Q. 101  Assuming that -

Upper Specification Limit = 24,
Lower Specification Limit = 18, 
The average = 21.5 and
Standard Deviation (within) = 0.75, 

Cp and Cpk can be calculated using the calculator at https://www.benchmarksixsigma.com/calculators/process-capability-index/ 

 

Explain the inference of the Cp and Cpk values that you get for the given data and give some inputs on the kind of action required. 

 

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

• All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/
• Please visit the forum home page at https://www.benchmarksixsigma.com/forum/ to respond to the latest question open till the next Tuesday/ Friday evening as per Indian Standard Time.  
• The best answer is always shown at the top among responses and the author finds honorable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term.

CP:1.33 and Cpk: 1.11 for the given values. 

This indicates process data points are in distributed with in 3 Sigma limits, but process mean has shifted towards the upper limit. this means chances of getting the deviations above specification.  

Process mean has to be adjusted to 21 from 21.5 to achieve process will be stable .

Cp =1.33 and Cpk = 1.11 are the value which we derived from the above mentioned data.

 

We can draw the following inferences about the process from this data:

 

·         Process is capable of producing with specification limits  but the process average is off target

·         The process complies to a 4 sigma process capability

·         99.73% of the products produced comply with the set customer standards

·         Cpk needs improvement to bring it at par with Cp

·      Cpk can be improved by examining the current process flowchart, removing areas of duplication / unnecessary steps, prioritize improvement and draw new control charts to evaluate how the effect of these changes

Cp and Cpk are two capability indices which are used as baseline measurement in measure and control phase. These indices measures common cause variation within subgroup and short term only. These are the statistical measurements to report process capability. 

 

Cp is a short term index and straightforward  indicator of process capability which doesn't account for its mean being centred. Higher the values reflects narrower the spread an d more precise. And it does not tell the location of spread, only tells about whether the process is capable or not. 

Ideally for a good process capability, its value considered as 2 means specification limits are two times of process limits. 

Cp = SW/PW = 2 but generally people accepts value more than 1.33.

 

This is very unrealistic to tell about process capability by considering only one factor only. Because sometimes a process can be more precise but it can be shifted towards any if its limits. And if it is shifted towards any upr or lwr limits which means there is probability to produces products which are out of specifications limits. And Cp doesn't provide any Information regarding the same so it is necessary to see other factors to find out exact scenario of the process that where we should do improvement like improve standard deviation or shifts mean. 

 

So here comes factor of Cpk which uses mean and SD to estimate probability. k is the amount of which a distribution is centred and Cpk tells the information about process mean. Cpk can never exceeds Cp value 

Cpk = Cp(1-k)

range of k varies from 0 to 1.

 

In other words we can say that Cpk is the adjustment of Cp for the effect of non standard distribution. 

 

It measures how much close to your target and how much consistent to your average. A person can perform more precisely with minimum variation but can be away from its centre and go towards one specification limits and it can be at exactly on average target value but can have more variation. 

 

So for any process to know about process capability and its position we should now both Cp and Cpk. 

 

In above example 

After calculations  we see that Cp is 1.33 and Cpk is 1.11 which is less than value of Cp. 

Generally accepted value for Cp is greater than equal to 1.33. So the process is capable but we don't know where its location is whether its shifted or not. Here we are seeing that Cpk is not equal to Cp. It means it has some value of k which is the amount a distribution is centred. So it shows that process is shifted and we need to work on to shift mean back to its centre position rather than to reduce standard deviation, because its easy to shift mean rather than reducing Standard deviation. 

 

Cp is 1.33 - which means variance in the process is upto 6 sigma.

 

and Cpk is 1.11 - Mean of the process is shifted towards upper limit.

 

By understanding both Cp and Cpk, First thing to do is - Variation is in control and still there is an opportunity to shift the Mean of the process to center for improving accuracy.

 

Higher the Cp value, lesser the variation in process.

 

Before we dwelve upon explaining the inferences of these Cp, Cpk values, let us quickly revisit Cp, Cpk definition.

