Six Sigma Black Belt Study Guides - PMTUTOR1)_Process_Capability... · Six Sigma Black Belt – Study Guides . ... • Cp is a measure, ... • The formula for Cpk is given by x=

1 www.pmtutor.org Powered by POeT Solvers Limited.

Six Sigma Black Belt –

Study Guides


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.



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



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

S

R

EJE

CT

S



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).



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



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

•



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



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.



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.



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



This process is short-term capable but not long-term capable



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.



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%



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



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


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




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



• 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)



• Testing of normality: Stat > basic stat > Normality test

• Since the p-value = 0.895 > 0.05, data is normal.



Process capability analysis

• Select Stat > Quality Tools > Capability Sixpack > Normal

• Enter the subgroup size as 1, and enter the given LSL and USL



Output



Alternative Minitab capability analysis for normally distributed data:

Stat > Quality Tools > Capability Analysis > Normal



• 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:



• 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.



• 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.



• 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).



• The following figure shows a Box-Cox plot.

• Lambda = -1.69



• Testing of normality of the transformed data: The transformed data looks

well behaved.



• 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.



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

Documents

Six Sigma Black Belt Study Guides - PMTUTOR1)_Process_Capability... · Six Sigma Black Belt – Study Guides . ... • Cp is a measure, ... • The formula for Cpk is given by x=