Section 07. CapabilityContinuous

07. CapabilityContinuous

Overview

Early in any project (in Define for goal setting or more formally in Measure) it is crucial to understand the current level of performance of the process prior to making any changes. Many Champions and Belts mistakenly believe this is purely to show how much the project saves or how big an improvement is made to justify continuation of the Lean Sigma program. These things are important but are only a small piece of the picture. The primary use of the measure is to ensure the gains are sustained after the improvements are in place. If the change is unmeasured, it often is undone later (with all the best intentions) because it is not fully understood. However, if a measured and verified performance change is made, then there is less likelihood for future damaging "tweaks."

There are many performance metrics available; for example throughput, OEE, quality, and so on, are typically represented as single numbers based on conformance to some goal. A better performance metric is to look at performance versus a specification(s), known as Capability.^[10]

^[10] For more details see How To Construct Fractional Factorial Experiments, Vol. 14 by Larry Barrentine.

The simplest form of Capability for continuous data is known as C_p and is calculated as the ratio of the specification range divided by the process width:

where:

s is the short-term process standard deviation
USL is the Upper Specification Limit
LSL is the Lower Specification Limit

The process width denominator is chosen as 6 standard deviations because this is deemed to a reasonable representation of the width of the process.^[11]

^[11] 99.73% of data points lie between ±3 standard deviations in any normally distributed data.

C_p suffers from one obvious flaw as depicted in Figure 7.07.1; it doesn't take into account the centering of the process. Figure 7.07.1 shows 3 graphs with same C_p but different process centering.

Figure 7.07.1. The effect of process centering on C_p.

A second metric needs to be introduced to counter this; known as C_pk, it is defined as

C_pk represents the distance of the center of the process to the nearest specification limit in units of process width (in fact it is half the width because only one side of the process curve is considered at a time).

C_pk is positive when the mean of the process is inside the specifications; it drops to zero as the mean hits the USL or LSL. In fact, it becomes negative as the process mean moves outside the specification range, as shown in Figure 7.07.2.

Figure 7.07.2. The effect of process centering on C_p and C_pk.

Roadmap

The roadmap to calculating the Capability for continuous data is as follows:

Step 1.	For the metrics in question, define the goals and specifications.
Step 2.	Collect process data. At least 30 data points are required.

Step 3.	Check the process stability (see "Control Charts" in this chapter). To be considered capable, the process needs to be stable. The definition of stability is having Consistent centering (mean) Predictable variation It becomes clear that a process could not consistently meet Customer requirements if its mean were moving around and its variation were changing unpredictably. To check stability, Belts can simply apply the appropriate Control Chart to the data prior to calculating its capability.
Step 4.	Check for Normality. The Capability calculations of C_p and C_pk rely on the data being normal. To check the normality, use a Normality Test as described in the section "Normality Test" later in this chapter. For non-normal data see "Other Options" in this section.
Step 5.	Calculate C_p and C_pk as per the preceding equations. Most statistical software packages do this readily.

Interpreting the Output

Example output for a Capability Study is shown in Figure 7.07.3. The key metrics to focus on are

Lower Specification Limit (LSL) as entered by the user
Target value if entered
Upper Specification Limit (USL) as entered by the user
Sample Mean
Potential Capability C_p and C_pk as defined by the equations in "Overview" in this section

Capability (expressed as C_p and C_pk) is intended to represent short-term behavior of the process. In reality processes tend to shift and drift over time; the variation stays reasonably consistent, but the mean moves to and fro.^[12] Taking this into account, the longer-term variation is actually larger than short-term and so the "capability" is lower in the long term than the short. Long-term "capability" is known as Performance and the equations are identical to those for Capability (short-term), but a longer-term standard deviation (σ instead of s) is used

^[12] Empirical process studies show that most processes tend to shift and drift about 1.5 standard deviations. Lean Sigma Belts really only need to know that it happens, rather than the equations to justify why.

