp = Cpk, then process mean is at the center of the
population
CR:
• Capability Ratio (CR) is the reciprocal of Cp
•
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Process Capability Analysis
LSL USL LSL USL
Cp = Cpk Cp ≠ Cpk
Centered Process Off Centered Process
Targ
et
= P
roc
es
s m
ea
n
Ta
rge
t
Pro
cess m
ean
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Process Capability Analysis
Process performance indices:
• This is used to analyze and compare the current process with improvement
efforts
• Like capability measures, performance measures are also effective when
• The process is stable
• The data is normal
• Three important performance indices are Pp, Ppk, Cpm
• The only difference in the formula of Pp, Ppk and the corresponding
capability indices is, in the performance indices long term sample standard
deviation(s) is used instead of short term process spread (σ).
The formulae are given by
Here ‘s’ refers to the sample standard deviation.
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Process Capability Analysis
Cpm:
• Cpm is considered as the most accurate index than other indices and it is
commonly known as Taguchi index.
• The formula for finding this index is
Here µ is the process average and T is the target value.
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Process Capability Analysis
Short-term and Long-term capability:
• The validity of the capability analysis increases with the time span of the
collected data increases.
• Process Capability Study is helpful to determine the ‘short’ term stability and
capability of the process, whereas the Process Performance Study is helpful
to determine the ‘long’ term stability and capability of a process
Short Long
Term Term
Performance Performance
Off – centered
Process
Centered
Process
Cp Pp
Cpk Ppk
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Process Capability Analysis
This process is short-term capable but not long-term capable
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Process Capability Analysis
Process Capability for non-normal data:
• In process capability study, it is assumed that the process data is normally
distributed. In the case of non- normal data the different indices of capability
study will not give valid results.
• For a non- normal data the capability studies are performed by
• Finding another distribution such as Exponential, chi-square and F
distribution, that fits the given data.
• Using nonlinear regression to fit a curve to the data.
• Using transformations like Box- Cox and Johnson’s transformations to
transform the data to another variable which is normally distributed.
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Process Capability Analysis
Process Capability for Attributes data:
• In the case of attributes data, the process capability is defined by using the
mean of nonconformity.
Example:
A electronic manufacturing process has a customer specification of
maximum two defectives per batch, produces lots of size 56 and the
customer specification is given by 2/56 = 0.035714. The value p for the
stable process is given by 0.03. Since the value of p is better than the
customer specification, the process is capable.
To find the percentage of batches having more than two defectives:
The probability of occurrence corresponding to each defective can be
calculated by using binomial distribution.
In the case of ‘0’ defective, the probability of occurrence is given by
P (0) = 56C0(0.03)0(1 – 0.03)56
= (0.97)56 = 0.182
Similarly we can find the probability of the other defective cases also using
binomial distribution and the values are given in the following table.
Defectives 0 1 2 Total
Probability of Occurrence
0.182 0.315 0.268 0.765
Hence the percentage of batches have more than 2 defective are 100-76.5
= 23.5%
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Process Capability Analysis
Calculation of process performance metrics:
There are numerous performance indicators used in Six Sigma projects.
• Percent defective
• Parts per million (PPM)
• Defects per million opportunities (DPMO)
• Defects per unit (DPU)
• Process Sigma
• Rolled throughput yield (RTY)
Consider the following example –
A process produces 20,000 books. Five types of defects can occur.
The number of occurrences of each defect type are given below:
Defect type Frequency
Typographical error 345
Missing pages 45
Incorrect ordering of pages 37
Feeding mistake 41
Hazing printing 25
Total number of defects 493
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Process Capability Analysis
1. Percent defective = Number of defects × 100 = 493 × 100 = 2.4%
Number of units 20,000
2. Parts per million (PPM) =
Number of defects × 1,000,000 = 493 × 1,000,000 = 24,650
Number of units 20,000
3. Defects per Million Opportunities (DPMO)
DPMO = Total number of defects × 1,000,000
Total number of opportunities
= 493 × 1,000,000
20000 × 5
= 4930
4. Defects per unit (DPU) = Number of defects = 493 = 0.02465
Number of units 20,000
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Process Capability Analysis
5. Process sigma =
Process sigma is calculated using excel.
