Process Capability (Cp, Cpk) and Process Performance (Pp, Ppk)

If you want to determine how a particular manufacturing process performs, there is a wide array of data about how many defective units it makes, how much variation there is from one unit to another, and so on. As a manager, you might wonder how to best make use of this data to understand and predict process performance. The two most common types of solutions to this problem are two different types of analysis: process capability and process performance.

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Basic Background

Process capability and process performance are two different ways to examine how a certain production process is performing, as the names imply. They have different contexts because process capability requires a process under statistical control, meaning mature and stable, while process performance is valid for a production process that has just been set up and is not yet under control. Within process capability there are two important statistical measures: Cp and Cpk. Process performance contains the analogous Pp and Ppk. In this post we will go through all of these in detail and explain why they matter and what their applications are.

An important part of management is understanding how well existing production lines are performing relative to their guidelines. The lines should be both consistent and on target. This is similar to the old grade-school analogy about precision and accuracy. Precision is having a small variance and producing consistently over and over again. Accuracy is being, on average, close to the target across many trials. If you are consistent without being accurate, the line will produce the same number of defectives over and over. If you are accurate but not precise, then your output will sometimes be above target and sometimes below but will average out to the right number. Only with both precision and accuracy do you get a fully functioning and efficient production line.

The point of this discussion is that Cp, Cpk, Pp, and Ppk are ways of measuring precision and accuracy. We will get into this in more detail later on, but the key is that Cp and Pp can tell you when your line is low in accuracy or precision and Cpk and Ppk are only high when the line output is both accurate and precise.

Statistical Control

Before going further it is worth explaining the concept of statistical control, because it separates process capability from process performance. Statistical control refers to a state where a process is capable of being measured and produces according to its potential, with as much accurate output and as little waste as possible. In other words, when a process is under statistical control it has reached its mature potential and is producing according to its specifications, with all kinks worked out. A process that is not yet under statistical control is likely to be a new one that has recently been completed. It may have hidden glitches or calibration problems that reduce its output, create too many defectives, or generate waste.

Process Performance and Control

When a process is new and not yet under control, managers use process performance to analyze it. Process performance and process capability use the same calculations, but different data. The new, raw state of the process under process performance analysis is subject to change as it is perfected. So the data and the analysis is useful to help point out problems, but it does not represent the ultimate potential of the process. Process capability does represent that potential and measures how well the process is doing compared to how well it should perform according to its specifications.

It is important for a process to perform at the correct level. If it is not precise and accurate enough, the parts it produces will be inconsistent, angering customers and leading to waste. If the opposite extreme happens, then the company may be spending too much on the process for a small gain, wasting money instead of materials. Both scenarios are bad because they can eat into profits. Complicating the matter, although the ideal is a happy medium, there is no consensus on exactly what that medium needs to be. Different outputs and types of equipment imply very different ideal Cpk and Ppk. A piece that will have a direct effect on people's safety needs higher standards than one that is decorative, for example. Once again, though, every industry and every process within each industry has its own set of standards, and the standards themselves tend to be arbitrary.

Indexes of Performance

These standards generally take the form of numbers that are used as a comparison basis for a process's Cpk or Ppk. Cpk is the process capability index and Ppk is the process performance index. Both of them will be a small number, almost always less than three. A higher number means more precision and accuracy because it means the output is close to its ideal and far from its specification limits, and that the variation from unit to unit is low. The Cpk or Ppk index can also be negative. That would mean that the process is producing outside of its specification limits, which is a very poor sign for its accuracy.

Each of Cp, Pp, Cpk, and Ppk has its own mathematical formula that you apply to find the final number. Cp and Pp are simpler. You take the upper specification limit, subtract the lower one, and then divide by six times the variance of output in your process. Having the gap between the USL and the LSL is a good thing because it means you have more space in which to produce and meet your targets, so you want the top of the fraction to be large. A low variance is good, because that is the same thing as saying high precision. If you can assume that the process output has a mean centered exactly in the middle between those two limits and also assume that the process output has a normal distribution, this formula is very useful. It produces a single number and the higher that number is, the more accurate and precise the process is, making for easy comparisons.

Mathematical Details of Cpk and Ppk

If you are not sure that the middle of the process should fall in the center, then you need to move up to Cpk and Ppk. These statistics, which you recall are called process capability index and process performance index, relax the assumption that the midpoint is the actual center of the process output. The new formula is composed of two parts. First, take the upper specification limit, subtract the sample mean of the output data for your process, and divide that by three times the variance. The second part is taking the sample mean, subtracting off the lower specification limit, and then dividing that by three times the variance. Whichever of those two parts is lower is your index number. The formula is a little more complicated but it has the added benefit of being more correct when the process is one where the distribution of outcomes is lopsided to one side or the other. For example, a steel beam might have a very important minimum strength and a less important maximum strength. If you produce these beams you want to make sure enough of your beams are at least as strong as the minimum, but you don't care as much about the maximum. That means your ideal output strength is closer to the maximum, the upper limit, than the minimum or lower limit.

Interpretation

The important thing to remember is not the details of the mathematical formulas, because you can always look those up, but what they mean. Cp and Pp help you measure the precision and accuracy of mean-center processes, and Cpk and Ppk do the same for processes where the mean may not be in the center. The other way to group it is that Cp and Cpk are measures of a process's ultimate potential performance level while Pp and Ppk instead measure how a process is doing right now after you have just built it and have not yet optimized it.

Keep in mind that Cp, Cpk, Pp, and Ppk are just four of several ways to measure the output of a process. There are other measures that use different assumptions that, for example, let you test how close your output is to a certain target level instead of using upper and lower limits. It all comes down to the process and which measure gives you the most useful interpretation of its quality. There is some judgement in choosing the right measure for the right process at the right time. Think of "performance" measures like Ppk as those based on right now while "capability" measures like Cpk are based on the best-case scenario for how a process will operate.

Takeaways

The bottom line is that tThere are many ways to measure how well a process is functioning, and even before you reach that step you must ensure that you have collected good data. Establish a consistent data collection system, which is a process on its own. Without descriptive data these formulas are useless. The saying is GIGO, for Garbage In Garbage Out. If the data isn't correct, truthful, and complete then the results of the formulas will be unrealistic and impossible to interpret. Once you have your data, you can select and apply an appropriate measure of performance to see how consistent and accurate your process is.

Author: Eric Raio

Eric Raio is one of the founders of Factory Solutions. When he isn't plotting new ways to create awesome software. He likes to geek out about flying drones and technology.

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