Cp index - AbsoluteAstronomy.com

Process improvement

In organizational development , process improvement is a series of actions taken by a process owner to identify, analyze and improve existing business processes within an organization to meet new goals and objectives. These actions often follow a specific methodology or strategy to create...

efforts, the process capability index or process capability ratio is a statistical measure of process capability

Process capability

A process is a unique combination of tools, materials, methods, and people engaged in producing a measurable output; for example a manufacturing line for machine parts...

: The ability of a process to produce output within specification

Specification (technical standard)

A specification is an explicit set of requirements to be satisfied by a material, product, or service. Should a material, product or service fail to meet one or more of the applicable specifications, it may be referred to as being out of specification;the abbreviation OOS may also be used...

limits. The concept of process capability only holds meaning for processes that are in a state of statistical control. Process capability indices measure how much "natural variation"

Common-cause and special-cause

Common- and special-causes are the two distinct origins of variation in a process, as defined in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming...

a process experiences relative to its specification limits and allows different processes to be compared with respect to how well an organization controls them. If the upper and lower specification

Specification (technical standard)

limits of the process are USL and LSL, the target process mean is T, the estimated mean of the process is

\hat{\mu}

and the estimated variability of the process (expressed as a standard deviation

Standard deviation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

) is

\hat{\sigma}

, then commonly-accepted process capability indices include: NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE

$\hat{C}_p = \frac{USL - LSL} {6 \hat{\sigma}}$	Estimates what the process is capable of producing if the process mean were to be centered between the specification limits. Assumes process output is approximately normally distributed.
$\hat{C}_{p,lower} = {\hat{\mu} - LSL \over 3 \hat{\sigma}}$	Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.
$\hat{C}_{p,upper} = {USL - \hat{\mu} \over 3 \hat{\sigma}}$	Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.
$\hat{C}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \hat{\sigma}} \Bigg]$	Estimates what the process is capable of producing, considering that the process mean may not be centered between the specification limits. (If the process mean is not centered, $\hat{C}_p$ overestimates process capability.) $\hat{C}_{pk} < 0$ if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.
$\hat{C}_{pm} = \frac{ \hat{C}_p } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }$	Estimates process capability around a target, T. $\hat{C}_{pm}$ is always greater than zero. Assumes process output is approximately normally distributed. $\hat{C}_{pm}$ is also known as the Taguchi Taguchi methods Taguchi methods are statistical methods developed by Genichi Taguchi to improve the quality of manufactured goods, and more recently also applied to, engineering, biotechnology, marketing and advertising... capability index.
$\hat{C}_{pkm} = \frac{ \hat{C}_{pk} } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }$	Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.

NEWLINENEWLINE

\hat{\sigma}

is estimated using the sample standard deviation

Unbiased estimation of standard deviation

The question of unbiased estimation of a standard deviation arises in statistics mainly as question in statistical theory. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the...

Recommended values

Process capability indices are constructed to express more desirable capability with increasingly higher values. Values near or below zero indicate processes operating off target (

\hat{\mu}

far from T) or with high variation. Fixing values for minimum "acceptable" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration. For example, in the automotive industry, the AIAG sets forth guidelines in the Production Part Approval Process, 4th edition for recommended C_pk minimum values for critical-to-quality process characteristics. However, these criteria are debatable and several processes may not be evaluated for capability just because they have not properly been assessed. Since the process capability is a function of the specification, the Process Capability Index is only as good as the specification . For instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification. The loss function of Taguchi

Taguchi methods

Taguchi methods are statistical methods developed by Genichi Taguchi to improve the quality of manufactured goods, and more recently also applied to, engineering, biotechnology, marketing and advertising...

better illustrates this concept. At least one academic expert recommends the following: NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE

Existing process	1.33	1.25
New process	1.50	1.45
Safety or critical parameter for existing process	1.50	1.45
Safety or critical parameter for new process	1.67	1.60
Six Sigma Six Sigma Six Sigma is a business management strategy originally developed by Motorola, USA in 1986. , it is widely used in many sectors of industry.Six Sigma seeks to improve the quality of process outputs by identifying and removing the causes of defects and minimizing variability in manufacturing and... quality process	2.00	2.00

NEWLINENEWLINE It should be noted though that where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive.

Relationship to measures of process fallout

The mapping from process capability indices, such as C_pk, to measures of process fallout is straightforward. Process fallout quantifies how many defects a process produces and is measured by DPMO

Defects per million opportunities

In process improvement efforts, defects per million opportunities or DPMO is a measure of process performance. It is defined as...

or PPM

Parts-per notation

In science and engineering, the parts-per notation is a set of pseudo units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement...

. Process yield is, of course, the complement of process fallout and is approximately equal to the area under the probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

\Phi(\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-\sigma}^\sigma e^{-t^2/2} \, dt

if the process output is approximately normally distributed. In the short term ("short sigma"), the relationships are:NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE

C_pk	Sigma level (σ)	Area under the probability density function Probability density function In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the... $\Phi(\sigma)$	Process yield	Process fallout (in terms of DPMO/PPM)
0.33	1	0.6826894921	68.27%	317311
0.67	2	0.9544997361	95.45%	45500
1.00	3	0.9973002039	99.73%	2700
1.33	4	0.9999366575	99.99%	63
1.67	5	0.9999994267	99.9999%	1
2.00	6	0.9999999980	99.9999998%	0.002

NEWLINENEWLINE In the long term, processes can shift or drift significantly (most control chart

Control chart

Control charts, also known as Shewhart charts or process-behaviour charts, in statistical process control are tools used to determine whether or not a manufacturing or business process is in a state of statistical control.- Overview :...

s are only sensitive to changes of 1.5σ or greater in process output), so process capability indices are not applicable as they require statistical control.

Example

Consider a quality characteristic with target of 100.00 μm

Micrometre

A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...

and upper and lower specification limits of 106.00 μm and 94.00 μm respectively. If, after carefully monitoring the process for a while, it appears that the process is in control and producing output predictably (as depicted in the run chart

Run Chart

A run chart, also known as a run-sequence plot is a graph that displays observed data in a time sequence. Often, the data displayed represent some aspect of the output or performance of a manufacturing or other business process.- Overview :...

below), we can meaningfully estimate its mean and standard deviation. If

\hat{\mu}

and

\hat{\sigma}

are estimated to be 98.94 μm and 1.03 μm, respectively, thenNEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE

$\hat{C}_p = \frac{USL - LSL} {6 \hat{\sigma}} = \frac{106.00 - 94.00} {6 \times 1.03} = 1.94$
$\hat{C}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \hat{\sigma}} \Bigg] = \min \Bigg[ {106.00 - 98.94 \over 3 \times 1.03}, { 98.94 - 94 \over 3 \times 1.03} \Bigg] = 1.60$
$\hat{C}_{pm} = \frac{ \hat{C}_p } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } } = \frac{ 1.94 } { \sqrt{ 1 + \left ( \frac{98.94 - 100.00} {1.03} \right )^2 } } = 1.35$
$\hat{C}_{pkm} = \frac{ \hat{C}_{pk} } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } } = \frac{ 1.60 } { \sqrt{ 1 + \left ( \frac{98.94 - 100.00} {1.03} \right )^2 } } = 1.11$

NEWLINENEWLINE The fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values for C_p, C_pk, C_pm, and C_pkm.

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