6. Throughput yield = e-DPU = e-0.02465 =0.976
7. Rolled Throughput Yield (RTY):
RTY applies to the yield from a series of processes and is calculated by
multiplying the individual process yields. If a product goes through five
processes whose yields are 0.993 0.987, 0.975, 0.969, and 0.957 then
RTY = 0.993 × 0.987 × 0.975 × 0.969 × 0.957 = 0.886
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Process Capability Analysis
• Example 1: Determine the process capability / performance
metrics of a process that had a lower specification of 2 and an
upper specification of 16 with responses: 3.4, 8.57, 2.42, 5.59,
9.92, 4.63, 7.48, 8.55, 6.1, 6.42, 4.46, 7.02, 5.86, 4.8, 9.6, 5.92
(data were collected sequentially).
• Testing of stability: The process is stable (under statistical control)
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Process Capability Analysis
• Testing of normality: Stat > basic stat > Normality test
• Since the p-value = 0.895 > 0.05, data is normal.
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Process Capability Analysis
Process capability analysis
• Select Stat > Quality Tools > Capability Sixpack > Normal
• Enter the subgroup size as 1, and enter the given LSL and USL
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Process Capability Analysis
Output
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Process Capability Analysis
Alternative Minitab capability analysis for normally distributed data:
Stat > Quality Tools > Capability Analysis > Normal
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Process Capability Analysis
• Example 2: A chemical process has a specification upper limit of 0.16
and physical lower limit of 0.10 for the level of contaminant. Determine the
estimated process capability / performance of the process, which has the
following output:
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Process Capability Analysis
• Phase 1: Check for special causes.
• 5 out of 119 points are out of control limits, which indicate that special
causes are present.
• Many a times we noticed that the LCL value becomes negative, which is
impossible in a real life situation. Apparently the individual chart is not the
best way to analyze data like this.
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Process Capability Analysis
• As we do not know whether special causes are present, we can’t
determine the proper distribution for the data. Likewise, if the distribution of
data is not known we can’t determine whether special causes are present
because the control limits may be in the wrong place. The central limit
theorem states that stable distributions produce normally distributed
averages even when the individual’s data are not normally distributed. We
create subgroups of size 5 in Minitab and test the stability (X-bar chart).
Instead of using individuals chart we use averages. The chart indicates
that the process is in statistical control.
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Process Capability Analysis
• Phase 2: Examine the distribution (Test for normality).
• The p-value indicates that the data is not normal.
• Hence, we use Box-Cox transformation (to be defined).
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Process Capability Analysis
• The following figure shows a Box-Cox plot.
• Lambda = -1.69
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Process Capability Analysis
• Testing of normality of the transformed data: The transformed data looks
well behaved.
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Process Capability Analysis
• Phase 3: Predicting long term defect rate for the process.
• Minitab calculates both the process performance indices (Ppk) as well as
the process capability indices (Cpk). The denominator for the process
performance indices is the overall standard deviation rather than the
standard deviation based only on within subgroup variability. This is called
the long term process capability, which Minitab labels as ‘Overall
Capability’.
• Minitab’s analysis indicates that the process is not capable (Ppk < 1). The
estimated long-term performance of the process is 193390.19 DPMO,
where the observed performance is even worse (201680.67 DPMO). The
difference is a reflection of lack of fit.
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Process Capability Analysis
Conclusion:
In this chapter we have learnt about:
• Process capability studies.
• Process capability indices.
• Process performance indices.
• Short term and long term capability.
• Process capability for non-normal data.
• Process capability for attributes data.
• Process performance vs. specification.
• Examples using Minitab