**Capability is undoubtedly one of the most well-known concepts in the management of processes. You still have to know how to measure it and interpret it correctly.**

## Introduction

By definition, the capability measures the ability of a process to perform a task. It measures the relationship between the requested performance and the actual performance of a process. It is translated in the following way:

It was Joseph Juran, one of the quality gurus, who was the first to develop an indicator of this type at the end of the Second World War. For him, the capability is defined by the ratio between the tolerance width and the capability of the process, which today corresponds to Cp. It will only thereafter be the other indicators Pp… were created.

There are now several standards explaining the concept, one will remember that of the QS9000, the most widespread. The standard distinguishes two types of capabilities:

- The
**long-term capability****:***It reflects the reality of the products delivered. We are talking about process performance*. - The
**short term capability****:***It translates dispersion over a very short time. The process is then referred to as capability, and the intrinsic capability of the process is translated into the current working conditions*.

## 1 – capability Yes, but with which data ?

Continuous data | Attributes data |
---|---|

Calculation of capabilities according to standard indicators that we describe below. | Capability control by comparison between the calculated DPMO and the acceptable DPMO by the quality. |

Calculation of Six Sigma Standard Indicators: DPU, DPO, DPMO and PPM | |

Sigma level identification |

### Why Can not Calculate Capability with Attribute Data ?

Calculating a capability with attribute data is in fact mathematically impossible. With this type of data, we get a %. This percentage of defect corresponds to the Z of the normal distribution (see below). But this Z, we can not know if it comes from a problem of decentering (Cpk) or a problem of width of our distribution (Cp).

However, in the literature, an approximation is found to identify the Cpk (which is the value used by convention):

(σ – 1,5) / 3

σ / 3

(% Customer-p)/(3 * √ ((p * (1-P))/n))

P: Proportion of defect we get

σ: Sigma level that we calculated via the DPMO

% customer: default% allowed by customer

n: Number of samples

**Example**

We want to calculate the capability of our equipment to properly recap bottles. Our only measure is to identify whether our cork is good or poorly put. We are in a binary reasoning “good/Not good”.

With such data, one calculates a percentage of defects, for example 5 faults on 100. With this data, we could not have a standard deviation or a tolerance interval.

Our only way is to compare this percentage to a percentage limit given by quality. This is equivalent to comparing DPMO.

**(σ – 1,5) / 3.**

## 2-Validate the normal law

The normality of the data is essential to have statistically correct results, since the formulas of calculations of the capability are built on the assumption of normality.

It is necessary to make a **Adjustment test** To make sure of this condition. One uses, for example, the **test of χ ^{2}**, which will be effective from the moment we have at minima 30 data. Below, we can only “

*assume*” The quality of the study.

It is noted that in some cases the non-normality of a process is ” *normal* “. For example :

- Process performing several actions at the same time: for a filling machine delivering a liquid in X bottles at the same time, it is most likely that all filling nozzles do not have the same wear
^{1}… The distribution of the filling will therefore not be ”*normal*“, but nevertheless this does not indicate a specific problem. - For data ”
*bounded*“, necessarily non-normal since we have a limit (example: roughness, flatness…)^{2}.

In the event that we have not obtained the normality of the data, several solutions can be envisaged:

**a transformation of the data:**We will then use the**transformed Box-Cox**.**an increase in control limits:**They can be set to 4 or 5 Sigma rather than 3^{3}.**Recalculate control Limits:**We recalculate the limits to + or-3 Sigma using then the law that applies to the distribution of our data.**“Do**experience shows that despite the data being non-normal, an interpretation is possible with all the conditions that must be put into it*nothing*“:^{4}(there is no way to know if there is a special cause and we will not be able Estimate the default percentage).**investigate:**Finally, we investigate the data coming out of the normal law to see if it is simply**aberrant**data that can be removed from the study.

## 3-Calculate the long-term capability

The long-term capability is measured on at least a week. It allows to include in the calculations, to take into account the variations of all the processes undertaken: Change of series, change of Team…

In a Long-term study, we will be careful to make the data survey randomly. For example, do the survey every day at 14hr.

One should not wait until an event occurs to carry out the survey (wait for a breakdown, or on the contrary wait for it to turn well…). We seek to represent reality not a specific situation.

### Intrinsic Performance of the process P_{p}

The P_{p} (*process Performance*) indicator represents the ratio of the target tolerance interval to the process dispersion.

