Q 494. It is generally said that Capability Metrics (Cp/Cpk) are more relevant for manufacturing industries where it is common to get 2 sided specification limits. However, there is another capability index - Cpm. Cpm is considered better than Cp/Cpk and is even relevant for service industries where we have 1 sided specification limits. Explain the concept of Cpm. Highlight with examples 
1. Why is it better than Cp/Cpk? 
2. How can it be used in service industries? 

 

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The Cp and Cpk are quite familiar capability indices commonly used to ensure that a process is running and how it is expanded compared to the tolerance (Cp), or that it stays well within specifications (Cpk).

 

One example for Cp and Cpm is that of an Airport Runaway. Cp tells us the width of the Runaway compared to the width of the airplane. If it is a personal plane, the runaway is fairly wide compared to the width, and i wouldn't worry much about staying on the road. This would yield high Cp value, but if it is a medium size plane, it would take moderate effort to keep closer to the center line, hence medium level os Cp value, However, if 

it is an international flight with maximum size plane, the pilot have to be extra careful to keep it within the safe zone on the Runaway, therefore resulting in low Cp value. 

We can relate above example with the process by assuming the plane size as the processess on production line and Runaway as the tolerance level. 

Now, Cpk tells us how close we are moving the plane to the shoulder of the road. You get a high Cpk when you drive in the center of the center lane. The closer you get to the shoulder, the lower the Cpk.

   

Alternatively , another sort of capability index exists: the Cpm, which is much less famous and used even less frequently. The main difference between the Cpm and the other capability indices is here the bias from the target is directly taken into consideration in the Cpm. This bias is mainly the difference between the process average and the target.

In Minitab, Cpm is calculated automatically, as soon as expected / targeted value is entered into a capability analysis . To obtain a best Cpm value, the process needs to be right on target, whereas to obtain a satisfactory Cpk value, a process simply needs to stay well within specifications. This might seem to lead to the same conclusion; however, there is an obvious and major difference: with the Cpm, a more inflicting drawback is put in place for not being right on target.

 

Cpm or Cpmk yields much better outcomes compared to cpk since it takes in account the target itself. In cases where only one specification is provided,  the calculation of cpk is rather insignificant, but the Cpm can be calculated.

Lets take an example here, 

 

Lets consider our subject is handling a process with a decent process capability index, where Cpk, has never dipped below 2 and typically averages beyond 3. Given this high degree of capability, subject might consider further eradicating the variation near the target. While the Cpk and Ppk are well informed and quite common indices, they may not provide as much information as our subject desires to continue to further enhance the process. This is moderately significant if the target is not the center-point of specifications. On the other hand, Cpm utilizes the target when measuing Std deviation. Like the sigma of individual formulae, compares each reading to a reference value, instead the data is compared to the traget value. These diffreneces are then squared. As a result , any observation that is farther from the traget value will raise the std deviation. Cpm is prefered to Cp because it measures process location and standard deviation.

 

In the below image, the process remains unchanged, as the process is more middled, Cpm gets better

,

Here, Cpm is good at top, better in mid and best at last.  

 

But once the subject keeps target as center, we even better Cpm"( last slope).

 

Boradly speaking, when we have both upper and lower specs in a process, mostly we get the target somewhere near the middle point. As per VOC, we should see the possibility of Target value being logical and feasable , before , then perhaps see to the possibility of moving the target value towards One specification . From excellence point of view, this makes more sense to have the least possible compromise with respect to the target value and Cpm capability indices helps us to measure and publish this part.

Cpm, also known as Taguchi Capability Measure, is yet another process capability metric, however is more robust than Cpk.

 

Before understanding Cpm fully, let us take a step back & understand Cpk.

 

Cpk is basically a measure of process centering, here the central value is the process mean, however it does not take into consideration the target value, which is in most cases the central value obtained from the customer specifications. The formula for calculating Cpk is as shown below:-

Cpk = min ( (USL - Mean / 3σ), (Mean - LSL / 3σ) )

 

Here it takes the lowest value out of the two i.e. USL - Process Mean & Process Mean - LSL. Let us see this through an example:-

 

 

 

Here in the above figure, we will look at the distance from the center of the plane’s nose tip to either side of the landing strip. This is however an indirect way of assessing the center of the process.

 

Cpm, on the other hand, contains the target value in its formula which is expressed as :-

 

Cpm = USL - LSL / 6*SQRT( σ^2 + (X̄ - Target))

Thus in addition to getting USL & LSL from customers, we are also getting from them the target value. Thus Cpm is a true measure of the process centering as it compares the difference between the process mean & target value. Let us see through the above example again:-

 

 

As you can see, we are now measuring the gap between the nose tip of the aircraft & the center of the landing strip.

