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Process Capability Index (Cpk) Formula

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  • Last Updated : 24 Feb, 2022

Process Capability (Cp) is an approach for determining the quantifiable attribute of a process in relation to a specification. Process Capability Index is abbreviated as Cpk (Cpk). It assesses a manufacturer’s capacity to create a product within the tolerance range of a client. Cpk is used to determine how near you are to achieving a goal and how consistent your performance is compared to your average. Cpk calculates the best-case scenario for your current procedure.

Process Capability Index

Cpk is a process capability index that determines how much a process can produce. Cpk, unlike Cp, does not assume the process mean is centred between the specified boundaries. Many people use Process Capability as a computation tool to estimate the output of a product they are manufacturing. It helps the manufacturers estimate the potential production and manage the resources to get the best results. The typical statistical analysis and distribution of data gathered are frequently used as inputs for the process capability index. It’s analogous to the mathematics terms mean and average value as well as standard deviation. However, it varies in that it employs a control chart analysis to assess the statistical control of the system.

The assumption that the measurements are regularly distributed is required by Cpk. 

Formula

Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)

where,

  • σ is the standard deviation.
  • upper specification limit = USL
  • lower specification limit = LSL

Interpreting CpK Values

  • If the Cp value is equal to the Cpk value, the process is functioning on the edge. The production capability is adequate and meets the design parameters for Six Sigma standards.
  • The process mean has violated one of the specification boundaries if Cpk is less than zero.
  • When Cpk is larger than zero but less than one, the process mean is within specification limitations, but a portion of the manufacturing output has exceeded them.
  • The process mean is properly centred and within the specification limits if Cpk is bigger than one.

Sample Problems

Question 1. The temperature of a room in a hospital where a patient is resting should be between 4°C and 20°C. If the mean temperature is 10 and the standard deviation is 2, find the process capability index.

Solution:

Given: USL = 20°C, LSL = 4°C, Mean = 10 and σ = 2°C

We know, Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)

⇒ \frac{USL-mean}{3\sigma}=\frac{20-10}{3(2)}\\=\frac{10}{6}\\=1.667

⇒ \frac{mean-LSL}{3\sigma}=\frac{10-4}{3(2)}\\=\frac{6}{6}\\=1

⇒ Cpk = min (1.667, 1)

Since 1 is the minimum value out of the two amounts, the process capability index is 1 .

Question 2. When food is given to a customer in a restaurant, it should be between 39°C and 49°C. The procedure utilized to keep the food at the proper temperature has a process standard variation of 2°C, with a mean temperature of 40. What is the process’s process capability index?

Solution:

Given: USL = 49°C, LSL = 39°C, Mean = 40 and σ = 2°C

We know, Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)

⇒ \frac{USL-mean}{3\sigma}=\frac{49-40}{3(2)}\\=\frac{9}{6}\\=1.5

⇒ \frac{mean-LSL}{3\sigma}=\frac{40-39}{3(2)}\\=\frac{1}{6}\\=0.166

⇒ Cpk = min (1.5, 0.166)

Since 0.166 is the minimum value out of the two amounts, the process capability index is 0.166 .

Question 3. The time spent on a task should be between 0 hours and 18 hours. If the meantime is 10 and the standard deviation is 2, find the process capability index.

Solution:

Given: USL = 18, LSL = 0, Mean = 10 and σ = 2°C

We know, Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)

⇒ \frac{USL-mean}{3\sigma}=\frac{18-10}{3(2)}\\=\frac{8}{6}\\=1.33

⇒ \frac{mean-LSL}{3\sigma}=\frac{10-6}{3(2)}\\=\frac{4}{6}\\=1.67

⇒ Cpk = min (1.33, 1.67)

Since 1.33 is the minimum value out of the two amounts, the process capability index is 1.33 .

Question 4. What does a Cpk value of 1.33 or more imply?

Solution:

Cpk is a standard index to state the capability of one process, the higher the Cpk value the better the process is. For instance, Machine 1 has a Cpk of 1.7 and machine 2 has a Cpk of 1.1. From the Cpk value, one can derive that Machine 1 is better than 2.

Cpk = or >1.33 indicates that the process is capable and meets specification limits. Any value less than this may mean variation is too wide compared to the specification or the process average is away from the target.

Question 5. Given that mean = 10 cm, specifications = 5 – 13 cm. Also, upper specification limit = μ + 1σ and lower specification limit = μ – 1σ. Calculate the standard deviation and process capability index.

Solution:

Given: USL = μ + 1σ = 13 and 

LSL = μ – 1σ = 5

Subtracting the LSL from the USL, we have:

2σ = 8 cm

⇒ σ = 4 cm

We know, Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)

⇒ \frac{USL-mean}{3\sigma}=\frac{13-10}{3(4)}\\=\frac{3}{12}\\=0.25\\⇒ \frac{mean-LSL}{3\sigma}=\frac{10-5}{3(4)}\\=\frac{5}{12}\\=0.41

⇒ Cpk = min (0.25, 0.41)

Since 0.25 is the minimum value out of the two amounts, the process capability index is 0.25 .


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