Question

Please be detailed and give examples.   

1. Meanings and applications of C_p and C_pk.

Answer 1

Cp and Cpk

Cp and Cpk, commonly referred to as process capability indices, are used to define the ability of a process to produce a product that meets requirements. These indices, which are a fairly recent addition to the field of statistical process management, greatly simplify the management of statistically controlled processes.

Cp

The Cp index is calculated using specification limits and the standard deviation only. This index indicates, in general, whether the process is capable of producing products to specifications. No information on the ability of the process to adhere to the target value is included in this index.

The formula for Cp is as follows:-

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CpK

This Cpk index is calculated using specification limits, the standard deviation, and the mean. The index indicates whether the process is capable of producing within specification and is also an indicator of the ability of the process to adhere to the target specification.

The formula for Cpk is as follows:

Example:- Food served at a restaurant should be between 39°C and 49°C when it is delivered to the customer. The process used to keep the food at the correct temperature has a process standard deviation of 2°C and the mean value for these temperatures is 40. What is the process capability index of the process?
Solution:

USL (Upper Specification Limit) =49°C
LSL (Lower Specification Limit) =39°C
Standard Deviation =2°C
Mean = 40

Cpk is given by,

Cpk=min(USL−mean3σ,mean−LSL3σ)

Now, seperate the formula into two parts, and find the solution:

Solution of part 1: (USL – Mean)/ 3σ

Substitute the values:

= (49-40)/3 ×2

= 9/6

Solution of part 1= 1.5

Solution of part 2: (Mean – LSL)/ 3σ

= (40-39)/3 ×2

= 1/6

Solution of part 2= 0.166

Now, substitute the solutions in the formula, we have:

Cpk = min (part 1, part 2)

Cpk = min (1.5, 0.166)

Since the mininum value is 0.166,

The process capability index, Cpk is 0.166.

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Application of Cp and CpK:-

The following conditions must be met before Cp and CpK can be successfully used to evaluate the ability of a process:

• the sample size must be adequate (large enough)

• the data should be tested for normality

• the process being analyzed should be under statistical control

Caution: Only after a process is under statistical control, can one safely assume that the mean and standard deviation to have a stable values over time.

Cpk is more widely used than Cp, since it takes into account the mean and the standard deviation in its calculation. Please note that the difference between Cp and Cpk is an indicator of how far the average of the process is from the target specification. When the average of the process approaches the target value, the gap between Cpk and Cp closes. When the average of the specification is equal to the target value, then Cpk is equal to Cp. Cpk can never exceed Cp.

Both Cp and Cpk can be calculated with the generation of descriptive statistic views and histograms

Cp

The Cp index is calculated using specification limits and the standard deviation only. This index indicates, in general, whether the process is capable of producing products to specifications. No information on the ability of the process to adhere to the target value is included in this index.

The formula for Cp is as follows:

Cp

This Cpk index is calculated using specification limits, the standard deviation, and the mean. The index indicates whether the process is capable of producing within specification and is also an indicator of the ability of the process to adhere to the target specification.

The formula for Cpk is as follows:

The following conditions must be met before Cp and CpK can be successfully used to evaluate the ability of a process:

• the sample size must be adequate (large enough)

• the data should be tested for normality

• the process being analyzed should be under statistical control

Caution:Only after a process is under statistical control, can one safely assume that the mean and standard deviation to have a stable values over time.

Cpk is more widely used than Cp, since it takes into account the mean and the standard deviation in its calculation. Please note that the difference between Cp and Cpk is an indicator of how far the average of the process is from the target specification. When the average of the process approaches the target value, the gap between Cpk and Cp closes. When the average of the specification is equal to the target value, then Cpk is equal to Cp. Cpk can never exceed Cp.

Both Cp and Cpk can be calculated with the generation of descriptive statistic views and histograms.