**Pp = IT/6** **σ _{LT}**

With:

**IT:**client Voice tolerance interval**6 σ**long-term dispersion. It is calculated on a minimum of 100 measures to be representative._{LT}:

### Detuning indicator P_{PK}

It represents the decentring of the distribution relative to the tolerances. It is calculated in the following way:

**Ppk = k/3** **σ _{LT}**

With:

**k:**distance between the average of our process and the nearest tolerance limit**3 σ**Half of the long-term dispersion_{LT}:

## 4-Calculate the short-term capability

The capability short term translates a punctual level of performance to produce what is requested. In other words, it measures the capability to produce well under fixed conditions (same product, same team…).

The formulas of calculations are strictly the same as before. It is simply noted that PP is therefore named Cp and Ppk is called Cpk (Cp from *Process Capability*).

**Cp = IT/6** **σ** _{CT}

**Cpk = k / 3****σ**_{CT}

Only the calculation of the dispersion differs. To calculate it, we have two ways to proceed:

- Either we know to collect at minima 100 parts that we know to be produced under the same conditions. The actual standard deviation is then calculated. Jumps and other drifts will be more ”
*visible*” with this method. - Either we cannot collect as many parts under identical conditions. We will estimate the standard deviation from the extents of the measurements. On the other hand, we will see less jumps and other drifts. 2 cases are possible:

**We have several samples per sample : σ _{CT} = R_{cross} / d_{2}**

**We have 1 sample per sampling: σ _{CT} = mR_{cross} / 1,128**

With:

**R**average of the range of each subgroup._{cross}:**MR**moving range It is calculated by the difference between the value I and the value i + 1._{cross}:**d**Estimation coefficient identified from sample number_{2}:

It is not easy to measure a short-term capability for processes requiring long time. For example, if we have a five-minute time, it would take more than 500 minutes to collect the number of pieces sufficient. It is obvious that sufficient stability cannot be considered in this time frame. The calculations cannot then be carried out and the calculations will be limited to long-term capability.

## 5-Interpretation of indicators

The short term capability will be used to validate batches of productions, where the long-term capability will be more used to drive the production load, carry out progress actions or conceive new processes.

### 5.1 Interpretation of Cpk and Ppk

These two indicators represent the average position of our process with respect to the target being sought. The graph below shows different position of our process compared to different value of Cpk or Ppk.

Value | Graphic description | Cpk interpretation | Ppk interpretation |
---|---|---|---|

Cpkor Ppk<0,67 | The process is very off-center. | Even if our dispersion is very small, too much decentering puts us in a risky situation. A 100% control is probably necessary. | The process is clearly not designed for the purpose. Either settings may be sufficient to refocus but if necessary, it will redesign. |

Cpkor Ppk<1 | Decentering is relatively weak. | The situation is very dependent on our Cp value. Nevertheless, even if it is good, preventive actions are to be implemented. | Some improvement actions were enough to put in a more comfortable situation. Nevertheless, depending on the case, we can stay like that. |

Cpkor Ppk ≥ 1 | The process is perfectly centered. | Nothing to report at the new centering. We are perfectly at the target. |

It is considered that a process is said to be capable when Cpk is 1. We accept then that about 3 pieces out of a thousand are out of tolerance.

**It should be noted that Toyota requires its subcontractors a minimum capability Ppk > 1.67**

### 5.2 Interpretation of Cp and Pp

The Cp and Pp represent the relationship between the dispersion of our process and the tolerance interval of the **voice of customer**. The graph below shows the Shape of this dispersion according to the value of Cp or Pp.

Value | Graphical description | Cp interpretation Cp | Pp interpretation Pp |
---|---|---|---|

Cpou Pp<0,67 | Dispersion larger than the interval. | More than 5% of the defects are generated. A 100% control is probably necessary. | We clearly have a process that does not respond to the client's request. It is necessary to rethink the process or consult the client to increase tolerance. |

Cpor Pp<1 | Dispersion slightly larger than the interval. | It generates between 0.0063% and 5% of default. You have to look at Pp to know if it is a "punctual" or "sustainable" situation. If it's sustainable, more control is needed. | It seems complex to "live" with a process that does not guarantee sufficient sustainable performance. Some simple actions of improvement should allow to increase the performances to be more "serene". |

1or Pp<1,33 | Dispersion slightly below the range. | ||

1,33Pp<1,67 | Low dispersion with respect to the interval. | We generate very few defects. Just check the process on time and correct the special causes as you go | The process is very effective, maybe too much. We can continue like this or, in some cases, reduce performance to reduce costs. |

1,67Pp<2 | Very low dispersion | ||

Cpor Pp > 2 | Dispersion more than 2 times less than the interval. |

### 5.3. Calculate Cpm and Ppm indicators

With regard to the different indicators calculated above, it is essential to combine them to interpret them in the right way. It is not enough for the process to be centered so that the result is good, and it is not enough for the dispersion to be lower than the tolerance for the result to be good.

