Process Capability (C_p) & Process Capability Index (C_pk) with solved manufacturing example!

Hi friends, if you are looking for Process Capability (C_P) & Process Capability Index (C_pk) then you are on the right platform. In this content, we will explain it with an example of manufacturing industry and also share template of C_P & C_pk.

Process Capability (C_P) – Process capability represents the performance of a process in a state of statistical control (statistical control means absence of special causes). It is determined by the total variability that exists because of all common causes present in the system. It compares the process capability to the maximum allowable variation as indicated by the tolerance. This provides a measure of how well the process will satisfy the variability requirements. It is also called process capability ratio.  

Desirable value of Cp ≥ 1. Now it is very interesting that why Cp value is greater or equal to 1. To answer this question, if we look at the formula closely, it is actually the ratio of (USL- LSL/6σ ) and we can interpret in this way that “What customer is wanting / What we are providing”. To meet customer’s requirement, manufacturer should provide at least same requirement to customer and therefore, ratio of this must be equal to 1.

   USL – LSL = (Customer requirement)

         6σ     = (Process spread of manufacturer)  

Cp ≥1, now if we refer the below picture – 2, where Cp= USL-LSL/6σ  = 60-35/55-40=20/15 which is greater than 1. 

Process Capability Index (C_pk) – This is a capability index that measures the potential for a process to generate defective outputs relative to either upper or lower specifications. It takes the process location as well as the capability into account.

Now the question arises, If we have Cp value then why we need the value of Cpk?

Let’s understand with an example.  Suppose upper specification limit of a process of a manufacturing company is 30 and lower specification limit is 6. Process spread is from 40 to 52 and standard deviation 2.  So, we can calculate Cp in this case as

Cp = USL – LSL / 6σ = 30 – 6/ 6*2 = 24/ 12 = 2,

So, in this case Cp = 2, but when we see the actual Pic -3 (conditional case) as illustrated in above diagram, it is clearly evident that process spread of said process is out of LSL & USL range.  So, process is not capable.  This is the limitation of Cp.  Formula of Cp does not define the location of process spread (Bell curve) whether process spread should be within LSL & USL, outside LSL and USL or adjacent to LSL & USL.

that’s why we need to calculate Process Capability Index Cpk where location of process spread is defined.

As described in formula itself that value of Cp will be taken as minimum of two values, but why is it so? The answer of this question has been explained with one example below in pic -4.

If the value of (X double bar – LSL)/ 3σ   & (USL – X double bar)/ 3σ is same, then process is at the centre or mean.  Because X1 is equal to X2.

But in below picture -5

 it is clearly seen that process spread has shifted towards right and process mean has shifted from m to m1.  So, in left side there is no non confirming of the product but in the right side the shadow red part as shown in the figure is non – confirming product  From formula of Cpk we get two values as per formula, one from left side and the other one from the right side of process mean. If we go by formula

 As per diagram (X. double bar – LSL) /3σ =   X1/3σ

         & (USL – X. double bar)/3σ =  X2/3σ

So, Cpk min ( X1/3σ, X2/3σ )  In case of  X1/3σ , as per diagram  X1> X2 so we will get capable Cpk value, but in case of X2/3σ we  will barely get capable or incapable Cpk value along with defectives product in this region.

 As a manufacturer our main aim is to provide defect free product to customer. so we will choose the minimum Cpk  value  min of ( X1/3σ, X2/3σ )   where the chance of barely capable or incapable Cpk  value is found.  It means process in not at the centre or mean. Once barely capable or incapable is identified , we can find the root cause and can take the corrective action  therefore process capability index Cpk may improve and finally  defect free product can be produced.

We take the minimum of the two ratios because it gives the worst-case situation. It also give relative position of process mean with respect to LSL & USL.

Let’s understand with an example. Suppose there is a pen manufacturing company where casting of pen is done. We will take 5 samples in a group and check the weight.  We need to check 20 such groups.  USL =114 gm  LSL = 104 gm. Now we will Calculate Cp & Cpk value.

Sl. No	X1	X2	X3	X4	X5	sum of individual	Average	X double bar
1	108.5	108.4	109.3	108.5	109.6	544.3	108.86	108.9665
2	108.2	109.1	112.3	110.5	109.6	549.7	109.94
3	108.1	108.2	109.3	109.2	111.8	546.6	109.32
4	108.5	110.4	108.4	108.5	108.45	544.25	108.85
5	108.4	107.2	107.3	110.2	108.4	541.5	108.3
6	107.2	108	108.5	108.9	108.4	541	108.2
7	109.2	108.6	109.2	110	109.2	546.2	109.24
8	108.2	109.1	112.3	110.5	109.6	549.7	109.94
9	109.2	109	109.2	110	109.2	546.6	109.32
10	108.4	107.4	108.2	107	109.3	540.3	108.06
11	107.2	109.2	109.2	109.5	108.4	543.5	108.7
12	108.2	109.1	109.5	110.5	109.6	546.9	109.38
13	109.3	108.2	109.8	109.3	108.2	544.8	108.96
14	109.2	109	109.2	111.5	109.2	548.1	109.62
15	106.3	107.4	109.7	108.6	109.4	541.4	108.28
16	108.3	109	112.4	110.3	109.5	549.5	109.9
17	108.7	108	109.1	109	111.3	546.1	109.22
18	108.4	110.3	108.6	108.4	108.3	544	108.8
19	108.1	107	107.3	111.2	108.2	541.8	108.36
20	107.1	107.9	108.4	108.7	108.3	540.4	108.08

First, we will calculate the standard deviation.

In excel we will put =STDEV.S(data set) = In this case standard deviation =1.164

Cp = USL – LSL/ 6σ  = (114-104)/(6*1.164)= 1.43

Cpk = Min (USL- X. double bar)/3σ , ( X. double bar – LSL)/ 3σ

      = Min   (  (114 -108.9)/3*1.164      , ( 108.96 – 104)/(3*1.164) )

      =  Min (( 5.1/3.492), (4.96/3.492))

Cpk = Min ((1.46, 1.42)) = 1.42