Most software packages try to emulate this short-term versus long-term standard deviation by measuring it in two different ways; for long-term the regular standard deviation of all the data is used and for short-term the value comes from an equation involving the Moving Average across the data. In Figure 7.07.3, the within standard deviation represents short term and the overall standard deviation represents long term. The within value is used to calculate the C_p and C_pk, whereas the overall value is used to calculate the P_p and P_pk.

Figure 7.07.3. An example of a Capability Study (output from Minitab v14).

The target value for C_p in Lean Sigma is 2.0 and for P_pk it is 1.5. These are not absolute requirements in any way, but if a process exhibits Capability at this level then it can be considered to be performing very well.

At the bottom of Figure 7.07.3 are boxes explaining likely performance of the process in terms of Parts per Million defective (PPM):

The Observed Performance represents the PPMs of the actual data points below the LSL, above the USL, and the total of both. If no points fall outside of specification during the data collection, then the PPMs here are zero.
The Within Performance represents the PPMs as calculated from a normal curve with the sample mean and short-term standard deviation. The calculated curve hangs over the LSL and USL, and thus, the area under the curve outside of the specification limits gives the PPMs. These are the expected defectives on a short-term basis.
The Overall Performance represents the PPMs as calculated from a normal curve with the sample mean and long-term standard deviation. The calculated curve hangs over the LSL and USL, and thus, the area under the curves outside of the specification limits gives the PPMs. These are the expected defectives on a long-term basis.

Of course, all the values are calculated from the sample of data taken. If a subsequent sample were taken, it is almost certain that different answers will arise; thus, it is inappropriate to quote PPMs to anything more than two or possibly three significant figures. Belts often take great delight in including numbers in reports to many decimal places, which is acceptable provided that they realize that the numbers will be different next time.

Other Options

There are a number of variants to the preceding (standard) approach to calculating Capability that generally depend on the shape of the data and the behavior of the specification limits.

Non-Normal Data

As mentioned in the Roadmap, normality of the data is a key consideration and is often a potential failure point for Belts. If the data are non-normally distributed, for example they are skewed to one side, then the C_p calculated is fallacious and could be misleading to the Belt.

If the data is non-normal and unimodal (just one hump in the distribution), then the situation can be remedied by transforming the data. If the data is multimodal (more than one hump), it is likely that the process is unstable and hopefully there are simple special causes that can be identified and eliminated. Remember that in process improvement, a poor process is best seen as a big opportunity.

The approach of transforming data is not dealt with here, but there are a number of transformations available in most statistics software packages, such as Box-Cox or Johnson. In fact in the more user-friendly statistical software packages there are specific non-normal C_p tool options.^[13]

^[13] For more details of applying transformations to Capability, see How To Construct Fractional Factorial Experiments, Vol. 14 by Larry Barrentine

Single-Sided Specifications

Many processes have both an Upper Specification Limit (USL) and a Lower Specification Limit (LSL); some have just one of these. For example, a metric, such as strength, might have an LSL in that there is a minimum strength to be considered acceptable by the Customer, but a USL doesn't make sense. In these cases, the C_p is no longer available as a metric because both specifications are required to calculate it:

Instead the C_pk is more appropriate:

For a single-sided specification, only half of the C_pk equation makes sense, so new metrics, known as C_p(upper) and C_p(lower), are used

The same rules apply to C_pU and C_pL as they do to C_pk. The target value is 1.5 or above. If the mean of the data lies on the specification limit, then the associated value of C_pU or C_pL is zero. If the mean is outside of the specification, then the C_pU or C_pL is negative.

Bounds

For some processes a lower or upper bound exists on the performance metric, for example:

Scrap cannot go below zero
Yield cannot go above 100%
Radius cannot go below zero

In this case, the bound cannot be considered to be a specification limit, so the same approach is used here as is used for a single-sided specification and the bound is effectively omitted. Any predicted points below a Lower Bound are ignored, as are any above an Upper Bound.