To overcome this problem, two indicators allow to take into account both dispersion and centring. We find:

For the short term capability (also referred to *as ” Taguchi capability index*“:

For the long-term capability:

The interpretation is as follows:

**Cp = Cpk = 1:**In this position, we have a perfectly centered process with a dispersion equal to the tolerance interval (at the most accurate).**Cp = Cpk :**at the most the difference between the two quantities is large, and the more possibility of malfunction.

### 5.4 The rate of control of the process

The control rate of the process is defined as the ratio between short-term dispersion and long-term dispersion:

**Process Control rate = σ _{CT} /σ_{LT}**

The interpretation is as follows:

**control rate close to 1:**The process is considered controlled.Most often, wanting to improve the indicator is a change in technology.**control rate close to 0 :**The process is not mastered. Many losses are due to very long changes, difficulty in setting up equipment, poor organization…**control rate > 1.2 :**There is a short-term anomaly. We must investigate.

## 6 – Illustration

We all had problems parking our car in our garage or on a parking space. Below, we present various situations where we can illustrate the different indicators of capability.

Illustration | Capability | Observation |
---|---|---|

Cp > 1Cpk > 1 | Our car being smaller than our place and our garage being "free", we easily manage to park in the center of it. | |

Cp < 1Cpk ≥ 1 | I'm well centered, my car is wider than my garage, I can not go home. | |

Cp ≥ 1Cpk < 1 | My car is well in my garage, but I have a hard time aiming for the center of it. From time to time, I scratch my mirror ... | |

Pp > 1Ppk < 1 | Sometimes children leave their bikes do not matter how in the garage. So, even if it is, I have no worries to park, it happens that I have to maneuver to park and sometimes, I scratch the retro or the door ... This is the difference between the Cp, Cpk and the Pp, Ppk: we take values over a longer time to evaluate the overall performance of our process. |

## 7 – Decide actions

The unique analysis of the capability indices is not sufficient. It must be completed with the analysis of a **Control chart**. 3 cases may occur :

**the indices of capability are very good, and only special causes appear on the control Board:**The process is declared capable. It is considered that it is possible to fly directly using the appropriate control chart and to proceed to the ”” stage of the*control***DMAIC**or ”*Check*” of the**PDCA**.**The indices of capability are ”**The process is declared ”*correct*” and we find special and common causes on the control Board:*capable*“, but under control. It is quite possible and even desirable to steer the process from the natural limits of the process. A project to improve the performance of the process and to return to the ”” Steps of the*analysis***DMAIC**or ”*DO*” of the**PDCA**must be implemented in parallel.**the indices of capability are bad and many common causes appear on the Control Chart :**The process is declared not capable. The process generates many variations of itself without being able to find the causes. It is necessary to redesign the process or review the specifications with the customer.

## 8-Calculate the indicators of the 6 Sigma

**PPM or DPMO**

They are directly linked to the values of Cp and Cpk. By convention, it will be calculated with our Cpk. The formula is as follows:

**PPM or DPMO = (1-Z _{Cpk}) * 1 000 000**

However, Cp can be used, but it will be less restrictive than Cpk since it assumes that the distribution is centered.

Moreover, if we want to project ourselves in a long-term performance, we use Ppk.

PPM = DPMO? It all depends on whether the study deals with one defect in particular or several flaws. For example, for the manufacture of a bottle of syrup, there may be a problem of label, cap, filling…:

- We calculate DPMO if we consider the problems of label, cap and filling.
- One calculates PPM, if one considers only a problem of label or a problem of cap or filling.

### The level of Sigma

It is calculated in the same way as **normal** via to recall the following formula:

**Sigma level = 1.5 + law. Normal. Standard. Inverse (PPM/1000000)**

### The percentage of defects

The percentage of default is obviously the PPM/1 000 000 all set in percentage.

## Source

1 – R.R. Mortal, G. C. Runge (1995) – Statistical process control of multiple stream process

2 – N. L. Johnson, S. Kotz (1993) – Process capability Indices

3 – S. A. Yourstone, W. J. Zimmer (1992) – Non normality and the deisign of control

4 – H. A. David (1981) – Order statistics

T. Pyzdec (1992) – Psydek’s Guide to SPC

M. Pillai (1991) – The pitfalls of capability measures

Standard NF X06-033-capacity of production means and manufacturing processes