 

Cp, Cpk are Process Capability Indices. Process Capability is an inherent variability of a given characteristic in a stable process. It represents process performance

over a period of stable operations . A capable process is one in which the output always adheres to the customer specifications.

 

Let us see the definition of Cp. Cp is the ratio of the specification limits (of a characteristic) to the natural process variation of a process (under a state of

statistical control). In other words, Cp = Voice of the Customer/Voice of the Process = (USL-LSL)/(6*SD) where SD= Standard Deviation and denoted by 'Sigma' symbol. A Cp value of >= 1,  indicates that process is technically capable. A Cp value of 2 represents that process performs at 6 sigma level. A Cp value of < 1 indicates that the process is poor.

 

If Cp portrays the capability to meet the customer specification limits, then why do we need Cpk. The answer is that Cp does not consider how well the output is centred on the target value. Cpk addresses the proximity of the centring of a process related to the spec. limits. Cpk splits the Cp into 2 parts, Cpkl and Cpku.

 

Cpk takes the lesser of the two values. The formulas related to this are
Cpku=(USL-XBAR)/(3*SD); Cpkl=(XBAR-LSL)/(3*SD) where SD is Standard Deviation and is actually denoted by 'Sigma' symbol.  Taking the lesser of the two values,
the Cpk value would be Min(Cpku,Cpkl).  

 

To explain the concept of Cp, Cpk more - let us take a couple of imaginary examples mimicking real time sports.

Example 1: Imagine a football team practising penalty kicks. Assuming the goal posts on both the ends to be the specification(spec) limits. The players think that putting the ball in between the posts is a goal (which it is). But the coach would like his/her wards to be highly consistent. He/she would expect his/her wards to

consistently hit a target within the goal post. Imagine the coach wants the football to be hit on topmost of the centre of the goal posts (or any other part)? Why ? Because goalkeepers normally try to distract the kickers(football players) and makes a pre-meditated move. Therefore , it is imperative that players hit their

practice-penalty goals consistently at the same target position as much as possible. While Cp talks about slotting of goals , Cpk talks about how much variation or drift is there in hitting the goals at topmost of the centre of the posts.

 

When Cp/Cpk would be high :
Example 2: Imagine a practice session going for a shooting competition which is bound to happen in a week. A team of 5 is trying its luck , keeping in mind the competition would be stiffer in the actual competition. The team is aiming a point of 9.8 (out of 10), which could a potential medal winning score ,as per the team.
The Qualifying range for the competition is 9.3 and above.  If the team is consistently shooting 9.3 and beyond 9.3 say upto 9.5 , then the Cp value is good(high). But if the team is consistently shooting near say 9.7 or even 9.8, then the Cpk is good or high. 

 

Now , let us go to the given statistics for this question:

USL=24; LSL=18;Ave=21.5;Standard Deviation = 0.75. As per the given calculator, we get the values of Cp as 1.33 and Cpk as 1.11.

 

What do we infer from this ?
1.Cp value of 1.33 indicates that the process is quite capable.

2.However, Cpk value of 1.11 conveys that the process sigma is 3.33 . Formula for calculating Sigma Level = 3 *Cpk = 3 * 1.11 =3.33 .
So if the sigma level is 3.33, then it shows that there is plenty of variation and there are so much defects for a process operating just over 3 sigma.Therefore, the variation needs to be reduced. Thats the reason why we look for Cpk . Remember from CP perspective it looked like good until we found this now !!

3.Ideal value for a Cpk should be either at 1.33 or 1.66(close to 4 or 5 sigma) since there can be space for a process drift either to the left or right of the nominal value (target value).

4. Applying the formulas which we discussed above, for Cpku and Cpkl , we find the values for them as
Cpku=(24-21.5)/(3*0.75)= 2.5/2.25=1.1111; Cpkl=(21.5-18)/(3*.75)=1.5555; Cpk=Min(Cpku,Cpkl)=Min(1.11,1.55) = 1.11

 

Conclusion:

From this, we can infer that the process is less capable relative to the upper specification limit. Though the variation in the process is acceptable, in order to improve the process performance, the mean needs to be moved away from the USL or back to the middle of the specification limits.

