Skip to content
Related Articles
Open in App
Not now

Related Articles

Process Capability Cp Formula

Improve Article
Save Article
  • Last Updated : 14 Jul, 2022
Improve Article
Save Article

A process is a series of steps/tasks to convert some input to an output. Capability on the other hand means to produce something with some set of specifications consistently. Thus we say that Process Capability is a statistical measurement of a process’s ability to produce parts of the desired specification consistently. 

With the help of historical data we get the answer to the question – “Can this process produce parts of desired specifications on a daily basis”. Process capability studies control graph, which helps determine if a process is in a controlled or uncontrolled state. 

Before we dive deep into various concepts, let us know some basic definitions

  1. Upper Specification Limit – This is the highest limit or the upper limit of a measurement or a reading
  2. Lower Specification Limit – This is the lowest limit or the lower limit of a measurement or a reading
  3. Upper Control Limit – The upper limit of the control line, any reading above this should be investigated for unusual variation 
  4. Lower Control Limit – The lower limit of the control line, any reading below this should be investigated for unusual variation 

Cp  and the Cpk Index

For any process, the final aim is to have a control chart that is

  1. Centered around the mean or nominal value
  2. Spread over a narrow specification width

To achieve this we take the help of Cp and the Cpk index or also called the Process Capability indexes

The formula for both is as follows:

C_p = \frac{\text{Specification Width}}{\text{Process Width}} = \frac{\text{USL - LSL}}{\text{UCL-LCL}} = \frac{\text{USL - LSL}}{6\sigma}\\

C_{pk} = \frac{\text{Distance from mean to nearest spec limit}}{\text{Distance of mean to control limit}} = \min({\frac{|\bar{\bar{x}}-UCL|}{3\sigma},{\frac{|\bar{\bar{x}}-LCL|}{3\sigma}}})

Cp determines the spread of the control chart, how narrow or wide the specification limits are.

Cpk determines the spread of the control chart relative to the specification limits.

Now let us take an analogy that can help us understand the two indexes.

Let us consider a scenario where a bunch of drivers, all having a different experience in driving, are told to park cars in a garage. All cars are identical in width, i.e., their specification(at least the width) is the same and this width is less than that of the garage(but not by much).

In this Cp determines the width of the queue of cars and Cpk determines how central they are to the garage door

Case no.

Cp

Cpk

Result

Case 1

Cp < 0.7

Cpk < 0.7

Here the width of the queue is narrow (Cp<0.5) but the cars aren’t central (Cpk<0.5) thus the cars won’t fit inside making this case not capable.

Case 2

0.7<Cp<1.33

0.7<Cpk<1.33

Here the width of the queue is less than in the previous case, and the cars are relatively more central, thus the cars just fit inside making this case just capable

Case 3

Cp>1.33

Cpk < 0.7

Here the width of the queue is the least but the cars aren’t centra, thus the cars wouldn’t fit inside making this case not capable

Case 4

Cp>1.33

Cpk>1.33

Here the width of the queue is the least and the cars are as central as they can be, thus the cars would easily fit inside making this case capable

Note: that these are just arbitrary numbers just give an analogy. 

From the above table, we can conclude that having a higher Cp and Cpk is always a better option, as it narrows down the control chart bringing all the readings closer to the mean and also centralizing it between the two control limits. This ensures that the number of defective products/parts produces is as low as possible.  

Sample Problems

Problem 1: Food served at a restaurant should be between 40°C and 50°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 43. What is the process capability index of the process?

Solution:

\bar{x}= 43

USL = 50

LSL = 40

σ = 2

 C_{pk} = \min(\frac{\bar{x}-LSL}{3\sigma},\frac{USL-\bar x}{3\sigma})\\ C_{pk} = \min(\frac{43-40}{3\times 2},\frac{50-43}{3\times 2})

Cpk = min (0.5, 1.16)

      = 0.5

Problem 2: Calculate the Cp of a control chart whose USL, LSL, and SD are 15,5,2 respectively.

Solution:

USL = 15

LSL = 5

σ = 2

C_p = \frac{\text{USL} -\text{LSL}}{6\times \sigma}\\ C_p = \frac{15-5}{6\times 2}

Cp = 10/12

     = 0.833

Problem 3: A control chart of a company gives the following readings;

Parameter Reading
USL 117
UCL 71
LSL 37
LCL 43
Standard Deviation 4.68
Mean 53

Calculate the process capability indexes(Cp and Cpk)

Solution:

From the table we see that

USL = 117

UCL = 71

LCl = 43

LSL = 37

σ = 4.68

\bar{x}= 53

C_p = \frac{\text{USL}-\text{LSL}}{\text{UCL}-\text{LCL}}\\ C_p = \frac{117-37}{71-43} = 2.85\\ C_{pk} = \min(\frac{\bar{x}-LSL}{3\sigma},\frac{USL-\bar x}{3\sigma})\\ C_{pk} = \min(\frac{53-37}{3\times 4.68},\frac{117-53}{3\times4.68})

Cpk = min (1.14 , 4.55)

      = 1.14


My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!