 

The Cpm capability index is based on the Taguchi loss curve concept which states that, the increase in quality loss is directly proportional to the gap between the process mean & the target. Higher gap results in higher losses. This is also evident in the below visual:-

 

 

 

As you can see that the loss is minimal near the target value & increases as move towards the specification limits.

 

There are three specific scenarios where Cpm proved to be a better measure of process capability compared to Cpk. These are as listed below:-

 

• In the first case we have a process with a wide tolerance range however there is a slight shift from the target, in such scenario Cpk won’t change much but there will be a reduction in Cpm, thus showcasing that Cpm is very sensitive even to small drifts of the process mean from the target.

 

• In the second scenario we have only a specification on one side, Cpk loses its significance however we can still experience shift of the process mean from the target, which will thus impact Cpm thus showcasing that Cpm is a better measure for one - sided specification.

• Here we are having a target which isn’t the centre point of the tolerance limits. Here the target is intentionally set accordingly. Here also Cpm will be affected by the shift from the target, however there will be no significant change in Cpk. This makes Cpm a more robust measure for processes having non-centered specification.

 

 

Now as far as the application of Cpm as a measure of the process capability in service industry is concerned we can think of few examples as explained below:-

 

• In a pizza delivery process, we can create a one sided specification in terms of the overall delivery lead time which is let’s suppose set at 30 min as it upper specification limit & we can set an internal target of 20 min & track the process capability basis the target of 20 min.

 

• In a wire transfer process of a bank, we can intentionally set the process TAT closer to the LSL i.e. farther way from the USL, in order to ensure a consistent performance & less risk of delays in processing the wire transfer, here Cpm will play a significant role in assessing the process capability.

 

• In the accounts receivable process, we can assess the sales executive performance in terms of the deviation of the DSO from the target value & for this Cpm will be the best measure in order to assess an individual’s capability & also compare with its peers.

Yes, it is true that Capability Matrices (Cp & Cpk) are more relevant for manufacturing Industries where it is common to get 2 sided specification limits. These indices do not adequately address the issue of process centering. Foe addressing this issue, an alternative definition of Cp advocated by Taguchi. Taguchi introduced the name Cpm for the Taguchi Index and examined statistical properties of an inefficient estimator under the assumption that the process mean coincides with the target value. 

Cpm is now another interesting process capability metric that is not very popular. In fact it is more robust than Cpk. Cpm is an advanced measure and it corrects some deficiencies in Cpk. It is also called as Taguchi Capability Measure. 

 

We can easily understood the difference between Cpk and Cpm with this below Illustration and formula; 

Cpk 

 

Cpk is considered as a measure of the process centering for any process. If we closely observe the Cpk formula, it does not require the actual target value. target value is the center value obtained from the customer Given or design specification. However it takes the LSL, USL & Process mean. In this way, we can say that this is an indirect method of assessing if the process centered to the tolerance. 

In the above picture, in order to know if the plane is landing exactly on the middle of the landing strip, we look at the distance of the plane's nose from either edge of the strip. That is good but it's only an indirect way of assessing the process center. 

 

Cpm

If we look at the above picture and formula of Cpm, it contains the target value. Thus in addition to obtaining the USL and LSL from the customer or Design, we also obtain the Target Value. This means, Cpm is a true process capability index, because it compares the difference between process mean and target Value. To draw an image for plane landing, we are comparing the plane's nose to the median line of the strip. 

 

Lets take an example, where there is only a specification on one side. In most business processes, service levels have only an upper or lower specification. In some cases, Cpk losses its significance. However, in all service scenarios, there is a target value. For example, Customers are told that payment will be updated in 4 hours but the internal target is to get it done in 2 hours. Process is designed for 2 hours. In such scenarios, Cpm will be a relevant centering metric. 

 

 

In this scenario, Cpk is high, but Cpm 0.54 is very less value as comparison to Cpk 21.40. Here Cpm is very sensitive and good capability measure while Cpk is not. 

 

To get some hands on experience, lets calculate the value of Cpk and Cpm for another service industry; 

 

Below are the few examples of use of Cpm (Taguchi Capability Measure) in Service Industry, where we generally have one sided specification with Target Value; 

- Resolution time less than X hours 

- Call time less than X minutes 

- Payment update in less than X hours 

- Number of min order places per hours 

- Number of max. calls in waiting and so on ......... 

 

While all published answers are correct, the winning answer has been provided by Chandra Shekhar Chauhan for demonstrating how Cpm can be calculated.