 

 

 

Cp and cpk are used to study the process which is stable otherwise the samples will not represent the process. about the inference Cp talks about spread of the values of a sample collected from the process compared with the specification limit. Smaller the spread better the Cp but Cpk talks about how far is the sample mean to spec mean. Sample mean closer to spec mean better the Cpk. In other words Cp is Precision Cpk is Accuracy 

Cpk of value 1.1111 means its barely capable process, because it produces 64 ppm but less than 2700 non-conforming ppm. In this case, the process width is touching the specification width.

 

Cp of value 1.3333 means its a capable process, but not highly capable process (which should have been near 1.5). 

 

The overall sigma value is between 3.3 & 3.4, meaning it has 967 non confirming parts per million, if we take sigma value as ~3.3.

 

Overall this looks like an existing process which has been just satisfying the needs but has potential to increase its capability & maturity.

Cp= 1.3333

Cpk=1.1111

In the given example ,Cp =1.33 and Cpk =1.11.

                       It means process is capable of  making parts within specification limit as Cp =1.33 whereas Cpk=1.11 we can say that process  is not centered as Cpk not equal to Cp i.e it is  closure to one of the specification limit i.e either USL or LSL 

                        In our example as Cp is good ,variation is acceptable.As Cpk is bad  (not equal to Cp) and it is less than Cp value, we need to shift the the Mean  to center the process within specification limit without increasing variation

The Industry guidelines provide the insights whether a process is capable or not. The general accpetable Minimum value for Cp is 1.33 and for the given data the Cp is 1.33, this is an indication of how the process could perform relative to the specification limits (USL and LSL). 

In the given case study the Cp is 1.33 & Cpk is 1.11.

This Cp relates to the process spread. it is a normally distributed population, 99.73% of the variation is within the + 3 std deviations of the process average. The Cp compares the specification width to the Process Width and is the capability the process could acheive.

the Cpk value 1.11 infers both the process spread and the process mean, Cpk cosiders the location of the Process mean and as Cp is > Cpk it infers that the process is not centered and it is recommended to reduce the variation in the process.

 

Cp is 1.33

Cpk is 1.11

Cp is Potential of the process and Cpk is process capability index, i.e., Location of the Process.

 

In the above process Cp=1.33 and Cpk=1.11

and Capability Ratio Cr=0.75

Since Cp is 1.33 and Cr = 0.75, The process is capable with strict monitoring.

 

Cpk(Lower) is 1.55, the process center slightly shifts to the lower side

which has to shifted to center as nearly as possible for having better Cpk near 1.33

 

Also reducing the spread, improves the Cp and decreases the Cr Value,

which give Robust process. 

 

 

 

Upper Specification Limit = 24,
Lower Specification Limit = 18, 
The average = 21.5 and
Standard Deviation (within) = 0.75

 

The Cp & Cpk Values for the above data's are as follows:

- Cp = 1.33 & Cpk = 1.11

- The Cpk Value is less than the desired level, i.e: In actual Condition Cpk>/=1.33.

- We generally want the Cpk Value at least 1.33 or higher than that to meet the Customer requirements

 

Inputs on the Kind of action:

- In the above mentioned data's, the average is mentioned as 21.5. But as per the actual calculation based on the USL & LSL, the average must be as 21.

- So it is clear that Given average[21.5] has been moved above the target mean [21] & so we have to bring it towards the Mean value

- We must first identify the Process variations that are the causes for the increased Average condition 

- when the remove the Variations in the Process,  the process will be in control & also the mean value can be obtained close to the target mean.

- Once the Process is centered - Mean average value is obtained, the Cp & Cpk value will be improved & Standard Deviation will be reduced

R Rajesh has provided clear definitions, examples and inference from the given date explaining the concept of Cp and Cpk in a complete manner. 

 

Vastupal's answer is a close second. It is good to see the competition increasing everyday.